@article{vogels2011inhibitory,
  title={Inhibitory plasticity balances excitation and inhibition in sensory pathways and memory networks},
  author={Vogels, Tim P and Sprekeler, Henning and Zenke, Friedemann and Clopath, Claudia and Gerstner, Wulfram},
  journal={Science},
  volume={334},
  number={6062},
  pages={1569--1573},
  year={2011},
  publisher={American Association for the Advancement of Science}
}
@article{katz1965measurement,
  title={The measurement of synaptic delay, and the time course of acetylcholine release at the neuromuscular junction},
  author={Katz, Bernard and Miledi, Ricardo},
  journal={Proceedings of the Royal Society of London. Series B. Biological Sciences},
  volume={161},
  number={985},
  pages={483--495},
  year={1965},
  publisher={The Royal Society London}
}

@article{lee2003hierarchical,
  title={Hierarchical Bayesian inference in the visual cortex},
  author={Lee, Tai Sing and Mumford, David},
  journal={JOSA a},
  volume={20},
  number={7},
  pages={1434--1448},
  year={2003},
  publisher={Optica Publishing Group}
}

@article{jacobs1999optimal,
  title={Optimal integration of texture and motion cues to depth},
  author={Jacobs, Robert A},
  journal={Vision research},
  volume={39},
  number={21},
  pages={3621--3629},
  year={1999},
  publisher={Elsevier}
}

@article{knill2003humans,
  title={Do humans optimally integrate stereo and texture information for judgments of surface slant?},
  author={Knill, David C and Saunders, Jeffrey A},
  journal={Vision research},
  volume={43},
  number={24},
  pages={2539--2558},
  year={2003},
  publisher={Elsevier}
}

@article{hillis2004slant,
  title={Slant from texture and disparity cues: Optimal cue combination},
  author={Hillis, James M and Watt, Simon J and Landy, Michael S and Banks, Martin S},
  journal={Journal of vision},
  volume={4},
  number={12},
  pages={1--1},
  year={2004},
  publisher={The Association for Research in Vision and Ophthalmology}
}

@article{weiss2002motion,
  title={Motion illusions as optimal percepts},
  author={Weiss, Yair and Simoncelli, Eero P and Adelson, Edward H},
  journal={Nature neuroscience},
  volume={5},
  number={6},
  pages={598--604},
  year={2002},
  publisher={Nature Publishing Group}
}

@incollection{behrens2014mr,
  title={MR diffusion tractography},
  author={Behrens, Timothy EJ and Sotiropoulos, Stamatios N and Jbabdi, Saad},
  booktitle={Diffusion MRI},
  pages={429--451},
  year={2014},
  publisher={Elsevier}
}

@article{friston2003dynamic,
  title={Dynamic causal modelling},
  author={Friston, Karl J and Harrison, Lee and Penny, Will},
  journal={Neuroimage},
  volume={19},
  number={4},
  pages={1273--1302},
  year={2003},
  publisher={Elsevier}
}

@article{beck2008probabilistic,
  title={Probabilistic population codes for Bayesian decision making},
  author={Beck, Jeffrey M and Ma, Wei Ji and Kiani, Roozbeh and Hanks, Tim and Churchland, Anne K and Roitman, Jamie and Shadlen, Michael N and Latham, Peter E and Pouget, Alexandre},
  journal={Neuron},
  volume={60},
  number={6},
  pages={1142--1152},
  year={2008},
  publisher={Elsevier}
}

@article{bayes1763essay,
  abstract = {reprinted in Biometrika 45 296-315 1958 Thomas Bayes
                 1702-1761
                 http://www.cs.monash.edu.au/\~{}lloyd/tildeImages/People/Bayes/index.html
                 ([about Bayes])},
  added-at = {2008-10-07T16:03:39.000+0200},
  author = {Bayes, T.},
  biburl = {https://www.bibsonomy.org/bibtex/21f1017bf1b4fb8840f01e1649d1f0804/brefeld},
  interhash = {a7bc6ffe9fc49d87aff583b0f1dda401},
  intrahash = {1f1017bf1b4fb8840f01e1649d1f0804},
  journal = {Phil. Trans. of the Royal Soc. of London},
  keywords = {Bayes, Bayesian, MDL, MML, c1763 conditional, joint, jrnl, probability, theorem,},
  pages = {370--418},
  timestamp = {2008-10-07T16:03:44.000+0200},
  title = {An essay towards solving a problem in the doctrine of
                 chances},
  volume = 53,
  year = 1763
}

@article{kullback1951information,
  title={On information and sufficiency},
  author={Kullback, Solomon and Leibler, Richard A},
  journal={The annals of mathematical statistics},
  volume={22},
  number={1},
  pages={79--86},
  year={1951},
  publisher={JSTOR}
}

@book{Feynman1972,
  author    = {Feynman, Richard},
  title     = {Statistical Mechanics; A Set of Lectures},
  year      = {1972},
  publisher = {Addison-Wesley},
  address   = {Massachusetts}
}

@article{mcculloch1943logical,
  title={A logical calculus of the ideas immanent in nervous activity},
  author={McCulloch, Warren S and Pitts, Walter},
  journal={The bulletin of mathematical biophysics},
  volume={5},
  pages={115--133},
  year={1943},
  publisher={Springer}
}

@article{hornik1989multilayer,
  title={Multilayer feedforward networks are universal approximators},
  author={Hornik, Kurt and Stinchcombe, Maxwell and White, Halbert},
  journal={Neural networks},
  volume={2},
  number={5},
  pages={359--366},
  year={1989},
  publisher={Elsevier}
}

@article{pavlov1906scientific,
  title={The scientific investigation of the psychical faculties or processes in the higher animals},
  author={Pavlov, Ivan P},
  journal={Science},
  volume={24},
  number={620},
  pages={613--619},
  year={1906},
  publisher={American Association for the Advancement of Science}
}

@article{ogawa1990brain,
  title={Brain magnetic resonance imaging with contrast dependent on blood oxygenation.},
  author={Ogawa, Seiji and Lee, Tso-Ming and Kay, Alan R and Tank, David W},
  journal={proceedings of the National Academy of Sciences},
  volume={87},
  number={24},
  pages={9868--9872},
  year={1990},
  publisher={National Acad Sciences}
}

@article{bernstein1902untersuchungen,
  title={Untersuchungen zur Thermodynamik der bioelektrischen Str{\"o}me},
  author={Bernstein, Julius},
  journal={Pfl{\"u}gers Archiv European Journal of Physiology},
  volume={92},
  number={10},
  pages={521--562},
  year={1902},
  publisher={Springer Berlin/Heidelberg}
}

@article{hodgkin1952quantitative,
  title={A quantitative description of membrane current and its application to conduction and excitation in nerve},
  author={Hodgkin, Alan L and Huxley, Andrew F},
  journal={The Journal of physiology},
  volume={117},
  number={4},
  pages={500},
  year={1952},
  publisher={Wiley}
}

@incollection{sherrington2023integrative,
  title={The integrative action of the nervous system},
  author={Sherrington, Charles Scott},
  booktitle={Scientific and Medical Knowledge Production, 1796-1918},
  pages={217--253},
  year={2023},
  publisher={Routledge}
}

@misc{dale1935pharmacology,
  title={Pharmacology and nerve-endings},
  author={Dale, Henry},
  year={1935},
  publisher={SAGE Publications}
}

@article{friston2005theory,
  title={A theory of cortical responses},
  author={Friston, Karl},
  journal={Philosophical transactions of the Royal Society B: Biological sciences},
  volume={360},
  number={1456},
  pages={815--836},
  year={2005},
  publisher={The Royal Society London}
}

@article{becker1992self,
  title={Self-organizing neural network that discovers surfaces in random-dot stereograms},
  author={Becker, Suzanna and Hinton, Geoffrey E},
  journal={Nature},
  volume={355},
  number={6356},
  pages={161--163},
  year={1992},
  publisher={Nature Publishing Group UK London}
}

@inproceedings{chen2020simple,
  title={A simple framework for contrastive learning of visual representations},
  author={Chen, Ting and Kornblith, Simon and Norouzi, Mohammad and Hinton, Geoffrey},
  booktitle={International conference on machine learning},
  pages={1597--1607},
  year={2020},
  organization={PmLR}
}

@inproceedings{zbontar2021barlow,
  title={Barlow twins: Self-supervised learning via redundancy reduction},
  author={Zbontar, Jure and Jing, Li and Misra, Ishan and LeCun, Yann and Deny, St{\'e}phane},
  booktitle={International conference on machine learning},
  pages={12310--12320},
  year={2021},
  organization={PMLR}
}

@article{tang2022biologically,
  title={Biologically plausible training mechanisms for self-supervised learning in deep networks},
  author={Tang, Mufeng and Yang, Yibo and Amit, Yali},
  journal={Frontiers in computational neuroscience},
  volume={16},
  pages={789253},
  year={2022},
  publisher={Frontiers Media SA}
}

@article{siddiqui2023blockwise,
  title={Blockwise self-supervised learning at scale},
  author={Siddiqui, Shoaib Ahmed and Krueger, David and LeCun, Yann and Deny, St{\'e}phane},
  journal={arXiv preprint arXiv:2302.01647},
  year={2023}
}

@article{pettersen2024self,
  title={Self-supervised grid cells without path integration},
  author={Pettersen, Markus and Sch{\o}yen, Vemund Sigmundson and {\O}stby, Mattis Dals{\ae}tra and Malthe-S{\o}renssen, Anders and Lepper{\o}d, Mikkel Elle},
  journal={bioRxiv},
  pages={2024--05},
  year={2024},
  publisher={Cold Spring Harbor Laboratory}
}

@article{xu2022conformal,
  title={Conformal isometry of lie group representation in recurrent network of grid cells},
  author={Xu, Dehong and Gao, Ruiqi and Zhang, Wen-Hao and Wei, Xue-Xin and Wu, Ying Nian},
  journal={arXiv preprint arXiv:2210.02684},
  year={2022}
}

@article{greedy2022single,
  title={Single-phase deep learning in cortico-cortical networks},
  author={Greedy, Will and Zhu, Heng Wei and Pemberton, Joseph and Mellor, Jack and Ponte Costa, Rui},
  journal={Advances in neural information processing systems},
  volume={35},
  pages={24213--24225},
  year={2022}
}

@article {PPR:PPR843203,
	Title = {Self-supervised predictive learning accounts for cortical layer-specificity},
	Author = {Kermani Nejad, Kevin and Anastasiades, Paul and Hertäg, Loreen and Costa, Rui Ponte},
	DOI = {10.1101/2024.04.24.590916},
	Abstract = {The neocortex constructs an internal representation of the world, but the underlying circuitry and computational principles remain unclear. Inspired by self-supervised learning algorithms, we introduce a computational theory wherein layer 2/3 (L2/3) learns to predict incoming sensory stimuli by comparing previous sensory inputs, relayed via layer 4, with current thalamic inputs arriving at layer 5 (L5). We demonstrate that our model accurately predicts sensory information in context-dependent temporal tasks, and that its predictions are robust to noisy and occluded sensory input. Additionally, our model generates layer-specific sparsity and latent representations, consistent with experimental observations. Next, using a sensorimotor task, we show that the model’s L2/3 and L5 prediction errors mirror mismatch responses observed in awake, behaving mice. Finally, through manipulations, we offer testable predictions to unveil the computational roles of various cortical features. In summary, our findings suggest that the multi-layered neocortex empowers the brain with self-supervised predictive learning.},
	Journal = {bioRxiv},
	Year = {2024},
	URL = {https://europepmc.org/article/PPR/PPR843203},
}

@article{levenstein2024sequential,
  title={Sequential predictive learning is a unifying theory for hippocampal representation and replay},
  author={Levenstein, Daniel and Efremov, Aleksei and Eyono, Roy Henha and Peyrache, Adrien and Richards, Blake},
  journal={bioRxiv},
  pages={2024--04},
  year={2024},
  publisher={Cold Spring Harbor Laboratory}
}

@article{bogacz2017tutorial,
  title={A tutorial on the free-energy framework for modelling perception and learning},
  author={Bogacz, Rafal},
  journal={Journal of mathematical psychology},
  volume={76},
  pages={198--211},
  year={2017},
  publisher={Elsevier}
}


@article{pinchetti2022predictive,
  title={Predictive Coding beyond Gaussian Distributions},
  author={Pinchetti, Luca and Salvatori, Tommaso and Yordanov, Yordan and Millidge, Beren and Song, Yuhang and Lukasiewicz, Thomas},
  journal={arXiv preprint arXiv:2211.03481},
  year={2022}
}

@article{salvatori2022incremental,
  title={Incremental Predictive Coding: A Parallel and Fully Automatic Learning Algorithm},
  author={Salvatori, Tommaso and Song, Yuhang and Millidge, Beren and Xu, Zhenghua and Sha, Lei and Emde, Cornelius and Bogacz, Rafal and Lukasiewicz, Thomas},
  journal={arXiv preprint arXiv:2212.00720},
  year={2022}
}

@article{byiringiro2022robust,
  title={Robust Graph Representation Learning via Predictive Coding},
  author={Byiringiro, Billy and Salvatori, Tommaso and Lukasiewicz, Thomas},
  journal={arXiv preprint arXiv:2212.04656},
  year={2022}
}


@inproceedings{sacramento2018dendritic,
 author = {Sacramento, Jo\~{a}o and Ponte Costa, Rui and Bengio, Yoshua and Senn, Walter},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {S. Bengio and H. Wallach and H. Larochelle and K. Grauman and N. Cesa-Bianchi and R. Garnett},
 publisher = {Curran Associates, Inc.},
 title = {Dendritic cortical microcircuits approximate the backpropagation algorithm},
 url = {https://proceedings.neurips.cc/paper/2018/file/1dc3a89d0d440ba31729b0ba74b93a33-Paper.pdf},
 volume = {31},
 year = {2018},
 pages={8721--8732}
}

@article{salvatori2021associative,
  title={Associative Memories via Predictive Coding},
  author={Salvatori, Tommaso and Song, Yuhang and Hong, Yujian and Sha, Lei and Frieder, Simon and Xu, Zhenghua and Bogacz, Rafal and Lukasiewicz, Thomas},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  year={2021}
}

@article{vincent2010stacked,
  title={Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion.},
  author={Vincent, Pascal and Larochelle, Hugo and Lajoie, Isabelle and Bengio, Yoshua and Manzagol, Pierre-Antoine and Bottou, L{\'e}on},
  journal={Journal of machine learning research},
  volume={11},
  number={12},
  year={2010}
}

@article{scoville1957loss,
  title={Loss of recent memory after bilateral hippocampal lesions},
  author={Scoville, William Beecher and Milner, Brenda},
  journal={Journal of neurology, neurosurgery, and psychiatry},
  volume={20},
  number={1},
  pages={11},
  year={1957},
  publisher={BMJ Publishing Group}
}

@article{squire1991medial,
  title={The medial temporal lobe memory system},
  author={Squire, Larry R and Zola-Morgan, Stuart},
  journal={Science},
  volume={253},
  number={5026},
  pages={1380--1386},
  year={1991},
  publisher={American Association for the Advancement of Science}
}

@article{squire1992memory,
  title={Memory and the hippocampus: a synthesis from findings with rats, monkeys, and humans.},
  author={Squire, Larry R},
  journal={Psychological review},
  volume={99},
  number={2},
  pages={195},
  year={1992},
  publisher={American Psychological Association}
}

@article{kanerva1992sparse,
  title={Sparse distributed memory and related models, volume 92. NASA Ames Research Center},
  author={Kanerva, Pentti},
  journal={Research Institute for Advanced Computer Science},
  year={1992}
}

@book{kanerva1988sparse,
  title={Sparse distributed memory},
  author={Kanerva, Pentti},
  year={1988},
  publisher={MIT press}
}

@book{hertz2018introduction,
  title={Introduction to the theory of neural computation},
  author={Hertz, John and Krogh, Anders and Palmer, Richard G},
  year={2018},
  publisher={CRC Press}
}

@article{ramsauer2020hopfield,
  title={Hopfield networks is all you need},
  author={Ramsauer, Hubert and Sch{\"a}fl, Bernhard and Lehner, Johannes and Seidl, Philipp and Widrich, Michael and Adler, Thomas and Gruber, Lukas and Holzleitner, Markus and Pavlovi{\'c}, Milena and Sandve, Geir Kjetil and others},
  journal={arXiv preprint arXiv:2008.02217},
  year={2020}
}

@article{bricken2021attention,
  title={Attention Approximates Sparse Distributed Memory},
  author={Bricken, Trenton and Pehlevan, Cengiz},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  year={2021}
}

@article{whittington2021relating,
  title={Relating transformers to models and neural representations of the hippocampal formation},
  author={Whittington, James CR and Warren, Joseph and Behrens, Timothy EJ},
  journal={arXiv preprint arXiv:2112.04035},
  year={2021}
}

@article{keeler1988comparison,
  title={Comparison between Kanerva's SDM and Hopfield-type neural networks},
  author={Keeler, James D},
  journal={Cognitive Science},
  volume={12},
  number={3},
  pages={299--329},
  year={1988},
  publisher={Elsevier}
}

@article{radhakrishnan2020overparameterized,
  title={Overparameterized neural networks implement associative memory},
  author={Radhakrishnan, Adityanarayanan and Belkin, Mikhail and Uhler, Caroline},
  journal={Proceedings of the National Academy of Sciences},
  volume={117},
  number={44},
  pages={27162--27170},
  year={2020},
  publisher={National Acad Sciences}
}

@article{benna2021place,
  title={Place cells may simply be memory cells: Memory compression leads to spatial tuning and history dependence},
  author={Benna, Marcus K and Fusi, Stefano},
  journal={Proceedings of the National Academy of Sciences},
  volume={118},
  number={51},
  year={2021},
  publisher={National Acad Sciences}
}

@article{lisman2009prediction,
  title={Prediction, sequences and the hippocampus},
  author={Lisman, John and Redish, A David},
  journal={Philosophical Transactions of the Royal Society B: Biological Sciences},
  volume={364},
  number={1521},
  pages={1193--1201},
  year={2009},
  publisher={The Royal Society London}
}

@article{stachenfeld2017hippocampus,
  title={The hippocampus as a predictive map},
  author={Stachenfeld, Kimberly L and Botvinick, Matthew M and Gershman, Samuel J},
  journal={Nature neuroscience},
  volume={20},
  number={11},
  pages={1643--1653},
  year={2017},
  publisher={Nature Publishing Group}
}

@article{barron2020prediction,
  title={Prediction and memory: A predictive coding account},
  author={Barron, Helen C and Auksztulewicz, Ryszard and Friston, Karl},
  journal={Progress in neurobiology},
  volume={192},
  pages={101821},
  year={2020},
  publisher={Elsevier}
}

@article{rao1999predictive,
  title={Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects},
  author={Rao, Rajesh PN and Ballard, Dana H},
  journal={Nature neuroscience},
  volume={2},
  number={1},
  pages={79--87},
  year={1999},
  publisher={Nature Publishing Group}
}

@article{friston2003learning,
  title={Learning and inference in the brain},
  author={Friston, Karl},
  journal={Neural Networks},
  volume={16},
  number={9},
  pages={1325--1352},
  year={2003},
  publisher={Elsevier}
}

@article{lecun2010mnist,
  title={MNIST handwritten digit database. 2010},
  author={LeCun, Yann and Cortes, Corinna and Burges, Christopher J},
  journal={URL http://yann. lecun. com/exdb/mnist},
  volume={7},
  number={23},
  pages={6},
  year={2010}
}

@article{krizhevsky2009learning,
  title={The {CIFAR-10} dataset},
  author={Krizhevsky, Alex and Nair, Vinod and Hinton, Geoffrey},
  journal={online: http://www. cs. toronto. edu/kriz/cifar. html},
  volume={55},
  number={5},
  year={2014}
}

@article{rumelhart1986learning,
  title={Learning representations by back-propagating errors},
  author={Rumelhart, David E and Hinton, Geoffrey E and Williams, Ronald J},
  journal={nature},
  volume={323},
  number={6088},
  pages={533--536},
  year={1986},
  publisher={Nature Publishing Group}
}

@article{mikulasch2022dendritic,
  title={Dendritic predictive coding: A theory of cortical computation with spiking neurons},
  author={Mikulasch, Fabian A and Rudelt, Lucas and Wibral, Michael and Priesemann, Viola},
  journal={arXiv preprint arXiv:2205.05303},
  year={2022}
}

@techreport{haynsworth1968schur,
  title={ON THE SCHUR COMPLEMENT.},
  author={Haynsworth, Emilie V},
  year={1968},
  institution={BASEL UNIV (SWITZERLAND) MATHEMATICS INST}
}

@article{song2020can,
  title={Can the Brain Do Backpropagation?---Exact Implementation of Backpropagation in Predictive Coding Networks},
  author={Song, Yuhang and Lukasiewicz, Thomas and Xu, Zhenghua and Bogacz, Rafal},
  journal={Advances in neural information processing systems},
  volume={33},
  pages={22566--22579},
  year={2020}
}


@inproceedings{salvatori2022reverse,
  title = {Reverse Differentiation via Predictive Coding},
  author = {Tommaso Salvatori and Yuhang Song and Zhenghua Xu and Thomas Lukasiewicz and Rafal Bogacz},
  year = {2022},
  booktitle = {Proceedings of the 36th AAAI Conference on Artificial Intelligence},
  month = {February},
  volume={10177},
  pages={507--524},
  publisher = {AAAI Press},
}

@article{millidge2022predictive,
  title={Predictive Coding: Towards a Future of Deep Learning beyond Backpropagation?},
  author={Millidge, Beren and Salvatori, Tommaso and Song, Yuhang and Bogacz, Rafal and Lukasiewicz, Thomas},
  journal={arXiv preprint arXiv:2202.09467},
  year={2022}
}

@InProceedings{Millidge22universal,
  title = 	 {Universal Hopfield Networks: A General Framework for Single-Shot Associative Memory Models},
  author =       {Millidge, Beren and Salvatori, Tommaso and Song, Yuhang and Lukasiewicz, Thomas and Bogacz, Rafal},
  booktitle = 	 {Proceedings of the 39th International Conference on Machine Learning},
  pages = 	 {15561--15583},
  year = 	 {2022},
  editor = 	 {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
  volume = 	 {162},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {17--23 Jul},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v162/millidge22a/millidge22a.pdf},
  url = 	 {https://proceedings.mlr.press/v162/millidge22a.html},
  abstract = 	 {A large number of neural network models of associative memory have been proposed in the literature. These include the classical Hopfield networks (HNs), sparse distributed memories (SDMs), and more recently the modern continuous Hopfield networks (MCHNs), which possess close links with self-attention in machine learning. In this paper, we propose a general framework for understanding the operation of such memory networks as a sequence of three operations: similarity, separation, and projection. We derive all these memory models as instances of our general framework with differing similarity and separation functions. We extend the mathematical framework of Krotov et al (2020) to express general associative memory models using neural network dynamics with local computation, and derive a general energy function that is a Lyapunov function of the dynamics. Finally, using our framework, we empirically investigate the capacity of using different similarity functions for these associative memory models, beyond the dot product similarity measure, and demonstrate empirically that Euclidean or Manhattan distance similarity metrics perform substantially better in practice on many tasks, enabling a more robust retrieval and higher memory capacity than existing&nbsp;models.}
}


@article{millidge2022predictive2,
  title={Predictive coding approximates backprop along arbitrary computation graphs},
  author={Millidge, Beren and Tschantz, Alexander and Buckley, Christopher L},
  journal={Neural Computation},
  volume={34},
  number={6},
  pages={1329--1368},
  year={2022},
  publisher={MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}
}

@article{alonso2022theoretical,
  title={A Theoretical Framework for Inference Learning},
  author={Alonso, Nick and Millidge, Beren and Krichmar, Jeff and Neftci, Emre},
  journal={arXiv preprint arXiv:2206.00164},
  year={2022}
}

@article{amit1987information,
  title={Information storage in neural networks with low levels of activity},
  author={Amit, Daniel J and Gutfreund, Hanoch and Sompolinsky, Haim},
  journal={Physical Review A},
  volume={35},
  number={5},
  pages={2293},
  year={1987},
  publisher={APS}
}

@book{hebb1949organization,
  title={The {O}rganization of {B}ehavior.},
  author={Hebb, Donald},
  year={1949},
  publisher={Wiley},
  address={New York}
}

@article{whittington2019theories,
  title={Theories of error back-propagation in the brain},
  author={Whittington, James CR and Bogacz, Rafal},
  journal={Trends in cognitive sciences},
  volume={23},
  number={3},
  pages={235--250},
  year={2019},
  publisher={Elsevier}
}

@article{chavlis2021drawing,
  title={Drawing inspiration from biological dendrites to empower artificial neural networks},
  author={Chavlis, Spyridon and Poirazi, Panayiota},
  journal={Current Opinion in Neurobiology},
  volume={70},
  pages={1--10},
  year={2021},
  publisher={Elsevier}
}

@article{gluck1993hippocampal,
  title={Hippocampal mediation of stimulus representation: A computational theory},
  author={Gluck, Mark A and Myers, Catherine E},
  journal={Hippocampus},
  volume={3},
  number={4},
  pages={491--516},
  year={1993},
  publisher={Wiley Online Library}
}

@article{richards2019dendritic,
  title={Dendritic solutions to the credit assignment problem},
  author={Richards, Blake A and Lillicrap, Timothy P},
  journal={Current opinion in neurobiology},
  volume={54},
  pages={28--36},
  year={2019},
  publisher={Elsevier}
}

@article{kanai2015cerebral,
  title={Cerebral hierarchies: predictive processing, precision and the pulvinar},
  author={Kanai, Ryota and Komura, Yutaka and Shipp, Stewart and Friston, Karl},
  journal={Philosophical Transactions of the Royal Society B: Biological Sciences},
  volume={370},
  number={1668},
  pages={20140169},
  year={2015},
  publisher={The Royal Society}
}

@article{bastos2012canonical,
  title={Canonical microcircuits for predictive coding},
  author={Bastos, Andre M and Usrey, W Martin and Adams, Rick A and Mangun, George R and Fries, Pascal and Friston, Karl J},
  journal={Neuron},
  volume={76},
  number={4},
  pages={695--711},
  year={2012},
  publisher={Elsevier}
}

@article{shipp2016neural,
  title={Neural elements for predictive coding},
  author={Shipp, Stewart},
  journal={Frontiers in psychology},
  volume={7},
  pages={1792},
  year={2016},
  publisher={Frontiers Media SA}
}

@article{millidge2021predictive,
  title={Predictive coding: a theoretical and experimental review},
  author={Millidge, Beren and Seth, Anil and Buckley, Christopher L},
  journal={arXiv preprint arXiv:2107.12979},
  year={2021}
}

@article{wirth2009trial,
  title={Trial outcome and associative learning signals in the monkey hippocampus},
  author={Wirth, Sylvia and Avsar, Emin and Chiu, Cindy C and Sharma, Varun and Smith, Anne C and Brown, Emery and Suzuki, Wendy A},
  journal={Neuron},
  volume={61},
  number={6},
  pages={930--940},
  year={2009},
  publisher={Elsevier}
}

@article{mehta2000experience,
  title={Experience-dependent asymmetric shape of hippocampal receptive fields},
  author={Mehta, Mayank R and Quirk, Michael C and Wilson, Matthew A},
  journal={Neuron},
  volume={25},
  number={3},
  pages={707--715},
  year={2000},
  publisher={Elsevier}
}


@article{ku2021contributions,
  title={The contributions of entorhinal cortex and hippocampus to error driven learning},
  author={Ku, Shih-pi and Hargreaves, Eric L and Wirth, Sylvia and Suzuki, Wendy A},
  journal={Communications biology},
  volume={4},
  number={1},
  pages={1--12},
  year={2021},
  publisher={Nature Publishing Group}
}

@article{lisman1999relating,
  title={Relating hippocampal circuitry to function: recall of memory sequences by reciprocal dentate--CA3 interactions},
  author={Lisman, John E},
  journal={Neuron},
  volume={22},
  number={2},
  pages={233--242},
  year={1999},
  publisher={Elsevier}
}

@inproceedings{bogacz1999high,
  title={High Capacity Neural Networks for Familiarity Discrimination},
  author={Bogacz, R and Brown, MW and Giraud-Carrier, C},
  booktitle={Proceedings of the Ninth International Conference on Artificial Neural Networks (ICANN99)},
  pages={773--778},
  year={1999},
  organization={Institution of Engineering and Technology (IET)}
}

@article{palacios2021acetylcholine,
  title={Acetylcholine prioritises direct synaptic inputs from entorhinal cortex to CA1 by differential modulation of feedforward inhibitory circuits},
  author={Palacios-Filardo, Jon and Udakis, Matt and Brown, Giles A and Tehan, Benjamin G and Congreve, Miles S and Nathan, Pradeep J and Brown, Alastair JH and Mellor, Jack R},
  journal={Nature communications},
  volume={12},
  number={1},
  pages={1--16},
  year={2021},
  publisher={Nature Publishing Group}
}

@inproceedings{deng2009imagenet,
  title={{ImageNet: A} large-scale hierarchical image database},
  author={Deng, Jia and Dong, Wei and Socher, Richard and Li, Li-Jia and Li, Kai and Fei-Fei, Li},
  booktitle={2009 IEEE conference on computer vision and pattern recognition},
  pages={248--255},
  year={2009},
  organization={Ieee}
}

@book{arnold1992ordinary,
  title={Ordinary differential equations},
  author={Arnold, Vladimir I.},
  year={1992},
  publisher={Springer Science \& Business Media}
}

@article{rao1997correlates,
  title={Correlates of attention in a model of dynamic visual recognition},
  author={Rao, Rajesh},
  journal={Advances in neural information processing systems},
  volume={10},
  year={1997}
}

@article{rosenbaum2022relationship,
  title={On the relationship between predictive coding and backpropagation},
  author={Rosenbaum, Robert},
  journal={Plos one},
  volume={17},
  number={3},
  pages={e0266102},
  year={2022},
  publisher={Public Library of Science San Francisco, CA USA}
}

@article{salvatori2021predictive,
  title={Predictive coding can do exact backpropagation on convolutional and recurrent neural networks},
  author={Salvatori, Tommaso and Song, Yuhang and Lukasiewicz, Thomas and Bogacz, Rafal and Xu, Zhenghua},
  journal={arXiv preprint arXiv:2103.03725},
  year={2021}
}

@article{millidge2021neural,
  title={Neural kalman filtering},
  author={Millidge, Beren and Tschantz, Alexander and Seth, Anil and Buckley, Christopher},
  journal={arXiv preprint arXiv:2102.10021},
  year={2021}
}

@article{huang2011predictive,
  title={Predictive coding},
  author={Huang, Yanping and Rao, Rajesh PN},
  journal={Wiley Interdisciplinary Reviews: Cognitive Science},
  volume={2},
  number={5},
  pages={580--593},
  year={2011},
  publisher={Wiley Online Library}
}

@article{mcclelland1995there,
  title={Why there are complementary learning systems in the hippocampus and neocortex: insights from the successes and failures of connectionist models of learning and memory.},
  author={McClelland, James L and McNaughton, Bruce L and O'Reilly, Randall C},
  journal={Psychological review},
  volume={102},
  number={3},
  pages={419},
  year={1995},
  publisher={American Psychological Association}
}

@article{olshausen1996emergence,
  title={Emergence of simple-cell receptive field properties by learning a sparse code for natural images},
  author={Olshausen, Bruno A and Field, David J},
  journal={Nature},
  volume={381},
  number={6583},
  pages={607--609},
  year={1996},
  publisher={Nature Publishing Group}
}

@article{elias1955predictive,
  title={Predictive coding--I},
  author={Elias, Peter},
  journal={IRE transactions on information theory},
  volume={1},
  number={1},
  pages={16--24},
  year={1955},
  publisher={IEEE}
}

@article{srinivasan1982predictive,
  title={Predictive coding: a fresh view of inhibition in the retina},
  author={Srinivasan, Mandyam Veerambudi and Laughlin, Simon Barry and Dubs, Andreas},
  journal={Proceedings of the Royal Society of London. Series B. Biological Sciences},
  volume={216},
  number={1205},
  pages={427--459},
  year={1982},
  publisher={The Royal Society London}
}

@article{kiebel2009recognizing,
  title={Recognizing sequences of sequences},
  author={Kiebel, Stefan J and Von Kriegstein, Katharina and Daunizeau, Jean and Friston, Karl J},
  journal={PLoS computational biology},
  volume={5},
  number={8},
  pages={e1000464},
  year={2009},
  publisher={Public Library of Science San Francisco, USA}
}

@article{friston2009predictive,
  title={Predictive coding under the free-energy principle},
  author={Friston, Karl and Kiebel, Stefan},
  journal={Philosophical transactions of the Royal Society B: Biological sciences},
  volume={364},
  number={1521},
  pages={1211--1221},
  year={2009},
  publisher={The Royal Society London}
}

@incollection{limanowski2017dis,
  title={(Dis-) Attending to the body},
  author={Limanowski, Jakub},
  booktitle={PPP-Philosophy and Predictive Processing},
  year={2017},
  publisher={Open MIND. Frankfurt am Main: MIND Group}
}

@article{clark2013many,
  title={The many faces of precision (Replies to commentaries on “Whatever next? Neural prediction, situated agents, and the future of cognitive science”)},
  author={Clark, Andy},
  journal={Frontiers in psychology},
  volume={4},
  pages={270},
  year={2013},
  publisher={Frontiers Media SA}
}

@book{gray2003neuropsychology,
  title={The Neuropsychology of Anxiety: An Enquiry Into the Function of the Septo-hippocampal System},
  author={Gray, Jeffrey Alan and McNaughton, Neil},
  year={2003},
  publisher={Oxford University Press}
}

@article{ororbia2022neural,
  title={The neural coding framework for learning generative models},
  author={Ororbia, Alexander and Kifer, Daniel},
  journal={Nature communications},
  volume={13},
  number={1},
  pages={1--14},
  year={2022},
  publisher={Nature Publishing Group}
}

@article{oord2018representation,
  title={Representation learning with contrastive predictive coding},
  author={Oord, Aaron van den and Li, Yazhe and Vinyals, Oriol},
  journal={arXiv preprint arXiv:1807.03748},
  year={2018},
  url={https://arxiv.org/pdf/1807.03748.pdf},
  keywords={ml}
}
@article{rainer1998selective,
  title={Selective representation of relevant information by neurons in the primate prefrontal cortex},
  author={Rainer, Gregor and Asaad, Wael F and Miller, Earl K},
  journal={Nature},
  volume={393},
  number={6685},
  pages={577--579},
  year={1998},
  publisher={Nature Publishing Group}
}


@article{spratling2017review,
  title={A review of predictive coding algorithms},
  author={Spratling, Michael W},
  journal={Brain and cognition},
  volume={112},
  pages={92--97},
  year={2017},
  publisher={Elsevier},
  keywords={survey}
}
@String(spratling2017review="A short and concise review of predictive coding algorithms up to 2017.")


@article{remondes2004role,
  title={Role for a cortical input to hippocampal area CA1 in the consolidation of a long-term memory},
  author={Remondes, Miguel and Schuman, Erin M},
  journal={Nature},
  volume={431},
  number={7009},
  pages={699--703},
  year={2004},
  publisher={Nature Publishing Group}
}

@article{urbanczik2014learning,
  title={Learning by the dendritic prediction of somatic spiking},
  author={Urbanczik, Robert and Senn, Walter},
  journal={Neuron},
  volume={81},
  number={3},
  pages={521--528},
  year={2014},
  publisher={Elsevier}
}

@article{mikulasch2023error,
  title={Where is the error? Hierarchical predictive coding through dendritic error computation},
  author={Mikulasch, Fabian A and Rudelt, Lucas and Wibral, Michael and Priesemann, Viola},
  journal={Trends in Neurosciences},
  volume={46},
  number={1},
  pages={45--59},
  year={2023},
  publisher={Elsevier}
}

@article{vanrullen2021deep,
  title={Deep learning and the Global Workspace Theory},
  author={VanRullen, Rufin and Kanai, Ryota},
  journal={Trends in Neurosciences},
  year={2021},
  publisher={Elsevier}
}

@String(huang2011predictive="A nice review of simple predictive coding architectures with a focus on their potential implementation in the brain.")
@book{sutton2018reinforcement,
  title={Reinforcement learning: An introduction},
  author={Sutton, Richard S and Barto, Andrew G},
  year={2018},
  publisher={MIT press}
}

@article{hosoya2005dynamic,
  title={Dynamic predictive coding by the retina},
  author={Hosoya, Toshihiko and Baccus, Stephen A and Meister, Markus},
  journal={Nature},
  volume={436},
  number={7047},
  pages={71--77},
  year={2005},
  publisher={Nature Publishing Group},
  keywords={process},
  url={https://www.nature.com/articles/nature03689}
}

@article{spratling2008predictive,
  title={Predictive coding as a model of biased competition in visual attention},
  author={Spratling, Michael W},
  journal={Vision research},
  volume={48},
  number={12},
  pages={1391--1408},
  year={2008},
  publisher={Elsevier},
  keywords={process},
  url={https://www.sciencedirect.com/science/article/pii/S0042698908001466}
}

@String(spratling2008predictive="Demonstrates that predictive coding is equivalent to popular biased competition models of neural function.")
@article{spratling2008reconciling,
  title={Reconciling predictive coding and biased competition models of cortical function},
  author={Spratling, Michael W},
  journal={Frontiers in computational neuroscience},
  volume={2},
  pages={4},
  year={2008},
  publisher={Frontiers},
  keywords={process},
  url={https://www.frontiersin.org/articles/10.3389/neuro.10.004.2008/full}
}



@article{feldman2010attention,
  title={Attention, uncertainty, and free-energy},
  author={Feldman, Harriet and Friston, Karl},
  journal={Frontiers in human neuroscience},
  volume={4},
  pages={215},
  year={2010},
  publisher={Frontiers},
  keywords={classic},
  url={https://www.frontiersin.org/articles/10.3389/fnhum.2010.00215/full}
}
@String(feldman2010attention="Makes a conjectured link between precision in predictive coding and attention in the brain.")

@article{friston2008variational,
  title={DEM: a variational treatment of dynamic systems},
  author={Friston, Karl and Trujillo-Barreto, N and Daunizeau, Jean},
  journal={Neuroimage},
  volume={41},
  number={3},
  pages={849--885},
  year={2008},
  publisher={Elsevier},
  keywords={classic}
}
@String(friston2008variational="Extends predictive coding to generalised coordinates, and derives the necessary inference algorithms for working with them -- i.e. DEM, dynamic expectation maximisation.")
@article{friston2010generalised,
  title={Generalised filtering},
  author={Friston, Karl and Stephan, Klaas and Li, Baojuan and Daunizeau, Jean},
  journal={Mathematical Problems in Engineering},
  volume={2010},
  year={2010},
  publisher={Hindawi},
  keywords={classic},
  url={https://www.hindawi.com/journals/mpe/2010/621670/}
}

@String(friston2008variational2="Foundational treatment of variational inference for dynamical systems, as represented in generalised coordinates. Also relates variational filtering to other non-variational schemes like particle filtering and Kalman filtering.")
@article{friston2008variational2,
  title={Variational filtering},
  author={Friston, Karl},
  journal={NeuroImage},
  volume={41},
  number={3},
  pages={747--766},
  year={2008},
  publisher={Elsevier},
  keywords={classic},
  url={https://www.sciencedirect.com/science/article/pii/S1053811908002462?casa_token=bzK7h_aIzY0AAAAA:rg1CzE6vNo-cktIHO_9EAoqmR5Zpy89klEn-Wy3NAzMoR8NcWgaF5_zEzyhrRB76N5RPyCZTIlY}
}

@article{shipp2013reflections,
  title={Reflections on agranular architecture: predictive coding in the motor cortex},
  author={Shipp, Stewart and Adams, Rick A and Friston, Karl J},
  journal={Trends in neurosciences},
  volume={36},
  number={12},
  pages={706--716},
  year={2013},
  publisher={Elsevier},
  keywords={process},
  url={https://www.sciencedirect.com/science/article/pii/S0166223613001604}
}
@String(shipp2013reflections="A process theory of predictive coding including action predictions which implement active inference (continuous version).")

@article{tschantz2022hybrid,
  title={Hybrid Predictive Coding: Inferring, Fast and Slow},
  author={Tschantz, Alexander and Millidge, Beren and Seth, Anil K and Buckley, Christopher L},
  journal={arXiv preprint arXiv:2204.02169},
  year={2022}
}
@article{heilbron2018great,
  title={Great expectations: is there evidence for predictive coding in auditory cortex?},
  author={Heilbron, Micha and Chait, Maria},
  journal={Neuroscience},
  volume={389},
  pages={54--73},
  year={2018},
  publisher={Elsevier},
  keywords={process},
  url={https://www.sciencedirect.com/science/article/pii/S030645221730547X}
}
@incollection{kok2015predictive,
  title={Predictive coding in sensory cortex},
  author={Kok, Peter and de Lange, Floris P},
  booktitle={An introduction to model-based cognitive neuroscience},
  pages={221--244},
  year={2015},
  publisher={Springer},
  keywords={process},
  url={https://link.springer.com/chapter/10.1007/978-1-4939-2236-9_11}
}

@article{sterzer2018predictive,
  title={The predictive coding account of psychosis},
  author={Sterzer, Philipp and Adams, Rick A and Fletcher, Paul and Frith, Chris and Lawrie, Stephen M and Muckli, Lars and Petrovic, Predrag and Uhlhaas, Peter and Voss, Martin and Corlett, Philip R},
  journal={Biological psychiatry},
  volume={84},
  number={9},
  pages={634--643},
  year={2018},
  publisher={Elsevier},
  keywords={applications},
  url={https://www.sciencedirect.com/science/article/pii/S0006322318315324}
}
@article{seth2012interoceptive,
  title={An interoceptive predictive coding model of conscious presence},
  author={Seth, Anil K and Suzuki, Keisuke and Critchley, Hugo D},
  journal={Frontiers in psychology},
  volume={2},
  pages={395},
  year={2012},
  publisher={Frontiers},
  keywords={applications},
  url={https://www.frontiersin.org/articles/10.3389/fpsyg.2011.00395/full}
}
@article{seth2013extending,
  title={Extending predictive processing to the body: emotion as interoceptive inference},
  author={Seth, Anil K and Critchley, Hugo D},
  journal={Behav. Brain Sci},
  volume={36},
  number={3},
  pages={227--228},
  year={2013},
  keywords={applications},
  url={https://pdfs.semanticscholar.org/d8ab/0cbc439db9c0a5783bc4c51a7bb6454ff711.pdf}
}
@book{clark2015surfing,
  title={Surfing uncertainty: Prediction, action, and the embodied mind},
  author={Clark, Andy},
  year={2015},
  publisher={Oxford University Press},
  keywords={classic},
  url={https://books.google.co.uk/books?hl=en&lr=&id=TnqECgAAQBAJ&oi=fnd&pg=PP1&dq=andy+clark+surfing+uncertainty&ots=aurm6iDbJR&sig=A5uoJIteAk4JDCEpnQaa2KAbfg4&redir_esc=y#v=onepage&q=andy%20clark%20surfing%20uncertainty&f=false}
}
@String(clark2015surfing="Andy's book is great for a high level overview, strong intuition pumps for understanding the theory, and a fantastic review of potential evidence and neuropyschiatric applications.")

@article{millidge2020predictive,
  title={Predictive Coding Approximates Backprop along Arbitrary Computation Graphs},
  author={Millidge, Beren and Tschantz, Alexander and Buckley, Christopher L},
  journal={arXiv preprint arXiv:2006.04182},
  year={2020},
  keywords={backprop},
  url={https://arxiv.org/abs/2006.04182}
}

@article{orchard2019making,
  title={Making Predictive Coding Networks Generative},
  author={Orchard, Jeff and Sun, Wei},
  journal={arXiv preprint arXiv:1910.12151},
  year={2019},
  keywords={extensions},
  url={https://arxiv.org/abs/1910.12151}
}
@article{sun2020predictive,
  title={A Predictive-Coding Network That Is Both Discriminative and Generative},
  author={Sun, Wei and Orchard, Jeff},
  journal={Neural Computation},
  volume={32},
  number={10},
  pages={1836--1862},
  year={2020},
  publisher={MIT Press},
  keywords={extensions},
  url={https://www.mitpressjournals.org/doi/abs/10.1162/neco_a_01311}
}
@article{bogacz2020dopamine,
  title={Dopamine role in learning and action inference},
  author={Bogacz, Rafal},
  journal={Elife},
  volume={9},
  pages={e53262},
  year={2020},
  publisher={eLife Sciences Publications Limited},
  keywords={extensions},
  url={https://elifesciences.org/articles/53262}
}
@article{lotter2016deep,
  title={Deep predictive coding networks for video prediction and unsupervised learning},
  author={Lotter, William and Kreiman, Gabriel and Cox, David},
  journal={arXiv preprint arXiv:1605.08104},
  year={2016},
  keywords={ml},
  url={https://arxiv.org/abs/1605.08104}
}
@inproceedings{han2019video,
  title={Video representation learning by dense predictive coding},
  author={Han, Tengda and Xie, Weidi and Zisserman, Andrew},
  booktitle={Proceedings of the IEEE International Conference on Computer Vision Workshops},
  pages={0--0},
  year={2019},
  keywords={ml},
  url={https://openaccess.thecvf.com/content_ICCVW_2019/html/HVU/Han_Video_Representation_Learning_by_Dense_Predictive_Coding_ICCVW_2019_paper.html}
}
@article{song2024inferring,
  title={Inferring neural activity before plasticity as a foundation for learning beyond backpropagation},
  author={Song, Yuhang and Millidge, Beren and Salvatori, Tommaso and Lukasiewicz, Thomas and Xu, Zhenghua and Bogacz, Rafal},
  journal={Nature Neuroscience},
  volume={27},
  number={2},
  pages={348--358},
  year={2024},
  publisher={Nature Publishing Group US New York}
}
@article{hafting2005microstructure,
  title={Microstructure of a spatial map in the entorhinal cortex},
  author={Hafting, Torkel and Fyhn, Marianne and Molden, Sturla and Moser, May-Britt and Moser, Edvard I},
  journal={Nature},
  volume={436},
  number={7052},
  pages={801--806},
  year={2005},
  publisher={Nature Publishing Group UK London}
}
@article{o1976place,
  title={Place units in the hippocampus of the freely moving rat},
  author={O'Keefe, John},
  journal={Experimental neurology},
  volume={51},
  number={1},
  pages={78--109},
  year={1976},
  publisher={Elsevier}
}
@article{wills2012abrupt,
  title={The abrupt development of adult-like grid cell firing in the medial entorhinal cortex},
  author={Wills, Thomas J and Barry, Caswell and Cacucci, Francesca},
  journal={Frontiers in neural circuits},
  volume={6},
  pages={21},
  year={2012},
  publisher={Frontiers Media SA}
}
@article{krupic2015grid,
  title={Grid cell symmetry is shaped by environmental geometry},
  author={Krupic, Julija and Bauza, Marius and Burton, Stephen and Barry, Caswell and O’Keefe, John},
  journal={Nature},
  volume={518},
  number={7538},
  pages={232--235},
  year={2015},
  publisher={Nature Publishing Group UK London}
}
@article{sorscher2023unified,
  title={A unified theory for the computational and mechanistic origins of grid cells},
  author={Sorscher, Ben and Mel, Gabriel C and Ocko, Samuel A and Giocomo, Lisa M and Ganguli, Surya},
  journal={Neuron},
  volume={111},
  number={1},
  pages={121--137},
  year={2023},
  publisher={Elsevier}
}
@article{werbos1990backpropagation,
  title={Backpropagation through time: what it does and how to do it},
  author={Werbos, Paul J},
  journal={Proceedings of the IEEE},
  volume={78},
  number={10},
  pages={1550--1560},
  year={1990},
  publisher={IEEE}
}
@article{wen2018deep,
  title={Deep predictive coding network for object recognition},
  author={Wen, Haiguang and Han, Kuan and Shi, Junxing and Zhang, Yizhen and Culurciello, Eugenio and Liu, Zhongming},
  journal={arXiv preprint arXiv:1802.04762},
  year={2018},
  keywords={ml},
  url={https://arxiv.org/abs/1802.04762}
}
@article{chalasani2013deep,
  title={Deep predictive coding networks},
  author={Chalasani, Rakesh and Principe, Jose C},
  journal={arXiv preprint arXiv:1301.3541},
  year={2013},
  keywords={ml},
  url={https://arxiv.org/abs/1301.3541}
}
@article{han2018deep,
  title={Deep predictive coding network with local recurrent processing for object recognition},
  author={Han, Kuan and Wen, Haiguang and Zhang, Yizhen and Fu, Di and Culurciello, Eugenio and Liu, Zhongming},
  journal={Advances in neural information processing systems},
  volume={31},
  pages={9201--9213},
  year={2018},
  keywords={ml},
  url={https://arxiv.org/pdf/1802.04762.pdf}
}
@article{boutin2019meaningful,
  title={Meaningful representations emerge from sparse deep predictive coding},
  author={Boutin, Victor and Franciosini, Angelo and Ruffier, Franck and Perrinet, Laurent},
  journal={arXiv preprint arXiv:1902.07651},
  year={2019},
  keywords={ml},
  url={https://www.researchgate.net/profile/Laurent_Perrinet/publication/331246369_Meaningful_representations_emerge_from_Sparse_Deep_Predictive_Coding/links/5cbe0d1ba6fdcc1d49a86405/Meaningful-representations-emerge-from-Sparse-Deep-Predictive-Coding.pdf}
}
@article{hosseini2020hierarchical,
  title={Hierarchical Predictive Coding Models in a Deep-Learning Framework},
  author={Hosseini, Matin and Maida, Anthony},
  journal={arXiv preprint arXiv:2005.03230},
  year={2020},
  keywords={ml},
  url={https://arxiv.org/abs/2005.03230}
}
@inproceedings{rane2020prednet,
  title={PredNet and Predictive Coding: A Critical Review},
  author={Rane, Roshan Prakash and Sz{\"u}gyi, Edit and Saxena, Vageesh and Ofner, Andr{\'e} and Stober, Sebastian},
  booktitle={Proceedings of the 2020 International Conference on Multimedia Retrieval},
  pages={233--241},
  year={2020},
  keywords={ml},
  url={https://dl.acm.org/doi/pdf/10.1145/3372278.3390694?casa_token=-IdjNoI837MAAAAA:XUBgc0lCY-iximHZdHy2lmspktIwT62dW9fYn4UzFyvA2O1JJFtTlCC2e_mKJaEr6nsYNacjnOJ_Rg}
}
@article{walsh2020evaluating,
  title={Evaluating the neurophysiological evidence for predictive processing as a model of perception},
  author={Walsh, Kevin S and McGovern, David P and Clark, Andy and O'Connell, Redmond G},
  journal={Annals of the New York Academy of Sciences},
  volume={1464},
  number={1},
  pages={242},
  year={2020},
  publisher={Wiley-Blackwell},
  keywords={process},
  url={https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7187369/}
}
@String(walsh2020evaluating="A great review delving deep into the evidence for predictive coding being implemented in the brain. Evidence is currently somewhat lacking, although the flexibility of the predictive coding framework allows it to encompass a lot of the findings here.")

@article{lawson2014aberrant,
  title={An aberrant precision account of autism},
  author={Lawson, Rebecca P and Rees, Geraint and Friston, Karl J},
  journal={Frontiers in human neuroscience},
  volume={8},
  pages={302},
  year={2014},
  publisher={Frontiers},
  keywords={applications},
  url={https://www.frontiersin.org/articles/10.3389/fnhum.2014.00302/full}
}
@article{van2013predictive,
  title={A predictive coding perspective on autism spectrum disorders},
  author={Van Boxtel, Jeroen JA and Lu, Hongjing},
  journal={Frontiers in psychology},
  volume={4},
  pages={19},
  year={2013},
  publisher={Frontiers},
  keywords={applications},
  url={https://www.frontiersin.org/articles/10.3389/fpsyg.2013.00019/full}
}
@article{kilner2007predictive,
  title={Predictive coding: an account of the mirror neuron system},
  author={Kilner, James M and Friston, Karl J and Frith, Chris D},
  journal={Cognitive processing},
  volume={8},
  number={3},
  pages={159--166},
  year={2007},
  publisher={Springer},
  keywords={applications},
  url={https://pubmed.ncbi.nlm.nih.gov/17429704/}
}
@article{spratling2010predictive,
  title={Predictive coding as a model of response properties in cortical area V1},
  author={Spratling, Michael W},
  journal={Journal of neuroscience},
  volume={30},
  number={9},
  pages={3531--3543},
  year={2010},
  publisher={Soc Neuroscience},
  keywords={applications},
  url={https://www.jneurosci.org/content/30/9/3531.short}
}
@article{hohwy2008predictive,
  title={Predictive coding explains binocular rivalry: An epistemological review},
  author={Hohwy, Jakob and Roepstorff, Andreas and Friston, Karl},
  journal={Cognition},
  volume={108},
  number={3},
  pages={687--701},
  year={2008},
  publisher={Elsevier},
  keywords={applications}
}
@article{spratling2019fitting,
  title={Fitting predictive coding to the neurophysiological data},
  author={Spratling, MW},
  journal={Brain research},
  volume={1720},
  pages={146313},
  year={2019},
  publisher={Elsevier},
  keywords={process},
  url={https://www.sciencedirect.com/science/article/pii/S0006899319303592?casa_token=Mq4I-lQm6OMAAAAA:qGWBzjDg799T4lW-YLpp6Mm6Hkh5WhNO1D4teQVE6fRuRCY_2Uv3q2d3xpgXtFCzdhzBy9mzSjg}
}

@article{boerlin2013predictive,
  title={Predictive coding of dynamical variables in balanced spiking networks},
  author={Boerlin, Martin and Machens, Christian K and Den{\`e}ve, Sophie},
  journal={PLoS computational biology},
  volume={9},
  number={11},
  pages={e1003258},
  year={2013},
  publisher={Public Library of Science San Francisco, USA}
}
@article{brendel2020learning,
  title={Learning to represent signals spike by spike},
  author={Brendel, Wieland and Bourdoukan, Ralph and Vertechi, Pietro and Machens, Christian K and Den{\'e}ve, Sophie},
  journal={PLoS computational biology},
  volume={16},
  number={3},
  pages={e1007692},
  year={2020},
  publisher={Public Library of Science San Francisco, CA USA}
}
@article{barrett2015interoceptive,
  title={Interoceptive predictions in the brain},
  author={Barrett, Lisa Feldman and Simmons, W Kyle},
  journal={Nature reviews neuroscience},
  volume={16},
  number={7},
  pages={419--429},
  year={2015},
  publisher={Nature Publishing Group}
}
@article{tschantz2021simulating,
  title={Simulating homeostatic, allostatic and goal-directed forms of interoceptive control using Active Inference},
  author={Tschantz, Alexander and Barca, Laura and Maisto, Domenico and Buckley, Christopher L and Seth, Anil and Pezzulo, Giovanni},
  journal={bioRxiv},
  year={2021},
  publisher={Cold Spring Harbor Laboratory}
}
@article{fountas2020predictive,
  title={A predictive processing model of episodic memory and time perception},
  author={Fountas, Zafeirios and Sylaidi, Anastasia and Nikiforou, Kyriacos and Seth, Anil K and Shanahan, Murray and Roseboom, Warrick},
  journal={bioRxiv},
  year={2020},
  publisher={Cold Spring Harbor Laboratory}
}
@article{thomson2010neocortical,
  title={Neocortical layer 6, a review},
  author={Thomson, Alex M},
  journal={Frontiers in neuroanatomy},
  volume={4},
  pages={13},
  year={2010},
  publisher={Frontiers}
}
@article{fiser2016experience,
  title={Experience-dependent spatial expectations in mouse visual cortex},
  author={Fiser, Aris and Mahringer, David and Oyibo, Hassana K and Petersen, Anders V and Leinweber, Marcus and Keller, Georg B},
  journal={Nature neuroscience},
  volume={19},
  number={12},
  pages={1658--1664},
  year={2016},
  publisher={Nature Publishing Group}
}
@article{takahashi2020active,
  title={Active dendritic currents gate descending cortical outputs in perception},
  author={Takahashi, Naoya and Ebner, Christian and Sigl-Gl{\"o}ckner, Johanna and Moberg, Sara and Nierwetberg, Svenja and Larkum, Matthew E},
  journal={Nature Neuroscience},
  volume={23},
  number={10},
  pages={1277--1285},
  year={2020},
  publisher={Nature Publishing Group}
}
@article{dhariwal2020jukebox,
  title={Jukebox: A generative model for music},
  author={Dhariwal, Prafulla and Jun, Heewoo and Payne, Christine and Kim, Jong Wook and Radford, Alec and Sutskever, Ilya},
  journal={arXiv preprint arXiv:2005.00341},
  year={2020}
}

@inproceedings{kok2015role,
  title={On the role of expectation in visual perception: A top-down view of early visual cortex},
  author={Kok, Peter and De Lange, FP},
  booktitle={Conference Abstract: XII International Conference on Cognitive Neuroscience (ICON-XII). doi: 10.3389/conf. fnhum},
  volume={427},
  year={2015}
}
@article{bell2016encoding,
  title={Encoding of stimulus probability in macaque inferior temporal cortex},
  author={Bell, Andrew H and Summerfield, Christopher and Morin, Elyse L and Malecek, Nicholas J and Ungerleider, Leslie G},
  journal={Current Biology},
  volume={26},
  number={17},
  pages={2280--2290},
  year={2016},
  publisher={Elsevier}
}

@article{amorim2010whose,
  title={Whose cortical column would that be?},
  author={Amorim Da Costa, Nuno Miguel M and Martin, Kevan},
  journal={Frontiers in neuroanatomy},
  volume={4},
  pages={16},
  year={2010},
  publisher={Frontiers}
}
@article{harris2018organization,
  title={The organization of intracortical connections by layer and cell class in the mouse brain},
  author={Harris, Julie A and Mihalas, Stefan and Hirokawa, Karla E and Whitesell, Jennifer D and Knox, Joseph and Bernard, Amy and Bohn, Philip and Caldejon, Shiella and Casal, Linzy and Cho, Andrew and others},
  journal={BioRxiv},
  pages={292961},
  year={2018},
  publisher={Cold Spring Harbor Laboratory}
}

@article{friston2008hierarchical,
  title={Hierarchical models in the brain},
  author={Friston, Karl},
  journal={PLoS computational biology},
  volume={4},
  number={11},
  year={2008},
  publisher={Public Library of Science}
}
@article{millidge2020relaxing,
  title={Relaxing the constraints on predictive coding models},
  author={Millidge, Beren and Tschantz, Alexander and Seth, Anil and Buckley, Christopher L},
  journal={arXiv preprint arXiv:2010.01047},
  year={2020}
}
@book{buzsaki2006rhythms,
  title={Rhythms of the Brain},
  author={Buzsaki, Gyorgy},
  year={2006},
  publisher={Oxford University Press}
}

@article{buckley2017free,
  title={The free energy principle for action and perception: A mathematical review},
  author={Buckley, Christopher L and Kim, Chang Sub and McGregor, Simon and Seth, Anil K},
  journal={Journal of Mathematical Psychology},
  volume={81},
  pages={55--79},
  year={2017},
  publisher={Elsevier}
}

@article{lillicrap2014random,
  title={Random feedback weights support learning in deep neural networks},
  author={Lillicrap, Timothy P and Cownden, Daniel and Tweed, Douglas B and Akerman, Colin J},
  journal={arXiv preprint arXiv:1411.0247},
  year={2014}
}
@article{lillicrap2016random,
  title={Random synaptic feedback weights support error backpropagation for deep learning},
  author={Lillicrap, Timothy P and Cownden, Daniel and Tweed, Douglas B and Akerman, Colin J},
  journal={Nature communications},
  volume={7},
  number={1},
  pages={1--10},
  year={2016},
  publisher={Nature Publishing Group}
}
@inproceedings{nokland2016direct,
  title={Direct feedback alignment provides learning in deep neural networks},
  author={N{\o}kland, Arild},
  booktitle={Advances in neural information processing systems},
  pages={1037--1045},
  year={2016}
}
@article{ororbia2020continual,
  title={Continual learning of recurrent neural networks by locally aligning distributed representations},
  author={Ororbia, Alexander and Mali, Ankur and Giles, C Lee and Kifer, Daniel},
  journal={IEEE Transactions on Neural Networks and Learning Systems},
  year={2020},
  publisher={IEEE}
}
@inproceedings{ororbia2019biologically,
  title={Biologically motivated algorithms for propagating local target representations},
  author={Ororbia, Alexander G and Mali, Ankur},
  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
  volume={33},
  pages={4651--4658},
  year={2019}
}
@inproceedings{balduzzi2015kickback,
  title={Kickback cuts backprop's red-tape: biologically plausible credit assignment in neural networks},
  author={Balduzzi, David and Vanchinathan, Hastagiri and Buhmann, Joachim},
  booktitle={Twenty-Ninth AAAI Conference on Artificial Intelligence},
  year={2015}
}
@inproceedings{lee2015difference,
  title={Difference target propagation},
  author={Lee, Dong-Hyun and Zhang, Saizheng and Fischer, Asja and Bengio, Yoshua},
  booktitle={Joint european conference on machine learning and knowledge discovery in databases},
  pages={498--515},
  year={2015},
  organization={Springer}
}
@article{zenke2018superspike,
  title={Superspike: Supervised learning in multilayer spiking neural networks},
  author={Zenke, Friedemann and Ganguli, Surya},
  journal={Neural computation},
  volume={30},
  number={6},
  pages={1514--1541},
  year={2018},
  publisher={MIT Press}
}
@article{mostafa2018deep,
  title={Deep supervised learning using local errors},
  author={Mostafa, Hesham and Ramesh, Vishwajith and Cauwenberghs, Gert},
  journal={Frontiers in neuroscience},
  volume={12},
  pages={608},
  year={2018},
  publisher={Frontiers}
}
@article{scellier2018generalization,
  title={Generalization of equilibrium propagation to vector field dynamics},
  author={Scellier, Benjamin and Goyal, Anirudh and Binas, Jonathan and Mesnard, Thomas and Bengio, Yoshua},
  journal={arXiv preprint arXiv:1808.04873},
  year={2018}
}

@article{bengio2015early,
  title={Early inference in energy-based models approximates back-propagation},
  author={Bengio, Yoshua and Fischer, Asja},
  journal={arXiv preprint arXiv:1510.02777},
  year={2015}
}
@inproceedings{liao2016important,
  title={How important is weight symmetry in backpropagation?},
  author={Liao, Qianli and Leibo, Joel Z and Poggio, Tomaso},
  booktitle={Thirtieth AAAI Conference on Artificial Intelligence},
  year={2016}
}
@article{baldi2016theory,
  title={A theory of local learning, the learning channel, and the optimality of backpropagation},
  author={Baldi, Pierre and Sadowski, Peter},
  journal={Neural Networks},
  volume={83},
  pages={51--74},
  year={2016},
  publisher={Elsevier}
}
@article{sporea2013supervised,
  title={Supervised learning in multilayer spiking neural networks},
  author={Sporea, Ioana and Gr{\"u}ning, Andr{\'e}},
  journal={Neural computation},
  volume={25},
  number={2},
  pages={473--509},
  year={2013},
  publisher={MIT Press}
}
@article{schiess2016somato,
  title={Somato-dendritic synaptic plasticity and error-backpropagation in active dendrites},
  author={Schiess, Mathieu and Urbanczik, Robert and Senn, Walter},
  journal={PLoS computational biology},
  volume={12},
  number={2},
  year={2016},
  publisher={Public Library of Science}
}

@article{bengio2017stdp,
  title={STDP-compatible approximation of backpropagation in an energy-based model},
  author={Bengio, Yoshua and Mesnard, Thomas and Fischer, Asja and Zhang, Saizheng and Wu, Yuhuai},
  journal={Neural computation},
  volume={29},
  number={3},
  pages={555--577},
  year={2017},
  publisher={MIT Press}
}
@article{o1996biologically,
  title={Biologically plausible error-driven learning using local activation differences: The generalized recirculation algorithm},
  author={O'Reilly, Randall C},
  journal={Neural computation},
  volume={8},
  number={5},
  pages={895--938},
  year={1996},
  publisher={MIT Press}
}
@article{guerguiev2017towards,
  title={Towards deep learning with segregated dendrites},
  author={Guerguiev, Jordan and Lillicrap, Timothy P and Richards, Blake A},
  journal={Elife},
  volume={6},
  pages={e22901},
  year={2017},
  publisher={eLife Sciences Publications Limited}
}
@article{marblestone2016toward,
  title={Toward an integration of deep learning and neuroscience},
  author={Marblestone, Adam H and Wayne, Greg and Kording, Konrad P},
  journal={Frontiers in computational neuroscience},
  volume={10},
  pages={94},
  year={2016},
  publisher={Frontiers}
}
@article{hassabis2017neuroscience,
  title={Neuroscience-inspired artificial intelligence},
  author={Hassabis, Demis and Kumaran, Dharshan and Summerfield, Christopher and Botvinick, Matthew},
  journal={Neuron},
  volume={95},
  number={2},
  pages={245--258},
  year={2017},
  publisher={Elsevier}
}
 @article{richards2019deep,
  title={A deep learning framework for neuroscience},
  author={Richards, Blake A and Lillicrap, Timothy P and Beaudoin, Philippe and Bengio, Yoshua and Bogacz, Rafal and Christensen, Amelia and Clopath, Claudia and Costa, Rui Ponte and de Berker, Archy and Ganguli, Surya and others},
  journal={Nature neuroscience},
  volume={22},
  number={11},
  pages={1761--1770},
  year={2019},
  publisher={Nature Publishing Group}
}
@book{hawkins2007intelligence,
  title={On intelligence: How a new understanding of the brain will lead to the creation of truly intelligent machines},
  author={Hawkins, Jeff and Blakeslee, Sandra},
  year={2007},
  publisher={Macmillan}
}
@article{george2009towards,
  title={Towards a mathematical theory of cortical micro-circuits},
  author={George, Dileep and Hawkins, Jeff},
  journal={PLoS computational biology},
  volume={5},
  number={10},
  year={2009},
  publisher={Public Library of Science}
}

@article{o2017deep,
  title={Deep predictive learning: a comprehensive model of three visual streams},
  author={O'Reilly, Randall C and Wyatte, Dean R and Rohrlich, John},
  journal={arXiv preprint arXiv:1709.04654},
  year={2017}
}
@article{seung2003learning,
  title={Learning in spiking neural networks by reinforcement of stochastic synaptic transmission},
  author={Seung, H Sebastian},
  journal={Neuron},
  volume={40},
  number={6},
  pages={1063--1073},
  year={2003},
  publisher={Elsevier}
}
@article{williams1992simple,
  title={Simple statistical gradient-following algorithms for connectionist reinforcement learning},
  author={Williams, Ronald J},
  journal={Machine learning},
  volume={8},
  number={3-4},
  pages={229--256},
  year={1992},
  publisher={Springer}
}
@article{mazzoni1991more,
  title={A more biologically plausible learning rule for neural networks.},
  author={Mazzoni, Pietro and Andersen, Richard A and Jordan, Michael I},
  journal={Proceedings of the National Academy of Sciences},
  volume={88},
  number={10},
  pages={4433--4437},
  year={1991},
  publisher={National Acad Sciences}
}
@article{amit2019deep,
  title={Deep learning with asymmetric connections and Hebbian updates},
  author={Amit, Yali},
  journal={Frontiers in computational neuroscience},
  volume={13},
  pages={18},
  year={2019},
  publisher={Frontiers}
}
@article{crick1989recent,
  title={The recent excitement about neural networks},
  author={Crick, Francis},
  journal={Nature},
  volume={337},
  number={6203},
  pages={129--132},
  year={1989},
  publisher={Springer}
}
@article{rumelhart1985feature,
  title={Feature discovery by competitive learning},
  author={Rumelhart, David E and Zipser, David},
  journal={Cognitive science},
  volume={9},
  number={1},
  pages={75--112},
  year={1985},
  publisher={Wiley Online Library}
}
@incollection{werbos1982applications,
  title={Applications of advances in nonlinear sensitivity analysis},
  author={Werbos, Paul J},
  booktitle={System modeling and optimization},
  pages={762--770},
  year={1982},
  publisher={Springer}
}

@article{friston2019free,
  title={A free energy principle for a particular physics},
  author={Friston, Karl},
  journal={arXiv preprint arXiv:1906.10184},
  year={2019}
}
@book{doya2007bayesian,
  title={Bayesian brain: Probabilistic approaches to neural coding},
  author={Doya, Kenji and Ishii, Shin and Pouget, Alexandre and Rao, Rajesh PN},
  year={2007},
  publisher={MIT press}
}
@article{pouget2013probabilistic,
  title={Probabilistic brains: knowns and unknowns},
  author={Pouget, Alexandre and Beck, Jeffrey M and Ma, Wei Ji and Latham, Peter E},
  journal={Nature neuroscience},
  volume={16},
  number={9},
  pages={1170},
  year={2013},
  publisher={Nature Publishing Group}
}
@article{knill2004bayesian,
  title={The Bayesian brain: the role of uncertainty in neural coding and computation},
  author={Knill, David C and Pouget, Alexandre},
  journal={TRENDS in Neurosciences},
  volume={27},
  number={12},
  pages={712--719},
  year={2004},
  publisher={Elsevier}
}
@inproceedings{krizhevsky2012imagenet,
  title={Imagenet classification with deep convolutional neural networks},
  author={Krizhevsky, Alex and Sutskever, Ilya and Hinton, Geoffrey E},
  booktitle={Advances in neural information processing systems},
  pages={1097--1105},
  year={2012}
}
@inproceedings{szegedy2015going,
  title={Going deeper with convolutions},
  author={Szegedy, Christian and Liu, Wei and Jia, Yangqing and Sermanet, Pierre and Reed, Scott and Anguelov, Dragomir and Erhan, Dumitru and Vanhoucke, Vincent and Rabinovich, Andrew},
  booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition},
  pages={1--9},
  year={2015}
}
@article{kaplan2020scaling,
  title={Scaling laws for neural language models},
  author={Kaplan, Jared and McCandlish, Sam and Henighan, Tom and Brown, Tom B and Chess, Benjamin and Child, Rewon and Gray, Scott and Radford, Alec and Wu, Jeffrey and Amodei, Dario},
  journal={arXiv preprint arXiv:2001.08361},
  year={2020}
}
@inproceedings{vaswani2017attention,
  title={Attention is all you need},
  author={Vaswani, Ashish and Shazeer, Noam and Parmar, Niki and Uszkoreit, Jakob and Jones, Llion and Gomez, Aidan N and Kaiser, {\L}ukasz and Polosukhin, Illia},
  booktitle={Advances in neural information processing systems},
  pages={5998--6008},
  year={2017}
}
@article{radford2019language,
  title={Language models are unsupervised multitask learners},
  author={Radford, Alec and Wu, Jeffrey and Child, Rewon and Luan, David and Amodei, Dario and Sutskever, Ilya},
  journal={OpenAI Blog},
  volume={1},
  number={8},
  pages={9},
  year={2019}
}
@article{mnih2015human,
  title={Human-level control through deep reinforcement learning},
  author={Mnih, Volodymyr and Kavukcuoglu, Koray and Silver, David and Rusu, Andrei A and Veness, Joel and Bellemare, Marc G and Graves, Alex and Riedmiller, Martin and Fidjeland, Andreas K and Ostrovski, Georg and others},
  journal={Nature},
  volume={518},
  number={7540},
  pages={529--533},
  year={2015},
  publisher={Nature Publishing Group}
}
@article{silver2017mastering,
  title={Mastering the game of go without human knowledge},
  author={Silver, David and Schrittwieser, Julian and Simonyan, Karen and Antonoglou, Ioannis and Huang, Aja and Guez, Arthur and Hubert, Thomas and Baker, Lucas and Lai, Matthew and Bolton, Adrian and others},
  journal={Nature},
  volume={550},
  number={7676},
  pages={354--359},
  year={2017},
  publisher={Nature Publishing Group}
}
@article{schrittwieser2019mastering,
  title={Mastering atari, go, chess and shogi by planning with a learned model},
  author={Schrittwieser, Julian and Antonoglou, Ioannis and Hubert, Thomas and Simonyan, Karen and Sifre, Laurent and Schmitt, Simon and Guez, Arthur and Lockhart, Edward and Hassabis, Demis and Graepel, Thore and others},
  journal={arXiv preprint arXiv:1911.08265},
  year={2019}
}
@article{vinyals2019grandmaster,
  title={Grandmaster level in StarCraft II using multi-agent reinforcement learning},
  author={Vinyals, Oriol and Babuschkin, Igor and Czarnecki, Wojciech M and Mathieu, Micha{\"e}l and Dudzik, Andrew and Chung, Junyoung and Choi, David H and Powell, Richard and Ewalds, Timo and Georgiev, Petko and others},
  journal={Nature},
  volume={575},
  number={7782},
  pages={350--354},
  year={2019},
  publisher={Nature Publishing Group}
}
@article{griewank1989automatic,
  title={On automatic differentiation},
  author={Griewank, Andreas and others},
  journal={Mathematical Programming: recent developments and applications},
  volume={6},
  number={6},
  pages={83--107},
  year={1989},
  publisher={Amsterdam}
}
@article{baydin2017automatic,
  title={Automatic differentiation in machine learning: a survey},
  author={Baydin, At{\i}l{\i}m G{\"u}nes and Pearlmutter, Barak A and Radul, Alexey Andreyevich and Siskind, Jeffrey Mark},
  journal={The Journal of Machine Learning Research},
  volume={18},
  number={1},
  pages={5595--5637},
  year={2017},
  publisher={JMLR. org}
}

@article{elliott2017compiling,
  title={Compiling to categories},
  author={Elliott, Conal},
  journal={Proceedings of the ACM on Programming Languages},
  volume={1},
  number={ICFP},
  pages={1--27},
  year={2017},
  publisher={ACM New York, NY, USA}
}
@article{revels2016forward,
  title={Forward-mode automatic differentiation in Julia},
  author={Revels, Jarrett and Lubin, Miles and Papamarkou, Theodore},
  journal={arXiv preprint arXiv:1607.07892},
  year={2016}
}
@article{innes2019zygote,
  title={Zygote: A differentiable programming system to bridge machine learning and scientific computing},
  author={Innes, Mike and Edelman, Alan and Fischer, Keno and Rackauckus, Chris and Saba, Elliot and Shah, Viral B and Tebbutt, Will},
  journal={arXiv preprint arXiv:1907.07587},
  year={2019}
}
@inproceedings{chen2018neural,
  title={Neural ordinary differential equations},
  author={Chen, Tian Qi and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David K},
  booktitle={Advances in neural information processing systems},
  pages={6571--6583},
  year={2018}
}
@article{tzen2019neural,
  title={Neural stochastic differential equations: Deep latent gaussian models in the diffusion limit},
  author={Tzen, Belinda and Raginsky, Maxim},
  journal={arXiv preprint arXiv:1905.09883},
  year={2019}
}
@article{rackauckas2019diffeqflux,
  title={Diffeqflux. jl-A julia library for neural differential equations},
  author={Rackauckas, Chris and Innes, Mike and Ma, Yingbo and Bettencourt, Jesse and White, Lyndon and Dixit, Vaibhav},
  journal={arXiv preprint arXiv:1902.02376},
  year={2019}
}
@article{degrave2019differentiable,
  title={A differentiable physics engine for deep learning in robotics},
  author={Degrave, Jonas and Hermans, Michiel and Dambre, Joni and Wyffels, Francis},
  journal={Frontiers in neurorobotics},
  volume={13},
  pages={6},
  year={2019},
  publisher={Frontiers}
}
@inproceedings{heiden2019real2sim,
  title={Real2sim transfer using differentiable physics},
  author={Heiden, Eric and Millard, David and Sukhatme, Gaurav},
  booktitle={Workshop on Closing the Reality Gap in Sim2real Transfer for Robotic Manipulation},
  year={2019}
}
@article{pal2019raytracer,
  title={RayTracer. jl: A Differentiable Renderer that supports Parameter Optimization for Scene Reconstruction},
  author={Pal, Avik},
  journal={arXiv preprint arXiv:1907.07198},
  year={2019}
}
@article{amos2019differentiable,
  title={The Differentiable Cross-Entropy Method},
  author={Amos, Brandon and Yarats, Denis},
  journal={arXiv preprint arXiv:1909.12830},
  year={2019}
}

@article{parr2018discrete,
  title={The discrete and continuous brain: from decisions to movement—and back again},
  author={Parr, Thomas and Friston, Karl J},
  journal={Neural computation},
  volume={30},
  number={9},
  pages={2319--2347},
  year={2018},
  publisher={MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}
}

@article{okada2017path,
  title={Path integral networks: End-to-end differentiable optimal control},
  author={Okada, Masashi and Rigazio, Luca and Aoshima, Takenobu},
  journal={arXiv preprint arXiv:1706.09597},
  year={2017}
}
@book{boyd2004convex,
  title={Convex optimization},
  author={Boyd, Stephen and Boyd, Stephen P and Vandenberghe, Lieven},
  year={2004},
  publisher={Cambridge university press}
}

@article{hochreiter1997long,
  title={Long short-term memory},
  author={Hochreiter, Sepp and Schmidhuber, J{\"u}rgen},
  journal={Neural computation},
  volume={9},
  number={8},
  pages={1735--1780},
  year={1997},
  publisher={MIT Press}
}
@article{auksztulewicz2016repetition,
  title={Repetition suppression and its contextual determinants in predictive coding},
  author={Auksztulewicz, Ryszard and Friston, Karl},
  journal={cortex},
  volume={80},
  pages={125--140},
  year={2016},
  publisher={Elsevier}
}
@article{weilnhammer2017predictive,
  title={A predictive coding account of bistable perception-a model-based fMRI study},
  author={Weilnhammer, Veith and Stuke, Heiner and Hesselmann, Guido and Sterzer, Philipp and Schmack, Katharina},
  journal={PLoS computational biology},
  volume={13},
  number={5},
  pages={e1005536},
  year={2017},
  publisher={Public Library of Science}
}
@article{watanabe2018illusory,
  title={Illusory motion reproduced by deep neural networks trained for prediction},
  author={Watanabe, Eiji and Kitaoka, Akiyoshi and Sakamoto, Kiwako and Yasugi, Masaki and Tanaka, Kenta},
  journal={Frontiers in psychology},
  volume={9},
  pages={345},
  year={2018},
  publisher={Frontiers}
}
@inproceedings{akrout2019deep,
  title={Deep learning without weight transport},
  author={Akrout, Mohamed and Wilson, Collin and Humphreys, Peter and Lillicrap, Timothy and Tweed, Douglas B},
  booktitle={Advances in Neural Information Processing Systems},
  pages={974--982},
  year={2019}
}
@article{scellier2017equilibrium,
  title={Equilibrium propagation: Bridging the gap between energy-based models and backpropagation},
  author={Scellier, Benjamin and Bengio, Yoshua},
  journal={Frontiers in computational neuroscience},
  volume={11},
  pages={24},
  year={2017},
  publisher={Frontiers}
}
@book{beal2003variational,
  title={Variational algorithms for approximate Bayesian inference},
  author={Beal, Matthew James and others},
  year={2003},
  publisher={university of London London}
}
@article{blei2017variational,
  title={Variational inference: A review for statisticians},
  author={Blei, David M and Kucukelbir, Alp and McAuliffe, Jon D},
  journal={Journal of the American statistical Association},
  volume={112},
  number={518},
  pages={859--877},
  year={2017},
  publisher={Taylor \& Francis}
}
@article{lillicrap2020backpropagation,
  title={Backpropagation and the brain},
  author={Lillicrap, Timothy P and Santoro, Adam and Marris, Luke and Akerman, Colin J and Hinton, Geoffrey},
  journal={Nature Reviews Neuroscience},
  pages={1--12},
  year={2020},
  publisher={Nature Publishing Group}
}
@article{tallec2017unbiased,
  title={Unbiased online recurrent optimization},
  author={Tallec, Corentin and Ollivier, Yann},
  journal={arXiv preprint arXiv:1702.05043},
  year={2017}
}
@article{ollivier2015training,
  title={Training recurrent networks online without backtracking},
  author={Ollivier, Yann and Tallec, Corentin and Charpiat, Guillaume},
  journal={arXiv preprint arXiv:1507.07680},
  year={2015}
}
@inproceedings{steil2004backpropagation,
  title={Backpropagation-decorrelation: online recurrent learning with O (N) complexity},
  author={Steil, Jochen J},
  booktitle={2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No. 04CH37541)},
  volume={2},
  pages={843--848},
  year={2004},
  organization={IEEE}
}
@article{williams1989learning,
  title={A learning algorithm for continually running fully recurrent neural networks},
  author={Williams, Ronald J and Zipser, David},
  journal={Neural computation},
  volume={1},
  number={2},
  pages={270--280},
  year={1989},
  publisher={MIT Press}
}
@article{lillicrap2019backpropagation,
  title={Backpropagation through time and the brain},
  author={Lillicrap, Timothy P and Santoro, Adam},
  journal={Current opinion in neurobiology},
  volume={55},
  pages={82--89},
  year={2019},
  publisher={Elsevier}
}
@article{lamme2000distinct,
  title={The distinct modes of vision offered by feedforward and recurrent processing},
  author={Lamme, Victor AF and Roelfsema, Pieter R},
  journal={Trends in neurosciences},
  volume={23},
  number={11},
  pages={571--579},
  year={2000},
  publisher={Elsevier}
}
@article{bullier2001integrated,
  title={Integrated model of visual processing},
  author={Bullier, Jean},
  journal={Brain research reviews},
  volume={36},
  number={2-3},
  pages={96--107},
  year={2001},
  publisher={Elsevier}
}
@article{serre2007feedforward,
  title={A feedforward architecture accounts for rapid categorization},
  author={Serre, Thomas and Oliva, Aude and Poggio, Tomaso},
  journal={Proceedings of the national academy of sciences},
  volume={104},
  number={15},
  pages={6424--6429},
  year={2007},
  publisher={National Acad Sciences}
}
@article{vanrullen2007power,
  title={The power of the feed-forward sweep},
  author={VanRullen, Rufin},
  journal={Advances in Cognitive Psychology},
  volume={3},
  number={1-2},
  pages={167},
  year={2007},
  publisher={University of Finance and Management in Warsaw}
}
@article{hung2005fast,
  title={Fast readout of object identity from macaque inferior temporal cortex},
  author={Hung, Chou P and Kreiman, Gabriel and Poggio, Tomaso and DiCarlo, James J},
  journal={Science},
  volume={310},
  number={5749},
  pages={863--866},
  year={2005},
  publisher={American Association for the Advancement of Science}
}
@inproceedings{he2016deep,
  title={Deep residual learning for image recognition},
  author={He, Kaiming and Zhang, Xiangyu and Ren, Shaoqing and Sun, Jian},
  booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition},
  pages={770--778},
  year={2016}
}
@article{yamins2014performance,
  title={Performance-optimized hierarchical models predict neural responses in higher visual cortex},
  author={Yamins, Daniel LK and Hong, Ha and Cadieu, Charles F and Solomon, Ethan A and Seibert, Darren and DiCarlo, James J},
  journal={Proceedings of the National Academy of Sciences},
  volume={111},
  number={23},
  pages={8619--8624},
  year={2014},
  publisher={National Acad Sciences}
}
@article{tacchetti2017invariant,
  title={Invariant recognition drives neural representations of action sequences},
  author={Tacchetti, Andrea and Isik, Leyla and Poggio, Tomaso},
  journal={PLoS computational biology},
  volume={13},
  number={12},
  pages={e1005859},
  year={2017},
  publisher={Public Library of Science}
}
@article{eickenberg2017seeing,
  title={Seeing it all: Convolutional network layers map the function of the human visual system},
  author={Eickenberg, Michael and Gramfort, Alexandre and Varoquaux, Ga{\"e}l and Thirion, Bertrand},
  journal={NeuroImage},
  volume={152},
  pages={184--194},
  year={2017},
  publisher={Elsevier}
}
@article{khaligh2014deep,
  title={Deep supervised, but not unsupervised, models may explain IT cortical representation},
  author={Khaligh-Razavi, Seyed-Mahdi and Kriegeskorte, Nikolaus},
  journal={PLoS computational biology},
  volume={10},
  number={11},
  year={2014},
  publisher={Public Library of Science}
}
@article{lindsay2020convolutional,
  title={Convolutional neural networks as a model of the visual system: past, present, and future},
  author={Lindsay, Grace},
  journal={Journal of Cognitive Neuroscience},
  pages={1--15},
  year={2020},
  publisher={MIT Press}
}
@article{merolla2014million,
  title={A million spiking-neuron integrated circuit with a scalable communication network and interface},
  author={Merolla, Paul A and Arthur, John V and Alvarez-Icaza, Rodrigo and Cassidy, Andrew S and Sawada, Jun and Akopyan, Filipp and Jackson, Bryan L and Imam, Nabil and Guo, Chen and Nakamura, Yutaka and others},
  journal={Science},
  volume={345},
  number={6197},
  pages={668--673},
  year={2014},
  publisher={American Association for the Advancement of Science}
}
@article{furber2014spinnaker,
  title={The spinnaker project},
  author={Furber, Steve B and Galluppi, Francesco and Temple, Steve and Plana, Luis A},
  journal={Proceedings of the IEEE},
  volume={102},
  number={5},
  pages={652--665},
  year={2014},
  publisher={IEEE}
}
@article{davies2018loihi,
  title={Loihi: A neuromorphic manycore processor with on-chip learning},
  author={Davies, Mike and Srinivasa, Narayan and Lin, Tsung-Han and Chinya, Gautham and Cao, Yongqiang and Choday, Sri Harsha and Dimou, Georgios and Joshi, Prasad and Imam, Nabil and Jain, Shweta and others},
  journal={IEEE Micro},
  volume={38},
  number={1},
  pages={82--99},
  year={2018},
  publisher={IEEE}
}
@article{amari1995information,
  title={Information geometry of the EM and em algorithms for neural networks},
  author={Amari, Shun-Ichi},
  journal={Neural networks},
  volume={8},
  number={9},
  pages={1379--1408},
  year={1995},
  publisher={Elsevier}
}
@article{linnainmaa1970representation,
  title={The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors},
  author={Linnainmaa, Seppo},
  journal={Master's Thesis (in Finnish), Univ. Helsinki},
  pages={6--7},
  year={1970}
}
@article{ruck1992comparative,
  title={Comparative analysis of backpropagation and the extended Kalman filter for training multilayer perceptrons},
  author={Ruck, Dennis W. and Rogers, Steven K. and Kabrisky, Matthew and Maybeck, Peter S. and Oxley, Mark E.},
  journal={IEEE Transactions on Pattern Analysis \& Machine Intelligence},
  number={6},
  pages={686--691},
  year={1992},
  publisher={IEEE}
}
@article{ollivier2019extended,
  title={The Extended Kalman Filter is a Natural Gradient Descent in Trajectory Space},
  author={Ollivier, Yann},
  journal={arXiv preprint arXiv:1901.00696},
  year={2019}
}
@inproceedings{chen2014stochastic,
  title={Stochastic gradient hamiltonian monte carlo},
  author={Chen, Tianqi and Fox, Emily and Guestrin, Carlos},
  booktitle={International conference on machine learning},
  pages={1683--1691},
  year={2014}
}
@article{mandt2017stochastic,
  title={Stochastic gradient descent as approximate bayesian inference},
  author={Mandt, Stephan and Hoffman, Matthew D and Blei, David M},
  journal={The Journal of Machine Learning Research},
  volume={18},
  number={1},
  pages={4873--4907},
  year={2017},
  publisher={JMLR. org}
}

@inproceedings{deneve2005bayesian,
  title={Bayesian inference in spiking neurons},
  author={Deneve, Sophie},
  booktitle={Advances in neural information processing systems},
  pages={353--360},
  year={2005}
}
@article{friston2009reinforcement,
  title={Reinforcement learning or active inference?},
  author={Friston, Karl and Daunizeau, Jean and Kiebel, Stefan J},
  journal={PloS one},
  volume={4},
  number={7},
  year={2009},
  publisher={Public Library of Science}
}
@article{friston2011optimal,
  title={What is optimal about motor control?},
  author={Friston, Karl},
  journal={Neuron},
  volume={72},
  number={3},
  pages={488--498},
  year={2011},
  publisher={Elsevier}
}

@inproceedings{baltieri2017active,
  title={An active inference implementation of phototaxis},
  author={Baltieri, Manuel and Buckley, Christopher L},
  booktitle={Artificial Life Conference Proceedings 14},
  pages={36--43},
  year={2017},
  organization={MIT Press}
}
@inproceedings{baltieri2018probabilistic,
  title={A probabilistic interpretation of PID controllers using active inference},
  author={Baltieri, Manuel and Buckley, Christopher L},
  booktitle={International Conference on Simulation of Adaptive Behavior},
  pages={15--26},
  year={2018},
  organization={Springer}
}
@article{calvo2017predicting,
  title={Predicting green: really radical (plant) predictive processing},
  author={Calvo, Paco and Friston, Karl},
  journal={Journal of The Royal Society Interface},
  volume={14},
  number={131},
  pages={20170096},
  year={2017},
  publisher={The Royal Society}
}
@article{friston2011action,
  title={Action understanding and active inference},
  author={Friston, Karl and Mattout, J{\'e}r{\'e}mie and Kilner, James},
  journal={Biological cybernetics},
  volume={104},
  number={1-2},
  pages={137--160},
  year={2011},
  publisher={Springer}
}
@article{friston2016active,
  title={Active inference and learning},
  author={Friston, Karl and FitzGerald, Thomas and Rigoli, Francesco and Schwartenbeck, Philipp and Pezzulo, Giovanni and others},
  journal={Neuroscience \& Biobehavioral Reviews},
  volume={68},
  pages={862--879},
  year={2016},
  publisher={Elsevier}
}
@article{friston2017active,
  title={Active inference: a process theory},
  author={Friston, Karl and FitzGerald, Thomas and Rigoli, Francesco and Schwartenbeck, Philipp and Pezzulo, Giovanni},
  journal={Neural computation},
  volume={29},
  number={1},
  pages={1--49},
  year={2017},
  publisher={MIT Press}
}
@article{friston2012active,
  title={Active inference and agency: optimal control without cost functions},
  author={Friston, Karl and Samothrakis, Spyridon and Montague, Read},
  journal={Biological cybernetics},
  volume={106},
  number={8-9},
  pages={523--541},
  year={2012},
  publisher={Springer}
}
@article{friston2015active,
  title={Active inference and epistemic value},
  author={Friston, Karl and Rigoli, Francesco and Ognibene, Dimitri and Mathys, Christoph and Fitzgerald, Thomas and Pezzulo, Giovanni},
  journal={Cognitive neuroscience},
  volume={6},
  number={4},
  pages={187--214},
  year={2015},
  publisher={Taylor \& Francis}
}
@article{friston2017curiosity,
  title={Active inference, curiosity and insight},
  author={Friston, Karl and Lin, Marco and Frith, Christopher D and Pezzulo, Giovanni and Hobson, J Allan and Ondobaka, Sasha},
  journal={Neural computation},
  volume={29},
  number={10},
  pages={2633--2683},
  year={2017},
  publisher={MIT Press}
}
@article{tschantz2019learning,
  title={Learning action-oriented models through active inference},
  author={Tschantz, Alexander and Seth, Anil K and Buckley, Christopher L},
  journal={bioRxiv},
  pages={764969},
  year={2019},
  publisher={Cold Spring Harbor Laboratory}
}
@article{itti2009bayesian,
  title={Bayesian surprise attracts human attention},
  author={Itti, Laurent and Baldi, Pierre},
  journal={Vision research},
  volume={49},
  number={10},
  pages={1295--1306},
  year={2009},
  publisher={Elsevier}
}
@article{baldi2010bits,
  title={Of bits and wows: A Bayesian theory of surprise with applications to attention},
  author={Baldi, Pierre and Itti, Laurent},
  journal={Neural Networks},
  volume={23},
  number={5},
  pages={649--666},
  year={2010},
  publisher={Elsevier}
}
@article{ostwald2012evidence,
  title={Evidence for neural encoding of Bayesian surprise in human somatosensation},
  author={Ostwald, Dirk and Spitzer, Bernhard and Guggenmos, Matthias and Schmidt, Timo T and Kiebel, Stefan J and Blankenburg, Felix},
  journal={NeuroImage},
  volume={62},
  number={1},
  pages={177--188},
  year={2012},
  publisher={Elsevier}
}
@article{still2012information,
  title={An information-theoretic approach to curiosity-driven reinforcement learning},
  author={Still, Susanne and Precup, Doina},
  journal={Theory in Biosciences},
  volume={131},
  number={3},
  pages={139--148},
  year={2012},
  publisher={Springer}
}
@inproceedings{sun2011planning,
  title={Planning to be surprised: Optimal bayesian exploration in dynamic environments},
  author={Sun, Yi and Gomez, Faustino and Schmidhuber, J{\"u}rgen},
  booktitle={International Conference on Artificial General Intelligence},
  pages={41--51},
  year={2011},
  organization={Springer}
}
@article{houthooft2016variational,
  title={Variational information maximizing exploration},
  author={Houthooft, Rein and Chen, Xi and Duan, Yan and Schulman, John and De Turck, Filip and Abbeel, Pieter},
  journal={Advances in Neural Information Processing Systems (NIPS)},
  year={2016}
}
@article{tschantz2020reinforcement,
  title={Reinforcement Learning through Active Inference},
  author={Tschantz, Alexander and Millidge, Beren and Seth, Anil K and Buckley, Christopher L},
  journal={arXiv preprint arXiv:2002.12636},
  year={2020}
}
@article{parr2017uncertainty,
  title={Uncertainty, epistemics and active inference},
  author={Parr, Thomas and Friston, Karl J},
  journal={Journal of The Royal Society Interface},
  volume={14},
  number={136},
  pages={20170376},
  year={2017},
  publisher={The Royal Society}
}
@article{schwartenbeck2013exploration,
  title={Exploration, novelty, surprise, and free energy minimization},
  author={Schwartenbeck, Philipp and FitzGerald, Thomas and Dolan, Ray and Friston, Karl},
  journal={Frontiers in psychology},
  volume={4},
  pages={710},
  year={2013},
  publisher={Frontiers}
}
@article{schwartenbeck2019computational,
  title={Computational mechanisms of curiosity and goal-directed exploration},
  author={Schwartenbeck, Philipp and Passecker, Johannes and Hauser, Tobias U and FitzGerald, Thomas HB and Kronbichler, Martin and Friston, Karl J},
  journal={Elife},
  volume={8},
  pages={e41703},
  year={2019},
  publisher={eLife Sciences Publications Limited}
}
@article{oudeyer2009intrinsic,
  title={What is intrinsic motivation? A typology of computational approaches},
  author={Oudeyer, Pierre-Yves and Kaplan, Frederic},
  journal={Frontiers in neurorobotics},
  volume={1},
  pages={6},
  year={2009},
  publisher={Frontiers}
}
@inproceedings{pathak2017curiosity,
  title={Curiosity-driven exploration by self-supervised prediction},
  author={Pathak, Deepak and Agrawal, Pulkit and Efros, Alexei A and Darrell, Trevor},
  booktitle={Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops},
  pages={16--17},
  year={2017}
}
@inproceedings{mohamed2015variational,
  title={Variational information maximisation for intrinsically motivated reinforcement learning},
  author={Mohamed, Shakir and Rezende, Danilo Jimenez},
  booktitle={Advances in neural information processing systems},
  pages={2125--2133},
  year={2015}
}
@article{burda2018large,
  title={Large-scale study of curiosity-driven learning},
  author={Burda, Yuri and Edwards, Harri and Pathak, Deepak and Storkey, Amos and Darrell, Trevor and Efros, Alexei A},
  journal={arXiv preprint arXiv:1808.04355},
  year={2018}
}
@article{millidge2019deep,
  title={Deep active inference as variational policy gradients},
  author={Millidge, Beren},
  journal={arXiv preprint arXiv:1907.03876},
  year={2019}
}
@article{tschantz2019scaling,
  title={Scaling active inference},
  author={Tschantz, Alexander and Baltieri, Manuel and Seth, Anil and Buckley, Christopher L and others},
  journal={arXiv preprint arXiv:1911.10601},
  year={2019}
}
@article{ccatal2020learning,
  title={Learning Perception and Planning with Deep Active Inference},
  author={{\c{C}}atal, Ozan and Verbelen, Tim and Nauta, Johannes and De Boom, Cedric and Dhoedt, Bart},
  journal={arXiv preprint arXiv:2001.11841},
  year={2020}
}
@article{van2019simulating,
  title={Simulating active inference processes by message passing},
  author={van de Laar, Thijs W and de Vries, Bert},
  journal={Frontiers in Robotics and AI},
  volume={6},
  number={20},
  year={2019},
  publisher={Frontiers Media}
}
@article{ueltzhoffer2018deep,
  title={Deep active inference},
  author={Ueltzh{\"o}ffer, Kai},
  journal={Biological cybernetics},
  volume={112},
  number={6},
  pages={547--573},
  year={2018},
  publisher={Springer}
}
@article{pezzulo2016active,
  title={Active Inference, epistemic value, and vicarious trial and error},
  author={Pezzulo, Giovanni and Cartoni, Emilio and Rigoli, Francesco and Pio-Lopez, L{\'e}o and Friston, Karl},
  journal={Learning \& Memory},
  volume={23},
  number={7},
  pages={322--338},
  year={2016},
  publisher={Cold Spring Harbor Lab}
}
@article{fitzgerald2015active,
  title={Active inference, evidence accumulation, and the urn task},
  author={FitzGerald, Thomas HB and Schwartenbeck, Philipp and Moutoussis, Michael and Dolan, Raymond J and Friston, Karl},
  journal={Neural computation},
  volume={27},
  number={2},
  pages={306--328},
  year={2015},
  publisher={MIT Press}
}
@article{mirza2016scene,
  title={Scene construction, visual foraging, and active inference},
  author={Mirza, M Berk and Adams, Rick A and Mathys, Christoph D and Friston, Karl J},
  journal={Frontiers in computational neuroscience},
  volume={10},
  pages={56},
  year={2016},
  publisher={Frontiers}
}
@article{parr2017active,
  title={The active construction of the visual world},
  author={Parr, Thomas and Friston, Karl J},
  journal={Neuropsychologia},
  volume={104},
  pages={92--101},
  year={2017},
  publisher={Elsevier}
}
@article{parr2018active,
  title={Active inference and the anatomy of oculomotion},
  author={Parr, Thomas and Friston, Karl J},
  journal={Neuropsychologia},
  volume={111},
  pages={334--343},
  year={2018},
  publisher={Elsevier}
}
@article{friston2018deep,
  title={Deep temporal models and active inference},
  author={Friston, Karl and Rosch, Richard and Parr, Thomas and Price, Cathy and Bowman, Howard},
  journal={Neuroscience \& Biobehavioral Reviews},
  volume={90},
  pages={486--501},
  year={2018},
  publisher={Elsevier}
}
@article{cullen2018active,
  title={Active inference in openai gym: A paradigm for computational investigations into psychiatric illness},
  author={Cullen, Maell and Davey, Ben and Friston, Karl J and Moran, Rosalyn J},
  journal={Biological psychiatry: cognitive neuroscience and neuroimaging},
  volume={3},
  number={9},
  pages={809--818},
  year={2018},
  publisher={Elsevier}
}
@article{mirza2019impulsivity,
  title={Impulsivity and active inference},
  author={Mirza, M Berk and Adams, Rick A and Parr, Thomas and Friston, Karl},
  journal={Journal of cognitive neuroscience},
  volume={31},
  number={2},
  pages={202--220},
  year={2019},
  publisher={MIT Press}
}
@article{parr2018computational,
  title={The computational anatomy of visual neglect},
  author={Parr, Thomas and Friston, Karl J},
  journal={Cerebral Cortex},
  volume={28},
  number={2},
  pages={777--790},
  year={2018},
  publisher={Oxford University Press}
}
@article{friston2017process,
  title={Active inference: a process theory},
  author={Friston, Karl and FitzGerald, Thomas and Rigoli, Francesco and Schwartenbeck, Philipp and Pezzulo, Giovanni},
  journal={Neural computation},
  volume={29},
  number={1},
  pages={1--49},
  year={2017},
  publisher={MIT Press}
}
@article{olshausen2004sparse,
  title={Sparse coding of sensory inputs},
  author={Olshausen, Bruno A and Field, David J},
  journal={Current opinion in neurobiology},
  volume={14},
  number={4},
  pages={481--487},
  year={2004},
  publisher={Elsevier}
}

@article{babadi2014sparseness,
  title={Sparseness and expansion in sensory representations},
  author={Babadi, Baktash and Sompolinsky, Haim},
  journal={Neuron},
  volume={83},
  number={5},
  pages={1213--1226},
  year={2014},
  publisher={Elsevier}
}

@article{brunel2004optimal,
  title={Optimal information storage and the distribution of synaptic weights: perceptron versus Purkinje cell},
  author={Brunel, Nicolas and Hakim, Vincent and Isope, Philippe and Nadal, Jean-Pierre and Barbour, Boris},
  journal={Neuron},
  volume={43},
  number={5},
  pages={745--757},
  year={2004},
  publisher={Elsevier}
}
@article{schweighofer2001unsupervised,
  title={Unsupervised learning of granule cell sparse codes enhances cerebellar adaptive control},
  author={Schweighofer, Nicolas and Doya, K and Lay, F},
  journal={Neuroscience},
  volume={103},
  number={1},
  pages={35--50},
  year={2001},
  publisher={Elsevier}
}
@article{litwin2017optimal,
  title={Optimal degrees of synaptic connectivity},
  author={Litwin-Kumar, Ashok and Harris, Kameron Decker and Axel, Richard and Sompolinsky, Haim and Abbott, LF},
  journal={Neuron},
  volume={93},
  number={5},
  pages={1153--1164},
  year={2017},
  publisher={Elsevier}
}
@book{sterling2015principles,
  title={Principles of neural design},
  author={Sterling, Peter and Laughlin, Simon},
  year={2015},
  publisher={MIT press}
}
@article{buzsaki2004neuronal,
  title={Neuronal oscillations in cortical networks},
  author={Buzsaki, Gyorgy and Draguhn, Andreas},
  journal={science},
  volume={304},
  number={5679},
  pages={1926--1929},
  year={2004},
  publisher={American Association for the Advancement of Science}
}
@article{hochreiter1998vanishing,
  title={The vanishing gradient problem during learning recurrent neural nets and problem solutions},
  author={Hochreiter, Sepp},
  journal={International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems},
  volume={6},
  number={02},
  pages={107--116},
  year={1998},
  publisher={World Scientific}
}
@article{millidge2022theoretical,
  title={A Theoretical Framework for Inference and Learning in Predictive Coding Networks},
  author={Millidge, Beren and Song, Yuhang and Salvatori, Tommaso and Lukasiewicz, Thomas and Bogacz, Rafal},
  journal={arXiv preprint arXiv:2207.12316},
  year={2022}
}
@article{song2022inferring,
  title={Inferring Neural Activity Before Plasticity: A Foundation for Learning Beyond Backpropagation},
  author={Song, Yuhang and Millidge, Beren Gray and Salvatori, Tommaso and Lukasiewicz, Thomas and Xu, Zhenghua and Bogacz, Rafal},
  journal={bioRxiv},
  year={2022},
  publisher={Cold Spring Harbor Laboratory}
}
@article{millidge2022backpropagation,
  title={Backpropagation at the Infinitesimal Inference Limit of Energy-Based Models: Unifying Predictive Coding, Equilibrium Propagation, and Contrastive Hebbian Learning},
  author={Millidge, Beren and Song, Yuhang and Salvatori, Tommaso and Lukasiewicz, Thomas and Bogacz, Rafal},
  journal={arXiv preprint arXiv:2206.02629},
  year={2022}
}
@article{salvatori2022learning,
  title={Learning on arbitrary graph topologies via predictive coding},
  author={Salvatori, Tommaso and Pinchetti, Luca and Millidge, Beren and Song, Yuhang and Bogacz, Rafal and Lukasiewicz, Thomas},
  journal={arXiv preprint arXiv:2201.13180},
  year={2022}
}

@article{parr2019neuronal,
  title={Neuronal message passing using Mean-field, Bethe, and Marginal approximations},
  author={Parr, Thomas and Markovic, Dimitrije and Kiebel, Stefan J and Friston, Karl J},
  journal={Scientific reports},
  volume={9},
  number={1},
  pages={1--18},
  year={2019},
  publisher={Nature Publishing Group}
}
@article{fox2012tutorial,
  title={A tutorial on variational Bayesian inference},
  author={Fox, Charles W and Roberts, Stephen J},
  journal={Artificial intelligence review},
  volume={38},
  number={2},
  pages={85--95},
  year={2012},
  publisher={Springer}
}
@article{wainwright2008graphical,
  title={Graphical models, exponential families, and variational inference},
  author={Wainwright, Martin J and Jordan, Michael I and others},
  journal={Foundations and Trends{\textregistered} in Machine Learning},
  volume={1},
  number={1--2},
  pages={1--305},
  year={2008},
  publisher={Now Publishers, Inc.}
}
@article{da2020active,
  title={Active inference on discrete state-spaces: a synthesis},
  author={Da Costa, Lancelot and Parr, Thomas and Sajid, Noor and Veselic, Sebastijan and Neacsu, Victorita and Friston, Karl},
  journal={arXiv preprint arXiv:2001.07203},
  year={2020}
}
@article{friston2008DEM,
  title={DEM: a variational treatment of dynamic systems},
  author={Friston, Karl and Trujillo-Barreto, N and Daunizeau, Jean},
  journal={Neuroimage},
  volume={41},
  number={3},
  pages={849--885},
  year={2008},
  publisher={Elsevier}
}
@article{friston2019physics,
  title={A free energy principle for a particular physics},
  author={Friston, Karl},
  journal={arXiv preprint arXiv:1906.10184},
  year={2019}
}
@article{parr2020markov,
  title={Markov blankets, information geometry and stochastic thermodynamics},
  author={Parr, Thomas and Da Costa, Lancelot and Friston, Karl},
  journal={Philosophical Transactions of the Royal Society A},
  volume={378},
  number={2164},
  pages={20190159},
  year={2020},
  publisher={The Royal Society Publishing}
}
@article{karl2012free,
  title={A free energy principle for biological systems},
  author={Karl, Friston},
  journal={Entropy},
  volume={14},
  number={11},
  pages={2100--2121},
  year={2012},
  publisher={Multidisciplinary Digital Publishing Institute}
}
@article{friston2012ao,
  title={Free energy, value, and attractors},
  author={Friston, Karl and Ao, Ping},
  journal={Computational and mathematical methods in medicine},
  volume={2012},
  year={2012},
  publisher={Hindawi}
}
@inproceedings{rawlik2013stochastic,
  title={On stochastic optimal control and reinforcement learning by approximate inference},
  author={Rawlik, Konrad and Toussaint, Marc and Vijayakumar, Sethu},
  booktitle={Twenty-Third International Joint Conference on Artificial Intelligence},
  year={2013}
}

@article{theodorou2010generalized,
  title={A generalized path integral control approach to reinforcement learning},
  author={Theodorou, Evangelos and Buchli, Jonas and Schaal, Stefan},
  journal={journal of machine learning research},
  volume={11},
  number={Nov},
  pages={3137--3181},
  year={2010}
}
@inproceedings{theodorou2012relative,
  title={Relative entropy and free energy dualities: Connections to path integral and kl control},
  author={Theodorou, Evangelos A and Todorov, Emanuel},
  booktitle={2012 IEEE 51st IEEE Conference on Decision and Control (CDC)},
  pages={1466--1473},
  year={2012},
  organization={IEEE}
}
@article{levine2018reinforcement,
  title={Reinforcement learning and control as probabilistic inference: Tutorial and review},
  author={Levine, Sergey},
  journal={arXiv preprint arXiv:1805.00909},
  year={2018}
}
@inproceedings{kappen2007introduction,
  title={An introduction to stochastic control theory, path integrals and reinforcement learning},
  author={Kappen, Hilbert J},
  booktitle={AIP conference proceedings},
  volume={887},
  number={1},
  pages={149--181},
  year={2007},
  organization={American Institute of Physics}
}
@article{kappen2005path,
  title={Path integrals and symmetry breaking for optimal control theory},
  author={Kappen, Hilbert J},
  journal={Journal of statistical mechanics: theory and experiment},
  volume={2005},
  number={11},
  pages={P11011},
  year={2005},
  publisher={IOP Publishing}
}
@article{toussaint2009probabilistic,
  title={Probabilistic inference as a model of planned behavior.},
  author={Toussaint, Marc},
  journal={KI},
  volume={23},
  number={3},
  pages={23--29},
  year={2009}
}
@inproceedings{attias2003planning,
  title={Planning by Probabilistic Inference.},
  author={Attias, Hagai},
  booktitle={AISTATS},
  year={2003},
  organization={Citeseer}
}
@article{williams2017model,
  title={Model predictive path integral control: From theory to parallel computation},
  author={Williams, Grady and Aldrich, Andrew and Theodorou, Evangelos A},
  journal={Journal of Guidance, Control, and Dynamics},
  volume={40},
  number={2},
  pages={344--357},
  year={2017},
  publisher={American Institute of Aeronautics and Astronautics}
}
@article{biehl2018expanding,
  title={Expanding the active inference landscape: more intrinsic motivations in the perception-action loop},
  author={Biehl, Martin and Guckelsberger, Christian and Salge, Christoph and Smith, Sim{\'o}n C and Polani, Daniel},
  journal={Frontiers in neurorobotics},
  volume={12},
  pages={45},
  year={2018},
  publisher={Frontiers}
}

@article{parr2019generalised,
  title={Generalised free energy and active inference},
  author={Parr, Thomas and Friston, Karl J},
  journal={Biological cybernetics},
  volume={113},
  number={5-6},
  pages={495--513},
  year={2019},
  publisher={Springer}
}
@article{schwobel2018active,
  title={Active inference, belief propagation, and the bethe approximation},
  author={Schw{\"o}bel, Sarah and Kiebel, Stefan and Markovi{\'c}, Dimitrije},
  journal={Neural computation},
  volume={30},
  number={9},
  pages={2530--2567},
  year={2018},
  publisher={MIT Press}
}
@inproceedings{yedidia2001generalized,
  title={Generalized belief propagation},
  author={Yedidia, Jonathan S and Freeman, William T and Weiss, Yair},
  booktitle={Advances in neural information processing systems},
  pages={689--695},
  year={2001}
}
@article{yedidia2005constructing,
  title={Constructing free-energy approximations and generalized belief propagation algorithms},
  author={Yedidia, Jonathan S and Freeman, William T and Weiss, Yair},
  journal={IEEE Transactions on information theory},
  volume={51},
  number={7},
  pages={2282--2312},
  year={2005},
  publisher={IEEE}
}
@article{millidge2020relationship,
  title={On the Relationship Between Active Inference and Control as Inference},
  author={Millidge, Beren and Tschantz, Alexander and Seth, Anil K and Buckley, Christopher L},
  journal={arXiv preprint arXiv:2006.12964},
  year={2020}
}
@inproceedings{gal2015modern,
  title={On modern deep learning and variational inference},
  author={Gal, Yarin and Ghahramani, Zoubin},
  booktitle={Advances in Approximate Bayesian Inference workshop, NIPS},
  volume={2},
  year={2015}
}

@article{dayan1995helmholtz,
  title={The helmholtz machine},
  author={Dayan, Peter and Hinton, Geoffrey E and Neal, Radford M and Zemel, Richard S},
  journal={Neural computation},
  volume={7},
  number={5},
  pages={889--904},
  year={1995},
  publisher={MIT Press}
}
@book{ghahramani2000graphical,
  title={Graphical models and variational methods},
  author={Ghahramani, Zoubin and Beal, Matthew J and others},
  year={2000},
  publisher={Advanced mean field methods-theory and practice. MIT Press}
}
@inproceedings{hinton1994autoencoders,
  title={Autoencoders, minimum description length and Helmholtz free energy},
  author={Hinton, Geoffrey E and Zemel, Richard S},
  booktitle={Advances in neural information processing systems},
  pages={3--10},
  year={1994}
}
@article{clark2013whatever,
  title={Whatever next? Predictive brains, situated agents, and the future of cognitive science},
  author={Clark, Andy},
  journal={Behavioral and brain sciences},
  volume={36},
  number={3},
  pages={181--204},
  year={2013},
  publisher={Cambridge University Press}
}
@article{brown2020language,
  title={Language models are few-shot learners},
  author={Brown, Tom B and Mann, Benjamin and Ryder, Nick and Subbiah, Melanie and Kaplan, Jared and Dhariwal, Prafulla and Neelakantan, Arvind and Shyam, Pranav and Sastry, Girish and Askell, Amanda and others},
  journal={arXiv preprint arXiv:2005.14165},
  year={2020}
}
@article{mumford1992computational,
  title={On the computational architecture of the neocortex},
  author={Mumford, David},
  journal={Biological cybernetics},
  volume={66},
  number={3},
  pages={241--251},
  year={1992},
  publisher={Springer}
}
@article{helmholtz1866concerning,
  title={Concerning the perceptions in general},
  author={Helmholtz, H von},
  journal={Treatise on physiological optics,},
  year={1866}
}
@article{shannon1948mathematical,
  title={A mathematical theory of communication},
  author={Shannon, Claude E},
  journal={The Bell system technical journal},
  volume={27},
  number={3},
  pages={379--423},
  year={1948},
  publisher={Nokia Bell Labs}
}
@article{barlow1961coding,
  title={The coding of sensory messages},
  author={Barlow, Horace B},
  journal={Current problems in animal behavior},
  year={1961},
  publisher={Cambridge University Press}
}
@article{barlow1989unsupervised,
  title={Unsupervised learning},
  author={Barlow, Horace B},
  journal={Neural computation},
  volume={1},
  number={3},
  pages={295--311},
  year={1989},
  publisher={MIT Press}
}
@article{barlow1961possible,
  title={Possible principles underlying the transformation of sensory messages},
  author={Barlow, Horace B and others},
  journal={Sensory communication},
  volume={1},
  pages={217--234},
  year={1961}
}
@incollection{neal1998view,
  title={A view of the EM algorithm that justifies incremental, sparse, and other variants},
  author={Neal, Radford M and Hinton, Geoffrey E},
  booktitle={Learning in graphical models},
  pages={355--368},
  year={1998},
  publisher={Springer}
}
@incollection{jordan1998introduction,
  title={An introduction to variational methods for graphical models},
  author={Jordan, Michael I and Ghahramani, Zoubin and Jaakkola, Tommi S and Saul, Lawrence K},
  booktitle={Learning in graphical models},
  pages={105--161},
  year={1998},
  publisher={Springer}
}
@article{rosenblatt1958perceptron,
  title={The perceptron: a probabilistic model for information storage and organization in the brain.},
  author={Rosenblatt, Frank},
  journal={Psychological review},
  volume={65},
  number={6},
  pages={386},
  year={1958},
  publisher={American Psychological Association}
}
@article{van2018simple,
  title={A simple introduction to Markov Chain Monte--Carlo sampling},
  author={Van Ravenzwaaij, Don and Cassey, Pete and Brown, Scott D},
  journal={Psychonomic bulletin \& review},
  volume={25},
  number={1},
  pages={143--154},
  year={2018},
  publisher={Springer}
}

@article{kalman1960new,
    author = {Kalman, R. E.},
    title = "{A New Approach to Linear Filtering and Prediction Problems}",
    journal = {Journal of Basic Engineering},
    volume = {82},
    number = {1},
    pages = {35-45},
    year = {1960},
    month = {03},
    abstract = "{The classical filtering and prediction problem is re-examined using the Bode-Shannon representation of random processes and the “state-transition” method of analysis of dynamic systems. New results are: (1) The formulation and methods of solution of the problem apply without modification to stationary and nonstationary statistics and to growing-memory and infinite-memory filters. (2) A nonlinear difference (or differential) equation is derived for the covariance matrix of the optimal estimation error. From the solution of this equation the co-efficients of the difference (or differential) equation of the optimal linear filter are obtained without further calculations. (3) The filtering problem is shown to be the dual of the noise-free regulator problem. The new method developed here is applied to two well-known problems, confirming and extending earlier results. The discussion is largely self-contained and proceeds from first principles; basic concepts of the theory of random processes are reviewed in the Appendix.}",
    issn = {0021-9223},
    doi = {10.1115/1.3662552},
    url = {https://doi.org/10.1115/1.3662552},
    eprint = {https://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/82/1/35/5518977/35\_1.pdf},
}
@book{wiener2019cybernetics,
  title={Cybernetics or Control and Communication in the Animal and the Machine},
  author={Wiener, Norbert},
  year={2019},
  publisher={MIT press}
}
@article{conant1970every,
  title={Every good regulator of a system must be a model of that system},
  author={Conant, Roger C and Ross Ashby, W},
  journal={International journal of systems science},
  volume={1},
  number={2},
  pages={89--97},
  year={1970},
  publisher={Taylor \& Francis}
}
@book{johnson2005pid,
  title={PID control},
  author={Johnson, Michael A and Moradi, Mohammad H},
  year={2005},
  publisher={Springer}
}
@article{gershman2019does,
  title={What does the free energy principle tell us about the brain?},
  author={Gershman, Samuel J},
  journal={arXiv preprint arXiv:1901.07945},
  year={2019},
  keywords={survey},
  url={https://arxiv.org/abs/1901.07945}
}
@String(gershman2019does="This provides a great high level introduction to the basic ideas and intuitions of the FEP, with a small amount of crucial mathematical background.")


@String(friston2019free="This is Karl's magisterial monograph, and contains the most comprehensive description of the FEP to date")

@String(friston2011action="Goes deep into the neuroscientific intuitions behind why you might want to think about action as a predicted observation and not a latent variable for biological brains. Presents Karl's view that action happens primarily at the periphery through simple 'reflex arcs' while all the real work is done by the generative models generating predictions." )

@article{fortier2018woodlice,
  title={Of woodlice and men},
  author={Fortier, Martin and Friedman, Daniel A},
  journal={The editors wish to express their gratitude to Cordelia Erickson-Davis, Brendan Fleig-Goldstein, Ella Letort, Olivia Marcus, and Sarvin Tafazoli for their valuable help in proofreading the interviews, and to all the contributors for accepting to participate to this issue.},
  pages={17},
  year={2018},
  keywords={classic},
  url={https://www.aliusresearch.org/uploads/9/1/6/0/91600416/alius_bulletin_n%C2%B02__2018_.pdf#page=27}
}
@String(fortier2018woodlice="A great interview with Karl. Goes into a lot of his personal motivations underlying his work on the FEP. I would recommend this perhaps as an initial place to start out if you know nothing of the FEP to grasp the underlying motivations of *what* it is trying to explain.")
@article{friston2012history,
  title={The history of the future of the Bayesian brain},
  author={Friston, Karl},
  journal={NeuroImage},
  volume={62},
  number={2},
  pages={1230--1233},
  year={2012},
  publisher={Elsevier},
  keywords={classic},
  url={https://www.sciencedirect.com/science/article/pii/S1053811911011657}
}
@article{friston2012free,
  title={Free energy, value, and attractors},
  author={Friston, Karl and Ao, Ping},
  journal={Computational and mathematical methods in medicine},
  volume={2012},
  year={2012},
  publisher={Hindawi},
  keywords={classic},
  url={https://www.hindawi.com/journals/cmmm/2012/937860/}
}

@String(friston2008variational="Extends predictive coding to generalised coordinates, and derives the necessary inference algorithms for working with them -- i.e. DEM, dynamic expectation maximisation.")
@article{ramstead2020tale,
  title={A tale of two densities: Active inference is enactive inference},
  author={Ramstead, Maxwell JD and Kirchhoff, Michael D and Friston, Karl J},
  journal={Adaptive Behavior},
  volume={28},
  number={4},
  pages={225--239},
  year={2020},
  publisher={SAGE Publications Sage UK: London, England},
  keywords={philosophy},
  url={https://journals.sagepub.com/doi/pdf/10.1177/1059712319862774}
}
@article{ramstead2018answering,
  title={Answering Schr{\"o}dinger's question: A free-energy formulation},
  author={Ramstead, Maxwell James D{\'e}sormeau and Badcock, Paul Benjamin and Friston, Karl John},
  journal={Physics of life reviews},
  volume={24},
  pages={1--16},
  year={2018},
  publisher={Elsevier},
  keywords={philosophy},
  url={https://www.sciencedirect.com/science/article/pii/S1571064517301409}
}
@article{veissiere2020thinking,
  title={Thinking through other minds: A variational approach to cognition and culture},
  author={Veissi{\`e}re, Samuel PL and Constant, Axel and Ramstead, Maxwell JD and Friston, Karl J and Kirmayer, Laurence J},
  journal={Behavioral and Brain Sciences},
  volume={43},
  year={2020},
  publisher={Cambridge University Press},
  keywords={philosophy},
  url={https://royalsocietypublishing.org/doi/full/10.1098/rsif.2017.0685}
}
@article{veissiere2020ttom,
  title={TTOM in action: Refining the variational approach to cognition and culture},
  author={Veissi{\`e}re, Samuel PL and Constant, Axel and Ramstead, Maxwell JD and Friston, Karl J and Kirmayer, Laurence J},
  journal={Behavioral and Brain Sciences},
  volume={43},
  year={2020},
  publisher={Cambridge University Press},
  keywords={philosophy},
  url={https://www.cambridge.org/core/journals/behavioral-and-brain-sciences/article/abs/ttom-in-action-refining-the-variational-approach-to-cognition-and-culture/ADD060A9EE6937A3104FA23290F2C519}
}

@inproceedings{bouchard2004tradeoff,
  title={The tradeoff between generative and discriminative classifiers},
  author={Bouchard, Guillaume and Triggs, Bill},
  booktitle={16th IASC International Symposium on Computational Statistics (COMPSTAT'04)},
  pages={721--728},
  year={2004}
}
@article{parkhurst2002modeling,
  title={Modeling the role of salience in the allocation of overt visual attention},
  author={Parkhurst, Derrick and Law, Klinton and Niebur, Ernst},
  journal={Vision research},
  volume={42},
  number={1},
  pages={107--123},
  year={2002},
  publisher={Elsevier}
}
@article{henderson2017gaze,
  title={Gaze control as prediction},
  author={Henderson, John M},
  journal={Trends in cognitive sciences},
  volume={21},
  number={1},
  pages={15--23},
  year={2017},
  publisher={Elsevier}
}
@article{kanan2009sun,
  title={SUN: Top-down saliency using natural statistics},
  author={Kanan, Christopher and Tong, Mathew H and Zhang, Lingyun and Cottrell, Garrison W},
  journal={Visual cognition},
  volume={17},
  number={6-7},
  pages={979--1003},
  year={2009},
  publisher={Taylor \& Francis}
}
@article{torralba2003statistics,
  title={Statistics of natural image categories},
  author={Torralba, Antonio and Oliva, Aude},
  journal={Network: computation in neural systems},
  volume={14},
  number={3},
  pages={391--412},
  year={2003},
  publisher={Taylor \& Francis}
}
@article{kanizsa1955margini,
  title={Margini quasi-percettivi in campi con stimolazione omogenea},
  author={Kanizsa, Gaetano},
  journal={Rivista di psicologia},
  volume={49},
  number={1},
  pages={7--30},
  year={1955}
}
@article{turrigiano1999homeostatic,
  title={Homeostatic plasticity in neuronal networks: the more things change, the more they stay the same},
  author={Turrigiano, Gina G},
  journal={Trends in neurosciences},
  volume={22},
  number={5},
  pages={221--227},
  year={1999},
  publisher={Elsevier}
}
@article{watt2010homeostatic,
  title={Homeostatic plasticity and STDP: keeping a neuron's cool in a fluctuating world},
  author={Watt, Alanna J and Desai, Niraj S},
  journal={Frontiers in synaptic neuroscience},
  volume={2},
  pages={5},
  year={2010},
  publisher={Frontiers}
}
@article{cichocki2010families,
  title={Families of alpha-beta-and gamma-divergences: Flexible and robust measures of similarities},
  author={Cichocki, Andrzej and Amari, Shun-ichi},
  journal={Entropy},
  volume={12},
  number={6},
  pages={1532--1568},
  year={2010},
  publisher={Molecular Diversity Preservation International}
}
@techreport{dellaert2002expectation,
  title={The expectation maximization algorithm},
  author={Dellaert, Frank},
  year={2002},
  institution={Georgia Institute of Technology}
}
@book{gupta2011theory,
  title={Theory and use of the EM algorithm},
  author={Gupta, Maya R and Chen, Yihua},
  year={2011},
  publisher={Now Publishers Inc}
}
@article{boyles1983convergence,
  title={On the convergence of the EM algorithm},
  author={Boyles, Russell A},
  journal={Journal of the Royal Statistical Society: Series B (Methodological)},
  volume={45},
  number={1},
  pages={47--50},
  year={1983},
  publisher={Wiley Online Library}
}
@article{wasserman2000bayesian,
  title={Bayesian model selection and model averaging},
  author={Wasserman, Larry},
  journal={Journal of mathematical psychology},
  volume={44},
  number={1},
  pages={92--107},
  year={2000},
  publisher={Elsevier}
}
@article{olah2017feature,
  title={Feature visualization},
  author={Olah, Chris and Mordvintsev, Alexander and Schubert, Ludwig},
  journal={Distill},
  volume={2},
  number={11},
  pages={e7},
  year={2017}
}

@article{dempster1977maximum,
  title={Maximum likelihood from incomplete data via the EM algorithm},
  author={Dempster, Arthur P and Laird, Nan M and Rubin, Donald B},
  journal={Journal of the Royal Statistical Society: Series B (Methodological)},
  volume={39},
  number={1},
  pages={1--22},
  year={1977},
  publisher={Wiley Online Library}
}

@article{banerjee2005clustering,
  title={Clustering with Bregman divergences.},
  author={Banerjee, Arindam and Merugu, Srujana and Dhillon, Inderjit S and Ghosh, Joydeep and Lafferty, John},
  journal={Journal of machine learning research},
  volume={6},
  number={10},
  year={2005}
}

@article{bruineberg2018anticipating,
  title={The anticipating brain is not a scientist: the free-energy principle from an ecological-enactive perspective},
  author={Bruineberg, Jelle and Kiverstein, Julian and Rietveld, Erik},
  journal={Synthese},
  volume={195},
  number={6},
  pages={2417--2444},
  year={2018},
  publisher={Springer},
  keywords={philosophy},
  url={https://link.springer.com/article/10.1007/s11229-016-1239-1}
}
@article{williams2018predictive,
  title={Predictive processing and the representation wars},
  author={Williams, Daniel},
  journal={Minds and Machines},
  volume={28},
  number={1},
  pages={141--172},
  year={2018},
  publisher={Springer},
  keywords={philosophy},
  url={https://link.springer.com/article/10.1007/s11023-017-9441-6}
}

@inproceedings{baltieri2020predictions,
  title={Predictions in the eye of the beholder: an active inference account of Watt governors},
  author={Baltieri, Manuel and Buckley, Christopher L and Bruineberg, Jelle},
  booktitle={Artificial Life Conference Proceedings},
  pages={121--129},
  year={2020},
  organization={MIT Press},
  keywords={philosophy},
  url={https://www.mitpressjournals.org/doi/abs/10.1162/isal_a_00288}
}

@String(friston2013life="A heuristic demonstration of the concept that Karl will later refer to as 'Bayesian mechanics', this paper surveys the notion that any random dynamical systems with the right kind of coupling among its sub-systems (i.e. a Markov blanket), will naturally appear as if it's performing a kind of approximate Bayesian inference. This argument is motivated by appeal to the existence of a non-equilibrium steady-state density, to which the system's probability distribution converges over time.")
@article{friston2013life,
  title={Life as we know it},
  author={Friston, Karl},
  journal={Journal of the Royal Society Interface},
  volume={10},
  number={86},
  pages={20130475},
  year={2013},
  publisher={The Royal Society},
  keywords={self-organisation},
  url={https://royalsocietypublishing.org/doi/full/10.1098/rsif.2013.0475}
}
@article{friston2015knowing,
  title={Knowing one's place: a free-energy approach to pattern regulation},
  author={Friston, Karl and Levin, Michael and Sengupta, Biswa and Pezzulo, Giovanni},
  journal={Journal of the Royal Society Interface},
  volume={12},
  number={105},
  pages={20141383},
  year={2015},
  publisher={The Royal Society},
  keywords={self-organisation},
  url={https://royalsocietypublishing.org/doi/full/10.1098/rsif.2014.1383}
}
@article{kuchling2019morphogenesis,
  title={Morphogenesis as Bayesian inference: A variational approach to pattern formation and control in complex biological systems},
  author={Kuchling, Franz and Friston, Karl and Georgiev, Georgi and Levin, Michael},
  journal={Physics of life reviews},
  year={2019},
  publisher={Elsevier},
  keywords={self-organisation},
  url={https://www.sciencedirect.com/science/article/pii/S1571064519300909?casa_token=IrMsxOkLhfYAAAAA:2agLQQPi8aeYTxxqotIPCyEzsHbpvOLwf_0eK5oW2Li0Gi5THHn2XRpWYfNT99M0Xy5pPD_S9CA}
}
@article{ramstead2020neural,
  title={Neural and phenotypic representation under the free-energy principle},
  author={Ramstead, Maxwell JD and Hesp, Casper and Tschantz, Alexander and Smith, Ryan and Constant, Axel and Friston, Karl},
  journal={Neuroscience \& Biobehavioral Reviews},
  year={2020},
  publisher={Elsevier},
  keywords={self-organisation},
  url={https://www.sciencedirect.com/science/article/pii/S0149763420306643?casa_token=16rC0ManFBUAAAAA:3mbntn5I7fObnA_Y397rvZbWrnUzkqmALD1LtS88tGrIRxbw9RQvU55XJuH-zKdBi6tPaN9faDM}
}
@article{friston2020parcels,
  title={Parcels and particles: Markov blankets in the brain},
  author={Friston, Karl and Fagerholm, Erik D and Zarghami, Tahereh S and Parr, Thomas and Hip{\'o}lito, In{\^e}s and Magrou, Lo{\"\i}c and Razi, Adeel},
  journal={arXiv preprint arXiv:2007.09704},
  year={2020},
  keywords={self-organisation},
  url={https://arxiv.org/abs/2007.09704}
}
@article{hipolito2020markov,
  title={Markov blankets in the brain},
  author={Hipolito, Ines and Ramstead, Maxwell and Convertino, Laura and Bhat, Anjali and Friston, Karl and Parr, Thomas},
  journal={arXiv preprint arXiv:2006.02741},
  year={2020},
  keywords={self-organisation},
  url={https://arxiv.org/abs/2006.02741}
}
@String(parr2020modules="The 'free energy' response to the Fodorian notion of 'modularity' as an explanation of functional segregation, here motivated by an appeal to the stochastic dynamics of Markov-blanketed systems. Parr et al. argue that, given a particular conditional independency structure among the components that comprise a random dynamical system, one can interpret the system and its dynamics as entertaining a mean-field factorised generative model of its local environment, as opposed to appealing to philosophically or otherwise-unsatisfying notions such as 'modularity'.")
@article{parr2020modules,
  title={Modules or Mean-Fields?},
  author={Parr, Thomas and Sajid, Noor and Friston, Karl J},
  journal={Entropy},
  volume={22},
  number={5},
  pages={552},
  year={2020},
  publisher={Multidisciplinary Digital Publishing Institute},
  keywords={self-organisation},
  url={https://www.mdpi.com/1099-4300/22/5/552},
}
@article{palacios2017biological,
  title={Biological self-organisation and Markov blankets},
  author={Palacios, Ensor Rafael and Razi, Adeel and Parr, Thomas and Kirchhoff, Michael and Friston, Karl},
  journal={BioRxiv},
  pages={227181},
  year={2017},
  publisher={Cold Spring Harbor Laboratory},
  keywords={self-organisation},
  url={https://www.biorxiv.org/content/10.1101/227181v1.abstract}
}
@String(parr2020markov="This paper gives succinct and schematic treatments of several of the main concepts explored in a Free Energy Principle for a Particular Physics, particularly those related to Bayesian mechanics and information geometry. This work importantly delineates some of the conditions required of a system, so that its internal states approximately parameterise beliefs about external states. Fluctuation theorems are also invoked to relate the probability of trajectories or sequences of states to existing concepts in the active inference world, such as information gain, risk, and ambiguity resolution.")

@String(da2020active="This is a great and thorough tutorial on discrete-state-space active inference. I would reccomend it to everybody new to the field.")


@String(friston2012active="The first (I think) discrete-state-space paper on active inference. Notable for using the standard variational free energy as objective function and not the expected free energy. Describes some of the intuitions behind active inference.")


@article{friston2017graphical,
  title={The graphical brain: belief propagation and active inference},
  author={Friston, Karl and Parr, Thomas and de Vries, Bert},
  journal={Network Neuroscience},
  volume={1},
  number={4},
  pages={381--414},
  year={2017},
  publisher={MIT Press},
  keywords={free_energy},
  url={https://www.mitpressjournals.org/doi/full/10.1162/NETN_a_00018}
}
@String(friston2017graphical="Introduces the general factor-graph message passing viewpoint on active inference. Also introduces hierarchical active inference models.")


@String(schwobel2018active="Introduces the Bethe free energy, as a result of making the Bethe approximation instead of the mean-field variational assumption to derive the message passing algorithms.")

@article{millidge2020whence,
  title={Whence the Expected Free Energy?},
  author={Millidge, Beren and Tschantz, Alexander and Buckley, Christopher L},
  journal={arXiv preprint arXiv:2004.08128},
  year={2020},
  keywords={free_energy},
  url={https://arxiv.org/abs/2004.08128}
}
@article{stein2005neuronal,
  title={Neuronal variability: noise or part of the signal?},
  author={Stein, Richard B and Gossen, E Roderich and Jones, Kelvin E},
  journal={Nature Reviews Neuroscience},
  volume={6},
  number={5},
  pages={389--397},
  year={2005},
  publisher={Nature Publishing Group}
}
@article{markov2014anatomy,
  title={Anatomy of hierarchy: feedforward and feedback pathways in macaque visual cortex},
  author={Markov, Nikola T and Vezoli, Julien and Chameau, Pascal and Falchier, Arnaud and Quilodran, Ren{\'e} and Huissoud, Cyril and Lamy, Camille and Misery, Pierre and Giroud, Pascale and Ullman, Shimon and others},
  journal={Journal of Comparative Neurology},
  volume={522},
  number={1},
  pages={225--259},
  year={2014},
  publisher={Wiley Online Library}
}
@article{uhlhaas2006neural,
  title={Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology},
  author={Uhlhaas, Peter J and Singer, Wolf},
  journal={neuron},
  volume={52},
  number={1},
  pages={155--168},
  year={2006},
  publisher={Elsevier}
}
@article{bastos2015visual,
  title={Visual areas exert feedforward and feedback influences through distinct frequency channels},
  author={Bastos, Andre Moraes and Vezoli, Julien and Bosman, Conrado Arturo and Schoffelen, Jan-Mathijs and Oostenveld, Robert and Dowdall, Jarrod Robert and De Weerd, Peter and Kennedy, Henry and Fries, Pascal},
  journal={Neuron},
  volume={85},
  number={2},
  pages={390--401},
  year={2015},
  publisher={Elsevier}
}
@article{ororbia2019spiking,
  title={Spiking Neural Predictive Coding for Continual Learning from Data Streams},
  author={Ororbia, Alexander},
  journal={arXiv preprint arXiv:1908.08655},
  year={2019}
}
@article{kaiser2020synaptic,
  title={Synaptic plasticity dynamics for deep continuous local learning (DECOLLE)},
  author={Kaiser, Jacques and Mostafa, Hesham and Neftci, Emre},
  journal={Frontiers in Neuroscience},
  volume={14},
  pages={424},
  year={2020},
  publisher={Frontiers}
}
@article{neftci2019surrogate,
  title={Surrogate gradient learning in spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks},
  author={Neftci, Emre O and Mostafa, Hesham and Zenke, Friedemann},
  journal={IEEE Signal Processing Magazine},
  volume={36},
  number={6},
  pages={51--63},
  year={2019},
  publisher={IEEE}
}
@article{zenke2021remarkable,
  title={The remarkable robustness of surrogate gradient learning for instilling complex function in spiking neural networks},
  author={Zenke, Friedemann and Vogels, Tim P},
  journal={Neural Computation},
  volume={33},
  number={4},
  pages={899--925},
  year={2021},
  publisher={MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}
}
@article{rockland2019we,
  title={What do we know about laminar connectivity?},
  author={Rockland, Kathleen S},
  journal={Neuroimage},
  volume={197},
  pages={772--784},
  year={2019},
  publisher={Elsevier}
}
@article{kietzmann2019recurrence,
  title={Recurrence is required to capture the representational dynamics of the human visual system},
  author={Kietzmann, Tim C and Spoerer, Courtney J and S{\"o}rensen, Lynn KA and Cichy, Radoslaw M and Hauk, Olaf and Kriegeskorte, Nikolaus},
  journal={Proceedings of the National Academy of Sciences},
  volume={116},
  number={43},
  pages={21854--21863},
  year={2019},
  publisher={National Acad Sciences}
}

@book{stengel1986stochastic,
  title={Stochastic optimal control: theory and application},
  author={Stengel, Robert F},
  year={1986},
  publisher={John Wiley \& Sons, Inc.}
}
@book{jazwinski2007stochastic,
  title={Stochastic processes and filtering theory},
  author={Jazwinski, Andrew H},
  year={2007},
  publisher={Courier Corporation}
}
@article{radford2021learning,
  title={Learning transferable visual models from natural language supervision},
  author={Radford, Alec and Kim, Jong Wook and Hallacy, Chris and Ramesh, Aditya and Goh, Gabriel and Agarwal, Sandhini and Sastry, Girish and Askell, Amanda and Mishkin, Pamela and Clark, Jack and others},
  journal={arXiv preprint arXiv:2103.00020},
  year={2021}
}

@String(millidge2020whence="Discusses whether we can derive the expected free energy objective function on principled ground from the FEP, and discusses different potential objective functions for active inference.")

@article{friston2020sophisticated,
  title={Sophisticated Inference},
  author={Friston, Karl and Da Costa, Lancelot and Hafner, Danijar and Hesp, Casper and Parr, Thomas},
  journal={arXiv preprint arXiv:2006.04120},
  year={2020},
  keywords={discrete},
  url={https://arxiv.org/abs/2006.04120}
}
@String(friston2020sophisticated="Introduces the next stage of active inference. 'Sophisticated' active inference, where agents make decisions not just on their beliefs about the future, but on how their beliefs will change in the future. Allows the simulation of real epistemic value -- i.e. act so as to change your beliefs in the future.")
@article{sajid2019active,
  title={Active inference: demystified and compared},
  author={Sajid, Noor and Ball, Philip J and Friston, Karl J},
  journal={arXiv},
  pages={arXiv--1909},
  year={2019},
  keywords={discrete},
  url={https://ui.adsabs.harvard.edu/abs/2019arXiv190910863S/abstract}
}
@article{da2020relationship,
  title={The relationship between dynamic programming and active inference: The discrete, finite-horizon case},
  author={Da Costa, Lancelot and Sajid, Noor and Parr, Thomas and Friston, Karl and Smith, Ryan},
  journal={arXiv preprint arXiv:2009.08111},
  year={2020},
  keywords={discrete},
  url={https://arxiv.org/abs/2009.08111}
}
@String(da2020relationship="Discusses the relationship between active inference and dynamic programming solutions to reinforcement learning problems (i.e. Q learning, value functions etc). Shows that they are largely equivalent except with different objectives (Expected Free Energy vs Expected Discounted Reward).")

@inproceedings{tschantz2020scaling,
  title={Scaling active inference},
  author={Tschantz, Alexander and Baltieri, Manuel and Seth, Anil K and Buckley, Christopher L},
  booktitle={2020 International Joint Conference on Neural Networks (IJCNN)},
  pages={1--8},
  year={2020},
  organization={IEEE},
  keywords={deep},
  url={https://ieeexplore.ieee.org/abstract/document/9207382/?casa_token=TfYG9cq3UvwAAAAA:Fn2oE7PCTGHlEN0jPQQE4P-qBiO_V-7xRFXXqYn7ubVoZeoiBd6ViBAxWf1L7j-R1wiKEOKvaXQ}
}
@String(tschantz2020scaling="Implements Deep Active Inference in a model-based RL setting using explicit planning with a transition model.")
@article{millidge2020deep,
  title={Deep active inference as variational policy gradients},
  author={Millidge, Beren},
  journal={Journal of Mathematical Psychology},
  volume={96},
  pages={102348},
  year={2020},
  publisher={Elsevier},
  keywords={deep},
  url={https://www.sciencedirect.com/science/article/pii/S0022249620300298?casa_token=GQLxvJzk3zMAAAAA:uotM5EqPWP9SIUV-5N8vvNnkVWSqFlS8W03MZ_W9GYCoyhWGRAhN9YmGjiTIaxCGHd4iwrjzElg}
}
@String(millidge2020deep="Implements deep active inference in a model-free policy gradient setting by amortising the learning of the expected-free-energy value function. Uses a transition model for the state-information gain term in the expected free energy.")

@String(ueltzhoffer2018deep="The first paper to try combining active inference with deep neural networks. Demonstrates the importance of the exploratory terms of the EFE to solve the mountain-car problem.")

@article{fountas2020deep,
  title={Deep active inference agents using Monte-Carlo methods},
  author={Fountas, Zafeirios and Sajid, Noor and Mediano, Pedro AM and Friston, Karl},
  journal={arXiv preprint arXiv:2006.04176},
  year={2020},
  keywords={deep},
  url={https://arxiv.org/abs/2006.04176}
}

@String(baltieri2017active="Active inference in plants!!!")

@article{baltieri2019pid,
  title={PID control as a process of active inference with linear generative models},
  author={Baltieri, Manuel and Buckley, Christopher L},
  journal={Entropy},
  volume={21},
  number={3},
  pages={257},
  year={2019},
  publisher={Multidisciplinary Digital Publishing Institute},
  keywords={continuous},
  url={https://www.mdpi.com/1099-4300/21/3/257}
}
@String(baltieri2019pid="Active inference under a linear gaussian generative model can replicate PID, but also provide a natural method for learning the tuning coefficients (by understanding them as precisions).")
@article{baltieri2020kalman,
  title={On Kalman-Bucy filters, linear quadratic control and active inference},
  author={Baltieri, Manuel and Buckley, Christopher L},
  journal={arXiv preprint arXiv:2005.06269},
  year={2020},
  keywords={continuous},
  url={https://arxiv.org/abs/2005.06269}
}
@String(baltieri2020kalman="A key step towards understanding how active inference relates to classical control theory methods such as Kalman Filters and LQR control.")
@article{watson2020active,
  title={Active Inference or Control as Inference? A Unifying View},
  author={Watson, Joe and Imohiosen, Abraham and Peters, Jan},
  journal={arXiv preprint arXiv:2010.00262},
  year={2020},
  keywords={continous},
  url={https://arxiv.org/abs/2010.00262}
}
@article{van2019application,
  title={Application of the Free Energy Principle to Estimation and Control},
  author={van de Laar, Thijs and {\"O}z{\c{c}}elikkale, Ay{\c{c}}a and Wymeersch, Henk},
  journal={arXiv preprint arXiv:1910.09823},
  year={2019},
  keywords={continuous},
  url={https://arxiv.org/abs/1910.09823}
}
@String(van2019application="Another approach to understanding how active inference relates to and extends classical control theory methods.")

@String(grimbergen2019state="An excellent overview and fantastic piece of work on the linear time-indepenent formulation of active inference and its relation to classical control theory.")
@article{pio2016active,
  title={Active inference and robot control: a case study},
  author={Pio-Lopez, L{\'e}o and Nizard, Ange and Friston, Karl and Pezzulo, Giovanni},
  journal={Journal of The Royal Society Interface},
  volume={13},
  number={122},
  pages={20160616},
  year={2016},
  publisher={The Royal Society},
  keywords={robotics},
  url={https://royalsocietypublishing.org/doi/full/10.1098/rsif.2016.0616}
}
@article{oliver2019active,
  title={Active inference body perception and action for humanoid robots},
  author={Oliver, Guillermo and Lanillos, Pablo and Cheng, Gordon},
  journal={arXiv preprint arXiv:1906.03022},
  year={2019},
  keywords={robotics},
  url={https://arxiv.org/abs/1906.03022}
}
@article{sancaktar2019end,
  title={End-to-end pixel-based deep active inference for body perception and action},
  author={Sancaktar, Cansu and Lanillos, Pablo},
  journal={arXiv preprint arXiv:2001.05847},
  year={2019},
  keywords={robotics},
  url={https://arxiv.org/abs/2001.05847}
}
@article{vanderbroeck2019active,
  title={Active inference for robot control: A factor graph approach},
  author={Vanderbroeck, Mees and Baioumy, Mohamed and van der Lans, Daan and de Rooij, Rens and van der Werf, Tiis},
  journal={Student Undergraduate Research E-journal!},
  volume={5},
  pages={1--5},
  year={2019},
  keywords={robotics},
  url={https://pdfs.semanticscholar.org/e177/b21b0a7f43ad3969ceb42bd8f1d912ea8d43.pdf}
}
@article{pezzato2020novel,
  title={A novel adaptive controller for robot manipulators based on active inference},
  author={Pezzato, Corrado and Ferrari, Riccardo and Corbato, Carlos Hern{\'a}ndez},
  journal={IEEE Robotics and Automation Letters},
  volume={5},
  number={2},
  pages={2973--2980},
  year={2020},
  publisher={IEEE},
  keywords={robotics},
  url={https://ieeexplore.ieee.org/abstract/document/9000729/?casa_token=OBL93SqhogUAAAAA:6w99NtC-Fk7P0tIiXx6QmTNsWnPS-GKK0OtsJEz-HWgniXwpY0Rtue_qlc5Fe9HQ2vj0ropx7hM}
}
@article{pezzulo2018hierarchical,
  title={Hierarchical active inference: A theory of motivated control},
  author={Pezzulo, Giovanni and Rigoli, Francesco and Friston, Karl J},
  journal={Trends in cognitive sciences},
  volume={22},
  number={4},
  pages={294--306},
  year={2018},
  publisher={Elsevier},
  keywords={continuous},
  url={https://www.sciencedirect.com/science/article/pii/S1364661318300226},
}
@article{corcoran2020allostatic,
  title={From allostatic agents to counterfactual cognisers: active inference, biological regulation, and the origins of cognition},
  author={Corcoran, Andrew W and Pezzulo, Giovanni and Hohwy, Jakob},
  journal={Biology \& Philosophy},
  volume={35},
  number={3},
  pages={1--45},
  year={2020},
  publisher={Springer},
  keywords={philosophy},
  url={https://link.springer.com/article/10.1007/s10539-020-09746-2},
}
@article{seth2013interoceptive,
  title={Interoceptive inference, emotion, and the embodied self},
  author={Seth, Anil K},
  journal={Trends in cognitive sciences},
  volume={17},
  number={11},
  pages={565--573},
  year={2013},
  publisher={Elsevier},
  keywords={philosophy},
  url={https://www.sciencedirect.com/science/article/pii/S1364661313002118}
}
@article{seth2016active,
  title={Active interoceptive inference and the emotional brain},
  author={Seth, Anil K and Friston, Karl J},
  journal={Philosophical Transactions of the Royal Society B: Biological Sciences},
  volume={371},
  number={1708},
  pages={20160007},
  year={2016},
  publisher={The Royal Society},
  keywords={philosophy},
  url={https://royalsocietypublishing.org/doi/full/10.1098/rstb.2016.0007}
}
@book{seth2014cybernetic,
  title={The cybernetic Bayesian brain},
  author={Seth, Anil K},
  year={2014},
  publisher={Open MIND. Frankfurt am Main: MIND Group},
  keywords={philosophy},
  url={https://open-mind.net/papers/the-cybernetic-bayesian-brain},
}
@article{seth2015presence,
  title={Presence, objecthood, and the phenomenology of predictive perception},
  author={Seth, Anil K},
  journal={Cognitive neuroscience},
  volume={6},
  number={2-3},
  pages={111--117},
  year={2015},
  publisher={Taylor \& Francis},
  keywords={philosophy},
  url={https://www.tandfonline.com/doi/full/10.1080/17588928.2015.1026888?casa_token=IA7GL_oM2VAAAAAA%3AXkJ7HbhxHcW7qCdSWyRSerOo8iZevM3x_8QV1b7C7qAvAqJMveeq4IeGRBTCCbQOCJ6Ix9p70VkKww}
}


@String(friston2008variational2="Foundational treatment of variational inference for dynamical systems, as represented in generalised coordinates. Also relates variational filtering to other non-variational schemes like particle filtering and Kalman filtering.")


@article{smith2020recent,
  title={Recent advances in the application of predictive coding and active inference models within clinical neuroscience},
  author={Smith, Ryan and Badcock, Paul and Friston, Karl J},
  journal={Psychiatry and Clinical Neurosciences},
  year={2020},
  publisher={Wiley Online Library},
  keywords={applications},
  url={https://onlinelibrary.wiley.com/doi/abs/10.1111/pcn.13138}
}
@String(smith2020recent="A comprehensive review of neuroscientific and computational psychiatry applications of the FEP and Active Inference.")

@article{whyte2020predictive,
  title={The Predictive Global Neuronal Workspace: A Formal Active Inference Model of Visual Consciousness},
  author={Whyte, Christopher J and Smith, Ryan},
  journal={bioRxiv},
  year={2020},
  publisher={Cold Spring Harbor Laboratory},
  keywords={applications},
  url={https://www.sciencedirect.com/science/article/pii/S0301008220301738}
}
@article{smith2019neurocomputational,
  title={Neurocomputational mechanisms underlying emotional awareness: insights afforded by deep active inference and their potential clinical relevance},
  author={Smith, Ryan and Lane, Richard D and Parr, Thomas and Friston, Karl J},
  journal={Neuroscience \& Biobehavioral Reviews},
  volume={107},
  pages={473--491},
  year={2019},
  publisher={Elsevier},
  keywords={applications},
  url={https://www.sciencedirect.com/science/article/pii/S014976341930541X?casa_token=c1UILIbIsKYAAAAA:tQbclbvyieFsSPf_oqulTP8Manv6fU6CeI1iHGZe5Iq4TryLR4pGRjK0Y7RD4gZCfjJo02uWyvw}
}

@article{smith2019simulating,
  title={Simulating emotions: An active inference model of emotional state inference and emotion concept learning},
  author={Smith, Ryan and Parr, Thomas and Friston, Karl J},
  journal={Frontiers in psychology},
  volume={10},
  pages={2844},
  year={2019},
  publisher={Frontiers},
  keywords={applications},
  url={https://www.frontiersin.org/articles/10.3389/fpsyg.2019.02844/full}
}

@article{smith2017hierarchical,
  title={The hierarchical basis of neurovisceral integration},
  author={Smith, Ryan and Thayer, Julian F and Khalsa, Sahib S and Lane, Richard D},
  journal={Neuroscience \& biobehavioral reviews},
  volume={75},
  pages={274--296},
  year={2017},
  publisher={Elsevier},
  keywords={applications},
  url={https://www.sciencedirect.com/science/article/pii/S014976341630673X?casa_token=V0kmqFxZWg4AAAAA:QE5VJB7jp6k22u8L7S7iWYh2_EiLV9a4gyXGUfSMIXOte_zYxGsH2YXUkvKY6PeDKriYy8nu11o}
}
@inproceedings{lanillos2018adaptive,
  title={Adaptive robot body learning and estimation through predictive coding},
  author={Lanillos, Pablo and Cheng, Gordon},
  booktitle={2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)},
  pages={4083--4090},
  year={2018},
  organization={IEEE},
  keywords={robotics},
  url={https://ieeexplore.ieee.org/document/8593684/}
}
@article{friston2010action,
  title={Action and behavior: a free-energy formulation},
  author={Friston, Karl and Daunizeau, Jean and Kilner, James and Kiebel, Stefan J},
  journal={Biological cybernetics},
  volume={102},
  number={3},
  pages={227--260},
  year={2010},
  publisher={Springer},
  keywords={classic},
  url={https://link.springer.com/article/10.1007/s00422-010-0364-z}
}

@article{marino2020predictive,
  title={Predictive Coding, Variational Autoencoders, and Biological Connections},
  author={Marino, Joseph},
  journal={arXiv preprint arXiv:2011.07464},
  year={2020}
}
@article{marino2018iterative,
  title={Iterative amortized inference},
  author={Marino, Joseph and Yue, Yisong and Mandt, Stephan},
  journal={arXiv preprint arXiv:1807.09356},
  year={2018}
}
@article{hubel1962receptive,
  title={Receptive fields, binocular interaction and functional architecture in the cat's visual cortex},
  author={Hubel, David H and Wiesel, Torsten N},
  journal={The Journal of physiology},
  volume={160},
  number={1},
  pages={106},
  year={1962},
  publisher={Wiley-Blackwell}
}

@book{brooks2011handbook,
  title={Handbook of markov chain monte carlo},
  author={Brooks, Steve and Gelman, Andrew and Jones, Galin and Meng, Xiao-Li},
  year={2011},
  publisher={CRC press}
}
@article{metropolis1953equation,
  title={Equation of state calculations by fast computing machines},
  author={Metropolis, Nicholas and Rosenbluth, Arianna W and Rosenbluth, Marshall N and Teller, Augusta H and Teller, Edward},
  journal={The journal of chemical physics},
  volume={21},
  number={6},
  pages={1087--1092},
  year={1953},
  publisher={American Institute of Physics}
}


@misc{welch1995introduction,
  title={An introduction to the Kalman filter},
  author={Welch, Greg and Bishop, Gary and others},
  year={1995},
  publisher={Citeseer}
}
@article{carandini2012normalization,
  title={Normalization as a canonical neural computation},
  author={Carandini, Matteo and Heeger, David J},
  journal={Nature Reviews Neuroscience},
  volume={13},
  number={1},
  pages={51--62},
  year={2012},
  publisher={Nature Publishing Group}
}
@article{ditchburn1955eye,
  title={Eye-movements in relation to retinal action.},
  author={Ditchburn, RW},
  journal={Optica Acta},
  year={1955}
}
@article{gerrits1970artificial,
  title={Artificial movements of a stabilized image},
  author={Gerrits, HJM and Vendrik, AJH},
  journal={Vision Research},
  volume={10},
  number={12},
  pages={1443--1456},
  year={1970},
  publisher={Elsevier}
}
@article{riggs1953disappearance,
  title={The disappearance of steadily fixated visual test objects},
  author={Riggs, Lorrin A and Ratliff, Floyd and Cornsweet, Janet C and Cornsweet, Tom N},
  journal={JOSA},
  volume={43},
  number={6},
  pages={495--501},
  year={1953},
  publisher={Optical Society of America}
}

@article{coppola1996extraordinarily,
  title={The extraordinarily rapid disappearance of entopic images},
  author={Coppola, David and Purves, Dale},
  journal={Proceedings of the National Academy of Sciences},
  volume={93},
  number={15},
  pages={8001--8004},
  year={1996},
  publisher={National Acad Sciences}
}
@article{papamakarios2019normalizing,
  title={Normalizing flows for probabilistic modeling and inference},
  author={Papamakarios, George and Nalisnick, Eric and Rezende, Danilo Jimenez and Mohamed, Shakir and Lakshminarayanan, Balaji},
  journal={arXiv preprint arXiv:1912.02762},
  year={2019}
}
@article{rezende2015variational,
  title={Variational inference with normalizing flows},
  author={Rezende, Danilo Jimenez and Mohamed, Shakir},
  journal={arXiv preprint arXiv:1505.05770},
  year={2015}
}
@article{tishby2000information,
  title={The information bottleneck method},
  author={Tishby, Naftali and Pereira, Fernando C and Bialek, William},
  journal={arXiv preprint physics/0004057},
  year={2000}
}
@article{desimone1998visual,
  title={Visual attention mediated by biased competition in extrastriate visual cortex},
  author={Desimone, Robert},
  journal={Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences},
  volume={353},
  number={1373},
  pages={1245--1255},
  year={1998},
  publisher={The Royal Society}
}
@article{desimone1995neural,
  title={Neural mechanisms of selective visual attention},
  author={Desimone, Robert and Duncan, John},
  journal={Annual review of neuroscience},
  volume={18},
  number={1},
  pages={193--222},
  year={1995},
  publisher={Annual Reviews 4139 El Camino Way, PO Box 10139, Palo Alto, CA 94303-0139, USA}
}
@article{reynolds1999competitive,
  title={Competitive mechanisms subserve attention in macaque areas V2 and V4},
  author={Reynolds, John H and Chelazzi, Leonardo and Desimone, Robert},
  journal={Journal of Neuroscience},
  volume={19},
  number={5},
  pages={1736--1753},
  year={1999},
  publisher={Soc Neuroscience}
}
@book{harpur1994experiments,
  title={Experiments with simple Hebbian-based learning rules in pattern classification tasks},
  author={Harpur, George F and Prager, Richard W},
  year={1994},
  publisher={Citeseer}
}
@article{kingma2013auto,
  title={Auto-encoding variational bayes},
  author={Kingma, Diederik P and Welling, Max},
  journal={arXiv preprint arXiv:1312.6114},
  year={2013}
}
@article{goodfellow2014generative,
  title={Generative adversarial nets},
  author={Goodfellow, Ian and Pouget-Abadie, Jean and Mirza, Mehdi and Xu, Bing and Warde-Farley, David and Ozair, Sherjil and Courville, Aaron and Bengio, Yoshua},
  journal={Advances in neural information processing systems},
  volume={27},
  pages={2672--2680},
  year={2014}
}
@article{child2020very,
  title={Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Images},
  author={Child, Rewon},
  journal={arXiv preprint arXiv:2011.10650},
  year={2020}
}
@inproceedings{van2017neural,
  title={Neural discrete representation learning},
  author={Van Den Oord, Aaron and Vinyals, Oriol and others},
  booktitle={Advances in Neural Information Processing Systems},
  pages={6306--6315},
  year={2017}
}
@article{oord2016wavenet,
  title={Wavenet: A generative model for raw audio},
  author={Oord, Aaron van den and Dieleman, Sander and Zen, Heiga and Simonyan, Karen and Vinyals, Oriol and Graves, Alex and Kalchbrenner, Nal and Senior, Andrew and Kavukcuoglu, Koray},
  journal={arXiv preprint arXiv:1609.03499},
  year={2016}
}
@article{baltieri2020bayesian,
  title={A Bayesian perspective on classical control},
  author={Baltieri, Manuel},
  journal={arXiv preprint arXiv:2004.10288},
  year={2020}
}
@book{alon2019introduction,
  title={An introduction to systems biology: design principles of biological circuits},
  author={Alon, Uri},
  year={2019},
  publisher={CRC press}
}

@article{millidge2020activation,
  title={Activation Relaxation: A Local Dynamical Approximation to Backpropagation in the Brain},
  author={Millidge, Beren and Tschantz, Alexander and Buckley, Christopher L and Seth, Anil},
  journal={arXiv preprint arXiv:2009.05359},
  year={2020}
}
@article{millidge2020investigating,
  title={Investigating the Scalability and Biological Plausibility of the Activation Relaxation Algorithm},
  author={Millidge, Beren and Tschantz, Alexander and Seth, Anil and Buckley, Christopher L},
  journal={arXiv preprint arXiv:2010.06219},
  year={2020}
}
@article{adams2013predictions,
  title={Predictions not commands: active inference in the motor system},
  author={Adams, Rick A and Shipp, Stewart and Friston, Karl J},
  journal={Brain Structure and Function},
  volume={218},
  number={3},
  pages={611--643},
  year={2013},
  publisher={Springer}
}
@article{williams1989experimental,
  title={Experimental analysis of the real-time recurrent learning algorithm},
  author={Williams, Ronald J and Zipser, David},
  journal={Connection science},
  volume={1},
  number={1},
  pages={87--111},
  year={1989},
  publisher={Taylor \& Francis}
}
@article{bellec2020solution,
  title={A solution to the learning dilemma for recurrent networks of spiking neurons},
  author={Bellec, Guillaume and Scherr, Franz and Subramoney, Anand and Hajek, Elias and Salaj, Darjan and Legenstein, Robert and Maass, Wolfgang},
  journal={bioRxiv},
  pages={738385},
  year={2020},
  publisher={Cold Spring Harbor Laboratory}
}

@article{murray2002shape,
  title={Shape perception reduces activity in human primary visual cortex},
  author={Murray, Scott O and Kersten, Daniel and Olshausen, Bruno A and Schrater, Paul and Woods, David L},
  journal={Proceedings of the National Academy of Sciences},
  volume={99},
  number={23},
  pages={15164--15169},
  year={2002},
  publisher={National Academy of Sciences}
}
@article{choi2018predictive,
  title={Predictive coding in area V4: dynamic shape discrimination under partial occlusion},
  author={Choi, Hannah and Pasupathy, Anitha and Shea-Brown, Eric},
  journal={Neural computation},
  volume={30},
  number={5},
  pages={1209--1257},
  year={2018},
  publisher={MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}
}
@article{garrido2009mismatch,
  title={The mismatch negativity: a review of underlying mechanisms},
  author={Garrido, Marta I and Kilner, James M and Stephan, Klaas E and Friston, Karl J},
  journal={Clinical neurophysiology},
  volume={120},
  number={3},
  pages={453--463},
  year={2009},
  publisher={Elsevier}
}
@article{wolpert2003unifying,
  title={A unifying computational framework for motor control and social interaction},
  author={Wolpert, Daniel M and Doya, Kenji and Kawato, Mitsuo},
  journal={Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences},
  volume={358},
  number={1431},
  pages={593--602},
  year={2003},
  publisher={The Royal Society}
}
@article{oliviers2024learning,
  title={Learning probability distributions of sensory inputs with Monte Carlo predictive coding},
  author={Oliviers, Gaspard and Bogacz, Rafal and Meulemans, Alexander},
  journal={PLOS Computational Biology},
  volume={20},
  number={10},
  pages={e1012532},
  year={2024},
  publisher={Public Library of Science San Francisco, CA USA}
}
@article{lecun2015deep,
  title={Deep learning},
  author={LeCun, Yann and Bengio, Yoshua and Hinton, Geoffrey},
  journal={nature},
  volume={521},
  number={7553},
  pages={436--444},
  year={2015},
  publisher={Nature Publishing Group UK London}
}
@article{kogo2015predictive,
  title={Is predictive coding theory articulated enough to be testable?},
  author={Kogo, Naoki and Trengove, Chris},
  journal={Frontiers in computational neuroscience},
  volume={9},
  pages={111},
  year={2015},
  publisher={Frontiers}
}
@article{keller2018predictive,
  title={Predictive processing: a canonical cortical computation},
  author={Keller, Georg B and Mrsic-Flogel, Thomas D},
  journal={Neuron},
  volume={100},
  number={2},
  pages={424--435},
  year={2018},
  publisher={Elsevier}
}
@article{van2020going,
  title={Going in circles is the way forward: the role of recurrence in visual inference},
  author={van Bergen, Ruben S and Kriegeskorte, Nikolaus},
  journal={arXiv preprint arXiv:2003.12128},
  year={2020}
}

@inproceedings{elsayed2019reduced,
  title={Reduced-Gate Convolutional LSTM Architecture for Next-Frame Video Prediction Using Predictive Coding},
  author={Elsayed, Nelly and Maida, Anthony S and Bayoumi, Magdy},
  booktitle={2019 International Joint Conference on Neural Networks (IJCNN)},
  pages={1--9},
  year={2019},
  organization={IEEE}
}
@inproceedings{lee2001algorithms,
  title={Algorithms for non-negative matrix factorization},
  author={Lee, Daniel D and Seung, H Sebastian},
  booktitle={Advances in neural information processing systems},
  pages={556--562},
  year={2001}
}
@article{radford2015unsupervised,
  title={Unsupervised representation learning with deep convolutional generative adversarial networks},
  author={Radford, Alec and Metz, Luke and Chintala, Soumith},
  journal={arXiv preprint arXiv:1511.06434},
  year={2015}
}
@article{rao1999optimal,
  title={An optimal estimation approach to visual perception and learning},
  author={Rao, Rajesh PN},
  journal={Vision research},
  volume={39},
  number={11},
  pages={1963--1989},
  year={1999},
  publisher={Elsevier}
}
@article{williams1995gradient,
  title={Gradient-based learning algorithms for recurrent},
  author={Williams, Ronald J and Zipser, David},
  journal={Backpropagation: Theory, architectures, and applications},
  volume={433},
  pages={17},
  year={1995}
}
@incollection{jordan1990attractor,
  title={Attractor dynamics and parallelism in a connectionist sequential machine},
  author={Jordan, Michael I},
  booktitle={Artificial neural networks: concept learning},
  pages={112--127},
  year={1990}
}
@article{werbos1988generalization,
  title={Generalization of backpropagation with application to a recurrent gas market model},
  author={Werbos, Paul J},
  journal={Neural networks},
  volume={1},
  number={4},
  pages={339--356},
  year={1988},
  publisher={Elsevier}
}
@article{dong1995temporal,
  title={Temporal decorrelation: a theory of lagged and nonlagged responses in the lateral geniculate nucleus},
  author={Dong, Dawei W and Atick, Joseph J},
  journal={Network: Computation in neural systems},
  volume={6},
  number={2},
  pages={159--178},
  year={1995},
  publisher={Taylor \& Francis}
}
@article{raymond2018computational,
  title={Computational principles of supervised learning in the cerebellum},
  author={Raymond, Jennifer L and Medina, Javier F},
  journal={Annual review of neuroscience},
  volume={41},
  pages={233--253},
  year={2018},
  publisher={Annual Reviews}
}
@article{thorpe1996speed,
  title={Speed of processing in the human visual system},
  author={Thorpe, Simon and Fize, Denis and Marlot, Catherine},
  journal={nature},
  volume={381},
  number={6582},
  pages={520--522},
  year={1996},
  publisher={Nature Publishing Group}
}
@article{keysers2001speed,
  title={The speed of sight},
  author={Keysers, Christian and Xiao, D-K and F{\"o}ldi{\'a}k, Peter and Perrett, David I},
  journal={Journal of cognitive neuroscience},
  volume={13},
  number={1},
  pages={90--101},
  year={2001},
  publisher={MIT Press}
}
@article{carlson2013representational,
  title={Representational dynamics of object vision: the first 1000 ms},
  author={Carlson, Thomas and Tovar, David A and Alink, Arjen and Kriegeskorte, Nikolaus},
  journal={Journal of vision},
  volume={13},
  number={10},
  pages={1--1},
  year={2013},
  publisher={The Association for Research in Vision and Ophthalmology}
}
@article{thunell2019memory,
  title={Memory for repeated images in rapid-serial-visual-presentation streams of thousands of images},
  author={Thunell, Evelina and Thorpe, Simon J},
  journal={Psychological science},
  volume={30},
  number={7},
  pages={989--1000},
  year={2019},
  publisher={Sage Publications Sage CA: Los Angeles, CA}
}
@article{lyon2010disynaptic,
  title={A disynaptic relay from superior colliculus to dorsal stream visual cortex in macaque monkey},
  author={Lyon, David C and Nassi, Jonathan J and Callaway, Edward M},
  journal={Neuron},
  volume={65},
  number={2},
  pages={270--279},
  year={2010},
  publisher={Elsevier}
}
@article{soares2017fast,
  title={Fast detector/first responder: interactions between the superior colliculus-pulvinar pathway and stimuli relevant to primates},
  author={Soares, Sandra C and Maior, Rafael S and Isbell, Lynne A and Tomaz, Carlos and Nishijo, Hisao},
  journal={Frontiers in Neuroscience},
  volume={11},
  pages={67},
  year={2017},
  publisher={Frontiers}
}
@article{garvert2014subcortical,
  title={Subcortical amygdala pathways enable rapid face processing},
  author={Garvert, Mona M and Friston, Karl J and Dolan, Raymond J and Garrido, Marta I},
  journal={Neuroimage},
  volume={102},
  pages={309--316},
  year={2014},
  publisher={Elsevier}
}

@inproceedings{felleman1991distributed,
  title={Distributed hierarchical processing in the primate cerebral cortex},
  author={Felleman, Daniel J and Van Essen, David C},
  booktitle={Cereb cortex},
  year={1991},
  organization={Citeseer}
}

@article{masreliez1977robust,
  title={Robust Bayesian estimation for the linear model and robustifying the Kalman filter},
  author={Masreliez, Cl and Martin, R},
  journal={IEEE transactions on Automatic Control},
  volume={22},
  number={3},
  pages={361--371},
  year={1977},
  publisher={IEEE}
}
@article{yuen2007selection,
  title={Selection of noise parameters for Kalman filter},
  author={Yuen, Ka-Veng and Hoi, Ka-In and Mok, Kai-Meng},
  journal={Earthquake Engineering and Engineering Vibration},
  volume={6},
  number={1},
  pages={49--56},
  year={2007},
  publisher={Springer}
}
@incollection{maybeck1990kalman,
  title={The Kalman filter: An introduction to concepts},
  author={Maybeck, Peter S},
  booktitle={Autonomous robot vehicles},
  pages={194--204},
  year={1990},
  publisher={Springer}
}
@inproceedings{oussalah2001adaptive,
  title={Adaptive Kalman filter for noise identification},
  author={Oussalah, M and De Schutter, Joris},
  booktitle={Proceedings of the international Seminar on Modal Analysis},
  volume={3},
  pages={1225--1232},
  year={2001},
  organization={KU Leuven; 1998}
}
@article{schultz1998predictive,
  title={Predictive reward signal of dopamine neurons},
  author={Schultz, Wolfram},
  journal={Journal of neurophysiology},
  volume={80},
  number={1},
  pages={1--27},
  year={1998},
  publisher={American Physiological Society Bethesda, MD}
}
@article{schultz1997neural,
  title={A neural substrate of prediction and reward},
  author={Schultz, Wolfram and Dayan, Peter and Montague, P Read},
  journal={Science},
  volume={275},
  number={5306},
  pages={1593--1599},
  year={1997},
  publisher={American Association for the Advancement of Science}
}
@article{lindquist1973feedback,
  title={On feedback control of linear stochastic systems},
  author={Lindquist, Anders},
  journal={SIAM Journal on Control},
  volume={11},
  number={2},
  pages={323--343},
  year={1973},
  publisher={SIAM}
}
@article{tallec2018can,
  title={Can recurrent neural networks warp time?},
  author={Tallec, Corentin and Ollivier, Yann},
  journal={arXiv preprint arXiv:1804.11188},
  year={2018}
}
@inproceedings{koutnik2014clockwork,
  title={A clockwork rnn},
  author={Koutnik, Jan and Greff, Klaus and Gomez, Faustino and Schmidhuber, Juergen},
  booktitle={International Conference on Machine Learning},
  pages={1863--1871},
  year={2014},
  organization={PMLR}
}

@book{jaynes2003probability,
  title={Probability theory: The logic of science},
  author={Jaynes, Edwin T},
  year={2003},
  publisher={Cambridge university press}
}
@article{cox1946probability,
  title={Probability, frequency and reasonable expectation},
  author={Cox, Richard T},
  journal={American journal of physics},
  volume={14},
  number={1},
  pages={1--13},
  year={1946},
  publisher={American Association of Physics Teachers}
}

@article{cao2020new,
  title={New labels for old ideas: Predictive processing and the interpretation of neural signals},
  author={Cao, Rosa},
  journal={Review of Philosophy and Psychology},
  volume={11},
  number={3},
  pages={517--546},
  year={2020},
  publisher={Springer}
}

@article{grill2004human,
  title={The human visual cortex},
  author={Grill-Spector, Kalanit and Malach, Rafael},
  journal={Annu. Rev. Neurosci.},
  volume={27},
  pages={649--677},
  year={2004},
  publisher={Annual Reviews}
}

@article{seth2020preface,
  title={Preface: The Brain as a Prediction Machine},
  author={Seth, Anil},
  journal={The Philosophy and Science of Predictive Processing},
  year={2020},
  publisher={Bloomsbury Publishing}
}

@article{hinton2006reducing,
  title={Reducing the dimensionality of data with neural networks},
  author={Hinton, Geoffrey E and Salakhutdinov, Ruslan R},
  journal={science},
  volume={313},
  number={5786},
  pages={504--507},
  year={2006},
  publisher={American Association for the Advancement of Science}
}
@article{millidge2019fixational,
  title={Fixational Eye Movements: Data Augmentation for the Brain?},
  author={Millidge, Beren and Shillcock, Richard},
  year={2019},
  journal={PsyArXiv},
  publisher={PsyArXiv}
}
@article{huys2016computational,
  title={Computational psychiatry as a bridge from neuroscience to clinical applications},
  author={Huys, Quentin JM and Maia, Tiago V and Frank, Michael J},
  journal={Nature neuroscience},
  volume={19},
  number={3},
  pages={404},
  year={2016},
  publisher={Nature Publishing Group}
}

@article{olshausen1997sparse,
  title={Sparse coding with an overcomplete basis set: A strategy employed by V1?},
  author={Olshausen, Bruno A and Field, David J},
  journal={Vision research},
  volume={37},
  number={23},
  pages={3311--3325},
  year={1997},
  publisher={Elsevier}
}

@article{rao1997dynamic,
  title={Dynamic model of visual recognition predicts neural response properties in the visual cortex},
  author={Rao, Rajesh PN and Ballard, Dana H},
  journal={Neural computation},
  volume={9},
  number={4},
  pages={721--763},
  year={1997},
  publisher={MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}
}

@article{moeller2022uncertainty,
  title={Uncertainty--guided learning with scaled prediction errors in the basal ganglia},
  author={Moeller, Moritz and Manohar, Sanjay and Bogacz, Rafal},
  journal={PLoS computational biology},
  volume={18},
  number={5},
  pages={e1009816},
  year={2022},
  publisher={Public Library of Science San Francisco, CA USA}
}

@article{navajas2017idiosyncratic,
  title={The idiosyncratic nature of confidence},
  author={Navajas, Joaquin and Hindocha, Chandni and Foda, Hebah and Keramati, Mehdi and Latham, Peter E and Bahrami, Bahador},
  journal={Nature human behaviour},
  volume={1},
  number={11},
  pages={810--818},
  year={2017},
  publisher={Nature Publishing Group}
}

@article{daw2006cortical,
  title={Cortical substrates for exploratory decisions in humans},
  author={Daw, Nathaniel D and O'Doherty, John P and Dayan, Peter and Seymour, Ben and Dolan, Raymond J},
  journal={Nature},
  volume={441},
  number={7095},
  pages={876--879},
  year={2006},
  publisher={Nature Publishing Group}
}
@article{howard2010neural,
  title={The neural mechanisms of learning from competitors},
  author={Howard-Jones, Paul A and Bogacz, Rafal and Yoo, Jee H and Leonards, Ute and Demetriou, Skevi},
  journal={Neuroimage},
  volume={53},
  number={2},
  pages={790--799},
  year={2010},
  publisher={Elsevier}
}
@article{beck2007probabilistic,
  title={Probabilistic population codes and the exponential family of distributions},
  author={Beck, J and Ma, WJ and Latham, PE and Pouget, A},
  journal={Progress in brain research},
  volume={165},
  pages={509--519},
  year={2007},
  publisher={Elsevier}
}
@inproceedings{wilson2009neural,
  title={A neural implementation of the Kalman filter},
  author={Wilson, Robert and Finkel, Leif},
  booktitle={Advances in neural information processing systems},
  pages={2062--2070},
  year={2009}
}
@article{deneve2007optimal,
  title={Optimal sensorimotor integration in recurrent cortical networks: a neural implementation of Kalman filters},
  author={Deneve, Sophie and Duhamel, Jean-Ren{\'e} and Pouget, Alexandre},
  journal={Journal of Neuroscience},
  volume={27},
  number={21},
  pages={5744--5756},
  year={2007},
  publisher={Soc Neuroscience}
}

@article{jiang2022dynamic,
  title={Dynamic Predictive Coding: A New Model of Hierarchical Sequence Learning and Prediction in the Cortex},
  author={Jiang, Linxing Preston and Rao, Rajesh PN},
  journal={bioRxiv},
  year={2022},
  publisher={Cold Spring Harbor Laboratory}
}
@article{kutschireiter2017nonlinear,
  title={Nonlinear Bayesian filtering and learning: a neuronal dynamics for perception},
  author={Kutschireiter, Anna and Surace, Simone Carlo and Sprekeler, Henning and Pfister, Jean-Pascal},
  journal={Scientific reports},
  volume={7},
  number={1},
  pages={1--13},
  year={2017},
  publisher={Nature Publishing Group}
}

@article{ruzzoli2019relevance,
  title={The relevance of alpha phase in human perception},
  author={Ruzzoli, Manuela and Torralba, Mireia and Fern{\'a}ndez, Luis Mor{\'\i}s and Soto-Faraco, Salvador},
  journal={Cortex},
  volume={120},
  pages={249--268},
  year={2019},
  publisher={Elsevier}
}

@article{tang2023recurrent,
  title={Recurrent predictive coding models for associative memory employing covariance learning},
  author={Tang, Mufeng and Salvatori, Tommaso and Millidge, Beren and Song, Yuhang and Lukasiewicz, Thomas and Bogacz, Rafal},
  journal={PLOS Computational Biology},
  volume={19},
  number={4},
  pages={e1010719},
  year={2023},
  publisher={Public Library of Science San Francisco, CA USA}
}

@article{chen2003bayesian,
  title={Bayesian filtering: From Kalman filters to particle filters, and beyond},
  author={Chen, Zhe and others},
  journal={Statistics},
  volume={182},
  number={1},
  pages={1--69},
  year={2003},
  publisher={Citeseer}
}

@article{brown1992introduction,
  title={Introduction to random signals and applied Kalman filtering(Book)},
  author={BROWN, ROBERTG and HWANG, PATRICKYC},
  journal={New York, John Wiley \& Sons, Inc., 1992. 512},
  year={1992}
}

@book{bishop2006pattern,
  title={Pattern recognition and machine learning},
  author={Bishop, Christopher M and Nasrabadi, Nasser M},
  volume={4},
  number={4},
  year={2006},
  publisher={Springer}
}

@article{singer2018sensory,
  title={Sensory cortex is optimized for prediction of future input},
  author={Singer, Yosef and Teramoto, Yayoi and Willmore, Ben DB and Schnupp, Jan WH and King, Andrew J and Harper, Nicol S},
  journal={elife},
  volume={7},
  pages={e31557},
  year={2018},
  publisher={eLife Sciences Publications, Ltd}
}

@book{dayan2005theoretical,
  title={Theoretical neuroscience: computational and mathematical modeling of neural systems},
  author={Dayan, Peter and Abbott, Laurence F},
  year={2005},
  publisher={MIT press}
}

@article{ohzawa1996encoding,
  title={Encoding of binocular disparity by simple cells in the cat's visual cortex},
  author={Ohzawa, Izumi and DeAngelis, Gregory C and Freeman, Ralph D},
  journal={Journal of Neurophysiology},
  volume={75},
  number={5},
  pages={1779--1805},
  year={1996},
  publisher={American Physiological Society Bethesda, MD}
}

@article{baker1990spatial,
  title={Spatial-and temporal-frequency selectivity as a basis for velocity preference in cat striate cortex neurons},
  author={Baker, Curtis L},
  journal={Visual neuroscience},
  volume={4},
  number={2},
  pages={101--113},
  year={1990},
  publisher={Cambridge University Press}
}

@article{livingstone1987psychophysical,
  title={Psychophysical evidence for separate channels for the perception of form, color, movement, and depth},
  author={Livingstone, Margaret S and Hubel, David H},
  journal={Journal of Neuroscience},
  volume={7},
  number={11},
  pages={3416--3468},
  year={1987},
  publisher={Soc Neuroscience}
}

@article{bogacz2000emergence,
  title={Emergence of movement sensitive neurons' properties by learning a sparse code for natural moving images},
  author={Bogacz, Rafal and Brown, Malcolm and Giraud-Carrier, Christophe},
  journal={Advances in neural information processing systems},
  volume={13},
  year={2000}
}

@article{olshausen2002sparse,
  title={Sparse coding of time-varying natural images},
  author={Olshausen, Bruno A},
  journal={Journal of Vision},
  volume={2},
  number={7},
  pages={130--130},
  year={2002},
  publisher={The Association for Research in Vision and Ophthalmology}
}

@article{baltieri2021kalman,
  title={Kalman filters as the steady-state solution of gradient descent on variational free energy},
  author={Baltieri, Manuel and Isomura, Takuya},
  journal={arXiv preprint arXiv:2111.10530},
  year={2021}
}

@article{nowak2003electrophysiological,
  title={Electrophysiological classes of cat primary visual cortical neurons in vivo as revealed by quantitative analyses},
  author={Nowak, Lionel G and Azouz, Rony and Sanchez-Vives, Maria V and Gray, Charles M and McCormick, David A},
  journal={Journal of neurophysiology},
  volume={89},
  number={3},
  pages={1541--1566},
  year={2003},
  publisher={American Physiological Society Bethesda, MD}
}

@article{singer2023hierarchical,
  title={Hierarchical temporal prediction captures motion processing along the visual pathway},
  author={Singer, Yosef and Taylor, Luke and Willmore, Ben DB and King, Andrew J and Harper, Nicol S},
  journal={Elife},
  volume={12},
  pages={e52599},
  year={2023},
  publisher={eLife Sciences Publications Limited}
}

@article{van1998independent,
  title={Independent component analysis of natural image sequences yields spatio-temporal filters similar to simple cells in primary visual cortex},
  author={van Hateren, J Hans and Ruderman, Dan L},
  journal={Proceedings of the Royal Society of London. Series B: Biological Sciences},
  volume={265},
  number={1412},
  pages={2315--2320},
  year={1998},
  publisher={The Royal Society}
}

@article{berkes2005slow,
  title={Slow feature analysis yields a rich repertoire of complex cell properties},
  author={Berkes, Pietro and Wiskott, Laurenz},
  journal={Journal of vision},
  volume={5},
  number={6},
  pages={9--9},
  year={2005},
  publisher={The Association for Research in Vision and Ophthalmology}
}

@article{hyvarinen2003bubbles,
  title={Bubbles: a unifying framework for low-level statistical properties of natural image sequences},
  author={Hyv{\"a}rinen, Aapo and Hurri, Jarmo and V{\"a}yrynen, Jaakko},
  journal={JOSA A},
  volume={20},
  number={7},
  pages={1237--1252},
  year={2003},
  publisher={Optica Publishing Group}
}

@article{palm2012prediction,
  title={Prediction as a candidate for learning deep hierarchical models of data},
  author={Palm, Rasmus Berg},
  journal={Technical University of Denmark},
  volume={5},
  year={2012}
}

@article{friedrich2021neural,
  title={Neural optimal feedback control with local learning rules},
  author={Friedrich, Johannes and Golkar, Siavash and Farashahi, Shiva and Genkin, Alexander and Sengupta, Anirvan and Chklovskii, Dmitri},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={16358--16370},
  year={2021}
}

@article{auksztulewicz2023omission,
  title={Omission responses in local field potentials in rat auditory cortex},
  author={Auksztulewicz, Ryszard and Rajendran, Vani Gurusamy and Peng, Fei and Schnupp, Jan Wilbert Hendrik and Harper, Nicol Spencer},
  journal={BMC biology},
  volume={21},
  number={1},
  pages={130},
  year={2023},
  publisher={Springer}
}

@article{pachitariu2017visual,
  title={Visual motion computation in recurrent neural networks},
  author={Pachitariu, Marius and Sahani, Maneesh},
  journal={bioRxiv},
  pages={099101},
  year={2017},
  publisher={Cold Spring Harbor Laboratory}
}

@article{pachitariu2012learning,
  title={Learning visual motion in recurrent neural networks},
  author={Pachitariu, Marius and Sahani, Maneesh},
  journal={Advances in Neural Information Processing Systems},
  volume={25},
  year={2012}
}

@article{julier2004unscented,
  title={Unscented filtering and nonlinear estimation},
  author={Julier, Simon J and Uhlmann, Jeffrey K},
  journal={Proceedings of the IEEE},
  volume={92},
  number={3},
  pages={401--422},
  year={2004},
  publisher={IEEE}
}

@article{ribeiro2004kalman,
  title={Kalman and extended kalman filters: Concept, derivation and properties},
  author={Ribeiro, Maria Isabel},
  journal={Institute for Systems and Robotics},
  volume={43},
  number={46},
  pages={3736--3741},
  year={2004},
  publisher={Lisboa, Portugal:}
}

@article{tulving1972episodic,
  title={Episodic and semantic memory.},
  author={Tulving, Endel},
  year={1972},
  publisher={Academic press}
}

@article{whittington2017approximation,
  title={An approximation of the error backpropagation algorithm in a predictive coding network with local hebbian synaptic plasticity},
  author={Whittington, James CR and Bogacz, Rafal},
  journal={Neural computation},
  volume={29},
  number={5},
  pages={1229--1262},
  year={2017},
  publisher={MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}
}


@article{sompolinsky1986temporal,
  title={Temporal association in asymmetric neural networks},
  author={Sompolinsky, Haim and Kanter, Ido},
  journal={Physical review letters},
  volume={57},
  number={22},
  pages={2861},
  year={1986},
  publisher={APS}
}

@inproceedings{millidge2022universal,
  title={Universal hopfield networks: A general framework for single-shot associative memory models},
  author={Millidge, Beren and Salvatori, Tommaso and Song, Yuhang and Lukasiewicz, Thomas and Bogacz, Rafal},
  booktitle={International Conference on Machine Learning},
  pages={15561--15583},
  year={2022},
  organization={PMLR}
}

@article{iatropoulos2022kernel,
  title={Kernel Memory Networks: A Unifying Framework for Memory Modeling},
  author={Iatropoulos, Georgios and Brea, Johanni and Gerstner, Wulfram},
  journal={arXiv preprint arXiv:2208.09416},
  year={2022}
}

@article{hopfield1982neural,
  title={Neural networks and physical systems with emergent collective computational abilities},
  author={Hopfield, John J},
  journal={Proceedings of the national academy of sciences},
  volume={79},
  number={8},
  pages={2554--2558},
  year={1982},
  publisher={National Acad Sciences}
}

@article{krotov2016dense,
  title={Dense associative memory for pattern recognition},
  author={Krotov, Dmitry and Hopfield, John J},
  journal={Advances in neural information processing systems},
  volume={29},
  year={2016}
}

@article{yoo2022bayespcn,
  title={BayesPCN: A Continually Learnable Predictive Coding Associative Memory},
  author={Yoo, Jinsoo and Wood, Frank},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={29903--29914},
  year={2022}
}


@misc{rao2023predictive,
	author = {Rajesh P.N. Rao and Linxing Preston Jiang},
	doi = {10.1093/acrefore/9780190264086.013.328},
	month = {11},
	publisher = {Oxford University Press},
	title = {Predictive Coding Theories of Cortical Function},
	url = {https://oxfordre.com/neuroscience/view/10.1093/acrefore/9780190264086.001.0001/acrefore-9780190264086-e-328},
	year = {2022},
	Bdsk-Url-1 = {https://oxfordre.com/neuroscience/view/10.1093/acrefore/9780190264086.001.0001/acrefore-9780190264086-e-328},
	Bdsk-Url-2 = {https://doi.org/10.1093/acrefore/9780190264086.013.328}}

@article{bellmund2020sequence,
  title={Sequence memory in the hippocampal--entorhinal region},
  author={Bellmund, Jacob LS and Polti, Ignacio and Doeller, Christian F},
  journal={Journal of Cognitive Neuroscience},
  volume={32},
  number={11},
  pages={2056--2070},
  year={2020},
  publisher={MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}
}

@article{honey1998hippocampal,
  title={Hippocampal lesions disrupt an associative mismatch process},
  author={Honey, Robert Colin and Watt, A and Good, M},
  journal={Journal of Neuroscience},
  volume={18},
  number={6},
  pages={2226--2230},
  year={1998},
  publisher={Soc Neuroscience}
}

@article{eichenbaum2014time,
  title={Time cells in the hippocampus: a new dimension for mapping memories},
  author={Eichenbaum, Howard},
  journal={Nature Reviews Neuroscience},
  volume={15},
  number={11},
  pages={732--744},
  year={2014},
  publisher={Nature Publishing Group UK London}
}

@article{kohonen1972correlation,
  title={Correlation matrix memories},
  author={Kohonen, Teuvo},
  journal={IEEE transactions on computers},
  volume={100},
  number={4},
  pages={353--359},
  year={1972},
  publisher={IEEE}
}


@article{o1994hippocampal,
  title={Hippocampal conjunctive encoding, storage, and recall: Avoiding a trade-off},
  author={O'Reilly, Randall C and McClelland, James L},
  journal={Hippocampus},
  volume={4},
  number={6},
  pages={661--682},
  year={1994},
  publisher={Wiley Online Library}
}

@article{treves1994computational,
  title={Computational analysis of the role of the hippocampus in memory},
  author={Treves, Alessandro and Rolls, Edmund T},
  journal={Hippocampus},
  volume={4},
  number={3},
  pages={374--391},
  year={1994},
  publisher={Wiley Online Library}
}

@article{eichenbaum2013memory,
  title={Memory on time},
  author={Eichenbaum, Howard},
  journal={Trends in cognitive sciences},
  volume={17},
  number={2},
  pages={81--88},
  year={2013},
  publisher={Elsevier}
}

@article{wallenstein1998hippocampus,
  title={The hippocampus as an associator of discontiguous events},
  author={Wallenstein, Gene V and Hasselmo, Michael E and Eichenbaum, Howard},
  journal={Trends in neurosciences},
  volume={21},
  number={8},
  pages={317--323},
  year={1998},
  publisher={Elsevier}
}

@article{rolls2010computational,
  title={A computational theory of episodic memory formation in the hippocampus},
  author={Rolls, Edmund T},
  journal={Behavioural brain research},
  volume={215},
  number={2},
  pages={180--196},
  year={2010},
  publisher={Elsevier}
}

@article{hawkins2009sequence,
  title={Sequence memory for prediction, inference and behaviour},
  author={Hawkins, Jeff and George, Dileep and Niemasik, Jamie},
  journal={Philosophical Transactions of the Royal Society B: Biological Sciences},
  volume={364},
  number={1521},
  pages={1203--1209},
  year={2009},
  publisher={The Royal Society London}
}

@article{jensen1996physiologically,
  title={Physiologically realistic formation of autoassociative memory in networks with theta/gamma oscillations: role of fast NMDA channels.},
  author={Jensen, Ole and Idiart, MA and Lisman, John E},
  journal={Learning \& Memory},
  volume={3},
  number={2-3},
  pages={243--256},
  year={1996},
  publisher={Cold Spring Harbor Lab}
}

@article{kanter1987associative,
  title={Associative recall of memory without errors},
  author={Kanter, I and Sompolinsky, Haim},
  journal={Physical Review A},
  volume={35},
  number={1},
  pages={380},
  year={1987},
  publisher={APS}
}


@article{chaudhry2023modern,
  title={Long Sequence Hopfield Memory},
  author={Chaudhry, Hamza Tahir and Zavatone-Veth, Jacob A and Krotov, Dmitry and Pehlevan, Cengiz},
  journal={arXiv preprint arXiv:2306.04532},
  year={2023}
}

@article{duong2023statistical,
  title={Statistical whitening of neural populations with gain-modulating interneurons},
  author={Duong, Lyndon R and Lipshutz, David and Heeger, David J and Chklovskii, Dmitri B and Simoncelli, Eero P},
  journal={arXiv preprint arXiv:2301.11955},
  year={2023}
}

@article{whitehead1991learning,
  title={Learning to perceive and act by trial and error},
  author={Whitehead, Steven D and Ballard, Dana H},
  journal={Machine Learning},
  volume={7},
  pages={45--83},
  year={1991},
  publisher={Springer}
}

@article{olafsdottir2018role,
  title={The role of hippocampal replay in memory and planning},
  author={{\'O}lafsd{\'o}ttir, H Freyja and Bush, Daniel and Barry, Caswell},
  journal={Current Biology},
  volume={28},
  number={1},
  pages={R37--R50},
  year={2018},
  publisher={Elsevier}
}

@article{roweis1999unifying,
  title={A unifying review of linear Gaussian models},
  author={Roweis, Sam and Ghahramani, Zoubin},
  journal={Neural computation},
  volume={11},
  number={2},
  pages={305--345},
  year={1999},
  publisher={MIT Press}
}

@article{barak2014working,
  title={Working models of working memory},
  author={Barak, Omri and Tsodyks, Misha},
  journal={Current opinion in neurobiology},
  volume={25},
  pages={20--24},
  year={2014},
  publisher={Elsevier}
}


@article{pehlevan2019neuroscience,
  title={Neuroscience-inspired online unsupervised learning algorithms: Artificial neural networks},
  author={Pehlevan, Cengiz and Chklovskii, Dmitri B},
  journal={IEEE Signal Processing Magazine},
  volume={36},
  number={6},
  pages={88--96},
  year={2019},
  publisher={IEEE}
}

@article{golkar2022constrained,
  title={Constrained Predictive Coding as a Biologically Plausible Model of the Cortical Hierarchy},
  author={Golkar, Siavash and Tesileanu, Tiberiu and Bahroun, Yanis and Sengupta, Anirvan and Chklovskii, Dmitri},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={14155--14169},
  year={2022}
}

@inproceedings{srivastava2015unsupervised,
  title={Unsupervised learning of video representations using lstms},
  author={Srivastava, Nitish and Mansimov, Elman and Salakhudinov, Ruslan},
  booktitle={International conference on machine learning},
  pages={843--852},
  year={2015},
  organization={PMLR}
}


@article{soomro2012ucf101,
  title={UCF101: A dataset of 101 human actions classes from videos in the wild},
  author={Soomro, Khurram and Zamir, Amir Roshan and Shah, Mubarak},
  journal={arXiv preprint arXiv:1212.0402},
  year={2012}
}

@article{chen2022hippocampus,
  title={Hippocampus as a generative circuit for predictive coding of future sequences},
  author={Chen, Yusi and Zhang, Huanqiu and Sejnowski, Terrence J},
  journal={bioRxiv},
  pages={2022--05},
  year={2022},
  publisher={Cold Spring Harbor Laboratory}
}

@article{dai2019transformer,
  title={Transformer-xl: Attentive language models beyond a fixed-length context},
  author={Dai, Zihang and Yang, Zhilin and Yang, Yiming and Carbonell, Jaime and Le, Quoc V and Salakhutdinov, Ruslan},
  journal={arXiv preprint arXiv:1901.02860},
  year={2019}
}

@article{demircigil2017model,
  title={On a model of associative memory with huge storage capacity},
  author={Demircigil, Mete and Heusel, Judith and L{\"o}we, Matthias and Upgang, Sven and Vermet, Franck},
  journal={Journal of Statistical Physics},
  volume={168},
  pages={288--299},
  year={2017},
  publisher={Springer}
}

@article{kessy2018optimal,
  title={Optimal whitening and decorrelation},
  author={Kessy, Agnan and Lewin, Alex and Strimmer, Korbinian},
  journal={The American Statistician},
  volume={72},
  number={4},
  pages={309--314},
  year={2018},
  publisher={Taylor \& Francis}
}

@article{bogacz2003comparison,
  title={Comparison of computational models of familiarity discrimination in the perirhinal cortex},
  author={Bogacz, Rafal and Brown, Malcolm W},
  journal={Hippocampus},
  volume={13},
  number={4},
  pages={494--524},
  year={2003},
  publisher={Wiley Online Library}
}

@article {millidge2023temporal,
	author = {Beren Millidge and Mufeng Tang and Mahyar Osanlouy and Rafal Bogacz},
	title = {Predictive Coding Networks for Temporal Prediction},
	elocation-id = {2023.05.15.540906},
	year = {2023},
	doi = {10.1101/2023.05.15.540906},
	publisher = {Cold Spring Harbor Laboratory},
	abstract = {One of the key problems the brain faces is inferring the state of the world from a sequence of dynamically changing stimuli, and it is not yet clear how the sensory system achieves this task. A well-established computational framework for describing perceptual processes in the brain is provided by the theory of predictive coding. Although the original proposals of predictive coding have discussed temporal prediction, later work developing this theory mostly focused on static stimuli, and key questions on neural implementation and computational properties of temporal predictive coding networks remain open. Here, we address these questions and present a formulation of the temporal predictive coding model that can be naturally implemented in recurrent networks, in which activity dynamics rely only on local inputs to the neurons, and learning only utilises local Hebbian plasticity. Additionally, we show that predictive coding networks can approximate the performance of the Kalman filter in predicting behaviour of linear systems, and behave as a variant of a Kalman filter which does not track its own subjective posterior variance. Importantly, predictive coding networks can achieve similar accuracy as the Kalman filter without performing complex mathematical operations, but just employing simple computations that can be implemented by biological networks. Moreover, we demonstrate how the model can be effectively generalized to non-linear systems. Overall, models presented in this paper show how biologically plausible circuits can predict future stimuli and may guide research on understanding specific neural circuits in brain areas involved in temporal prediction.Competing Interest StatementBM and RB are shareholders in Neu Edge Ltd, which designs AI accelerator hardware.},
	URL = {https://www.biorxiv.org/content/early/2023/05/16/2023.05.15.540906},
	eprint = {https://www.biorxiv.org/content/early/2023/05/16/2023.05.15.540906.full.pdf},
	journal = {bioRxiv}
}

@incollection{levy1989computational,
  title={A computational approach to hippocampal function},
  author={Levy, William B},
  booktitle={Psychology of learning and motivation},
  volume={23},
  pages={243--305},
  year={1989},
  publisher={Elsevier}
}

@article{howard2002distributed,
  title={A distributed representation of temporal context},
  author={Howard, Marc W and Kahana, Michael J},
  journal={Journal of mathematical psychology},
  volume={46},
  number={3},
  pages={269--299},
  year={2002},
  publisher={Elsevier}
}

@article{howard2014unified,
  title={A unified mathematical framework for coding time, space, and sequences in the hippocampal region},
  author={Howard, Marc W and MacDonald, Christopher J and Tiganj, Zoran and Shankar, Karthik H and Du, Qian and Hasselmo, Michael E and Eichenbaum, Howard},
  journal={Journal of Neuroscience},
  volume={34},
  number={13},
  pages={4692--4707},
  year={2014},
  publisher={Soc Neuroscience}
}

@article{botvinick2006short,
  title={Short-term memory for serial order: a recurrent neural network model.},
  author={Botvinick, Matthew M and Plaut, David C},
  journal={Psychological review},
  volume={113},
  number={2},
  pages={201},
  year={2006},
  publisher={American Psychological Association}
}

@article{eliasmith2012large,
  title={A large-scale model of the functioning brain},
  author={Eliasmith, Chris and Stewart, Terrence C and Choo, Xuan and Bekolay, Trevor and DeWolf, Travis and Tang, Yichuan and Rasmussen, Daniel},
  journal={science},
  volume={338},
  number={6111},
  pages={1202--1205},
  year={2012},
  publisher={American Association for the Advancement of Science}
}

@article{amari1972learning,
  title={Learning patterns and pattern sequences by self-organizing nets of threshold elements},
  author={Amari, S-I},
  journal={IEEE Transactions on computers},
  volume={100},
  number={11},
  pages={1197--1206},
  year={1972},
  publisher={IEEE}
}

@article{whittington2022build,
  title={How to build a cognitive map},
  author={Whittington, James CR and McCaffary, David and Bakermans, Jacob JW and Behrens, Timothy EJ},
  journal={Nature neuroscience},
  volume={25},
  number={10},
  pages={1257--1272},
  year={2022},
  publisher={Nature Publishing Group US New York}
}

@article{whittington2018generalisation,
  title={Generalisation of structural knowledge in the hippocampal-entorhinal system},
  author={Whittington, James and Muller, Timothy and Mark, Shirely and Barry, Caswell and Behrens, Tim},
  journal={Advances in neural information processing systems},
  volume={31},
  year={2018}
}

@article{whittington2020tolman,
  title={The Tolman-Eichenbaum machine: unifying space and relational memory through generalization in the hippocampal formation},
  author={Whittington, James CR and Muller, Timothy H and Mark, Shirley and Chen, Guifen and Barry, Caswell and Burgess, Neil and Behrens, Timothy EJ},
  journal={Cell},
  volume={183},
  number={5},
  pages={1249--1263},
  year={2020},
  publisher={Elsevier}
}

@article{crannell1957comparison,
  title={A comparison of immediate memory span for digits, letters, and words},
  author={Crannell, CW and Parrish, JM},
  journal={The Journal of Psychology},
  volume={44},
  number={2},
  pages={319--327},
  year={1957},
  publisher={Taylor \& Francis}
}

@article{henson1998short,
  title={Short-term memory for serial order: The start-end model},
  author={Henson, Richard NA},
  journal={Cognitive psychology},
  volume={36},
  number={2},
  pages={73--137},
  year={1998},
  publisher={Elsevier}
}

@article{tolman1948cognitive,
  title={Cognitive maps in rats and men.},
  author={Tolman, Edward C},
  journal={Psychological review},
  volume={55},
  number={4},
  pages={189},
  year={1948},
  publisher={American Psychological Association}
}

@article{behrens2018cognitive,
  title={What is a cognitive map? Organizing knowledge for flexible behavior},
  author={Behrens, Timothy EJ and Muller, Timothy H and Whittington, James CR and Mark, Shirley and Baram, Alon B and Stachenfeld, Kimberly L and Kurth-Nelson, Zeb},
  journal={Neuron},
  volume={100},
  number={2},
  pages={490--509},
  year={2018},
  publisher={Elsevier}
}

@article{fyhn2008grid,
  title={Grid cells in mice},
  author={Fyhn, Marianne and Hafting, Torkel and Witter, Menno P and Moser, Edvard I and Moser, May-Britt},
  journal={Hippocampus},
  volume={18},
  number={12},
  pages={1230--1238},
  year={2008},
  publisher={Wiley Online Library}
}

@article{yartsev2011grid,
  title={Grid cells without theta oscillations in the entorhinal cortex of bats},
  author={Yartsev, Michael M and Witter, Menno P and Ulanovsky, Nachum},
  journal={Nature},
  volume={479},
  number={7371},
  pages={103--107},
  year={2011},
  publisher={Nature Publishing Group UK London}
}

@article{doeller2010evidence,
  title={Evidence for grid cells in a human memory network},
  author={Doeller, Christian F and Barry, Caswell and Burgess, Neil},
  journal={Nature},
  volume={463},
  number={7281},
  pages={657--661},
  year={2010},
  publisher={Nature Publishing Group UK London}
}

@article{o2005dual,
  title={Dual phase and rate coding in hippocampal place cells: theoretical significance and relationship to entorhinal grid cells},
  author={O'Keefe, John and Burgess, Neil},
  journal={Hippocampus},
  volume={15},
  number={7},
  pages={853--866},
  year={2005},
  publisher={Wiley Online Library}
}

@article{hasselmo2007grid,
  title={Grid cell firing may arise from interference of theta frequency membrane potential oscillations in single neurons},
  author={Hasselmo, Michael E and Giocomo, Lisa M and Zilli, Eric A},
  journal={Hippocampus},
  volume={17},
  number={12},
  pages={1252--1271},
  year={2007},
  publisher={Wiley Online Library}
}

@article{fuhs2006spin,
  title={A spin glass model of path integration in rat medial entorhinal cortex},
  author={Fuhs, Mark C and Touretzky, David S},
  journal={Journal of Neuroscience},
  volume={26},
  number={16},
  pages={4266--4276},
  year={2006},
  publisher={Soc Neuroscience}
}

@article{burak2009accurate,
  title={Accurate path integration in continuous attractor network models of grid cells},
  author={Burak, Yoram and Fiete, Ila R},
  journal={PLoS computational biology},
  volume={5},
  number={2},
  pages={e1000291},
  year={2009},
  publisher={Public Library of Science San Francisco, USA}
}

@article{cueva2018emergence,
  title={Emergence of grid-like representations by training recurrent neural networks to perform spatial localization},
  author={Cueva, Christopher J and Wei, Xue-Xin},
  journal={arXiv preprint arXiv:1803.07770},
  year={2018}
}

@article{langston2010development,
  title={Development of the spatial representation system in the rat},
  author={Langston, Rosamund F and Ainge, James A and Couey, Jonathan J and Canto, Cathrin B and Bjerknes, Tale L and Witter, Menno P and Moser, Edvard I and Moser, May-Britt},
  journal={Science},
  volume={328},
  number={5985},
  pages={1576--1580},
  year={2010},
  publisher={American Association for the Advancement of Science}
}

@article{wills2010development,
  title={Development of the hippocampal cognitive map in preweanling rats},
  author={Wills, Tom J and Cacucci, Francesca and Burgess, Neil and O'Keefe, John},
  journal={science},
  volume={328},
  number={5985},
  pages={1573--1576},
  year={2010},
  publisher={American Association for the Advancement of Science}
}

@article{weber2018learning,
  title={Learning place cells, grid cells and invariances with excitatory and inhibitory plasticity},
  author={Weber, Simon Nikolaus and Sprekeler, Henning},
  journal={Elife},
  volume={7},
  pages={e34560},
  year={2018},
  publisher={eLife Sciences Publications, Ltd}
}

@article{tang2024sequential,
  title={Sequential memory with temporal predictive coding},
  author={Tang, Mufeng and Barron, Helen and Bogacz, Rafal},
  journal={Advances in Neural Information Processing Systems},
  volume={36},
  year={2024}
}

@article{millidge2024predictive,
  title={Predictive coding networks for temporal prediction},
  author={Millidge, Beren and Tang, Mufeng and Osanlouy, Mahyar and Harper, Nicol S and Bogacz, Rafal},
  journal={PLOS Computational Biology},
  volume={20},
  number={4},
  pages={e1011183},
  year={2024},
  publisher={Public Library of Science San Francisco, CA USA}
}

@article{williams1990efficient,
  title={An efficient gradient-based algorithm for on-line training of recurrent network trajectories},
  author={Williams, Ronald J and Peng, Jing},
  journal={Neural computation},
  volume={2},
  number={4},
  pages={490--501},
  year={1990},
  publisher={MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}
}

@article{zilli2010coupled,
  title={Coupled noisy spiking neurons as velocity-controlled oscillators in a model of grid cell spatial firing},
  author={Zilli, Eric A and Hasselmo, Michael E},
  journal={Journal of Neuroscience},
  volume={30},
  number={41},
  pages={13850--13860},
  year={2010},
  publisher={Soc Neuroscience}
}

@article{giocomo2011computational,
  title={Computational models of grid cells},
  author={Giocomo, Lisa M and Moser, May-Britt and Moser, Edvard I},
  journal={Neuron},
  volume={71},
  number={4},
  pages={589--603},
  year={2011},
  publisher={Elsevier}
}

@article{ocko2018emergent,
  title={Emergent elasticity in the neural code for space},
  author={Ocko, Samuel A and Hardcastle, Kiah and Giocomo, Lisa M and Ganguli, Surya},
  journal={Proceedings of the National Academy of Sciences},
  volume={115},
  number={50},
  pages={E11798--E11806},
  year={2018},
  publisher={National Acad Sciences}
}

@article{dordek2016extracting,
  title={Extracting grid cell characteristics from place cell inputs using non-negative principal component analysis},
  author={Dordek, Yedidyah and Soudry, Daniel and Meir, Ron and Derdikman, Dori},
  journal={Elife},
  volume={5},
  pages={e10094},
  year={2016},
  publisher={eLife Sciences Publications, Ltd}
}

@article{dorrell2022actionable,
  title={Actionable neural representations: Grid cells from minimal constraints},
  author={Dorrell, William and Latham, Peter E and Behrens, Timothy EJ and Whittington, James CR},
  journal={arXiv preprint arXiv:2209.15563},
  year={2022}
}

@article{schaeffer2024self,
  title={Self-supervised learning of representations for space generates multi-modular grid cells},
  author={Schaeffer, Rylan and Khona, Mikail and Ma, Tzuhsuan and Eyzaguirre, Cristobal and Koyejo, Sanmi and Fiete, Ila},
  journal={Advances in Neural Information Processing Systems},
  volume={36},
  year={2024}
}

@article{banino2018vector,
  title={Vector-based navigation using grid-like representations in artificial agents},
  author={Banino, Andrea and Barry, Caswell and Uria, Benigno and Blundell, Charles and Lillicrap, Timothy and Mirowski, Piotr and Pritzel, Alexander and Chadwick, Martin J and Degris, Thomas and Modayil, Joseph and others},
  journal={Nature},
  volume={557},
  number={7705},
  pages={429--433},
  year={2018},
  publisher={Nature Publishing Group UK London}
}

@article{schaeffer2022no,
  title={No free lunch from deep learning in neuroscience: A case study through models of the entorhinal-hippocampal circuit},
  author={Schaeffer, Rylan and Khona, Mikail and Fiete, Ila},
  journal={Advances in neural information processing systems},
  volume={35},
  pages={16052--16067},
  year={2022}
}

@article{sanger1989optimal,
title = {Optimal unsupervised learning in a single-layer linear feedforward neural network},
journal = {Neural Networks},
volume = {2},
number = {6},
pages = {459-473},
year = {1989},
issn = {0893-6080},
doi = {https://doi.org/10.1016/0893-6080(89)90044-0},
url = {https://www.sciencedirect.com/science/article/pii/0893608089900440},
author = {Terence D. Sanger},
keywords = {Neural network, Unsupervised learning, Hebbian learning, Feedforward, Karhunen-Loeve Transform, Image coding, Texture, Cortical receptive fields},
abstract = {A new approach to unsupervised learning in a single-layer linear feedforward neural network is discussed. An optimality principle is proposed which is based upon preserving maximal information in the output units. An algorithm for unsupervised learning based upon a Hebbian learning rule, which achieves the desired optimality is presented. The algorithm finds the eigenvectors of the input correlation matrix, and it is proven to converge with probability one. An implementation which can train neural networks using only local “synaptic” modification rules is described. It is shown that the algorithm is closely related to algorithms in statistics (Factor Analysis and Principal Components Analysis) and neural networks (Self-supervised Backpropagation, or the “encoder” problem). It thus provides an explanation of certain neural network behavior in terms of classical statistical techniques. Examples of the use of a linear network for solving image coding and texture segmentation problems are presented. Also, it is shown that the algorithm can be used to find “visual receptive fields” which are qualitatively similar to those found in primate retina and visual cortex.}
}

@article{widloski2014model,
  title={A model of grid cell development through spatial exploration and spike time-dependent plasticity},
  author={Widloski, John and Fiete, Ila R},
  journal={Neuron},
  volume={83},
  number={2},
  pages={481--495},
  year={2014},
  publisher={Elsevier}
}

@article{kropff2008emergence,
  title={The emergence of grid cells: Intelligent design or just adaptation?},
  author={Kropff, Emilio and Treves, Alessandro},
  journal={Hippocampus},
  volume={18},
  number={12},
  pages={1256--1269},
  year={2008},
  publisher={Wiley Online Library}
}

@article{bredenberg2021impression,
  title={Impression learning: Online representation learning with synaptic plasticity},
  author={Bredenberg, Colin and Lyo, Benjamin and Simoncelli, Eero and Savin, Cristina},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={11717--11729},
  year={2021}
}

@article{george2024generative,
  title={A generative model of the hippocampal formation trained with theta driven local learning rules},
  author={George, Tom M and Stachenfeld, Kimberly L and Barry, Caswell and Clopath, Claudia and Fukai, Tomoki},
  journal={Advances in Neural Information Processing Systems},
  volume={36},
  year={2024}
}

@article{bush2014grid,
  title={What do grid cells contribute to place cell firing?},
  author={Bush, Daniel and Barry, Caswell and Burgess, Neil},
  journal={Trends in neurosciences},
  volume={37},
  number={3},
  pages={136--145},
  year={2014},
  publisher={Elsevier}
}

@article{sargolini2006conjunctive,
  title={Conjunctive representation of position, direction, and velocity in entorhinal cortex},
  author={Sargolini, Francesca and Fyhn, Marianne and Hafting, Torkel and McNaughton, Bruce L and Witter, Menno P and Moser, May-Britt and Moser, Edvard I},
  journal={Science},
  volume={312},
  number={5774},
  pages={758--762},
  year={2006},
  publisher={American Association for the Advancement of Science}
}

@article{mcnaughton2006path,
  title={Path integration and the neural basis of the'cognitive map'},
  author={McNaughton, Bruce L and Battaglia, Francesco P and Jensen, Ole and Moser, Edvard I and Moser, May-Britt},
  journal={Nature Reviews Neuroscience},
  volume={7},
  number={8},
  pages={663--678},
  year={2006},
  publisher={Nature Publishing Group UK London}
}

@article{winter2015passive,
  title={Passive transport disrupts grid signals in the parahippocampal cortex},
  author={Winter, Shawn S and Mehlman, Max L and Clark, Benjamin J and Taube, Jeffrey S},
  journal={Current Biology},
  volume={25},
  number={19},
  pages={2493--2502},
  year={2015},
  publisher={Elsevier}
}

@article{krupic2012neural,
  title={Neural representations of location composed of spatially periodic bands},
  author={Krupic, Julija and Burgess, Neil and O’Keefe, John},
  journal={Science},
  volume={337},
  number={6096},
  pages={853--857},
  year={2012},
  publisher={American Association for the Advancement of Science}
}

@article{navratilova2016grids,
  title={Grids from bands, or bands from grids? An examination of the effects of single unit contamination on grid cell firing fields},
  author={Navratilova, Zaneta and Godfrey, Keith B and McNaughton, Bruce L},
  journal={Journal of neurophysiology},
  volume={115},
  number={2},
  pages={992--1002},
  year={2016},
  publisher={American Physiological Society Bethesda, MD}
}

@inproceedings{snow2022biological,
  title={Biological softmax: Demonstrated in modern Hopfield networks},
  author={Snow, Mallory A and Orchard, Jeff},
  booktitle={Proceedings of the Annual Meeting of the Cognitive Science Society},
  volume={44},
  number={44},
  year={2022}
}

@article{jones1993basket,
  title={Basket-like interneurones in layer II of the entorhinal cortex exhibit a powerful NMDA-mediated synaptic excitation},
  author={Jones, RSG and B{\"u}hl, EH},
  journal={Neuroscience letters},
  volume={149},
  number={1},
  pages={35--39},
  year={1993},
  publisher={Elsevier}
}

@article{witter2006spatial,
  title={Spatial representation and the architecture of the entorhinal cortex},
  author={Witter, Menno P and Moser, Edvard I},
  journal={Trends in neurosciences},
  volume={29},
  number={12},
  pages={671--678},
  year={2006},
  publisher={Elsevier}
}

@article{ouchi2024predictive,
  title={Predictive grid coding in the medial entorhinal cortex},
  author={Ouchi, Ayako and Fujisawa, Shigeyoshi},
  journal={Science},
  volume={385},
  number={6710},
  pages={776--784},
  year={2024},
  publisher={American Association for the Advancement of Science}
}

@article{canto2008does,
  title={What does the anatomical organization of the entorhinal cortex tell us?},
  author={Canto, Cathrin B and Wouterlood, Floris G and Witter, Menno P},
  journal={Neural plasticity},
  volume={2008},
  number={1},
  pages={381243},
  year={2008},
  publisher={Wiley Online Library}
}

@article{duncan2012evidence,
  title={Evidence for area CA1 as a match/mismatch detector: A high-resolution fMRI study of the human hippocampus},
  author={Duncan, Katherine and Ketz, Nicholas and Inati, Souheil J and Davachi, Lila},
  journal={Hippocampus},
  volume={22},
  number={3},
  pages={389--398},
  year={2012},
  publisher={Wiley Online Library}
}

@article{tscshantz2023hybrid,
  title={Hybrid predictive coding: Inferring, fast and slow},
  author={Tscshantz, Alexander and Millidge, Beren and Seth, Anil K and Buckley, Christopher L},
  journal={PLoS computational biology},
  volume={19},
  number={8},
  pages={e1011280},
  year={2023},
  publisher={Public Library of Science San Francisco, CA USA}
}

@article{buzsaki2015hippocampal,
  title={Hippocampal sharp wave-ripple: A cognitive biomarker for episodic memory and planning},
  author={Buzs{\'a}ki, Gy{\"o}rgy},
  journal={Hippocampus},
  volume={25},
  number={10},
  pages={1073--1188},
  year={2015},
  publisher={Wiley Online Library}
}

@article{o1993phase,
  title={Phase relationship between hippocampal place units and the EEG theta rhythm},
  author={O'Keefe, John and Recce, Michael L},
  journal={Hippocampus},
  volume={3},
  number={3},
  pages={317--330},
  year={1993},
  publisher={Wiley Online Library}
}

@article{huxter2008theta,
  title={Theta phase--specific codes for two-dimensional position, trajectory and heading in the hippocampus},
  author={Huxter, John R and Senior, Timothy J and Allen, Kevin and Csicsvari, Jozsef},
  journal={Nature neuroscience},
  volume={11},
  number={5},
  pages={587--594},
  year={2008},
  publisher={Nature Publishing Group US New York}
}

@article{palmer2015predictive,
  title={Predictive information in a sensory population},
  author={Palmer, Stephanie E and Marre, Olivier and Berry, Michael J and Bialek, William},
  journal={Proceedings of the National Academy of Sciences},
  volume={112},
  number={22},
  pages={6908--6913},
  year={2015},
  publisher={National Academy of Sciences}
}


@article{yang2019task,
  title={Task representations in neural networks trained to perform many cognitive tasks},
  author={Yang, Guangyu Robert and Joglekar, Madhura R and Song, H Francis and Newsome, William T and Wang, Xiao-Jing},
  journal={Nature neuroscience},
  volume={22},
  number={2},
  pages={297--306},
  year={2019},
  publisher={Nature Publishing Group US New York}
}

@article{stensola2012entorhinal,
  title={The entorhinal grid map is discretized},
  author={Stensola, Hanne and Stensola, Tor and Solstad, Trygve and Fr{\o}land, Kristian and Moser, May-Britt and Moser, Edvard I},
  journal={Nature},
  volume={492},
  number={7427},
  pages={72--78},
  year={2012},
  publisher={Nature Publishing Group UK London}
}

@article{krupic2014environment,
  title={How environment geometry affects grid cell symmetry and what we can learn from it},
  author={Krupic, Julija and Bauza, Marius and Burton, Stephen and Lever, Colin and O'Keefe, John},
  journal={Philosophical Transactions of the Royal Society B: Biological Sciences},
  volume={369},
  number={1635},
  pages={20130188},
  year={2014},
  publisher={The Royal Society}
}

@inproceedings{graves2013speech,
  title={Speech recognition with deep recurrent neural networks},
  author={Graves, Alex and Mohamed, Abdel-rahman and Hinton, Geoffrey},
  booktitle={2013 IEEE international conference on acoustics, speech and signal processing},
  pages={6645--6649},
  year={2013},
  organization={Ieee}
}

@article{dabaghian2012topological,
  title={A topological paradigm for hippocampal spatial map formation using persistent homology},
  author={Dabaghian, Yuri and M{\'e}moli, Facundo and Frank, Loren and Carlsson, Gunnar},
  year={2012},
  publisher={Public Library of Science San Francisco, USA}
}

@misc{pinchetti2025benchmarkingpredictivecodingnetworks,
      title={Benchmarking Predictive Coding Networks -- Made Simple}, 
      author={Luca Pinchetti and Chang Qi and Oleh Lokshyn and Gaspard Olivers and Cornelius Emde and Mufeng Tang and Amine M'Charrak and Simon Frieder and Bayar Menzat and Rafal Bogacz and Thomas Lukasiewicz and Tommaso Salvatori},
      year={2025},
      eprint={2407.01163},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2407.01163}, 
}

@article{graves2013generating,
  title={Generating sequences with recurrent neural networks},
  author={Graves, Alex},
  journal={arXiv preprint arXiv:1308.0850},
  year={2013}
}