Journal article
Nonlinear partial differential equations in neuroscience: from modelling to mathematical theory
- Abstract:
- Many systems of partial differential equations have been proposed as simplified representations of complex collective behaviours in large networks of neurons. In this survey, we briefly discuss their derivations and then review the mathematical methods developed to handle the unique features of these models, which are often nonlinear and non-local. The first part focuses on parabolic FokkerPlanck equations: the Nonlinear Noisy Leaky Integrate and Fire neuron model, stochastic neural fields in PDE form with applications to grid cells, and rate-based models for decision-making. The second part concerns hyperbolic transport equations, namely the model of the Time Elapsed since the last discharge and the jump-based Leaky Integrate and Fire model. The last part covers some kinetic mesoscopic models, with particular attention to the kinetic Voltage-Conductance model and FitzHugh-Nagumo kinetic Fokker-Planck systems.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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- Files:
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(Preview, Accepted manuscript, pdf, 3.5MB, Terms of use)
-
- Publisher copy:
- 10.1142/S0218202525400044
Authors
+ European Research Council
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- Funder identifier:
- https://ror.org/0472cxd90
- Grant:
- 883363
+ Engineering and Physical Sciences Research Council
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- Funder identifier:
- https://ror.org/0439y7842
- Grant:
- EP/V051121/1
- Publisher:
- World Scientific Publishing
- Journal:
- Mathematical Models and Methods in Applied Sciences More from this journal
- Volume:
- 35
- Issue:
- 2
- Pages:
- 403-584
- Publication date:
- 2025-03-04
- Acceptance date:
- 2025-01-10
- DOI:
- EISSN:
-
1793-6314
- ISSN:
-
0218-2025
- Language:
-
English
- Keywords:
- Pubs id:
-
2077406
- Local pid:
-
pubs:2077406
- Deposit date:
-
2025-01-12
- ARK identifier:
Terms of use
- Copyright holder:
- World Scientific Publishing Company
- Copyright date:
- 2025
- Rights statement:
- © World Scientific Publishing Company
- Notes:
- The author accepted manuscript (AAM) of this paper has been made available under the University of Oxford's Open Access Publications Policy, and a CC BY public copyright licence has been applied.
- Licence:
- CC Attribution (CC BY)
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