https://doi.org/10.1140/epjs/s11734-021-00172-3
Regular Article
Fokker–Planck approach to neural networks and to decision problems
A unique method for stochastic models in computational and cognitive neuroscience
1
Bernstein Center for Computational Neuroscience, 10115, Berlin, Germany
2
Physics Department, Humboldt University Berlin, 12489, Berlin, Germany
a
sebastian.vellmer@bccn-berlin.de
Received:
30
November
2020
Accepted:
23
April
2021
Published online:
22
June
2021
We review applications of the Fokker–Planck equation for the description of systems with event trains in computational and cognitive neuroscience. The most prominent example is the spike trains generated by integrate-and-fire neurons when driven by correlated (colored) fluctuations, by adaptation currents and/or by other neurons in a recurrent network. We discuss how for a general Gaussian colored noise and an adaptation current can be incorporated into a multidimensional Fokker–Planck equation by Markovian embedding for systems with a fire-and-reset condition and how in particular the spike-train power spectrum can be determined by this equation. We then review how this framework can be used to determine the self-consistent correlation statistics in a recurrent network in which the colored fluctuations arise from the spike trains of statistically similar neurons. We then turn to the popular drift-diffusion models for binary decisions in cognitive neuroscience and demonstrate that very similar Fokker–Planck equations (with two instead of only one threshold) can be used to study the statistics of sequences of decisions. Specifically, we present a novel two-dimensional model that includes an evidence variable and an expectancy variable that can reproduce salient features of key experiments in sequential decision making.
© The Author(s) 2021
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
