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Population density equations for stochastic processes with memory kernels

Lai, Yi Ming; de Kamps, Marc

Authors

Yi Ming Lai

Marc de Kamps



Abstract

We present a method for solving population density equations (PDEs)–-a mean-field technique describing homogeneous populations of uncoupled neurons—where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation—a recent result from random network theory—describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process.

Citation

Lai, Y. M., & de Kamps, M. (2017). Population density equations for stochastic processes with memory kernels. Physical Review E, 95, Article 062125. https://doi.org/10.1103/PhysRevE.95.062125

Journal Article Type Article
Acceptance Date May 9, 2017
Publication Date Jun 20, 2017
Deposit Date Jul 4, 2017
Publicly Available Date Mar 29, 2024
Journal Physical Review E
Print ISSN 2470-0045
Electronic ISSN 2470-0053
Publisher American Physical Society
Peer Reviewed Peer Reviewed
Volume 95
Article Number 062125
DOI https://doi.org/10.1103/PhysRevE.95.062125
Public URL https://nottingham-repository.worktribe.com/output/867314
Publisher URL https://journals.aps.org/pre/abstract/10.1103/PhysRevE.95.062125

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