Population Density Equations for Stochastic Processes with Memory Kernels

TL;DR

This paper introduces a flexible numerical method for solving population density equations with non-Markovian noise, combining neuroscience and network theory to model neuron populations with arbitrary jump distributions.

Contribution

It develops a modular approach using geometric binning and the generalized Montroll-Weiss equation to handle non-Markovian stochastic processes in neural population models.

Findings

Accurately models jump responses in neuron populations.

Handles arbitrary distributions of spike intervals.

Separates deterministic and stochastic dynamics effectively.

Abstract

We present a novel method for solving population density equations (PDEs), 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- (LIF) and quadratic-integrate and fire (QIF) neurons subject to spike trains with Poisson and gamma distributed spike 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.