Coupled stochastic systems of Skorokhod type: well-posedness of a mathematical model and its applications

TL;DR

This paper introduces a new well-posed mathematical model using coupled Skorokhod-type stochastic differential equations with jumps to describe complex population dynamics within bounded domains, including neuronal systems.

Contribution

It develops a novel functional framework for modeling population interactions with stochastic boundary behavior, extending existing methods to systems with jumps and bounded domains.

Findings

Proved well-posedness of the coupled Skorokhod SDE system.

Provided numerical examples illustrating the model.

Applied the model to neuronal population dynamics.

Abstract

Population dynamics with complex biological interactions, accounting for uncertainty quantification, is critical for many application areas. However, due to the complexity of biological systems, the mathematical formulation of the corresponding problems faces the challenge that the corresponding stochastic processes should, in most cases, be considered in bounded domains. We propose a model based on a coupled system of reflecting Skorokhod-type stochastic differential equations with jump-like exit from a boundary. The setting describes the population dynamics of active and passive populations. As main working techniques, we use compactness methods and Skorokhod's representation of solutions to SDEs posed in bounded domains to prove the well-posedness of the system. This functional setting is a new point of view in the field of modelling and simulation of population dynamics. We provide the details of the model, as well as representative numerical examples, and discuss the applications of a Wilson-Cowan-type system, modelling the dynamics of two interacting populations of excitatory and inhibitory neurons. Furthermore, the presence of random input current, reflecting factors together with Poisson jumps, increases firing activity in neuronal systems.