Spatio-Temporal Hybrid (PDMP) Models: Central Limit Theorem and Langevin Approximation for Global Fluctuations. Application to Electrophysiology

TL;DR

This paper establishes a Central Limit Theorem for Hilbert-valued PDMP models, describing their fluctuations around deterministic limits, with applications to neural membrane models in neuroscience.

Contribution

It introduces a CLT for Hilbert-valued PDMPs and characterizes their fluctuations via stochastic PDEs, advancing theoretical understanding and applications in neuroscience modeling.

Findings

Derived a CLT for Hilbert-valued PDMP models.

Formulated the fluctuations as stochastic PDEs.

Applied results to neural membrane models.

Abstract

In the present work we derive a Central Limit Theorem for sequences of Hilbert-valued Piecewise Deterministic Markov process models and their global fluctuations around their deterministic limit identified by the Law of Large Numbers. We provide a version of the limiting fluctuations processes in the form of a distribution valued stochastic partial differential equation which can be the starting point for further theoretical and numerical analysis. We also present applications of our results to two examples of hybrid models of spatially extended excitable membranes: compartmental-type neuron models and neural fields models. These models are fundamental in neuroscience modelling both for theory and numerics.