Infinite systems of interacting chains with memory of variable length - a stochastic model for biological neural nets

TL;DR

This paper introduces a new class of non-Markovian stochastic processes with interacting components that model biological neural networks, extending existing models and providing a method to construct unique stationary processes with explicit correlation bounds.

Contribution

It develops a novel non-Markovian process framework with variable memory length for neural systems and constructs a unique stationary version using a Kalikow-type decomposition.

Findings

Explicit upper-bound for correlation between inter-spike intervals

Model extends interacting particle systems and variable-length memory chains

Framework applicable to large random neural networks

Abstract

We consider a new class of non Markovian processes with a countable number of interacting components. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. The system evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian, and the stochastic chains with memory of variable length which have finite state space. These features make it suitable to describe the time evolution of biological neural systems. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time. This construction implies uniqueness of the stationary process. Finally we consider the case where the interactions between components are given by a critical directed Erd\"os-R\'enyi-type random graph with a large but finite number of components. In this framework we obtain an explicit upper-bound for the correlation between successive inter-spike intervals which is compatible with previous empirical findings.