A Markovian event-based framework for stochastic spiking neural networks

TL;DR

This paper introduces a Markovian event-based framework for stochastic spiking neural networks, demonstrating that spike times form a Markov chain and deriving explicit transition probabilities for common neuron models.

Contribution

It provides a rigorous, event-based Markovian description of stochastic neural networks, extending previous deterministic models to include noise, delays, and refractory periods.

Findings

Spike times form a Markov chain in stochastic neural networks.

Explicit transition probabilities are derived for linear integrate-and-fire models.

The framework encompasses models with synaptic noise, delays, and refractory periods.

Abstract

In spiking neural networks, the information is conveyed by the spike times, that depend on the intrinsic dynamics of each neuron, the input they receive and on the connections between neurons. In this article we study the Markovian nature of the sequence of spike times in stochastic neural networks, and in particular the ability to deduce from a spike train the next spike time, and therefore produce a description of the network activity only based on the spike times regardless of the membrane potential process. To study this question in a rigorous manner, we introduce and study an event-based description of networks of noisy integrate-and-fire neurons, i.e. that is based on the computation of the spike times. We show that the firing times of the neurons in the networks constitute a Markov chain, whose transition probability is related to the probability distribution of the interspike interval of the neurons in the network. In the cases where the Markovian model can be developed, the transition probability is explicitly derived in such classical cases of neural networks as the linear integrate-and-fire neuron models with excitatory and inhibitory interactions, for different types of synapses, possibly featuring noisy synaptic integration, transmission delays and absolute and relative refractory period. This covers most of the cases that have been investigated in the event-based description of spiking deterministic neural networks.