Replica-mean-field limits for intensity-based neural networks

Speaker : Prof. François Baccelli
Date: 09/06/2021
Time: 2:00 pm - 3:00 pm
Location: Zoom + LINCS


Computational neuroscience features a variety of stochastic network models where both the input and the output of each neuron are point processes. In these models, each neuron has a state which evolves as an integral with respect to its input point processes and which resets when the neuron spikes. In intensity-based neural networks, the spikes of a given neuron have a stochastic intensity which is a linear function of the neuron state. Spiking events define the output point processes of the neuron. These, together with the geometry of the network connections, define in turn the input point processes of other neurons.

Due to the inherent complexity of such intensity-based neural models, relating the spiking activity of a network to its structure currently requires simplifying assumptions, such as considering models in the thermodynamic mean-field limit. In this limit, an infinite number of neurons interact via vanishingly small interactions, thereby erasing the finite size geometry of interactions.

To better capture the geometry in question, this paper analyzes the activity of intensity-based neural networks in the replica-mean-field limit regime. Such systems are made of infinitely many replicas which have the same basic structure as that of the finite network of interest and interact through randomized connections.

The main contribution is an analytical characterization of the stationary dynamics of intensity-based neural networks excitatory synapses in this replica-mean-field limit. Specifically, the stationary dynamics of these limiting networks is functionally characterized via ordinary or partial differential equations derived from the Poisson Hypothesis of stochastic network theory. This functional characterization is reduced to a system of self-consistency equations specifying the stationary neuronal spiking rates. The approach combines the rate-conservation principle of Palm calculus, analytical considerations from generating-function methods, and propagation of chaos techniques.

Such limits can be used for first-order models, whereby elementary replica constituents are single neurons with independent Poisson inputs, and in second-order models, where these constituents are pairs of neurons with exact pairwise interactions. In both cases, these replica-mean-field networks provide tractable versions that retain important features of the finite network structure of interest.

Joint work with T. Taillefumier.