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.
|