IS29: Probabilistic and Statistical Study of Systems of Interacting Neurons

Organizer: Guilherme Ost & Patricia Reynaud Bouret (Federal University of Rio de Janeiro & Université Côte d’Azur)

Nonparametric estimation of the jump rate in mean field interacting systems of neurons

Eva Löcherbach

We consider finite systems of \(N\) interacting neurons described by non-linear Hawkes processes in a mean field frame. Neurons are described by their membrane potential. They spike randomly, at a rate depending on their potential. In between successive spikes, their membrane potential follows a deterministic flow. We estimate the spiking rate function based on the observation of the system of \(N\) neurons over a fixed time interval \([0,t]\). Asymptotic are taken as \(N,\) the number of neurons, tends to infinity. We introduce a kernel estimator of Nadaraya-Watson type and discuss its asymptotic properties with help of the deterministic dynamical system describing the mean field limit. We compute the minimax rate of convergence in an \(L^2 -\)error loss over a range of Hölder classes and obtain the classical rate of convergence \(N^{ - 2\beta/ ( 2 \beta + 1)} ,\) where \(\beta\) is the regularity of the unknown spiking rate function.

Asymptotic behaviour of networks of Hopfield-like neurons

Etienne Tanré

We consider a system of \(N\) neurons in interaction. The membrane potential of the neuron \(i\) follows the SDE \[dX^{i,N}_t = \left( g(X^{i,N}_t)+\sum_ {j =1}^N J_{i\leftarrow j} f(X^{j,N}_t) \right)\,dt+\lambda dW^i_t \quad \text{ for } i = 1, \cdots, N,\] where the Brownian motion \(W^i\) models the noise governing neuron \(i\) and \(g\) gives the intrinsic activity of an isolated neuron. The interaction between neurons are modeled by the activation function \(f\), typically a sigmoid, and the synaptic weights \(J_{i\leftarrow j}\), that is the effect of the neuron \(j\) on the membrane potential of neuron \(i\)

In this talk, we extend the results of \([2]\) to general coefficients \(f\), \(g\) and we only assume that the synaptic weights are random, i.i.d., with mean of order \(J/N\) and standard deviation of order \(\sigma/\sqrt{N}\). We give the asymptotic dynamics of a typical neuron as the size \(N\) of the network goes to infinity. The theory of Volterra equations allows us to give an explicit expression of the solution. Finally, we develop numerical methods of approximation of the solution.

Bibliography

\([1]\) Faugeras, O. and Tanré, E. (2024). Universality of the mean-field equations of networks of Hopfield-like neurons https://arxiv.org/abs/2408.14290.

\([2]\) Faugeras, O. Soret, É. and Tanré, E. (2020). Asymptotic behavior of a network of neurons with random linear interactions https://arxiv.org/abs/2006.03083.