Spiking neurons : interacting Hawkes processes, mean field limits and oscillations

Abstract

Abstract. This paper gives a short survey of some aspects of the study of Hawkes processes in high dimensions, in view of modeling large systems of interacting neurons. We first present a perfect simulation result allowing for a graphical construction of the process in infinite dimension. Then we turn to the study of mean field limits and show how in some cases the delays present in the Hawkes intensities give rise to oscillatory behavior of the limit process.Résumé. Cet article résume brièvement certains aspects de l'étude de processus de Hawkes en grande dimension, en vue d'une modélisation de systèmes de neurones en interactions. Nous présentons d'abord un résultat de simulation parfaite qui donne lieu à une construction graphique du processus en dimension infinie. Ensuite nous étudions des limites de champ moyen de processus de Hawkes et montrons comment les délais apparaissant dans le processus d'intensité engendrent des oscillations du processus limite.

Heterogeneous Neuronal and Synaptic Dynamics for Spike-Efficient Unsupervised Learning: Theory and Design Principles

“…A point process is a collection of random points on some underlying mathematical space, such as the real line, the Cartesian plane, or more abstract spaces. The notion of using point process models, especially the interactive Hawkes processes, to model the spiking dynamics of LIF network dynamics has been studied in the literature previously [61,62,63,64]. We leverage these results to prove that heterogeneity in the synaptic dynamics can help reduce the spike count, as already discussed in the paper.…”

“…C. We apologize if there is still confusion, and we will add more in-depth discussion in the final manuscript as discussed below. In their paper, Locherbach et al [61] survey some aspects of the study of Hawkes processes in high dimensions to model biological neural systems and study their long-term behavior. Galves et al [62] provided an overview of point processes used as stochastic models for interacting neurons in discrete and continuous time.…”

Limit theorems for Hawkes processes including inhibition

“…Hawkes processes have been introduced by Hawkes [15] and are widely used for modeling purposes: originally as models for the appearances of earthquakes [15,17], but now in finance [16,3] and econometrics or in neuroscience as models of spike trains of neurons [21,14]. We refer to the bibliography of our references for more details.…”