Multiplicatively interacting point processes and applications to neural modeling

We introduce a nonlinear modification of the classical Hawkes process, which allows inhibitory couplings between units without restrictions. The resulting system of interacting point processes provides a useful mathematical model for recurrent networks of spiking neurons with exponential transfer functions. The expected rates of all neurons in the network are approximated by a first-order differential system. We study the stability of the solutions of this equation, and use the new formalism to implement a winner-takes-all network that operates robustly for a wide range of parameters. Finally, we discuss relations with the generalised linear model that is widely used for the analysis of spike trains.

💡 Research Summary

The paper introduces a novel modification of the classical Hawkes point‑process framework by inserting a multiplicative interaction term that allows both excitatory and inhibitory couplings without the usual constraints. In the proposed model each neuron i has a conditional intensity (instantaneous firing rate) λ_i(t) defined as

 λ_i(t)=exp( μ_i + Σ_j w_{ij}·h_j(t) ),

where μ_i is a baseline log‑rate, w_{ij} is a coupling coefficient that can be positive (excitation) or negative (inhibition), and h_j(t) is a filtered version of neuron j’s past spikes (typically an exponential kernel). This formulation preserves the exponential link function familiar from generalized linear models (GLMs) but replaces the linear sum of past activity with a product of the coupling weight and the filtered spike history. Consequently, inhibitory effects are represented directly by negative w_{ij} without the need for additional non‑linear transformations.

To obtain a tractable description of network dynamics the authors take expectations over the point processes and derive a first‑order differential system for the mean firing rates m_i(t)=E