Exact finite-dimensional description for networks of globally coupled spiking neurons

TL;DR

This paper derives an exact low-dimensional mathematical description of the collective behavior of large networks of globally coupled spiking neurons, connecting microscopic neuron models to macroscopic observables.

Contribution

It provides a novel six-dimensional reduction for stochastic neuron networks and links it to known low-dimensional manifolds like Ott-Antonsen and Watanabe-Strogatz.

Findings

Exact six-dimensional dynamics for stochastic neuron networks.

Reduction to three dimensions for noise-free identical neurons.

Demonstration of nontrivial basins of attraction in bistable regimes.

Abstract

We consider large networks of globally coupled spiking neurons and derive an exact low-dimensional description of their collective dynamics in the thermodynamic limit. Individual neurons are described by the Ermentrout-Kopell canonical model that can be excitable or tonically spiking, and interact with other neurons via pulses. Utilizing the equivalence of the quadratic integrate- and-fire and the theta neuron formulations, we first derive the dynamical equations in terms of the Kuramoto-Daido order parameters (Fourier modes of the phase distribution) and relate them to two biophysically relevant macroscopic observables, the firing rate and the mean voltage. For neurons driven by Cauchy white noise or for Cauchy-Lorentz distributed input currents, we adapt the results by Cestnik and Pikovsky [arXiv:2207.02302 (2022)] and show that for arbitrary initial conditions the collective dynamics reduces to six dimensions. We also prove that in this case the dynamics asymptotically converges to a two-dimensional invariant manifold first discovered by Ott and Antonsen. For identical, noise-free neurons, the dynamics reduces to three dimensions, becoming equivalent to the Watanabe-Strogatz description. We illustrate the exact six-dimensional dynamics outside the invariant manifold by calculating nontrivial basins of different asymptotic regimes in a bistable situation.