# Coherent dynamics in soft-threshold integrate-and-fire networks

## TL;DR

This paper investigates bifurcations and spatiotemporal patterns in stochastic integrate-and-fire neural networks using a mean-field approach, revealing mechanisms for oscillations, localized activity, and wave phenomena.

## Contribution

It introduces a mean-field framework to analyze complex bifurcations and spatiotemporal dynamics in neural networks with delays and structured interactions.

## Key findings

- Uniform oscillations arise via subcritical Hopf bifurcation.
- Turing bifurcation leads to localized sustained activity.
- Spatiotemporal patterns like standing and traveling waves are observed.

## Abstract

We study bifurcations in networks of integrate-and-fire neurons with stochastic spike emission, focusing on the effects of the spatial and temporal structure of the synaptic interactions. Using a deterministic mean-field approximation of the population dynamics, we characterize spatial, temporal, and spatiotemporal patterns of macroscopic activity. In the mean-field theory, synaptic delays give rise to uniform oscillations across the population through a subcritical Hopf bifurcation of the stationary uniform equilibrium. With local excitation and long-range inhibition the network undergoes a Turing bifurcation, resulting in a localized area of sustained activity, or stationary bump. When the coupling has both delays, local inhibition, and long range excitation, the network undergoes a Turing-Hopf bifurcation leading to spatiotemporal dynamics, such as standing and traveling waves. When multiple instabilities are excited, we observe other complex spatiotemporal dynamics. We confirm all these predictions of the mean-field theory in simulations of the underlying stochastic model.