# Dynamics of neural fields with exponential temporal kernel

## TL;DR

This paper investigates the bifurcation behavior of neural fields with an exponential temporal kernel, revealing the absence of static bifurcations and characterizing conditions for dynamic patterns like traveling waves.

## Contribution

It provides a detailed bifurcation analysis of neural fields with exponential kernels, highlighting the unique dynamic behaviors and explicit conditions for pattern formation.

## Key findings

- Exponential kernels do not allow static bifurcations.
- Hopf bifurcations produce temporal oscillations.
- Turing-Hopf bifurcations generate traveling waves.

## Abstract

We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green's function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing-Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.