Synchrony-induced modes of oscillation of a neural field model

TL;DR

This paper analyzes how oscillatory modes emerge in a neural field model of heterogeneous neurons, revealing the relationship between connectivity patterns and oscillation frequencies, and demonstrating their presence in bump states.

Contribution

It derives an exact low-dimensional neural field model for heterogeneous networks, linking connectivity Fourier components to oscillation frequencies and analyzing stability and bifurcations.

Findings

Oscillatory modes correspond to specific wave numbers and frequencies.

Decay rates are uniform across modes due to heterogeneity-induced desynchronization.

Similar oscillatory modes are present in neural bump states away from bifurcation.

Abstract

We investigate the modes of oscillation of heterogeneous ring-networks of quadratic integrate-and-fire neurons with non-local, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific frequency, analogous to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network's oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially-homogeneous state, and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead, is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially-inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states, and are maintained away from onset.