Abstract
We study localized patterns in an exact mean-field description of a spatially extended network of quadratic integrate-and-fire neurons. We investigate conditions for the existence and stability of localized solutions, so-called bumps, and give an analytic estimate for the parameter range, where these solutions exist in parameter space, when one or more microscopic network parameters are varied. We develop Galerkin methods for the model equations, which enable numerical bifurcation analysis of stationary and time-periodic spatially extended solutions. We study the emergence of patterns composed of multiple bumps, which are arranged in a snake-and-ladder bifurcation structure if a homogeneous or heterogeneous synaptic kernel is suitably chosen. Furthermore, we examine time-periodic, spatially localized solutions (oscillons) in the presence of external forcing, and in autonomous, recurrently coupled excitatory and inhibitory networks. In both cases, we observe period-doubling cascades leading to chaotic oscillations.
Highlights
Localized states in neuronal networks, so-called bumps, are related to working memory1,2 and feature selectivity,3 whereby neurons encoding similar stimuli or features show an increased firing rate for the duration of the related cognitive task
We introduced a framework to study localized solutions in a neural field model that was recently derived as an exact representation of the mean field dynamics of networks of spiking neurons
This model does not permit closed-form solutions such as the Amari model with Heaviside firing rates, we show that it is possible to give an analytical estimate for the range of model parameters for which stable localized solutions exist
Summary
Localized states in neuronal networks, so-called bumps, are related to working memory and feature selectivity, whereby neurons encoding similar stimuli or features show an increased firing rate for the duration of the related cognitive task. Some limitations can be overcome if the microscopic model is a heterogeneous network of synaptically coupled θ or QIF neurons, subject to random, Cauchy-distributed background currents. A neural field model that describes without approximation of the average firing rate r(x, t) and the average membrane potential v(x, t) of the spatially extended networks presented above has been developed recently,. W ⊗ r is a convolution and w(y)dy = 1 This neural field model is related to mean-field descriptions of networks of theta neurons and was obtained using the Ott–Antonsen ansatz.. This neural field model is related to mean-field descriptions of networks of theta neurons and was obtained using the Ott–Antonsen ansatz.29 It retains the transient dynamics of the microscopic network, including spike synchrony, and has, a richer dynamic repertoire than purely rate-based models..
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