Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression

We study the spatiotemporal dynamics of a two-dimensional excitatory neuronal network with synaptic depression. Coupling between populations of neurons is taken to be nonlocal, while depression is taken to be local and presynaptic. We show that the network supports a wide range of spatially structured oscillations, which are suggestive of phenomena seen in cortical slice experiments and in vivo. The particular form of the oscillations depends on initial conditions and the level of background noise. Given an initial, spatially localized stimulus, activity evolves to a spatially localized oscillating core that periodically emits target waves. Low levels of noise can spontaneously generate several pockets of oscillatory activity that interact via their target patterns. Periodic activity in space can also organize into spiral waves, provided that there is some source of rotational symmetry breaking due to external stimuli or noise. In the high gain limit, no oscillatory behavior exists, but a transient stimulus can lead to a single, outward propagating target wave

Stationary bumps in a piecewise smooth neural field model with synaptic depression

We analyze the existence and stability of stationary pulses or bumps in a one–dimensional piecewise smooth neural field model with synaptic depression. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Synaptic depression dynamically reduces the strength of synaptic weights in response to increases in activity. We show that in the case of a Mexican hat weight distribution, there exists a stable bump for sufficiently weak synaptic depression. However, as synaptic depression becomes stronger, the bump became unstable with respect to perturbations that shift the boundary of the bump, leading to the formation of a traveling pulse. The local stability of a bump is determined by the spectrum of a piecewise linear operator that keeps track of the sign of perturbations of the bump boundary. This results in a number of differences from previous studies of neural field models with Heaviside firing rate functions, where any discontinuities appear inside convolutions so that the resulting dynamical system is smooth. We also extend our results to the case of radially symmetric bumps in two–dimensional neural field models

On the spectra of certain integro-differential-delay problems with applications in neurodynamics

We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a= 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs

Existence and Stability of Standing Pulses in Neural Networks : I Existence

We consider the existence of standing pulse solutions of a neural network integro-differential equation. These pulses are bistable with the zero state and may be an analogue for short term memory in the brain. The network consists of a single-layer of neurons synaptically connected by lateral inhibition. Our work extends the classic Amari result by considering a non-saturating gain function. We consider a specific connectivity function where the existence conditions for single-pulses can be reduced to the solution of an algebraic system. In addition to the two localized pulse solutions found by Amari, we find that three or more pulses can coexist. We also show the existence of nonconvex ``dimpled'' pulses and double pulses. We map out the pulse shapes and maximum firing rates for different connection weights and gain functions.Comment: 31 pages, 29 figures, submitted to SIAM Journal on Applied Dynamical System

Traveling waves of excitation in neural field models

Field models provide an elegant mathematical framework to analyze large-scale patterns of neural activity. On the microscopic level, these models are usually based on either a firing-rate picture or integrate-and-fire dynamics. This article shows that in spite of the large conceptual differences between the two types of dynamics, both generate closely related plane-wave solutions. Furthermore, for a large group of models, estimates about the network connectivity derived from the speed of these plane waves only marginally depend on the assumed class of microscopic dynamics. We derive quantitative results about this phenomenon and discuss consequences for the interpretation of experimental data