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

Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay

The work introduces a model for reciprocal connections in neural fields by a nonlocal feedback mechanism, while the neural field exhibits nonlocal interactions and intra-areal transmission delays. We study the speed of traveling fronts with respect to the transmission delay, the spatial feedback range and the feedback delay for general axonal and feedback connectivity kernels. In addition, we find a novel shape of traveling fronts due to the applied feedback and criteria for its occurence are derived

Synaptically generated wave propagation in excitable neural media

We study the propagation of solitary waves in a one-dimensional network of excitable integrate-and-fire neurons with axo-dendritic synaptic coupling. We show that for small axonal delays there exists a stable solitary wave, and derive a power scaling law for the velocity as a function of the coupling. In the case of large axonal delays and fast synapses we establish that the solitary wave can destabilize via a Hopf bifurcation in the firing times

Waves and Oscillations in Model Neuronal Networks

In this thesis methods from nonlinear dynamical systems, pattern formation and bifurcation theory, combined with numerical simulations, are applied to three models in neuroscience. In Chapter 1 we analyze the Wilson-Cowan equations for a single self-excited population of cells with absolute refractory period. We construct the normal form for a Hopf bifurcation, and prove that by increasing the refractory period the network switches from a steady state to an oscillatory behavior. Numerical simulations indicate that for large values of refractoriness the oscillation converges to a relaxation-like pattern, the period of which we estimate. Chapter 2 brings new results for the rate model introduced by Hansel and Sompolinsky who study feature selectivity in local cortical circuits. We study their model with a more general, nonlinear sigmoid gain function, and prove that the system can exhibit different kind of patterns such as stationary states, traveling waves and standing waves. Standing waves can be obtained only if the threshold is sufficiently high and only for intermediate values of the strength of adaptation. A large adaptation strength destabilizes the pattern. Therefore the localized activity starts to propagate along the network, resulting in a traveling wave. We construct the normal form for Hopf and Takens-Bogdanov with O(2)-symmetry bifurcations and study the interactions between spatial and spatio-temporal patterns in the neural network. Numerical simulations are provided.Chapter 3 addresses several questions with regard to the traveling wave propagation in a leaky-integrate-and-fire model for a network with purely excitatory (exponentially decaying) synaptic coupling. We analyze the case when the neurons fire multiple spikes and derive a formula for the voltage. We compute in a certain parameter space, the curves that delineate the region where single-spike traveling wave solutions exist, and show that there is a region of parameter space where neurons can propagate a two-spike traveling wave

What can a mean-field model tell us about the dynamics of the cortex?

In this chapter we examine the dynamical behavior of a spatially homogeneous two-dimensional model of the cortex that incorporates membrane potential, synaptic flux rates and long- and short-range synaptic input, in two spatial dimensions, using parameter sets broadly realistic of humans and rats. When synaptic dynamics are included, the steady states may not be stable. The bifurcation structure for the spatially symmetric case is explored, identifying the positions of saddle–node and sub- and supercritical Hopf instabilities. We go beyond consideration of small-amplitude perturbations to look at nonlinear dynamics. Spatially-symmetric (breathing mode) limit cycles are described, as well as the response to spatially-localized impulses. When close to Hopf and saddle–node bifurcations, such impulses can cause traveling waves with similarities to the slow oscillation of slow-wave sleep. Spiral waves can also be induced. We compare model dynamics with the known behavior of the cortex during natural and anesthetic-induced sleep, commenting on the physiological significance of the limit cycles and impulse responses