arXiv subjects

Alan Dyson

Publications and source records attributed to Alan Dyson.

3 recordsLinked to original sources

arXiv2025

Existence and Uniqueness of Fast Traveling Pulses in Singularly Perturbed Nonlocal Neural Fields With Heaviside Nonlinearities: a Complete Proof

We rigorously prove the existence and uniqueness of fast traveling pulse solutions to the singularly perturbed neural field system with linear feedback and Heaviside nonlinearity structure within a spatial convolution. Although a long-standing open problem, the pulse is well-accepted to often exist based on its original singular construction, closed form when it exists, and follow-ups, but prior to this study, there has not been a proof that overcomes the difficulties of (i) solving for the fast speed and width functions using the implicit function theorem at $\epsilon=0$ and (ii) tracking the resultant formal homoclinic orbit near its singular orbit during fast, slow, and mixed time scales. First, we provide new, first-order approximations of the pulse speed and width. We then show that the formal pulse is close to the front and back at threshold crossing points and that the Hausdorff distance between the formal pulse and the singular homoclinic orbit converges to zero, demonstrating that the formal pulse is a true solution to the original system. Broadly, our methods provide insight into nonlocal geometric singular perturbation theory.

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arXiv2018

Existence and Uniqueness of Traveling Fronts in Lateral Inhibition Neural Fields with Sigmoidal Firing Rates

We rigorously prove the existence of traveling fronts in neural field models with lateral inhibition coupling types and smooth sigmoidal firing rates. With Heaviside firing rates as our base point (where unique traveling fronts exist), we repeatedly apply the implicit function theorem in Banach spaces to provide a non-monotone version of the homotopy approach originally proposed by Ermentrout and McLeod (1993) in their seminal study of monotone fronts in purely excitatory models. By comparing smooth and Heaviside firing rates, we develop global wave speed and profile comparisons that guide our analysis, leading to uniqueness (modulo translation) in the perturbative case. Moreover, we establish a meaningful a priori existence result; we prove existence holds for a range of firing rates, independent of continuation path.

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arXiv2018

Traveling Wave Solutions to a Neural Field Model With Oscillatory Synaptic Coupling Types

In this paper, we investigate the existence, uniqueness, and spectral stability of traveling waves arising from a single threshold neural field model with one spatial dimension, a Heaviside firing rate function, axonal propagation delay, and biologically motivated oscillatory coupling types. Neuronal tracing studies show that long-ranged excitatory connections form stripe-like patterns throughout the mammalian cortex; thus, we aim to generalize the notions of pure excitation, lateral inhibition, and lateral excitation by allowing coupling types to spatially oscillate between excitation and inhibition. In turn, we hope to analyze traveling fronts and pulses with novel shapes. With fronts as our main focus, we exploit Heaviside firing rate functions in order to establish existence and utilize speed index functions with at most one critical point as a tool for showing uniqueness of wave speed. We are able to construct Evans functions, the so-called stability index functions, in order to provide positive spectral stability results. Finally, we show that by incorporating slow linear feedback, we can compute fast pulses numerically with phase space dynamics that are similar to their corresponding singular homoclinical orbits; hence, our work answers open problems and provides insight into new ones.

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