Traveling Bumps and Their Collisions in a Two-Dimensional Neural Field

Abstract

A neural field is a continuous version of a neural network model accounting for dynamical pattern forming from populational firing activities in neural tissues. These patterns include standing bumps, moving bumps, traveling waves, target waves, breathers, and spiral waves, many of them observed in various brain areas. They can be categorized into two types: a wave-like activity spreading over the field and a particle-like localized activity. We show through numerical experiments that localized traveling excitation patterns (traveling bumps), which behave like particles, exist in a two-dimensional neural field with excitation and inhibition mechanisms. The traveling bumps do not require any geometric restriction (boundary) to prevent them from propagating away, a fact that might shed light on how neurons in the brain are functionally organized. Collisions of traveling bumps exhibit rich phenomena; they might reveal the manner of information processing in the cortex and be useful in various applications. The trajectories of traveling bumps can be controlled by external inputs.

Interface dynamics in planar neural field models

“…Rather to give a feel of the shape of a traveling spot we plot the level set where $ufalse({\xi }_1,{\xi }_{2}false)=h$ using (79) in Figure 11D including terms up to $c^3$. This nicely illustrates that spots contract in the direction of propagation and widen in the orthogonal direction, and provides a theoretical explanation for the shape of traveling spots recently reported in [28]. With the aid of direct numerical simulations we have also explored the scattering properties of traveling spots.…”

Transient Localized Wave Patterns and Their Application to Migraine

“…Similar ideas to obtain localized traveling, though not transient, waves have been introduced in various contexts, for instance, an integral negative feedback or a third, fast diffusing inhibitory component for moving spots in semiconductor materials, gas discharge phenomena, and chemical systems [31,44-46]. Furthermore, in neural field models [47], localized two-dimensional bumps are studied [48-50] in integrodifferential equations (without diffusion) in the context, for example, of memory formation [51]. Localized structures have also been discussed in the context of cortical spreading depression (SD) in migraine before, in particular a model with narrowly tuned parameters that shows transient waves [13,52,53] and a model with mean field feedback control that allows for localized waves [30].…”

Response of traveling waves to transient inputs in neural fields

“…In addition, spatially structured solutions arising in two-dimensional models like spiral waves [30] or traveling spots [57] could also be analyzed in this framework. Such an approach may even predict the angle of deflection of moving spots resulting from a transient input [58].…”