Dynamic patterns in a two-dimensional neural field with refractoriness

Abstract

The formation of dynamic patterns such as localized propagating waves is a fascinating self-organizing phenomenon that happens in a wide range of spatially extended systems including neural systems, in which they might play important functional roles. Here we derive a type of two-dimensional neural-field model with refractoriness to study the formation mechanism of localized waves. After comparing this model with existing neural-field models, we show that it is able to generate a variety of localized patterns, including stationary bumps, localized waves rotating along a circular path, and localized waves with longer-range propagation. We construct explicit bump solutions for the two-dimensional neural field and conduct a linear stability analysis on how a stationary bump transitions to a propagating wave under different spatial eigenmode perturbations. The neural-field model is then partially solved in a comoving frame to obtain localized wave solutions, whose spatial profiles are in good agreement with those obtained from simulations. We demonstrate that when there are multiple such propagating waves, they exhibit rich propagation dynamics, including propagation along periodically oscillating and irregular trajectories; these propagation dynamics are quantitatively characterized. In addition, we show that these waves can have repulsive or merging collisions, depending on their collision angles and the refractoriness parameter. Due to its analytical tractability, the two-dimensional neural-field model provides a modeling framework for studying localized propagating waves and their interactions.

Unified Analysis of Global and Focal Aspects of Absence Epilepsy via Neural Field Theory of the Corticothalamic System

“…Here, we focus on the interplay of the cortical propagation and the underlying corticothalamic loop, which is beyond most previous work about spatially extended heterogeneous cortical model, employed to investigate the effect of cortical diffusion on the seizure activity and their transitions as well as transient dynamics [42][43][44][45]. The emergence of breathers in seizure localization has also been explored in previous work, which was discussed in an abstract neural field model with Maxico hat connection profile and also some adaptive feedback, such as spike-frequency adaptation and so on [46][47][48]. Theoretically, the sigmoid function is simplified to a step function, easing linear stability analysis and feasibly employing the Evans function [35,49].…”

“…Their spatiotemporal properties such as spatial extents and temporal 59 frequencies are comparable with experimental observations in humans and rats. The 60 resulting cortical waves have robust frequencies, and their underlying dynamical 61 mechanisms are discovered by employing eigenvalue spectra and corresponding 62 eigenmodes at critical states. Thus we uncover a unified dynamical mechanism for the 63 global and focal aspects of absence epilepsy.…”

“…In this work, we have focused on the interplay of cortical propagation and the 454 underlying corticothalamic loop, which goes beyond most previous work about spatially 455 extended heterogeneous but purely cortical models [56][57][58][59]. The emergence of breathers 456 during seizure localization was explored in previous work, but only in an abstract neural 457 field model with Mexican hat connection profile and feedbacks, such as spike-frequency 458 adaptation [42,60,61]. In most previous theoretical work, the sigmoid function was 459 simplified to a step function, easing linear stability analysis [38,62]; however, the step 460 function is unrealistic, whereas our corticothalamic model is physiologically justifiable, 461 and relies on parameters that are experimentally measurable.…”

Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis

“…Numerical Bifurcation Analysis. Gong and Robinson [36], and Qi and Gong [62] found wandering spots and propagating ensembles using direct numerical simulations on the plane. Here, we perform a numerical bifurcation analysis with various control parameters for the structures found in Section 3 on a one-dimensional domain.…”

“…Further simulation shows that the model also supports coherent states in the form of stochastic travelling waves. In two spatial dimensions, the system is known to support travelling spots [36,62]. In Figure 4, we show a time simulation of the stochastic model with initial condition…”