Spots: Breathing, drifting and scattering in a neural field model
Spots: Breathing, drifting and scattering in a neural field model
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斑点：神经场模型中的呼吸、漂移和散射
DOI：
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发表时间：
2014
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通讯作者：
D. Avitabile
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作者：
S. Coombes;Helmut Schmidt;D. Avitabile
具有短程激发和长程抑制的二维神经场模型可以以点的形式展示局部解。此外，由于包含尖峰频率适应电流，这些模型还可以支持呼吸和旅行点。在本章中，我们将展示如何通过线性尖峰频率自适应来分析神经场模型中斑点的特性。对于赫维赛德发射率函数，我们使用接口描述来推导一组四个非线性常微分方程来描述光斑的宽度，并展示稳态解如何经历霍普夫不稳定性，从而导致周期解的分支（呼吸）。为了实现平滑的发射率函数，我们开发了用于完整时空模型演化的数值代码，并对径向对称解进行数值分岔分析。确定用于分析分叉点附近的呼吸行为的幅度方程。还导出了漂移不稳定性的条件，并使用中心流形简化来描述该分叉附近的缓慢移动点。该分析扩展到涵盖两个缓慢移动的点的情况，并确定它们在正面碰撞中会相互反射。
Two dimensional neural field models with short range excitation and long range inhibition can exhibit localised solutions in the form of spots. Moreover, with the inclusion of a spike frequency adaptation current, these models can also support breathers and travelling spots. In this chapter we show how to analyse the properties of spots in a neural field model with linear spike frequency adaptation . For a Heaviside firing rate function we use an interface description to derive a set of four nonlinear ordinary differential equations to describe the width of a spot, and show how a stationary solution can undergo a Hopf instability leading to a branch of periodic solutions (breathers). For smooth firing rate functions we develop numerical codes for the evolution of the full space-time model and perform a numerical bifurcation analysis of radially symmetric solutions. An amplitude equation for analysing breathing behaviour in the vicinity of the bifurcation point is determined. The condition for a drift instability is also derived and a center manifold reduction is used to describe a slowly moving spot in the vicinity of this bifurcation. This analysis is extended to cover the case of two slowly moving spots, and establishes that these will reflect from each other in a head-on collision.