论文标题
球体上神经场的分叉
Bifurcations of Neural Fields on the Sphere
论文作者
论文摘要
在大量神经元中研究模式形成的常见模型是神经场。我们研究了具有兴奋性和抑制性神经元的神经场，例如Wilson和Cowan（1972），并具有传播延迟和间隙连接。我们以Visser等人的工作为基础。 （2017）通过在球体上研究这些模型中的模式形成。具体而言，我们研究了以扩散项模型的不同数量的间隙 - 界面如何影响神经场的行为。我们详细介绍了在存在球形对称性的情况下通过HOPF分叉生成的周期轨道。为此，我们得出了一般公式，以计算这些分叉的正常形式系数高达三阶，并预测所得分支的稳定性，并制定了一种新的数值方法来求解球体上的延迟方程。
A common model to study pattern formation in large groups of neurons is the neural field. We investigate a neural field with excitatory and inhibitory neurons, like Wilson and Cowan (1972), with transmission delays and gap junctions. We build on the work of Visser et al. (2017) by investigating pattern formation in these models on the sphere. Specifically, we investigate how different amounts of gap-junctions, modelled by a diffusion term, influence the behaviour of the neural field. We look in detail at the periodic orbits which are generated by Hopf bifurcation in the presence of spherical symmetry. For this end, we derive general formulas to compute the normal form coefficients of these bifurcations up to third order and predict the stability of the resulting branches, and formulate a novel numerical method to solve delay equations on the sphere.
