论文标题
在平均峰值神经元模型中的HOPF分叉
Hopf bifurcation in a Mean-Field model of spiking neurons
论文作者
论文摘要
我们研究了一个由泊松度量驱动的非线性麦基·维拉索夫（McKean-Vlasov SDE）家族，对一般性集成和开火神经元网络的平均渐近造型进行了建模。我们提供了足够的条件，可以通过HOPF分叉进行定期解决方案。我们的光谱条件涉及显式全态功能的根部位置。证明依赖于两种主要成分。首先，我们引入了一个离散的时间马尔可夫链，对神经元连续峰的阶段进行建模。马尔可夫链的不变度量与周期溶液的形状有关。其次，我们使用lyapunov-schmidt方法获得自洽振荡。我们通过玩具模型来说明结果，可以分析所有光谱条件。
We study a family of non-linear McKean-Vlasov SDEs driven by a Poisson measure, modelling the mean-field asymptotic of a network of generalized Integrate-and-Fire neurons. We give sufficient conditions to have periodic solutions through a Hopf bifurcation. Our spectral conditions involve the location of the roots of an explicit holomorphic function. The proof relies on two main ingredients. First, we introduce a discrete time Markov Chain modeling the phases of the successive spikes of a neuron. The invariant measure of this Markov Chain is related to the shape of the periodic solutions. Secondly, we use the Lyapunov-Schmidt method to obtain self-consistent oscillations. We illustrate the result with a toy model for which all the spectral conditions can be analytically checked.
