Canards in a minimal piecewise-linear square-wave burster.
Canards in a minimal piecewise-linear square-wave burster.
复制标题
最小分段线性方波爆发器中的鸭翼。
作者：
Mathieu Desroches;S. Fernández;Martin Krupa
我们构建了一个分段线性（PWL）近似的Hindmarsh-Rose（HR）神经元模型，这是最小的，在这个意义上说，向量场具有最少数量的线性区域，以重现所有的动态存在于原始HR模型与经典的参数值。这包括方波爆发和称为鸭式的特殊轨迹，其具有长的排斥段，并组织具有n和n + 1个尖峰的稳定爆发模式之间的过渡，也称为尖峰添加鸭式爆炸。我们提出了一个第一近似的光滑HR模型，使用连续PWL系统，并表明其快速子系统不能拥有同宿分支，这是必要的，以获得适当的方波爆发。然后，我们放松的假设的连续性的向量场在所有区域，我们表明，我们可以得到一个同宿分支的快速子系统。我们使用最近发展的PWL系统鸭翼理论，以再现所研究的HR模型的尖峰加鸭翼爆炸特征，例如，在Desroches等人中，Chaos 23（4），046106（2013）.
We construct a piecewise-linear (PWL) approximation of the Hindmarsh-Rose (HR) neuron model that is minimal, in the sense that the vector field has the least number of linearity zones, in order to reproduce all the dynamics present in the original HR model with classical parameter values. This includes square-wave bursting and also special trajectories called canards, which possess long repelling segments and organise the transitions between stable bursting patterns with n and n + 1 spikes, also referred to as spike-adding canard explosions. We propose a first approximation of the smooth HR model, using a continuous PWL system, and show that its fast subsystem cannot possess a homoclinic bifurcation, which is necessary to obtain proper square-wave bursting. We then relax the assumption of continuity of the vector field across all zones, and we show that we can obtain a homoclinic bifurcation in the fast subsystem. We use the recently developed canard theory for PWL systems in order to reproduce the spike-adding canard explosion feature of the HR model as studied, e.g., in Desroches et al., Chaos 23(4), 046106 (2013).