Desroches M, Fernández-García S, Krupa M
Inria Sophia-Antipolis Méditerranée Research Centre, MathNeuro Project-Team 2004 route des Lucioles BP 93, 06902 Valbonne Cedex, France.
Departamento EDAN, University of Seville, Facultad de Matemáticas C/ Tarfia, s/n., 41012 Sevilla, Spain.
Chaos. 2016 Jul;26(7):073111. doi: 10.1063/1.4958297.
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).
我们构建了Hindmarsh-Rose（HR）神经元模型的分段线性（PWL）近似，这种近似是最小化的，即向量场具有最少数量的线性区域，以便用经典参数值再现原始HR模型中存在的所有动力学。这包括方波爆发以及称为鸭轨的特殊轨迹，鸭轨具有长的排斥段，并组织具有n个和n + 1个尖峰的稳定爆发模式之间的转换，也称为加尖峰鸭轨爆炸。我们使用连续的PWL系统提出了光滑HR模型的一阶近似，并表明其快速子系统不可能具有同宿分岔，而这是获得适当方波爆发所必需的。然后，我们放宽了向量场在所有区域的连续性假设，并表明我们可以在快速子系统中获得同宿分岔。我们使用最近为PWL系统开发的鸭轨理论，以再现如Desroches等人在《混沌》23(4), 046106 (2013)中所研究的HR模型的加尖峰鸭轨爆炸特征。
