Desroches M, Krupa M, Rodrigues S
INRIA Paris-Rocquencourt Research Centre, Domaine de Voluceau, Rocquencourt BP 105, Le Chesnay Cedex, 78153, France,
J Math Biol. 2013 Oct;67(4):989-1017. doi: 10.1007/s00285-012-0576-z. Epub 2012 Sep 4.
A technique is presented, based on the differential geometry of planar curves, to evaluate the excitability threshold of neuronal models. The aim is to determine regions of the phase plane where solutions to the model equations have zero local curvature, thereby defining a zero-curvature (inflection) set that discerns between sub-threshold and spiking electrical activity. This transition can arise through a Hopf bifurcation, via the so-called canard explosion that happens in an exponentially small parameter variation, and this is typical for a large class of planar neuronal models (FitzHugh-Nagumo, reduced Hodgkin-Huxley), namely, type II neurons (resonators). This transition can also correspond to the crossing of the stable manifold of a saddle equilibrium, in the case of type I neurons (integrators). We compute inflection sets and study how well they approximate the excitability threshold of these neuron models, that is, both in the canard and in the non-canard regime, using tools from invariant manifold theory and singularity theory. With the latter, we investigate the topological changes that inflection sets undergo upon parameter variation. Finally, we show that the concept of inflection set gives a good approximation of the threshold in both the so-called resonator and integrator neuronal cases.
本文提出了一种基于平面曲线微分几何的技术，用于评估神经元模型的兴奋性阈值。目的是确定相平面中模型方程解的局部曲率为零的区域，从而定义一个零曲率（拐点）集，该集可区分阈下电活动和爆发性电活动。这种转变可以通过霍普夫分岔产生，通过在指数小参数变化中发生的所谓鸭式爆炸，这对于一大类平面神经元模型（菲茨休 - 纳古莫模型、简化的霍奇金 - 赫胥黎模型），即II型神经元（谐振器）来说是典型的。在I型神经元（积分器）的情况下，这种转变也可以对应于鞍点平衡稳定流形的交叉。我们计算拐点集，并使用不变流形理论和奇点理论的工具研究它们对这些神经元模型兴奋性阈值的近似程度，即在鸭式和非鸭式两种情况下。利用后者，我们研究了拐点集在参数变化时所经历的拓扑变化。最后，我们表明拐点集的概念在所谓的谐振器和积分器神经元情况下都能很好地近似阈值。
