The Bogdanov–Takens Normal Form: A Minimal Model for Single Neuron Dynamics

Entropy

Ulises Pereira²  Pierre Coullet¹  Enrique Tirapegui³ 

[1] INLN, UMR 7335 CNRS, Universite de Nice Sophia-Antipolis, 1361 routes des Lucioles, Sophia-Antipolis, 06560 Valbonne, France;;Department of Statistics, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60615, USADepartamento de Física, FCFM, Universidad de Chile, Casilla 487-3, Santiago 6511226, Chile;

关键词: neuron dynamics;    Bogdanov–Takens;    minimal equation;    normal form;    conductance-based model;    single neuron model;    Morris–Lecar;   

DOI  :  10.3390/e17127850

来源: mdpi

【 摘 要 】

Conductance-based (CB) models are a class of high dimensional dynamical systems derived from biophysical principles to describe in detail the electrical dynamics of single neurons. Despite the high dimensionality of these models, the dynamics observed for realistic parameter values is generically planar and can be minimally described by two equations. In this work, we derive the conditions to have a Bogdanov–Takens (BT) bifurcation in CB models, and we argue that it is plausible that these conditions are verified for experimentally-sensible values of the parameters. We show numerically that the cubic BT normal form, a two-variable dynamical system, exhibits all of the diversity of bifurcations generically observed in single neuron models. We show that the Morris–Lecar model is approximately equivalent to the cubic Bogdanov–Takens normal form for realistic values of parameters. Furthermore, we explicitly calculate the quadratic coefficient of the BT normal form for a generic CB model, obtaining that by constraining the theoretical I-V curve’s curvature to match experimental observations, the normal form appears to be naturally cubic. We propose the cubic BT normal form as a robust minimal model for single neuron dynamics that can be derived from biophysically-realistic CB models.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland.