Next Article in Journal
Distributed Vector Quantization Based on Kullback-Leibler Divergence
Next Article in Special Issue
Energy Flows in Low-Entropy Complex Systems
Previous Article in Journal
The Fisher Thermodynamics of Quasi-Probabilities
Previous Article in Special Issue
Minimum Dissipation Principle in Nonlinear Transport
Open AccessArticle

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

1
Department of Statistics, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60615, USA
2
INLN, UMR 7335 CNRS, Universite de Nice Sophia-Antipolis, 1361 routes des Lucioles, Sophia-Antipolis, 06560 Valbonne, France
3
Departamento de Física, FCFM, Universidad de Chile, Casilla 487-3, Santiago 6511226, Chile
*
Author to whom correspondence should be addressed.
Academic Editor: Giorgio Sonnino
Entropy 2015, 17(12), 7859-7874; https://doi.org/10.3390/e17127850
Received: 21 October 2015 / Revised: 20 November 2015 / Accepted: 23 November 2015 / Published: 30 November 2015
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. View Full-Text
Keywords: neuron dynamics; Bogdanov–Takens; minimal equation; normal form; conductance-based model; single neuron model; Morris–Lecar neuron dynamics; Bogdanov–Takens; minimal equation; normal form; conductance-based model; single neuron model; Morris–Lecar
Show Figures

Figure 1

MDPI and ACS Style

Pereira, U.; Coullet, P.; Tirapegui, E. The Bogdanov–Takens Normal Form: A Minimal Model for Single Neuron Dynamics. Entropy 2015, 17, 7859-7874.

Show more citation formats Show less citations formats

Article Access Map by Country/Region

1
Search more from Scilit
 
Search
Back to TopTop