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

Abstract

Abstract: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.

Qualitative changes in phase-response curve and synchronization at the saddle-node-loop bifurcation

“…Starting at a SNIC bifurcation in planar general neuron models, we demonstrate that a variation in the separation of time scales provokes a generic sequence of firing onset bifurcations. Compared to other bifurcation studies, which rely on a local unfolding of a codimension-three bifurcation [36,37], our approach proves the generic bifurcation structure including appearance and ordering of codimension-two bifurcations on a global scale not restricted to local analysis. The composed bifurcation diagram hence predicts the behavior of a class of neurons over the whole range of time-scale parameters, and thereby warrants a direct comparison with biological neurons.…”

Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes

“…Submanifolds of the three-parameter unfolding of the deg. TB singularity can be identified in several neuron models and this unfolding has been proposed as a key element to understand neural excitability [ 30 – 32 , 37 , 39 ]. Our results allow one to extend the understanding of the dynamical repertoire hosted in this unfolding, by giving a complete description of the planar bursting activity that can be obtained thanks to a coupling of the planar unfolding with a slower system.…”

“…These unfoldings are very rich, containing saddle-node, SNIC, saddle-homoclinic, supercritical Hopf, subcritical Hopf and fold limit cycle bifurcations [ 23 ]. It has been proposed that the unfolding of this singularity could provide a minimal model to understand neuronal excitability and its modulation [ 30 , 31 ], or a qualitative model for a cortical mass [ 32 ]. In addition, the deg.…”

M-current induced Bogdanov–Takens bifurcation and switching of neuron excitability class

“…The authors in [21] considered a general conductancebased neuron model and studied the existence of the BTC point in the parameter space of the applied current, leak conductance and capacitance. In [22], the authors give general conditions for the existence of the BT bifurcation in any conductance-based model. Our work builds on these latter two papers and extends them to the situation where an M-current is present in the model.…”