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.
Highlights
Our purpose here is to present a robust minimal model for single neuron dynamics
Using as a starting point a generic CB model, we looked for a critical point that encloses the diversity of local bifurcations observed in single neuron dynamics
It robustly displays much of the features observed in single neuron dynamics
Summary
Our purpose here is to present a robust minimal model for single neuron dynamics. We have a dynamical systems approach, and the detailed calculations can be found in [1]. To describe neuronal dynamics for different types of neurons, there is an overwhelming diversity of thousands of high dimensional nonlinear models This class of models is called conductance-based models (CB models), and they are considered the most biophysically realistic models [4,5,6,7]. We present here the BT cubic normal form in Arnold’s choice, which leads to a second order in the time differential equation for the variable u, which has a very appealing mechanical interpretation as a Hamiltonian system with a nonlinear friction. We investigate numerically this equation, and we find that it robustly exhibits the local and global bifurcations observed in single neurons
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