M-current Induced Bogdanov-Takens Bifurcation and Switching of Neuron Excitability Class

TL;DR

This paper analyzes how the inclusion of M-current in conductance-based neuron models leads to bifurcations, specifically Bogdanov-Takens points, causing neurons to switch from Class-I to Class-II excitability.

Contribution

It provides precise conditions for the occurrence of Bogdanov-Takens and Bogdanov-Takens-Cusp points in neuron models with M-current, revealing how excitability class transitions occur.

Findings

Identification of conditions for BT and BTC points in neuron models.

Demonstration of excitability class switching from I to II.

Validation of bifurcation analysis with numerical simulations.

Abstract

In this work, we consider a general conductance-based neuron model with the inclusion of the acetycholine sensitive, M-current. We study bifurcations in the parameter space consisting of the applied current, $I_{app}$ the maximal conductance of the M-current, $g_M$, and the conductance of the leak current, $g_L$. We give precise conditions for the model that ensure the existence of a Bogdanov-Takens (BT) point and show such a point can occur by varying $I_{app}$ and $g_{M}$. We discuss the case when the BT point becomes a Bogdanov-Takens-Cusp (BTC) point and show that such a point can occur in the three dimensional parameter space. The results of the bifurcation analysis are applied to different neuronal models and are verified and supplemented by numerical bifurcation diagrams generated using the package MATCONT. We conclude that there is a transition in the neuronal excitability type organized by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.