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

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 $I_{app}$ I a p p , the maximal conductance of the M-current $g_{M}$ g M and the conductance of the leak current $g_{L}$ g L . We give precise conditions for the model that ensure the existence of a Bogdanov–Takens (BT) point and show that such a point can occur by varying $I_{app}$ I a p p and $g_{M}$ 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 . We conclude that there is a transition in the neuronal excitability type organised by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.

How neuronal morphology impacts the synchronisation state of neuronal networks

“…Altogether this gives K active variables, and we will hereafter index all the active variables a i from i = 1,…, K for simplicity. As in prior research regarding conductance-based models ( Kirst et al, 2017 ; Schleimer and Schreiber, 2018 ; Al-Darabsah and Campbell, 2021 ), we will assume that the active variables are independent of each other and that their steady state distributions a i ,∞ ( υ ) saturate as υ → ±∞. Each active current has a reversal potential E j , a maximal conductance G j , and depends on the product of its active variables where p i is the gating exponent associated with active variable a i .…”

“…For finite-dimensional dynamical systems, such as conductance-based point-neuron models, the Jacobian of the system has a finite number of eigenvalues. This gives a clear, but by no means trivial, approach to calculating the Hopf, BT and BTC bifurcations ( Hesse et al, 2017 ; Kirst et al, 2017 ; Al-Darabsah and Campbell, 2021 ). However, for a spatially continuous system, the number of dimensions is not finite.…”

“…The BTC point has codimension 3, meaning that its location must be expressed in terms of three bifurcation parameters. In the work of Kirst et al Kirst et al (2017) , an equation for the BTC point for point-neuron conductance-based models is given with ( I ext , g L , C m ) as the bifurcation parameters, while in ( Al-Darabsah and Campbell, 2021 ), Al-Darabsah and Campbell use the M-current conductance instead of C m . Here we use the bifurcation parameters ( I ext , ρ, τ δ ) as they are common to all conductance-based neuron models and it includes two dendritic parameters.…”

“…Thus far, examples of the SNIC to HOM switch are primarily found in modelling studies (though see Hesse et al (2022) ). Previous research has shown this switch can be induced via changes to the membrane capacitance ( Hesse et al, 2017 ), extracellular potassium concentration ( Contreras et al, 2021 ), increasing the leak conductance ( Kirst et al, 2017 ), increasing the M-current conductance ( Al-Darabsah and Campbell, 2021 ), and increasing the temperature (Hesse et al, 2022 ).…”

How neuronal morphology impacts the synchronisation state of neuronal networks

“…Along these lines, weakly coupled inhibitory neurons with class 1 cell-intrinsic excitability do not foster synchronous network states [ 9 ], while neurons arranged in the same network topology with homoclinic-type action-potential generation favour in-phase synchronisation [ 7 , 10 ]. Among the parameters that alter the cellular excitability class, we find those that directly affect ion channel dynamics, including channel composition, extracellular ion concentration, and modulators such as temperature [ 6 , 7 , 10 – 14 ]. However, these are not all.…”