Inflection, canards and excitability threshold in neuronal models

Abstract

A technique is presented, based on the differential geometry of planar curves, to evaluate the excitability threshold of neuronal models. The aim is to determine regions of the phase plane where solutions to the model equations have zero local curvature, thereby defining a zero-curvature (inflection) set that discerns between sub-threshold and spiking electrical activity. This transition can arise through a Hopf bifurcation, via the so-called canard explosion that happens in an exponentially small parameter variation, and this is typical for a large class of planar neuronal models (FitzHugh–Nagumo, reduced Hodgkin–Huxley), namely, type II neurons (resonators). This transition can also correspond to the crossing of the stable manifold of a saddle equilibrium, in the case of type I neurons (integrators). We compute inflection sets and study how well they approximate the excitability threshold of these neuron models, that is, both in the canard and in the non-canard regime, using tools from invariant manifold theory and singularity theory. With the latter, we investigate the topological changes that inflection sets undergo upon parameter variation. Finally, we show that the concept of inflection set gives a good approximation of the threshold in both the so-called resonator and integrator neuronal cases.

Appendix: The full Hodgkin–Huxley equations

We recall here the four-dimensional Hodgkin–Huxley model from Hodgkin and Huxley (1952) together with the original parameter values. The equations read

$$\begin{aligned} \dot{V}&= (I -\bar{g}_{Na}m^3h(V-V_{Na})-\bar{g}_Kn^4(V-V_K) -g_L(V-V_L))/C, \end{aligned}$$

(67)

$$\begin{aligned} \dot{n}&= \alpha _n(V)(1-n)-\beta _n(V)n, \end{aligned}$$

(68)

$$\begin{aligned} \dot{m}&= \alpha _m(V)(1-m)-\beta _m(V)m, \end{aligned}$$

(69)

$$\begin{aligned} \dot{h}&= \alpha _h(V)(1-h)-\beta _h(V)h, \end{aligned}$$

(70)

where

$$\begin{aligned} \alpha _n(V)&= \frac{0.01(V+55)}{1-\exp (-0.1(V+55))},\quad \quad \beta _n(V) = 0.125\exp (-(V+65)/80), \\ \alpha _m(V)&= \frac{0.1(V+40)}{1-\exp (-0.1(V+40))},\quad \quad \beta _m(V) = 4\exp (-(V+65)/18), \\ \alpha _h(V)&= 0.07\exp (-0.05(V+65)),\quad \quad \beta _h(V) = \frac{1}{1+\exp (-0.1(V+35))}. \end{aligned}$$

The original parameter values for the Hodgkin–Huxley equation are given in the table below

Ionic conductances ($\text{ mmho}/\text{ cm}^2$)			Reversal potentials ($\text{ m}V$)			Membrane capacitance (${\mu }F/\text{ cm}^2$)
$\bar{g}_{Na}$	$\bar{g}_K$	$g_L$	$V_{Na}$	$V_K$	$V_L$	$C$
$120$	$36$	$0.3$	$50$	$-77$	$-54.4$	$1$