Toward a universal descripcion for single neuron dynamics

Abstract

Nea¡ a bifurcation the behaviour of any physical system is universal (i.e. not depend of its specific details) and is described by a uniuersal equation c led, normal form. These equations are uniuersality classes, and very different dynamical systems (near to a giveu bifurcation) will be desc¡ibed for the same normal form. Although this is mathematically strictly true infinitesimally near of the bifurcation, it occurs often that the qua.litative aspects of the behaviour of the system is still given by the uormal form even outside of the infinitesimal neighbourhood of the bifurcation point. In this work we will show that the electrical dynamics of a single neuron is a beautiful example of this fact. At the present time, to describe neuronal dynamics for different types of neurons there exist an overwhelming diversity of thousands of high dimensional nonlinear models. These class of models are called couductance based models (CB models) and a¡e considered the most biophysically rea.listic models. Despite the huge diversitv of types of neurons and models, they display a universal dynamics generically captured by phenomenological models of two variables (for spiking dynamics), and in non generic cases by three variables (for bursting dynamics). But the mathematical mechanisms by which single neurons display this universal behaviour are still not clear. We analytically show that a CB models that meet the biophysi cal conditions for spiking are generically in the neighbourhood of the Bogdanov-Takens bifurcation. We numerically confinn that the dynamics displayed by spiking CB models is qualitatively described by the subcritical Bogdanov-Takens normal form (a two variable equation). tr\rthermore, we found an anal¡,tic method to reduce, or transform in the two variable cases, either CB or phenomenological models to an equation with the same form of the subcdtical Bogdanov-Takens normal form. We a.nalltically and numerically show that the reduced most famous CB and phenomenological models (Morris-Lecar, Hodgkin and Huxley and generalised FitzHugh-Nagumo model) are actually equivalent to the subcritical Bogdanov-Takens normal form and retain all the qualitative dynamics of the original equations. We also show that the Tliple Zero bifurcation is not generic for CB models, but if these models meet the biophysical conditions for bursting, then they will be in its neighbourhood. F\rrthermore, we analvtically show that the Hindmarsh-Rose model the most famous phenomenological model for neuronal bursting- can be transformed by a trivial change of variable in almost the Tiiple Zero normal form. This results make an advance toward ari universal description of single neuron dynamics. Moreover the relevant experimenta,l quantities measured in experiments have a clear link with the proposed mathematical description.