Estimation of Synaptic Conductance in the Spiking Regime for the McKean Neuron Model

Abstract

In this work, we aim at giving a first proof of concept to address the estimation of synaptic conductances when a neuron is spiking, a complex inverse non-linear problem which is an open challenge in neuroscience. Our approach is based on a simplified model of neuronal activity, namely a piecewise linear version of the FitzHugh-Nagumo model. This simplified model allows a precise knowledge of the non-linear f-I curve by using standard techniques of non-smooth dynamical systems. In the regular firing regime of the neuron model, we obtain an approximation of the period which, in addition, improves previous approximations given in the literature up-to-date. By knowing both this expression of the period and the current applied to the neuron, and then solving an inverse problem with a unique solution, we are able to estimate the steady synaptic conductance of the cell's oscillatory activity. Moreover, the method gives also good estimations when the synaptic conductance varies slowly in time.

Estimation of Synaptic Activity during Neuronal Oscillations

“…These traces are approximated by calibrating the membrane potential using dynamic clamp in in vitro experiments of isolated neurons and a quadratic polynomial approximation. More recently, in [26] we introduce a mathematical approach to estimate conductances by characterizing the period of the membrane potential, and in [27] estimations of the conductances of spike trains of the membrane potential are performed.…”

“…In [26], we present a first proof-of-concept to estimate synaptic conductances in the spiking regime based on information about the f − I curve of the neuron. The idea was exemplified with the McKean model, a piecewise linear system that allows for a quasi-analytic expression of the f − I curve.…”

“…By interpolation, a continuous estimation of g syn,est (t) can be obtained, which is proved to be an accurate estimation of the prescribed conductances provided that they have slower variation (in time) than the membrane potential. Although the results obtained in [26] show that using the f − I to estimate synaptic conductances in spiking regimes can be a suitable strategy, there were only applied to the McKean model, a very specific and non-realistic model, which is nothing but a caricature of the Fitzhugh-Nagumo system.…”

“…In this paper, following the ideas introduced in [26], we propose a procedure to estimate conductances under a regular spiking regime. This procedure is based on a modification of the standard frequency-input curve, which consists of replacing the input constant current by a constant synaptic conductance ( f − g syn curve).…”

Saddle-node of limit cycles in planar piecewise linear systems and applications

“…Together with the property of mimicking the richness of nonlinear dynamics, PWL systems exhibit a simpler analytical treatment. This property allows, in many cases, to obtain quantitative information of the analyzed dynamical objects (for instance, the period and amplitude of limit cycles, [18,22,28]) and to explain the way some bifurcations take place [6]. Saddle-node bifurcation of limit cycles in the PWL context has been reported in different publications.…”