Non-parametric estimation of the spiking rate in systems of interacting neurons

Abstract

We consider a model of interacting neurons where the membrane potentials of the neurons are described by a multidimensional piecewise deterministic Markov process (PDMP) with values in R N , where N is the number of neurons in the network. A deterministic drift attracts each neuron's membrane potential to an equilibrium potential m. When a neuron jumps, its membrane potential is reset to a resting potential, here 0, while the other neurons receive an additional amount of potential 1 N . We are interested in the estimation of the jump (or spiking) rate of a single neuron based on an observation of the membrane potentials of the N neurons up to time t. We study a Nadaraya-Watson type kernel estimator for the jump rate and establish its rate of convergence in L 2 . This rate of convergence is shown to be optimal for a given Hölder class of jump rate functions. We also obtain a central limit theorem for the error of estimation. The main probabilistic tools are the uniform ergodicity of the process and a fine study of the invariant measure of a single neuron.

Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

“…On the contrary of [24], a Lyapunov-type inequality allows us to get rid of the compact state-space assumption. Due to the deterministic and degenerate nature of the jumps, the process does not have a density continuous with respect to the Lebesgue measure.…”

Absolute continuity of the invariant measure in piecewise deterministic Markov Processes having degenerate jumps

“…The purpose of the present paper is not to establish recurrence conditions ensuring that Assumption 3 holds. We refer the reader to Costa and Dufour (2008) [11] for a general treatment of the stability properties of PDMP's and to Duarte and Ost (2015) [14] or to Hodara et al (2016) [18] for examples of processes that follow our model assumptions, which are systems of interacting neurons where the Harris recurrence has been proven. 1 Of course, a finer study of conditions ensuring the existence of a non-exploding solution to (1.1) can be conducted, but this is outside the scope of the present paper.…”

“…We call such processes house-ofcards-like interacting particle systems. Systems of this type are good models for systems of interacting neurons as introduced by Galves and Löcherbach (2016) [17], see also Duarte and Ost (2016) [14] and Hodara et al (2016) [18].…”

“…The principal application we have in mind is given by systems of interacting neurons as in [14], [17] and [18]. For these systems, it can be shown that our strategy works and that the system possesses an invariant probability measure which is absolutely continuous with respect to the Lebesgue measure.…”

Long time behavior of a mean-field model of interacting neurons

“…This strategy has been developed in [30,2] for diffusive problems. The long time behavior of the particles system (1) has been studied in [16,22] (again in a slightly different setting but the methods could be adapted to our case): the authors proved that the particles system is Harris-ergodic and consequently converges weakly to its unique invariant probability measure. However, transferring the long time behavior of the particles system to the McKean-Vlasov equation is possible if the propagation of chaos holds uniformly in time.…”