## Bernoulli

• Bernoulli
• Volume 25, Number 1 (2019), 771-792.

### Estimating the interaction graph of stochastic neural dynamics

#### Abstract

In this paper, we address the question of statistical model selection for a class of stochastic models of biological neural nets. Models in this class are systems of interacting chains with memory of variable length. Each chain describes the activity of a single neuron, indicating whether it spikes or not at a given time. The spiking probability of a given neuron depends on the time evolution of its presynaptic neurons since its last spike time. When a neuron spikes, its potential is reset to a resting level and postsynaptic current pulses are generated, modifying the membrane potential of all its postsynaptic neurons. The relationship between a neuron and its pre- and postsynaptic neurons defines an oriented graph, the interaction graph of the model. The goal of this paper is to estimate this graph based on the observation of the spike activity of a finite set of neurons over a finite time. We provide explicit exponential upper bounds for the probabilities of under- and overestimating the interaction graph restricted to the observed set and obtain the strong consistency of the estimator. Our result does not require stationarity nor uniqueness of the invariant measure of the process.

#### Article information

Source
Bernoulli, Volume 25, Number 1 (2019), 771-792.

Dates
Revised: June 2017
First available in Project Euclid: 12 December 2018

https://projecteuclid.org/euclid.bj/1544605263

Digital Object Identifier
doi:10.3150/17-BEJ1006

Mathematical Reviews number (MathSciNet)
MR3892336

Zentralblatt MATH identifier
07007224

#### Citation

Duarte, Aline; Galves, Antonio; Löcherbach, Eva; Ost, Guilherme. Estimating the interaction graph of stochastic neural dynamics. Bernoulli 25 (2019), no. 1, 771--792. doi:10.3150/17-BEJ1006. https://projecteuclid.org/euclid.bj/1544605263

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