arXiv subjects

Aline Duarte

Publications and source records attributed to Aline Duarte.

7 recordsLinked to original sources

arXiv2026

LAN property for the parameter of the jump rate in mean field interacting systems of neurons

In the context of a large system of $N$ neurons interacting through spike events in a mean-field regime as $N\to \infty$, we characterize the estimation of a multidimensional parameter in the spiking rate, when the neural states are observed over a fixed time horizon. We first prove the local asymptotic normality (LAN) property and leverage classical theory to establish the asymptotic efficiency of the maximum likelihood estimator. While the theory of Ibragimov and Hasminskii yields strong results, up to global asymptotic minimax bound, its applicability appears currently limited to models without state resets at spike times. Following then H\"opfner's classical approach, we nevertheless derive, in a general setting including neuron reset, the consistency, asymptotic normality and local asymptotic minimax optimality of the estimator. Keywords: Local Asymptotic Normality (LAN); Mean-field regime; Interacting particle system; Multidimensional parameter estimation; Jump rate estimation; Maximum likelihood estimator (MLE); Asymptotic minimax optimality

math.ST↗

arXiv2025

Nonparametric estimation of the jump rate in mean field interacting systems of neurons

We consider finite systems of $N$ interacting neurons described by non-linear Hawkes processes in a mean field frame. Neurons are described by their membrane potential. They spike randomly, at a rate depending on their potential. In between successive spikes, their membrane potential follows a deterministic flow. We estimate the spiking rate function based on the observation of the system of $N$ neurons over a fixed time interval $[0,t]$. Asymptotic are taken as $N,$ the number of neurons, tends to infinity. We introduce a kernel estimator of Nadaraya-Watson type and discuss its asymptotic properties with help of the deterministic dynamical system describing the mean field limit. We compute the minimax rate of convergence in an $L^2 -$error loss over a range of H\"older classes and obtain the classical rate of convergence $ N^{ - 2\beta/ ( 2 \beta + 1)} , $ where $ \beta $ is the regularity of the unknown spiking rate function.

math.PR↗

arXiv2020

Retrieving the structure of probabilistic sequences of auditory stimuli from EEG data

Using a new probabilistic approach we model the relationship between sequences of auditory stimuli generated by stochastic chains and the electroencephalographic (EEG) data acquired while 19 participants were exposed to those stimuli. The structure of the chains generating the stimuli are characterized by rooted and labeled trees whose leaves, henceforth called contexts, represent the sequences of past stimuli governing the choice of the next stimulus. A classical conjecture claims that the brain assigns probabilistic models to samples of stimuli. If this is true, then the context tree generating the sequence of stimuli should be encoded in the brain activity. Using an innovative statistical procedure we show that this context tree can effectively be extracted from the EEG data, thus giving support to the classical conjecture.

q-bio.NC↗

arXiv2020

SeqROCTM: A Matlab toolbox for the analysis of Sequence of Random Objects driven by Context Tree Models

In several research problems we deal with probabilistic sequences of inputs (e.g., sequence of stimuli) from which an agent generates a corresponding sequence of responses and it is of interest to model the relation between them. A new class of stochastic processes, namely \textit{sequences of random objects driven by context tree models}, has been introduced to model such relation in the context of auditory statistical learning. This paper introduces a freely available Matlab toolbox (SeqROCTM) that implements this new class of stochastic processes and three model selection procedures to make inference on it. Besides, due to the close relation of the new mathematical framework with context tree models, the toolbox also implements several existing model selection algorithms for context tree models.

cs.AI↗

arXiv2018

Stability, convergence to equilibrium and simulation of non-linear Hawkes Processes with memory kernels given by the sum of Erlang kernels

Non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels are considered. It is shown that their stability properties can be studied in terms of an associated class of piecewise deterministic Markov processes, called Markovian cascades of successive memory terms. Explicit conditions implying the positive Harris recurrence of these processes are presented. The proof is based on integration by parts with respect to the jump times. A crucial property is the non-degeneracy of the transition semigroup which is obtained thanks to the invertibility of an associated Vandermonde matrix. For Lipschitz continuous rate functions we also show that these Markovian cascades converge to equilibrium exponentially fast with respect to the Wasserstein distance. Finally, an extension of the classical thinning algorithm is proposed to simulate such Markovian cascades.

math.PR↗

arXiv2016

A model for neural activity in the absence of external stimuli

We study a stochastic process describing the continuous time evolution of the membrane potentials of finite system of neurons in the absence of external stimuli. The values of the membrane potentials evolve under the effect of {\it chemical synapses}, {\it electrical synapses} and a \textit{leak current}. The evolution of the process can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. When a neuron spikes, its membrane potential is immediately reset to a resting value. Simultaneously, the membrane potential of the neurons which are influenced by it receive an additional positive value. Furthermore, between consecutive spikes, the system follows a deterministic motion due both to electrical synapses and the leak current. Electrical synapses push the system towards its average potential, while the leak current attracts the membrane potential of each neuron to the resting value. We show that in the absence leakage the process converges exponentially fast to an unique invariant measure, whenever the initial configuration is non null. More interesting, when leakage is present, we proved the system stops spiking after a finite amount of time almost surely. This implies that the unique invariant measure is supported only by the null configuration.

math.PR↗

arXiv2015

Hydrodynamic Limits for Spatially Structured Interacting Neurons

In this paper we study the hydrodynamic limit for a stochastic process describing the time evolution of the membrane potentials of a system of neurons with spatial dependency. We do not impose on the neurons mean-field type interactions. The values of the membrane potentials evolve under the effect of chemical and electrical synapses and leak currents. The system consists of $ε^{-2}$ neurons embedded in $[0,1)^2$, each spiking randomly according to a point process with rate depending on both its membrane potential and position. When neuron $i$ spikes, its membrane potential is reset to a resting value while the membrane potential of $j$ is increased by a positive value $ε^2 a(i,j)$, if $i$ influences $j$. Furthermore, between consecutive spikes, the system follows a deterministic motion due both to electrical synapses and leak currents. The electrical synapses are involved in the synchronization of neurons. For each pair of neurons $(i,j)$, we modulate this synchronizing strength by $ε^2 b(i,j)$, where $b(i,j)$ is a nonnegative symmetric function. On the other hand, the leak currents inhibit the activity of all neurons, attracting simultaneously their membrane potentials to the resting value. In the main result of this paper is shown that the empirical distribution of the membrane potentials converges, as the parameter $ε$ goes to zero, to a probability density $ρ_t(u,r)$ which is proved to obey a non linear PDE of Hyperbolic type.

math.PR↗