Scientific Research

3D Vizualization of Galves–Löcherbach model

The Goalkeeper Game, a NeuroMat tool

Antonio Galves, NeuroMat director

Eva Löcherbach, NeuroMat associate investigator

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A main scientific achievement was the introduction by the NeuroMat team of a new class of stochastic processes aimed at a realistic description of nets of spiking neurons. These processes are systems with infinitely many interacting chains with memory of variable length. Since their introduction, these stochastic processes have become part of the research agenda of several centers in the world.

Our contributions to the investigation of this new class of stochastic processes include:

1. the identification of mathematical conditions assuring the existence of the processes together with the design of a perfect simulation algorithm for their numerical implementation;
2. results on the hydrodynamical limit of processes belonging to the class. This is an important step to relate different scales of description of the system, from the microscopic level, modelling systems of spiking neurons, to the mesoscopic and macroscopic levels, describing EEG and fMRI data;
3. existence of phase transition for a specific instantiation of these models with leakage, setting a new framework for the rigorous investigation of spontaneous transitions of brain activity states, e.g. healthy to seizure-like activity. This is the first phase transition result rigorously proved for a mathematical model describing a system of interacting spiking neurons;
4. introduction of a novel estimator of the interaction graph for models in this class and the proof of its strong consistency, not requiring the usual assumptions of stationarity and uniqueness of the invariant measure. This contribution addresses an important issue in contemporary neurobiology, namely the question of how to infer neural interactions from the activity of an ensemble of neurons.

A second major achievement is the introduction of a new mathematical approach to address the classical conjecture that the brain retrieves statistical regularities from sequences of stimuli. This approach is based on a new class of stochastic processes, namely sequences of random objects driven by chains with memory of variable length. These processes appear as good candidates to model the relationship between sequences of stimuli and sequences of suitably parsed brain signals and behavioral states registered while exposed to stimuli.

This framework offers a new way to model structural learning and memory in the brain, including the following promising directions of research:

1. It provides an effective way to identify brain sensitivity and reaction to sequences of stimuli which goes much beyond the possibilities offered by current averaging-based methods;
2. As a consequence, it allows the introduction of a entire new class of experimental protocols in which physiological or behavioral data are recorded while a volunteer is exposed to sequences of stimuli generated by a stochastic chain with memory of variable length;
3. This approach offers promising perspectives in clinical neuroscience by identifying different signatures in response to structured sequences of stimuli in neurological disorders;
4. From a purely statistical point of view, sequences of random objects driven by chains with memory of variable length constitute an innovative tool in functional data analysis and high dimension statistics.