The project that created the Research, Innovation and Dissemination Center for Neuromathematics (RIDC NeuroMat) in 2013 has been renewed by the São Paulo Research Foundation for a six-year term, and in this context we launch a series to connect our past and current trajectory. This series has been called "First 5, Next 6", a direct reference to content produced by our team within the center renewal process, especially a presentation to FAPESP's International Assessment Committee in 2017. The first part of this text is informed by NeuroMat's Statement of Impact, again a document prepared in the renewal process.
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:
the identification of mathematical conditions assuring the existence of the processes together with the design of a perfect simulation algorithm for their numerical implementation;
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;
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;
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.
The direction of NeuroMat’s ongoing research is to model brain plasticity, structural learning and memory constitution in the brain, including the following promising lines of research:
Modeling plasticity in the framework of the new class of models describing systems of spiking neurons introduced in Galves & Loecherbach 2013. A first step is a paper by Galves, Loecherbach, Pouzat & Presutti to be submitted on short-term plasticity. This model generalizes the 2013 model by allowing the synaptic weights to evolve in time as a function of the spiking activity of each neuron. In this framework, it is possible to study rigorously transient behaviors which are reminiscent of short-term memory.
This work on plasticity is part of a broader effort to study phase transition, metastability and criticality in stochastic neural networks.
Introducing of a entire new class of experimental protocols in which physiological or behavioral data are recorded while a volunteer is exposed to structured sequences of stimuli. This is done by identifying different signatures in response to structured sequences of stimuli in healthy subjects and in patients with neurological disorders. A paper by Duarte et al. presenting the mathematical framework underlying this proposal will be submitted soon.