Estimation d'état et de paramètres de modèles de masses neuronales : une approche par la théorie du contrôle

State and parameter estimation of neural mass models: a control-theoretic approach

• Auteur / Autrice : Adel Annabi
• Direction : Jean-Baptiste Pomet
• Type : Projet de thèse
• Discipline(s) : Doctorat mathematiques
• Date : Inscription en doctorat le 01/10/2022
• Etablissement(s) : Université Côte d'Azur
• Ecole(s) doctorale(s) : École doctorale Sciences fondamentales et appliquées (Nice ; 2000-....)
• Partenaire(s) de recherche : Laboratoire : INRIA - MCTAO - Mathématiques pour le Contrôle, le Transport et leurs Applications

Mots clés

Mots clés libres

Théorie du contrôle

Observateurs

Modèles d'activité neuronale

Observabilité

Mathématiques appliquées

State and Parameter estimation

Control Theory

Mdelling neuronal activity

Observer Design

Identification

Estimation

Résumé

Ce sujet porte sur l'observabilité des équations de Wilson Cowan, et la construction d'observateurs pour ces dernières. La motivation est la reconstruction de l'état dans les modèles le splus populaires du cortex visuel V1, les entrées du système étant des stimulations visuelles, et les sorties étant des mesures partielles ou indirectes d'activité (électrodes...). Une description plus détaillée est disponible en anglais.

The primary visual cortex V1 is an area of the brain responsible for a first treatment of images arriving from the retina. High level geometric models of V1 proved to be remarkable in their predictive capability. Wilson-Cowan equations, based on averaging the discrete structure of the cortex, have been successfully used, in particular, to identify recurring visual hallucinatory patterns. These nonlinear integro-differential equations, where an integral kernel encodes mutual activity inhibition and excitation between neurons based on their relative positions, are used to model neuronal activity in cortical areas. The thesis will be devoted to the introduction of control theory methods for the analysis of these dynamics. More precisely, we propose to study the evolution of cortical activity from an input-output perspective. In the visual cortex for instance, activity is partly generated by an exterior input in the form of a visual stimulation. The observability point of view asks if one can recover the activity in such a cortical area based only on knowledge of the stimulating input and a partial measurement of the activity. Even when observability conditions are met, there still remains to propose fast and efficient methods for activity estimation based on observer methods. We are especially interested in this exploration in the context of the visual cortex. Indeed, in the right circumstances, exterior stimulation can lead to the emergence of new self induced activity in the visual cortex that we understand as geometric visual hallucinations. We are interested in detecting the occurence of such phenomena in the context of in vivo measuring of the visual cortex activity, and rely on tools from estimation of dynamical systems to achieve this goal. The PhD student will have to tackle the following challenges: --- Study observability of Wilson-Cowan (WC) equations, starting from low dimensional reductions of the problem, and progressing to the more complete infinite dimensional cases. An approch by finite dimensional approximation, as well as direct inifnite dimensional case studies should allow this extension to the more general case. Multiple measurement setups should be explored,relying on the appropriate modelling the student will have to develop. --- In continuation of the observability results, the student will have to develop efficient observers for the WC equations considered. A first effort will be devoted to the weakly contracting case, allowing to consider linear observers. Next nonlinear observers will be proposed. A particular care will be devoted to the question of modeling of saturation in WC equations, where its influence on observability is still an open question in the control theory community. These developments should be suported by numerical implementation of the developed estimation strategies. --- In tandem with this work on the mathematical side, the student is expected that to explore the litterature on mathematical neurosciences, neural fields and dynamical models for cortical activity. The students will synthesize this knowledge in order to refine the modeling, in particular on the measurment side. The student will be brought to encounter researchers from the field of neuroscience and be able to propose modelisation and algorithmic solutions in the context of perceptual illusions in particular. Finally, the question of identification, tangential to the observability problem, should also be explored in order to extend the observer methods to the estimation of various paramters in WC equations. A case of particular interest being estimation of elements of the interaction kernel in the dynamics.