Détail du poste
Établissement : Université Paris-Saclay GS Sciences de l'ingénierie et des systèmes École doctorale : Sciences et Technologies de l'Information et de la Communication Laboratoire de recherche : Laboratoire des Signaux et Systèmes Direction de la thèse : Luca GRECO ORCID 0000000239615219 Début de la thèse : 2026-10-01 Date limite de candidature : 2026-05-11T23:59:59 Comprendre comment le cerveau organise et régule sa propre activité constitue l'un des principaux défis des neurosciences modernes. La dynamique corticale est façonnée par l'interaction entre la structure des réseaux à grande échelle, la plasticité synaptique et les propriétés intrinsèques des populations neuronales, donnant naissance à des schémas spatio-temporels complexes qui sous-tendent la cognition, la perception et le comportement. Malgré des décennies d'efforts de modélisation, les approches existantes restent prises entre deux extrêmes : des modèles phénoménologiques qui sont mathématiquement traitables mais biologiquement opaques, et des modèles mécanistes qui sont fondés sur la biophysique mais analytiquement intraitables.
Cette thèse propose un cadre unifié, ancré dans la théorie du contrôle et les systèmes dynamiques non linéaires, afin de faire le pont entre ces deux niveaux de description. L'idée centrale est de coupler un modèle mécaniste de l'activité corticale à haute dimension (HD) avec un modèle phénoménologique à basse dimension (LD). Grâce à ce couplage, le modèle HD hérite de l'interprétabilité du modèle LD, tout en restant biologiquement plausible. De plus, en définissant des lois de plasticité qualitatives dans le cadre LD puis en les transposant au modèle HD, nous pouvons dériver des règles de plasticité synaptique biologiquement plausibles.
Nous souhaitons appliquer ce cadre à deux phénomènes spécifiques. Le premier concerne les schémas d'activation séquentielle : la dynamique corticale évolue souvent à travers des séquences ordonnées d'états métastables, un motif observé dans les paradigmes moteurs, perceptifs et méditatifs. S'appuyant sur un modèle d'activité neuronale séquentielle basé sur Lotka-Volterra, cette thèse développera des lois de plasticité codant la modulation des temps de transition entre les états (comme on l'observe, par exemple, chez les pratiquants de la méditation par attention focalisée) et en déduira leurs équivalents en haute dimension.
Le deuxième phénomène est la criticité auto-organisée (SOC) : l'hypothèse, étayée empiriquement, selon laquelle le cortex s'autorégule de manière autonome à proximité d'un point critique, maximisant ainsi sa gamme dynamique et sa flexibilité computationnelle. La thèse étudiera les stratégies de contrôle par rétroaction qui conduisent un paramètre de champ moyen vers sa valeur critique (inconnue), puis transposera la loi de plasticité qui en résulte dans un cadre de haute dimension, tout en tenant compte de la nature statistique de l'approximation par champ moyen.
Ces deux problèmes nécessitent de faire face à la non-linéarité intrinsèque de la dynamique et à la nature non standard des objectifs de contrôle impliqués, qui ne s'inscrivent ni dans les paradigmes classiques de stabilisation ni dans ceux de suivi de trajectoire. Les résultats attendus comprennent des résultats mathématiques novateurs sur la conception de la plasticité basée sur l'observateur, ainsi qu'une validation numérique par rapport à des signatures comportementales et électrophysiologiques connues. In this thesis, we focus on two qualitative properties of cortical dynamics: sequential activation patterns, and self-organized criticality.
Sequential activation patterns.
Neural dynamics often exhibit metastable states: transient, weakly stable configurations in which the system remains for a finite time before transitioning to another state [6, 7, 8, 9]. Rather than isolated events, these states appear as ordered sequences that can be tracked across different tasks, measurement modalities, and levels of analysis, suggesting a common dynamical motif. Sequential metastable patterns have been reported across several cognitive and perceptual processes, including motor memory consolidation [10], binocular rivalry [11], and focused attention meditation [12]. The recurrence of such patterns across domains indicates that sequential switching is not task-specific but reflects a broader organizing principle of neural activity [13].
In a recent study [1], we proposed a biologically plausible model of sequential population activity, by imposing that the population activity follows a Lotka-Volterra dynamics, as in [14].
Self-organized criticality.
Self-organized criticality (SOC) is a phenomenon observed in many physical and biological systems. It characterizes dynamical systems that autonomously evolve toward a critical equilibrium, that is, a state located at the boundary between two qualitatively distinct behaviors. In other words, SOC corresponds to a situation in which one or more parameters self-regulate toward a bifurcation point. SOC plays a crucial role in the emergence of complexity, since in such systems, small perturbations can generate radically different responses. A classic example of SOC is a sandpile to which grains are added one by one: the pile eventually reaches a critical state, and the addition of a single grain may (or may not) trigger an avalanche of varying magnitude.
Theoretical studies [5, 15, 16] and experimental work [4, 17] suggest that the brain may operate near criticality, which could contribute to its optimal computational capabilities relative to its metabolic constraints. Indicators of SOC have been identified in in vitro neuronal cultures as well as in the brains of curarized animals. Despite these findings, the hypothesis of a brain operating near criticality remains highly debated [18, 19]. The main reason for this debate is that markers traditionally associated with critical systems, such as avalanches or power-law distributions of event occurrences, can also emerge in systems operating far from criticality. Traditional approaches to studying cortical dynamics have often relied on phenomenological models encoding the qualitative behavior of neural activity, but that typically lack a biological basis, or mechanistic ones that can be fitted to data, but that usually lack interpretability.
Our goal is twofold:
i. To produce mechanistic and phenomenological models of cortical dynamics that are both biologically plausible and interpretable, by lifting low-dimensional phenomenological models to high-dimensional mechanistic models.
ii. To tackle problems where plasticity plays a crucial role, by defining synaptic plasticity laws in the low-dimensional setting and then lifting them to the high-dimensional biological model.
General framework
Our starting point is a high-dimensional model of cortical dynamics of the form
\begin{equation}\tag{HD}
\dot{x} = F(x, \theta), \qquad x \in \mathbb{R}^n, \theta \in \mathbb{R}^p,
\end{equation}
where $x$ is the state of the system (e.g., $x_1, \ldots, x_n$ represents the activities of $n$ neurons), and $\theta$ is a vector of parameters. For example, $F(\cdot, \theta)$ could be the nonlinear function encoding the dynamics of an artificial neural network.
At the same time, we consider a low-dimensional (phenomenological) model of the form
\begin{equation}\tag{LD}
\begin{cases}
\dot{z} = f(z, u)\\
\dot u = h(z, u),
\end{cases}
z \in \mathbb{R}^m, u \in \mathbb{R}^p,
\end{equation}
where $z$ is the state of the low-dimensional model (e.g., $z_1, \ldots, z_m$ represents the activities of $m$ macroscopic neural populations), and $u$ is a parameter. Plasticity is encoded in the dynamics of $u$.
Our first goal is to find $\theta = \theta(u)$ such that the dynamics (HD) of the high-dimensional model is consistent with the dynamics (LD) of the low-dimensional model. The exact meaning of 'consistency' depends on the specific problem at hand, but it could be defined, for example, by requiring that $z=Px$ for some projection $P$ and that $PF(x,\theta) = f(Px,u)$.
Once this is achieved, in order to obtain biologically plausible plasticity laws, we need to be able to derive an equation
\begin{equation*}
\dot{\theta} = H(x, \theta),
\end{equation*}
such that $\theta(u)$ is compliant with the dynamics of $u$ in (LD). Once again, the exact meaning of 'compliant' depends on the specific problem at hand, but it could be defined, for example, by requiring that $\theta(u)$ is a solution of the above equation when $u$ evolves according to (LD).
Work plan
========
We propose to focus on two specific problems: sequential activation patterns, and self-organized criticality.
We will first focus on the problem of sequential activation patterns, where the low-dimensional variable $z$ represents aggregated neural activities in different populations, and is obtained by projecting the high-dimensional variable $x$ through a projection $P$. The perspective student will then be expected to:
SAP1. Starting from the Lotka-Volterra-based model of sequential activity proposed in [1], propose a dynamics $\dot u = h(z,u)$ encoding the changes in the residence times in different states that are due to plasticity (e.g., in the case of focused attention meditation practitioners [2]). This will require techniques from control theory, and in particular observer design.
SAP2. Once the above is achieved, exploit the aggregated state-space representation of the model to derive a plasticity law $\dot{\theta} = H(x, \theta)$.
SAP3. Finally, numerically simulate the derived plasticity law in the high-dimensional model, and verify that it is able to reproduce the desired behavioral patterns (see [3]).
Secondly, we will consider the problem of self-organized criticality. Here, the relationship between the high-dimensional model and the low-dimensional one is more complex, since the low-dimensional model is assumed to be a mean-field model of the high-dimensional one, derived independently of the plasticity dynamics. The perspective student will then be expected to:
SOC1. Study the literature concerning the possible low-dimensional mean field models, where $u$ is treated as a bifurcation parameter. Determine feedback control strategies $\dot u = h(z,u)$ that enable it to converge toward its (a priori unknown) critical value.
SOC2. Derive a plasticity law $\dot{\theta} = H(x, \theta)$ that preserves the relationship between the high-dimensional model and its low-dimensional mean-field approximation. Observe that this point is more complex than SAP2, since the low-dimensional model is not defined as a projection of the high-dimensional one, but rather as a non-linear mean-field approximation. In particular, the latter requires statistical tools, with which the student will need to become familiar.
SOC3. Simulate the derived plasticity law in the high-dimensional model, and verify that it is able to reproduce classical statistical hallmarks of criticality (e.g., avalanches with power-law distributions [4, 5]). The proposed research will be primarily theoretical, and will involve the development of mathematical models and computational simulations.
The main theoretical tools will come from the theory of nonlinear dynamical systems and control theory. In particular, observer design techniques will be used to derive plasticity laws for the low-dimensional model. This control-theoretic perspective has received limited attention so far, with the notable exception of [16], which, however, is restricted to very specific classes of systems. The difficulty stems from the intrinsically nonlinear nature of the dynamics involved, as well as from the non-standard nature of the control objective, which cannot be framed as either stabilization or trajectory tracking.
Le profil recherché
Publiée le 30/04/2026 - Réf : 597c2ad158dcc6c5222c10ea1abd34b4
