Gaëtan Vignoud

Étude de la plasticité synaptique dans les réseaux stochastiques de neurones: La plasticité synaptique est un mécanisme biologique intégrant l'activité neuronale dans l'évolution des connections récurrentes entre différents neurones. La spike-timing dependent plasticity (STDP) représente l'ensemble des mécanismes de plasticité dépendant des instants de potentiels d'action pré- et post-synaptiques. Nous avons développé une classe générale de modèles pour reproduire différentes règles de STDP dans un cadre stochastique, en modélisant notamment les ensembles de potentiels d'action par des processus ponctuels. La plasticité synaptique est un processus lent comparé à l'activité neuronale, justifiant une analyse de ce système avec des arguments de la théorie lent-rapide. En ajoutant des hypothèses de régularité, un théorème de convergence dans la limite lent-rapide est énoncé et appliqué à différentes règles de STDP. Enfin, une analyse théorique et numérique de l'évolution des poids synaptiques nous a amené à conclure sur l'impact de la STDP sur un système neuronal simple. En parallèle, nous avons étudié l'influence de la STDP de type anti-Hebienne dans des réseaux reproduisant certaines propriétés du striatum, un noyau sous-cortical impliqué dans l'apprentissage procédural. Des résultats expérimentaux ont récemment démontré que deux régions du striatum dorsal, le striatum dorsolatéral (DLS) et le striatum dorsomédial (DMS) sont caractérisées par deux types de STDP de polarités différentes. Les conclusions de cette étude mettent en avant que la plasticité présente dans le DMS permet au réseau d'oublier rapidement tous motifs précédemment acquis, alors que la STDP du DLS participe à la maintenance de ces mêmes motifs.

A Hawkes process on R is a point process whose intensity function at time t is a functional of its past activity before time t. It is defined by its activation function Φ and its memory function h. In this paper, the Hawkes property is expressed as an operator on the sub-space of non-negative sequences associated to distances between its points. By using the classical correspondence between a stationary point process and its Palm measure, we establish a characterization of the corresponding Palm measure as an invariant distribution of a Markovian kernel. We prove that if Φ is continuous and its growth rate is at most linear with a rate below some constant, then there exists a stationary Hawkes point process. The classical Lipschitz condition of the literature for an unbounded function Φ is relaxed. Our proofs rely on a combination of coupling methods, monotonicity properties of linear Hawkes processes and classical results on Palm distributions. An investigation of the Hawkes process starting from the null measure, the empty state, on R− plays also an important role. The linear case of Hawkes and Oakes is revisited at this occasion. If the memory function h is an exponential function, under a weak condition it is shown that there exists a unique stationary Hawkes point process. In this case, its Palm measure is expressed in terms of the invariant distribution of a one-dimensional Harris ergodic Markov chain. When the activation function is a polynomial Φ with degree >1, there does not exist a stationary Hawkes process and if the Hawkes process starts from the empty state, a scaling result for the accumulation of its points is obtained.

We investigate spike-timing dependent plasticity (STPD) in the case of a synapse connecting two neuronal cells. We develop a theoretical analysis of several STDP rules using Markovian theory. In this context there are two different timescales, fast neuronal activity and slower synaptic weight updates. Exploiting this timescale separation, we derive the long-time limits of a single synaptic weight subject to STDP. We show that the pairing model of presynaptic and postsynaptic spikes controls the synaptic weight dynamics for small external input on an excitatory synapse. This result implies in particular that mean-field analysis of plasticity may miss some important properties of STDP. Anti-Hebbian STDP favors the emergence of a stable synaptic weight. In the case of an inhibitory synapse the pairing schemes matter less, and we observe convergence of the synaptic weight to a nonnull value only for Hebbian STDP. We extensively study different asymptotic regimes for STDP rules, raising interesting questions for future work on adaptative neuronal networks and, more generally, on adaptative systems.

In neuroscience, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. A Spike-Timing Dependent Plasticity (STDP) rule is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. A general mathematical framework has been introduced in~arXiv:2010.08195. In this paper we develop and investigate a scaling approach of these models based on several biological assumptions. Experiments show that long-term synaptic plasticity evolves on a much slower timescale than the cellular mechanisms driving the activity of neuronal cells, like their spiking activity or the concentration of various chemical components created/suppressed by this spiking activity. For this reason, a scaled version of the stochastic model of~arXiv:2010.08195 is introduced and a limit theorem, an averaging principle, is stated for a large class of plasticity kernels. A companion paper~arXiv:2010.08790 is entirely devoted to the tightness properties used to prove these convergence results. These averaging principles are used to study two important STDP models: pair-based rules and calcium-based rules. Our results are compared with the approximations of neuroscience STDP models. A class of discrete models of STDP rules is also investigated for the analytical tractability of its limiting dynamical system.

In neuroscience, learning and memory are usually associated to long-term changes of neuronal connectivity. In this context, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called {\em synapses} and represented by a scalar value, the synaptic weight. Spike-Timing Dependent Plasticity (STDP) is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. If numerous models of neuronal cells have been proposed in the mathematical literature, few of them include a variable for the time-varying strength of the connection. A new, general, mathematical framework is introduced to study synaptic plasticity associated to different STDP rules. The system composed of two neurons connected by a single synapse is investigated and a stochastic process describing its dynamical behavior is presented and analyzed. The notion of plasticity kernel is introduced as a key component of plastic neural networks models, generalizing a notion used for pair-based models. We show that a large number of STDP rules from neuroscience and physics can be represented by this formalism. Several aspects of these models are discussed and compared to canonical models of computational neuroscience. An important sub-class of plasticity kernels with a Markovian formulation is also defined and investigated. In these models, the time evolution of cellular processes such as the neuronal membrane potential and the concentrations of chemical components created/suppressed by spiking activity has the Markov property.

In neuroscience, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. A spike-timing-dependent plasticity (STDP) rule is a biologically based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. A general mathematical framework has been introduced in [P. Robert and G. Vignoud, SIAM. J. Appl. Math., 81 (2021), pp. 1821--1846]. In this paper, we develop and investigate a scaling approach of these models based on several biological assumptions. Experiments show that long-term synaptic plasticity evolves on a much slower timescale than the cellular mechanisms driving the activity of neuronal cells, like their spiking activity or the concentration of various chemical components created/suppressed by this spiking activity. For this reason, a scaled version of the stochastic model of Robert and Vignoud [SIAM. J. Appl. Math., 81 (2021), pp. 1821--1846] is introduced and a limit theorem, an averaging principle, is stated for a large class of plasticity kernels. A companion paper [P. Robert and G. Vignoud, J. Stat. Phys. 183 (2021), 47] is entirely devoted to the tightness properties used to prove these convergence results. These averaging principles are used to study two important STDP models: pair-based rules and calcium-based rules. Our results are compared with the approximations of neuroscience STDP models. A class of discrete models of STDP rules is also investigated for the analytical tractability of its limiting dynamical system.

In neuroscience, learning and memory are usually associated with long-term changes in neuronal connectivity. In this context, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. Spike-timing-dependent plasticity (STDP) is a biologically based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. Although numerous models of neuronal cells have been proposed in the mathematical literature, few of them include a variable for the time-varying strength of the connection. A new, general, mathematical framework is introduced to study synaptic plasticity associated to different STDP rules. The system composed of two neurons connected by a single synapse is investigated, and a stochastic process describing its dynamical behavior is presented and analyzed. The notion of plasticity kernel is introduced as a key component of plastic neural networks models, generalizing a notion used for pair-based models. We show that a large number of STDP rules from neuroscience and physics can be represented by this formalism. Several aspects of these models are discussed and compared to canonical models of computational neuroscience. An important subclass of plasticity kernels with a Markovian formulation is also defined and investigated. In these models, the time evolution of cellular processes such as the neuronal membrane potential and the concentrations of chemical components created/suppressed by spiking activity has the Markov property.