Young Researchers Workshop: Kinetic descriptions in theory and applications

Filippov trajectories and measure-valued solutions in the Kuramoto model with Hebbian singular couplings

David Poyato

University of Granada

Since Kuramoto proposed a model of coupled oscillators, the study of synchronization has attracted attention from different point of views: Biology, Chemistry, Neuroscience, etc. Such a phenomenon consists in the natural emergent behavior of an ensemble of agents that interact through periodic rules. Those patterns can be observed in several biological systems like the flashing of fireflies, the beating of heart cells and the synaptic firing of neurons in the brain. In the latter setting, Hebbian learning proposes an explanation of the mechanisms governing the evolution of the synaptic connections between neurons. In this talk we will focus on the Kuramoto model with non-uniform weights. Specifically, we will explore the fast learning regime of the coupling weights towards a Hebbian plasticity function with singularities. First, we shall introduce the agent-based singular system of N coupled oscillators and the three associated regimes of singularity (subcritical, critical and supercritical). Because of the presence of singularities, we will propose a well-posedness theory in the sense of Filippov giving rise to solutions that exhibit the finite-time fase synchronization under certain assumptions. We will also explore the phenomenon of clustering into distinguished groups. Our second goal will be to introduce the associated macroscopic model in terms of a Vlasov-McKean equation that corresponds to the singular counterpart of the Kuramoto-Sakaguchi equation. We will propose a well posedness theory that extends that of the agent-based system via the concept of weak measure-valued solutions in the sense of the Filippov flow. Such solutions emerge as rigorous mean field limit when the number N of particles tends to infinity. Finally, we will conclude by remarking the analogies and differences with other models in the literature like the singular Cucker-Smale model. This is a joint work with Jinyeong Park (Hanyang University, Korea) and Juan Soler (University of Granada, Spain).