Since Kuramoto proposed a model of coupled oscillators,
the study of synchronization has pulled the attention from
different point of views: biology, chemistry, neuroscience, etc.
Such a phenomenon can be observed very often in biological
systems like the flashing of fireflies, the beating of heart
cells and the synaptic firing of neurons in the brain.
Mathematically, those patterns arise as the natural collective
behavior of an ensemble of agents that interact via periodic
rules. Although the discrete agent-based models are interesting
by themselves, real life situations involve a large amount $N$
of agents. In those cases, the system is described by $N$
coupled ODEs, what becomes very hard to tackle both from the
analytical and numerical point of views.
Fortunately, we can sometimes approximate with a single PDE that
governs the macroscopic/fluid description of the system.
In this talk we will focus on the Kuramoto model with
non-uniform and singular coupling weights obeying Hebb’s rule
in neuroscience. First, we shall introduce the agent-based
system of $N$ coupled oscillators and the three associated
regimes with regard to singularity: subcritical, critical and
supercritical. We will propose a well-posedness theory in the
sense of Filippov to tackle the presence of singularities,
giving rise to global solutions with new rich behaviour:
finite-time phase synchronization and clustering into
distinguished groups. Later, we will introduce the corresponding
kinetic Vlasov equation. It consists in a macroscopic (fluid)
type model governed by a continuity equation for the probability
density of oscillators along the manifold $\mathbb{T}\times
\mathbb{R}$, where the (compressible) velocity field is nonlocal
and self-generated. Since the kernel is singular, we will
propose a well posedness theory via the concept of weak
measure-valued solutions in the sense of Filippov’s flows that
remains valid after eventual collisions. Indeed, such solutions
emerge as rigorous mean field limit when the number $N$ of
agents tends to infinity. Finally, we will conclude by stating
some rates of convergence for those solutions and remarking some
analogies and differences with other related models in the
literature like the singular Cucker—Smale model.
