The Kuramoto model with Hebbian singular couplings: from agent based to Vlasov equations

par David Poyato (Université Lyon 1)

mardi 10 déc. 2019, 15:15 → 16:15 Europe/Paris

435 (UMPA)

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. 
 

 

 

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.