**The seminar meets regularly on Wednesdays** at **4pm** in **Eckhart 202**. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Tianling Jin.

Analysis of `Integrate and Fire' models for neural networks

Neurons exchange informations via discharges, propagated by membrane potential, which trigger firing of the many connected neurons. How to describe large networks of such neurons? How can such a network generate a spontaneous activity? Such questions can be tackled using nonlinear integro-differential equations. These are now classically used to describe neuronal networks or neural assemblies. Among them, the Wilson-Cowan equations are the best known and describe spiking rates in different brain locations. Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. Several mathematical results will be presented concerning existence, blow-up, convergence to steady state, for the excitatory and inhibitory neurons, with or without refractory states. Conditions for the transition to spontaneous activity (periodic solutions) will be discussed. It has also been proposed to describe directly the spike time distribution which seems to encode more directly the neuronal information. This leads to a structured population equation that describes at time \(t\) the probability to find a neuron with time \(s\) elapsed since its last discharge. Here, we can show that small or large connectivity leads to desynchronization. For intermediate regimes, sustained periodic activity occurs. A common mathematical tool is the use of the relative entropy method. This talk is based on works with K. Pakdaman and D. Salort, M. Caceres, J. A. Carrillo and D. Smets.

Lecture 1: The low density limit: formal derivation.

Wednesday, January 14, 4-5pm, Ryerson 251.

Lecture 2: A short time convergence result.

Thursday, January 15, 4:30-5:30pm, Eckhart 202 .

Lecture 3: The case of fluctuations around a global equilibrium.

Friday, January 16, 4-5pm, Eckhart 202 .

More on ergodicity and parabolic equations (I)

More on ergodicity and parabolic equations (II)

Nonlinear equations in \(L^2\) and applications to MFG

The Talbot effect in a non-linear dynamics

In the first part of the talk I shall present a linear model based on the Schrodinger equation with constant coefficient and periodic boundary conditions that explains the so-called Talbot effect in optics. In the second part I will make a connection of this Talbot effect with turbulence through the Schrodinger map which is a geometric non-linear partial differential equation.

Boundary regularity for fully nonlinear integro-differential equations

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order \(2s\), with \(s\in(0,1)\). We consider the class of nonlocal operators \(\mathcal L_*\subset\mathcal L_0\), which consists of infinitesimal generators of stable Levy processes belonging to the class \(\mathcal L_0\) of Caffarelli-Silvestre. For fully nonlinear operators \(I\) elliptic with respect to \(\mathcal L_*\), we prove that solutions to \(Iu = f\) in \(\Omega\), \(u = 0\) in \(\mathbb R^n \setminus \Omega\), satisfy \(u/d^s \in C^{s+\gamma}(\overline\Omega)\), where \(d\) is the distance to \(\partial\Omega\) and \(f \in C^{\gamma}\). The constants in all our estimates remain bounded as the order of the equation approaches 2. Thus, in the limit \(s \uparrow 1\) we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.

Homogenization for periodic systems with defects

The talk will report on joint works with Xavier Blanc (Paris 7) and Pierre-Louis Lions (College de France). The setting is that of homogenization of linear elliptic equations. We consider local defects embedded in periodic microstructures, or interfaces between two different microstructures. We prove well-posedness of the corrector problem for these environments and show how its solution allows to better approximate the solution of the original oscillatory problem.

Some theoretical issues arising in practical numerical random homogenization

The talk will report on a series of joint works with F. Legoll, S. Lemaire, W. Minvielle (Ecole des Ponts). Numerical random homogenization is a challenging topic, computationally. We discuss two practical situations where the state of the art is still unsatisfactory. In the first case, the difficulty is that the coefficient describing the medium is not completely known. In the second case, the computation of a corrector on a truncated domain is prohibitively expensive. In both cases, we present some practical techniques that raise new theoretical questions.

Metastability for parabolic equations with drift.

I will outline pde methods analyzing the exponentially long time behavior of solutions to linear uniformly parabolic equations which are small perturbations of a transport equation with vector field having a globally stable point. The results say that the solutions converge to a constant, which is either the initial value at the stable point or the boundary value at the minimum points of the associated quasi-potential, which are due to Freidlin and Wentzell and Freidlin and Koralov, and applies partially to nonlinear parabolic equations. This is based on a joint work with Prof. Souganidis.

For questions, contact Tianling Jin at: tj