Spatio-temporal dynamics in nonlocal excitable systems
with multiple timescales
Advisors:
Grégoy Faye: Institut de Mathemématiques de Toulouse, Gregory.Faye@math.univ-toulouse.fr
Jonathan Touboul: Collège de France, CIRB & Inria Mycenae, jonathan.touboul@college-de-france.fr
Subject
The topic of the internship is the study of spatio-temporal pattern formation in excitable
systems. Such systems are ubiquitous in applications, and the motivation of the models to
be studied arises from the modeling of brain’s activity. In neuroscience, there are at least
two spatial scales at which such excitable spatially extended systems arise: (i) in the
modeling of the axonal transmission in cells, and (ii) in the modeling of collective patterns
of activity along the cortex. In these two situations, diverse patterns appear, including
stationary localized activity (bumps), oscillating bumps (breathers or sloshers) [1], fronts
and waves of different types (see spiraling wave recorded experimentally on neuronal
cultures, JY Wu, G. Washington University).
The modeling of these phenomena generally involves functional equations of different types, such as
partial differential equations, integro-differential equations and McKean-Vlasov equations, generally
governing the evolution of several variables at very different timescales. This is an active domain in
mathematics and in applications. In the neuroscience context, we recently established the existence
of pulses [2]. Slow-fast systems are very interesting and display very specific dynamical structures,
well understood in low dimensions, but still lacking theoretical
understanding in infinite dimensions. We recently undertook this
development to delayed dynamical systems [3]. This yields very complex
dynamics including chaos (see the complex structure of the Hopf
bifurcations curves on the right).
The object of the internship will be to combine the methods developed in
[2] for the existence of pulses to regimes in which the separation of timescales fails and where
dynamical systems methods in the flavor of [3] may be used.
To tackle this exciting problem, we are looking for a highly motivated student trained in
mathematics, familiar with functional analysis, with a keen interest in dynamical systems and
biology.
The student will receive a compensation.
This project is expected to be followed up with a co-advised thesis.
References:
[1] G. Faye and J. Touboul. Pulsatile localized dynamics in delayed neural field equations in arbitrary
dimension. SIAM Appl Math, 74(5) :1657–1690, 2014.
[2] G. Faye and A. Scheel. Existence of pulses in excitable media with nonlocal coupling. Adv. In Maths.,
vol 270 (2015), pp. 400-456.
[3] M. Krupa and J. Touboul. Canard explosion in delayed equations with multiple timescales. J. of Dyn.
And Diff. Equ., (2015), in press.