nonlinear dynamics

Dora Matzakos-Karvouniari
Dora Matzakos-Karvouniari

Modelling spontaneous propagating waves in the early retina

Dora Matzakos-Karvouniari
theodora.karvouniari@univ-cotedazur.fr
LJAD, Université Côte d'Azur, Campus Valrose

During early retina development, waves of activity propagate across the retina and play a key role in building the early visual system. In vertebrates species, upon maturation and before eye-opening, transient networks of cells generate these waves, characterized by $3$ consecutive stages. Here, we focus on the biophysical detailed modelling of the second stage (stage II), during which waves are controlled by directly interconnected specific cells, the cholinergic starburst amacrine cells (SACs) which are able to burst autonomously. We propose plausible underlying mechanisms for: i) waves generation at the single neuron level, ii) propagation at the network level in a landscape marked by previous waves prints and iii) waves termination. Based on a bifurcation analysis we show how biophysical parameters control retinal waves characteristics and we provide a theoretical condition for waves propagation and disappearance. Moreover, we show that the continuous decrease of the strength of the acetylcholine synaptic coupling, associated with the crossing of a synchronization transition, impacts dramatically the waves distribution. We report especially on the existence of power law distributions of the avalanche size not only at the synchronization threshold, but also for a whole range of coupling strength. This may play a key role in the ability of the retina to respond to visual stimuli by maximizing its dynamical range.

Etienne Farcot
Etienne Farcot

Steady and wave-like patterns in flux-based auxin transport models

Etienne Farcot
etienne.farcot@nottingham.ac.uk
School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
Auxin is a major plant hormone, and its spatial distribution in plant tissues is a key driver of plant structure and geometry. Auxin transport is a complex process, combining cell-to-cell diffusion and active transport. The latter is mediated by membrane-bound transporters whose inhomogeneous distribution is controlled by auxin itself. The details of this process are still largely unknown, despite numerous recent advances. In this work, the focus is on a mathematical model implementing one of the current biological assumptions, which is that auxin flux is the variable controlling transporters' distribution. We show that identical auxin patterns can be achieved by distinct transporters distributions, and characterize these in graph theoretical terms. Under a condition of regularity of the dependence of transporters on the flux, we can prove that one of these steady states, with zero flux everywhere, is asymptotically stable for any choice of parameters. When the condition of regularity is not satisfied the same steady state may undergo bifurcations and become unstable. In particular, we can observe stable oscillations taking the form of a travelling wave of auxin, on a row of cells.