Wednesday 25 May 2016Workshop on fast-slow systems

Bernd Krauskopf, Vadim Biktashev, Hinke Osinga - Universities of Auckland and Exeter

Harrison LT04 14:00-17:00

There will be a three talks on fast-slow systems associated with the Centre for Predictive Modelling in Healthcare.

14.00 Bernd Krauskopf (University of Auckland) "Canard orbits and mixed-mode oscillations in a chemical reaction model"

To study how mixed-mode oscillations are organised in Koper's three-dimensional chemical reaction model, we compute the two-dimensional attracting and repelling slow manifolds and their intersection curves, known as canard orbits. We also show how a tangency between the repelling slow manifold and the two-dimensional unstable manifold of a saddle-focus equilibrium shapes the dynamics locally and globally. (Joint work with: Jose Mujica and Hinke Osinga (both University of Auckland))

14.45 Vadim Biktashev (University of Exeter) "Drift of scroll waves in thin layers caused by thickness features: asymptotic theory and numerical simulations"

A scroll wave in a very thin layer of excitable medium is similar to a spiral wave, but its behaviour is affected by the layer geometry. We identify the effect of sharp variations of the layer thickness, which is separate from filament tension and curvature-induced drifts described earlier. We outline a two-step asymptotic theory describing this effect, including asymptotics in the layer thickness and calculation of the drift of so perturbed spiral waves using response functions. As specific examples, we consider drift of scrolls along thickness steps, ridges, ditches, and disk-shaped thickness variations. Asymptotic predictions agree with numerical simulations.

15.30 break

16.00 Hinke Osinga (University of Auckland) "Intrinsic excitability and the role of saddle slow manifolds"

Excitable cells, such as neurons, exhibit complex oscillations in response to external input, e.g., from other neurons in a network. We consider the effect of a brief stimulation from the rest state of a minimal neuronal model with multiple time scales. The transient dynamics arising from such short current injections brings out the intrinsic bursting capabilities of the system.  We focus on transient bursts, that is, the transient generation of one or more spikes, and use a simple polynomial model to illustrate our analysis. We take a geometric approach to explain how spikes arise and how their number changes as parameters are varied. We discuss how the onset of new spikes is controlled by stable manifolds of a slow manifold of saddle type. We give a precise definition of such a stable manifold and use numerical continuation of suitable two-point boundary value problems to approximate them. (Joint work with: Krasimira Tsaneva-Atanasova (University of Exeter), Vivien Kirk and Saeed Farjami (both University of Auckland))

16.45 end

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