Past

 I will discuss the dynamics of synaptically coupled model neurons that undergo a form of accommodation in the presence of sustained activity. The basic model is an integral equation for synaptic activity that depends upon the non-local network connectivity, synaptic response, and firing rate of a single neuron. A phenomenological model of accommodation is examined whereby the firing rate is taken to be a simple state-dependent threshold function. As in the case without threshold accommodation classical Mexican-Hat connectivity is shown to allow for the existence of spatially localised states (bumps). Importantly an analysis of bump stability (in both one and two spatial dimensions) using recent Evans function techniques shows that bumps may undergo instabilities leading to the emergence of both breathers and travelling waves. Numerical simulations show that bifurcations in this model have the same generic properties as those seen in many other dissipative systems that support localised structures, and in particular those of coupled cubic complex Ginzburg-Landau equations, and three component reaction diffusion equations. Interestingly, travelling pulses in this model truly have a discrete character in the sense that they scatter as auto-solitons. /notices/events/abstracts/differential-equations/ht06/Coombes.shtml    

We will survey the topological interpretation of modal languages, with some modern features, such as the appropriate bisimulations and model comparison games. Then we move to an epistemic version of this, showing how it provides a finer set of epistemic distinctions for group behaviour, including different notions of common knowledge. We explain the background for this in an epistemic MU-calculus. Finally, if we can pull this off within the time limit, we will discuss how topological models also show up in current dynamic-epistemic systems of belief revision.

 

 

In the year 1858, Herman Ludwig Ferdinand von Helmholtz published in Crelle's Journal a deep and pioneering paper on vortex motions where the topological properties of vortex lines in a fluid motion were emphasised. This work has been a strong source of inspiration for P G Tait who settled down the foundation of knot theory and for H Poincare, the father of geometric theory of dynamical systems. As a matter of fact, by the end of the 19^th century, three topics, knots, flows and fluids were closely related. In the last decades, the topic has been boosted by a series of new appealing problems and interesting results gathered under the name Topological Methods in Hydrodynamics. Our talk will start with a short trip around the pioneering works. Then we will focus on two essential recent topics: