Department of Applied Mathematics University of Leeds, Leeds ls2 9jt, UK

UK Nonlinear News Review

Workshop on mathematical neuroscience
Royal Society of Edinburgh, 21-23 March 2005

Yulia Timofeeva

The first speaker was Dr Severine Mahon (UCL, UK) who described recent in vivo experimental results from cerebellar Purkinje cells exhibiting bistability. Dr Alice Yew (Surrey, UK) next discussed the mechanisms of signal transduction and information processing in the olfactory system as well as the modelling of signal adaptation in olfactory receptor neurons. Professor Steven Cox (Houston, USA) presented a mathematical approach for the determination of channel localisation in a neural system with nonuniformly distributed membrane conductance and leakage by solving an appropriate inverse problem. An interesting experiment demonstrating how rats, using their whisker sensory system, localise objects and judge the size, shape and texture of surfaces was presented by Dr Rasmus Petersen (Manchester, UK). He further discussed the neural coding of tactile stimuli in the vibrissal somatosensory cortex system. Dr Jonathan Rubin (Pittsburgh, USA) presented new results on the dynamic range of bursting in a network of synaptically-coupled cells. Using geometric singular perturbation theory he showed how the network supports two qualitatively distinct forms of synchronised bursting. The final speaker of the day, Dr John Terry (Loughborough, UK), demonstrated how a bifurcation analysis of macroscopic neural models might assist in understanding the onset of epileptic seizures. The first day concluded with a well attended poster session and reception at the International Centre for Mathematical Science (14 India St, Edinburgh).

The second day began with a talk by Dr William Troy (Pittsburgh, USA) who examined pattern formation, spiral waves and bump formation in partial integro-differential population equations used extensively in neuronal modelling. Dr Carlo Laing (Massey, NZ) appropriately continued the discussion on spatial wave propagation in neural field equations and described the solution dependence on system domain boundaries. Next, Dr Markus Owen (Nottingham, UK) showed a variety of exotic solutions including bumps, breathers and replicators in a neural field model with nonlinear threshold accommodation. Moreover, he made extensive use of Evans function techniques to uncover the fundamental bifurcation mechanisms in this model. Dr David Pinto (Rochester, USA) looked at the issue of modelling assumptions in theoretical neuroscience. In particular, the differences between modelling approaches with fast inhibition and more realistic delayed inhibition were explained. Dr Peter Ashwin (Exeter, UK) presented a study of pulse-coupled oscillator networks and showed how unstable attractors naturally and robustly arise. Finally, Dr Kresimir Josic (Houston, USA) examined networks of phase oscillators and used symmetry and groupoid ideas to show that the architecture of a network can have a large effect on system dynamics. Informal discussions were continued later that day during the workshop dinner.

The last day of the meeting was started by the talk of Professor Jack Cowan (Chicago, USA). He explained that stochastic neural networks have a natural mathematical description that corresponds closely with that used in quantum field theory. Based on a 3-state model of a neuron he developed a statistical neural field formalism, going beyond the standard mean-field approach, able to describe spike train correlations. Professor Roman Borisyuk (Plymouth, UK) then discussed aspects of cortical microcircuit modelling relating to the phenomenon of partial synchronisation that appears as a short coherent response of a neural subpopulation to a specific stimulus. Dr Carson Chow (Bethesda, USA) looked at the mechanism of bump wandering in spiking neural networks, and gave a mathematical analysis of non-stationary travelling bumps in the Lighthouse model of Hermann Haken. Finally, Professor David Willshaw (Edinburgh, UK) described electrophysiological mapping techniques and described how label-matching rather than electrical activity may underlie the formation of ordered nerve connections.

This was a very vibrant international meeting, attended by over 70 participants, with many productive exchanges. The meeting was organised under the auspices of the ICMS and financially supported by the EPSRC through the Mathematical Biomedical Network, the London Mathematical Society, the ICIAM 99 Fund and the Leverhulme Trust Theoretical Neuroscience Network.