Interdisciplinary Seminar in Nonlinear Science

Title: Nonlinear dynamics of synaptically coupled neural oscillators
Speaker: Prof. Paul Bressloff
Speaker Info: Loughborough University (Visiting U.Chicago)
Brief Description:
Special Note:

A dynamical theory of pulse-coupled "integrate-and-fire" oscillators is presented that unifies two distinct approaches to mathematical neuroscience and provides the foundations for a general theory of spiking neurons. We begin by deriving conditions for $1:1$ frequency locking in a network of IF oscillators with non-instantaneous interactions. This leads to a set of phase equations determining the relative firing times of the oscillators and the self-consistent collective period. Methods for analyzing these phase equations are described including symmetric bifurcation theory. An application to locomotion in simple vertebrates is described. We then investigate the stability of phase-locked solutions by constructing a linearized map of the firing times and analyzing its spectrum. We establish that previous results concerning the stability properties of IF oscillator networks are incomplete since they only take into account the effects of weak coupling instabilities. We show how strong coupling instabilities can induce transitions to non-phase locked states characterized by periodic or quasiperiodic variations of the inter-spike intervals on attracting invariant circles. The resulting spatio-temporal pattern of network activity is compared with the behaviour of a corresponding firing rate (analog) model obtained by performing a short-term time-average of the IF dynamics. It is shown that for sufficiently slow interactions there is good agreement between the two models on an appropriately defined time-scale. We conclude with some numerical examples of strong coupling instabilities including the formation of bursting states and spatially periodic patterns of activity. The latter is the starting point for the development of detailed cortical models.
Date: Friday, January 21, 2000
Time: 2:00pm
Where: Tech M416
Contact Person: Prof. Riecke
Contact email: h-riecke@nwu.edu
Contact Phone: 847-491-8316
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