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SUMMARY:Networks of Nonsmooth Oscillators &amp\; Applications in Neuroscie
 nce  - Stephen Coombes\, University of Nottingham
DTSTART:20200123T140000Z
DTEND:20200123T150000Z
UID:TALK134380@talks.cam.ac.uk
CONTACT:Alberto Padoan
DESCRIPTION:The tools of weakly coupled phase oscillator theory have had a
  profound impact on the neuroscience community\, providing insight into a 
 variety of network behaviours ranging from central pattern generation to s
 ynchronisation. However\, there are many instances where this theory is ex
 pected to break down\, say in the presence of strong coupling.  To gain in
 sight into the behaviour of neural networks when phase-oscillator descript
 ions are not appropriate we turn instead to the study of tractable piece-w
 ise linear (pwl) systems.  There has been an appreciation for some time in
  the applied sciences\, and particularly in electrical engineering\, of th
 e benefits of studying caricatures of complex systems built from pwl and p
 ossibly discontinuous dynamical systems.  Although a beautifully simplisti
 c modelling perspective the necessary loss of smoothness precludes the use
  of many results from the standard toolkit of smooth dynamical systems\, a
 nd one must be careful to correctly determine conditions for existence\, u
 niqueness and stability of solutions.  \n\nIn this talk I will describe a 
 variety of pwl neural oscillators and show how to analyse periodic orbits.
   Building on this approach I will show how to analyse network states\, wi
 th a focus on synchrony.  I will make use of an extension of the master st
 ability function (MSF) approach utilising saltation matrices\, and show ho
 w this framework is very amenable to explicit calculations when considerin
 g networks of pwl oscillators.  These can include pwl integrate-and-fire (
 IF) systems with smooth synaptic interactions\, for which synchrony is ubi
 quitous in the case of a balance between excitation and inhibition.  Moreo
 ver\, the MSF approach is readily generalised to treat other phase-locked 
 states such as clusters.  For the case of nonsmooth synaptic interactions 
 there is a further mathematical challenge that requires a careful treatmen
 t of the order in which perturbations cross the IF threshold.  A similar i
 ssue arises in networks of switch-like elements\, as exemplified by Glass 
 networks and neural mass models with a Heaviside nonlinearity.  Finally\, 
 I will discuss the dynamics of the famous Wilson-Cowan model posed on a re
 alistic large-scale brain atlas and\, time permitting\, the important role
  that axonal delays can have on emergent network brain states and rhythms.
LOCATION: Cambridge University Engineering Department\, TBD
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