PhD Thesis: Dynamics of Synaptically Interacting Integrate-and-Fire Neurons

[Matthew James]

Dear Colleagues,

I would like to draw your attention to my PhD thesis "Dynamics of Synaptically 
Interacting Integrate-and-Fire Neurons" supervised by Professor Paul Bressloff 
and Dr Steve Coombes at the Department of Mathematical Sciences, Loughborough 
University, UK.

Available at: http://www.lboro.ac.uk/departments/ma/pg/theses/mampj-abs.html 

Abstract:

Travelling waves of activity have been experimentally observed in many neural 
systems. The functional significance of such travelling waves is not always 
clear. Elucidating the mechanisms of wave initiation, propagation and 
bifurcation may therefore have a role to play in ascertaining the function of 
such waves. Previous treatments of travelling waves of neural activity have
focussed on the mathematical analysis of travelling pulses and numerical studies 
of travelling waves. It is the aim of this thesis to provide insight into the 
propagation and bifurcation of travelling waveforms in biologically realistic 
systems. 

There is a great deal of experimental evidence which suggests that the response 
of a neuron is strongly dependent upon its previous activity. A simple model of 
this synaptic adaptation is incorporated into an existing theory of strongly 
coupled discrete integrate-and-fire (IF) networks. Stability boundaries for 
synchronous firing shift in parameter space according to the level of 
adaptation, but the qualitative nature of solutions is unaffected. The level of 
synaptic adaptation is found to cause a switch between bursting states and those 
which display temporal coherence. 

Travelling waves are analysed within a framework for a one-dimensional continuum 
of integrate-and-fire neurons. Self-consistent speeds and periods are determined 
from integro-differential equations. A number of synaptic responses
(alpha-function and passive and quasi-active dendrites) produce qualitatively 
similar results in the travelling pulse case. For IF neurons, an additional 
refractory mechanism needs to be introduced in order to prevent arbitrarily high 
firing rates. Different mathematical formulations are considered with each 
producing similar results. Dendrites are extensions of a neuron which branch 
repeatedly and the electrical properties may vary. Under certain conditions, 
this active membrane gives rise to a membrane impedance that displays a 
prominent maximum at some nonzero resonant frequency. Dispersion curves which
relate the speed of a periodic travelling wave to its period are constructed for 
the different synaptic responses with additional oscillatory behaviour apparent 
in the quasi-active dendritic regime. These stationary points are shown to be 
critical for the formation of multi-periodic wave trains. It is found that 
periodic travelling waves with two periods bifurcate from trains with a
single period via a drift in the spike times at stationary points in the 
dispersion curve. 

Some neurons rebound and fire after release from sustained inhibition. Many 
previous mathematical treatments have not included the effect of this activity. 
Analytical studies of a simple model which exhibits post-inhibitory rebound show 
that these neurons can support half-centre oscillations and periodic travelling 
waves. In contrast to IF networks, only a single travelling pulse wavespeed is 
possible in this network. Simulations of this biophysical model show broad 
agreement with the analytical solutions and provide insight into more complex 
waveforms. 

Results of the thesis are presented in a discussion along with possible 
directions for future study. Noise, inhomogeneous media and higher spatial 
dimensions are suggested. 

Keywords: biophysical models, dendrites, integrate-and-fire, neural coding, 
neural networks, post-inhibitory rebound, synaptic adaptation, travelling waves 



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Dr. Matthew P. James
Max-Planck-Institute for Mathematics in the Sciences
Inselstrasse 22 - 26
04103 Leipzig / Germany

Phone: +49-341-9959-531
Fax: +49-341-9959-658
Email: james at mis.mpg.de
URL: http://personal-homepages.mis.mpg.de/james/