In 2007, Modolo and colleagues derived a population density equation for a population
of Izhekevich neurons. This population density equation can describe oscillations in
the brain that occur in Parkinson’s disease. Numerical simulations of the population
density equation showed bursting behaviour even though the individual neurons had
parameters that put them in the tonic firing regime. The bursting comes from neuron
interactions but the mechanism producing this behaviour was not clear. In this thesis
we study numerical behaviour of the population density equation and then use a
combination of analysis and numerical simulation to analyze the basic qualitative
behaviour of the population model by means of a simplifying assumption: that the
initial density is a Dirac function and all neurons are identical, including the number
of inputs they receive, so they remain as a point mass over time. This leads to a new
ODE model for the population. For the new ODE system, we define a Poincaré map
and then to describe and analyze it under conditions on model parameters that are
met by the typical values adopted by Modolo and colleagues. We show that there is a
unique fixed point for this map and that under changes in a bifurcation parameter, the
system transitions from fast tonic firing, through an interval where bursting occurs,
the number of spikes decreasing as the bifurcation parameter increases, and finally to
slow tonic firing. / Graduate
| Identifer | oai:union.ndltd.org:uvic.ca/oai:dspace.library.uvic.ca:1828/11700 |
| Date | 29 April 2020 |
| Creators | Xie, Rongzheng |
| Contributors | Ibrahim, Slim, Edwards, Roderick |
| Source Sets | University of Victoria |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
| Rights | Available to the World Wide Web |
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