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Analysis of electrophysiological models of spontaneous secondary spiking and triggered activity

University of British Columbia

We have examined two mathematical models describing the electrophysiology of a neuron and a cardiac cell, respectively, which exhibit an unusual response to high frequency stimulation. For certain parameter sets, both models behave qualitatively like the classic Hodgkin-Huxley equations for a squid giant axon; several brief depolarizing current pulses give rise to a corresponding number of action potentials followed by a return to rest. However, if the parameters are adjusted to reflect certain experimental conditions, a few "spontaneous" action potentials sometimes follow the directly induced action potentials. The number of spontaneous action potentials depends on the number and frequency of action potentials in the original spike train. Our objective was to gain a qualitative understanding of the mechanisms involved in this phenomenon and the effects of certain experimental interventions in promoting or suppressing the occurrence of the spontaneous action potentials. We first studied the Kepler and Marder (KM) model of spontaneous secondary spiking in a crab neuron. Then we examined the DiFrancesco-Noble (DN) equations for a mammalian cardiac Purkinje fiber which can exhibit a behaviour analogous to spontaneous secondary spiking, referred to in cardiac literature as triggered activity. Using a combination of bifurcation analysis and numerical computation, we showed that spontaneous action potentials are likely to occur in'both models when a critical bifurcation parameter is just to the left of a saddle-node of periodics (SNP) bifurcation. In the K M model, the neurotransmitter, serotonin, promoted spontaneous secondary spiking by shifting the bifurcation parameter closer to the SNP. Similarly, in the DN equations, application of digitalis increased the bifurcation parameter (intracellular [Na⁺]) while high extracellular [Ca²⁺] shifted the SNP to a lower value, both effects promoting triggered activity. Both models consist of an excitable subsystem and another subsystem that can build up slowly in response to action potentials. Spontaneous action potentials result from the bidirectional feedback between the two subsystems. By simplifying the K M model, we showed that a 3D model can exhibit spontaneous action potentials and that while the shape of the action potentials is unimportant, the relative time constants of the two subsystems are crucial.

Thesis/Dissertation

application/pdf

2009-05-28

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http://hdl.handle.net/2429/8438

Mathematics

University of British Columbia

UBCV

DSpace