6 research outputs found
Analysis of electrophysiological models of spontaneous secondary spiking and triggered activity
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.Science, Faculty ofMathematics, Department ofGraduat