Date of Award

Summer 2008

Document Type


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Farzan Nadim

Second Advisor

Amitabha Koshal Bose

Third Advisor

Jorge P. Golowasch

Fourth Advisor

Victor Victorovich Matveev

Fifth Advisor

Andrew Hill


Bursting oscillations are prevalent in neurons of central pattern generators (CPGs) that produce rhythmic motor activity, and the activity phase plays an important role in determining the normal or dysfunctional network output. The activity phase is the delay time-with respect to some reference time in each cycle and normalized by the oscillation cycle period-of the onset of action potentials by a neuron. This dissertation investigates how the A-current, in conjunction with other intrinsic properties, sets the activity phase of a neuron driven by inhibition.

This dissertation is divided into two major components. In the first component, methods of dynamical systems are applied to explore the transient properties of the activity phase of the follower neuron, which is modeled using a simplified three-variable model based on the Morris-Lecar equations. Based on the analysis of the effect of the A-current in determining the phase, recursive equations are derived to calculate the activity phase of the follower neuron, following a single inhibitory input as well as its steady state phase in response to a rhythmic input. The modeling findings are compared with experimental data from follower PY neurons in the pyloric CPG of the crab C. borealis. In these experiments, the Dynamic Clamp technique is used to produce artificial intrinsic and synaptic currents in the follower PY neurons. It is found that the activity phase can be determined by the period and duty cycle of the pacemaker, and the recursive equations provide faithful predictions of the activity phase when the cycle period of the pacemaker is varied under different protocols.

In the second component of the dissertation, a five-compartment model is built based on the morphology of the PY neuron to produce a realistic representation of the biological PY neurons in order to investigate how the distribution of the A-current affects the activity phase. This model involves a set of 53 coupled nonlinear ordinary differential equations which are numerically integrated using a 4th order Runge-Kutta method. A Genetic Algorithm is applied to recursively optimize the possible parameters for all the intrinsic currents in each compartment. The results show that different distributions of the A-current lead to different bursting behaviors even if the total A-current conductance is kept constant.

These results show that the activity phase of the follower neurons can be affected significantly by the strength and the distribution of the A-current, together with other intrinsic and synaptic properties. The activity phase can be predicted by the results of a low-dimensional model, and the possible distribution of the intrinsic currents can be computed by developing more realistic models based on the shape of biological neurons.

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Mathematics Commons