Document Type


Date of Award

Fall 1-31-2014

Degree Name

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


Mathematical Sciences

First Advisor

Farzan Nadim

Second Advisor

Jorge P. Golowasch

Third Advisor

Victor Victorovich Matveev

Fourth Advisor

Amitabha Koshal Bose

Fifth Advisor

Dirk Bucher


Conduction delay in an axon is the time required for an action potential to propagate between two positions. It is a function of the axon’s passive membrane properties, voltage-gated ion channels and the Na+/K+ pump, and can be substantially affected by neuromodulators. The conduction delay of action potential, generated by the pyloric dilator (PD) neuron unmyel i nated motor axon in the stomatogastric nervous system, shows significant variability with ongoing bursting or Poisson stimulation. When the axon is stimulated, the mean value (Dmean) and coefficient variation of conduction delay (CV-D) slowly increase with time (slow timescale effect), and the relationship between delay and instantaneous stimulus frequency (Fi nst) is non-monotonic (fast timescale effect). This dissertation investigates how the history-dependence of conduction delay is generated and the contributions of different ionic currents to conduction delay.

This dissertation is comprised of three parts. In the first part, we build a biophysical model that includes several characterized ionic currents and the Na+/K+ pump in order to unmask the mechanisms underlying the history dependence of conduction delay. This model captures both the slow and fast timescale effects of conduction delay obtained from the realistic burst stimulation and Poisson stimulation at different mean frequencies. Additionally, the effects of a neuromodulator (dopamine) and a channel blocker (CsCl) on the history-dependence of conduction delay were also accurately captured by the biophysical model. Specifically, the Na+/K+ pump plays a critical role in the slow increase of Dmean and CV-D. At the fast timescale, the non-monotonic relationship between conduction delay and Finst is captured by the dynamical properties of INa. Furthermore, we systematically investigated the contributions of different ionic currents on conduction delay and spike shape parameters (i.e., duration, trough and peak voltages) with realistic burst stimulation protocols. Specifically, we found that only INa substantially affects the variability of conduction delay.

Based on this observation, in the second part of the dissertation, we intended to use the dynamical parameters of INa to build an equation to accurately predict the variability of conduction delay. We found that conduction delay is mostly determined by the opening rate of the Na+ activation variable prior to the action potential (αm(VT)), and the closing rate of its inactivation variable at the peak (flh(VP)). Consequently, we developed an empirical equation for conduction delay in our model using multivariate linear regression of the Poisson stimulation data. The resulting equation accurately predicted the history-dependence of conduction delay on novel data. In our model data both αm and βh are almost linear functions of their respective voltage variables (VT and VP) in the voltage ranges observed. We, therefore, simplified our empirical equation and the new equation can also accurately predict the history dependence of conduction delayin the model. More importantly, it provides accurate predictions of conduction delay from experimental measurements of action potential voltage trajectories in the motor axon without need of computational modeling.

In the third and final part of the dissertation, I will develop a decoding technique to investigate the functional relationship between conduction delay and the history activity in the PD axon. Using biological data obtained from representative experiments of the PD axon with Poisson stimulation, all the parameters in the decoding technique are determined after a routine optimization process. With these optimized parameters, the decoding model can accurately predict the conduction delay only from the stimulus time. A similar technique is developed and applied to explore and predict the voltage facilitation exposed by the cpv2-a muscle.

These results show that conduction delay is affected by the short- and long-term history activity in the PD axon. The conductance-based biophysical model, the empirical equations and the decoding technique, which were developed in this dissertation, provide quantitative tools to explore the mechanisms of history-dependence of conduction delay, and predict conduction delay both in the model results and in the experimental measurements.

Included in

Mathematics Commons



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