Characterizing the Neural Response to Electric Shocks Using Discrete-Time Delay Maps

...And How Advanced Features of Mathematica Can Improve the Accuracy of the Maps and the Speed With Which They Are Computed.

David O'Gorman

Research Associate
Department of Biomedical Engineering and the Hearing Research Center

Abstract

In cases of profound hearing loss, auditory sensation can be restored by the cochlear implant neural prosthesis. This implanted prosthesis excites the otherwise non-responsive auditory nerve fibers with electrical current pulses. These pulses of electrical current activate voltage-gated ion channels embedded in lipid membrane of the nerve fiber, which, in turn, trigger the generation of action potentials—brief electrochemical impulses—which then propagate to the brain stem. This sequence of pulse-triggered action potentials encodes all the information about sound that is available to the brain. Systematically optimizing cochlear implants, then, depends upon understanding the relationship between the stimulating current pulses and the evoked sequence of action potentials, and yet achieving this understanding is difficult because the differential equations that describe action potential generation are multi-dimensional and highly nonlinear [Hodgkin and Huxley, 1952]. Phase-space analysis, however, suggests a simplification. Specifically, motion of the system state through phase-space can be decomposed into two components, a very fast component in which a transient attractor is approached and a slower component which tracks the transient attractor toward a fixed point [Rabinovitch et al., 1994, O'Gorman et al., in preparation]. To a very good approximation, the effect of a pulse can be summarized by the delay it introduces into the system's evolution along the transient attractor. Thus, the neural response to a sequence of pulses can be accurately summarized by a delay map which specifies the pulse-induced delay as a function of the pre-pulse state. In this talk, I will give an overview of the biophysics of pulse-triggered action potential generation, describe the delay-map simplification, and show how advanced features of Mathematica—namely, the event-handling framework in NDSolve and the Nearest function—can be harnessed to increase the accuracy of delay maps, as well as the speed with which they are computed.

Acknowledgments:

The map generation algorithm was developed in conjunction with Andrew Moylan of Wolfram Research at the Advanced Mathematica Summer School 2009.  I would also like to acknowledge Roger Germundsson of Wolfram Research for helpful suggestions.

References:

Hodgkin, A. L., and Huxley, A. F. (1952). "A quantitative description of membrane current and its application to conduction and excitation in nerve," Journal of Physiology 117, 500-544.

Rabinovitch, A., Thieberger, R., and Friedman, M. (1994). "Forced Bonhoeffer - van der Pol oscillator in its excited mode," Physical Review E 50, 1572-1578.