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George L. Wilcox, Fellow

Axonal Propagation of Impulses in Peripheral Afferents: Models Based on Molecular Information

This project is a long-term multidisciplinary endeavor involving the fields of molecular and cellular biology, computer science, pharmacology, and mathematics. Biophysically detailed models of impulse propagation in neurons have been studied with mathematical models since the pioneering work of Hodgkin and Huxley in 1952. The equations they formulated were a system of four partly dissipative semilinear parabolic partial differential equations with Neumann boundary conditions. Over the years, more complexity has been added to the basic model, and as a result, typical systems today may comprise 20 to 30 coupled nonlinear differential equations. Such equations have been studied by biologists, mathematicians, and computer scientists for many decades. Due to the nonlinear nature of the equations, there is very little hope of obtaining analytical solutions. The only option available is to compute numerical solutions.

In the recent past, widespread use of the techniques of molecular biology has made it possible to sequence and clone the genes responsible for the proteins that cause cellular excitability such as cell membrane ion channels and ion pumps. In addition, by expressing genes that code for ion channels in cells such as the Xenopus laevis oocyte (as has been done in the past by the Professor Wilcox), it has become possible to study the detailed kinetics of these channels and dissect the components of cellular excitability at a molecular level. In addition, the result of mutations in the genetic code can also be studied, thus allowing the investigation of harmful genetic mutations. Finally, it is also possible to construct mathematical models of the action of antiarrhythmic drugs at the molecular, cellular, and tissue levels.

In this work, these researchers are simulating electrical activity of two types of cells-myelinated A-d sensory afferent neurons and unmyelinated C-type afferent neurons responsible for transmission of pain. Collaborators in the University of Leicester have constructed cDNA libraries for such cells and have started constructing databases of ion channels with information about kinetics and pharmacology. With this molecular information, these researchers are in a position to construct models of propagation of electrical activity in single neurons (cell soma, axon, and nerve terminals). In addition to numerous questions about normal physiology of these systems, the effects of mutations and drugs can be effectively studied.

Currently, these researcher's codes on the IBM SP are implemented in the C++ language. They are taking full advantage of the optimization tools in the new xlC compiler. The KAI C++ compiler is currently being beta tested to parallelize the codes. Parallelization efficacy is being tested by running the codes by varying the number of CPUs used and comparing run times.

Current runs are being made with both a variable-step, variable-order implicit method as well as a fourth order explicit time-integrator. With these methods, typical run times are projected to be eight hours on a single CPI for each case. With approximately thirty parameters, this implies that in order to study parameter sensitivities (e.g., doubling and halving parameter values are anticipated one at a time), roughly six hundred cases need to be run in order to understand parameter sensitivities. The independence of these evaluations indicates that explicit parallelization (running multiple jobs on multiple processors) are possible during these sensitivity analyses. In addition, by adding better formulations of potassium channels and additional temperature-, ligand-, and pH-sensitive cation channels, an additional thirty parameters need to be tested. Simulations of the effects of ion channel blocking drugs and interactions with intrinsic nerve properties introduce a further ninety parameters. Finally, a model of neuropathic pain introduces another sixty parameters.

Current runtimes on the SGI Origin 2000 are approximately sixteen hours on a single CPU. While the code did run in parallel, changes to the compilers have forced crucial parts of the code to run in serial mode with significant degradation in performance. With the imminent release of an SGI C++ compiler with OpenMP directives, the work will be free of the above problems and performance of the codes should improve considerably.

This information is available in alternative formats upon request by individuals with disabilities. Please send email to alt-format@msi.umn.edu or call 612-624-0528.

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