Abstract: Medical research has indicated that abnormalities of skeletal muscles, myotonia and periodic paralysis, are caused by alteration in the voltage-gated sodium channels. This assumption led to studies of channel behavior based on the dynamics of membrane potentials. Cannon, et al developed a two-compartment Hodgkin-Huxley type model that had a reformulation of the sodium current term and did some simulations to compare with experiment. Here we discuss a geometric perturbation analysis on the model system, reducing it from an eight-order system to a third-order system. The conditions on the system parameters under which the model exhibits dynamic behavior that resembles clinical observations are derived. We are able to detect slow-fast limit cycles which generate bursts of action potentials characteristic of the clinical case where active and non-active phases are observed to alternate in a pulsatile fashion, such as that in patients with Hyperkalemic periodic paralysis. Relying on the observation that the state variables possess drastically diversified dynamics, we explain the differences between the action potential dynamics of a normal subject and those of myotonia or periodic paralysis cases. The model seems to display mixed-mode oscillations that need further analysis.

Comments Concerning Models of Myelinated Fibers

Abstract: I will introduce three relatively simple models for myelinated neural fibers, and discuss what has, and has not, been done on developing and analyzing traveling wave solutions to such problems. Such solutions must satisfy nonlinear functional differential equations with both forward and backward delays that must be determined along with the wave solution.

Introduction to the Boundary Control Method and its Application to Inverse Problems

Abstract: The boundary control method is an approach to inverse problems based on the relationship between control and systems theory. I will first give some motivation for studying certain inverse problems, then reduce the problem to a "simple" case. Then I will develop aspects of the boundary control method in a way that leads to an algorithmic approach for estimating a certain distributed parameter. I will wrap up with comments about other problems I am, or would like to attack. This project is joint with S. Avdonin, U. Alaska, Fairbanks.

Neuronal Cable Theory on Dendritic Trees

Abstract: We are interested in the qualitative behavior of diffusion problems on metric tree graphs. In this talk we extend neuronal cable theory to tree graphs that represent (idealized) dendritic trees, and discuss analytical results concerning threshold conditions, traveling wave solutions, bounds on conduction speed, and conduction block. As time permits we will mention work on an (inverse) problem in linear cable theory on tree graphs of recovering a parameter, namely the conductance on each branch.

Persistence and Competition: A Review of these Ideas in various Environments

Abstract: In this presentation I will start with models of a single population, concentrating on historic models in a non-spatial setting. Next I move to addressing population dynamics when there is mobility via a diffusion mechanism. After presenting some solution behavior, we move on to an advection-driven setting, like a simple creek environment, then a branched environment (river network). After this I return to basics of adding a second, competitive, species, first discussing the competitive exclusion principle in a single compartment setting, then discussing how the picture changes in the presence of diffusion and advection. I will finish with presenting some problems worth pursuing. The presentation is designed to be reasonably accessible to students with some differential equations background, but should raise some interesting, but unresolved questions in dynamics of populations.