Myotonia Modeling presentation
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.



Neuronal Cable Theory on Graphs presentation
Abstract: Dendritic tree morphology is basically a finite, compact metric tree graph. We extend neuronal cable theory to certain types of graphs, and attack a number of forward and inverse problems. For inverse problems, given certain types of (biologically relevant) boundary measurements, we not only want to determine lengths and radii of branches, or conductances given the other parameters, but also to recover the tree morphology. One of the techniques we will employ here is the Boundary Control method, which has been effectively used for inverse problems in mathematical physics. For forward problems, we have formulated energy and comparison principle methods in order to examine nonlinear behavior, particularly threshold and conduction properties for use on graph domains. Along with these analytical approaches an important goal will be to develop effective numerical algorithms to recover (spatially distributed) parameters from graph boundary measurements and solution behavior on our graph domains. We have partial results on all subproblems discussed, and are actively pursuing further development of the ideas.



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.



Determining a Distributed Parameter for a Neuronal Cable Model on a Metric Tree Graph
Abstract: The goal in this talk is to discuss the inverse problem of recovering a single spatially distributed conductance parameter in a cable theory model (one-dimensional diffusion equation) defined on a finite tree graph. We employ a boundary control method that gives a unique reconstruction and an algorithmic approach. The motivation for this work is that dendrites of nerve cells are tree-like graphs, which have non-uniformly distributed physical parameters, one being channel conductance. It is also one of the first studies of the application of boundary control methods to inverse problems of parabolic problems on graphs, and one of the first uses of the method in this application area. This is collaborative work with Sergei Avdonin, U. Tennessee at Chattanooga.