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.

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.

Extending dynamics to graph domains: two examples from biology

Abstract: This presentation introduces two different scenarios where it is appropriate to consider partial differential equations on (quantum) tree graph domains. The first example concerns threshold and conduction properties from neuronal cable theory on a nerve's dendritic tree. The second example concerns species persistence in a river network. A variety of unanswered questions will also be mentioned.

Atherosclerotic plaque development: strategies for modeling the growth and degradation of the fibrous cap

Abstract: Cardiovascular disease is a leading cause of death in the US and many developed countries. Atherosclerosis is a major contributor to this disease profile. Atherosclerosis is an inflammatory disease of major arteries due to fatty lesions forming in arterial walls, causing stenosis (contracting blood flow) and thrombosis (blood clots, blockage). Certain lesions, called vulnerable plaques, are responsible for most deaths from atherosclerosis. The growth and degradation of these plaques is very dynamic, involving complex biochemical, hemodynamic, and mechanical interactions. But the present experimental means for studying arterial plaque development is limited, calling for augmenting such studies by mathematical modeling, analysis, and simulation. In this talk I will give a background to the biology and outline a strategy for model development, starting with an ODE model of principle chemical and cellular processes, and progressing to more complicated PDE models that include more mechanisms. At this stage little is proved, so the talk must be viewed as a possible roadmap for approaching a variety of questions.

Persistence and Competition in River Networks

Abstract: Starting with work of Speirs, Gurney, Carlson, and others, we review some work done on persistence in population models in advection-driven environments, which includes river networks as metric tree graphs. Then we discuss the competition model work of Vasilyeva-Lutscher and extensions to a tree graph. Of particular interest here is the nature of the competitive exclusion principle in these environments. We'll also mention further questions to be explored.

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.