Research interests: Mathematical Neurobiology

A major interest of mine is the development, and analysis, of mathematical models for describing electrical behavior in nerve cells. This has included, for example,

1. Analyzing simple models of closely spaced fibers in the study of ephaptic transmission of action potentials;

2. Various questions about propagation and threshold phenomena for models of myelinated axons;

3. Propagation behavior in models of dendrites with active spines;

4. Modeling sparse persistent sodium channels in certain fibers;

5. We have been involved in modeling studies of mechanical-to-electrical transduction (mechanoreception). Mechanoreception is a prominent feature of sensory systems, from hearing (hair cells in the cochlea), to sense of touch (somatosensory system). A particular interest was the modeling of the largest of skin receptors, the Pacinian corpuscle, which is an encapsulated nerve ending that senses high frequency mechanical stimuli;

6. I have also been interested in how non-uniform properties of cable models of dendrites can affect signal propagation behavior. This continues to be one of my projects, along with structure of waves and patterns in neural field theories. One recent project involved incorporating more geometric features in cable theory models;

7. More recent collaborations have involved inverse problems of estimating spatially distributed parameter values in cable models, given certain types of electrical measurements. For example, it is well documented that pyramidal cells in the hippocampus have lots of different ion conductances that are not uniformly distributed along the axon and dendritic branches. I have worked with collaborators to develop a numerical way of estimating any one of these conductances from voltage and current data using a recording electrode;

8. I also have explored necessary conditions needed by degenerate selective associative systems to function in certain ways.

9. I am starting to investigate forward and inverse problems associated with neuronal cable theory on networks (metric tree graphs). This is motivated by interest in recovering physical parameters (conductances, diameters, etc.) in dendritic trees. I am also interested in whether one can determine the graph morphology by knowing appropriate voltage measurements at the boundary vertices of the graph.

>From a mathematical standpoint, much of my work involves analyzing reaction-diffusion equations, and I have a general interest in all types of nonlinear diffusion equations.We also work with models of nonlinear differential and difference equations.

Some Randomly Selected Research Work:

1.Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph, Inverse Problems and Imaging, 9(3)(2015), 645-659 (with S. Avdonin).

2."Dynamical analysis of a model of skeletal muscles with myotonia or periodic paralysis", Nonlinear Studies, 1(2)(2010),1-20 (with Y. Lenbury and Kamonwan Kocharoen).

3."Degeneracy-driven dynamics of selective repertoires", Bull. Math. Biol. 71(6), 2009, 1349-1365 (with S. Atamas).

4. "Wave front solution and their shape for continuous neuronal networks with lateral inhibition", IMA J. Appl. Math., 71 (2006), 544-564 (with S. Ruktamatakul, Y. Lenbury).

5. "A distributed parameter identification problem in neuronal cable theory models", Math. Biosciences 194(1), 2005, 1-19 (with G. Craciun).

6. "Neuronal integrative analysis of the ~Qdumbell~R model for passive neurons", Integrative Neuroscience 1(2) 2002, 217-239 (with W. Krzyzanski, R.R. Poznanski).

7. "Theoretical analysis of the amplification of synaptic potentials by small clusters of persistent sodium channels in dendrites", Math. Biosciences 166, 2000, 123-147 (with R.R. Poznanski).

8. "Analysis of a model of membrane potential for a skin receptor", Math Biosciences 158, 1999, 1-45 (with W. Krzyzanski).