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).