Stochastic Phenomena, Spike Time Patterns, and Control in Neural Dynamics
Ion channel fluctuations, irregular synaptic barrages and other sources of ``noise'' limit the precision and reliability with which nerve cells produce action potentials. But highly precise and reliable patterns of spike times have been observed experimentally both in vitro and in vivo. What is the origin and functional significance of precise temporal patterns in the ``neural code''? Problems of current interest include (1) Relation of noise spectrum and intensity and input shape and amplitude to spike time precision in single cell models (integrate-and-fire, conductance based models). (2) Genericity of spike time convergence in simple deterministic neural oscillator models. (3) Construction of biophysical models for single cell and network activity including the effects of noise in comparison with experimental data from the sea hare Aplysia (collaboration with the Chiel laboratory). (4) Development of novel statistical techniques for identifying patterns in multiunit recordings in both Aplysia and in mammalian hippocampus.
Gradient Sensing, Signal Transduction, and Information Theory
Signal transduction networks are the biochemical systems by which living cells sense their environments, make and act on decisions -- all without the benefit of a nervous system. How do cells use networks of chemical reactions to process information? We are combining mathematical ideas from the theory of stochastic point processes and Brownian motion with information theory to develop a framework for understanding information processing in biochemical systems. As a case study we are studying gradient sensing in neutrophils (white blood cells) and the social amoebae (Dictyostelium) from the points of view of information theory and statistical signal detection theory. Projects range from highly theoretical (devising information measures for time varying continuous time Markov processes) to highly computational (simulation of gradient sensing networks using explicit Monte Carlo techniques such as MCell. Gradient sensing work is being pursued in collaboration with the Baskaran laboratory.
Pattern Formation in the Visual Cortex
The pathway from the eyes to the visual cortex organizes spontaneously during development using a combination of intrinsic chemical markers and correlation-based, activity-dependent (``Hebbian") mechanisms. The resulting cortical architecture shows fascinating quasiregular patterns with elements including pinwheel and other phase singularity lattices in the cortical maps representing orientation, ocular dominance, retinotopic position and other features of the visual world. Using methods from equivariant bifurcation theory -- the study of branching solutions in the presence of symmetry -- an elegant theory has been developed that accounts for many aspects of the structure of cortical maps. The same mathematical structure underlies the forms of geometric visual hallucinations reported by subjects experiencing sensory deprivation or treatment with mescal, cannabis and other hallucinogens.
Bioinformatics, Data Mining, and Malaria
Molecular diagnostic technology pioneered by collaborator P. Zimmerman in CWRU's Center for Global Health and Diseases promise faster, more accurate means for tracking and combatting the spread of drug resistance in endemic malaria populations worldwide. In order to achieve their potential the new methods require novel analysis tools. Mathematical ideas as simple as coordinate transformation and histogram segmentation have proven effective in boosting the accuracy of molecular genotyping techniques for discriminating drug-sensitive from drug-resistant infections.
Source: http://www.case.edu/math/thomas/
