Dynamical systems theory has been successful in describing pattern formation throughout chemistry and physics, but in biology, dynamics theory has been frequently overlooked. Epileptic seizures result from an enormous variety of causes. Nevertheless in EEGs they exhibit a small set of universal patterns. These patterns are ideally suited to be studied using a dynamical framework. Such a study has never been done. For this project, theoretical and numerical methods will lead to a dynamical characterization of patterns in EEG data for seizures. The work will give rise to testable hypotheses to determine the relevant system parameters for seizure dynamics. The mathematical framework is a system of integro-differential equations with nonlocal coupling terms. These equations model the interplay of excitatory and inhibitory neuronal ensembles. The nonlocal convolution terms add mathematical complexity, but are crucial for modeling long distance synaptic coupling in neuronal ensembles. In some cases seizure behavior can be modeled by stationary patterns, such as homogeneous neuronal activation (generalized status epilepticus), a single spatially localized stationary pulse (focal status epilepticus), and metastable multipeak solutions (potentially pre-seizure states). Most often, however, seizures display transient behavior, which is modeled as a series of transitions between stationary patterns. These patterns can be described in the following three stages (a seizure may or may not go through all stages): The transition of the neuronal ensemble from rest to a localized pulse (a focal seizure), the propagation from a localized pulse to a homogeneous activated state (a secondary generalization), and the return to the rest state. The transitional states in the model result from transient changes in the following system parameters: excitatory coupling strength, inhibitory coupling strength, excitatory and inhibitory footprints, neuronal firing threshold, and the inhibitory delay. In order to verify the effect of time variation of these parameters, it will be necessary to develop new theoretical techniques to study nonlocal reaction-diffusion systems with time varying parameters. The PI is a mathematician with a background in nonlinear dynamics. She has an established collaboration with the coinvestigators, who are the director and two members of a neural dynamics group. A small grant will allow her to develop these techniques for future neuroscience research. ? ?
