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Current Topics Workshop: Nonlocal Integro-Differential Equations in Mathematics and Biology (March 6-8, 2003)

(Cosponsored by the Mathematical Research Institute, OSU, Department of Mathematics)

Organizers: Bjorn Sandstede and David Terman

The goal of the workshop is to articulate the present and future trends in:

1. Modeling the neuronal networks
2. Analysis of related mathematical models, and to discuss
3. General mathematical techniques for integro-differential equations.

The nonlocal interaction of neurons plays a crucial role in the generation of waves and patterns in the brain. Each neuron may send excitatory or inhibitory synaptic input to other neurons, and the intrinsic and synaptic dynamics may involve multiple time scales. The spatio-temporal properties of patterns, such as whether neurons fire in synchrony or not, typically depend on the type and strength of the neuronal interactions.

The mathematical models that incorporate these effects are often integro-differential equations. These models account for the spatial interactions via convolution integrals whose kernels encode the specific interaction properties of neurons. Not much analytical work has been done for equations of this type, and special properties such as maximum principles or singular perturbation theory need to be exploited.

This workshop focuses on surveying the mathematical techniques used to model and analyze the inherently nonlocal interaction of neurons and on concrete applications in neuroscience.