Mathematics Colloquium - Remus Osan
Traveling waves of electrical activity in neural tissue are often modeled using large-scale neural networks comprised of simplified neuron models such as integrate and fire. Because synaptic connections among neurons can reach both near and distant neighbors, excitation from one neuron influences the dynamics of many other neurons. Consequently, the exact dynamics of these networks are potentially extremely complex. We show that in the case of two widely used choices of neural connectivity, namely the finite support and the exponential function, the global network dynamics for the traveling waves are instead described by local evolution equations. This simplified description yields valuable insights into the existence and stability of these traveling waves, and it allows one to compute exact solutions for the exponential case. Finally, we outline some ideas and future directions for how these results extend to higher dimensions and more complicated connectivity functions.