Institute for Mathematics and Its Applications
Philip Holmes, Princeton University
Chains of coupled oscillators of simple ``rotator'' type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere, where the details of neural architecture are yet unknown. One would like to include more than simple rhythmic oscillations in such models, without going to the full complexity of Hodgkin-Huxley type models.
I will describe a class of flows on tori which capture key features of excitable and oscillatory neural networks of CPGs. The model is motivated by experiments of A. Cohen et al. on lamprey CPG with brainstem attached, but may have wider applicability. The analysis involves pairs of coupled oscillators with both excitatory and inhibitory ``synaptic'' coupling, derived from a Fitzhugh-Nagumo type reduction of the Hodgkin-Huxley equations, and includes bifurcations describing locomotion onset and cessation. I will also discuss traveling wave patterns arising from chains of oscillators, including simulations of ``body shapes'' generated by a double chain providing input to a kinematic musculature model.
The talk will be based on joint work with D.A. Taylor (Iomega Corporation) and A.H. Cohen (University of Maryland).