  This work explores four nonlinear classical models of neural oscillators, the
Hodgkin-Huxley model, the Fitzhugh-Nagumo model, the Morris-Lecar model, and
the Hindmarsh-Rose model. Nonlinear contraction theory is used to develop
observers and perform synchronization analysis on these systems. Neural
oscillation and signaling models are based on the biological function of the
neuron, with behavior mediated through the channeling of ions across the cell
membrane. The variable assumed to be measured is the membrane potential, which
may be obtained empirically through the use of a neuronal force-clamp system,
or may be transmitted through the axon to other neurons. All other variables
are estimated by using partial state or full state observers. Basic observer
rate convergence analysis is performed for the Fitzhugh Nagumo system, partial
state observer design is performed for the Morris-Lecar system, and basic
synchronization analysis is performed for both the Fitzhugh-Nagumo and the
Hodgkin-Huxley systems.
