The Morris-Lecar equations with delay

In this thesis we study the Morris-Lecar equations with delay. We modify the original Morris-Lecar ODE model to simulate delayed recurrent (inhibitory and excitatory) feedback in a appropriate neuro-muscular feedback loop. The loop consists of two neurons and a muscle fibre, whereby a neuron excites a muscle fibre, which influences a second neuron which in turn excites or inhibits the first neuron. The effects of feedback are described in terms of the voltage across the membrane of the muscle fibre. The model is formulated in terms of delay differential equations with a single discrete time delay to account for impulse conduction and synaptic delays, and assumes the muscle fibre possesses two noninactivating conductances, one for Ca2+ and one for K+. Each single-ion conductance system with delay is studied before analysing the model with both conductances operational. A dynamical systems approach is used to analyse the bifurcation structure of the systems as the delay parameters (mu and tau) are varied, using both analytical and numerical techniques. The systems with delay are more complex than their non-delayed counterparts, and exhibit stability switching of equilibria, Hopf bifurcation leading to stable soft and hard oscillations, and multistability. The direction and stability of Hopf bifurcations are determined by applying the normal form theory and the centre manifold theorem. Biophysical interpretation of results are also included. Applicability and limitations of the model are discussed, along with suggestions for future research.