Quasi-discrete dynamics of a neural net: The lighthouse model

This paper studies the features of a net of pulse-coupled model neurons, taking into account the dynamics of dendrites and axons. The axonal pulses are modelled by delta-functions. In the case of small damping of dendritic currents, the model can be treated exactly and explicitly, Because of the delta-functions, the phase-equations can be converted into algebraic equations at discrete times. We first exemplify our procedure by two neurons, and then present the results for N neurons. We admit a general dependence of input and coupling strengths on the neuronal indices. In detail, the results are (1) exact solution of the phase-locked state; (2) stability of phase-locked state with respect to perturbations, such as phase jumps and random fluctuations, the correlation functions of the phases are calculated; (3) phase shifts due to spontaneous opening of vesicles or due to failure of opening; (4) effect of different sensory inputs on axonal pulse frequencies of coupled neurons.