Map Based Models in Neurodynamics

This tutorial reviews a new important class of mathematical phenomenological models of neural activity generated by iterative dynamical systems: the so-called map-based systems. We focus on 1-D and 2-D maps for the replication of many features of the neural activity of a single neuron. It was shown that such systems can reproduce the basic activity modes such as spiking, bursting, chaotic spiking-bursting, subthreshold oscillations, tonic and phasic spiking, normal excitability, etc. of the real biological neurons. We emphasize on the representation of chaotic spiking-bursting oscillations by chaotic attractors of 2-D models. We also explain the dynamical mechanism of formation of such attractors and transition from one mode to another. We briefly present some synchronization mehanisms of chaotic spiking-bursting activity for two coupled neurons described by 1-D maps.