ABSTRACT
ABSTRACT. A deterministic nonlinear dynamic systems approach to mathematical analysis of the integrate-and-fire (I & F) model neuron*, a monoionic simplification of the Hodgkin-Huxley model for action potential generation in the excitable biological membrane, is presented. It is shown that under periodic activation (driving signal), the firing behavior is described by an iterative map of the interval [0, 21t] onto itself which we call phase-transition map (PTM). Periodic activation of the I & F model neuron occurs when its dendrites receive correlated incident patterns percipitated by certain sensory stimulus or when a network of such dendritic patterns enters a synchronized (phase-locked) state. Like other maps of the interval onto itself, the PfM is studied employing the tools of nonlinear dynamics. This furnishes a novel way of viewing the microneurodynamics of neural networks and shows, that despite the simplifications made in the I & F model neuron *, it exhibits, in its spiking behavior, a high degree of functional complexity approaching that of the living neuron. This is manifested by a variety of firing modalities depending on parameters of the periodic activation, which include regular firing phase-locked to the periodic activation or a subharmonic of it, quasi-periodic firing, bursting, and possibly erratic firing, and can bifurcate (rapidly switch) between these firing modalities as the parameters of the periodic activation are altered, hence the name bifurcation model. Illustrative examples of this complex behavior are given in the form of bifurcation diagram, Arnold Tongues diagram and the Devil's Staircase diagram. These show the neuron is able to detect coherent episodes in its incident spike wavefront, the aggregate of spike trains incident on its synaptic inputs, and encodes such coherent activity whenever it occurs, in a complex manner depending on the parameters of the periodic activation potential produced by dendritic-tree processing of the incident wavefront. When the activation potential is not periodic, the bifurcation model reverts to the usual sigmoidal response and shows an upper limit on firing frequency which serves the useful function of containing the maximum firing activity in a network of such neurons in a manner analogous, but not exactly equivalent, to a similar limit imposed by refractoriness in the living neuron.
