Effect of duration of synaptic activity on spike rate of a Hodgkin-Huxley neuron with delayed feedback

A recurrent loop consisting of a single Hodgkin-Huxley neuron influenced by a chemical excitatory delayed synaptic feedback is considered. We show that the behavior of the system depends on the duration of the activity of the synapse, which is determined by the activation and deactivation time constants of the synapse. For the fast synapses, those for which the effect of the synaptic activity is small compared to the period of firing, depending on the delay time, spiking with single and multiple interspike intervals is possible and the average firing rate can be smaller or larger than that of the open loop neuron. For slow synapses for which the synaptic time constants are of order of the period of the firing, the self-excitation increases the firing rate for all values of the delay time. We also show that for a chain consisting of few similar oscillators, if the synapses are chosen from different time constants, the system will follow the dynamics imposed by the fastest element, which is the oscillator that receives excitations via a slow synapse. The generalization of the results to other types of relaxation oscillators is discussed and the results are compared to those of the loops with inhibitory synapses as well as with gap junctions.