Abstract

We examine the effects of paired delayed excitatory and inhibitory feedback on a single integrate--and--fire neuron with reversal potentials embedded within a feedback network. These effects are studied using bifurcation theory and numerical analysis. The feedback occurs through modulation of the excitatory and inhibitory conductances by the previous firing history of the neuron; as a consequence, the feedback also modifies the membrane time constant. Such paired feedback is ubiquitous in the nervous system. We assume that the feedback dynamics are slower than the membrane time constant, which leads to a rate model formulation. Our paper provides an extensive analysis of the possible dynamical behaviors of such simple yet realistic neural loops as a function of the balance between positive and negative feedback, with and without noise, and offers insight into the potential behaviors such loops can exhibit in response to time-varying external inputs. With excitatory feedback, the system can be quiescent, periodically firing, or exhibit bistability between these two states. With inhibitory feedback, quiescence, oscillatory firing rates and bistability between constant and oscillatory firing rate solutions are possible. The general case of paired feedback exhibits a blend of the behaviors seen in the aforementioned extreme cases, and can produce chaotic firing. We further derive a condition for a dynamically balanced paired feedback, in which there is neither bistability nor oscillations. We also show how a biophysically plausible smoothing of the firing function by noise can modify the existence and stability of fixed points and oscillations of the system. + Corresponding author.

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