Bifurcations and enhancement of neuronal firing induced by negative feedback

Bifurcations and enhancement of neuronal firing induced by negative feedback from an inhibitory autapse are investigated. A slow inhibitory autapse model that can exhibit dynamics similar to the inhibitory autapse of a rat interneuron is considered. Compared with a Morris-Lecar (ML) model without an autapse, enlargement of the parameter region of firing activities in the ML model with a slow inhibitory autapse can be identified with 1-parameter and 2-parameter bifurcations and is induced by a shift of an inverse Hopf bifurcation point. The right shift of the inverse Hopf bifurcation point from firing to resting state with a high membrane potential level causes the resting state in the ML model without an autapse to change to firing in the ML model with an autapse. This shows that the autapse can enhance neuronal firing and can be well interpreted by the dynamic responses of the resting state to inhibitory impulse current. In addition, many complex dynamics such as coexisting behaviors and codimension-2 bifurcations are also induced, and the relationship between the inverse Hopf bifurcation point and a physiological concept, depolarization block, a phenomenon in which a neuron enters from firing to resting state when it receives excessive excitatory or depolarizing current, is discussed. The results provide a novel viewpoint that the inhibitory autapse enhances rather than suppresses neuronal firing near the inverse Hopf bifurcation point, which is an important example showing that negative feedback can play a positive role in nonlinear dynamics.