Relationship between threshold and bifurcations for paradoxical firing responses along with seizure induced by inhibitory stimulation

Abstract Inhibition-induced enhancement or paradoxical response of firing was related to Hopf bifurcation instead of saddle-node bifurcation on an invariant cycle (SNIC), due to the “negative” threshold and rotated vector fields. In the present letter, by changing multiple parameters to adjust the vector fields, the condition for the paradoxical response and “negative” threshold is extended to SNIC near a codimension-2 bifurcation appearing prior to the Hopf bifurcation, which presents a comprehensive relationship between bifurcations and threshold. Especially, the result for a special current can well explain the enhanced firing along with seizure induced by inhibitory interneuron, implying that SNIC far from the codimension-2 bifurcation of pyramidal neuron is a potential candidate to avoid seizure.