Dynamical Systems in Neuroscience:

Excitability 263membrane potential, V (mV)0-20-40-60-800.050.040.030.02I=-100.01I=-10 I=4.30 10 20 30 40 50time (ms)K + activation gate, n0V-nullclineI=4.3-65 -60 -55 -50membrane potential, V (mV)Figure 7.40: Post-inhibitory spike of an integrator neuron near Bogdanov-Takens bifurcation;see Fig. 7.39.n-nullclineresonatorintegratorspikespike1 2focus1node212Figure 7.41: Post-inhibitory facilitation — enhancement of subthreshold depolarizingpulse (2) by preceding inhibitory pulse (1) — can occur in integrator neurons nearBogdanov-Takens bifurcation.the K + current to n < 0.005 pushes the point (V, n) to the shaded region, i.e., beyondthe threshold. Upon release from inhibition, the integrator neuron produces a reboundspike and then returns to the resting state.Integrator neurons can also exhibit frequency preference and resonance, illustratedin Fig. 7.41. The post-inhibitory facilitation in resonator neurons in Fig. 7.41a wasdescribed in Sect. 7.2.7. It may happen in integrator neurons when the node equilibriumhas nearly equal eigenvalues and nearly parallel eigenvectors, as in Fig. 7.41b. Theformer are about to become complex-conjugate resulting in rotation of the vector fieldaround the equilibrium, and hence in the post-inhibitory rebound response to the first(inhibitory) pulse.Resonator neurons near Bogdanov-Takens bifurcation can fire spikes with noticeablelatencies. This occurs because the V -nullcline follows the n-nullcline at the focusequilibrium in Fig. 7.39, bottom. Such a proximity creates a “tunnel” with smallvector field that slows down the spiking trajectory. Finally, the neuron can exhibitan oscillation (marked as P in Fig. 7.42, bottom) before firing a spike in response to