Dynamical Systems in Neuroscience:

254 Excitability(a) integrator(b) resonatorK + activation gateinhibitionexcitation-60 -40 -20membrane potential, mVinhibitionexcitation-60 -40 -20membrane potential, mVFigure 7.30: Direction of excitatory and inhibitory input in integrators (a) and resonators(b).hyperpolarizing steps, it does not depend on the bifurcation mechanism of excitability,and it can occur in integrators or resonators.Some neurons can exhibit rebound spikes after short and relatively weak hyperpolarizingcurrents, as we illustrate in Fig. 7.29. The negative pulse deactivates afast low-threshold resonant current, e.g., K + current, which is partially activated atrest. Upon release from the hyperpolarization, there is a deficit of the outward currentand the net membrane current results in rebound depolarization and possibly a spike.Such a response occurs on the fast time scale and it does depend on the bifurcationmechanism of excitability.In Fig. 7.30 we show why integrators cannot fire rebound spikes to short stimulation,while resonators typically can. A brief excitatory pulse of current depolarizes themembrane and brings it closer to the threshold manifold, as in Fig. 7.30a. Consequently,an inhibitory pulse hyperpolarizes the membrane and increases the distance to thethreshold manifold. The dynamics of such a neuron is consistent with our intuitionthat excitation facilitates spiking and inhibition prevents it.Contrary to our intuition, inhibition can also facilitate spiking in resonator neuronsbecause the threshold set may wrap around the resting state, as in Fig. 7.30b. A sufficientlystrong inhibitory pulse can push the state of the neuron beyond the thresholdset thereby evoking a rebound action potential. If the inhibitory pulse is not strong, itstill can have an excitatory effect, since it brings the state of the system closer to thethreshold set. For example, it can enhance the effect of subsequent excitatory pulses,as we illustrate in Fig. 7.31. The excitatory pulse here is subthreshold if applied alone.However, it becomes superthreshold if preceded by an inhibitory pulse. The timing ofpulses is important here, as we discussed in Sect. 7.2.2. John Rinzel suggested to callthis phenomenon a post-inhibitory facilitation.