Dynamical Systems in Neuroscience:

Excitability 25310 ms10 mV-45 mV0 pA-100 pAFigure 7.29: Rebound spikes in responseto a brief hyperpolarizing pulsein a brainstem mesV neuron having fastsubthreshold oscillations of membranepotential.to I = 0 (white square in the figure) to a new location (black circle). Whether theneuron fires or not depends on the location of the old equilibrium relative to the stablemanifold to the saddle, which plays the role of the new threshold. In case A theneuron does not fire, in case B it fires even though the resting state is still stable. Theneuronal rheobase is the amplitude of the current I that puts the threshold exactlyon the location of the old equilibrium. Such a value of I always exists, and it oftencorresponds to the saddle-node bifurcation value. Notice that the rheobase currentresults in a spike with infinite latency, at least theoretically.A resonator neuron may not have well-defined rheobase simply because it may nothave well-defined threshold. Indeed, the dotted line in Fig. 7.28b may correspond toa subthreshold or superthreshold response depending on where it is in the thresholdset. Stimulating such a neuron with “rheobase” current produces spikes with finitelatencies but partial amplitudes. A bistable resonator (near subcritical Andronov-Hopfbifurcation) may have a well-defined rheobase because it has a well-defined threshold— the small-amplitude unstable limit cycle.7.2.7 Post-inhibitory spikeProlonged injection of a hyperpolarizing current and then sudden release from hyperpolarizationcan produce a rebound post-inhibitory response in many neurons. Thehyperpolarizing current is often called anodal current, release from the hyperpolarizationis called anodal break, so rebound spiking is called anodal break excitation(FitzHugh 1976). Notice that firing of a neuron follows a sudden increase of injectedcurrent, whether it is a positive step or release from a negative step.Often, post-inhibitory responses are caused by the “hyperpolarization-activated” h-current, which slowly builds up and upon termination of the hyperpolarization drivesthe membrane potential over the threshold manifold (or threshold set). Alternatively,the rebound response can be caused by slow de-inactivation of Na + or Ca 2+ currents,or slow de-activation of a K + current that is partially activated at rest and preventsfiring. In any case, such a rebound response relies on slow currents and long or strong