Dynamical Systems in Neuroscience:

252 Excitability(a) integrator(b) resonator?new restingstatethresholdspikethresholdsetnew restingstatePSPspikeABold restingstate, I=0old restingstate, I=0Figure 7.28: Integrators have well-defined rheobase current while resonators may not.sponding to partial-amplitude action potentials, such as those in Fig. 7.26. No singlecurve separates small potentials from action potentials, so there is no well-definedthreshold manifold.FitzHugh (1955) noticed that the threshold set can be quite thin in some models,including the Hodgkin-Huxley model. In particular, the difference between the trajectoriescorresponding to small potentials and action potentials can be as small as 0.0001mV, which is smaller than the noisy fluctuations of the membrane potential. Thus,to observe an intermediate-amplitude spike in such models, one needs to simulate themodels with accuracy beyond the limits of uncertainty which appear when the physicalinterpretation of the model is considered. As a result, for any practical purposesuch models exhibit all-or-none behavior with the threshold set looking like a thresholdmanifold. FitzHugh referred to this as being a quasithreshold phenomenon.Quasi-thresholds are related to the special canard trajectory depicted in Fig. 7.27d.The trajectory follows the unstable branch of the cubic nullcline all the way to the rightknee point P. The flow near the trajectory is highly unstable; any small perturbationpushes the state of the system to the left or to the right, resulting in a “subthreshold”or “superthreshold” response. The solutions depicted in Fig. 7.26, right, try to followsuch a trajectory. An easy way to compute the trajectory in two-dimensional relaxationoscillators is to start with the point P and integrate the system backwards (t → −∞).We discuss canard (French duck) solutions in detail in Sect. 6.3.4.7.2.6 RheobaseNeuronal rheobase, i.e., the minimal amplitude of a current of infinite duration thatmakes the neuron fire, measures the “current threshold” of the neuron. Integrators havea well-defined rheobase while resonators may not. To see this, consider an integratorneuron in Fig. 7.28a receiving a current step that changes instantly its phase portrait.In particular, the current moves the equilibrium from the old location corresponding