Dynamical Systems in Neuroscience:

228 ExcitabilityAlternatively, the equilibrium may remain stable and co-exist with the newly bornlimit cycle, as it happens during saddle homoclinic orbit or fold limit cycle bifurcationsin Fig. 7.2. The dynamical system is no longer excitable, but bistable, though manyscientists still treat bistable systems as excitable. An appropriate synaptic input canswitch the behavior from resting to spiking and back. Notice that we considered onlybifurcations of a limit cycle so far.• Bifurcation of the equilibrium. Suppose the system is bistable, as in Fig. 7.2.Since the equilibrium is near the cycle, a small modification of the vector field inthe shaded neighborhood can make it disappear via saddle-node bifurcation, orlose stability via subcritical Andronov-Hopf bifurcation.In any case, excitable dynamical systems can bifurcate into oscillatory systems eitherdirectly or indirectly through bistable systems. All these cases are summarized inFig. 7.2.7.1.2 Hodgkin’s classificationAs we mentioned in the introduction chapter, the first one to study bifurcation mechanismsof excitability (years before mathematicians discovered such bifurcations) wasHodgkin (1948), who injected steps of currents of various amplitudes into excitablemembranes and looked at the resulting spiking behavior. We illustrate his experimentsin Fig. 7.3 using recordings of rat neocortical and brainstem neurons. When the currentstrength is small, the neurons are quiescent. When the current is strong, the neuronsfire trains of action potentials. Depending on the average frequency of such firing,Hodgkin identified two major classes of excitability:• Class 1 neural excitability. Action potentials can be generated with arbitrarilylow frequency, depending on the strength of the applied current.• Class 2 neural excitability. Action potentials are generated in a certainfrequency band that is relatively insensitive to changes in the strength of theapplied current.Class 1 neurons, sometimes called type I neurons, fire with a frequency that may varysmoothly over a broad range of about 2 to 100 Hz or even higher. The important observationhere is that the frequency can be changed tenfold. In contrast, the frequencyband of Class 2 neurons is quite limited, e.g., 150−200 Hz, but it can vary from neuronto neuron. The exact numbers are not important to us here. The qualitative distinctionbetween the classes noticed by Hodgkin is that the frequency-current relation (theF-I curve in Fig. 7.3, bottom) starts from zero and continuously increases for Class 1neurons, but is discontinuous for Class 2 neurons.Obviously, the two classes of excitability have different neuro-computational properties.Class 1 excitable neurons can smoothly encode the strength of an input, e.g.,