Dynamical Systems in Neuroscience:

Introduction 15neurons, is discontinuous for Class 2 neurons, and is not defined at all for Class 3neurons.Obviously, neurons belonging to different classes have different neuro-computationalproperties: Class 1 neurons, which include cortical excitatory pyramidal neurons, cansmoothly encode the strength of the input into the output firing frequency, as inFig. 1.11, left. In contrast, Class 2 neurons, such as fast-spiking (FS) cortical inhibitoryinterneurons, cannot do that; instead, they fire in a relatively narrow frequency band,as in Fig. 1.11, right. Class 3 neurons cannot exhibit sustained spiking activity, soHodgkin regarded them as “sick” or “unhealthy”. There are other distinctions betweenthe classes, which we discuss later.Different classes of excitability occur because neurons have different bifurcations ofresting and spiking states – a phenomenon first explained by Rinzel and Ermentrout(1989). If ramps of current are injected to measure the F-I curves, then Class 1 excitabilityoccurs when the neuron undergoes the saddle-node bifurcation on invariantcircle depicted in Fig. 1.12b. Indeed, the period of the limit cycle attractor is infiniteat the bifurcation point, and then it decreases as the bifurcation parameter – say, theinjected current – increases. The other three bifurcations result in Class 2 excitability.Indeed, the limit cycle attractor exists and has a finite period when the resting statein Fig. 1.12 undergoes a subcritical Andronov-Hopf bifurcation, so emerging spikinghas a non-zero frequency. The period of the small limit cycle attractor appearing viasupercritical Andronov-Hopf bifurcation is also finite, so the frequency of oscillationsis non-zero, but their amplitudes are small. In contrast to the common and erroneousfolklore, the saddle-node bifurcation (off limit cycle) also results in Class 2 excitabilitybecause the limit cycle has a finite period at the bifurcation. There is a considerablelatency (delay) to the first spike in this case, but the subsequent spiking has non-zerofrequency. Thus, the simple scheme “Class 1 = saddle-node, Class 2 = Hopf” thatpermeates many publications is incorrect.When pulses of current are used to measure the F-I curve, as in Hodgkin’s experiments,the firing frequency depends on other factors, and not only the type of thebifurcation of the resting state. In particular, low-frequency firing can be observed insystems near Andronov-Hopf bifurcations, as we show in Chap. 7. To avoid possibleconfusion, we define the class of excitability based only on slow ramp experiments.Hodgkin’s classification has an important historical value but it is of little use forthe dynamic description of a neuron, since naming a class of excitability of a neurondoes not tell much about the bifurcations of the resting state. Indeed, it only says thatsaddle-node on invariant circle bifurcation (Class 1) is different from the other threebifurcations (Class 2), and only when ramps are injected. Instead, dividing neurons intointegrators and resonators with bistable or monostable activity is more informative, sowe adopt the classification in Fig. 1.13 in this book. In this classification, Class 1 neuronis a monostable integrator, whereas Class 2 neuron could be a bistable integrator or aresonator.