Dynamical Systems in Neuroscience:

14 Introductionco-existence of resting and spiking statesYES(bistable)NO(monostable)subthreshold oscillationsNO(integrator)YES(resonator)saddle-nodesubcriticalAndronov-Hopfsaddle-node oninvariant circlesupercriticalAndronov-HopfFigure 1.13: Classification of neurons intomonostable/bistable integrators/resonatorsaccording to the bifurcation of the restingstate in Fig. 1.12.asymptotic firing frequency, Hz40302010Class 1 excitabilityF-I curve00 100 200 300injected dc-current, I (pA)asymptotic firing frequency, Hz25020015010050Class 2 excitabilityF-I curve00 500 1000 1500injected dc-current, I (pA)Figure 1.14: Frequency-current (F-I) curves of cortical pyramidal neuron and brainstemmesV neuron from Fig. 7.3. These are the same neurons used in the ramp experimentin Fig. 1.11.1.2.3 Hodgkin classificationHodgkin (1948) was the first to study bifurcations in neuronal dynamics, years beforethe mathematical theory of bifurcations was developed. He stimulated squid axonswith pulses of various amplitudes and identified three classes of responses:• 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 3 neural excitability. A single action potential is generated in responseto a pulse of current. Repetitive (tonic) spiking can be generated only forextremely strong injected currents or not at all.The qualitative distinction between the classes is that the frequency-current relation(the F-I curve in Fig. 1.14) starts from zero and continuously increases for Class 1