Dynamical Systems in Neuroscience:

240 Excitabilityproperties integrators resonatorsbifurcationsaddle-node oninvariant circlesaddle-nodesubcriticalAndronov-HopfsupercriticalAndronov-Hopfexcitability class 1class 2 class 2 class 2oscillatorypotentialsfrequencypreferencenonoyesyesI-V relationat restnon-monotonemonotonespike latency large smallthresholdand rheobasewell-definedmay not be definedall-or-noneaction potentials yes noco-existence ofresting and spikingpost-inhibitory spikeor facilitation(brief stimuli)inhibition-inducedspikingno yes yes nonoyesnopossibleFigure 7.15: Summary of neuro-computational properties.uniquely the type of bifurcation of the resting state, as we summarize in Fig. 7.14. Forexample, a bistable integrator corresponds to a saddle-node bifurcation whereas monostableresonator corresponds to a supercritical Andronov-Hopf bifurcation. Integratorsand resonators have drastically different neuro-computational properties, summarizedin Fig. 7.15 and discussed next (the I-V curves are discussed in the previous chapter).7.2.1 Fast subthreshold oscillationsAccording to the definition, resonators have oscillatory potentials whereas integratorsdo not. This feature is so important that many of the other neuronal properties discussedlater are just mere consequences of the existence or absence of such oscillations.Fast subthreshold oscillations, as in Fig. 7.16, are typically due to a fast lowthresholdpersistent K + current. At rest, there is a balance of all inward currentsand this partially activated K + current. A brief depolarization further activates K +current and results in fast after-hyperpolarization. While the cell is hyperpolarized,the current de-activates below its steady state level, the balance is shifted toward theinward currents, and the membrane potential depolarizes again, and so on.