Dynamical Systems in Neuroscience:

Bursting 385bifurcations of limit cyclessaddle-nodeon invariantcirclesaddlehomoclinicorbitsupercriticalAndronov-Hopffoldlimitcyclesaddle-node(fold)fold/circlefold/homoclinicfold/Hopffold/fold cyclebifurcations of equilibriasaddle-nodeon invariantcirclesupercriticalAndronov-HopfsubcriticalAndronov-Hopfcircle/circleHopf/circlesubHopf/circlecircle/homoclinicHopf/homoclinicsubHopf/homocliniccircle/HopfHopf/HopfsubHopf/Hopfcircle/fold cycleHopf/fold cyclesubHopf/fold cyclebistabilitybeforethe burstbistabilityat the endof the burstFigure 9.47: Bistability, i.e., co-existence of resting and spiking states, depends on thetopological type of bursting.the amplitude of emerging oscillations, which could be tricky. In models, slow passagethrough supercritical Andronov-Hopf bifurcations often results in a delayed transitionto oscillations with an intermediate or large amplitude, hence such a bifurcation maylook like subcritical. In recordings, like the one in Fig. 9.39a or in Fig. 9.46, noisedestabilizes the focus equilibrium before the subcritical Andronov-Hopf bifurcationtakes place and gives the impression that the amplitude increases gradually, i.e., as ifthe bifurcation were supercritical.The existence of fast subthreshold oscillations indicates that the bursting neuronacts as a resonator, at least right before the onset of a burst. In Sect. 7.2.2 we showedthat such neurons prefer a certain resonant frequency of stimulation that matches thefrequency of subthreshold oscillations. A resonant input may excite the neuron andinitiate a burst or it may delay the transition to the burst, depending on its phaserelative to the phase of subthreshold oscillations.In contrast, all bursters in the upper half of the table in Fig. 9.23, i.e., “fold/*” and“circle/*” types, do not have fast subthreshold oscillations, at least before the onsetof each burst (see Ex. 5). The fast subsystem of such bursters acts as an integrator:It prefers high-frequency inputs; the higher the frequency, the sooner the transition tothe spiking state. The phase of the input does not play any role here.