Dynamical Systems in Neuroscience:

384 Burstingaction potentials cut2 mV100 msFigure 9.46: The conductance noise destabilizes the focusequilibrium in a mesencephalic V neuron before subcriticalAndronov-Hopf bifurcation takes place, therebygiving an impression of a supercritical Andronov-Hopfbifurcation (data modified from Wu et al. 2001).types are defined for models of the fast-slow form (9.1) assuming that the ratio of timescales, µ, is sufficiently small. Not all neurons can be described adequately by suchmodels, hence extending the classification to those neurons may be worthless. Typicalexample when the classification fails is the model of bursting of the sensory processingneuron in weakly electric fish, known as the “ghostburster” (Doiron et al. 2002) inwhich µ > 0.1.If a bursting neuron can be described accurately by a model having fast-slow form(9.1), then there is no problem to determine its topological type — just freeze theslow subsystem, i.e., set µ = 0, and find bifurcations of the fast subsystem treatingu as a parameter. Software packages, such as XPPAUT, AUTO, or MATLAB basedMATCONT, are helpful in bifurcation analyses of such systems.What if a neuron has an apparent fast-slow dynamics but its model is not knownat present? To determine the types of bifurcations of the fast subsystem, we firstuse non-invasive observations: presence or absence of fast subthreshold oscillations,changes in intraburst (interspike) frequency, changes in spike amplitudes, etc. Eachpiece of information excludes some bifurcations and narrows the set of possible typesof bursting. Then, we can use invasive methods, e.g., small perturbations, to test theco-existence of resting and spiking states, and narrow down the choice of bifurcationsfurther. With some luck, we can exclude sufficiently many bifurcations and determineexactly the type of bursting without even knowing the details of the mathematicalmodel that describes it.9.4.2 Integrators vs. ResonatorsA conspicuous feature of neuronal systems near Andronov-Hopf bifurcation, whethersubcritical or supercritical, is the existence of fast subthreshold oscillations of themembrane potential. Quite often, these oscillations are visible in recordings of themembrane potential. If not, then they could be evoked by a brief small pulse ofcurrent. Apparently, a bursting neuron exhibiting such oscillations in the quiescentstate is either of the “Hopf/*” type of “subHopf/*” type, where the wildcard “*”denotes any appropriate bifurcation of the spiking state. All such bursters are in thelower half of the table in Fig. 9.23.To discern whether the bifurcation is supercritical or subcritical, one needs to study