Dynamical Systems in Neuroscience:

spikingrestingBursting 383saddle-node oninvariant circlebifurcationfoldbifurcationxufoldbifurcationSaddle-Node onInvariant CircleFoldFoldFigure 9.45: “Fold/circle” bursting: The resting state disappears via fold bifurcationand the spiking state disappears via saddle-node on invariant circle bifurcation (modifiedfrom Izhikevich 2000).9.4 Neuro-Computational PropertiesThere is more to the topological classification of bursters than just a mathematicalexercise. Indeed, in Chap. 7 we have shown that the neuro-computational propertiesof an excitable system depend on the type of bifurcation of the resting state. Thesame is valid for a burster: Its neuro-computational properties depend on the kind ofbifurcations of the resting and spiking states, that is, on the burster’s type. Knowingthe topological type of a given bursting neuron, we know what the neuron can doand more importantly what it cannot do, regardless of the model that describes itsdynamics.9.4.1 How to distinguish?First, we stress that the topological classification of bursters provided in the previoussection is defined for mathematical models, and not for real neurons. Moreover, the