Dynamical Systems in Neuroscience:

350 Burstingslow inactivation of inward current120slow activation of outward current100V(t)100V(t)806040806020400200 500 1000 1500 2000 0 500 1000 1500 2000 2500 3000 3500abFigure 9.9: Hodgkin-Huxley (1952) model with three gating variables is minimal forbursting (modified from Fig. 1.10 in Izhikevich 2001).0uncoupledhalf-centeroscillator-mutual inhibition-Figure 9.10: Central pattern generation by mutually inhibitory oscillators.may argue that the model in the figure is not Hodgkin-Huxley at all, since we changedthe kinetics of some currents by an order of magnitude.Thinking in terms of minimal models, we can understand what is essential forspiking and bursting and what is not. In addition, we can clearly see that some wellknownconductance-based models form partially-ordered set. For example, the chainof neuronal models Morris-Lecar (I Ca +I K ) ≺ Hodgkin-Huxley (I Na,t +I K ) ≺ Butera-Rinzel-Smith (I Na,t +I K +I K,slow ) is obtained by adding a conductance or gating variableto one model to get the next one. Here, A ≺ B means A is a subsystem of B.Understanding the ionic bases of bursting is an important step in analysis of burstingdynamics. However, such an understanding may not provide sufficient informationon why the bursting pattern looks as it does, what the neuro-computational propertiesof the neuron are, and how they depend on the parameters of the system. Indeed, weshowed in Chap. 5 that spiking models based on quite different ionic mechanisms canhave identical dynamics and vice versa. This is true for bursting models as well.