Dynamical Systems in Neuroscience:

Chapter 5Conductance-Based Models andTheir ReductionsIn this chapter we present examples of geometrical phase plane analysis of varioustwo-dimensional neural models. In particular, we consider minimal models, i.e., thosehaving minimal sets of currents that enable the models to generate action potentials.The remarkable fact is that all these models can be reduced to planar systems havingN-shaped V -nullclines. We will see that the behavior of the models depends not somuch on the ionic currents as on the relationship between (in)activation curves and thetime constants. That is, models involving completely different currents can have identicaldynamics, and conversely, models involving similar currents can have completelydifferent dynamics.5.1 Minimal ModelsThere are a few dozens of known voltage- and Ca 2+ -gated currents having diverseactivation and inactivation dynamics, and this number grows every year. Some ofthem are summarized in Sect. 2.3.5. Almost any combination of the currents wouldresult in interesting non-linear behavior, such as excitability. Therefore, there arebillions (more than 2 30 ) of different electrophysiological models of neurons. Here wesay that two models are “different” if for example one has the h-current I h and theother does not, without even considering how much of the I h is there. How can weclassify all such models?Let us do the following thought experiment: Consider a conductance-based modelcapable of exhibiting periodic spiking, i.e., having a limit cycle attractor. Let usremove completely a current or one of its gating variables, and ask the question “Doesthe reduced model have a limit cycle attractor, at least for some values of parameters?”If it does, we remove one more gating variable or current, and proceed until we arriveat a model that satisfies the following two properties:• It has a limit cycle attractor, at least for some values of parameters.133