Dynamical Systems in Neuroscience:

164 Conductance-Based Modelsmodels is mostly due to the seminal paper by John Rinzel and Bard Ermentrout “Analysisof Neural Excitability and Oscillations”, published as a chapter in Koch and Segev’sbook Methods in Neuronal Modeling (1989, second edition in 1999). Not only did theyintroduced the geometrical methods to a wide computational neuroscience audience,but also were able to explain a number of outstanding problems, such as the origin ofClass 1 and 2 excitability observed by Hodgkin in 1949.Rinzel and Ermentrout illustrated most of the concepts using the Morris-Lecar(1981) model, which is a I Ca + I K -minimal voltage-gated model equivalent to the I Na,p+ I K -model considered above. Due to its simplicity, the Morris-Lecar model is widelyused in computational neuroscience research. This is the reason we use its analogue,the I Na,p + I K -model, throughout the book.Hutcheon and Yarom (2000) suggested to classify all currents into amplifying andresonant. There have been no attempts to classify various electrophysiological mechanismsof excitability in neurons, though minimal models, such as the I Na,t -model orthe I Ca +I K(C) -model, would not surprise most researchers. The other models wouldprobably look bizarre for classical electrophysiologists, though they provide a goodopportunity to practice geometrical phase plane analysis and support FitzHugh’s observationthat N-shaped V -nullcline is the key characteristic of neuronal dynamics.Izhikevich (2003) took advantage of this observation and suggested the simple model(5.5, 5.6) that captures the spike-generation mechanism of many known neuronal types,see Chap. 8.Exercises1. Show that the I A -model cannot have a limit cycle attractor when I A has instantaneousactivation kinetics. (Hint: Use Bendixson criterion.)2. When the injected dc-current I or the Na + maximal conductance g Na in theI Na,p +I K -model have large values, the excited state (V ≈ −20 mV) becomesstable. Sketch possible intersections of nullclines of the model.3. Using I as a bifurcation parameter, determine the saddle-node bifurcation diagramof• the I Na,t -model with parameters as in Fig. 5.6a,• the I A -model with parameters as in Fig. 5.14a.4. Why is g in Fig. 5.22c negative when V is hyperpolarized?5. In Fig. 5.25 we plot the currents that constitute the right-hand side of the voltageequation (5.3),I − I fast (V ) and I slow (V ) = g(V − E K ) ,on the (V, I)-plane. The curves define fast and slow movements of the state ofthe system. Interpret the figure. (Hint: treat the curves as “sort-of-nullclines”).