Dynamical Systems in Neuroscience:

158 Conductance-Based Models5.2.2 Equivalent potentialsInspired by the reduction idea of Krinskii and Kokoz (1973), Kepler et al. (1992) suggesteda systematic method of reducing the complexity of conductance-based Hodgkin-Huxley-type modelsC ˙V = I − I(V, x 1 , . . . , x n )ẋ i = (m i,∞ (V ) − x i )/τ i (V ) , i = 1, . . . , n ,where x 1 , . . . , x n is a set of gating variables. The goal is to find certain patterns orcombinations of the gating variables that can be lumped to reduce the dimension ofthe system. For example, we want to combine all resonant variables operating on asimilar time scale into a “master” recovery variable, then do the same for amplifyingvariables.Let us convert each variable x i (t) to the equivalent potential v i (t) that satisfiesx i = m i,∞ (v i ) .In other words, the equivalent potential is the voltage which, in a voltage clamp, wouldgive the value x i when the model is at an equilibrium. Applying the chain rule tov i = m −1i,∞ (x i), we express the model above in terms of equivalent potentials:C ˙V = I − I(V, m 1,∞ (v 1 ), . . . , m n,∞ (v n )) ,˙v i = (m i,∞ (V ) − m i,∞ (v i ))/(τ i (V ) m ′ i,∞(v i )) .Since the Boltzmann functions m i,∞ (V ) are invertible, the denominators do not vanish.No approximations have been made yet; the new model is entirely equivalent to theoriginal one, it is just expressed in a different coordinate system. The new coordinates,however, expose many patterns among the equivalent voltage variables that were notobvious in the original, gating coordinate system.Kepler et al. (1992) developed an algorithm that substitutes resonant and amplifyingvariables by their weighted averages. The weights are found using Lagrangemultipliers and strictly local criteria aimed at preserving the bifurcation structure ofthe model. There is also a set of tests that informs the user when the method is likelyto fail. The method results in a lower-dimensional system that is easier to simulate,visualize, and understand.5.2.3 Nullclines and I-V recordingsWe saw that the form and the position of nullclines provided important informationabout the neuron dynamics, i.e., the number of equilibria, their stability, the existenceof limit cycle attractors, etc. The same information, in principle, can also be obtainedfrom the analysis of the neuronal current-voltage (I-V) relations. This is not a coincidence,since there is a profound relationship between nullclines and experimentallymeasured I-V curves.