Dynamical Systems in Neuroscience:

160 Conductance-Based Models1recovery variable, u0.80.60.40.2u-nullclineV-nullcline0.20.10-80 -60 -40 -20 0 20membrane potential, V (mV)a0-70 -60 -50 -40membrane potential, V (mV)bFigure 5.23: Phase portrait (a) and its magnification (b) of a typical neuronal modelhaving voltage variable V and a recovery variable u.of the Na + current, and it is responsible for the excitability property of the neuron. Itis also responsible for the N-shape of the V -nullcline, as we see next.Once the I-V relations are found, we can find the nullclines of the system (5.3, 5.4).From the equationI − I 0 (V ) − g(V − E K ) = 0we can easily find the V -nullclineg = {I − I 0 (V )}/(V − E K ) (V -nullcline) ,which has the inverted N-shape depicted in Fig. 5.22c because I 0 (V ) does. Whilemeasuring I ∞ (V ), we hold V long enough so that all conductances reach their steadystatevalues. The steady-state value g = g ∞ (V ) can be obtained from the equationI − I 0 (V ) − g(V − E K ) = −I ∞ (V ) ,which says that the asymptotic steady-state current is the sum of the steady-state fastcurrent and steady-state slow current. Thereforeg = {I + I ∞ (V ) − I 0 (V )}/(V − E K )(g-nullcline)depicted in Fig. 5.22c. Since we used the I Na,p +I K -model with parameters as inFig. 5.4b, top, we are not surprised that the V - and g-nullclines found here havethe same shape and relative position as those in Fig. 5.4b, top. In Ex. 5 we furtherexplore the relationship between the I-V curves and neuronal dynamics.