Dynamical Systems in Neuroscience:

Two-Dimensional Systems 970.7K + activation variable, n0.60.50.40.30.20.1(d)relativerefractory(c)absoluterefractoryn-nullcline(b)spikedownstroke(a)spikeupstroke(regenerative)peakof spikeV-nullcline0resting-80 -70 -60 -50 -40 -30 -20 -10 0 10 20membrane voltage, V (mV)Figure 4.4: Nullclines of the I Na,p +I K -model (4.1, 4.2) with low-threshold K + currentin Fig. 4.1b. (The vector field is slightly distorted for the sake of clarity of illustration).As another example, let us determine the nullclines of the I Na,p +I K -model (4.1,4.2). The V -nullcline is given by the equationwhich has the solutionI − g L (V −E L ) − g Na m ∞ (V ) (V −E Na ) − g K n (V −E K ) = 0 ,n = I − g L(V −E L ) − g Na m ∞ (V ) (V −E Na )g K (V − E K )(V -nullcline)depicted in Fig. 4.4. It typically has the form of a cubic parabola. The equationdefines the n-nullclinen = n ∞ (V )n ∞ (V ) − n = 0(n-nullcline),which coincides with the K + steady-state activation function n ∞ (V ), though only aninitial segment of this curve fits in Fig. 4.4. It is easy to see how the V - and n-nullclines partition the phase plane into four regions, in each of which the vector fieldhas a different direction:(a) Both V and n increase: Both Na + and K + currents activate and lead to theupstroke of the action potential.(b) V decreases but n still increases: Na + current deactivates but the slower K +current still activates and leads to the downstroke of the action potential.