Dynamical Systems in Neuroscience:

Two-Dimensional Systems 1110.60.5n-nullcline0.4K + activation variable, n0.30.2V-nullcline0.10-0.1-80 -70 -60 -50 -40 -30 -20 -10 0 10 20membrane voltage, V (mV)Figure 4.19: Phase portrait of the I Na,p +I K -model having high-threshold K + current.n < 0, we cannot interpret the result. As an exercise, prove that if all gating variablesof a model are initially in the range [0, 1], then they stay in the range for all t ≥ 0.4.2.6 Example: FitzHugh-Nagumo modelThe FitzHugh-Nagumo model (FitzHugh 1961, Nagumo et al. 1962)˙V = V (a − V )(V − 1) − w + I , (4.11)ẇ = bV − cw , (4.12)imitates generation of action potentials by Hodgkin-Huxley-type models having cubic(N-shaped) nullclines as in Fig. 4.4. Here V mimics the membrane voltage and the“recovery” variable w mimics activation of an outward current. Parameter I mimicsthe injected current, and for the sake of simplicity we set I = 0 in our analysis below.Parameter a describes the shape of the cubic parabola V (a−V )(V −1), and parametersb > 0 and c ≥ 0 describe the kinetics of the recovery variable w. When b and c aresmall, the model may exhibit relaxation oscillations.The nullclines of the FitzHugh-Nagumo model have the cubic and linear formw = V (a − V )(V − 1) + Iw = b/c V(V -nullcline),(w-nullcline),