Dynamical Systems in Neuroscience:

Bifurcations 1970.8-100.70.040.60.02-20K + activation gate, n0.50.40.30.2V-nullcline0saddleT 1-60 -55 -50n-nullclinelimit cyclemembrane voltage, V (mV)-30-40T 2T 20.1T 10-80 -70 -60 -50 -40 -30 -20 -10 0 10membrane voltage, V (mV)-50T 1 T 2-600 1 2 3 4 5 6time, msFigure 6.29: The period of the limit cycle is T = T 1 + T 2 with T 2 → ∞ as the cycleapproaches the saddle equilibrium. Shown is the I Na,p +I K -model with I = 3.5.part of the limit cycle in the figure) and T 2 denote the time spent in the small neighborhoodof the saddle equilibrium (continuous part of the limit cycle in the shadowedregion), so that the period of limit cycle is T = T 1 + T 2 . While T 1 is relatively constant,T 2 → ∞ as I approaches the bifurcation value I b = 3.08, and the limit cycleapproaches the saddle. In Ex. 11 we show thatT 2 = − 1 λ 1ln{τ(I − I b )} ,where λ 1 is the positive (unstable) eigenvalue of the saddle, and τ is a parameter thatdepends on the size of the neighborhood, global features of the vector field, etc. Wecan represent the period, T , in the formT (I) = − 1 λ 1ln{τ 1 (I − I b )} ,where a single parameter τ 1 = τe −λ 1T 1accounts for all global features of the model,including the width of the action potential, the shape of the limit cycle, etc. One caneasily determine τ 1 if the eigenvalue λ 1 and the period of the limit cycle is known for atleast one value of I. The I Na,p +I K -model has τ 1 = 0.2, as we show in Fig. 6.30. Noticethat the theoretical frequency 1000/T (I) matches the numerically found frequency ina broad range. Also, notice how imprecise the numerical results are (see inset in thefigure).