Fig. 21

ML model for subthreshold oscillations. (Adapted from [27].) (a) Bifurcation diagram of the deterministic ML model. As \(I_{\mathrm{app}}\) is increased, the system undergoes a supercritical Hopf bifurcation (H) at \(I_{\mathrm{app}}^{*}=183\) pA, which leads to the generation of stable oscillations. The maximum and minimum values of oscillations are plotted as black (solid) curves. Oscillations disappear via another supercritical Hopf bifurcation. (b), (c) Phase plane diagrams of the deterministic model for (b) \(I_{\mathrm{app}}=170~\mbox{pA}\) (below the Hopf bifurcation point) and (c) \(I_{\mathrm{app}}=190~\mbox{pA}\) (above the Hopf bifurcation point). The red (dashed) curve is the w-nullcline and the solid (gray) curve represents the v-nullcline. (d), (e) Corresponding voltage time courses. In contrast to Sect. 3.1, we now take \(\alpha_{\mathrm{K}}=\beta_{\mathrm{K}}\mathrm{e}^{2[v-v_{\mathrm{K},1}]/v_{\mathrm{K},2}}\). Sodium parameters: \(g_{\mathrm{Na}}= 4.4~\mbox{mS}\), \(V_{\mathrm{Na}}= 55~\mbox{mV}\), \(\beta_{\mathrm{Na}} = 100~\mbox{ms}^{-1}\), \(v_{\mathrm{Na},1}= -1.2~\mbox{mV}\), \(v_{\mathrm{Na},2}=18~\mbox{mV}\). Leak parameters: \(g_{\mathrm{L}}=2.2~\mbox{mS}\), \(V_{\mathrm{L}}= -60~\mbox{mV}\). Potassium parameters: \(g_{\mathrm{K}}=8~\mbox{mS}\), \(V_{\mathrm{K}}=-84~\mbox{mV}\), \(\beta_{\mathrm{K}}= 0.35~\mbox{ms}^{-1}\), \(v_{\mathrm{K},1}= 2~\mbox{mV}\), \(v_{\mathrm{K},2}= 30~\mbox{mV}\). Also \(C_{m}=1~\mbox{mF}\)