FIG. 9.

Generation of one-phase oscillations in the pre-BötC model. A: traces of output activities of pre-I and early-I neurons. B: bifurcation diagram for the full system given by Eqs. 1, 2, and 5, generated with dimensionless bifurcation parameter D₁. For each fixed D₁, the full system has a unique equilibrium point, at which the time derivatives of all variables are zero. The curve formed of these points is plotted, with a dashed component where these points are unstable and a solid component where they are stable. Also, for each D₁ value that is not too large, the system has a periodic solution, which is a one-phase oscillation. For each such D₁ value, the maximal and minimal values of V₁ along the one-phase oscillation are marked with circles. As D₁ grows, oscillations are terminated in an Andronov–Hopf (AH) bifurcation near D₁ = 0.03 and, for larger D₁, the only stable state is an equilibrium point of the full system. C: one-phase oscillations (solid) superimposed on V₁ nullclines (dashed) in the (V₁, h_NaP) plane, for various D₁ values as indicated. Recall that each V₁ nullcline is the collection of points on which dV₁/dt = 0 for the corresponding D₁. The h_NaP-nullcline, which consists of points where dh_NaP/dt = 0 and which is independent of D₁ (see Eq. 5), is also shown (dash-dotted). The oscillations evolve clockwise and the amplitude shrinks as D₁ increases such that the bifurcation is approached. D: changes of the oscillation period (T) and the durations of inspiration (T_I) and expiration (T_E) produced by changes in total (dimensionless) drive to pre-I neuron (D₁).