If the pacing is sufficiently rapid, say Bselleck is the average shortening of APD resulting from decreasing B below Bcrit, and an(x) is the amplitude of alternans at the nth beat. It is assumed that an(x) varies slowly from beat to beat, so that one may regard it as the discrete values of a smooth function a(x,t) of continuous time t, i.e., an(x)=a(x,tn) where tn=nB for n=0,1,2,��. Based on the above assumptions, a weakly nonlinear modulation equation for a(x,t) was derived in Ref. 18 which, after nondimensionalization with respect to time, is given by ?ta=��a+��2?xxa?w?xa?��?1��0xa(x��,t)dx��?ga3.

(2.3) Here ��, the bifurcation parameter may be obtained by18 ��=12(B?Bcrit)?f��(Dcrit), (2.4) where Dcrit=Bcrit?Acrit; ��,w,�� are positive parameters, each having the units of length that are derived from the equations of the cardiac model; and the nonlinear term ?ga3 limits growth after the onset of linear instability. Neumann boundary conditions ?xa(?,t)=0 (2.5) are imposed in Eq. 2.3. To complete the???xa(0,t)=0, nondimensionalization of Eq. 2.3, we define the following dimensionless ?��=??w��?2, (2.6) and we rescale the time??x��=x?w��?2,??variables: ����=��?w3��?4, g��=g?w?2��2. (2.7) In this??�ҡ�=��?w?2��2,??t and parameters �� and g, t��=t?w2��?2, notation, Eq. 2.3 may be rewritten ?t��a=�ҡ�a+La?g��a3, (2.

8) where L is the linear operator on the interval 0

[The figure is based on lengths =6 and 15, but the behavior is qualitatively similar for all sufficiently large . Note that all eigenvalues lie in the (stable) left half plane.] It may be seen from the figure that there is a critical value ��c?1, such that if ��?1<��c?1, Drug_discovery the real eigenvalue ��0 of L has largest real part (thus steady-state bifurcation occurs first) and if ��?1>��c?1, then the complex pair ��1,2 has the largest real part (thus Hopf bifurcation occurs first).