 Fast-slow matching for a calcium pulse

In a two-variable calcium-wave model, the fast front scale can reduce to the displayed dynamical system. Its equilibria are intersections of the kinetic nullcline $V=g(U)$ and a line of slope $-1/\gamma$. At a simple intersection with $f_V<0$, the Jacobian determinant is $f_V(1+\gamma g^{\prime })$; a negative determinant identifies a saddle point. Matching a rising front requires a heteroclinic orbit from one saddle to another, selected by a shooting method in the speed $s$. A falling connection must use the same speed. Between fronts the slow equation is $s(1+\gamma g^{\prime })U^{\prime }=\mu -U$. Recovery on the two outer rising branches requires the resting value $\mu$ to lie between the low landing value and the high takeoff value. The nullcline shape supplies this geometry but does not by itself prove existence of both connections.