= Fast-slow matching for a calcium pulse
{title2=$U_X=s[U-U_B+\gamma(V-V_B)],\quad sV_X=f(U,V)$}
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 $f_V(1+\gamma g')$; 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 $s(1+\gamma g')U'=\mu-U$. 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.
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