= Solution
For the <travelling wave> coordinate $\xi=x+st$, the <chain rule> gives
$$
\boxed{s(U'+\gamma V')=\mu-U+\varepsilon U'',\qquad\varepsilon sV'=f(U,V).}
$$
<Fast-slow matching for a calcium pulse> joins two fast transitions, $B\to C$ and $D\to A$, joined by slow portions $C\to D$ and $A\to B$. Write the <nullcline> as $V=g(U)$. The resting point has $U_B=\mu$, $V_B=g(\mu)$ in the leading matched description.
Put $\xi=\varepsilon X$. Multiplying the first equation by $\varepsilon$ gives $U_{XX}=s(U_X+\gamma V_X)+\varepsilon(U-\mu)$. At leading order, integrate and use $U_X\to0$, $(U,V)\to(U_B,V_B)$ at the start of the front. Together with the second equation this gives
$$
\boxed{U_X=s[U-U_B+\gamma(V-V_B)],\qquad sV_X=f(U,V).}
$$
The fast equilibria are intersections of $V=g(U)$ and $V=V_B-(U-U_B)/\gamma$. As the slope $-1/\gamma$ changes, there may be one or three simple intersections, with two distinct intersections at a tangency. At a simple point of the nullcline, $f_U=-f_Vg'$, and the sign convention above/below it gives $f_V<0$. The <Jacobian matrix> of the fast system has
$$
\det J=f_V-\gamma f_U=f_V(1+\gamma g'),\qquad\operatorname{tr}J=s+f_V/s.
$$
For the usual three-intersection configuration, the two outer points have $1+\gamma g'>0$, hence negative determinant and one positive and one negative <eigenvalue>: they are <saddle points>. The middle point has positive determinant; it is linearly stable if $s^2<-f_V$ and unstable if $s^2>-f_V$. Zero trace is a possible oscillatory stability threshold, with a nonlinear classification requiring further information about $f$.
To select the front speed, follow the appropriate branch of the <unstable manifold> of $B$ and vary $s$ until it meets the <stable manifold> of $C$. This is a <shooting method> for a <heteroclinic orbit>. Follow the high-$U$ slow branch from $C$ to $D$; the falling front obeys the same fast equations with the integration constant anchored at $D$. Choose $D$ so that its descending saddle connection reaches a low-$U$ point $A$ at that same speed. The remaining slow branch returns toward the resting point, completing the matched spike. The construction requires the indicated connections; a nullcline sketch alone does not specify their numerical speed or prove their existence for every possible kinetic function $f$.
\Image[/past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2007-calcium-pulse-phase-plane.png]
{title=Fast and slow portions of the calcium-wave phase-plane construction}
{description=A qualitative original sketch: the rising nullcline branches carry the slow return and recovery, while dashed paths indicate the saddle connections that a shooting calculation must find. The paths illustrate the matching construction and are not a numerical solution for a specified kinetic function.}
On the slow portions, $f(U,V)=0$ at leading order, so $V=g(U)$ and $V'=g'(U)U'$. Neglecting the small diffusion term yields
$$
\boxed{s(1+\gamma g'(U))U'=\mu-U.}
$$
On the outer rising branches the coefficient on the left is positive. Recovery from $C$ toward $D$ requires decreasing $U$, hence $\mu<U_D$; the low branch from $A$ back toward rest requires increasing $U$, hence $U_A<\mu$. Therefore $\boxed{U_A<\mu<U_D}$. These inequalities distinguish the recovery branches from a trajectory that would move away from the required matching points.
Back to article page
