Spiral core in singly diffusive excitable media

We formulate the problem of finding the spiral core which smoothly matches onto the asymptotic rotating solution of the FitzHugh-Nagumo model. We prove that the inner problem (with scale ε, the ratio of the reaction rates) has a solution for all possible outer solutions on scale ${\mathrm{\epsilon }}^{2/3}$; furthermore, we explicitly determine this solution via a simple numerical procedure. This completes the rigorous demonstration of the existence of rotating spiral solutions in singly diffusive excitable systems.