Recently, it has been shown that properties of excitable media stirred by two-dimensional chaotic flows can be properly studied in a one-dimensional framework, describing the transverse profile of the filament-like structures observed in the system. Here, we perform a bifurcation analysis of this one-dimensional approximation as a function of the Damkohler number, the ratio between the chemical and the strain rates. Different branches of stable solutions are calculated, and a Hopf bifurcation, leading to an oscillating filament, identified.