A periodic branch for the delayed noisy leaky integrate-and-fire equation

We propose a local branch of nonnegative periodic solutions of the delayed noisy leaky integrate-and-fire reset equation with inhibitory coupling and nonconstant firing rate. Translating both voltage thresholds with the coupling keeps a centered stationary reset profile fixed. An explicit boundary-response function gives a nondegenerate two-parameter reduction at high frequency. Resolvent bounds, time-Sobolev trace estimates, a negative-part energy argument and parameter reconstruction address the full PDE on its voltage half-line. The result uses the repository target's freedom to choose negative physical thresholds.