  We present a high-dimensional Winfree model in this paper. The Winfree model
is the first mathematical model for synchronization on the circle. We
generalize this model to the high-dimensional sphere and we call it "the
Winfree sphere model." We restricted the support of the influence function in
the neighborhood of the attraction point with a small diameter to mimic the
influence function as the Dirac-delta distribution. Restricting the support of
the influence function allows several new conditions of the complete
phase-locking states for the identical Winfree sphere model compare to previous
results. We also provide the exponential $\ell^1$-stability and the existence
of the equilibrium solution to obtain the complete oscillator death state of
the Winfree sphere model.
