  The Winfree model is a phase-coupled synchronization model which simplifies
pulse-coupled models such as the Peskin model on pacemaker cells. It is
well-known that the Winfree ensemble with the first-order coupling exhibits
discrete asymptotic patterns such as incoherence, locking and death depending
on the coupling strength and variance of natural frequencies. In this paper, we
further study higher-order couplings which makes the dynamics more close to the
behaviors of the Peskin model. For this, we propose several sufficient
frameworks for asymptotic patterns compared to the first-order coupling model.
Our proposed conditions on the coupling strength, natural frequencies and
initial data are independent of the number of oscillators so that they can be
applied to the corresponding mean-field model. We also provide several
numerical simulations and compare them with analytical results.
