Uniform-in-Time Continuum Limit of the Winfree Model on an Infinite Cylinder and Emergent Dynamics

We study a uniform-in-time continuum limit of the lattice Winfree model (LWM) on an inifinite cylinder under restricted initial datum and suitable assumptions on system functions such as natural frequency functions and the coupling strength function. Roughly speaking, the continuum Winfree model (CWM) is an integro-differential equation for the temporal evolution of a Winfree phase field over an infinite cylinder. For bounded measurable initial phase field and natural frequency functions, we can establish a unique global existence of classical solutions to the CWM in a large coupling regime. At the same time, we can also see that a classical solution to the CWM can be obtained as an L-1 -limit of a sequence of lattice solutions to the LWM under a suitable framework. This provides a good approximation of solution to the CWM as a sequence of solutions to the LWM. Moreover, we verify that the continuum limit of a sequence of equilibria to the LWM tends to a unique stationary profile of the CWM.