Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics

We study a uniform-in-time continuum limit of the lattice Winfree model(LWM) and its asymptotic dynamics which depends on system functions such as natural frequency function and coupling strength function. The continuum Winfree model(CWM) is an integro-differential equation for the temporal evolution of Winfree phase field. The LWM describes synchronous behavior of weakly coupled Winfree oscillators on a lattice lying in a compact region. For bounded measurable initial phase field, we establish a global well-posedness of classical solutions to the CWM under suitable assumptions on coupling function, and we also show that a classical solution to the CWM can be obtained as a \begin{document}$ L^1 $\end{document}-limit of a sequence of lattice solutions. Moreover, in the presence of frustration effect, we show that stationary states and bump states can emerge from some admissible class of initial data in a large and intermediate coupling regimes, respectively. We also provide several numerical examples and compare them with analytical results.