  From the action potentials of neurons and cardiac cells to the amplification
of calcium signals in oocytes, excitability is a hallmark of many biological
signalling processes. In recent years, excitability in single cells has been
related to multiple-timescale dynamics through canards, special solutions which
determine the effective thresholds of the all-or-none responses. However, the
emergence of excitability in large populations remains an open problem. Here,
we show that the mechanism of excitability in large networks and mean-field
descriptions of coupled quadratic integrate-and-fire (QIF) cells mirrors that
of the individual components. We initially exploit the Ott-Antonsen ansatz to
derive low-dimensional dynamics for the coupled network and use it to describe
the structure of canards via slow periodic forcing. We demonstrate that the
thresholds for onset and offset of population firing can be found in the same
way as those of the single cell. We combine theoretical analysis and numerical
computations to develop a novel and comprehensive framework for excitability in
large populations, applicable not only to models amenable to Ott-Antonsen
reduction, but also to networks without a closed-form mean-field limit, in
particular sparse networks.
