Abstract
We consider stochastic excitable units with three discrete states.
Each state is characterized by a waiting time density function. This
approach allows for a non-Markovian description of the dynamics of
separate excitable units and of ensembles of such units. We discuss
the emergence of oscillations in a globally coupled ensemble with
excitatory coupling. In the limit of a large ensemble we derive the
non-Markovian mean-field equations: nonlinear integral equations
for the populations of the three states. We analyze the stability
of their steady solutions. Collective oscillations are shown to persist
in a large parameter region beyond supercritical and subcritical
Hopf bifurcations. We compare the results with simulations of discrete
units as well as of coupled FitzHugh-Nagumo systems.
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