Non-Markovian approach to globally coupled excitable systems.
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.
Institute of Physics, Humboldt-University of Berlin, Newtonstrasse15, 12489 Berlin, Germany.
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T
Prager
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M
Falcke
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L
Schimansky-Geier
M A
MA
Zaks
D000200:Action Potentials / Q000502:physiology*; D000818:Animals; D001683:Biological Clocks / Q000502:physiology*; D003198:Computer Simulation; D006801:Humans; D008390:Markov Chains; D008959:Models, Neurological; D015233:Models, Statistical; D009415:Nerve Net / Q000502:physiology*; D009474:Neurons / Q000502:physiology*