%0 Journal Article %1 Prag_2007_011118 %A Prager, T. %A Falcke, M. %A Schimansky-Geier, L. %A Zaks, M. A. %D 2007 %J Phys. Rev. E Stat. Nonlin. Soft Matter Phys. %K Action Animals; Biological Chains; Clocks; Computer Humans; Markov Models, Nerve Net; Neurological; Neurons Potentials; Simulation; Statistical; %N 1 Pt 1 %P 011118 %T Non-Markovian approach to globally coupled excitable systems.. %V 76 %X 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.

@article{Prag_2007_011118, 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.}, added-at = {2009-06-03T11:20:58.000+0200}, author = {Prager, T. and Falcke, M. and Schimansky-Geier, L. and Zaks, M. A.}, biburl = {https://www.bibsonomy.org/bibtex/21f5e72e58d1cc2524623ee0ed6d0f3f0/hake}, description = {The whole bibliography file I use.}, institution = {Institute of Physics, Humboldt-University of Berlin, Newtonstrasse15, 12489 Berlin, Germany.}, interhash = {0a48e9d4703fa11bac95cb01b99779a9}, intrahash = {1f5e72e58d1cc2524623ee0ed6d0f3f0}, journal = {Phys. Rev. E Stat. Nonlin. Soft Matter Phys.}, keywords = {Action Animals; Biological Chains; Clocks; Computer Humans; Markov Models, Nerve Net; Neurological; Neurons Potentials; Simulation; Statistical;}, month = Jul, number = {1 Pt 1}, pages = 011118, pmid = {17677421}, timestamp = {2009-06-03T11:21:26.000+0200}, title = {Non-Markovian approach to globally coupled excitable systems..}, volume = 76, year = 2007 }