Abstract
We consider stochastic systems with m internal states in which discrete events (e.g. hopping events between metastable states or firing events of neurons) occur at a state-dependent rate. Transitions between states are possible with certain fixed rates. Because the state immediately after an event depends in general on the history of the process, the intervals between two consecutive events (“residence times”) are correlated among each other, i.e. the residence time sequence constitutes a nonrenewal process. We construct a general kinetic scheme that accounts for the number of events at a given time. The count statistics is used to derive a general expression for the correlation coefficient of residence times with a certain lag. We apply the theoretical result to a simple neuron model with discrete threshold states leading to negative interspike interval correlations.
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References
L. Gammaitoni, F. Marchesoni, S. Santucci, Phys. Rev. Lett. 74, 1052 (1995)
L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998)
M.H. Choi, R.F. Fox, P. Jung, Phys. Rev. E 57, 6335 (1998)
A. Neiman, A. Silchenko, V. Anishchenko, L. Schimansky-Geier, Phys. Rev. E 58, 7118 (1998)
A. Pikovsky, J. Kurths, Phys. Rev. Lett. 78, 775 (1997)
A. Longtin, Phys. Rev. E 55, 868 (1997)
B. Lindner, L. Schimansky-Geier, Phys. Rev. E 60, 7270 (1999)
B. Lindner, J. García-Ojalvo, A. Neiman, L. Schimansky-Geier, Phys. Rep. 392, 321 (2004)
O.V. Ushakov, H.J. Wünsche, F. Henneberger, I.A. Khovanov, L. Schimansky-Geier, M.A. Zaks, Phys. Rev. Lett. 95, 123903 (2005)
A. Neiman, L. Schimansky-Geier, A. Cornell-Bell, F. Moss, Phys. Rev. Lett. 83, 4896 (1999)
J.A. Freund, A.B. Neiman, L. Schimansky-Geier, Europhys. Lett. 50, 8 (2000)
R. Rozenfeld, J.A. Freund, A. Neiman, L. Schimansky-Geier, Phys. Rev. E 64, 051107 (2001)
L. Callenbach, P. Hänggi, S.J. Linz, J.A. Freund, L. Schimansky-Geier, Phys. Rev. E 65, 051110 (2002)
T. Prager, L. Schimansky-Geier, Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 71, 031112 (2005)
D.R. Cox, Renewal Theory (Methuen, London, 1962)
S.B. Lowen, M.C. Teich, J. Acoust. Soc. Am. 92, 803 (1992)
R. Ratnam, M.E. Nelson, J. Neurosci. 20, 6672 (2000)
M.J. Chacron, A. Longtin, M. St-Hilaire, L. Maler, Phys. Rev. Lett. 85, 1576 (2000)
Y.H. Liu, X.J. Wang, J. Comp. Neurosci. 10, 25 (2001)
B. Lindner, Phys. Rev. E 69, 022901 (2004)
T.A. Engel, L. Schimansky-Geier, A. Herz, S. Schreiber, I. Erchova, J. Neurophysiol. 100, 1576 (2008)
M.P. Nawrot, C. Boucsein, V. Rodriguez-Molina, A. Aertsen, S. Grun, S. Rotter, Neurocomp. 70, 1717 (2007)
A. Neiman, D.F. Russell, Phys. Rev. Lett. 86, 3443 (2001)
S. Bahar, J. Kantelhardt, A. Neiman, H. Rego, D. Russell, L. Wilkens, A. Bunde, F. Moss, Europhys. Lett. 56, 454 (2001)
A. Neiman, D.F. Russell, Phys. Rev. E 71, 061915 (2005)
T.A. Engel, B. Helbig, D.F. Russell, L. Schimansky-Geier, A. Neiman, Phys. Rev. E 80, 021919 (2009)
J.W. Middleton, M.J. Chacron, B. Lindner, A. Longtin, Phys. Rev. E 68, 021920 (2003)
T. Schwalger, L. Schimansky-Geier, Phys. Rev. E 77, 031914 (2008)
J. Benda, A.V.M. Herz, Neural Comp. 15, 2523 (2003)
M.J. Chacron, K. Pakdaman, A. Longtin, Neural Comp. 15, 253 (2003)
M.J. Chacron, A. Longtin, L. Maler, J. Neurosci. 21, 5328 (2001)
M.J. Chacron, B. Lindner, A. Longtin, Phys. Rev. Lett. 92, 080601 (2004)
B. Lindner, T. Schwalger, Phys. Rev. Lett. 98, 210603 (2007)
T. Schwalger, B. Lindner, Phys. Rev. E 78, 021121 (2008)
J.A. McFadden, J. Roy. Stat. Soc. B 24, 364 (1962)
X.J. Wang, J. Neurophysiol. 79, 1549 (1998)
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Schwalger, T., Lindner, B. Theory for serial correlations of interevent intervals. Eur. Phys. J. Spec. Top. 187, 211–221 (2010). https://doi.org/10.1140/epjst/e2010-01286-y
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DOI: https://doi.org/10.1140/epjst/e2010-01286-y