Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Journal of Computational Neuroscience

Volumn 31, Issue 2, 2011, Pages 401-418

  Woodman, Michael Marmaduke ^a,b   Canavier, Carmen C ^a  

^a LOUISIANA STATE UNIVERSITY HEALTH SCIENCES CENTER   (United States)

^b AIX MARSEILLE UNIVERSITY   (France)

Author keywords

Conduction delays; Oscillator; Pulse coupling; Synchrony

Indexed keywords

BRAIN; FEEDBACK; LOCKS (FASTENERS); OSCILLATORS (ELECTRONIC); OSCILLATORS (MECHANICAL); SYNCHRONIZATION;

CONDUCTION DELAY; DELAYED FEEDBACK; EXISTENCE AND STABILITY; FIRING PATTERNS; NUMBER OF CYCLES; PHASE-LOCKING; PULSE COUPLED OSCILLATORS; PULSE COUPLING; STABILITY CRITERIONS; SYNCHRONY;

STABILITY CRITERIA;

ALGORITHM; ARTICLE; COMPUTER SIMULATION; CONTROLLED STUDY; CORTICAL SYNCHRONIZATION; FEEDBACK SYSTEM; LINEAR SYSTEM; MATHEMATICAL COMPUTING; MATHEMATICAL MODEL; NERVE CELL NETWORK; NERVE CONDUCTION; NONLINEAR SYSTEM; OSCILLATORY POTENTIAL; POSTSYNAPTIC POTENTIAL; PREDICTION; PRESYNAPTIC POTENTIAL; PRIORITY JOURNAL; STIMULUS RESPONSE; TEMPORAL SUMMATION;

EID: 83055188516     PISSN: 09295313     EISSN: 15736873     Source Type: Journal    
DOI: 10.1007/s10827-011-0315-2     Document Type: Article

References (51)

1. S Achuthan CC Canavier 2009 Phase resetting curves determine synchronization, phase-locking, and clustering in networks of neural oscillators The Journal of Neuroscience 29 5218 33 19386918 10.1523/JNEUROSCI.0426-09.2009 1:CAS:528:DC%2BD1MXlsVWhurw%3D
2. CD Acker N Kopell JA White 2003 Synchronization of strongly coupled excitatory neurons: relating network behavior to biophysics Journal of Computational Neuroscience 15 71 90 12843696 10.1023/A:1024474819512 (Pubitemid 36992478)
3. P Andersen H Silfvenius SH Sundberg O Sveen H Wigström 1978 Functional characteristics of unmyelinated fibres in the hippocampal cortex Brain Research 144 1 11 8 638755 10.1016/0006-8993(78)90431-6 1:STN:280:DyaE1c7ksFyltw%3D%3D (Pubitemid 8297905)
4. CC Canavier SA Achuthan 2010 Pulse coupled oscillators and the phase resetting curve Mathematical Biosciences 226 77 96 20460132 10.1016/j.mbs.2010. 05.001
5. Canavier, C. C,, Fernandez, F., Kispersky, T., & White, J. A. (2009). Generic solutions for pulse coupled oscillatory neurons: Synchrony, antiphase, and leader/follower. Program No. 321.6. 2009 Neuroscience Meeting Planner. Chicago, IL: Society for Neuroscience, 2009. Online.
6. Chandrasekaran, L., Achuthan, S., & Canavier, C. C. (2011). Stability of two cluster solutions in pulse coupled networks of neural oscillators. J. Computational Neurosci.
7. SM Crook GB Ermentrout JM Bower 1998 Dendritic and synaptic effects in systems of coupled cortical oscillators Journal of Computational Neuroscience 5 315 329 9663554 10.1023/A:1008839112707 1:STN:280:DyaK1czis1KrtQ%3D%3D (Pubitemid 28274311)
8. O D'Huys R Vicente T Erneux J Danckaert I Fischer 2008 Synchronization properties of network motifs: influence of coupling delay and symmetry Chaos 18 037116 19045490 10.1063/1.2953582
9. M Dhamala VK Jirsa M Ding 2004 Enhancement of neural synchrony by time delay Physical Review Letters 92 074104 14995856 10.1103/PhysRevLett.92.074104
10. RO Dror CC Canavier RJ Butera JW Clark JH Byrne 1999 A mathematical criterion based on phase response curves for the stability of a ring network of oscillators Biological Cybernetics 80 11 23 20809292 10.1007/s004220050501 1:STN:280:DC%2BC3cjovVWgsQ%3D%3D (Pubitemid 129547932)
11. MG Earl SH Strogatz 2003 Synchronization in oscillator networks with delayed coupling: a stability criterion Physical Review E 67 036204 10.1103/PhysRevE.67.036204
12. AK Engel P Konig AK Kreitner W Singer 1991 Interhemispheric synchronization of oscillatory neural responses in cat visual cortex Science 252 1177 1179 10.1126/science.252.5009.1177 (Pubitemid 21917031)
13. GB Ermentrout N Kopell 1998 Fine structure of neural spiking and synchronization in the presence of conduction delays Proceedings of the National Academy of Sciences of the United States of America 95 1259 1264 9448319 10.1073/pnas.95.3.1259 1:CAS:528:DyaK1cXosFWjsA%3D%3D (Pubitemid 28124946)
14. U Ernst K Pawelzik T Geisel 1995 Synchronization induced by temporal delays in pulse-couple oscillators Physical Review Letters 74 1570 1573 10059062 10.1103/PhysRevLett.74.1570 1:CAS:528:DyaK2MXjvFKmtL0%3D
15. U Ernst K Pawelzik T Geisel 1998 Delay-induced multistable synchronization of biological oscillators Physical review E 57 2150 2162 10.1103/PhysRevE.57.2150 1:CAS:528:DyaK1cXhtFCntb4%3D
16. I Fischer R Vicente JM Buldu M Peil CR Mirasso MC Torrent Garcia-Ojalvo 2006 Zero-lag long range synchronization via dynamical relaying Physical Review Letters 97 123902 17025966 10.1103/PhysRevLett.97.123902 (Pubitemid 44414738)
17. Foss, J. (1999). Control of multistability in neural feedback systems with delay (PhD thesis). Chicago: The University of Chicago.
18. J Foss J Milton 2000 Multistability in recurrent neural loops arising from delay Journal of Neurophysiology 84 975 985 10938321 1:STN:280: DC%2BD3cvivFGgsQ%3D%3D (Pubitemid 30655617)
19. P Fries 2005 A mechanism for cognitive dynamics: neuronal communication through neuronal coherence Trends in Cognitive Neuroscience 9 1364 68 10.1016/j.tics.2005.08.011 (Pubitemid 41338822)
20. Galassi, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Alken, P., et al. (2009). GNU scientific library reference manual (3rd ed.). United Kingdom: Network Theory Ltd.
21. Golubitsky, M., Stewart, I. N., & Schaeffer, D. G. (1988). Singularities and Groups in Bifurcation Theory: Vol. II, Applied Mathematical Sciences 69. Springer-Verlag, New York.
22. EM Izhikevich 1998 Phase models with explicit time delays Physical Review E 58 905 908 10.1103/PhysRevE.58.905 1:CAS:528:DyaK1cXktlOksrs%3D (Pubitemid 128596119)
23. J Karbowski N Kopell 2000 Multispikes and synchronization in a large neural network with temporal delays Neural Computation 12 1573 1606 10935919 10.1162/089976600300015277 1:STN:280:DC%2BD3cvhvVGisg%3D%3D
24. TW Ko B Ermentrout 2009 Delays and weakly coupled neuronal oscillators Philosophical Transactions of the Royal Society 367 1097 1115 10.1098/rsta.2008.0259
25. P Konig TB Schillen 1991 Stimulus-dependent assembly formation of oscillatory responses: I. Synchronization Neural Computation 3 155 166 10.1162/neco.1991.3.2.155
26. P Konig AK Engel W Singer 1995 Relation between oscillatory activity and long range synchronization in cat visual cortex Proceedings of the National academy of Sciences of the United States of America 92 290 294 7816836 10.1073/pnas.92.1.290 1:STN:280:DyaK2M7hs1Gntg%3D%3D
27. L Melloni C Molina M Pena D Torres W Singer E Rodriguez 2007 Synchronization of neural activity across cortical areas correlates with conscious perception The Journal of Neuroscience 27 2858 2865 17360907 10.1523/JNEUROSCI.4623-06.2007 1:CAS:528:DC%2BD2sXjs1Wqu70%3D (Pubitemid 46438979)
28. RE Mirollo SH Strogatz 1990 Synchronization of pulse-coupled biological oscillators SIAM Journal on Applied Mathematics 50 1645 1662 10.1137/0150098
29. C Morris H Lecar 1981 Voltage oscillations in the barnacle giant muscle fiber Biophysical Journal 35 193 213 7260316 10.1016/S0006-3495(81)84782-0 1:STN:280:DyaL3M3lslSmtw%3D%3D (Pubitemid 11079105)
30. M Oh V Matveev 2009 Loss of phase-locking in non-weakly coupled inhibitory networks of type-I model neurons Journal of Computational Neuroscience 26 303 320 18690530 10.1007/s10827-008-0112-8
31. SA Oprisan AA Prinz CC Canavier 2004 Phase resetting and phase locking in hybrid circuits of one model and one biological neuron Biophysical J. 87 2283 2298 10.1529/biophysj.104.046193 1:CAS:528:DC%2BD2cXot12nu7s%3D (Pubitemid 41071330)
32. JL Perez Velasquez RF Galan LG Dominguez Y Leshchenko S Lo J Belkas R Guevara Erra 2007 Phase response curves in the characterization of epileptiform activity Physical Review E 76 061912 10.1103/PhysRevE.76.061912 (Pubitemid 351377703)
33. DD Pervouchine TI Netoff HG Rotstein JA White MO Cunningham MA Whittington NJ Kopell 2006 Low-dimensional maps encoding dynamics in entorhinal cortex and hippocampus Neural Computation 18 1 34 10.1162/neco.2006.18.11.2617 (Pubitemid 44684601)
34. Peskin, CS. (1975). Mathematical aspects of heart physiology. New York: Courant Institute of Mathematical Sciences, New York, 268-278.
35. A Prasad SK Dana R Kamatak J Kurths B Blasius R Ramaswamy 2008 Universal occurrence of the phase-flip bifurcation in time-delay coupled systems Chaos 18 023111 18601478 10.1063/1.2905146
36. MWH Remme M Lengyel BS Gutkin 2009 The role of ongoing dendritic oscillations in single-neuron dynamics PLoS Computational Biology 5 9 e1000493 19730677 10.1371/journal.pcbi.1000493
37. Rinzel, J., & Ermentrout, G. B. (1998). Analysis of neural excitability and oscillations. In C. Koch & I. Segev (Eds.), Methods in neuronal modeling from ions to networks. Cambridge: MIT.
38. HG Schuster P Wagner 1989 Mutual entrainment of two limit cycle oscillators with time delayed coupling Prog. Theor. Phys. 81 939 10.1143/PTP.81.939
39. HG Schuster P Wagner 1990 A model for neuronal oscillations in the visual cortex. 2. Phase description of the feature dependent synchronization Biological Cybernetics 64 83 85 2285763 10.1007/BF00203634 1:STN:280: DyaK3M7jslOgsQ%3D%3D
40. FH Sieling CC Canavier AA Prinz 2009 Predictions of phase-locking in excitatory hybrid networks: excitation does not promote phase-locking in pattern-generating networks as reliably as inhibition Journal of Neurophysiology 102 1 69 84 19357337 10.1152/jn.00091.2009
41. W Singer 1993 Synchronization of cortical activity and its putative role in information processing and learning Annual Review of Physiology 55 349 374 8466179 10.1146/annurev.ph.55.030193.002025 1:STN:280:DyaK3s3itFWjuw%3D%3D (Pubitemid 23102137)
42. W Singer 1999 Neural synchrony: a versatile code for definition of relations Neuron 24 49 65 10677026 10.1016/S0896-6273(00)80821-1 1:CAS:528:DyaK1MXmsVOrsr8%3D (Pubitemid 29466147)
43. M Timme F Wolf 2008 The simplest problem in the collective dynamics of neural networks: is synchrony stable? Nonlinearity 21 1579 1599 10.1088/0951-7715/21/7/011
44. M Timme F Wolf T Geisel 2002 Coexistence of regular and irregular dynamics in complex networks of pulse coupled oscillators Physical Review Letters 89 258701 12484926 10.1103/PhysRevLett.89.258701
45. ABL Tort HG Rotstein T Dugladze T Gloveli NJ Kopell 2007 On the formation of gamma-coherent cell assemblies by oriens lacunosum-moleculare interneurons in the hippocampus PNAS 104 13490 13495 17679692 10.1073/pnas.0705708104 1:CAS:528:DC%2BD2sXps1Kguro%3D (Pubitemid 47293972)
46. RD Traub MA Whittington IM Stanford JGR Jefferys 1996 A mechanism for the generation of long-range synchronous fast oscillations in the cortex Nature 383 621 624 8857537 10.1038/383621a0 1:CAS:528:DyaK28XmtlSqsbw%3D (Pubitemid 26347583)
47. PJ Uhlhass W Singer 2006 Neural synchrony in brain disorders: relevance for cognitive dysfunction and pathophysiology Neuron 52 155 158 10.1016/j.neuron.2006.09.020 (Pubitemid 44466360)
48. PJ Uhlhass G Pipa B Lima L Melloni S Neuenschwander D Nikolic W Singer 2009 Neural synchrony in cortical networks: history concept and current status Frontiers in Integrative Neuroscience 10.3389
49. R Vicente LL Gollo CR Mirasso I Fischer G Pipa 2008 Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays PNAS 105 17157 17162 18957544 10.1073/pnas.0809353105 1:CAS:528:DC%2BD1cXhtlKlt7nL
50. XJ Wang G Buzsaki 1996 Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model The Journal of Neuroscience 16 6402 6413 8815919 1:CAS:528:DyaK28Xnt1Ortbc%3D (Pubitemid 26337364)
51. T Womelsdorf J-M Schoeffelen R Oostenveld W Singer R Desimone AK Engel P Fries 2007 Modulation of neuronal interactions through neuronal synchronization Science 316 1609 1612 17569862 10.1126/science.1139597 1:CAS:528: DC%2BD2sXmtl2jsL8%3D (Pubitemid 46982032)