Pulse coupled oscillators and the phase resetting curve

Mathematical Biosciences

Volumn 226, Issue 2, 2010, Pages 77-96

  Canavier, Carmen C ^a,b   Achuthan, Srisairam ^a  

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

^b LOUISIANA STATE UNIVERSITY SCHOOL OF MEDICINE   (United States)

Author keywords

Clustering; Phase locking; Phase resetting; Pulse coupled oscillators; Splay; Synchronization

Indexed keywords

BIOLOGICAL NETWORKS; CIRCUIT ARCHITECTURES; COUPLED OSCILLATORS; DISCRETE MAPS; FIRING MODES; FIRING PATTERNS; FIXED POINTS; FIXED-DELAY; FORCED OSCILLATORS; GENERAL APPROACH; IN-NETWORK; LIMIT-CYCLE OSCILLATORS; PHASE RESETTING; PHASE RESETTING CURVES; PHASE-LOCKING; PULSE COUPLED OSCILLATORS;

DELAY CIRCUITS; OSCILLATORS (ELECTRONIC); OSCILLATORS (MECHANICAL); PHASE LOCKED LOOPS;

COUPLED CIRCUITS;

CLUSTER ANALYSIS; CURVATURE; NETWORK ANALYSIS; NETWORK DESIGN; NUMERICAL METHOD; OSCILLATION; PERTURBATION; PHASE VELOCITY; SYNCHRONY;

ACTION POTENTIAL; ANALYTICAL EQUIPMENT; ANALYZER; ARCHITECTURE; ARTIFICIAL NEURAL NETWORK; COGNITION; LOCOMOTION; MATHEMATICAL ANALYSIS; MATHEMATICAL MODEL; MATHEMATICAL PARAMETERS; MOTIVATION; PHASE RESETTING CURVE; PRESYNAPTIC MEMBRANE; PULSE COUPLED OSCILLATOR; REVIEW; RHYTHM; SIMULATION; WORKING MEMORY;

ACTION POTENTIALS; ALGORITHMS; MODELS, NEUROLOGICAL; NERVE NET; NEURONS;

EID: 77954144054     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mbs.2010.05.001     Document Type: Review

References (84)

1. Kuramoto Y. Chemical Oscillations, Waves and Turbulence 1984, Springer-Verlag, Berlin.
2. Winfree A.T. The Geometry of Biological Time 2001, Springer-Verlag, New York.
3. Mirollo R.E., Strogatz S.H. Synchronization of pulse coupled biological oscillators. SIAM J. Appl. Math. 1990, 50:1645.
4. Peskin C.S. Mathematical Aspects of Heart Physiology 1975, Courant Institute of Mathematical Sciences, New York University, New York, pp. 268-278.
5. Ernst U., Pawelzik K., Geisel T. Synchronization induced by temporal delays in pulse coupled oscillators. Phys. Rev. Lett. 1995, 74:1570.
6. Ernst U., Pawelzik K., Geisel T. Delay-induced multistable synchronization of biological oscillators. Phys. Rev. E 1998, 57:2150.
7. Canavier C.C., Butera R.J., Dror R.O., Baxter D.A., Clark J.W., Byrne J.H. Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation. Biol. Cybern. 1997, 77:367.
8. Canavier C.C., Baxter D.A., Clark J.W., Byrne J.H. Control of multistability in ring circuits of oscillators. Biol. Cybern. 1999, 80:87.
9. Canavier C.C., Baxter D.A., Byrne J.H. Repetitive Firing of Action Potentials: A Dynamic Analysis. Encyclopedia of Life Sciences 2002, vol. 16:271. Nature Publishing Group, London.
10. Luo C., Clark J.W., Canavier C.C., Baxter D.A., Byrne J.H. Multimodal behavior in a four neuron ring circuit: mode switching. IEEE Trans. Biomed. Eng. 2004, 51:205.
11. S. Achuthan, C.C. Canavier, Intrinsic and synaptic dynamics have separate effects on phase locking for weakly coupled oscillators, Conference Abstract: 2010 South East Nerve Net (SENN) and Georgia/South Carolina Neuroscience Consortium (GASCNC) conferences. doi:10.3389/conf.fnins.2010.04.00001.
12. Acker C.D., Kopell N., White J.A. Synchronization of strongly coupled excitatory neurons: relating network behavior to biophysics. J. Comput. Neurosci. 2003, 15:71.
13. Pervouchine D.D., Netoff T.I., Rostein H.G., White J.A., Cunningham M.O., Whittington M.A., Kopell N.J. Low dimensional maps encoding dynamics in entorhinal cortex and hippocampus. Neural Comput. 2006, 18:2617.
14. Wang X.J., Buzsaki G. Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. J. Neurosci. 1996, 16:6402.
15. Maran S.K., Canavier C.C. Using phase resetting to predict 1:1 and 2:2 locking in two neuron networks in which firing order is not always preserved. J. Comput. Neurosci. 2008, 24:37.
16. Oh M., Matveev V. Loss of phase-locking in non-weakly coupled inhibitory networks of type-I model neurons. J. Comput. Neurosci. 2009, 26:303.
17. Achuthan S., Canavier C.C. Phase resetting curves determine synchronization, phase-locking, and clustering in networks of neural oscillators. J. Neurosci. 2009, 29:5218.
18. Sieling F.H., Canavier C.C., Prinz A.A. Predictions of phase-locking in excitatory hybrid networks: excitation does not promote phase-locking in pattern generating networks as reliably as inhibition. J. Neurophysiol. 2009, 102:69.
19. Oprisan S.A., Canavier C.C. Stability analysis of entrainment by two periodic inputs with a fixed delay. Neurocomputing 2003, 52-54:59.
20. Perkel D.H., Schulman J.H., Bullock T.H., Moore G.P., Segundo J.P. Pacemaker neurons: effects of regularly spaced synaptic input. Science 1964, 145:61.
21. C.C. Canavier, S. Achuthan, History of the application of phase resetting curves to neurons coupled in a pulsatile manner, in: N.W. Schultheiss, A.A. Prinz, R.J. Butera (Eds.), Phase Response Curves in Neuroscience: Theory, Experiment and Analysis, Springer Series in Computational Neuroscience, Springer, New York, 2011, in press.
22. Foss J., Milton J. Multistability in recurrent neural loops arising from delay. J. Neurophysiol. 2000, 84:975.
23. Dror R.O., Canavier C.C., Butera R.J., Clark J.W., ByrnE J.H. A mathematical criterion based on phase response curves for stability in a ring of coupled oscillators. Biol. Cybern. 1999, 80:11.
24. Oprisan S.A., Prinz A.A., Canavier C.C. Phase resetting and phase locking in hybrid circuits of one model and one biological neuron. Biophys. J. 2004, 87:2283.
25. Oprisan S.A., Canavier C.C. Stability analysis of rings of pulse-coupled oscillators: the effect of phase resetting in the second cycle after the pulse is important at synchrony and for long pulses. Diff. Equat. Dynam. Syst. 2001, 9:242.
26. Goel P., Ermentrout G.B. Synchrony, stability, and firing patterns in pulse coupled oscillators. Physica D 2002, 163:191.
27. M. Woodman, C.C. Canavier, Phase locking of pulse coupled oscillators with delays is determined by the phase response curve, Frontiers in Systems Neuroscience, Computational and Systems Neuroscience, 2009. doi:10.3389/conf.neuro.06.2009.03.139.
28. Canavier C.C., Gurel Kazanci F., Prinz A.A. Phase resetting curves allow for simple and accurate prediction of robust N:1 phase locking for strongly coupled neural oscillators. Biophys. J. 2009, 97:59.
29. Morris C., Lecar H. Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 1981, 35:193.
30. L. Chandrasekaran, C.C. Canavier, S. Achuthan, Mutually pulse coupled neurons that do not synchronize in isolation can synchronize via reciprocal coupling with another neural population, Computational Neuroscience meeting abstract (Poster ID # 015), San Antonio 2010.
31. S. Achuthan, L. Chandrasekaran, C.C. Canavier, Phase resetting curve analysis of global synchrony, the splay mode and clustering in N neuron all to all pulse coupled networks, in: N.W. Schultheiss, A. Prinz, R. Butera (Eds.), Phase Response Curves in Neuroscience: Theory, Experiment and Analysis, Springer Series in Computational Neuroscience, Springer, New York, 2011, in press.
32. Dror R.O., Canavier C.C., Butera R.J., Clark J.W., Byrne J.H. A mathematical criterion based on phase response curves for stability in a ring of coupled oscillators. Biol. Cybern. 1999, 80:11.
33. S. Achuthan, F.H. Sieling, A.A. Prinz, C.C. Canavier, Phase resetting curves in presence of heterogeneity and noise, in: M. Ding, D. Glanzman (Eds.), Neuronal Variability and its Functional Significance, Oxford University Press, New York, 2011.
34. Winfree A.T. Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 1967, 16:15.
35. Buck J. Synchronous rhythmic flashing of fireflies II. Quart. Rev. Biol. 1988, 63:265.
36. de la Iglesia H.O., Cambras T., Schwartz W.J., Diez-Noguera A. Forced desynchronization of dual circadian oscillators within the rat suprachiasmatic nucleus. Curr. Biol. 2004, 14:796.
37. Buzsaki G. Rhythms of the Brain 2006, Oxford University Press Inc., New York.
38. Yuste R., MacLean J.N., Smith J., Lansner A. The cortex as a central pattern generator. Nat. Rev. Neurosci. 2005, 6:477.
39. Glass L., Mackey M.C. From Clocks to Chaos: The Rhythms of Life 1988, Princeton University Press, Princeton, NJ.
40. Strogatz S.H. Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life 2003, Hyperion, New York.
41. Pikovsky A., Rosenblum M., Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences 2001, Cambridge University Press, Cambridge, UK.
42. Fell J., Klaver P., Lehnertz K., Grunwald T., Schaller C., Elger C.E., Fernandez G. Human memory formation is accompanied by rhinal-hippocampal coupling and decoupling. Nat. Neurosci. 2001, 4:1259.
43. Rodriguez E., George N., Lachaux J.P., Martinerie J., Renault B., Varela F.J. Perception's shadow: long distance synchronization of human brain activity. Nature 1999, 397:430.
44. Huguenard J.R., McCormick D.A. Thalamic synchrony and dynamic regulation of global forebrain oscillations. Trends Neurosci. 2007, 30:350.
45. Traub R.D., Jefferys J.G. Are there unifying principles underlying the generation of epileptic after discharges in vitro?. Prog. Brain Res. 1994, 102:383.
46. Hammond C., Bergman H., Brown P. Pathological synchronization in Parkinson's disease: networks, models and treatments. Trends Neurosci. 2007, 30:357.
47. Uhlhaas P.J., Singer W. Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology. Neuron 2006, 52:155.
48. Neurons, Networks, and Motor Behavior 1997, MIT Press, Cambridge, MA. S.G. Stein, S. Grillner, A.I. Selverston, D.G. Stuart (Eds.).
49. Llinas R.R. I of the Vortex: From Neurons to Self 2002, MIT Press, Cambridge, MA.
50. Velazquez J.L.P., Galan R.F., Dominguez L.G., Leshchenko Y., Lo S., Belkas J., Erra R.G. Phase response curves in the characterization of epileptiform activity. Phys. Rev. E 2007, 76:061912.
51. Mancilla J.G., Lewis T.J., Pinto D.J., Rinzel J., Connors B.W. Synchronization of electrically coupled pairs of inhibitory interneurons in neocortex. J. Neurosci. 2007, 27:2058.
52. Cui J., Canavier C.C., Butera R.J. Functional phase response curves: a method for understanding synchronization of adapting neurons. J. Neurophysiol. 2009, 102:387.
53. Rubin J.E., Terman D. Geometric singular perturbation analysis of neuronal dynamics. Handbook of Dynamical Systems: Toward Applications 2002, vol. 2. Elsevier, Amsterdam.
54. Netoff T.I., Banks M.I., Dorval A.D., Acker C.D., Haas J.S., Kopell N.K., White J.A. Synchronization in hybrid neuronal networks of the hippocampal formation. J. Neurophysiol. 2005, 93:1197.
55. Rinzel J., Ermentrout G.B. Analysis of neural excitability and oscillations. Methods in Neuronal Modeling from Ions to Networks 1998, MIT Press, Cambridge, MA. C. Koch, I. Segev (Eds.).
56. Senn W., Urbanczik R. Similar nonleaky integrate and fire neurons with instantaneous couplings always synchronize. SIAM J. Appl. Math. 2000, 61:1143.
57. Timme M., Wolf F., Geisel T. Coexistence of regular and irregular dynamics in complex networks of pulse coupled oscillators. Phys. Rev. Lett. 2002, 89:258701.
58. Timme M., Wolf F. The simplest problem in the collective dynamics of neural networks: Is synchrony stable?. Nonlinearity 2008, 21:1579.
59. Sharp A.A., O'Neil M.B., Abbott L.F., Marder E. The dynamic clamp: artificial conductances in biological neurons. Trends Neurosci. 1993, 16:389.
60. Sharp A.A., O'Neil M.B., Abbott L.F., Marder E. Dynamic clamp: computer-generated conductances in real neurons. J. Neurophysiol. 1993, 69:992.
61. Talathi S.S., Hwang D.U., Miliotis A., Carney P.R., Ditto W. Predicting synchrony in a heterogeneous pulse coupled oscillators. Phys. Rev. E 2009, 80:021908.
62. Talathi S.S., Hwang D.U., Carney P.R., Ditto W. Synchrony with shunting inhibition in feed-forward inhibitory networks. J. Comput. Neurosci. 2010, 28:305.
63. Golomb D., Wang X.J., Rinzel J. Synchronization properties of spindle oscillations in a thalamic reticular nucleus model. J. Neurophysiol. 1994, 72:1109.
64. Ermentrout G.B., Chow C.C. Modeling neural oscillations. Physiol. Behav. 2002, 77:629.
65. Gerstner W. Rapid phase locking in systems of pulse coupled oscillators with delays. Phys. Rev. Lett. 1996, 76:1755.
66. Izhikevich E.M. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting 2007, The MIT Press, Cambridge, MA.
67. J. Foss, Control of Multistability in Neural Feedback Systems with Delay, Ph.D. Thesis, The University of Chicago, Chicago, IL, 1999.
68. Pavlidis T. Biological Oscillators: Their Mathematical Analysis 1973, Academic Press, New York, NY.
69. Prinz A.A. The dynamic clamp a decade after its invention. AxoBits 2004, 40:6.
70. Prinz A.A., Abbott L.F., Marder E. The dynamic clamp comes of age. Trends Neurosci. 2004, 27:218.
71. Destexhe A., Bal T. Dynamic-Clamp: From Principles to Applications, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting 2009, Springer, New York, NY.
72. Netoff T.I., Acker C.D., Bettencourt J.C., White J.A. Beyond two-cell networks: experimental measurement of neuronal responses to multiple synaptic inputs. J. Comput. Neurosci. 2005, 18:287.
73. G.B. Ermentrout, B. Beverlin II, T. Netoff, Phase response curves to measure ion channel effects on neurons, in: N.W. Schultheiss, A.A. Prinz, R.J. Butera (Eds.), Phase Response Curves in Neuroscience: Theory, Experiment and Analysis, Springer Series in Computational Neuroscience, Springer, New York, 2011, in press.
74. Gutkin B.S., Ermentrout G.B., Reyes A.D. Phase-response curves give the responses of neurons to transient inputs. J. Neurophysiol. 2005, 94:1623.
75. Hodgkin A.L. The local electric changes associated with repetitive action in a non-medullated axon. J. Physiol. 1948, 107:165.
76. Ermentrout G.B., Pascal M., Gutkin B.S. The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators. Neural Comput. 2001, 13:1285.
77. Hauptmann C., Roulet J.C., Niederhauser J.J., et al. External trial deep brain stimulation device for the application of desynchronizing stimulation techniques. J. Neural Eng. 2009, 6:066003.
78. Osorio I., Frei M.G. Seizure abatement with single dc pulses: Is phase resetting at play?. Intl. J. Neural Syst. 2009, 3:149.
79. Rizzuto D.S., Madsen J.R., Broomfield E.B., et al. Reset of human neocortical oscillations during a working memory task. Proc. Natl. Acad. Sci. 2003, 100:7931.
80. Llinás R.R. Inferior olive oscillation as the temporal basis for motricity and oscillatory reset as the basis for motor error correction. Neuroscience 2009, 162:797.
81. Rutishauser U., Ross I.B., Mamelak A.M., Schuman E.M. Human memory strength is predicted by theta-frequency phase-locking of single neurons. Nature 2010, 464:903.
82. Ermentrout G.B., Kopell N. Oscillator death in systems of coupled neural oscillators. SIAM J. Appl. Math. 1990, 50:125.
83. Ermentrout G.B., Kopell N. Multiple pulse interactions and averaging in coupled neural oscillators. J. Math. Biol. 1991, 29:195.
84. Pfeuty B., Mato G., Golomb D., Hansel D. Electrical synapses and synchrony; the role of intrinsic currents. J. Neurosci. 2003, 23:6280.