http://bactra.org/notebooks Cosma's Notebooks en http://bactra.org/notebooks/2009/04/10#synchronization <P>Synchronization in the brain probably <a href="neuro-synch.html">deserves its own notebook</a>. So does <a href="topology-and-synchronization.html">the influence of network toplogy on synchronization</a>. <P><em>Things to work on:</em> how well does the "generalized synchrony" notion itself generalize to systems with stochastic dynamics? <P>See also: <a href="complex-networks.html">Complex Networks</a>; <a href="chaos.html">Dynamics</a>; <a href="excitable-media.html">Excitable Media</a>; <a href="neural-coding.html">Neural Coding</a>; <a href="neuro-synch.html">Synchronization in Neural Systems</a> <ul>Recommended, big picture: <li>Arkady Pikovsky, Michael Rosenblum and J&uuml;rgen Kurths, <cite>Synchronization: A Universal Concept in Nonlinear Sciences</cite> <li>Arthur Winfree, <cite>The Geometry of Biological Time</cite> </ul> <ul>Recommended, close-ups: <li>Fatihcan M. Atay, T&uuml;rker Biyikoglu and J&uuml;rgen Jost, "On the synchronization of networks with prescribed degree distributions", <a href="http://arxiv.org/abs/nlin.AO/0407024">nlin.AO/0407024</a> [Networks with any degree distribution can be made arbitrarily hard to synchronize] <li>Fatihcan M. Atay, J&uuml;rgen Jost and Andreas Wende, "Delays, connection topology, and synchronization of coupled chaotic maps", <a href="http://arxiv.org/abs/cond-mat/0312177">cond-mat/0312177</a> <li>Roberto F. Gal&aacute;n, G. Bard Ermentrout, and Nathaniel N. Urban, "Efficient Estimation of Phase-Resetting Curves in Real Neurons and its Significance for Neural-Network Modeling", <a href="http://dx.doi.org/10.1103/PhysRevLett.94.15810"><cite>Physical Review Letters</cite> <strong>94</strong> (2005): 158101</a> <li>J&uuml;rgen Jost and M. P. Joy, "Spectral Properties and Synchronization in Coupled Map Lattices," <citE>Physical Review E</cite> <strong>65</strong> (2002): 016201 = <a href="http://arxiv.org/abs/nlin.CD/0110037">nlin.CD/0110037</a> <li><a href="http://www.cs.cas.cz/~mp">Milan Palus</a> [to whom thanks for reprints] <ul> <li>and Vladimir Komarek, Zbynek Hrncir and Katalin Sterbova, "Synchronization as adjustment of information rates: Detection from bivariate time series", <a href="http://dx.doi.org/10.1103/PhysRevE.63.046211"><cite>Physical Review E</cite> <strong>63</strong> (2001): 046211</a> <li>and Aneta Stefanovska, "Direction of coupling from phases of interacting oscillators: An information-theoretic approach", <a href="http://dx.doi.org/10.1103/PhysRevE.67.055201"><citE>Physical Review E</cite> <strong>67</strong> (2003): 055201</a> </ul> <li>R. Quian Quiroga, A. Kraskov, T. Kreuz and P. Grassberger, "Performance of different synchronization measures in real data: A case study on electroencephalographic signals," <cite>Physical Review E</cite> <strong>65</strong> (2002): 041903 = <a href="http://arxiv.org/abs/nlin.CD/0109023">nlin.CD/0109023</a> [See also the interesting "Comment" by R. B. Duckrow and A. M. Albano, <cite>PRE</cite> <strong>67</strong> (2003): 063901, and the reply by the original authors, 063902 in the same volume] </ul> <ul>Modesty forbids me to recommend: <li>Kristina Lisa Klinkner, CRS, and Marcelo F. Camperi, "Measuring Shared Information and Coordinated Activity in Neuronal Networks", <a href="http://arxiv.org/abs/q-bio.NC/0506009">q-bio.NC/0506009</a> </ul> <ul>To read: <li>Juan A. Acebr&oacute;n, L. L. Bonilla, Conrad J. P&eacute;rez Vicente, F&eacute;lix Ritort and Renato Spigler, "The Kuramoto model: A simple paradigm for synchronization phenomena ", <a href="http://dx.doi.org/10.1103/RevModPhys.77.137"><cite>Reviews of Modern Physics</cite> <strong>77</strong> (2005): 137--186</a> <li>R. E. Amritkar, Sarika Jalan and Chin-Kun Hu, "Synchronized clusters in coupled map networks. II. Stability analysis", <a href="http://dx.doi.org/10.1103/PhysRevE.72.016212"><cite>Physical Review E</cite> <strong>72</strong> (2005): 016212</a> <li>C. Anteneodo, A. M. Batista and R. L. viana, "Chaos synchronization in long-range coupled map lattices", <a href="http://arxiv.org/abs/nlin.CD/0401034">nlin.CD/0401034</a> <li>C. Anteneodo, S. E. de S. Pinto, A. M. Batista and R. L. Viana, "Analytical results for coupled map lattices with long-range interactions", <a href="http://arxiv.org/abs/nlin.CD/0308014">nlin.CD/0308014</a> <li>Fernando Antoneli, Ana Paula S Dias, Martin Golubitsky and Yunjiao Wang, "Patterns of synchrony in lattice dynamical systems", <a href="http://dx.doi.org/10.1088/0951-7715/18/5/016"><cite>Nonlinearity</cite> <strong>18</strong> (2005): 2193--2209</a> <li>Toru Aonishi and Masato Okada, "Dynamically-Coupled Oscillators: Cooperative Behavior via Dynamical Interaction," <a href="http://arxiv.org/abs/cond-mat/0207506">cond-mat/0207506</a> <li>L. Arnhold, Peter Grassberger, K. Lehnertz and C. E. Elger, <cite>Physica D</cite> <strong>134</strong> (1999): 419 <li>Fatihcan M. Atay <ul> <li>"Distributed Delays Facilitate Amplitude Death of Coupled Oscillators", <a href="http://arxiv.org/abs/nlin.AO/0312050">nlin.AO/0312050</a> = <cite>Physical Review Letters</cite> <strong>91</strong> (2003): 094101 <li>"Total and partial amplitude death in networks of diffusively coupled oscillators", <a href="http://arxiv.org/abs/nlin.AO/0401033">nlin.AO/0401033</a> = <cite>Physica D</cite> <strong>183</strong> (2003): 1--18 </ul> <li>Franco Bagnoli and Raul Rechtman, "Synchronization universality classes and stability of smooth, coupled map lattices", <a href="http://arxiv.org/abs/cond-mat/0507205">cond-mat/0507205</a> <li>M. S. Baptista, T. Pereira, J. C. Sartorelli, I. L. Caldas and J. Kurths, "Phase Synchronization and invariant measures in sinusoidally perturbed chaotic systems", <a href="http://arxiv.org/abs/cond-mat/0507715">cond-mat/0507715</a> <li>Stephan Bialonski and Klaus Lehnertz, "Identifying phase synchronization clusters in spatially extended dynamical systems", <a href="http://dx.doi.org/http://link.aps.org/abstract/PRE/v74/e051909"><cite>Physical Review E</cite> <strong>74</strong> (2006): 051909</a> <li>Juan C. Botero and Jean-Jacques E. Slotine, "Examples of Synchronization in Discrete Chaotic Systems", <a href="http://arxiv.org/abs/nlin.CD/0609064">nlin.CD/0609064</a> <li>Michael A. Buice and Carson C. Chow, "Correlations, fluctuations and stability of a finite-size network of coupled oscillators", <a href="http://arxiv.org/abs/0704.1650">arxiv:0704.1650</a> <li>A. Carpio, "Wave trains, self-oscillations and synchronization in discrete media", <a href="http://arxiv.org/abs/cond-mat/0506119">cond-mat/0506119</a> <li>Mario Chavez, Claude Adam, Vincent Navarro, Stefano Boccaletti and Jacques Martinerie, "On the intrinsic time scales involved in synchronization: A data-driven approach", <a href="http://dx.doi.org/10.1063/1.1938467"><cite>Chaos</cite> <strong>15</strong> (2005): 023904</a> <li>Lauren M. Childs and Steven H. Strogatz, "Stability diagram for the forced Kuramoto model", <a href="http://arxiv.org/abs/0807.4717">arxiv:0807.4717</a> <li>L. Cisneros, J. Jimenez, M. G. Cosenza, and A. Parravano, "Information transfer and nontrivial collective behavior in chaotic coupled map networks," <a href="http://arxiv.org/abs/nlin.CD/0202010">nlin.CD/0202010</a> <li>M. Escalona-Moran, M. G. Cosenza, P. Guillen and P. Coutin, "Synchronization and clustering in electroencephalographic signals", <a href="http://arxiv.org/abs/nlin.CD/0506014">nlin.CD/0506014</a> <li>Hirokazu Fujisaka and Masayoshi Inoue, "Statistical-physical theory of multivariate temporal fluctations: Global characterization and temporal correlation", <cite>Physical Review A</cite> <strong>41</strong> (1990): 5302--5319 [Finite-size scaling in the Kuramoto model, really] <li>Hirokazu Fujisaka, Satoki Uchiyama and Takehiko Horita, "Mapping Model of Chaotic Phase Synchronization", <a href="http://arxiv.org/abs/nlin.CD/0506010">nlin.CD/0506010</a> <li>P. Garcia, A. Parravano, M. G. Cosenza, J. Jimenez, and A. Marcano, "Coupled Map Networks as Communication Schemes," <a href="http://arxiv.org/abs/nlin.CD/0201042">nlin.CD/0201042</a> <li>Alexander E. Hramov and Alexey A. Koronovskii, "Generalized synchronization: a modified system approach", <a href="http://arxiv.org/abs/nlin.CD/0506023">nlin.CD/0506023</a> <li>Alexander E. Hramov, Alexey A. Koronovskii, Mariya K. Kurovskaya, and Olga I. Moskalenko, "Synchronization of spectral components and its regularities in chaotic dynamical systems", <a href="http://dx.doi.org/10.1103/PhysRevE.71.056204"><cite>Physical Review E</cite> <strong>71</strong> (2005): 056204</a> <li>Ali Jadbabaie, Nader Motee and Mauricio Barahona, "On the stability of the Kuramoto model of coupled nonlinear oscillators", <a href="http://arxiv.org/abs/math.OC/0504419">math.OC/0504419</a> <li>Sarika Jalan, R. E. Amritkar and Chin-Kun Hu, "Synchronized clusters in coupled map networks. I. Numerical studies", <a href="http://dx.doi.org/10.1103/PhysRevE.72.016211"><cite>Physical Review E</cite> <strong>72</strong> (2005): 016211</a> <li>Sarika Jalan, Fatihcan M. Atay and J&uuml;rgen Jost <ul> <li>"Detection of synchronised chaos in coupled map networks using symbolic dynamics", <a href="http://arxiv.org/abs/nlin.CD/0510057">nlin.CD/0510057</a> <li>"Symbolic synchronization and the detection of global properties of coupled dynamics from local information", <a href="http://dx.doi.org/10.1063/1.2336415"><cite>Chaos</cite> <strong>16</strong> (2006): 033124</a> = <a href="http://arxiv.org/abs/nlin.CD/0604016">nlin.CD/0604016</a> </ul> <li>J&uuml;rgen Jost and Kiran M. Kolwankar, "Global Analysis of Synchronization in Coupled Maps", <a href="http://dx.doi.org/10.1142/S0218127406017087">International Journal of Bifurcations and Chaos</cite> <strong>16</strong> (2006): 3695--3703</a> <li> I. Kanter, W. Kinzel and E. Kanter, "Secure exchange of information by synchronization of neural networks," <a href="http://arxiv.org/abs/cond-mat/0202112">cond-mat/0202112</a> <li>Y. Kuramoto, <cite>Chemical Oscillations, Waves, and Turbulence</cite> [Now re-issued as a cheap Dover paperback] <li>Susanna C. Manrubia, Alexander S. Mikhailov and Damian H. Zanette, <cite>Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems</cite> <li>Arturo C. Mart&iacute; and C. Masoller, "Delay-induced Synchronization Phenomena in an Array of Globally Coupled Logistic Maps," <a href="http://arxiv.org/abs/nlin.CD/0212037">nlin.CD/0212037</a> <li>Norbert Marwan, M. Thiel, N. R. Nowaczyk, "Cross Recurrence Plot Based Synchronization of Time Series," <a href="http://arxiv.org/abs/physics/0201062">physics/0201062</a> <li>M. S. O. Massunaga and M. Bahiana, "Synchronization in large populations of limit cycle oscillators with long-range interactions," <a href="http://arxiv.org/abs/cond-mat/0201508">cond-mat/0201508</a> <li>Patrick McGraw and Michael Menzinger, "Analysis of Nonlinear Synchronization Dynamics of Oscillator Networks by Laplacian Spectral Methods", <a href="http://arxiv.org/abs/cond-mat/0610522">cond-mat/0610522</a> <li>P. K. Mohanty and Antonio Politi, "Partial synchronization of globally coupled oscillators", <a href="http://arxiv.org/abs/cond-mat/0507037">cond-mat/0507037</a> <li>Ernest Montbri&oacute;, J&uuml;rgen Kurths and Bernd Blasius, "Synchronization of two interacting populations of oscillators", <a href="http://arxiv.org/abs/nlin.AO/0406004">nlin.AO/0406004</a> <li>Markus Muller, Gerold Baier, Andreas Galker, Ulrich Stephani and Hiltrud Muhle, "Detection and characterization of changes of the correlation structure in multivariate time series", <a href="http://dx.doi.org/10.1103/PhysRevE.71.046116"><cite>PRE</cite> <strong>71</strong> (2005): 046116</a> <li>Miguel A. Munoz and Romualdo Pastor-Satorras, "Stochastic theory of synchronization transitions in extended systems," <a href="http://arxiv.org/abs/cond-mat/0301059">cond-mat/0301059</a> <li>Amitabha Nandi, G. Santhosh, R. K. Brojen Singh, Ram Ramaswamy, "The synchronization of stochastic oscillators", <a href="http://arxiv.org/abs/q-bio.QM/0608022">q-bio.QM/0608022</a> <li>Grigory V. Osipov, Mikhail V. Ivanchenko, Jurgen Kurths, and Bambi Hu, "Synchronized chaotic intermittent and spiking behavior in coupled map chains", <a href="http://dx.doi.org/10.1103/PhysRevE.71.056209"><cite>Physical Review E</cite> <strong>71</strong> 92005): 056209</a> <li>Per Ostborn, S. Aberg and G. Ohlen, "Phase transitions towards frequency entrainment in large oscillator lattices," <a href="http://arxiv.org/abs/cond-mat/0302234">cond-mat/0302234</a> <li>Edward Ott, John H. Platig, Thomas M. Antonsen, Michelle Girvan, "Echo Phenomena in Large Systems of Coupled Oscillators", <a href="http://arxiv.org/abs/0807.4499">arxiv:0807.4499</a> [Sounds cool] <li>Diego Pazo, "Thermodynamic limit of the first-order phase transition in the Kuramoto model", <a href="http://arxiv.org/abs/nlin.AO/0509020">nlin.AO/0509020</a> <li>M. Pineda and M. G. Cosenza, "Synchronization in driven versus autonomous coupled chaotic maps", <a href="http://dx.doi.org/10.1103/PhysRevE.71.057201"><cite>Physical Review E</cite> <strong>71</strong> (2005): 057201</a> <li>R. Quian Quiroga, L. Arnhold and Peter Grassberger, <cite>Physical Review E</cite> <strong>61</strong> (2000): 5142 <li>R. Quian Quiroga, T. Kreuz and Peter Grassberger, "Event synchronization: a simple and fast method to measure synchronicity and time delay patterns," <a href="http://arxiv.org/abs/nlin.CD/0202065">nlin.CD/0202065</a> <li>Govindan Rangarajan and Mingzhou Ding, "Stability of Synchronized Chaos in Coupled Dynamical Systems," <a href="http://arxiv.org/abs/nlin.CD/0201037">nlin.CD/0201037</a> <li>Nikolai F. Rulkov and Valentin S. Afraimovich, "Detectability of non-differentiable generalized synchrony," <a href="http://arxiv.org/abs/nlin.CD/0303022">nlin.CD/0303022</a> <li>Nikolai F. Rulkov, M. S. Sushcik, Lev S. Tsimring and Henry D. I. Abarbanel, <cite>Physical Review E</cite> <strong>51</strong> (1995): 980 <li>Philip Seliger, Stephen C. Young, and Lev S. Tsimring, "Plasticity and learning in a network of coupled phase oscillators," <a href="http://arxiv.org/abs/nlin.AO/0110044">nlin.AO/0110044</a> <li>A. Shabunin, V. Astakhov and Jurgen Kurths, "Quantitative analysis of chaotic synchronization by means of coherence", <a href="http://dx.doi.org/10.1103/PhysRevE.72.016218"><cite>Physical Review E</cite> <strong>72</strong> (2005): 016218</a> <li>E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, "Inverse Anticipating Synchronization," <a href="http://arxiv.org/abs/nlin.CD/0111011">nlin.CD/0111011</a> <li>Michael Small and Kevin Judd, "Detecting periodicity in experimental data using linear modeling techniques", <a href="http://arxiv.org/abs/physics/9810021">physics/9810021</a> [Not strictly synchrony measurement, but closely related] <li>Hiromichi Suetani, Yukito Iba and Kazuyuki Aihara <ul> <li>"Detecting Generalized Synchronization Between Chaotic Signals: A Kernel-based Approach", <a href="http://arxiv.org/abs/nlin.CD/0507006">nlin.CD/0507006</a> = <cite>Journal of Physics A: Mathematical and General</cite> <strong>39</strong> (2006) 10723--10742 <li>"Detecting Generalized Synchronization of Chaotic Dynamical Systems: A Kernel-based Method and Choice of Its Parameter", <a href="http://arxiv.org/abs/nlin.CD/0507041">nlin.CD/0507041</a> </ul> <li>Jun-nosuke Teramae and Yoshiki Kuramoto, "Strong Desynchronizing Effects of Weak Noise in Globally Coupled Systems," <a href="http://arxiv.org/abs/nlin.CD/0102006">nlin.CD/0102006</a> <li>Xiao Fan Wang, "Slower Speed and Stronger Coupling: Adaptive Mechanisms of Self-Organized Chaos Synchronization," <a href="http://arxiv.org/abs/cond-mat/0109395">cond-mat/0109395</a> <li>Damian H. Zanette <ul> <li>"Disturbing synchronization: Propagation of perturbations in networks for coupled oscillators", <a href="http://arxiv.org/abs/cond-mat/0412356">cond-mat/0412356</a> <li>"Propagating structures in globally coupled systems with time delays," <cite>Physical Review E</cite> <strong>62</strong> (2000): 3167--3172 <li>"Propagation of small perturbations in synchronized oscillator networks", <a href="http://arxiv.org/abs/cond-mat/0409159">cond-mat/0409159</a> <li>"Synchronization and frustration in oscillator networks with attractive and repulsive interactions", <a href="http://arxiv.org/abs/cond-mat/0509063">cond-mat/0509063</a> </ul> <li>Damian H. Zanette and Alexander S. Mikhailov, "Dynamical systems with time-dependent coupling: clustering and critical behavior", <cite>Physica D</cite> <strong>194</strong> (2004): 203--218 </ul>