Typical trajectories of coupled degrade-and-fire oscillators: From dispersed populations to massive clustering

Journal of Mathematical Biology

Volumn 68, Issue 7, 2014, Pages 1627-1652

  Fernandez, Bastien ^a   Tsimring, Lev S ^b  

^a CENTRE DE PHYSIQUE THÉORIQUE   (France)

^b UNIVERSITY OF CALIFORNIA SAN DIEGO   (United States)

Author keywords

Coupled oscillators; Extensive clustering; Random initial conditions

Indexed keywords

REPRESSOR PROTEIN;

ANIMAL; BIOLOGICAL MODEL; BIOLOGICAL RHYTHM; CLUSTER ANALYSIS; HUMAN; MATHEMATICAL PHENOMENA; METABOLISM; NONLINEAR SYSTEM; PHYSIOLOGY;

ANIMALS; BIOLOGICAL CLOCKS; CLUSTER ANALYSIS; HUMANS; MATHEMATICAL CONCEPTS; MODELS, BIOLOGICAL; MODELS, NEUROLOGICAL; NONLINEAR DYNAMICS; REPRESSOR PROTEINS;

EID: 84900005083     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-013-0680-8     Document Type: Article

References (25)

1. Acebron JA, Bonilla LL, Perez-Vicente CJ, Ritort F, Spigler R (2005) The Kuramoto model: a simple paradigm for synchronization phenomena. Rev Mod Phys 77: 137-185.
2. Bottani S (1995) Pulse-coupled relaxation oscillators: from biological synchronization to self-organized criticality. Phys Rev Lett 74: 4189.
3. Brunel N, Hakim V (1999) Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput 11: 1621-1671.
4. Buck J (1988) Synchronous rhythmic flashing of fireflies. II. Q Rev Biol 63: 265-289.
5. Burkitt AN (2006) A review of the integrate-and-fire neuron model: I. homogeneous synaptic input. Biol Cybern 95: 1-19.
6. Coutinho R, Fernandez B (2004) Fronts in extended systems of bistable maps coupled via convolutions. Nonlinearity 17: 23-47.
7. Danino T, Mondragon-Palomino O, Tsimring LS, Hasty J (2010) A synchronized quorum of genetic clocks. Nature 463: 326-330.
8. Ernst U, Pawelzik K, Geisel T (1995) Synchronization induced by temporal delays in pulse-coupled oscillators. Phys Rev Lett 74: 1570.
9. Fernandez B, Tsimring LS (2011) Corepressive interaction and clustering of degrade-and-fire oscillators. Phys Rev E 84: 051916.
10. Gonze D, Bernard S, Waltermann C, Kramer A, Herzel H (2005) Spontaneous synchronization of coupled circadian oscillators. Biophys J 89: 120-129.
11. Golomb D, Hansel D, Shraiman B, Sompolinsky H (1992) Clustering in globally coupled phase oscillators. Phys Ref A 45: 3516-3530.
12. Kolmogorov AN, Fomin SV (1999) Elements of the theory of functions and functional analysis. Dover Publications, Mineaola.
13. Lee DeVille RE, Peskin CS, Spencer JH (2010) Dynamics of stochastic neuronal networks and the connections to random graph theory. Math Mod Nat Phenom 5: 26-66.
14. Mather W, Bennet MR, Hasty J, Tsimring LS (2009) Delay-induced degrade-and-fire oscillations in small genetic circuits. Phys Rev Lett 102: 068105.
15. Mirollo RE, Strogatz SH (1990) Synchronization of pulse-coupled biological oscillators. SIAM J Appl Math 50: 1645-1662.
16. Peskin CS (1975) Mathematical aspects of heart physiology. Courant Institute of Mathematical Science Publication, New York.
17. Pikovsky A, Rosenblum M, Kurths J (2003) Synchronization: a universal concept in nonlinear sciences. Cambridge University Press, Cambridge.
18. Prindle A, Samayoa P, Razinkov I, Danino T, Tsimring LS, Hasty J (2012) A sensing array of radically coupled genetic 'biopixels'. Nature 481: 39-44.
19. Seen W, Urbanczik R (2000) Similar nonleaky integrate-and-fire neurons with instantaneous couplings always synchronize. SIAM J Appl Math 61: 1143-1155.
20. Strogatz SH (2000) From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143: 1-20.
21. Tass PA (1999) Phase resetting in medicine and biology. Springer, Berlin.
22. Tsimring LS, Rulkov NF, Larsen ML, Gabbay M (2005) Repulsive synchronization in an array of phase oscillators. Phys Rev Lett 95: 14101.
23. van Vreeswijk C, Abbott LF, Ermentrout GB (1994) When inhibition not excitation synchronizes neural firing. J Comput Neurosci 1: 313-321.
24. Wiesenfeld K, Swift JW (1995) Averaged equations for Josephson junction series arrays. Phys Rev E 51: 1020-1025.
25. Yamaguchi S, Isejima H, Matsuo T, Okura, K. Yagita R, Kobayashi M, Okamura H (2003) Synchronization of cellular clocks in the suprachiasmatic nucleus. Science's STKE 302(5649): 1408.