×

Network architecture and spatio-temporally symmetric dynamics. (English) Zbl 1118.34032

The main question, which is addressed in this paper, is how network architecture can determine spatiotemporally structured periodic dynamics of the network. The considered setup is quite general: Let \(x=(x_1,\dotsm,x_K)\in \mathbb{R}^{mK}\) be the phase variables of the network, where \(x_j\in \mathbb{R}^m\) are the variables of \(j\)-th subsystem. The dynamical system of the network is described by the higher-order ordinary differential equation system \(x'=F(x)\). As a rule, the network architecture implies some symmetry properties, e.g. equivariance of \(F\) with respect to some group of permutations \(G\). Using the theory of equivariant differential equations and coupled oscillators theory, the authors show that a robust fully oscillatory dynamics, in which no cell is stationary, can appear only in the networks, where all cells interact with each other or which contain a single group of interacting cells which drive the reminder of the network. Another highlight of the paper is the algorithm for the construction of systems which possess stable spatio-temporally symmetric periodic solutions.

MSC:

34C14 Symmetries, invariants of ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
92B20 Neural networks for/in biological studies, artificial life and related topics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albert, R.; Barabási, A.-L., Statistical mechanics of complex networks, Rev. Modern Phys., 74, 47-97 (2002) · Zbl 1205.82086
[2] Ashwin, P.; Swift, J. W., The dynamics of \(n\) weakly coupled identical oscillators, J. Nonlinear Sci., 2, 1, 69-108 (1992) · Zbl 0872.58049
[3] Atay, F. M.; Biyikoglu, T.; Jost, J., Synchronization of networks with prescribed degree distributions, IEEE Trans. Circuits Syst. I, 53, 92-98 (2006) · Zbl 1374.05064
[4] Bressloff, P. C.; Coombes, S., Symmetry and phase-locking in a ring of pulse-coupled oscillators with distributed delays, Physica D, 126, 1-2, 99-122 (1999) · Zbl 0937.34060
[5] Brown, E.; Holmes, P.; Moehlis, J., (Globally Coupled Oscillator Networks. Globally Coupled Oscillator Networks, Perspectives and Problems in Nonlinear Science (2003), Springer: Springer New York), 183-215 · Zbl 1132.92316
[6] Buono, P. L.; Golubitsky, M., Models of central pattern generators for quadruped locomotion: I. Primary gaits, J. Math. Biol., 42, 291-326 (2001) · Zbl 1039.92007
[7] Canavier, C. C.; Baxter, D. A.; Clark, J. W.; Byrne, J. H., Control of multistability in ring circuits of oscillators, Biol. Cybern., 80, 87-102 (1999) · Zbl 0918.92008
[8] Edwards, R.; Glass, L., Combinatorial explosion in model gene networks, Chaos, 10, 3, 691-704 (2000) · Zbl 1033.92014
[9] Elmhirst, T., Spatio-temporal symmetries in \(S_N\)- and \(S_N \times Z_2\)-equivariant systems, Dyn. Syst., 18, 4, 279-293 (2003) · Zbl 1058.37021
[10] Elowitz, M. B.; Leibler, S., A synthetic oscillatory network of transcriptional regulators, Nature, 403, 335-338 (2000)
[11] Ermentrout, G. B., The behavior of rings of coupled oscillators, J. Math. Biol., 23, 1, 55-74 (1985) · Zbl 0583.92002
[12] Gardner, T.; Cantor, C. R.; Collins, J. J., Construction of a genetic toggle switch in Escherichia Coli, Nature, 403, 339-342 (2000)
[13] M. Golubitsky, K. Josić, E. Shea-Brown, Spatiotemporally symmetric solutions in networks of phase oscillators (Preprint); M. Golubitsky, K. Josić, E. Shea-Brown, Spatiotemporally symmetric solutions in networks of phase oscillators (Preprint)
[14] Golubitsky, M.; Josić, K.; Shea-Brown, E., Winding numbers and average frequencies in phase oscillator networks, J. Nonlinear Sci., 16, 201-231 (2006) · Zbl 1130.37345
[15] Golubitsky, M.; Stewart, I., (The Symmetry Perspective. The Symmetry Perspective, Progress in Mathematics, vol. 200 (2002), Birkhäuser Verlag: Birkhäuser Verlag Basel)
[16] Golubitsky, M.; Stewart, I., Nonlinear dynamics of networks: The groupoid formalism, Bull. Amer. Math. Soc., 43, 305-364 (2006) · Zbl 1119.37036
[17] Golubitsky, M.; Stewart, I.; Buono, P. L.; Collins, J. J., Symmetry in locomotor central pattern generators and animal gaits, Nature, 401, 6754, 693-695 (1999)
[18] Golubitsky, M.; Stewart, I.; Török, A., Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dyn. Syst., 4, 1, 78-100 (2005), (electronic) · Zbl 1090.34030
[19] Hasty, J.; McMillen, D.; Isaacs, F.; Collins, J. J., Computational studies of gene regulatory networks: in numero molecular biology, Nat. Rev. Neurosci., 2, 268-279 (2001)
[20] Hooper, S. L., Transduction of temporal patterns by single neurons, Nat. Neurosci., 1, 720-726 (1998)
[21] Hoppensteadt, F. C.; Izhikevich, E., Weakly Connected Neural Networks (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0887.92003
[22] Huxter, J.; Burgess, N.; O’Keefe, J., Independent rate and temporal coding in hippocampal pyramidal cells, Nature, 425, 828-832 (2003)
[23] Ikegaya, Y.; Aaron, G.; Cossart, R.; Aronov, D.; Lampl, I.; Ferster, D.; Yuste, R., Synfire chains and cortical songs: Temporal modules of cortical activity, Science, 304, 559 (2004)
[24] Izhikevich, E., Polychronization: Computation with spikes, Neural Comput., 18, 245-288 (2006) · Zbl 1090.92006
[25] Josić, K.; Peleš, S., Synchronization in networks of general, weakly non-linear oscillators, J. Phys. A, 37, 49, 11801-11818 (2004) · Zbl 1074.34043
[26] Kopell, N., We got rhythm: Dynamical systems of the nervous system, Notices Amer. Math. Soc., 47, 6-16 (2000) · Zbl 1016.91505
[27] Kopell, N.; Ermentrout, G. B., Symmetry and phaselocking in chains of weakly coupled oscillators, Comm. Pure Appl. Math., 39, 5, 623-660 (1986) · Zbl 0596.92011
[28] Kopell, N.; Ermentrout, G. B., Coupled oscillators and the design of central pattern generators, Math. Biosci., 90, 1-2, 87-109 (1988), Nonlinearity in biology and medicine, Los Alamos, NM, 1987 · Zbl 0649.92009
[29] Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence (1984), Springer-Verlag: Springer-Verlag Berlin · Zbl 0558.76051
[30] Marder, E.; Calabrese, R. L., Principles of rhythmic motor pattern generation, Physiol. Rev., 76, 3, 687(31) (1996)
[31] Milo, R.; Shen-Orr, S.; Itzkovitz, S.; Kashtan, N.; Chklovskii, D.; Alon, U., Network motifs: Simple building blocks of complex networks, Science, 298, 824-827 (2002)
[32] Park, J. H.; Bayliss, A.; Matkowsky, B. J., Dynamics in a rod model of solid flame waves, SIAM J. Appl. Math., 65, 2, 521-549 (2004/05), (electronic) · Zbl 1136.80305
[33] D.D. Pervouchine, T.I. Netoff, H.G. Rotstein, J.A. White, M.O. Cunningham, M.A. Whittington, N.J. Kopell, Low-dimensional maps encoding dynamics in entorhinal cortex and hippocampus, 2006 (Preprint); D.D. Pervouchine, T.I. Netoff, H.G. Rotstein, J.A. White, M.O. Cunningham, M.A. Whittington, N.J. Kopell, Low-dimensional maps encoding dynamics in entorhinal cortex and hippocampus, 2006 (Preprint) · Zbl 1102.92008
[34] Q.-C. Pham, J.-J. Slotine, Stable concurrent synchronization in dynamic system networks, 2005, arxiv:q-bio.NC/0510051; Q.-C. Pham, J.-J. Slotine, Stable concurrent synchronization in dynamic system networks, 2005, arxiv:q-bio.NC/0510051 · Zbl 1158.68449
[35] Pikovsky, A.; Rosenblum, M.; Kurths, J., Synchronization: A Universal Concept in Nonlinear Science (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1219.37002
[36] Prinz, A. A.; Bucher, D.; Marder, E., Similar network activity from disparate circuit parameters, Nat. Neurosci., 7, 1345-1352 (2004)
[37] Seipel, J. E.; Holmes, P.; Full, R. J., Dynamics and stability of insect locomotion: A hexapedal model for horizontal plane motions, Biol. Cybern., 91, 76-90 (2004) · Zbl 1082.92010
[38] Song, S.; Sjostrom, P. J.; Reigl, M.; Nelson, S.; Chklovskii, D. B., Highly nonrandom features of synaptic connectivity in local cortical circuits, PLoS Biol., 3, 0507-0519 (2005)
[39] Stewart, I.; Collins, J. J., A group-theoretic approach to rings of coupled biological oscillators, Biol. Cybern., 71, 95-103 (1994) · Zbl 0804.92009
[40] Terman, D.; Rubin, J. E.; Wilson, C. H., Activity patterns in a model for the subthalamopallidal network of the basal ganglia, J. Neurosci., 22, 7, 2963-2976 (2002)
[41] Terry, J. R.; Thornburg, K. S.; DeShazer, D. J.; VanWiggeren, G. D.; Zhu, S.; Ashwin, P.; Roy, R., Synchronization of chaos in an array of three lasers, Phys. Rev. E, 59, 4, 4036-4046 (1999)
[42] I. Vragović, E. Louis, C.D.E. Boschi, G.J. Ortega, Diversity-induced synchronized oscillation in close-to-threshold excitable elements arranged on regular networks, 2005, ArXiv: cond-mat/0410171; I. Vragović, E. Louis, C.D.E. Boschi, G.J. Ortega, Diversity-induced synchronized oscillation in close-to-threshold excitable elements arranged on regular networks, 2005, ArXiv: cond-mat/0410171 · Zbl 1200.34055
[43] Williams, T. L.; Bowtell, G., The calculation of frequency-shift functions for chains of coupled oscillators with application to a network model of the lamprey locomotor pattern generator, Comput. Neurosci., 4, 47-55 (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.