September  2013, 18(7): 1827-1844. doi: 10.3934/dcdsb.2013.18.1827

Phase-locked trajectories for dynamical systems on graphs

1. 

Department of Mathematics & Center for Dynamics, TU Dresden, 01069 Dresden, Germany, Germany

Received  March 2012 Revised  February 2013 Published  May 2013

We prove a general result on the existence of periodic trajectories of systems of difference equations with finite state space which are phase-locked on certain components which correspond to cycles in the coupling structure. A main tool is the new notion of order-induced graph which is similar in spirit to a Lyapunov function. To develop a coherent theory we introduce the notion of dynamical systems on finite graphs and show that various existing neural networks, threshold networks, reaction-diffusion automata and Boolean monomial dynamical systems can be unified in one parametrized class of dynamical systems on graphs which we call threshold networks with refraction. For an explicit threshold network with refraction and for explicit cyclic automata networks we apply our main result to show the existence of phase-locked solutions on cycles.
Citation: Jeremias Epperlein, Stefan Siegmund. Phase-locked trajectories for dynamical systems on graphs. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1827-1844. doi: 10.3934/dcdsb.2013.18.1827
References:
[1]

S. Ahn, "Transient and Attractor Dynamics in Models for Odor Discrimination,", Ph.D. thesis, (2010).   Google Scholar

[2]

S. Ahn and W. Just, Digraphs vs. dynamics in discrete models of neuronal networks,, Discrete and Continuous Dynamical Systems Series B (DCDS-B), 17 (2012), 1365.  doi: 10.3934/dcdsb.2012.17.1365.  Google Scholar

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S. Ahn, B. H. Smith, A. Borisyuk and D. Terman, Analyzing neuronal networks using discrete-time dynamics,, Physica D: Nonlinear Phenomena, 239 (2010), 515.  doi: 10.1016/j.physd.2009.12.011.  Google Scholar

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J. Bang-Jensen and G. Z. Gutin, "Digraphs. Theory, Algorithms and Applications,", Second edition, (2009).  doi: 10.1007/978-1-84800-998-1.  Google Scholar

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O. Colón-Reyes, R. Laubenbacher and B. Pareigis, Boolean monomial dynamical systems,, Annals of Combinatorics, 8 (2005), 425.  doi: 10.1007/s00026-004-0230-6.  Google Scholar

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R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Reprint of the second (1989) edition, (1989).   Google Scholar

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R. Durrett, "Random Graph Dynamics,", Cambridge Series in Statistical and Probabilistic Mathematics, (2010).   Google Scholar

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C. Espinosa-Soto, P. Padilla-Longoria and E. R. Alvarez-Buylla, A gene regulatory network model for cell-fate determination during arabidopsis thaliana flower development that is robust and recovers experimental gene expression profiles,, The Plant Cell Online, 16 (2004), 2923.  doi: 10.1105/tpc.104.021725.  Google Scholar

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E. Goles and S. Martínez, "Neural and automata Networks. Dynamical Behavior and Applications,", Kluwer Academic Publishers Group, (1990).  doi: 10.1007/978-94-009-0529-0.  Google Scholar

[11]

E. Goles and M. Matamala, Reaction-diffusion automata: Three states implies universality,, Theory of Computing Systems, 30 (1997), 223.  doi: 10.1007/BF02679460.  Google Scholar

[12]

E. Goles, F. Fogelman-Soulié and D. Pellegrin, Decreasing energy functions as a tool for studying threshold networks,, Discrete Applied Mathematics, 12 (1985), 261.  doi: 10.1016/0166-218X(85)90029-0.  Google Scholar

[13]

G. Grimmett, "Probability on Graphs. Random Processes on Graphs and Lattices,", Institute of Mathematical Statistics Textbooks, 1 (2010).   Google Scholar

[14]

W. Just, S. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks,, Physica D, 237 (2008), 3186.  doi: 10.1016/j.physd.2008.08.011.  Google Scholar

[15]

S. A. Kauffman, "The Origins of Order: Self-Organization and Selection in Evolution,", Oxford University Press, (1993).   Google Scholar

[16]

T. Luczak and J. E. Cohen, Stability of vertices in random Boolean cellular automata,, Random Structures & Algorithms, 2 (1991), 327.  doi: 10.1002/rsa.3240020307.  Google Scholar

[17]

J. F. Lynch, Dynamics of random Boolean networks,, in, 38 (2007), 15.  doi: 10.1142/9789812706799_0002.  Google Scholar

[18]

M. Matamala and E. Goles, Dynamic behavior of cyclic automata networks,, Discrete Applied Mathematics, 77 (1997), 161.  doi: 10.1016/S0166-218X(97)84104-2.  Google Scholar

[19]

W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity,, Bulletin of Mathematical Biology, 5 (1943), 115.  doi: 10.1007/BF02478259.  Google Scholar

show all references

References:
[1]

S. Ahn, "Transient and Attractor Dynamics in Models for Odor Discrimination,", Ph.D. thesis, (2010).   Google Scholar

[2]

S. Ahn and W. Just, Digraphs vs. dynamics in discrete models of neuronal networks,, Discrete and Continuous Dynamical Systems Series B (DCDS-B), 17 (2012), 1365.  doi: 10.3934/dcdsb.2012.17.1365.  Google Scholar

[3]

S. Ahn, B. H. Smith, A. Borisyuk and D. Terman, Analyzing neuronal networks using discrete-time dynamics,, Physica D: Nonlinear Phenomena, 239 (2010), 515.  doi: 10.1016/j.physd.2009.12.011.  Google Scholar

[4]

J. Bang-Jensen and G. Z. Gutin, "Digraphs. Theory, Algorithms and Applications,", Second edition, (2009).  doi: 10.1007/978-1-84800-998-1.  Google Scholar

[5]

E. Behrends, "Introduction to Markov Chains. With Special Emphasis on Rapid Mixing,", Friedr. Vieweg & Sohn, (2000).   Google Scholar

[6]

O. Colón-Reyes, R. Laubenbacher and B. Pareigis, Boolean monomial dynamical systems,, Annals of Combinatorics, 8 (2005), 425.  doi: 10.1007/s00026-004-0230-6.  Google Scholar

[7]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Reprint of the second (1989) edition, (1989).   Google Scholar

[8]

R. Durrett, "Random Graph Dynamics,", Cambridge Series in Statistical and Probabilistic Mathematics, (2010).   Google Scholar

[9]

C. Espinosa-Soto, P. Padilla-Longoria and E. R. Alvarez-Buylla, A gene regulatory network model for cell-fate determination during arabidopsis thaliana flower development that is robust and recovers experimental gene expression profiles,, The Plant Cell Online, 16 (2004), 2923.  doi: 10.1105/tpc.104.021725.  Google Scholar

[10]

E. Goles and S. Martínez, "Neural and automata Networks. Dynamical Behavior and Applications,", Kluwer Academic Publishers Group, (1990).  doi: 10.1007/978-94-009-0529-0.  Google Scholar

[11]

E. Goles and M. Matamala, Reaction-diffusion automata: Three states implies universality,, Theory of Computing Systems, 30 (1997), 223.  doi: 10.1007/BF02679460.  Google Scholar

[12]

E. Goles, F. Fogelman-Soulié and D. Pellegrin, Decreasing energy functions as a tool for studying threshold networks,, Discrete Applied Mathematics, 12 (1985), 261.  doi: 10.1016/0166-218X(85)90029-0.  Google Scholar

[13]

G. Grimmett, "Probability on Graphs. Random Processes on Graphs and Lattices,", Institute of Mathematical Statistics Textbooks, 1 (2010).   Google Scholar

[14]

W. Just, S. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks,, Physica D, 237 (2008), 3186.  doi: 10.1016/j.physd.2008.08.011.  Google Scholar

[15]

S. A. Kauffman, "The Origins of Order: Self-Organization and Selection in Evolution,", Oxford University Press, (1993).   Google Scholar

[16]

T. Luczak and J. E. Cohen, Stability of vertices in random Boolean cellular automata,, Random Structures & Algorithms, 2 (1991), 327.  doi: 10.1002/rsa.3240020307.  Google Scholar

[17]

J. F. Lynch, Dynamics of random Boolean networks,, in, 38 (2007), 15.  doi: 10.1142/9789812706799_0002.  Google Scholar

[18]

M. Matamala and E. Goles, Dynamic behavior of cyclic automata networks,, Discrete Applied Mathematics, 77 (1997), 161.  doi: 10.1016/S0166-218X(97)84104-2.  Google Scholar

[19]

W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity,, Bulletin of Mathematical Biology, 5 (1943), 115.  doi: 10.1007/BF02478259.  Google Scholar

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