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 and 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, Ohio State University, 2010.

[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), 1365-1381. doi: 10.3934/dcdsb.2012.17.1365.

[3]

S. Ahn, B. H. Smith, A. Borisyuk and D. Terman, Analyzing neuronal networks using discrete-time dynamics, Physica D: Nonlinear Phenomena, 239 (2010), 515-528. doi: 10.1016/j.physd.2009.12.011.

[4]

J. Bang-Jensen and G. Z. Gutin, "Digraphs. Theory, Algorithms and Applications," Second edition, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2009. doi: 10.1007/978-1-84800-998-1.

[5]

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

[6]

O. Colón-Reyes, R. Laubenbacher and B. Pareigis, Boolean monomial dynamical systems, Annals of Combinatorics, 8 (2005), 425-439. doi: 10.1007/s00026-004-0230-6.

[7]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Reprint of the second (1989) edition, Studies in Nonlinearity, Westview Press, Boulder, CO, 2003.

[8]

R. Durrett, "Random Graph Dynamics," Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010.

[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), 2923-2939. doi: 10.1105/tpc.104.021725.

[10]

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

[11]

E. Goles and M. Matamala, Reaction-diffusion automata: Three states implies universality, Theory of Computing Systems, 30 (1997), 223-229. doi: 10.1007/BF02679460.

[12]

E. Goles, F. Fogelman-Soulié and D. Pellegrin, Decreasing energy functions as a tool for studying threshold networks, Discrete Applied Mathematics, 12 (1985), 261-277. doi: 10.1016/0166-218X(85)90029-0.

[13]

G. Grimmett, "Probability on Graphs. Random Processes on Graphs and Lattices," Institute of Mathematical Statistics Textbooks, 1, Cambridge University Press, Cambridge, 2010.

[14]

W. Just, S. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks, Physica D, 237 (2008), 3186-3196. doi: 10.1016/j.physd.2008.08.011.

[15]

S. A. Kauffman, "The Origins of Order: Self-Organization and Selection in Evolution," Oxford University Press, New York, 1993.

[16]

T. Luczak and J. E. Cohen, Stability of vertices in random Boolean cellular automata, Random Structures & Algorithms, 2 (1991), 327-334. doi: 10.1002/rsa.3240020307.

[17]

J. F. Lynch, Dynamics of random Boolean networks, in "Current Developments in Mathematical Biology," Ser. Knots Everything, 38, World Scientific Publ., Hackensack, NJ, (2007), 15-38. doi: 10.1142/9789812706799_0002.

[18]

M. Matamala and E. Goles, Dynamic behavior of cyclic automata networks, Discrete Applied Mathematics, 77 (1997), 161-184. doi: 10.1016/S0166-218X(97)84104-2.

[19]

W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biology, 5 (1943), 115-133. doi: 10.1007/BF02478259.

show all references

References:
[1]

S. Ahn, "Transient and Attractor Dynamics in Models for Odor Discrimination," Ph.D. thesis, Ohio State University, 2010.

[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), 1365-1381. doi: 10.3934/dcdsb.2012.17.1365.

[3]

S. Ahn, B. H. Smith, A. Borisyuk and D. Terman, Analyzing neuronal networks using discrete-time dynamics, Physica D: Nonlinear Phenomena, 239 (2010), 515-528. doi: 10.1016/j.physd.2009.12.011.

[4]

J. Bang-Jensen and G. Z. Gutin, "Digraphs. Theory, Algorithms and Applications," Second edition, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2009. doi: 10.1007/978-1-84800-998-1.

[5]

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

[6]

O. Colón-Reyes, R. Laubenbacher and B. Pareigis, Boolean monomial dynamical systems, Annals of Combinatorics, 8 (2005), 425-439. doi: 10.1007/s00026-004-0230-6.

[7]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Reprint of the second (1989) edition, Studies in Nonlinearity, Westview Press, Boulder, CO, 2003.

[8]

R. Durrett, "Random Graph Dynamics," Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010.

[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), 2923-2939. doi: 10.1105/tpc.104.021725.

[10]

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

[11]

E. Goles and M. Matamala, Reaction-diffusion automata: Three states implies universality, Theory of Computing Systems, 30 (1997), 223-229. doi: 10.1007/BF02679460.

[12]

E. Goles, F. Fogelman-Soulié and D. Pellegrin, Decreasing energy functions as a tool for studying threshold networks, Discrete Applied Mathematics, 12 (1985), 261-277. doi: 10.1016/0166-218X(85)90029-0.

[13]

G. Grimmett, "Probability on Graphs. Random Processes on Graphs and Lattices," Institute of Mathematical Statistics Textbooks, 1, Cambridge University Press, Cambridge, 2010.

[14]

W. Just, S. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks, Physica D, 237 (2008), 3186-3196. doi: 10.1016/j.physd.2008.08.011.

[15]

S. A. Kauffman, "The Origins of Order: Self-Organization and Selection in Evolution," Oxford University Press, New York, 1993.

[16]

T. Luczak and J. E. Cohen, Stability of vertices in random Boolean cellular automata, Random Structures & Algorithms, 2 (1991), 327-334. doi: 10.1002/rsa.3240020307.

[17]

J. F. Lynch, Dynamics of random Boolean networks, in "Current Developments in Mathematical Biology," Ser. Knots Everything, 38, World Scientific Publ., Hackensack, NJ, (2007), 15-38. doi: 10.1142/9789812706799_0002.

[18]

M. Matamala and E. Goles, Dynamic behavior of cyclic automata networks, Discrete Applied Mathematics, 77 (1997), 161-184. doi: 10.1016/S0166-218X(97)84104-2.

[19]

W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biology, 5 (1943), 115-133. doi: 10.1007/BF02478259.

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