Gradient and Hamiltonian coupled systems on undirected networks

Manuela Aguiar; Ana Dias; Miriam Manoel; Manuela Aguiar; Ana Dias; Miriam Manoel

doi:10.3934/mbe.2019232

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 4622-4644. doi: 10.3934/mbe.2019232

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Gradient and Hamiltonian coupled systems on undirected networks

1.
Centro de Matemática, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
2.
Faculdade de Economia, Universidade do Porto, Rua Dr Roberto Frias, 4200-464 Porto, Portugal
3.
Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
4.
Departamento de Matemática, ICMC, Universidade de São Paulo, 13560-970 Caixa Postal 668, São Carlos, SP - Brazil

Received: 26 December 2018 Accepted: 06 May 2019 Published: 23 May 2019

Many real world applications are modelled by coupled systems on undirected networks. Two striking classes of such systems are the gradient and the Hamiltonian systems. In fact, within these two classes, coupled systems are admissible only by the undirected networks. For the coupled systems associated with a network, there can be flow-invariant spaces (synchrony subspaces where some subsystems evolve synchronously), whose existence is independent of the systems equations and depends only on the network topology. Moreover, any coupled system on the network, when restricted to such a synchrony subspace, determines a new coupled system associated with a smaller network (quotient). The original network is said to be a lift of the quotient network. In this paper, we characterize the conditions for the coupled systems property of being gradient or Hamiltonian to be preserved by the lift and quotient coupled systems. This characterization is based on determining necessary and sufficient conditions for a quotient (lift) network of an undirected network to be also undirected. We show that the extra gradient or Hamiltonian structure of a coupled system admissible by an undirected network can be lost by the systems admissible by a (directed) quotient network. Conversely, gradient (Hamiltonian) dynamics can appear for an undirected quotient network of a directed network or of an undirected network whose associated dynamics is not gradient (Hamiltonian). We illustrate with a neural network given by two groups of neurons that are mutually coupled through either excitatory or inhibitory synapses, which is modelled by a coupled system exhibiting both gradient and Hamiltonian structures.
- undirected network,
- admissible function,
- potential function,
- Hamiltonian function,
- quotient network,
- lift network,
- synchrony subspace
Citation: Manuela Aguiar, Ana Dias, Miriam Manoel. Gradient and Hamiltonian coupled systems on undirected networks[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4622-4644. doi: 10.3934/mbe.2019232

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Abstract

Many real world applications are modelled by coupled systems on undirected networks. Two striking classes of such systems are the gradient and the Hamiltonian systems. In fact, within these two classes, coupled systems are admissible only by the undirected networks. For the coupled systems associated with a network, there can be flow-invariant spaces (synchrony subspaces where some subsystems evolve synchronously), whose existence is independent of the systems equations and depends only on the network topology. Moreover, any coupled system on the network, when restricted to such a synchrony subspace, determines a new coupled system associated with a smaller network (quotient). The original network is said to be a lift of the quotient network. In this paper, we characterize the conditions for the coupled systems property of being gradient or Hamiltonian to be preserved by the lift and quotient coupled systems. This characterization is based on determining necessary and sufficient conditions for a quotient (lift) network of an undirected network to be also undirected. We show that the extra gradient or Hamiltonian structure of a coupled system admissible by an undirected network can be lost by the systems admissible by a (directed) quotient network. Conversely, gradient (Hamiltonian) dynamics can appear for an undirected quotient network of a directed network or of an undirected network whose associated dynamics is not gradient (Hamiltonian). We illustrate with a neural network given by two groups of neurons that are mutually coupled through either excitatory or inhibitory synapses, which is modelled by a coupled system exhibiting both gradient and Hamiltonian structures.

References

[1]	J. C. Bronski, L. De Ville and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model, Chaos: an Int. J. Non. Sci., 22 (2012), 033133.
[2]	M. Manoel and M. R. Roberts, Gradient systems on coupled cell networks, Nonlinearity, 28 (2015), 3487–3509.
[3]	M. Plank, On the dynamics of Lotka-Volterra equations having an invariant hyperplane, SIAM J. Appl. Math., 59 (1999), 1540–1551.
[4]	P.L. Buono, B. Chan and A. Palacios, Dynamics and Bifurcations in a D n -symmetric Hamiltonian Network. Application to Coupled Gyroscopes, Phys. D, 290 (2015), 8–23.
[5]	B. S. Chan, P. L. Buono and A. Palacios, Topology and Bifurcations in Hamiltonian Coupled Cell Systems, Dyn. Syst., 32 (2017), 23–45.
[6]	D. S Tourigny, Networks of planar Hamiltonian systems, Comm. in Non. Sci. and Num. Sim., 53 (2017), 263–277.
[7]	V. V. Gafiychuk and A. K. Prykarpatsky, Pattern formation in neural dynamical systems governed by mutually Hamiltonian and gradient vector field structures, Cond. Matt. Phys., 7 (2004), 551– 563.
[8]	E. Lee and D. Terman, Oscillatory rhythms in a model network of excitatory and inhibitory Neurons, SIAM J. Appl. Dyn. Syst, 18 (2019), 354–392.
[9]	P. J. Uhlhaas and W. Singer, Neural Synchrony in Brain Review Disorders: Relevance for Cognitive Dysfunctions and Pathophysiology, Neuron, 52 (2006), 155–168.
[10]	P. Duarte, R. L. Fernandes and W. M. Oliva, Dynamics of the attractor in the Lotka-Volterra equations, J. Diff. Eq., 149 (1998), 143–189.
[11]	J. M. Neuberger, N. Sieben and J. W. Swift, Synchrony and anti-synchrony for difference-coupled vector fields on graph network systems, preprint, arXiv:1805.04144.
[12]	R. E. Mirollo and S. H. Strogatz. The spectrum of the locked state for the Kuramoto model of coupled oscillators, Physica D, 205 (2005), 249–266.
[13]	S. E. Korshunov, Phase diagram of the antiferromagnetic XY model with a triangular lattice in an external magnetic field, J. Phys. C: Solid State Phys., 19 (1986), 5927–5935.
[14]	D. H. Lee, R. G. Caflisch, J. D. Joannopoulos, et al., Antiferromagnetic classical XFmodel: A mean-field analysis, Phys. Rev. B, 29 (1984), 2680–2684.
[15]	J. C. Walter and C. Chatelain, Numerical investigation of the ageing of the fully frustrated XY model, J. Stat. Mech., 10 (2009), P10017-1-17.
[16]	E. Goles and G. A. Ruz, Dynamics of neural networks over undirected graphs, Neural Networks, 63 (2015), 156–169.
[17]	M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: the groupoid formalism, B. Am. Math. Soc., 43 (2006), 305–364.
[18]	M. Golubitsky, I. Stewart and A. Török, Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dyn. Syst., 4 (2005), 78–100.
[19]	I. Stewart, M. Golubitsky and M. Pivato, Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dyn. Syst., 2 (2003), 609–646.
[20]	M. Denaxa, G. Neves, A. Rabinowitz, S. Kemlo, P. Liodis, J. Burrone and V. Pachnis, Modulation of apoptosis controls inhibitory interneuron number in the cortex, Cell Rep., 22 (2018), 1710–1721.
[21]	I. Stewart, The lattice of balanced equivalence relations of a coupled cell network, Math. Proc. Cambridge Philos. Soc., 143 (2007), 165–183.
[22]	M. A. D. Aguiar and A. P. S. Dias, The Lattice of Synchrony Subspaces of a Coupled Cell Network: Characterization and Computation Algorithm, J. Nonlinear Sci., 24 (2014), 949–996.
[23]	M. A. D. Aguiar, A. P. S. Dias., M. Golubitsky, et al., Bifurcations from regular quotient networks: a first insight, Physica D, 238 (2009), 137–155.
[24]	A. P. S. Dias and E. M. Pinho, Spatially Periodic Patterns of Synchrony in Lattice Networks, SIAM J. Appl. Dyn. Syst., 8 (2009), 641–675.
[25]	M. Aguiar, P. Ashwin, A. Dias, et al., Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation, J. Nonlinear Sci., 21 (2011), 271–323.
[26]	M. A. D. Aguiar and A. P. S. Dias, Heteroclinic network dynamics on joining coupled cell networks, Dyn. Syst Int. J., 32 (2017), 4–22.
[27]	S. Alford and M. Alpert, A synaptic mechanism for network synchrony, Front Cell Neurosci., 8 (2014), 290.

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