Abstract
We address the issue of dissipative vs. non-dissipative behavior in a mesoscopic set of coupled elements such as oscillators, with one half having gain and the other half having losses. We introduce a graph with coupling as the graph edges in given fixed number and gain/loss elements as its nodes. This relates to parity-time symmetry, notably in optics, e.g. set of coupled fibers, and more generally to the issue of taming divergence related to imaginary parts of eigenvectors in various network descriptions, for instance biochemical, neuronal, ecological. We thus look for the minimization of the imaginary part of all eigenvalues altogether, with a collective figure of merit. As more edges than gain/loss pairs are introduced, the unbroken cases , i.e., stable cases with real eigenvalues in spite of gain and loss, become statistically very scarce. A minimization from a random starting point by moving one edge at a time is studied, amounting to investigate how the hugely growing configuration number impedes the attainment of the desired minimally-dissipative target. The minimization path and its apparent stalling point are analyzed in terms of network connectivity metrics. We expand in the end on the relevance in biochemical signaling networks and the so-called “stability-optimized circuits” relevant to neural organization.
Graphical abstract
Similar content being viewed by others
References
L. Feng, R. El-Ganainy, L. Ge, Nat. Photon. 11, 752 (2017)
C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)
Y. Kim, S. Warren, F Favero, J. Sone, J. Clegg, M. Neil, C. Paterson, J. Knight, P. French, C. Dunsby, Opt. Express 36, 3661 (2018)
H. Chen, C. Jin, B. Huang, N.K. Fontaine, R. Ryf, K. Shang, N. Grégoire, S. Morency, R.-J. Essiambre, G. Li, Y. Messaddeq, S. LaRochelle, Nat. Photon. 10, 529 (2016)
Y. Jung, M. Wada, K. Shibahara, S. Jain, I.A. Davidson, P. Barua, J.R. Hayes, T. Sakamoto, T. Mizuno, Y. Miyamoto, Y. Sasaki, K. Saitoh, K. Nakajima, D.J. Richardson, IEEE J. Lightwave Technol. 38, 2938 (2020)
D. Lin, J. Carpenter, Y. Feng, S. Jain, Y. Jung, Y. Feng, M.N. Zervas, D.J. Richardson, Nat. Commun. 11, 3986 (2020)
M. Gaio, D. Saxena, J. Bertolotti, D. Pisignano, A. Camposeo, R. Sapienza, Nat. Commun. 10, 226 (2019)
S. Rotter, Nat. Photon. 13, 140 (2019)
H.A. Haus, Such modal propagation constants are those of the solution of the source-free Maxwell equation, usually cast into a wave equation when all fields of a given mode have an exp(iβ~z) dependence along the invariant z-axis, inWaves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, 1984)
A. Lupu, H. Benisty, A. Degiron, Opt. Express 21, 21651 (2013)
G. Oster, A.S. Perelson, A. Katchalsky, Q. Rev. Biophys. 6, 1 (1973)
D.C. Mikulecky, Comput. Chem. 25, 369 (2001)
J.C. Delvenne, H. Sandberg, Physica D 267, 123 (2014)
B. Maschke, A. van derSchaft, IFAC-Pap. OnLine 52, 418 (2019)
A. Lotka, Proc. Natl. Acad. Sci. U.S.A. 8, 147 (1922)
A. Lotka, Proc. Natl. Acad. Sci. U.S.A. 7, 168 (1921)
K.H. Jensen, M.A. Zwienieck, K. Berg-Sørensen, H. Bruus, N.M. Holbrook, J. Liesche, A. Schulz, T. Bohr, Rev. Mod. Phys. 88, 035007–1 (2016)
A.R. Zomorrodi, D. Segrè, J. Mol. Biol. 428, 837 (2016)
P. Ghisellini, C. Cialani, S. Ulgiati, J. Cleaner Prod. 114, 11 (2016)
R.E. May, Nature 238, 413 (1972)
I.V. Barashenkov, L. Baker, N.V. Alexeeva, Phys. Rev. A 87, 033819 (2013)
H. Benisty, A. Lupu, A. Degiron, Phys. Rev. A 91, 053825 (2015)
N.X.A. Rivolta, H. Benisty, B. Maes, Phys. Rev. A 96, 023864 (2017)
S. Lepri, C. Trono, G. Giacomelli, Phys. Rev. Lett. 118, 123901 (2017)
L. Ge, A.D. Stone, Phys. Rev. X 4, 031011 (2014)
S. Assawaworrarit, X. Yu, S. Fan, Nature 546, 387 (2017)
J.L. Gross, J. Yellen,Graph Theory and Its Applications, 2nd edn. (Chapman and Hall/CRC Press, Boca Raton, 2005)
F.J. Dyson, J. Math. Phys. 3, 140 (1962), and references therein
V. Brac de la Perrière, Q. Gaimard, H. Benisty, A. Ramdane, A. Lupu, J. Phys. D Appl. Phys. 52, 255103 (2019)
H. Benisty, C. Yan, A.T. Lupu, A. Degiron, IEEE J. Lightwave Technol. 30, 2675 (2012)
N.B. Nguyen, S.A. Maier, M. Hong, R. Oulton, New J. Phys. 18, 12502 (2016)
F.A. Rodrigues, T.K.D.M. Peron, P. Ji, J. Kurths, Phys. Rep. 610, 1 (2016)
J. Hofbauer, K. Sigmund,Evolutionary Games and Population Dynamics (Cambridge University Press, Cambridge, UK, 1998)
V. Yukalov, E. Yukalova, D. Sornette, Eur. Phys. J. Special Topics 205, 313 (2012)
S.E. Puliafito, J.L. Puliafito, M.C. Grand, Ecol. Econ. 65, 602 (2008)
T.L. Hill,Free Energy Transduction and Biochemical Cycle Kinetics (Springer-Verlag, New York, Inc., 1989)
J. Anderson, Y.C. Chang, A. Papachristodoulou, Automatica 47, 1165 (2011)
R. Breitling, D. Gilbert, M. Heiner, R. Orton, Brief. Bioinform. 9, 404 (2008)
R.N. Gutenkunst, J.J. Waterfall, F.P. Casey, K.S. Brown, C.R. Myers, J.P. Sethna, PLoS Comput. Biol. 3, e189 (2007)
G. Tiana, S. Krishna, S. Pigolotti, M.H. Jensen, K. Sneppen, Phys. Biol. 4, R1 (2007)
B.B. Aldridge, J.M. Burke, D.A. Lauffenburger, P.K. Sorger, Nat. Cell Biol. 8, 1195 (2006)
A. Ciliberto, F. Capuani, J.J. Tyson, PLoS Comput. Biol. 3, 0463 (2007)
S.R. Caplan, A. Essig, Proc. Natl. Acad. Sci. U.S.A. 64, 211 (1969)
S.R. Caplan, A. Essig,Bioenergetics and Linear Nonequilibrium Thermodynamics, The Steady State, Harvard Books in Biophysics Series (Harvard University Press, 2013), Vol. 3
E. Feliu, C. Wiuf, J.R. Soc. Interface 9, 1224 (2012)
S. di Santo, P. Villegas, R. Burioni, M.A. Muñoz, J. Stat. Mech. 2018, 073402 (2018)
B.B. Aldridge, G. Haller, P.K. Sorger, D.A. Lauffenburger, IEE Proc.-Syst. Biol. 153, 425 (2006)
J. Schaber, A. Lapytsko, A. Flockerzi, J.R. Soc. Interface 11, 20130971 (2013)
K.T. Dineley, E.J. Weeber, C. Atkins, J.P. Adams, A.E. Anderson, J.D. Sweatt, J. Neurochem. 77, 961 (2001)
J. Vera, J. Bachmann, A.C. Pfeifer, V. Becker, J.A. Hormiga, N.V. Torres Darias, J. Timmer, U. Klingmüller, O. Wolkenhauer, BMC Syst. Biol. 2, 38 (2008)
A. Semyanova, Cell Calc. 78, 15 (2019)
C. Goupil, H. Ouerdane, E. Herbert, G. Benenti, Y. D’Angelo, Ph. Lecoeur, Phys. Rev. E, 94, 032136 (2016)
C. Goupil, H. Ouerdane, E. Herbert, C. Goupil, Y. D’Angelo, New J. Phys. 21, 023021 (2019)
H. Vroylandt, A. Bonfils, G. Verley, Phys. Rev. E 93, 052123 (2016)
M. Polettini, G. Verley, M. Esposito, Phys. Rev. Lett. 114, 050601 (2015)
G. Hennequin, T.P. Vogels, W. Gerstner, Neuron 82, 1394 (2014)
B.K. Murphy, K.D. Miller, Neuron 61, 635 (2009)
J. Vanbiervliet, B. Vandereycken, W. Michiels, S. Vandewalle, M. Diehl, SIAM J. Optim. 20, 156 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supplementary material in the form of one pdf file available from the Journal web page at https://doi.org/10.1140/epjb/e2020-10219-x.
Publisher's Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Supplementary data
Rights and permissions
About this article
Cite this article
Benisty, H., Goupil, C. Configuration barrier towards parity-time symmetry in randomly connected mesoscopic sets on a graph. Eur. Phys. J. B 93, 192 (2020). https://doi.org/10.1140/epjb/e2020-10219-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjb/e2020-10219-x