Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
121 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Dynamically selected steady states and criticality in non-reciprocal networks (2312.12039v1)

Published 19 Dec 2023 in cond-mat.dis-nn and cond-mat.stat-mech

Abstract: Diverse equilibrium systems with heterogeneous interactions lie at the edge of stability. Such marginally stable states are dynamically selected as the most abundant ones or as those with the largest basins of attraction. On the other hand, systems with non-reciprocal (or asymmetric) interactions are inherently out of equilibrium, and exhibit a rich variety of steady states, including fixed points, limit cycles and chaotic trajectories. How are steady states dynamically selected away from equilibrium? We address this question in a simple neural network model, with a tunable level of non-reciprocity. Our study reveals different types of ordered phases and it shows how non-equilibrium steady states are selected in each phase. In the spin-glass region, the system exhibits marginally stable behaviour for reciprocal (or symmetric) interactions and it smoothly transitions to chaotic dynamics, as the non-reciprocity (or asymmetry) in the couplings increases. Such region, on the other hand, shrinks and eventually disappears when couplings become anti-symmetric. Our results are relevant to advance the knowledge of disordered systems beyond the paradigm of reciprocal couplings, and to develop an interface between statistical physics of equilibrium spin-glasses and dynamical systems theory.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (61)
  1. T. Castellani and A. Cavagna, “Spin-glass theory for pedestrians,” Journal of Statistical Mechanics: Theory and Experiment 2005, P05012 (2005).
  2. H. Nishimori, Statistical physics of spin glasses and information processing: an introduction, 111 (Clarendon Press, 2001).
  3. T. Aspelmeier, A. J. Bray,  and M. A. Moore, “Complexity of ising spin glasses,” Phys. Rev. Lett. 92, 087203 (2004).
  4. A. Cavagna, I. Giardina,  and G. Parisi, “Numerical study of metastable states in ising spin glasses,” Phys. Rev. Lett. 92, 120603 (2004).
  5. M. Müller and M. Wyart, “Marginal stability in structural, spin and electron glasses,” Annu. Rev. Condens. Matter Phys. 6, 177–200 (2015).
  6. A. V. Herz and J. J. Hopfield, “Earthquake cycles and neural reverberations: Collective oscillations in systems with pulse-coupled threshold elements,” Phys. Rev. Lett. 75, 1222–1225 (1995).
  7. M. M. D. Challet, A. Chessa and Y.-C. Zhang, “From minority games to real markets,” Quantitative Finance 1, 168–176 (2001).
  8. D. Stokić, R. Hanel,  and S. Thurner, “Inflation of the edge of chaos in a simple model of gene interaction networks,” Phys. Rev. E 77, 061917 (2008).
  9. J. M. Beggs and N. Timme, “Being critical of criticality in the brain,” Frontiers Physiol. 3 (2012).
  10. T. Mora and W. Bialek, “Are biological systems poised at criticality?” J. Stat. Phys. 144 (2011).
  11. M. A. Muñoz, “Colloquium: Criticality and dynamical scaling in living systems,” Reviews of Modern Physics 90, 031001 (2018).
  12. P. Moretti and M. A. Muñoz, “Griffiths phases and the stretching of criticality in brain networks,” Nature communications 4, 1–10 (2013).
  13. G. Biroli, G. Bunin,  and C. Cammarota, “Marginally stable equilibria in critical ecosystems,” New Journal of Physics 20, 083051 (2018).
  14. A. Altieri, F. Roy, C. Cammarota,  and G. Biroli, “Properties of equilibria and glassy phases of the random lotka-volterra model with demographic noise,” Phys. Rev. Lett. 126, 258301 (2021).
  15. T. M. Pham and K. Kaneko, “Double-replica theory for evolution of genotype-phenotype interrelationship,” Phys. Rev. Research 5, 023049 (2023).
  16. L. Correale, M. Leone, A. Pagnani, M. Weigt,  and R. Zecchina, “The computational core and fixed point organization in boolean networks,” Journal of Statistical Mechanics: Theory and Experiment 2006, P03002 (2006).
  17. Y. V. Fyodorov and B. A. Khoruzhenko, “Nonlinear analogue of the may-wigner instability transition,” Proc. Natl. Acad. Sci. 113, 6827–6832 (2016).
  18. S. Hwang, V. Folli, E. Lanza, G. Parisi, G. Ruocco,  and F. Zamponi, “On the number of limit cycles in asymmetric neural networks,” Journal of Statistical Mechanics: Theory and Experiment 2019, 053402 (2019).
  19. S. B. Fedeli, Y. V. Fyodorov,  and J. Ipsen, “Nonlinearity-generated resilience in large complex systems,” Phys. Rev. E 103, 022201 (2021).
  20. B. Lacroix-A-Chez-Toine and Y. V. Fyodorov, ‘‘Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates,” Journal of Physics A: Mathematical and Theoretical 55, 144001 (2022).
  21. V. Ros, F. Roy, G. Biroli, G. Bunin,  and A. M. Turner, “Generalized lotka-volterra equations with random, nonreciprocal interactions: The typical number of equilibria,” Phys. Rev. Lett. 130, 257401 (2023).
  22. H. Sompolinsky, A. Crisanti,  and H.-J. Sommers, “Chaos in random neural networks,” Phys. Rev. Lett. 61, 259 (1988).
  23. I. Ginzburg and H. Sompolinsky, “Theory of correlations in stochastic neural networks,” Phys. Rev. E 50, 3171–3191 (1994).
  24. M. Fruchart, R. Hanai, P. B. Littlewood,  and V. Vitelli, “Non-reciprocal phase transitions,” Nature 592, 363–369 (2021).
  25. G. Wainrib and J. Touboul, “Topological and dynamical complexity of random neural networks,” Phys. Rev. Lett. 110, 118101 (2013).
  26. P. Van Mieghem, Graph spectra for complex networks (Cambridge University Press, 2010).
  27. S. N. Dorogovtsev, A. V. Goltsev, J. F. Mendes,  and A. N. Samukhin, “Spectra of complex networks,” Phys. Rev. E 68, 046109 (2003).
  28. T. Tao, “Outliers in the spectrum of iid matrices with bounded rank perturbations,” Probab. Theory Relat. Fields 155, 231–263 (2013).
  29. K. Rajan and L. Abbott, “Eigenvalue spectra of random matrices for neural networks,” Phys. Rev. Lett. 97, 188104 (2006).
  30. A. Knowles and J. Yin, “The Isotropic Semicircle Law and Deformation of Wigner Matrices,” Communications on Pure and Applied Mathematics 66, 1663–1749 (2013).
  31. A. Knowles and J. Yin, “The outliers of a deformed Wigner matrix,” The Annals of Probability 42 (2014), 10.1214/13-AOP855.
  32. H. J. Sommers, A. Crisanti, H. Sompolinsky,  and Y. Stein, “Spectrum of Large Random Asymmetric Matrices,” Phys. Rev. Lett. 60, 1895–1898 (1988).
  33. S. O’Rourke and D. Renfrew, “Low rank perturbations of large elliptic random matrices,” Electronic Journal of Probability 19, 1–65 (2014), arXiv:1309.5326 [math-ph].
  34. D. Dahmen, S. Grün, M. Diesmann,  and M. Helias, “Second type of criticality in the brain uncovers rich multiple-neuron dynamics,” Proc. Natl. Acad. Sci. 116, 13051–13060 (2019).
  35. E. R. Garcia, M. J. Crumpton,  and T. Galla, “Niche overlap and hopfield-like interactions in generalized random lotka-volterra systems,” Phys. Rev. E 108, 034120 (2023).
  36. J. Schuecker, S. Goedeke,  and M. Helias, “Optimal sequence memory in driven random networks,” Phys. Rev. X 8, 041029 (2018).
  37. J. Kadmon and H. Sompolinsky, “Transition to chaos in random neuronal networks,” Phys. Rev. X 5, 041030 (2015).
  38. K. Rajan, L. F. Abbott,  and H. Sompolinsky, “Stimulus-dependent suppression of chaos in recurrent neural networks,” Phys. Rev. E 82, 011903 (2010).
  39. F. Mastrogiuseppe and S. Ostojic, “Intrinsically-generated fluctuating activity in excitatory-inhibitory networks,” PLoS computational biology 13, e1005498 (2017).
  40. P. C. Martin, E. D. Siggia,  and H. A. Rose, “Statistical dynamics of classical systems,” Phys. Rev. A 8, 423–437 (1973).
  41. A. Coolen, “Chapter 15 statistical mechanics of recurrent neural networks ii — dynamics,” in Neuro-Informatics and Neural Modelling, Handbook of Biological Physics, Vol. 4, edited by F. Moss and S. Gielen (North-Holland, 2001) pp. 619–684.
  42. J. P. L. Hatchett and A. C. C. Coolen, “Asymmetrically extremely dilute neural networks with langevin dynamics and unconventional results,” Journal of Physics A: Mathematical and General 37, 7199 (2004).
  43. A. Crisanti and H. Sompolinsky, “Path integral approach to random neural networks,” Phys. Rev. E 98, 062120 (2018).
  44. M. Helias and D. Dahmen, “Dynamic mean-field theory for random networks,” in Statistical Field Theory for Neural Networks (Springer, 2020) pp. 95–126.
  45. M. Opper and S. Diederich, “Phase transition and 1/ f noise in a game dynamical model,” Phys. Rev. Lett. 69, 1616–1619 (1992).
  46. L. Sidhom and T. Galla, “Ecological communities from random generalized Lotka-Volterra dynamics with nonlinear feedback,” Phys. Rev. E 101, 032101 (2020).
  47. D. Sherrington and S. Kirkpatrick, “Solvable Model of a Spin-Glass,” Phys. Rev. Lett. 35, 1792–1796 (1975).
  48. We use the “hat” notation for quantities measured in simulations, to distinguish them from their theoretical counterparts, as obtained from the path integral approach. In particular, single-realizations are denoted by a subscript s𝑠sitalic_s, e.g., M^ssubscript^𝑀𝑠\hat{M}_{s}over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT, whereas averages over many realizations are denoted with M^^𝑀\hat{M}over^ start_ARG italic_M end_ARG.
  49. S. Cure and I. Neri, “Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems,” SciPost Physics 14, 093 (2023).
  50. R. Hanel, M. Pöchacker,  and S. Thurner, “Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics,” Phil. Trans. R. Soc. A. 368, 5583–5596 (2010).
  51. G. B. Morales, S. Di Santo,  and M. A. Muñoz, “Quasiuniversal scaling in mouse-brain neuronal activity stems from edge-of-instability critical dynamics,” Proceedings of the National Academy of Sciences 120, e2208998120 (2023).
  52. G. B. Morales and M. A. Muñoz, “Optimal input representation in neural systems at the edge of chaos,” Biology 10, 702 (2021).
  53. T. Laffargue, K.-D. N. T. Lam, J. Kurchan,  and J. Tailleur, “Large deviations of lyapunov exponents,” Journal of Physics A: Mathematical and Theoretical 46, 254002 (2013).
  54. T. Laffargue, P. Sollich, J. Tailleur,  and F. van Wijland, “Large-scale fluctuations of the largest lyapunov exponent in diffusive systems,” Europhysics Letters 110, 10006 (2015).
  55. A. Wardak and P. Gong, “Extended anderson criticality in heavy-tailed neural networks,” Phys. Rev. Lett. 129, 048103 (2022).
  56. G. Torrisi, R. Kühn,  and A. Annibale, “Percolation on the gene regulatory network,” Journal of Statistical Mechanics: Theory and Experiment 2020, 083501 (2020).
  57. R. Toral and P. Colet, Stochastic numerical methods: an introduction for students and scientists (John Wiley & Sons, 2014).
  58. J. P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Reviews of Modern Physics 57, 617–656 (1985).
  59. A. Pikovsky and A. Politi, Lyapunov exponents: a tool to explore complex dynamics (Cambridge University Press, Cambridge, 2016).
  60. G. Benettin, L. Galgani, A. Giorgilli,  and J.-M. Strelcyn, “Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application,” Meccanica 15, 21–30 (1980).
  61. A. Wolf, J. B. Swift, H. L. Swinney,  and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D: Nonlinear Phenomena 16, 285–317 (1985).
Citations (5)

Summary

We haven't generated a summary for this paper yet.

X Twitter Logo Streamline Icon: https://streamlinehq.com