- Open Access
Weak-Correlation Universality and Macroscopic Fluctuations in Power-Law Recurrent Networks
Phys. Rev. Lett. 135, 107403 – Published 4 September, 2025
DOI: https://doi.org/10.1103/l6pc-y1x6
Abstract
Balanced network theory describes the dynamic state of randomly connected systems with intrinsic negative feedback, predicting the suppression of correlations as connectivity increases. We show that this weak-correlation universality holds broadly but breaks down in power-law networks where the largest out-degrees scale with system size. In such networks, residual correlations cross over to a regime that retains macroscopic fluctuations in the thermodynamic limit. Unlike Erdös-Rényi networks, the degree heterogeneity of power-law topologies amplifies coherent population-level fluctuations. Our results provide a theoretical foundation for the strong correlations observed in real-world networks with sparse and heterogeneous connectivity.
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References (20)
- C. van Vreeswijk and H. Sompolinsky, Chaos in neuronal networks with balanced excitatory and inhibitory activity, Science 274, 1724 (1996); A. Renart, J. d. l. Rocha, P. Bartho, L. Hollender, N. Parga, A. Reyes, and K. D. Harris, The asynchronous state in cortical circuits, 327, 587 (2010); G. Facchetti, G. Iacono, and C. Altafini, Computing global structural balance in large-scale signed social networks, Proc. Natl. Acad. Sci. U.S.A. 108, 20953 (2011); M. Saeedian, N. Azimi-Tafreshi, G. R. Jafari, and J. Kertesz, Epidemic spreading on evolving signed networks, Phys. Rev. E 95, 022314 (2017); A. Zenere, V. Kuzmanovski, P. Fontana, M. Vingron, and T. Zambelli, Using high-throughput multi-omics data to investigate structural balance in elementary gene regulatory network motifs, Bioinformatics 37, 2315 (2021).
- R. J. Glauber, Time-dependent statistics of the Ising model, J. Math. Phys. (N.Y.) 4, 294 (1963).
- C. van Vreeswijk and H. Sompolinsky, Chaotic balanced state in a model of cortical circuits, Neural Comput. 10, 1321 (1998).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/l6pc-y1x6 for details of calculations, additional results, and numerical methods, which includes Refs. [5–9].
- C. van Vreeswijk and H. Sompolinsky, Irregular activity in large networks of neurons, in Les Houches (Elsevier, New York, 2005), Vol. 80, pp. 341–406.
- M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64, 026118 (2001).
- J. L. Doob, Topics in the theory of Markoff chains, Trans. Am. Math. Soc. 52, 37 (1942).
- R. Brette, Exact simulation of integrate-and-fire models with synaptic conductances, Neural Comput. 19, 2608 (2007).
- R. Engelken, Sparseprop: Efficient event-based simulation and training of sparse recurrent spiking neural networks, Adv. Neural Inf. Process. Syst. 36, 3638 (2023).
- R. Darshan, C. van Vreeswijk, and D. Hansel, Strength of correlations in strongly recurrent neuronal networks, Phys. Rev. X 8, 031072 (2018).
- F. Farkhooi and W. Stannat, Complete mean-field theory for dynamics of binary recurrent networks, Phys. Rev. Lett. 119, 208301 (2017).
- D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys. 22, 403 (1976).
- J. B. Rundle, K. F. Tiampo, W. Klein, and J. S. Sá Martins, Self-organization in leaky threshold systems: The influence of near-mean field dynamics and its implications for earthquakes, neurobiology, and forecasting, Proc. Natl. Acad. Sci. U.S.A. 99, 2514 (2002).
- N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, J. Comput. Neurosci. 8, 183 (2000).
- D. Hansel and H. Sompolinsky, Chaos and synchrony in a model of a hypercolumn in visual cortex, J. Comput. Neurosci. 3, 7 (1996).
- M. Helias, T. Tetzlaff, and M. Diesmann, The correlation structure of local neuronal networks intrinsically results from recurrent dynamics, PLoS Comput. Biol. 10, e1003428 (2014); D. Dahmen, H. Bos, and M. Helias, Correlated fluctuations in strongly coupled binary networks beyond equilibrium, Phys. Rev. X 6, 031024 (2016).
- A. Y. Tan, Y. Chen, B. Scholl, E. Seidemann, and N. J. Priebe, Sensory stimulation shifts visual cortex from synchronous to asynchronous states, Nature (London) 509, 226 (2014); S. Arroyo, C. Bennett, and S. Hestrin, Correlation of synaptic inputs in the visual cortex of awake, behaving mice, Neuron 99, 1289 (2018); M. R. Cohen and A. Kohn, Measuring and interpreting neuronal correlations, Nat. Neurosci. 14, 811 (2011).
- J. O. Lloyd-Smith, S. J. Schreiber, P. E. Kopp, and W. M. Getz, Superspreading and the effect of individual variation on disease emergence, Nature (London) 438, 355 (2005); R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86, 3200 (2001).
- A. Attanasi, A. Cavagna, L. Del Castello, I. Giardina, T. S. Grigera, A. Jelić, S. Melillo, L. Parisi, E. Silvestri, M. Viale et al., Collective behaviour without collective order in wild swarms of midges, PLoS Comput. Biol. 10, e1003697 (2014).
- F. Farkhooi, Weak-correlation universality and macroscopic fluctuations in power-law recurrent networks, Zenodo, 10.5281/zenodo.16546375 (2025).
