Breakdown of random matrix universality in Markov models

Abstract

Biological systems need to react to stimuli over a broad spectrum of timescales. If and how this ability can emerge without external fine-tuning is a puzzle. We consider here this problem in discrete Markovian systems, where we can leverage results from random matrix theory. Indeed, generic large transition matrices are governed by universal results, which predict the absence of long timescales unless fine-tuned. We consider an ensemble of transition matrices and motivate a temperature-like variable that controls the dynamic range of matrix elements, which we show plays a crucial role in the applicability of the large matrix limit: as the dynamic range increases, a phase transition occurs whereby the random matrix theory result is avoided, and long relaxation times ensue, in the entire `ordered' phase. We furthermore show that this phase transition is accompanied by a drop in the entropy rate and a peak in complexity, as measured by predictive information (Bialek, Nemenman, Tishby Neural Computation 13(21) 2001). Extending the Markov model to a Hidden Markov model (HMM), we show that observable sequences inherit properties of the hidden sequences, allowing HMMs to be understood in terms of more accessible Markov models. We then apply our findings to fMRI data from 820 human subjects scanned at wakeful rest. We show that the data can be quantitatively understood in terms of the random model, and that brain activity lies close to the phase transition when engaged in unconstrained, task-free cognition -- supporting the brain criticality hypothesis in this context.

arXiv comment12 pages; v2: expanded introduction and added references, addition of eigenvalue scatter plots

References in corpus (22)

• Are biological systems poised at criticality?
• Dynamical synapses causing self-organized criticality in neural networks
• Colloquium: Criticality and dynamical scaling in living systems
• Entropy estimation of symbol sequences
• Spectra of complex networks
• Self-Organized Criticality model for Brain Plasticity
• Marginal Stability in Structural, Spin and Electron Glasses
• Marginally Stable Equilibria in Critical Ecosystems
• Zipf's law and criticality in multivariate data without fine-tuning
• Universality of Zipf's Law
• Evidence for quasicritical brain dynamics
• Self-organization toward criticality by synaptic plasticity
• Zipf's law for word frequencies: word forms versus lemmas in long texts
• Complexity Through Nonextensivity
• Circular Law Theorem for Random Markov Matrices
• Random Language Model
• Large Pseudo-Counts and $L_2$-Norm Penalties Are Necessary for the Mean-Field Inference of Ising and Potts Models
• Shannon Entropy Rate of Hidden Markov Processes
• Spectra of Random Stochastic Matrices and Relaxation in Complex Systems
• The ensemble of random Markov matrices
• Emergence of order in random languages
• Random stochastic matrices from classical compact Lie groups and symmetric spaces

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• Entangled criticality and irreversibility in random Markov dynamics
• Statistics of the non-zero eigenvalues and singular values of low-rank random matrices with non-negative entries