Griffiths phases and localization in hierarchical modular networks

We study variants of hierarchical modular network models suggested by Kaiser and Hilgetag [ Front. in Neuroinform., 4 (2010) 8] to model functional brain connectivity, using extensive simulations and quenched mean-field theory (QMF), focusing on structures with a connection probability that decays exponentially with the level index. Such networks can be embedded in two-dimensional Euclidean space. We explore the dynamic behavior of the contact process (CP) and threshold models on networks of this kind, including hierarchical trees. While in the small-world networks originally proposed to model brain connectivity, the topological heterogeneities are not strong enough to induce deviations from mean-field behavior, we show that a Griffiths phase can emerge under reduced connection probabilities, approaching the percolation threshold. In this case the topological dimension of the networks is finite, and extended regions of bursty, power-law dynamics are observed. Localization in the steady state is also shown via QMF. We investigate the effects of link asymmetry and coupling disorder, and show that localization can occur even in small-world networks with high connectivity in case of link disorder.

[1]  Yoshiki Kuramoto,et al.  Self-entrainment of a population of coupled non-linear oscillators , 1975 .

[2]  Thomas Vojta,et al.  Rare regions and Griffiths singularities at a clean critical point: the five-dimensional disordered contact process. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  M. Nicolelis,et al.  Spike Avalanches Exhibit Universal Dynamics across the Sleep-Wake Cycle , 2010, PloS one.

[4]  Angélica S. Mata,et al.  Heterogeneous pair-approximation for the contact process on complex networks , 2014, 1402.2832.

[5]  Joaquín J. Torres,et al.  Robust Short-Term Memory without Synaptic Learning , 2010, PloS one.

[6]  John M. Beggs,et al.  Being Critical of Criticality in the Brain , 2012, Front. Physio..

[7]  Edward T. Bullmore,et al.  Modular and Hierarchically Modular Organization of Brain Networks , 2010, Front. Neurosci..

[8]  Albert-László Barabási,et al.  Hierarchical organization in complex networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  M. A. Muñoz,et al.  Frustrated hierarchical synchronization and emergent complexity in the human connectome network , 2014, Scientific Reports.

[10]  D. Long.  Networks of the Brain , 2011 .

[11]  Alessandro Vespignani,et al.  Large-scale topological and dynamical properties of the Internet. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Zhen Jin,et al.  Influence of infection rate and migration on extinction of disease in spatial epidemics. , 2010, Journal of theoretical biology.

[13]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[14]  R. Dickman,et al.  Nonequilibrium Phase Transitions in Lattice Models , 1999 .

[15]  W. Singer,et al.  Neuronal avalanches in spontaneous activity in vivo. , 2010, Journal of neurophysiology.

[16]  Nikola T. Markov,et al.  Weight Consistency Specifies Regularities of Macaque Cortical Networks , 2010, Cerebral cortex.

[17]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[18]  M. A. Muñoz,et al.  Rounding of abrupt phase transitions in brain networks , 2014, 1407.7392.

[19]  R. Dickman,et al.  How to simulate the quasistationary state. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  M. A. Muñoz,et al.  Griffiths phases and the stretching of criticality in brain networks , 2013, Nature Communications.

[21]  B. M. Fulk.  MATH , 1992 .

[22]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[23]  M. A. Muñoz,et al.  Griffiths phases on complex networks. , 2010, Physical review letters.

[24]  Marcus Kaiser,et al.  A tutorial in connectome analysis: Topological and spatial features of brain networks , 2011, NeuroImage.

[25]  T. Prescott,et al.  The brainstem reticular formation is a small-world, not scale-free, network , 2006, Proceedings of the Royal Society B: Biological Sciences.

[26]  Mark S. Granovetter.  The Strength of Weak Ties , 1973, American Journal of Sociology.

[27]  Florentin Wörgötter,et al.  Self-Organized Criticality in Developing Neuronal Networks , 2010, PLoS Comput. Biol..

[28]  John M. Beggs,et al.  Neuronal Avalanches in Neocortical Circuits , 2003, The Journal of Neuroscience.

[29]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[30]  O. Sporns,et al.  Organization, development and function of complex brain networks , 2004, Trends in Cognitive Sciences.

[31]  David Bawden,et al.  Book Review: Evolution and Structure of the Internet: A Statistical Physics Approach. , 2006 .

[32]  Viola Priesemann,et al.  Neuronal Avalanches Differ from Wakefulness to Deep Sleep – Evidence from Intracranial Depth Recordings in Humans , 2013, PLoS Comput. Biol..

[33]  Norman T. J. Bailey,et al.  The Mathematical Theory of Infectious Diseases , 1975 .

[34]  Miguel A Muñoz,et al.  Quenched disorder forbids discontinuous transitions in nonequilibrium low-dimensional systems. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  P. Bak,et al.  Self-organized criticality. , 1988, Physical review. A, General physics.

[36]  Noam Berger,et al.  The diameter of long-range percolation clusters on finite cycles , 2001, Random Struct. Algorithms.

[37]  W. Marsden.  I and J , 2012 .

[38]  D. Turcotte,et al.  Self-organized criticality , 1999 .

[39]  M P Young,et al.  Anatomical connectivity defines the organization of clusters of cortical areas in the macaque monkey and the cat. , 2000, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[40]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[41]  S. Levin,et al.  Emergent trade-offs and selection for outbreak frequency in spatial epidemics , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[42]  Zach D. Haga,et al.  Avalanche Analysis from Multielectrode Ensemble Recordings in Cat, Monkey, and Human Cerebral Cortex during Wakefulness and Sleep , 2012, Front. Physio..

[43]  Romualdo Pastor-Satorras,et al.  Slow dynamics and rare-region effects in the contact process on weighted tree networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Christos Faloutsos,et al.  Epidemic thresholds in real networks , 2008, TSEC.

[45]  D. J. Felleman,et al.  Distributed hierarchical processing in the primate cerebral cortex. , 1991, Cerebral cortex.

[46]  C. Bédard,et al.  Does the 1/f frequency scaling of brain signals reflect self-organized critical states? , 2006, Physical review letters.

[47]  Géza Ódor,et al.  Slow, bursty dynamics as a consequence of quenched network topologies. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  Zhen Jin,et al.  Phase transition in spatial epidemics using cellular automata with noise , 2011, Ecological Research.

[49]  Marcus Kaiser,et al.  Optimal Hierarchical Modular Topologies for Producing Limited Sustained Activation of Neural Networks , 2009, Front. Neuroinform..

[50]  P. Latham,et al.  Synergy, Redundancy, and Independence in Population Codes, Revisited , 2005, The Journal of Neuroscience.

[51]  T. Liggett.  Interacting Particle Systems , 1985 .

[52]  G. Ódor,et al.  Spectral analysis and slow spreading dynamics on complex networks. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  Thomas Vojta,et al.  TOPICAL REVIEW: Rare region effects at classical, quantum and nonequilibrium phase transitions , 2006 .

[54]  R. Pastor-Satorras,et al.  Langevin approach for the dynamics of the contact process on annealed scale-free networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[55]  T. E. Harris.  Contact Interactions on a Lattice , 1974 .

[56]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[57]  Sergey N. Dorogovtsev,et al.  Localization and Spreading of Diseases in Complex Networks , 2012, Physical review letters.

[58]  A. Barabasi,et al.  Hierarchical Organization of Modularity in Metabolic Networks , 2002, Science.

[59]  Marcus Kaiser,et al.  Hierarchy and Dynamics of Neural Networks , 2010, Front. Neuroinform..

[60]  Robert B. Griffiths,et al.  Nonanalytic Behavior Above the Critical Point in a Random Ising Ferromagnet , 1969 .

[61]  Géza Odor.  Phase transition classes in triplet and quadruplet reaction-diffusion models. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[62]  Peter E. Latham,et al.  Computing and Stability in Cortical Networks , 2004, Neural Computation.

[63]  A. van Ooyen,et al.  A simple rule for axon outgrowth and synaptic competition generates realistic connection lengths and filling fractions. , 2009, Cerebral cortex.

[64]  G. Ódor.  Universality classes in nonequilibrium lattice systems , 2002, cond-mat/0205644.

[65]  C. Lee Giles,et al.  Efficient identification of Web communities , 2000, KDD '00.

[66]  Shan Yu,et al.  A Small World of Neuronal Synchrony , 2008, Cerebral cortex.