Brain Resting State is Critical SFN 2008 Washington, DC Dante R. Chialvo 798.5/UU90 Physiology, Feinberg School of Medicine, Northwestern University, Chicago, IL 60611. USA. Networks link’s statistics for different threshold ρ values Network’s statistical properties All you want to know T ρ <k> N C L D Cran Lran Dran Brain “rest” is defined - more or less unsuccessfully - as the state Brain (ρ +) Positive correlated networks in which there is no explicit brain input or output. Could it be “Friends Net” 2.0 0.1 133 40000 0.065 2.72 4 0.005 2.61 4 2.3 0.3 127 40000 0.516 6.83 31 0.048 2.71 5 comparable to any known dynamical state? 3.0 0.09 128 40000 0.064 2.73 4 0.003 2.65 4 Brain 0.62 128 26985 0.45 4.4 13 0.061 2.62 5 We have proposed that the brain stays near the critical point of a Supercritical Negative correlated networks Critical second order phase transition, where neuronal groups generate a Ising Subcritical (ρ -) 2.0 -0.06 26 29370 0.01 3.24 5 0.01 3.38 8 diversity of flexible collective behaviors, due to the known model “Enemy Net” 2.3 -0.38 26 4784 0 4.17 13 0.1 2.95 7 abundance of metastable states at the transition. It is from this 3.0 -0.10 23 40000 0 3.72 6 0.0 3.7 6 Brain -0.71 27 1684 0 3.54 11 0.14 2.69 5 viewpoint, that the dynamics of brain resting might correspond to a critical state. This conjecture is tested here comparing fMRI brain resting state data from healthy subjects with a paradigmatic critical system, the Ising model. Critical Ising networks ~ brain networks Identical behavior of degree, clustering Check list Subcritical Critical Supercritical correlation and path length in critical The analysis shows striking similarities between the two systems, Ising networks and brain networks lending support to the conjecture that the functioning brain Ising Voxel-based operates near a critical point. Critical Functional Brain Networks Clustering and Path Assortativity Manuscript and related papers: www.chialvo.net 1. Brain networks are small world, with dense local correlations. 2. The distribution of the number of links Voxel-based correlation networks Nodes spatial location measured with the appropriate resolution is BRAIN scale-free. k ρ + Nets 3. Brain networks are assortative, a tendency for nodes with similar number of links to be directly connected. Undirected Degree (k) 4. Positively correlated cortical domains Subcritical Critical Supercritical coexist with approximately equal size anti- correlated non-local domains. Steps to extract the networks (a, b) and computing (c - i) graph properties ([a-i] From van den Heuvel et al., Neuroimage, Still, so what? 2008; [k] from Eguiluz et al., Phys. Rev. Letters, 2005) Roughed balanced correlation landscape is Networks are collections of nodes joined by links. Sites are connected typical in critical Ising and brain networks by a link if their mutual correlation is larger than a certain value ρ . • A toy model with only local excitatory interactions exhibits, at the phase transition, all the relevant BRAIN correlation properties described for the brain resting orde r Critical Point ρ - Nets state large-scale spatio-temporal patterns. disorder • Anti-correlated patterns appearing in absence of TC “inhibitory connections” are related to emergent collective properties. T<TC T~TC T>TC ~1 • Being collective properties, most of these correlation Average of five subjects effects should be seen only at relatively low frequencies. Only local positive interactions • An “order parameter”, characterizing cortical balance Subcritical Critical SuperCritical Balance is disrupted in Chronic Pain might be derived from these concepts, able to E=−J Σ <i,j> Si Sj – B Σk Sk Snapshots of spins states in the Ising model. Patients (Baliki et al. , J. Neurosc. 2008). Long range correlations emerges at the critical point parametrize brain fMRI in health and disease. The Ising model is a toy model of a magnet, it simulates the local interactions between spins. Depending on the temperature, there is Work in collaboration with J. Foss (NWU), D. Fraiman and P. Balenzuela collective order or disorder; a continuous 2nd order phase transition. Funded by NIH NINDS NS58661 (Physics, FCEN, UBA and Mathematics and Sciences Univ. San Andres, Buenos Aires, Argentina).
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