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Frustrated hierarchical synchronization and emergent complexity in the human connectome network.

Villegas P, Moretti P, Muñoz MA - Sci Rep (2014)

Bottom Line: This novel phase stems from the hierarchical modular organization of the connectome.Where one would expect a hierarchical synchronization process, we show that the interplay between structural bottlenecks and quenched intrinsic frequency heterogeneities at many different scales, gives rise to frustrated synchronization, metastability, and chimera-like states, resulting in a very rich and complex phenomenology.We uncover the origin of the dynamic freezing behind these features by using spectral graph theory and discuss how the emerging complex synchronization patterns relate to the need for the brain to access -in a robust though flexible way- a large variety of functional attractors and dynamical repertoires without ad hoc fine-tuning to a critical point.

View Article: PubMed Central - PubMed

Affiliation: Departamento de Electromagnetismo y Física de la Materia e Instituto Carlos I de Física Teórica y Computacional. Universidad de Granada, E-18071 Granada, Spain.

ABSTRACT
The spontaneous emergence of coherent behavior through synchronization plays a key role in neural function, and its anomalies often lie at the basis of pathologies. Here we employ a parsimonious (mesoscopic) approach to study analytically and computationally the synchronization (Kuramoto) dynamics on the actual human-brain connectome network. We elucidate the existence of a so-far-uncovered intermediate phase, placed between the standard synchronous and asynchronous phases, i.e. between order and disorder. This novel phase stems from the hierarchical modular organization of the connectome. Where one would expect a hierarchical synchronization process, we show that the interplay between structural bottlenecks and quenched intrinsic frequency heterogeneities at many different scales, gives rise to frustrated synchronization, metastability, and chimera-like states, resulting in a very rich and complex phenomenology. We uncover the origin of the dynamic freezing behind these features by using spectral graph theory and discuss how the emerging complex synchronization patterns relate to the need for the brain to access -in a robust though flexible way- a large variety of functional attractors and dynamical repertoires without ad hoc fine-tuning to a critical point.

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Local synchronization in the human connectome.(a) Oscillations of the local order parameters (“chimera-like states”) in one particular modulus in the partitions of the HC into 12 (green, l = 1, and k = 3) and 2 (magenta, l = 2, and k = 10) moduli, respectively. The characteristic frequency of these oscillations is typically between 0.01 and 0.1 Hz (a range which coincides with slow modes detected in brain activity; see e.g.32). (b) Average of the local order parameter over all moduli and (c) chimera index for moduli at levels as in a), as a function of k. Global order (thin black line in b)) emerges only after local order is attained at lower levels. (d) Average decay of activity ρ for identical frequencies ω = 0 in the HC network and comparison with a single-level modular network (made up of 4 similar random moduli at a single hierarchical level) of the same size and average connectivity as the HC network. Symbols stand for different values of k. (e) Characteristic decay times corresponding to the inverse of the first 1000 non-trivial eigenvalues of the Laplacian matrix (x axis) as a function of their respective ordered indices (y axis), for networks as in (d). The stretched exponential behavior in (d) is the result of the convolution of slow time scales associated with small eigenvalues in (e).
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f3: Local synchronization in the human connectome.(a) Oscillations of the local order parameters (“chimera-like states”) in one particular modulus in the partitions of the HC into 12 (green, l = 1, and k = 3) and 2 (magenta, l = 2, and k = 10) moduli, respectively. The characteristic frequency of these oscillations is typically between 0.01 and 0.1 Hz (a range which coincides with slow modes detected in brain activity; see e.g.32). (b) Average of the local order parameter over all moduli and (c) chimera index for moduli at levels as in a), as a function of k. Global order (thin black line in b)) emerges only after local order is attained at lower levels. (d) Average decay of activity ρ for identical frequencies ω = 0 in the HC network and comparison with a single-level modular network (made up of 4 similar random moduli at a single hierarchical level) of the same size and average connectivity as the HC network. Symbols stand for different values of k. (e) Characteristic decay times corresponding to the inverse of the first 1000 non-trivial eigenvalues of the Laplacian matrix (x axis) as a function of their respective ordered indices (y axis), for networks as in (d). The stretched exponential behavior in (d) is the result of the convolution of slow time scales associated with small eigenvalues in (e).

Mentions: Fig. 3 shows numerical results for the local order parameter r(l) for some of the moduli at the 2 hierarchical levels, l = 1 and l = 2 in the HC network. It reveals that (Fig. 3a) local coherences exhibit oscillatory patterns in time (with characteristic frequencies typically between 0.01 and 0.1 Hz) and that (Fig. 3b) the transition to local coherence at progressively higher hierarchical level occurs at progressively larger values of k; i.e. coherence emerges out of a hierarchical bottom-up process as illustrated above for the for the two-block model (see3542). Observe, however, that local oscillations were not present in the two-block model. This suggests that the 12 moduli in the HC are on their turn composed of finer sub-moduli and that structural frustration, as introduced above, affects all hierarchical levels. The average variance of local coherences (called chimera index, χ)44 exhibits a marked peak –reflecting maximal configurational variability– at the transition point for the corresponding level (Fig. 3b–c and Methods section). Similar intra-modular oscillatory patterns –dubbed chimera states– have been recently found41434445 in Kuramoto models in which explicit phase lags induce a different kind of frustration, hindering global synchronization. Strictly speaking, chimeras are defined in systems of identical oscillators. In such a case, a non-zero phase lag term is essential for partial synchronization to occur. Realistic models of the brain, however, require oscillators to be heterogenous. States of partial synchronization in empirical brain networks with frequency heterogeneity have been found for Kuramoto models with explicit time delays31. In contrast, the chimera-like states put forward here have a purely structural origin, as they arise from the network topology. It was noted in the past that synchronization in a synthetic network with hubs could be limited to those hubs by tuning clustering properties, and global order could be attained in a monotonous step-like fashion upon increasing k46. Fig. 3b instead reveals that the ordering process in the hierarchical modular HC may be non-monotonous: coherence does not systematically grow with k. Indeed, the emergence of local order in some community may hinder or reduce coherence in others, inducing local “desynchronization” and reflecting the metastable nature of the explored states.


Frustrated hierarchical synchronization and emergent complexity in the human connectome network.

Villegas P, Moretti P, Muñoz MA - Sci Rep (2014)

Local synchronization in the human connectome.(a) Oscillations of the local order parameters (“chimera-like states”) in one particular modulus in the partitions of the HC into 12 (green, l = 1, and k = 3) and 2 (magenta, l = 2, and k = 10) moduli, respectively. The characteristic frequency of these oscillations is typically between 0.01 and 0.1 Hz (a range which coincides with slow modes detected in brain activity; see e.g.32). (b) Average of the local order parameter over all moduli and (c) chimera index for moduli at levels as in a), as a function of k. Global order (thin black line in b)) emerges only after local order is attained at lower levels. (d) Average decay of activity ρ for identical frequencies ω = 0 in the HC network and comparison with a single-level modular network (made up of 4 similar random moduli at a single hierarchical level) of the same size and average connectivity as the HC network. Symbols stand for different values of k. (e) Characteristic decay times corresponding to the inverse of the first 1000 non-trivial eigenvalues of the Laplacian matrix (x axis) as a function of their respective ordered indices (y axis), for networks as in (d). The stretched exponential behavior in (d) is the result of the convolution of slow time scales associated with small eigenvalues in (e).
© Copyright Policy - open-access
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4126002&req=5

f3: Local synchronization in the human connectome.(a) Oscillations of the local order parameters (“chimera-like states”) in one particular modulus in the partitions of the HC into 12 (green, l = 1, and k = 3) and 2 (magenta, l = 2, and k = 10) moduli, respectively. The characteristic frequency of these oscillations is typically between 0.01 and 0.1 Hz (a range which coincides with slow modes detected in brain activity; see e.g.32). (b) Average of the local order parameter over all moduli and (c) chimera index for moduli at levels as in a), as a function of k. Global order (thin black line in b)) emerges only after local order is attained at lower levels. (d) Average decay of activity ρ for identical frequencies ω = 0 in the HC network and comparison with a single-level modular network (made up of 4 similar random moduli at a single hierarchical level) of the same size and average connectivity as the HC network. Symbols stand for different values of k. (e) Characteristic decay times corresponding to the inverse of the first 1000 non-trivial eigenvalues of the Laplacian matrix (x axis) as a function of their respective ordered indices (y axis), for networks as in (d). The stretched exponential behavior in (d) is the result of the convolution of slow time scales associated with small eigenvalues in (e).
Mentions: Fig. 3 shows numerical results for the local order parameter r(l) for some of the moduli at the 2 hierarchical levels, l = 1 and l = 2 in the HC network. It reveals that (Fig. 3a) local coherences exhibit oscillatory patterns in time (with characteristic frequencies typically between 0.01 and 0.1 Hz) and that (Fig. 3b) the transition to local coherence at progressively higher hierarchical level occurs at progressively larger values of k; i.e. coherence emerges out of a hierarchical bottom-up process as illustrated above for the for the two-block model (see3542). Observe, however, that local oscillations were not present in the two-block model. This suggests that the 12 moduli in the HC are on their turn composed of finer sub-moduli and that structural frustration, as introduced above, affects all hierarchical levels. The average variance of local coherences (called chimera index, χ)44 exhibits a marked peak –reflecting maximal configurational variability– at the transition point for the corresponding level (Fig. 3b–c and Methods section). Similar intra-modular oscillatory patterns –dubbed chimera states– have been recently found41434445 in Kuramoto models in which explicit phase lags induce a different kind of frustration, hindering global synchronization. Strictly speaking, chimeras are defined in systems of identical oscillators. In such a case, a non-zero phase lag term is essential for partial synchronization to occur. Realistic models of the brain, however, require oscillators to be heterogenous. States of partial synchronization in empirical brain networks with frequency heterogeneity have been found for Kuramoto models with explicit time delays31. In contrast, the chimera-like states put forward here have a purely structural origin, as they arise from the network topology. It was noted in the past that synchronization in a synthetic network with hubs could be limited to those hubs by tuning clustering properties, and global order could be attained in a monotonous step-like fashion upon increasing k46. Fig. 3b instead reveals that the ordering process in the hierarchical modular HC may be non-monotonous: coherence does not systematically grow with k. Indeed, the emergence of local order in some community may hinder or reduce coherence in others, inducing local “desynchronization” and reflecting the metastable nature of the explored states.

Bottom Line: This novel phase stems from the hierarchical modular organization of the connectome.Where one would expect a hierarchical synchronization process, we show that the interplay between structural bottlenecks and quenched intrinsic frequency heterogeneities at many different scales, gives rise to frustrated synchronization, metastability, and chimera-like states, resulting in a very rich and complex phenomenology.We uncover the origin of the dynamic freezing behind these features by using spectral graph theory and discuss how the emerging complex synchronization patterns relate to the need for the brain to access -in a robust though flexible way- a large variety of functional attractors and dynamical repertoires without ad hoc fine-tuning to a critical point.

View Article: PubMed Central - PubMed

Affiliation: Departamento de Electromagnetismo y Física de la Materia e Instituto Carlos I de Física Teórica y Computacional. Universidad de Granada, E-18071 Granada, Spain.

ABSTRACT
The spontaneous emergence of coherent behavior through synchronization plays a key role in neural function, and its anomalies often lie at the basis of pathologies. Here we employ a parsimonious (mesoscopic) approach to study analytically and computationally the synchronization (Kuramoto) dynamics on the actual human-brain connectome network. We elucidate the existence of a so-far-uncovered intermediate phase, placed between the standard synchronous and asynchronous phases, i.e. between order and disorder. This novel phase stems from the hierarchical modular organization of the connectome. Where one would expect a hierarchical synchronization process, we show that the interplay between structural bottlenecks and quenched intrinsic frequency heterogeneities at many different scales, gives rise to frustrated synchronization, metastability, and chimera-like states, resulting in a very rich and complex phenomenology. We uncover the origin of the dynamic freezing behind these features by using spectral graph theory and discuss how the emerging complex synchronization patterns relate to the need for the brain to access -in a robust though flexible way- a large variety of functional attractors and dynamical repertoires without ad hoc fine-tuning to a critical point.

Show MeSH
Related in: MedlinePlus