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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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Synchronization in hierarchical modular networks.Top panel: sketch of the HMN model. At hierarchical level 1, 2s basal fully connected blocks of size M are linked pairwise into super-blocks by establishing a fixed number α of random unweighted links between the elements of each (α = 2 in the Fig.). Newly formed blocks are then linked iteratively with the same α up to level s, until the network becomes connected. (a), (b), (c) as in Fig. 3, but for a HMN with N = 512, s = 5, and α = 4. Hierarchical levels are i = 1 → 5 in black, blue, green, magenta and red respectively (not all shown in a) for clarity). (d) Time relaxation of activity ρ for homogeneous characteristic frequencies ω = 0, for logarithmically equally spaced values of k. Averages over 106 realizations of HMNs with N = 4096 and s = 11. Inset: as in the main plot (d), but representing as a function of t1/2 and confirming the predicted stretched exponential behavior. (e) Inverse tail-eigenvalues (as in Fig. 3) for a HMN as in e).
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f4: Synchronization in hierarchical modular networks.Top panel: sketch of the HMN model. At hierarchical level 1, 2s basal fully connected blocks of size M are linked pairwise into super-blocks by establishing a fixed number α of random unweighted links between the elements of each (α = 2 in the Fig.). Newly formed blocks are then linked iteratively with the same α up to level s, until the network becomes connected. (a), (b), (c) as in Fig. 3, but for a HMN with N = 512, s = 5, and α = 4. Hierarchical levels are i = 1 → 5 in black, blue, green, magenta and red respectively (not all shown in a) for clarity). (d) Time relaxation of activity ρ for homogeneous characteristic frequencies ω = 0, for logarithmically equally spaced values of k. Averages over 106 realizations of HMNs with N = 4096 and s = 11. Inset: as in the main plot (d), but representing as a function of t1/2 and confirming the predicted stretched exponential behavior. (e) Inverse tail-eigenvalues (as in Fig. 3) for a HMN as in e).

Mentions: To shed additional light on the previous findings for the HC –i.e. the emergence of chimera-like states and anomalously slow dynamics– we suggest to go beyond the single-level modular network model and study hierarchical modular networks (HMN) in which moduli exists within moduli in a nested way at various scales345689. HMN are assembled in a bottom-up fashion: local fully-connected moduli (e.g. of 16 nodes) are used as building blocks. They are recursively grouped by establishing additional inter-moduli links in a level-dependent way as sketched in Fig. 4(top)1549.


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

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

Synchronization in hierarchical modular networks.Top panel: sketch of the HMN model. At hierarchical level 1, 2s basal fully connected blocks of size M are linked pairwise into super-blocks by establishing a fixed number α of random unweighted links between the elements of each (α = 2 in the Fig.). Newly formed blocks are then linked iteratively with the same α up to level s, until the network becomes connected. (a), (b), (c) as in Fig. 3, but for a HMN with N = 512, s = 5, and α = 4. Hierarchical levels are i = 1 → 5 in black, blue, green, magenta and red respectively (not all shown in a) for clarity). (d) Time relaxation of activity ρ for homogeneous characteristic frequencies ω = 0, for logarithmically equally spaced values of k. Averages over 106 realizations of HMNs with N = 4096 and s = 11. Inset: as in the main plot (d), but representing as a function of t1/2 and confirming the predicted stretched exponential behavior. (e) Inverse tail-eigenvalues (as in Fig. 3) for a HMN as in e).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f4: Synchronization in hierarchical modular networks.Top panel: sketch of the HMN model. At hierarchical level 1, 2s basal fully connected blocks of size M are linked pairwise into super-blocks by establishing a fixed number α of random unweighted links between the elements of each (α = 2 in the Fig.). Newly formed blocks are then linked iteratively with the same α up to level s, until the network becomes connected. (a), (b), (c) as in Fig. 3, but for a HMN with N = 512, s = 5, and α = 4. Hierarchical levels are i = 1 → 5 in black, blue, green, magenta and red respectively (not all shown in a) for clarity). (d) Time relaxation of activity ρ for homogeneous characteristic frequencies ω = 0, for logarithmically equally spaced values of k. Averages over 106 realizations of HMNs with N = 4096 and s = 11. Inset: as in the main plot (d), but representing as a function of t1/2 and confirming the predicted stretched exponential behavior. (e) Inverse tail-eigenvalues (as in Fig. 3) for a HMN as in e).
Mentions: To shed additional light on the previous findings for the HC –i.e. the emergence of chimera-like states and anomalously slow dynamics– we suggest to go beyond the single-level modular network model and study hierarchical modular networks (HMN) in which moduli exists within moduli in a nested way at various scales345689. HMN are assembled in a bottom-up fashion: local fully-connected moduli (e.g. of 16 nodes) are used as building blocks. They are recursively grouped by establishing additional inter-moduli links in a level-dependent way as sketched in Fig. 4(top)1549.

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