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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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Global synchronization dynamics in the human connectome.(a) Time average of the order parameter R(t), for Kuramoto dynamics on the HC network for a specific and fixed set of frequencies extracted from a N(0, 1) Gaussian distribution. A broad intermediate regime separates the incoherent phase (low k) from the synchronous one (high k). In this regime, coherence increases with k in an intermittent fashion, and with strong dependence on the frequency realization. (b) Raster plot of individual phases (vertical axis) showing local rather than global synchrony and illustrating the coexistence of coherent and incoherent nodes (k = 2.7) as time runs. (c) R(t) for 4 values of k (arrows in the main plot). (d) Adjacency matrix of the HC network with nodes ordered to emphasize its modular structure as highlighted by a community detection algorithm (main text), keeping the partition into the 2 hemispheres (dashed lines). Intra-modular connections (shown in color) are dense while inter-modular ones (grey) are limited to tiny subsets, acting as interfaces between moduli. Integration between hemispheres is mostly carried out by the 3 central moduli. This plot visually illustrates the hierarchical modular organization of the human connectome network.
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f1: Global synchronization dynamics in the human connectome.(a) Time average of the order parameter R(t), for Kuramoto dynamics on the HC network for a specific and fixed set of frequencies extracted from a N(0, 1) Gaussian distribution. A broad intermediate regime separates the incoherent phase (low k) from the synchronous one (high k). In this regime, coherence increases with k in an intermittent fashion, and with strong dependence on the frequency realization. (b) Raster plot of individual phases (vertical axis) showing local rather than global synchrony and illustrating the coexistence of coherent and incoherent nodes (k = 2.7) as time runs. (c) R(t) for 4 values of k (arrows in the main plot). (d) Adjacency matrix of the HC network with nodes ordered to emphasize its modular structure as highlighted by a community detection algorithm (main text), keeping the partition into the 2 hemispheres (dashed lines). Intra-modular connections (shown in color) are dense while inter-modular ones (grey) are limited to tiny subsets, acting as interfaces between moduli. Integration between hemispheres is mostly carried out by the 3 central moduli. This plot visually illustrates the hierarchical modular organization of the human connectome network.

Mentions: We have performed a computational study of the Kuramoto model running on top of the HC network (details are given in the Methods section). Our results reveal the existence of an intermediate regime placed between the coherent and the incoherent phase (see Fig. 1). This is characterized by broad quasi-periodic temporal oscillations of R(t) which wildly depend upon the realization of intrinsic frequencies3637. Anomalously large sampling times would be required to extract good statistics for the actual mean values and variances. Collective oscillations of R(t) are a straightforward manifestation of partial synchronization and they are robust against changes in the frequency distribution (e.g. Gaussian, Lorentzian, uniform, etc.) whereas the location and width of the intermediate phase depend upon details. As this phenomenology is reminiscent of Griffiths phases –posed in between order and disorder and stemmig from the existence of semi-isolated regions151617– it is natural to investigate how the HC hierarchical modular structure affects synchronization dynamics.


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

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

Global synchronization dynamics in the human connectome.(a) Time average of the order parameter R(t), for Kuramoto dynamics on the HC network for a specific and fixed set of frequencies extracted from a N(0, 1) Gaussian distribution. A broad intermediate regime separates the incoherent phase (low k) from the synchronous one (high k). In this regime, coherence increases with k in an intermittent fashion, and with strong dependence on the frequency realization. (b) Raster plot of individual phases (vertical axis) showing local rather than global synchrony and illustrating the coexistence of coherent and incoherent nodes (k = 2.7) as time runs. (c) R(t) for 4 values of k (arrows in the main plot). (d) Adjacency matrix of the HC network with nodes ordered to emphasize its modular structure as highlighted by a community detection algorithm (main text), keeping the partition into the 2 hemispheres (dashed lines). Intra-modular connections (shown in color) are dense while inter-modular ones (grey) are limited to tiny subsets, acting as interfaces between moduli. Integration between hemispheres is mostly carried out by the 3 central moduli. This plot visually illustrates the hierarchical modular organization of the human connectome network.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f1: Global synchronization dynamics in the human connectome.(a) Time average of the order parameter R(t), for Kuramoto dynamics on the HC network for a specific and fixed set of frequencies extracted from a N(0, 1) Gaussian distribution. A broad intermediate regime separates the incoherent phase (low k) from the synchronous one (high k). In this regime, coherence increases with k in an intermittent fashion, and with strong dependence on the frequency realization. (b) Raster plot of individual phases (vertical axis) showing local rather than global synchrony and illustrating the coexistence of coherent and incoherent nodes (k = 2.7) as time runs. (c) R(t) for 4 values of k (arrows in the main plot). (d) Adjacency matrix of the HC network with nodes ordered to emphasize its modular structure as highlighted by a community detection algorithm (main text), keeping the partition into the 2 hemispheres (dashed lines). Intra-modular connections (shown in color) are dense while inter-modular ones (grey) are limited to tiny subsets, acting as interfaces between moduli. Integration between hemispheres is mostly carried out by the 3 central moduli. This plot visually illustrates the hierarchical modular organization of the human connectome network.
Mentions: We have performed a computational study of the Kuramoto model running on top of the HC network (details are given in the Methods section). Our results reveal the existence of an intermediate regime placed between the coherent and the incoherent phase (see Fig. 1). This is characterized by broad quasi-periodic temporal oscillations of R(t) which wildly depend upon the realization of intrinsic frequencies3637. Anomalously large sampling times would be required to extract good statistics for the actual mean values and variances. Collective oscillations of R(t) are a straightforward manifestation of partial synchronization and they are robust against changes in the frequency distribution (e.g. Gaussian, Lorentzian, uniform, etc.) whereas the location and width of the intermediate phase depend upon details. As this phenomenology is reminiscent of Griffiths phases –posed in between order and disorder and stemmig from the existence of semi-isolated regions151617– it is natural to investigate how the HC hierarchical modular structure affects synchronization dynamics.

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