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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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Related in: MedlinePlus

Two-block model.(a) Sketch of the two-block model. (b) Global order parameter for the two-block model with M = 128 and two interfacial nodes. Results of the numerical integration of the 258 Kuramoto equations (blue points) are in strikingly good agreement with the integration of Eqs.(2) (solid blue line). Local block-wise order parameters are shown for comparison (small symbols; dashed lines are guides to the eye). A first transition, where local order emerges, occurs at k ≈ 0.02, while global coherence is reached at k ≈ 90. In the intermediate region, R(t) oscillates (inset), revealing the lack of global coherence. Despite the simplicity of this toy model, these results constitute the essential building-block upon which further levels of complexity rely (see main text).
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f2: Two-block model.(a) Sketch of the two-block model. (b) Global order parameter for the two-block model with M = 128 and two interfacial nodes. Results of the numerical integration of the 258 Kuramoto equations (blue points) are in strikingly good agreement with the integration of Eqs.(2) (solid blue line). Local block-wise order parameters are shown for comparison (small symbols; dashed lines are guides to the eye). A first transition, where local order emerges, occurs at k ≈ 0.02, while global coherence is reached at k ≈ 90. In the intermediate region, R(t) oscillates (inset), revealing the lack of global coherence. Despite the simplicity of this toy model, these results constitute the essential building-block upon which further levels of complexity rely (see main text).

Mentions: To shed further light on the properties of synchronization on the HC, we consider a very simple network model –allowing for analytical understanding– which will constitute the elementary “building-block” for subsequent more complex analyses. This consists of a few blocks with very large internal connectivity and very sparse inter-connectivity. Each block is composed by a bulk of nodes that share no connection with the outside and a relatively small “interfacial” set that connects with nodes in other blocks. For instance, in the simplest realization, consisting of just two blocks connected by a single pair of nodes (Fig. 2), each block is endowed with local coherence rA,B, average phase ψA,B, and average characteristic frequency ωA,B, while 1-node interfaces have perfect coherence r = 1, phase φA,B, and characteristic frequency νA,B. In this case, N = 2M + 2, and the OA ansatz can be safely applied to each block (large M) but not to single-node interfaces. In the particular case (convenient for analytical treatment) in which g(w) are zero-mean Lorentz distributions with spreads δA,B, the resulting set of OA equations can be easily shown to be: (together with r = 1 for each 1-node interface), and a symmetric set (A ↔ B) for block B. The solution of Eq.(2) –displayed in Fig. 2– reveals a transition to local coherence within each block at a certain threshold value of k ≈ 0.02. As soon as local order is attained, rA,B ≈ 1 and , from Eq.(2) the mutual synchronization process obeys and a symmetrical equation for . For small k, the right-hand side is dominated by νA + MωA: whereas the average value ωA becomes arbitrarily small within blocks (assuming that M is large), the frequency νA does not. Consequently, synchronization between the two blocks through the interfacial link is frustrated: each block remains internally synchronized but is unable to achieve coherence with the other over a broad interval of coupling strengths. This interval is delimited above by a second transition at k ~ max{M/ωA,B/, νA,B}, where k is large enough as to overcome frustration and generate global coherence. This picture is confirmed by numerical integration of the full system of N coupled Kuramoto equations as well as by its OA approximation (Eq.(2)), both in remarkably good agreement. Therefore, local and global coherences have their onsets at two well-separated transition points35 and –similarly to the much more complex HC case– R oscillates in the intermediate regime (Fig. 2). Similar results hold for versions of the model with more than two moduli (e.g. 4; see below). The existence of two distinct (local and global) transitions had already been reported in a recent study of many blocks with much stronger inter-moduli connections than here35 (even if, owing to this difference, no sign of an intermediate oscillatory phase was reported). In particular, the value of two-block models has already been explored in the past, for systems of identical oscillators with non-zero phase lags, in which each node is coupled equally to all the others in its community, and less strongly to those in the other41. In such systems, local coherence emerged for large enough values of the phase lag. Our two-block model shows that the presence of “structural bottlenecks” between moduli combined with heterogeneous frequencies at their contact nodes (interfaces) are essential ingredients to generate a broad region of global oscillations in R, even in the absence of phase lag. Still, it is obviously a too-simplistic model to account for all the rich phenomenology emerging on the HC, as we show now.


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

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

Two-block model.(a) Sketch of the two-block model. (b) Global order parameter for the two-block model with M = 128 and two interfacial nodes. Results of the numerical integration of the 258 Kuramoto equations (blue points) are in strikingly good agreement with the integration of Eqs.(2) (solid blue line). Local block-wise order parameters are shown for comparison (small symbols; dashed lines are guides to the eye). A first transition, where local order emerges, occurs at k ≈ 0.02, while global coherence is reached at k ≈ 90. In the intermediate region, R(t) oscillates (inset), revealing the lack of global coherence. Despite the simplicity of this toy model, these results constitute the essential building-block upon which further levels of complexity rely (see main text).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f2: Two-block model.(a) Sketch of the two-block model. (b) Global order parameter for the two-block model with M = 128 and two interfacial nodes. Results of the numerical integration of the 258 Kuramoto equations (blue points) are in strikingly good agreement with the integration of Eqs.(2) (solid blue line). Local block-wise order parameters are shown for comparison (small symbols; dashed lines are guides to the eye). A first transition, where local order emerges, occurs at k ≈ 0.02, while global coherence is reached at k ≈ 90. In the intermediate region, R(t) oscillates (inset), revealing the lack of global coherence. Despite the simplicity of this toy model, these results constitute the essential building-block upon which further levels of complexity rely (see main text).
Mentions: To shed further light on the properties of synchronization on the HC, we consider a very simple network model –allowing for analytical understanding– which will constitute the elementary “building-block” for subsequent more complex analyses. This consists of a few blocks with very large internal connectivity and very sparse inter-connectivity. Each block is composed by a bulk of nodes that share no connection with the outside and a relatively small “interfacial” set that connects with nodes in other blocks. For instance, in the simplest realization, consisting of just two blocks connected by a single pair of nodes (Fig. 2), each block is endowed with local coherence rA,B, average phase ψA,B, and average characteristic frequency ωA,B, while 1-node interfaces have perfect coherence r = 1, phase φA,B, and characteristic frequency νA,B. In this case, N = 2M + 2, and the OA ansatz can be safely applied to each block (large M) but not to single-node interfaces. In the particular case (convenient for analytical treatment) in which g(w) are zero-mean Lorentz distributions with spreads δA,B, the resulting set of OA equations can be easily shown to be: (together with r = 1 for each 1-node interface), and a symmetric set (A ↔ B) for block B. The solution of Eq.(2) –displayed in Fig. 2– reveals a transition to local coherence within each block at a certain threshold value of k ≈ 0.02. As soon as local order is attained, rA,B ≈ 1 and , from Eq.(2) the mutual synchronization process obeys and a symmetrical equation for . For small k, the right-hand side is dominated by νA + MωA: whereas the average value ωA becomes arbitrarily small within blocks (assuming that M is large), the frequency νA does not. Consequently, synchronization between the two blocks through the interfacial link is frustrated: each block remains internally synchronized but is unable to achieve coherence with the other over a broad interval of coupling strengths. This interval is delimited above by a second transition at k ~ max{M/ωA,B/, νA,B}, where k is large enough as to overcome frustration and generate global coherence. This picture is confirmed by numerical integration of the full system of N coupled Kuramoto equations as well as by its OA approximation (Eq.(2)), both in remarkably good agreement. Therefore, local and global coherences have their onsets at two well-separated transition points35 and –similarly to the much more complex HC case– R oscillates in the intermediate regime (Fig. 2). Similar results hold for versions of the model with more than two moduli (e.g. 4; see below). The existence of two distinct (local and global) transitions had already been reported in a recent study of many blocks with much stronger inter-moduli connections than here35 (even if, owing to this difference, no sign of an intermediate oscillatory phase was reported). In particular, the value of two-block models has already been explored in the past, for systems of identical oscillators with non-zero phase lags, in which each node is coupled equally to all the others in its community, and less strongly to those in the other41. In such systems, local coherence emerged for large enough values of the phase lag. Our two-block model shows that the presence of “structural bottlenecks” between moduli combined with heterogeneous frequencies at their contact nodes (interfaces) are essential ingredients to generate a broad region of global oscillations in R, even in the absence of phase lag. Still, it is obviously a too-simplistic model to account for all the rich phenomenology emerging on the HC, as we show now.

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