Limits...
Kuramoto model simulation of neural hubs and dynamic synchrony in the human cerebral connectome.

Schmidt R, LaFleur KJ, de Reus MA, van den Berg LH, van den Heuvel MP - BMC Neurosci (2015)

Bottom Line: Furthermore, suppressing structural connectivity among hub nodes resulted in an elevated modular state (p < 4.1 × l0(-3), 0.015 < λ < 0.04), indicating that hub-to-hub connections are critical in intermodular synchronization.Finally, perturbing the oscillatory behavior of hub nodes prevented functional modules from synchronizing, implying that synchronization of functional modules is dependent on the hub nodes' behavior.Our results converge on anatomical hubs having a leading role in intermodular synchronization and integration in the human brain.

View Article: PubMed Central - PubMed

Affiliation: Department of Neurology, Brain Center Rudolf Magnus, University Medical Center Utrecht, Heidelberglaan 100, PO Box 85500, 3508 GA, Utrecht, Netherlands. r.schmidt@umcutrecht.nl.

ABSTRACT

Background: The topological structure of the wiring of the mammalian brain cortex plays an important role in shaping the functional dynamics of large-scale neural activity. Due to their central embedding in the network, high degree hub regions and their connections (often referred to as the 'rich club') have been hypothesized to facilitate intermodular neural communication and global integration of information by means of synchronization. Here, we examined the theoretical role of anatomical hubs and their wiring in brain dynamics. The Kuramoto model was used to simulate interaction of cortical brain areas by means of coupled phase oscillators-with anatomical connections between regions derived from diffusion weighted imaging and module assignment of brain regions based on empirically determined resting-state data.

Results: Our findings show that synchrony among hub nodes was higher than any module's intramodular synchrony (p < 10(-4), for cortical coupling strengths, λ, in the range 0.02 < λ < 0.05), suggesting that hub nodes lead the functional modules in the process of synchronization. Furthermore, suppressing structural connectivity among hub nodes resulted in an elevated modular state (p < 4.1 × l0(-3), 0.015 < λ < 0.04), indicating that hub-to-hub connections are critical in intermodular synchronization. Finally, perturbing the oscillatory behavior of hub nodes prevented functional modules from synchronizing, implying that synchronization of functional modules is dependent on the hub nodes' behavior.

Conclusion: Our results converge on anatomical hubs having a leading role in intermodular synchronization and integration in the human brain.

No MeSH data available.


Structural binary input with functional output. The binary undirected adjacency matrix that describes the cortical brain network in the lower triangle, and the edgewise weighted synchronization output (simulated functional connectivity) in the upper triangle (generated at cortical coupling factor λ = 0.02). The histogram at the bottom shows the degree (i.e. number of connections) of each node. Each module is color-coded and rich club nodes, distributed across the functional modules, are shown in black. The eleven functional modules are: (1) Default Mode, (2) Primary Visual, (3) Extrastriate Visual, (4) Motor, (5) Sensory, (6) Bilateral Parietal, (7) Left Parietal Frontal, (8) Right Parietal Frontal, (9) Auditory, (10) Salience, and (11) Frontal
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License 1 - License 2
getmorefigures.php?uid=PMC4556019&req=5

Fig1: Structural binary input with functional output. The binary undirected adjacency matrix that describes the cortical brain network in the lower triangle, and the edgewise weighted synchronization output (simulated functional connectivity) in the upper triangle (generated at cortical coupling factor λ = 0.02). The histogram at the bottom shows the degree (i.e. number of connections) of each node. Each module is color-coded and rich club nodes, distributed across the functional modules, are shown in black. The eleven functional modules are: (1) Default Mode, (2) Primary Visual, (3) Extrastriate Visual, (4) Motor, (5) Sensory, (6) Bilateral Parietal, (7) Left Parietal Frontal, (8) Right Parietal Frontal, (9) Auditory, (10) Salience, and (11) Frontal

Mentions: The Kuramoto model simulates dynamic behavior of a set of coupled oscillators. Applying the Kuramoto model to the group-averaged anatomical brain network, each node was assigned an oscillator with a fixed, random internal angular frequency and an initial random phase. The model is defined as,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \dot{\theta }_{i} = \omega_{i} + \lambda \mathop \sum \nolimits_{{{\text{j = }}1}}^{\text{N}} {W}_{ji} \sin (\theta_{j} - \theta_{i} ) $$\end{document}θ˙i=ωi+λ∑j =1NWjisin(θj-θi)where θi(t) and ωi are the phase and internal angular frequency of oscillator i respectively. The cortical coupling strength that is applied to the edges is denoted by λ. Wij is the symmetrical group matrix containing all connections between cortical nodes (Fig. 1) and N is the total number of nodes. The set of differential equations (Eq. 1) was solved numerically using a Runge–Kutta solver. During every run of the model, each node was assigned a random initial phase, θ, uniformly distributed between [−π, π] and a random internal frequency, ω, uniformly distributed between [0, 1]. Simulations were carried out for T = 700 with a transient time of τ = 300, keeping these parameters the same as in previous work [3]. Explorative longer runs of T = 5000 did not change the nature of our findings. Output data from the last 400 time points was then used in the analyses.Fig. 1


Kuramoto model simulation of neural hubs and dynamic synchrony in the human cerebral connectome.

Schmidt R, LaFleur KJ, de Reus MA, van den Berg LH, van den Heuvel MP - BMC Neurosci (2015)

Structural binary input with functional output. The binary undirected adjacency matrix that describes the cortical brain network in the lower triangle, and the edgewise weighted synchronization output (simulated functional connectivity) in the upper triangle (generated at cortical coupling factor λ = 0.02). The histogram at the bottom shows the degree (i.e. number of connections) of each node. Each module is color-coded and rich club nodes, distributed across the functional modules, are shown in black. The eleven functional modules are: (1) Default Mode, (2) Primary Visual, (3) Extrastriate Visual, (4) Motor, (5) Sensory, (6) Bilateral Parietal, (7) Left Parietal Frontal, (8) Right Parietal Frontal, (9) Auditory, (10) Salience, and (11) Frontal
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4556019&req=5

Fig1: Structural binary input with functional output. The binary undirected adjacency matrix that describes the cortical brain network in the lower triangle, and the edgewise weighted synchronization output (simulated functional connectivity) in the upper triangle (generated at cortical coupling factor λ = 0.02). The histogram at the bottom shows the degree (i.e. number of connections) of each node. Each module is color-coded and rich club nodes, distributed across the functional modules, are shown in black. The eleven functional modules are: (1) Default Mode, (2) Primary Visual, (3) Extrastriate Visual, (4) Motor, (5) Sensory, (6) Bilateral Parietal, (7) Left Parietal Frontal, (8) Right Parietal Frontal, (9) Auditory, (10) Salience, and (11) Frontal
Mentions: The Kuramoto model simulates dynamic behavior of a set of coupled oscillators. Applying the Kuramoto model to the group-averaged anatomical brain network, each node was assigned an oscillator with a fixed, random internal angular frequency and an initial random phase. The model is defined as,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \dot{\theta }_{i} = \omega_{i} + \lambda \mathop \sum \nolimits_{{{\text{j = }}1}}^{\text{N}} {W}_{ji} \sin (\theta_{j} - \theta_{i} ) $$\end{document}θ˙i=ωi+λ∑j =1NWjisin(θj-θi)where θi(t) and ωi are the phase and internal angular frequency of oscillator i respectively. The cortical coupling strength that is applied to the edges is denoted by λ. Wij is the symmetrical group matrix containing all connections between cortical nodes (Fig. 1) and N is the total number of nodes. The set of differential equations (Eq. 1) was solved numerically using a Runge–Kutta solver. During every run of the model, each node was assigned a random initial phase, θ, uniformly distributed between [−π, π] and a random internal frequency, ω, uniformly distributed between [0, 1]. Simulations were carried out for T = 700 with a transient time of τ = 300, keeping these parameters the same as in previous work [3]. Explorative longer runs of T = 5000 did not change the nature of our findings. Output data from the last 400 time points was then used in the analyses.Fig. 1

Bottom Line: Furthermore, suppressing structural connectivity among hub nodes resulted in an elevated modular state (p < 4.1 × l0(-3), 0.015 < λ < 0.04), indicating that hub-to-hub connections are critical in intermodular synchronization.Finally, perturbing the oscillatory behavior of hub nodes prevented functional modules from synchronizing, implying that synchronization of functional modules is dependent on the hub nodes' behavior.Our results converge on anatomical hubs having a leading role in intermodular synchronization and integration in the human brain.

View Article: PubMed Central - PubMed

Affiliation: Department of Neurology, Brain Center Rudolf Magnus, University Medical Center Utrecht, Heidelberglaan 100, PO Box 85500, 3508 GA, Utrecht, Netherlands. r.schmidt@umcutrecht.nl.

ABSTRACT

Background: The topological structure of the wiring of the mammalian brain cortex plays an important role in shaping the functional dynamics of large-scale neural activity. Due to their central embedding in the network, high degree hub regions and their connections (often referred to as the 'rich club') have been hypothesized to facilitate intermodular neural communication and global integration of information by means of synchronization. Here, we examined the theoretical role of anatomical hubs and their wiring in brain dynamics. The Kuramoto model was used to simulate interaction of cortical brain areas by means of coupled phase oscillators-with anatomical connections between regions derived from diffusion weighted imaging and module assignment of brain regions based on empirically determined resting-state data.

Results: Our findings show that synchrony among hub nodes was higher than any module's intramodular synchrony (p < 10(-4), for cortical coupling strengths, λ, in the range 0.02 < λ < 0.05), suggesting that hub nodes lead the functional modules in the process of synchronization. Furthermore, suppressing structural connectivity among hub nodes resulted in an elevated modular state (p < 4.1 × l0(-3), 0.015 < λ < 0.04), indicating that hub-to-hub connections are critical in intermodular synchronization. Finally, perturbing the oscillatory behavior of hub nodes prevented functional modules from synchronizing, implying that synchronization of functional modules is dependent on the hub nodes' behavior.

Conclusion: Our results converge on anatomical hubs having a leading role in intermodular synchronization and integration in the human brain.

No MeSH data available.