Kuramoto dynamics, glassy synchronization and rare

regions in the human connectome

Pablo Villegas

Quantitative Laws II

(University of Granada)

The Kuramoto model

Intrinsic frecuency

·

θi = ωi + k

N∑

j=1

Wijsin (θj − θi) + ηi (t) i = 1, ...,N

Coupling term

Noise

z = re iψ = 1N

N∑j=1

e iθj

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The Human Connectome

Figure : Adjacency matrix and modular structure

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The Human Connectome

A hierarchical synchronization process appears, with local phase transitions

and a new intermediate phase between order and disorder.

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A model: Hierarchical Modular Network

We can see the same phenomena in synthetic Hierarchic Modular Networks

(HMNs), with local phase transitions and a bottom to up synchronization

process.

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Metastability in HMNs

Switching behavior in the HC and HMNs. This behavior closely resembles

’up and down’ states.

200 400 600 8000.3

0.5

0.7

0.3

0.5

0.7

0.3

0.5

0.7HC(a)

ord

erp

ara

met

er

(b)

0.2

0.3

0.4

0.2

0.3

0.4

20000 40000 60000 80000

time

0.2

0.3

0.4

HMN

time

0 200 400 600 800 1000

time

0.20

0.22

0.24

0.26

0.28

0.30

ord

erp

aram

eter

ord

erp

aram

eter

(s

tead

y)

(a) (b)

10−1 100 101 102

noise coeff.

0.200

0.205

0.210

0.215

0.220

0.225

0.230

Resonant peaks for some levels of intermediate noise in HMN networks.

Perturbations lead the system to more coherent attractors in the

intermediate phase.

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Thanks for your attention

Collaborators

We acknowledge financial support from MINECO (National Plan of I+D+i), grant FIS2013-43201-P.

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