**by Géza Ódor (EKMFA Complex Systems Department, Budapest), Jeffrey Kelling (HZDR Dresden), Gustavo Deco (UPF Barcelona) **

Email me!

*We have extended the study of the Kuramoto model with additive Gaussian noise running on the KKI-18 large human connectome graph. We determined the dynamical behavior of this model by solving it numerically in an assumed homeostatic state, below the synchronization crossover point we determined previously. The desynchronization duration distributions exhibit power-law tails, characterized by the exponent in the range 1.1 < τ _{t} <2, overlapping the in vivo human brain activity experiments by Palva et al. We show that these scaling results remain valid, by a transformation of the ultra-slow eigenfrequencies to Gaussian with unit variance.We also compare the connectome results with those, obtained on a regular cube with N = 106 nodes, related to the embedding space, and show that the quenched internal frequencies themselves can cause frustrated synchronization scaling in an extended coupling space.*

**Fig.1 **– We obtained the **KKI-18 **graph from the Open Connectome project repository. This is based on the diffusion tensor image, approximating the structural connectivity of the white matter of a human brain. The graph version we downloaded in 2015 comprises a large component with

N=804 092 nodes, connected via 41 523 908 undirected edges and several small sub- components, which were ignored here. The topological dimension of this graph is: d = 3.05.

This graph allowed us to run extensive dynamical studies on present day CPU/GPU clusters, large enough to draw conclusions on the scaling behavior without too strong finite size effects, hindering scaling regions. These large connectomes of the human brain possess 1 mm3 resolution, obtained by a combination of diffusion weighted, functional and structural magnetic resonance imaging scans. These are symmetric, weighted networks, where the weights measure the number of fiber tracts between nodes. The **KKI-18 **graph is generated via the MIGRAINE method. It exhibits a hierarchical modular structure by the construction from the Desikan cerebral regions with (at least) two quite different scales. The graph topology is displayed on Fig. 1, in which modules were identified by the Leiden algorithm and the network of modules generated and visualized using the python igraph library. This identified 153 modules, with sizes varying between 7 and 35 332 nodes in the displayed case, however since this is a heuristic approach, these numbers vary by about 10 %. A recent experimental study has provided confirmation for the connectome generation used here. This suggests that diffusion MRI tractography is a powerful tool for exploring the structural connection architecture of the brain. The size of circles is proportional with the number of nodes. The network of modules itself shows modularity arising from the hierarchical structure of the **KKI- 18 **connectome. The each circle’s color indicates its membership in one of seven modules obtained through Leiden community analysis of the displayed cluster graph. In the simulations we normalized the incoming weights: w_{ij}‘= w_{ij }/ Σ w_{ij }to provide local homeostasis. In reality such local homeostasis is the consequence of the competition of exhibitory and inhibitory neurons.

**Fig 2. **– Having determined the transition point, using local slopes method of <R(t)>, we run the numerical solver at control parameter values below Kc , by starting with thousands of random initial states representing ensemble of different quenched disorder of ω_{i }we measureed the first crossing times defined by R_{c} of the random state. The figure shows the evolution of R(t) of single realizations on the **KKI-18 **graph at K = 1.7. The dashed line shows the threshold value Rc = 1/√(N) = 0.001094, where we measure the characteristic times: t_{x} of the first crossing.

**Fig 3. **– Following a histogramming procedure, with PL growing bin sizes in t_{x} , we obtained the duration distributions p(t_{x}), of t_{x} on the **KKI-18 **model. The almost ~1/t singular decay distribution for K_{c} ≥ 1.7 marks synchronized phase, where due to the strong coupling arbitrary large avalanches can occur; at K_{c} one obtains τ_{t} ≃ 1.2. The τ_{t} = 1.2(1) is out of the range of neuro experiments, but in the sub-threshold region we find good agreement/overlap with the experiments: 1.5 < τ_{t} < 2.4. Legends: K = 1.7 (boxes), 1.6 (stars), 1.4 (bullets), 1.3 (+), 1.2 (up triangles). The dashed lines shows PL fits for the tail region t_{x} > 20.

**Fig. 4 –** The 4-th order Runge-Kutta algorithm, speed up by a factor of ~ 40-100 on GPU cards with respect to Xeon Gold CPUs, has been found to be sensitive to the annealed noise, as the

derivatives can change a lot from site to site. Duration distribution of tx on the **KKI-18 **model at K = 1.4 and s = 1 noise amplitude using different numerical precisions: Δ = 0.1 (bullets), Δ = 0.01 (boxes), Δ = 0.001 (diamonds). The dashed line shows a PL fit for the tail region: t_{x} > 20 of the Δ = 0.01 data. The stars show former results, obtained for the noiseless case.

Very nice phase analysis – we did use the kuramoto phase measure at the population level to study phase bundling during avalanche condition and pharmacological perturbation outside the avalanche regime. We found that only when the system shows avalanches you have maximal entropy in phase locking and phase decoupling (Hongdian et al. 2012 J Neurosci Maximal variability of phase synchrony in cortical networks with neuronal avalanches). Wonder how this would show up in your Fig. 3/4