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Robustness of Griffiths effects in homeostatic connectome models

I provide numerical evidence for the robustness of the Griffiths phase (GP) reported previously in dynamical threshold model simulations on a large human brain network with N=836733 connected nodes. The model, with equalized network sensitivity, is extended in two ways: introduction of refractory states or by randomized time dependent thresholds. The non-universal power-law dynamics in an extended control parameter region survives these modifications for a short refractory state and weak disorder. In case of temporal disorder the GP shrinks and for stronger heterogeneity disappears, leaving behind a mean-field type of critical transition. Activity avalanche size distributions below the critical point decay faster than in the original model, but the addition of inhibitory interactions sets it back to the range of experimental values.

I provide numerical evidence for the robustness of the Griffiths phase (GP) reported previously in dynamical threshold model simulations on a large human brain network with N = 836733 connected nodes. The model, with equalized network sensitivity, is extended in two ways: introduction of refractory states or by randomized time dependent thresholds. The non-universal power-law dynamics in an extended control parameter region survives these modifications for a short refractory state and weak disorder. In case of temporal disorder the GP shrinks and for stronger heterogeneity disappears, leaving behind a mean-field type of critical transition. Activity avalanche size distributions below the critical point decay faster than in the original model, but the addition of inhibitory interactions sets it back to the range of experimental values.

I. INTRODUCTION
Recent experimental evidence suggests [1,2], that our brain operates near criticality, where spatial and temporal correlations diverge [3], a dynamics, that enhances the brain information-processing capabilities, increasing sensitivity and optimizing the dynamic range [4]. It might be the case that the brain tunes itself near criticality via self-organization (SOC) [5], or alternatively, could benefit from the same properties if sufficient heterogenities [6] (that is disorder) are present to induce an extended semicritical region, known as Griffiths Phase (GP) [7]. Alternatively, it has recently been proposed that living systems may also self-tune to criticality as the consequence of evolution and adaptation [8]. The theory of phase transitions of statistical physics applied for these biological models can provide a clue to better understanding [9].
While SOC models [10] are usually homogeneous, real systems are highly inhomogeneous and one must consider if the heterogeneity is weak enough to apply "clean models" for the description of experiments. Strong disorder can lead to so called rare-region (RR) effects [11] that smear the phase transitions [13,14]. RR-s are very slowly relaxing domains, remaining in the opposite phase than the whole system for a long time, causing slow evolution of the order parameter. In the entire GP, which is an extended control parameter region around the critical point, fluctuations diverge and auto-correlations exhibit fat tailed, power-law (PL) behavior, causing burstyness [15]. Therefore, here one can observe dynamical critical behavior without the need of fine tuning. This was proposed to be the reason for the working memory in the brain [16] and considered to be a possible explanation for the observed criticality in neuroscience [17].
The seminal experiments by Beggs and Plenz [1] reported neuronal avalanches with size (s) dependence, defined as either the number of electrodes with suprathreshold activity or the sum of the potentials, according to a power law, p(s) ∝ s −1.5 . For the duration distribu-tion of such events P (t) ∝ t −2 PL tails were observed. These exponents are in agreement with the mean-field (MF) exponents of the Directed Percolation (DP) criticality: τ = 3/2, τ t = 2, as well as with many other universality classes sharing the same MF avalanche exponents (see [18,19]). Mean-field exponents are expected to occur if the fluctuation effects are weak, when the system dimensionality is above the upper critical dimension D c . For DP D c = 4, but recent graph dimension measurements of large human connectomes suggest lower values [20]. Other experimental setups have revealed somewhat different exponents [21][22][23] suggesting different universal behaviors. Furthermore, Palva et al [24] have found that source-reconstructed M/EEG data exhibit robust powerlaw long-range time correlations as well as scale-invariant avalanches with a broad range of exponents: 1 < τ < 1.6 and 1.5 < τ t < 2.4. An obvious explanation for this wide spread of critical exponents can be heterogenities, which in the GP cause non-universal dynamical exponents [7,11,12].
It has been conjectured that network heterogeneity can cause GP-s if the topological (graph) dimension D, defined by N r ∼ r D , where N r is the number of (j) nodes within topological distance r = d(i, j) from an arbitrary origin (i), is finite [25]. This hypothesis has been supported numerically for various spreading models on scalefree graphs [26][27][28]. In case of modular topological structure RR-s are enhanced and Griffiths effects have been reported in synthetic brain networks with finite D [29][30][31] as well as in scale-free networks [32,33]. However, extended simulations of activity avalanches on very large human connectomes, possessing broad link weight distributions, did not support critical like phase transition [34]. It turned out that in those graphs the weight heterogenities are too strong to allow the occurrence of RR-s. This means that only the strongly connected hubs play a role in the activation/deactivation processes and weak nodes just follow them. This seems to be rather unrealistic and uneconomic in a brain of billions of neurons, thus some kind of input sensitivity equilibration was assumed via variable, node dependent thresholds. This makes the system homeostatic and simulations proved the occurrence of criticality, as well as Griffiths effects. Indeed there is some evidence that neurons have a certain adaptation to their input excitation levels [35] and can be modeled by variable thresholds [36]. Recent theoretical studies have also suggested that homeostatic plasticity mechanisms may play a role in facilitating criticality [37][38][39][40]. Very recently comparison of modeling and experiments arrived at a similar conclusion: equalized network sensitivity improves the predicting power of a model at criticality in agreement with the FMRI correlations [41].
A two-state (active/inactive) stochastic threshold model with normalized input sensitivity [34], was investigated by simulations on a large, weighted human CC and the spatio-temporal behavior of activity avalanches was determined in the inactive phase. Numerical evidence has been found for a generic scale invariance in an extended control parameter space. Since this model may look to be too simplistic, especially for neuroscientists, now I extend it and perform numerical investigations of its dynamical behavior. The extensions are done in two directions. The addition of refractory states to the binary active/inactive model and the possibility of time dependent thresholds. The refractory states happen for one time step following the active state of a node, thus an activated neighbor cannot activate back immediately, altering the spreading process. This is a common feature of brain models [42].
Time dependency of the network, which can be the consequence of some external fields, chemicals or some cellular material distributed by the blood flow has been simulated by altering the thresholds randomly, in a synchronous way for all nodes. Such temporal disorder, has been shown to lead to temporal Griffiths phases, resulting in algebraic size dependency (space-time flipped) [43,44] and enormous density fluctuations, characterized by an infinitely broad probability distribution at criticality. The combination of uncorrelated spatial and temporal disorder is irrelevant for the universal critical behavior [19], but correlated, diffusive spatio-temporal disorder has been found to be a relevant perturbation [45,46]. In this case I investigate the effect of long-range correlated spatio-temporal heterogeneity in the form of the addition of a time dependent global sensitivity threshold to the quenched spatial disorder of the network.

II. MODELS AND METHODS
This study uses data from the Open Connectome project (OCP) [47], obtained by Diffusion Tensor Imaging (DTI) [48] showing structural brain connectivity of the white matter. Other research on the structural networks have been much smaller sized. For example the network obtained by Sporns and collaborators, using diffusion imaging techniques [49,50], consists of a highly coarsegrained mapping of anatomical connections in the human brain, comprising N = 998 brain areas and the fiber tract densities between them. The graph used here comprises N = 848848 nodes, allowing one to run extensive dynamical simulations on present days CPU/GPU clusters, still large enough to draw conclusions on the scaling behavior without the effects of finite sizes. Smaller systems near a critical phase transition point, where correlations diverge, suffer from finite size corrections, which can hide true scaling region and RR effects.
These DTI connectomes of the human brain possess 1 mm 3 resolution, using a combination of diffusion weighted, functional and structural magnetic resonance imaging scans. These graphs are symmetric, weighted networks, where the weights measure the number of fiber tracts between nodes. The large graph "KKI-18" used here is generated by the MIGRAINE method, described in [51]. These OCP graphs exhibit hierarchical levels by construction from the Desikan cerebral regions with (at least) two quite different scales. As it was discussed in [34] such coarse grained CC-s suffer possible sources of errors, like unknown noise in the data generation; underestimation of long connections; radial accuracy, influencing endpoints of the tracts and hierarchical levels of the cortical organization; or transverse accuracy, determining which cortical area is connected to another. Still important modifications, such as inhibitory links, directedness, or random loss of connections up to 20% confirmed the robustness of Griffiths effects, suggesting that fine network details may not play an important role. It is also important, that the PL tail in the weight distribution is similar to what was obtained by a synaptic learning algorithm in an artificial neural network [52].
The investigated graph exhibits a single giant component of size N = 836733 nodes (out of N = 848848), 41523931 undirected edges and several small sub-components, ignored here. In [20] we found that contrary to the small world network coefficients OCP graphs exhibit topological dimension slightly above D = 3 [20] and certain amount of universality, supporting the selection of KKI-18 as a representative of the large human CC-s available. This dimensionality suggests weak longrange connections, in addition to the D = 3 dimensional embedding ones and warrants to see heterogeneity effects in dynamical models defined on them. Further details are discussed in [20] and [34].
In my previous study [34] a two-state (x i = 0 or 1) spreading model was used to describe the propagation, branching and annihilation of activity on this network. This model is similar to those of Refs. [3,54], a discrete time stochastic cellular automata (SCA). The dynamical process, mimicking neuronal avalanches, is started by activating a randomly selected node. At each network update every (i) node is visited and tested if the sum of incoming weights (w i,j ) of active neighbors reaches a given threshold value If this condition is met a node activation is attempted with probability λ. Alternatively, an active node is deactivated with probability ν. New states of the nodes are overwritten only after a full network update, i.e. a synchronous update is performed at discrete time steps. The updating process continues as long as there are active sites or up to a maximum time limit t = 10 6 Monte Carlo sweeps (MCs). In case the system has fallen into inactivity, the actual time step is stored to calculate the survival probability p(t) of runs. The average activity: x i and the number of activated nodes during the avalanche s = N i=1 T t=1 x i of duration is calculated at the end of the simulations. This SCA type of synchronous updating is not expected to affect the dynamical scaling behavior [31] and provides a possibility for parallel algorithms. In fact the code was implemented on GPU-s, with 12× speedup with respect to good CPU cores [53]. Measurements on 10 6 to 10 7 independent runs, started from randomly selected, active initial sites were averaged over for each control parameter value investigated.
In [34] I showed that this model running on the KKI-18 graph does not exhibit criticality, but due to the large weight differences and graph dimensionality hubs cause a catastrophic, discontinuous like transition. To keep the local sustained activity requirement for the brain [54] and avoid nodes, that practically do not affect anything I modified the model with variable thresholds of equalized sensitivity, by normalizing the incoming weights as Now I extend this model with the addition of a refractory state, which occurs with nodes between an activation and deactivation event for one time step. This prevents activated neighbors from immediate reactivation of the source and one can expect to find propagating fronts, resembling dynamical percolation [19].
Another modification is the possibility for allowing a stochastic time dependence in the threshold value, by lowering K to K − ∆K, with probability 1/2 at each time MC step for all nodes. I tested here ∆K = 0.01, 0.02, 0.04, 0.05 disorder strengths.
By varying the control parameters, K, λ and ν one can find a critical transition between an active and an absorbing steady state. At a critical point the survival probability of the activity avalanches (measured by the total number of active sites) is expected to scale asymptotically as where δ is the survival probability exponent [55]. This is related to the avalanche duration scaling P (t) ∝ t −τt , via the relation [18]: In case of single active node initial conditions the number of active sites (measured by the total number of active sites during the spreading experiment) initially grows as with the exponent η, which can be related to the avalanche size distribution via the scaling relation This was derived in [18], assuming a bell shaped distribution p(s, t), which is the conditional probability of an avalanche having size s, given it dies at time t.

III. DYNAMICAL SIMULATION RESULTS
In [34] I compared results obtained on directed, randomly diluted edge variants of the KKI-18 network with those of the undirected graph and showed robustness of the GP for those changes. For simplicity I consider here the original KKI-18 network, which can be obtained from the graph, generated by the MIGRANE algorithm, after removing components, which do not belong to the largest connected one. The control parameter space was restricted by fixing λ ≃ 1, which mimics an efficient brain model and ν = 1 or ν = 0.95, corresponding to weak stochasticity in the deactivation transition. To see critical phase transition this requires a threshold value K ≃ 0.2. The time dependent simulations were performed by selecting active initial sites randomly and by averaging over 10 7 realizations up to t max = 3 × 10 5 Monte Carlo sweeps (MCs). Throughout the paper time is measured in MCs.

A. Refractory state variable threshold model
Spreading simulations were run at K = 0.2 for different λ values. Fig. 1 suggests a phase transition at λ ≃ 0.985(4), above which p(t) curves evolve to finite constant values. It is very hard to locate the transition clearly, since the evolution slows down and log-periodic oscillations also add, as the consequence of the modular network. Below the transition point we can find p(t) decay curves with truncated PL tails, characterized by the exponents 0.7 < δ < 1.6, as we vary λ between ≃ 0.84 and 0.98. However, modulation of these PL-s due to the modular network topology is also clearly visible. The lower-bound of the scaling region is rather uncertain and estimated by the λ value, where p(t) and ρ(t) seems to decay faster than a PL all for all times. Using the scaling relation (3) this means 1.7 < τ t < 2.6, falling in the range of brain experiments: 1.5 < τ t < 2.4 [24].
One can obtain even clearer PL tails for the density of active sizes ρ(t) with smaller modulations (see the Fig. 2). The GP seems to stretch from λ = 0.88 with a linear deity decay to λ = 0.98, characterized by a PL fit for the tail t −0.47(1) . Finally, the avalanche size distributions (Fig. 3), exhibit clear PL tails, with continuously changing exponents in the range 1.4 < τ < 1.91, again in agreement with the brain experiments 1 < τ < 1.6 [24]. The results for the exponent τ can be crosschecked using the relation (6). For λ = 0.98 one obtains: τ = 1.34 (15), close to, but a little below, the direct measurements of avalanches: τ = 1.50(5).

B. Time dependent thresholds
In the time dependent variable threshold model I fixed ν = 0.95, K = 0.25 and performed spreading simulations using GPU-s, up to t max = 10 6 MCs. Before the application of the time dependent thresholds the GPU accelerated algorithm was tested by the reproduction and extension of the results of [34] up to 10 7 MCs, up to two orders of higher values (see Sect. III C and [53]). Precise estimates for the critical point have been achieved for the model with inhibitory links (see Sect. III C).
In the case of the undirected graph, for ν = 0.95, K = 0.25 I estimated a GP in the region: 0.85 ≤ λ ≤ 0.95 [34]. Now I investigate what happens if a weak temporal disorder is added. Binary distributed variable thresholds have been applied, first with ∆ = 0.02 as shown on Fig. 4.
The average density decays in a PL manner for λ = 0.85 and λ = 0.86 with slopes α = 0.317(1), α = 0.245(1) respectively. The λ = 0.84 curve seems to be subcritical, while the λ = 0.87 one is super-critical. Lowering the threshold occasionally results in supercritical control for λ > 0.86, which in the time independent case was inside the GP. But part of the PL region, seems to survive this perturbation.
The same conclusion can be drawn by considering the survival probabilities on Fig. 5, and we can see a narrower GP as before.
In contrast, for stronger disorder: ∆ = 0.04 one can see a critical transition without the signs of any GP region. At the critical point λ c ≃ 0.78(1) one can fit decay: p(t) ∝ t −0.9(1) close to the MF value of DP (see Fig. 6). The same is true for the average density of activity. This suggests that this temporally applied "disorder" is strong enough to move all previous curves lying in the GP into the active phase, while the previously sub-critical ones remain in the inactive.
The avalanche size distributions for ∆ = 0.02 also ex-  hibit non-universal PL-s in the GP, as shown in Fig. 7, with somewhat higher slope values: p(s) ∝ s −2.0(3) . For clarity I plotted results for just three λ's, but the others fit well to them. This result for the exponent τ can be compared with the value we get via the scaling relation (6). For λ = 0.86 we can obtain: τ = 1.31 (15), away from the direct measurement. The reason behind it must be an asymmetric shaped p(s, t) distribution, which has already been  showed in [34]. Without this assumption inconsistent results were obtained in [58].

C. Time dependent thresholds with inhibition
In real brain inhibitory neurons, suppressing communications of connections exist. I did not simulate such nodes explicitly but considered negative connections by fast inhibitory inter-neurons. To model this in [34] I changed 30% of the link weights as : w ′ i,j = w i,j randomly. This produces further heterogeneity, thus stronger RR effects. Figure 8 shows the survival probabilities, for K = 0.1 and ν = 0.95. In this work I show the extension of the results presented in [34] by two orders of magnitude in time (from t max = 10 5 to t max = 10 7 ) using a GPU code, with the primary aim to confirm the parallel program, and with the secondary aim to provide more precise results at the critical point and for the GP. The critical point, above which p(t) signals persistent activity, is around ν = 0.56(1), hard to locate clearly, since the evolution slows down and exhibit strong (oscillating) corrections. Here ultra-slow, log. scaling tail can be observed of the form: p(t) ∝ ln(t) −0.84(1) as the left inset of Fig. 8 shows. This value is smaller than what was found in [34], for the original, positive weighted graph: δ = 3.5(3), suggesting a lower effective topological dimension due to the inhibitions. For comparison, in case of spatially quenched disordered models belonging to the same universality class one dimension:δ = 0.38187 [56], while in two-dimensions:δ = 1.9(2) [57].
Below the transition point the survival exponent changes continuously in the range 0 < δ < 0.5 as a response for the variation of ν between 0.5 and 0.57 (right inset of Fig. 8). To see corrections to scaling I determined the local slopes of the dynamical exponents δ as the discretized, logarithmic derivative of (2). The effective exponent of δ is measured as using t − t ′ = 8. A remarkable stabilization of δ eff can be observed for 10 4 < t < 10 7 in the GP. As in Section III B I applied time dependent thresholds by lowering K = 0.1 to K = 0.09 at random time steps with probability 0.5. In this case the GP shrunk to the region: 0.483 < λ < 0.513 approximately (see Fig 9). Thus the critical point moved down to λ ≃ 0.51(1) with respect to the time independent model. Near the criti-  cal point the avalanche size distributions exhibit roughly p(s) ∝ t −1.5 PL tails in agreement with experiments as shown on Fig. 10. Furthermore, the exponents decrease slightly by increasing λ.
Similarly to the non-inhibitory case, for larger temporal disorder, i. e. for ∆K = 0.05 one cannot see clear PLs, or maybe just in a narrow region, hindered by strong periodic modulations. Location of the phase transition point in this case is hard, but one can estimate it to be λ ≃ 0.40 (1). The avalanche size distributions exhibit p(s) ∝ s −1.4 PL tails here, but extended GP cannot be observed now.

IV. DISCUSSION AND CONCLUSIONS
I have extended the dynamical scaling study of the activity behavior of the connectome model started in [34] with the addition of refractory states and time dependent threshold values. High precision simulations showed the survival of GP in both cases, albeit in case of temporal disorder the GP shrinks. By lowering or increasing the threshold occasionally, RR-s of the original GP feel super-or sub-critical "quenches". Thus their evolutions do not follow the slow dynamics with large characteristic times. In the case of the addition of refractory states the GP seems to be even wider as in the two-state model. The avalanche size distributions exhibit fat tails with control parameter dependent PL exponents near the MF value, but the τ exponent does not satisfy the scaling law (6) connecting it to δ and η in general, suggesting asymmetric p(s|t) distributions like in [34].
In summary, while in the case of quasi-static, nonequalized threshold models fat tail distributed weights exclude rare-region effects (thus a GP), in equalized ones, simulations provide numerical evidence for robust occurrence of Griffiths effects for susceptible-excitedsusceptible type of models [34], for susceptible-excitedrefractory-susceptible models, even in the case of weak temporal noise. Inhibitory weights do not alter this scenario qualitatively, but provide homeostatic states in which the scaling behavior is quantitatively more similar to the brain experiments.
In particular, for the susceptible-excited-refractorysusceptible model the duration exponents in the GP are: 1.7 < τ t < 2.6, while the avalanche size exponents are in the range: 1.4 < τ < 1.91. For the weakly time-dependent (∆ = 0.02) susceptible-excitedsusceptible model one can estimate: 1.5 < τ t < 1.7 and 2 < τ < 2.28. In the case of inhibition, the scaling exponents of the weakly time-dependent, susceptible-excitedsusceptible model least squares fitting and (6) results in: 1 < τ t < 1.5, 1.1 < τ < 1.6. In general, for the durations a minimum value may emerge as: τ t = 1 + δ > 1, as a consequence of a non-conventional critical point, at the top of the GP in case of logarithmic scaling. For the avalanche size distributions there is no such minimum value of τ = 2.2, reported in a fuse model, which is defined on hierarchical modular networks [59]. Instead, the simulations on this connectome, or in the case of nonhierarchical modular networks [33] τ < 1 can be observed only.
Simultaneous, interacting avalanches should not alter the global universal dynamical critical exponents in statistical physics, although one can speculate how to define an avalanche in this case. If one could measure the activity spreading causally, one could decide if activated nodes are distinguishable or not. In the former case one can apply a neutral theory and obtain scale-free dynamics even away from the edge of a phase transition [60]. In the latter case I believe one should not obtain different critical behavior, like in case of identical multi-component reaction-diffusion systems in higher dimensions [19]. In a more realistic model, exhibiting memory of nodes in accumulating action potentials, interacting avalanches can influence their mutual spreading behavior, like for a paircontact process [19] and initial condition dependent scaling may also emerge.
It is important to stress that this study does not aim to exclude SOC generating synaptic learning and plasticity mechanisms, leading to homeostatic states and closeness of a criticality in brain models. Instead of that I show how heterogeneities alter the dynamical behavior, causing critical like PL-s in an extended control parameter region even below the singular phase transition point and provide a possible explanation for the spread of exponents measured. Experimental justification of GP would require to measuring the avalanche and autocorrelation exponents in high resolution and show how the scattered exponents depend on some control parameters (sensitivity, inhibition, ... etc.) of the brain. An interesting continuation of this work would be to test partially equalized input sensitivities, where first order phase transition was shown [34], in agreement with brain experiments, showing abrupt transition between up and down states see [61].
The codes and the graph used here are available on request from the author.