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Conceived and designed the experiments: AD. Performed the experiments: AD. Analyzed the data: JDT AD. Contributed reagents/materials/analysis tools: JDT AD. Wrote the paper: JDT AD.

The authors have declared that no competing interests exist.

The presence of self-organized criticality in biology is often evidenced by a power-law scaling of event size distributions, which can be measured by linear regression on logarithmic axes. We show here that such a procedure does not necessarily mean that the system exhibits self-organized criticality. We first provide an analysis of multisite local field potential (LFP) recordings of brain activity and show that event size distributions defined as negative LFP peaks can be close to power-law distributions. However, this result is not robust to change in detection threshold, or when tested using more rigorous statistical analyses such as the Kolmogorov–Smirnov test. Similar power-law scaling is observed for surrogate signals, suggesting that power-law scaling may be a generic property of thresholded stochastic processes. We next investigate this problem analytically, and show that, indeed, stochastic processes can produce spurious power-law scaling without the presence of underlying self-organized criticality. However, this power-law is only apparent in logarithmic representations, and does not survive more rigorous analysis such as the Kolmogorov–Smirnov test. The same analysis was also performed on an artificial network known to display self-organized criticality. In this case, both the graphical representations and the rigorous statistical analysis reveal with no ambiguity that the avalanche size is distributed as a power-law. We conclude that logarithmic representations can lead to spurious power-law scaling induced by the stochastic nature of the phenomenon. This apparent power-law scaling does not constitute a proof of self-organized criticality, which should be demonstrated by more stringent statistical tests.

Many natural complex systems, such as earthquakes or sandpile avalanches, permanently evolve at a phase transition point, a type of dynamics called self-organized criticality (SOC)

In neuroscience, it is of obvious interest to determine if the recruitment of activity in neural networks occurs in power-law distributed avalanches. This would be evidence that the brain may function according to critical states, rather than the usual wave-type, oscillatory or stochastic dynamics. Moreover, power-law relations are often associated with long-lasting correlations in the system, as with the behavior near critical points. Indeed, the presence of self-organized criticality was inferred for several biological systems, including spontaneous brain activity

To investigate if criticality is important for brain function, the same type of analysis was also investigated

In the present paper, we attempt to resolve these contradictory observations by first performing the same analysis on negative LFP peaks in cats, and using different statistical tests and models to explain these observations. We study the statistical distribution of avalanche sizes, as well as the distribution of the amplitude of negative peaks in the LFPs (linked to neuronal firings), positive peaks, and surrogate data. We then study similar stochastic problems, and investigate whether the results obtained by the experimental data analysis can also be observed in purely stochastic systems without the presence of underlying self-organized criticality. Eventually, we compare the results obtained to the analysis of avalanche data produced by a neural network known to present self-organized criticality

The experimental data used in the analysis consist of simultaneous recordings of multisite local field potentials (LFPs) and unit activity in the parietal cortex of awake cats (see

Eight pairs of tungsten electrodes (placement illustrated on top) were inserted in cat cerebral cortex (area 5–7, parietal) as described in detail in

Negative or positive peaks were detected from the LFPs as follows. Signals were mean-subtracted and divided by their standard deviations to obtain comparable amplitude statistics. To avoid artifactual peak detection because of occasional slow components or drifts, the signals were digitally filtered below 15 Hz (high-pass), and the peaks were detected using an adjustable fixed threshold. The peak was defined as the extremum of the ensemble of data points that exceeded the threshold. The detected peaks were then repositioned in the intact original signal (see

Top: detection of negative LFP peaks. The LFP signal is shown together with the detected nLFPs (circles). Middle: nLFP-based wave-triggered average (WTA) of unit activity, showing that the negative peaks were associated with an increase of neuronal firing. Bottom: rasters of nLFP activity. The same procedure is compared for high threshold (left panels) and low threshold (right panels).

Avalanches were defined by binning the raster of negative peaks of the LFP (nLFPs) into time bins of size

Surrogate signals were generated from the nLFP data sets by shuffling the occurrence times of the different peaks, while keeping the same distribution of peak amplitudes. The occurrence times were replaced by random numbers taken from a flat distribution. The avalanche analysis was then performed on this shuffled data set. Note that, because shuffling changed the timing of the peaks, the whole set of avalanches changed.

The results of neuronal avalanche analysis recorded in the cat cerebral cortex will be compared to two types of artificial data sets. From the nature of the LFPs and the links between unit firing and LFP peaks above a certain threshold (see the

The stochastic processes studied are based on the following two simple models: the shot noise and the Ornstein-Uhlenbeck model.

The first stochastic model considered is a high-frequency shot-noise process consisting of exponential events convolved with a Poisson process. This process,

In the limit of a high number of Poisson processes with summable intensities (or in the limit of a finite number of Poisson process with high firing rate and suitable scaling on the jump amplitude), the solution of equation (1) converges in law towards the solution of the equation:

We finally performed the statistical avalanche size analysis in a situation where self-organized criticality was known to be present. We used a model proposed by Levina and colleagues, which consists of a network of spiking neurons with dynamical synapses, in which the neuronal avalanches are characterized by a typical and robust self-organized critical behavior

Mathematically, a continuous random variable

In this paper, we are interested in discriminating power-laws from another type of distribution: the exponentially-tailed distribution. Random variables with such distributions are characterized for

Taking the logarithm of the probability density of a power-law random variable, we obtain

Assume that

For the continuous power-law distribution the log-likelihood of the data for the estimated parameter value is:

Therefore, given the samples

For a given data set, we now know how to evaluate the best power-law and best exponential-law fits. But is either fit plausible and accurate? In order to answer this question, we use a standard goodness-of-fit test which generates a p-value quantifying the likelihood of obtaining a fit as good or better than that observed, if the hypothesized distribution is correct. This method involves sampling the fitted distribution to generate artificial data sets of size

The methods described above provide the better possible fit for a data set with different probability laws and and the statistical relevance of the model fitted to explain the data set. However, in the case where neither model is rejected by the p-value test, these procedures do not allow to quantify which model provides the better fit.

Several methods have been proposed to directly compare models, such as cross validation

Note that this method compares fits on a given same data set, which requires in particular the use of the same

We start by analyzing the power-law scaling from experimental data. To analyze the power-law relations from LFP activity, we exploited the well-known relation between negative LFP peaks and neuronal firing. We identified the negative peaks of the LFPs (nLFPs), corresponding to events exceeding a fixed threshold, as shown in

We next performed an avalanche analysis based on the occurrence of nLFPs. Similar to previous studies

The nLFP avalanche size distributions were computed according to an avalanche analysis (see text). For a high detection threshold, the avalanche distribution is better fit by a power-law (left panels); for a low detection threshold, it is better explained by an exponential distribution (right panels).

To assess the significance of this result, we performed a Kolmogorov- Smirnov test to the same data. The results of this test are presented in

Data Type and threshold | Exponential fit | Power-Law fit | Log Likelihood ratio | ||||||||

KS | p-val | % | KS | p-val | % | LLR | p-val | Result | |||

Neg. Low | .18 | 0.028 | 0.07 | 38 | 5.32 | 0.050 | 0.94 | 4 | −1211 | 0.0 | Exp |

Neg. High | 0.13 | 0.042 | 0.07 | 20 | 2.01 | 0.077 | 0 | 88 | −133 | 0.0 | Exp |

Pos. Low | 0.079 | 0.052 | 0 | 2.7 | 2.97 | 0.041 | 0.70 | 9 | −1275 | 0.0 | Exp |

Pos High | 0.18 | 0.033 | 0.27 | 31 | 1.93 | 0.091 | 0 | 93 | −351 | 0.0 | Exp |

The statistical avalanche analysis performed when the avalanche size was defined as the total number of events (peaks) within each avalanche give an even more ambiguous result. Indeed, both the exponential and the power-law distributions provide a good fit to the data, and the log likelihood indicates that the exponential law provides a better fit but it has a null significance, so does not give any information on the law that best fits the data (see

Data Type and threshold | Exponential fit | Power-Law fit | Log-Likelihood ratio | ||||||||

KS | p-val | % | KS | p-val | % | LLR | p-val | Result | |||

Neg. Low | 0.19 | 0.023 | 0.64 | 54 | 1.26 | 0.020 | 0.83 | 18 | −77 | 1.0 | |

Neg. High | 0.27 | 0.045 | 0.27 | 29 | 1.74 | 0.009 | 0.97 | 100 | −61 | 1.0 | |

Pos Low | 0.23 | 0.030 | 0.19 | 70 | 1.20 | 0.021 | 0.60 | 54 | −232 | 1.0 | |

Pos. High | 0.36 | 0.067 | 0.14 | 50 | 1.54 | 0.012 | 0.91 | 100 | −110 | 1.0 |

While these findings suggest that the the nLFP avalanches may also be exponentially distributed, this exponential scaling may be artifactual. Although the underlying neural activity may follow a power-law distribution, the low-threshold condition could add spurious peaks unrelated to neuronal activity, and that would give an exponential trend to the distribution. This increased “noise” is evident in the WTA in

To further test the dependence on unit activity, we have repeated the same avalanche analysis, but using positive peaks of the LFP (pLFP;

A. Detection of positive LFP peaks using identical procedures as for nLFPs. B. The WTA indicates no relation between pLFPs and unit activity. C. Scaling of avalanche size distribution, showing similar behavior as observed for nLFPs (compare with

Another essential test is to generate surrogate data sets. These were produced by taking the nLFP data sets, and randomly shuffling the occurrence times of the different peaks, while keeping the same distribution of peak amplitudes (see

A. Shuffled peaks obtained from randomizing the timing of nLFP peaks. B. The WTA indicates that shuffling removes the relationship between nLFPs and neural activity C. Scaling of avalanche peak size distribution, showing similar behavior as for nLFPs (compare with

The power-law scaling of nLFP size distributions was also apparent when representing graphically the peak distributions from single LFP channels, as illustrated in

The peak distribution is shown on log-linear (A) and logarithmic scale (B).

Data Type and threshold | Exponential fit | Power-Law fit | Log-Likelihood ratio | ||||||||

KS | p-val | % | KS | p-val | % | LLR | p-val | Result | |||

Neg. Low | 2.39 | 0.029 | 0.055 | 39 | 6.17 | 0.056 | 0.00 | 33 | −47 | 0.0 | Exp |

Neg. High | 2.82 | 0.030 | 0.68 | 80 | 9.05 | 0.048 | 0.53 | 34 | −4.4 | 0.04 | Exp |

Pos. Low | 2.07 | 0.022 | 0.25 | 98 | 6.15 | 0.041 | 0.06 | 26 | −37.7 | 0.0 | Exp |

Pos. High | 2.21 | 0.038 | 0.29 | 56 | 6.85 | 0.044 | 0.10 | 100 | −1.29 | 0.66 |

These results suggest that the power-law scaling seen in log-log representations is not necessarily related to neuronal activity, but could rather represent a generic property of these signals. To test this hypothesis, we now turn to the analysis and simulation of stochastic processes.

We first investigate computationally whether a power-law relation can be obtained from the peak size distribution of a purely stochastic process. To this end, we generate a high-frequency shot-noise process (as described in

The peaks were detected on the shot noise process

A. Sample of the stochastic process and detected peaks. B. Peak size distribution on a log-linear scale. C. Same distribution on a log-log scale. Straight lines indicate the best fit obtained using linear regression.

We now investigate this problem analytically. We treat the case where the number of independent Poisson processes

We are interested in the probability that the supremum of this process reaches a certain threshold value

The maxima of this process occur at the event times of the Poisson process,

Furthermore, conditionally on

Let us now consider the distribution of the maxima of the process (7) given that the process does an excursion above a certain threshold. This case can be treated in a similar fashion, but considering the distribution of local minima also. These local minima are reached at times

Simulation results of these distributions are presented in

Single-barrier case (A,B) on a log-linear scale (A) and on a log-log scale (B) show a globally linear trend. Excursions (C,D) show exactly the same profile. Simulation parameters: intensity of the process

Data type | Exponential fit | Power-Law fit | ||||

KS | p-val | KS | p-val | |||

Shot-Noise | 0.70 | 0.103 | 0.12 | 10.08 | 0.185 | 0.00 |

single-barrier | ||||||

Shot-Noise | 0.72 | 0.014 | 1.00 | 15.00 | 0.094 | 0.28 |

excursion | ||||||

Ornstein–Uhlenbeck | 2.40 | 0.042 | 0.97 | 44 | 0.077 | 0.62 |

single-barrier | ||||||

Ornstein–Uhlenbeck | 2.42 | 0.0051 | 1.00 | 48.00 | 0.012 | 0.92 |

excursion |

In the case of the Ornstein-Uhlenbeck model, the stochastic process modelling the LFP has the same regularity as the Brownian motion, and therefore is is nowhere differentiable, and has a dense countable set of local maxima. In that case, peaks are no more defined as local maxima of the process, and the problem is reduced to determining the probability that the process exceeds a certain value. This probability can be deduced from the law of the first hitting time of the Ornstein–Uhlenbeck process. Indeed, let us denote by

The excursion case continuous equivalent consists in considering the probability of exceeding a certain quantity

Therefore, the repartition function of the maxima, and that of the maxima above a certain threshold, can be deduced from the repartition function of the first hitting time of the process

In this case again, the same remarks apply: we observe (see

(A,B): single-barrier peaks, on a log-linear scale (A) and on a log-log scale (B), and excursions (C,D), on a log-linear scale (C) and on a log-log scale (D). Both case present the same profile and a globally linear trend for both axis. Simulation parameters: intensity of the process

We finally performed the above statistical analysis on the avalanche data generated by the artificial network in the critical state of Levina and colleagues

The power-law distribution provides a very good graphical fit (A), whereas the exponential distribution provides a poor fit (B). Data from ref.

Data type | Exponential fit | Power-Law fit | Log-Likelihood ratio | ||||||

KS | p-val | KS | p-val | LLR | p-val | Result | |||

Full data set | 0.10 | 0.2820 | 0.00 | 1.44 | 0.0027 | 0.85 | 1645 | 0.0 | PL |

0.10 | 0.2806 | 0.00 | 1.42 | 0.0061 | 0.80 | 2483 | 0.0 | PL |

In this paper, we have provided an analysis of multisite LFP recordings in awake cats, using the detection of negative LFP peaks (nLFPs), as done in a previous study

Because the exponential scaling could be interpreted as a spurious result due to the addition of a large number of peaks unrelated to neuronal activity, we considered two controls: first, positive LFP peaks, which are not related to neuronal activity, and randomly shuffled peak times, which makes the system equivalent to a stochastic process with the same peak amplitude distribution as the data. The two cases show similar apparent power-law scaling and dependency to threshold as for nLFPs.

These results suggest that the spurious power-law scaling could be a generic property of thresholded stochastic processes. To investigate this point in more depth, we studied a similar peak detection paradigm applied to two simple stochastic models, one corresponding to LFPs arising from a linear summation of spikes arriving at the times of a Poisson process (a shot-noise process) and the diffusion limit of this phenomenon (an Ornstein–Uhlenbeck process). The former case can be solved in a closed integral form while the latter case is solved using the laws of the first hitting times of the Ornstein-Uhlenbeck process. Both models demonstrate the same ambiguity: when only looking at the log-linear and log-log plots, and both power-laws and exponential laws can be fitted. However, the application of the more rigorous Kolmogorov–Smirnov test demonstrated that some apparent power-law scaling (as seen from log-log representations) is not based on solid statistical grounds, in real data as well as in the theoretical laws computed, in agreement with previous studies (see e.g.

This analysis therefore confirms that thresholded stochastic processes can display power-law scaling, but only when performing simple line fitting in log-log representations. Indeed, we observe that it is always possible to fit a power-law distribution to the tail of the distribution with a quite good agreement, but these fits do not hold for large threshold values (see

The same analysis applied to a network presenting self-organized criticality confirms with no ambiguity that the distribution of avalanche size presents a clear power-law distribution, whereas in cortical LFPs the power-law scaling in log-log representations was not supported by statistical analyses. We conclude that power-law scaling, particularly when deduced from log-log representations, does not constitute a proof of self-organized criticality, but should be complemented by more sophisticated statistical analyses.

Thus, contrary to a previous study in monkey

We thank Anna Levina and colleagues for kindly providing us avalanche data from their neural network model published in