Time-series thresholding and the definition of avalanche size

PHYSICAL REVIEW E 100, 012133 (2019) Time-series thresholding and the definition of avalanche size Pablo Villegas,1Serena di Santo,1,2,3Raffaella Burioni,2,3and Miguel A. Muñoz1,2 1Departamento de Electromagnetismo y Física de la Materia and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071, Granada, Spain 2Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, via G.P. Usberti, 7/A-43124, Parma, Italy 3INFN, Gruppo Collegato di Parma, via G.P. Usberti, 7/A-43124, Parma, Italy (Received 27 February 2019; published 23 July 2019) Avalanches whose sizes and durations are distributed as power laws appear in many contexts, from physics to geophysics and biology. Here we show that there is a hidden peril in thresholding continuous times series—from either empirical or synthetic data—for the identification of avalanches. In particular, we consider two possible alternative definitions of avalanche size used, e.g., in the empirical determination of avalanche exponents in the analysis of neural-activity data. By performing analytical and computational studies of an Ornstein-Uhlenbeck process (taken as a guiding example) we show that (1) if relatively large threshold values are employed to determine the beginning and ending of avalanches and (2) if—as sometimes done in the literature—avalanche sizes are defined as the total area (above zero) of the avalanche, then true asymptotic scaling behavior is not seen, instead the observations are dominated by transient effects. This problem—that we have detected in some recent works—leads to misinterpretations of the resulting scaling regimes. DOI: 10.1103/PhysRevE.100.012133 I. INTRODUCTION Episodic outbursts of activity or “avalanches” of highly variable durations and sizes are observed in a large variety of scenarios in condensed matter physics (vortices of type II superconductors [1] and Barkhaussen noise [2,3]), high-energy astrophysics (x-ray flares [4]), geophysics (earthquakes [5]), meteorology (rainfall [6]), and neuroscience (neuronal avalanches [7]), as well as in other biological systems (gene knock-out cascades) [8]) and man-made systems (failures on electrical power grids [9]). The probability distributions of sizes and durations of such avalanches often exhibit a “fat-tail” that can be fitted as a power-law distribution, i.e., the fingerprint of scaling behavior. Such scaling or scale invariance is often considered as evidence of underlying criticality, and many of the above systems are claimed to operate at (tuned or self-organized) critical points [10–14]. In particular, in the context of biology the idea that living systems (parts, aspects, or groups of them) may extract important functional advantages from operating at criticality—i.e., at the edge of two different phases—has been deeply explored in recent years [15,16]. In this regard, ground-breaking experimental evidence by Beggs and Plenz [7] revealed the existence of scaleinvariant episodes of electrochemical activity in neural tissues thereafter named neural avalanches. Subsequently, neural avalanches were robustly detected across a large variety of experimental settings, tissues, and species [7,17–23]. In particular, neuronal avalanche sizes, S, were robustly observed to be distributed as a power law P(S)∼S−τwith τ≈3/2 up to some upper cutoff; similarly, avalanche durations T were well fitted by P(T)∼T−αwith α≈2uptosome characteristic maximum time [7]. Furthermore, fundamental scaling relationships [24] were observed to be fulfilled: e.g., the averaged avalanche size scales as S∼Tγand the set of exponents obey γ=(α−1)/(τ−1) [25]. This set of empirically reported exponent values is in agreement with that of the well-known critical (or “unbiased”) branching processes, also called Galton-Watson process, originally introduced to describe the statistics of the extinction of family names) [26–29]. Actually, the set of exponent values τ=3/2, α=2, and γ=2 are extremely universal as they are shared by many different propagation processes in high-dimensional systems as well as in many types of networks [2,30]. In particular, they are the mean-field exponents shared by models such as the contact process, directed and isotropic percolation, susceptible-infected-susceptible, and a large list of other prototypical models for spreading or propagation dynamics above their respective upper critical dimensions [24,31–34]. Thus, it was conjectured that neuronal systems might operate close to the edge of marginal propagation of neural (electro-chemical) activity [7,35], opening the door to exciting theoretical perspectives and some debate (see Ref. [16]for a recent review). However, as extensively discussed in the literature, diverse generative processes for the emergence of power laws exist [36–38], and not all power-law distributions can be taken as a signature of criticality. For instance, a diverging correlation length needs to be identified to assign a given phenomenon to criticality. In recent years, some authors have suggested that the origin of the observed power-law scaling in neural systems might stem from other types of criticality (rather than marginal propagation) [39–41]oreven be unrelated to critical behavior [42,43]. In the present brief paper—leaving aside the putative connection with criticality—we contribute with an additional piece of information to the already controversial discussion about the statistics of neuronal avalanches. In particular, we 2470-0045/2019/100(1)/012133(6) 012133-1 ©2019 American Physical Society PABLO VILLEGAS et al. PHYSICAL REVIEW E 100, 012133 (2019) show that some of the reported empirical evidence in favor of the value τ=3/2—and, thus, seemingly in favor of the existence of an underlying critical branching process—might be misleading as there is a technical problem in the way avalanches are measured, which hinders the observation of the true asymptotic behavior. More in general, we underline that particular attention needs to be taken when avalanches of activity—defined by thresholding—are inferred from a continuous time series of activity. Our findings, add to the recent literature warning on the “perils” associated with thresholding in time series [44–46]. II. DEFINITION OF AVALANCHES As discussed in the introduction, avalanching phenomena are best described, at least in mean field, as branching processes. Such processes can reach the value 0, which is an absorbing state: Avalanches are naturally defined as excursions away from such a state caused by small perturbations [47]. However, in many contexts—including neuroscience but not only—the term “avalanche” is used to refer to excursions of time series above some given (arbitrary) threshold, regardless of absorbing states the existence of any absorbing state. In this section we discuss such avalanches and their statistics. A. Avalanches in the Wiener and Ornstein-Uhlenbeck processes Let us consider, for argument’s sake, a time series for a stochastic real variable x—as illustrated in Fig. 1—generated by a Wiener Process, i.e., by a continuous-time unbiased random walk (RW) defined by the following Langevin equation [48]: ˙x(t)=ση(t),(1) where η(t) is a Gaussian white noise with zero mean and unit variance, and σis the noise amplitude. For such a time series (which can be thought of as describing the time course of the activity of some arbitrary system) the duration Tof an avalanche is the amount of time for which xstays (a) (b) (c) (d) FIG. 1. Illustration of how avalanche duration Tand size Sare defined for an unbiased RW. (a) Illustration of a particular time series, in which two avalanches of durations T1and T2and sizes S1and S2, respectively, are emphasized. The threshold is set to 0 in this case (red dashed line). Lower panels show the probability distributions of (b) sizes, (c) durations, and (d) average size for a fixed given duration (straight lines correspond to the well-known analytical predictions, and symbols stand for computational results). TABLE I. Summary of the avalanche (mean-field) exponents: Size (τ), duration (α), and averaged avalanche size (γ) for the (unbiased) branching process (BP) and the (unbiased) random walk (RW); see, e.g., Ref. [47]. P(S)∼S−τP(T)∼S−αP(S|T)∼Tγ BP τ=3/2α=2γ=2 RW τ=4/3α=3/2γ=3/2 above a given threshold, i.e., an avalanche begins and ends when the activity signal crosses the threshold from below and above; the avalanche size Sis the area covered between the walk trajectory and the threshold reference line (see Fig. 1). Observe that similarly, given the symmetry of the process, one could also define avalanches as excursions below threshold. The probability distribution of avalanche durations Tcan be straightforwardly identified with the first-return time statistics of RWs (see Fig. 1), which is well known to scale with an exponent α=3/2. Similarly, also the size-distribution exponent τ=4/3 and the remaining exponent γ=3/2are well known for RWs (see Fig. 1and Table I); pedagogical derivations of these results, as well as a comparison with the branching process class, can be found in, e.g., Refs. [47,49]. Importantly, these results for the RW do not depend of the value of the chosen threshold. In the more general case in which the walker is confined to hover around a given mean value, one can describe the problem, in first approximation, as an Ornstein-Uhlenbeck process [48]: ˙x(t)=−ax(t)+ση(t),(2) where there is an additional linear force term, −ax (corresponding to the negative derivative of the parabolic potential bounding the walker close to x=0). Such a force introduces an upper cutoff in the first-return times statistics of the unbiased RW (see, e.g., Ref. [49] for a detailed derivation). Thus, avalanches intended as excursions above a given threshold in a process with a well-defined steady-state value have powerlaw-distributed sizes and durations, with the exponents of the RW class (as in Table Iand as in Fig. 1), but only up to an upper-cutoff scale controlled by 1/a, such that it goes to infinity, i.e., it disappears when avanishes. As a corollary of all this, let us remark that many real time series describing (e.g., biological) problems in which some stochastic variable fluctuates symmetrically around a given mean value exhibit effective avalanching behavior that—up to a certain scale of size and time—can be described by the exponent values of the RW. Let us stress again that, as argued above, it can be a matter of debate whether this type of behavior—describable in terms of RW excursions above a threshold—can be called “avalanching.” Actually, for most of the examples of interest in physics, as discussed at the beginning of this paper, this does not constitute an adequate description because it does not include any absorbing state. B. On the definition of avalanche size In many other circumstances time series exhibit asymmetric excursions around a mean value and/or may 012133-2 TIME-SERIES THRESHOLDING AND THE DEFINITION … PHYSICAL REVIEW E 100, 012133 (2019) FIG. 2. Sketch of a nonsymmetric stochastic process for a positive definite variable (describing, e.g., density of neural activity). θ(red dashed line) signals the arbitrarily fixed threshold employed to define avalanches. For the large avalanche in the center of the graph, Sis the avalanche size using criterion A (area above threshold, colored in orange) and Tis its duration. On the other hand, using criterion B, =S+s∗(where s∗is the area of the rectangle between zero and the threshold, colored in blueish color, with s∗∝T)isan often-used alternative definition of avalanche size. As discussed in the text this definition may induce misleading interpretations of the resulting exponents. become trapped at some absorbing state. An example of this is obtained when the variable under scrutiny is a positive-definite density (e.g., of neural activity), which, by definition, is constrained to take positive values, x(t)>0. In such cases, especially in the ones when there is always some lingering activity so that the zero value is hardly reached (see Fig. 2) a threshold θ>0 is often employed to define avalanches as periods during which the activity remains above such a threshold. In these situations, two alternative possibilities are often used in the literature to measure the size of so-defined avalanches [50]: (A) Following the RW analogy, as done above, for a given avalanche, one can define its size Sas the area in between the time-series curve and the threshold (θ) reference line. (B) Alternatively, one can define the avalanche size as the overall integral of the time series during the avalanche, i.e., above the reference line x=0 (see, e.g., Refs. [51,52], but there are other works making this choice). The difference between the two criteria to define avalanche sizes is sketched in Fig. 2., the total integral of the activity is equal to =s∗+S, where Sis the integral of the signal above threshold, and s∗is the area of the rectangle under the threshold. In what follows, we compare the statistics of avalanches obtained using these two alternative definitions of size A and B for an Ornstein-Uhlenbeck process. This will serve as an illustration of a more general phenomenon that may also occur for other processes, such as the one sketched in Fig. 2. First, we discuss computational results, and then we employ scaling arguments to explain the findings. On the one hand, as already shown in Fig. 1using S, i.e., criterion A, one reproduces the expected theoretical results for all three avalanche exponents. On the other hand, as illustrated in Fig. 3, the statistics of avalanche sizes, as determined employing for an Ornstein-Uhlenbeck process, is anomalous and does not match the expectations for the theoretically known values, i.e., the measured value τ≈3/2 does not coincide with the expected value τ=4/3. In particular, the numerical (a) (b) (c) log10 log10 log10 log10 log10 FIG. 3. Statistics of avalanches of activity in a stochastic process (Ornstein-Uhlenbeck with a=0.1) employing as a measure of the avalanche size. Observe that both (a) avalanche-size and (b) avalanche-duration distributions obey scaling with the same exponent value for many orders of magnitude (a base-10 logarithmic scale is used for both axes in all three plots). However, the exponent values τ=α=3/2 and, consequently, as depicted in panel (c), γ=1.0 [satisfying the important scaling relation γ=(α−1)/(τ− 1)] do not coincide with the expectations for an Ornstein-Uhlenbeck process. observation of, τ≈3/2 could be (wrongly) taken as evidence of branching process-like scaling [51,52]. The fact that there is something suspicious with definition (2) can be noticed observing that both sizes and durations scale in the same way, entailing γ=(α−1)/(τ−1) =1, which would imply a locally linear (i.e., “tentlike”) shape of avalanches [53–55]. C. Scaling arguments The correction s∗for a given avalanche (such that = S+s∗) is nothing but s∗=θT, where Tis the avalanche duration. The distribution of first-passage times for the Wiener process is given by P(T)∼T−3/2. As said above, the same result holds for the Ornstein-Uhlenbeck case up to an upper cutoff. Thus, has a correction s∗with respect to Sthat scales as the avalanche duration: P(s∗)∼s∗−3/2. Assuming that the probability to observe a given size S conditioned toagiven avalanche duration T,P(S|T), is a peaked function around its mean value (as usually occurs for avalanches [24]) and using the fact that S∼Tγ, with γ=3/2, then (T)=S(T)+s∗(T)=˜cT 3/2+θT,(3) from where it follows that d=(cT 1/2+θ)dT.(4) Thus, we can readily write (using the implicit function theorem): P[(T)] =P(T)dT d=NT−3/2 cT 1/2+θ.(5) From this, in the limit of vanishing threshold θin Eq. (4), one has P()≈N cT−2≈N c[(/c)2/3]−2∼−4/3,(6) which is the correct result for the avalanche size distribution of an Ornstein-Uhlenbeck process. On the other hand, for larger 012133-3 PABLO VILLEGAS et al. PHYSICAL REVIEW E 100, 012133 (2019) values of the threshold θand relatively small values of T(and, thus, also typically small values of ), one has P()≈N θT−3/2≈N θ[/θ]−3/2∼−3/2,(7) in agreement with the numerical observation above. In other words, the additional contribution s∗dominates the scaling behavior of the avalanche size when θis relatively large. It is important to emphasize that, in any case, one should recover the correct asymptotic value—i.e., the behavior for large values of —of the avalanche size exponent (τ=4/3) for any value of θ, but this requires going to larger and larger avalanche sizes as θis chosen larger and larger. In particular, Fig. 4illustrates that there is a crossover from the value τ≈3/2 measured for small avalanche sizes to the true asymptotic scaling τ=4/3, for larger sizes. The crossover point grows with θ, so that the effect is not observed for θ≈0, but may extend for many scales even for moderate values of θ. In particular, given that an upper cutoff to scaling may exist (controlled, e.g., by 1/ain the case of an Ornstein-Uhlenbeck process that we are considering here or by finite-size effects), the transient behavior usually extends all the way up to the (a) (b) log10 log10 log10 FIG. 4. Distribution of avalanche sizes for a stochastic (OrnsteinUhlenbeck) process employing for the measure of avalanche sizes. (a) Avalanche-size distribution for the case θ=1 (with a=0and σ=1 in this case): Observe that the true exponent value τ=4/3is asymptotically recovered for large avalanche sizes. (b) Distribution of avalanche sizes for different values of the threshold parameter, θ. The associated exponent changes continuously between the two limiting exponents 3/2 (for large thresholds) and 4/3forsufficiently small ones (parameter values: a=0.1, σ=0.5; thresholds as marked in the legend). cutoff, so that the true asymptotic behavior can be unobservable with criterion B if large values of θare considered. Thus, summing up, considering criterion B for the definition of avalanche sizes together with relatively large threshold values or not sufficiently large statistics may lead to the observation of an effective value τ≈3/2; this may induce a misinterpretation of the scaling universality class, suggesting it is branching-process-like rather than what it actually is: A RW-like process. Let us emphasize that the previous discussion has been done for an Ornstein-Uhlenbeck process. However, it perfectly illustrates the problem associated with criterion B in more general circumstances, e.g., it also applies to asymmetric processes as the one sketched in Fig. 2. In any case, criterion B mixes the scalings of actual sizes and times, leading to potential interpretation errors in general stochastic processes. This problem is at the origin of misclassification of scaling behavior in existing works analyzing neuronal avalanches (see, e.g., Refs. [51,52]). III. CONCLUSION In this brief paper we have shown that an inappropriate definition of avalanche sizes as measured as excursions above a given threshold in continuous time series can lead to misleading conclusions. To illustrate this, we have studied a simple Ornstein-Uhlenbeck process (representing, e.g., the time course of activity in a mesoscopic model of neural activity) and have measured avalanches sizes in two possible ways: (1) as the integrated activity Sover a given threshold and (2) integrating the total activity signal between two threshold crossings, as illustrated in Fig. 2.Wehaveshown both computationally and using scaling arguments that this latest definition can induce strong biases in the determination of the avalanche-size exponent τ. In particular, if large values of the threshold θare considered, then—for relatively small avalanches—one observes the exponent value τ≈3/2, which could lead to the erroneous interpretation that an effective unbiased branching process dynamics exists. On the other hand, for sufficiently small threshold values and for sufficiently large avalanche sizes the correct scaling τ=4/3 is recovered. As discussed above the problem associated with criterion B extends to any type of stochastic process because it mixes up the scaling of actual sizes with that of durations, giving rise to misleading results. This is the underlying reason why recent analyses of avalanches in mesoscopic models of neural activity that consider relatively large thresholds [50–52]) obtain τ≈3/2— compatible with the scaling of a critical branching process—a result that our analysis reveals is not asymptotic. Their corresponding underlying dynamics describe fluctuations around a given mean value, and, thus, the associated avalanches should be related to excursions of random walkers and not to critical branching processes. As an important final remark, let us stress that it is essential—and it should always be done—to consider the full set of avalanche exponents, i.e., τ,α, and γas well as the scaling relations between them, in order to avoid possible errors and misleading interpretations and properly identify the type of scaling behavior. 012133-4 TIME-SERIES THRESHOLDING AND THE DEFINITION … PHYSICAL REVIEW E 100, 012133 (2019) ACKNOWLEDGMENTS We acknowledge the Spanish Ministry and Agencia Estatal de investigación (AEI) through Grant No. FIS2017-84256-P (European Regional Development Fund (ERDF)) for financial support. This study has been partially financed by the Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía and European Regional Development Fund (ERDF), ref. SOMM17/6105/UGR as well as to Teach in Parma for support to M.A.M. The study is supported by Fondazione Cariparma, under TeachInParma Project. We thank V. 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