Time-series thresholding and the definition of avalanche size
Pablo Villegas Serena di Santo Raffaella Burioni Miguel A. Muñoz




 Departamento de Electromagnetismo
  y Física de la Materia e Instituto Carlos I de Física
  Teórica y Computacional. Universidad de Granada.  E-18071,
  Granada, Spain 
 Departamento de
  Electromagnetismo y Física de la Materia e Instituto Carlos I
  de Física Teórica y Computacional. Universidad de Granada.
  E-18071, Granada, Spain Dipartimento di Scienze
  Matematiche, Fisiche e Informatiche, Università di Parma, via
  G.P. Usberti, 7/A - 43124, Parma, Italy INFN, Gruppo
  Collegato di Parma, via G.P. Usberti, 7/A - 43124, Parma, Italy
 Dipartimento
  di Scienze Matematiche, Fisiche e Informatiche, Università di
  Parma, via G.P. Usberti, 7/A - 43124, Parma, Italy
INFN, Gruppo Collegato di Parma, via G.P. Usberti, 7/A -
  43124, Parma, Italy 
Departamento de Electromagnetismo y Física de la
  Materia e Instituto Carlos I de Física Teórica y
  Computacional. Universidad de Granada.  E-18071, Granada,
  SpainDipartimento di Scienze Matematiche, Fisiche e
  Informatiche, Università di Parma, via G.P. Usberti, 7/A - 43124,
  Parma, Italy

  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 –either from 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 (i) if relatively
  large threshold values are employed to determine the beginning and
  ending of avalanches and (ii) 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.  

Timeseries thresholding and the definition of avalanche size
    Miguel A. Muñoz
    December 30, 2023
============================================================







§ 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
<cit.> and Barkhaussen noise <cit.>),
high-energy astrophysics (X-ray flares <cit.>), geophysics
(earthquakes <cit.>), meteorology (rainfall <cit.>),
neuroscience (neuronal avalanches <cit.>), as well as in other
biological systems (gene knock-out cascades) <cit.>) and
man-made systems (failures on electrical power grids
<cit.>).  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
<cit.>.  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 <cit.>.
 
In this regard, groundbreaking experimental evidence by Beggs and
Plenz <cit.>, revealed the existence of scale-invariant
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
<cit.>.
In particular, neuronal avalanche sizes, S, were robustly observed
to be distributed as a power-law P(S) ∼ S^-τ with
τ≈ 3/2 up to some upper cut-off; similarly, avalanche
durations T were well fitted by P(T)∼ T^-α with
α≈ 2 up to some characteristic maximum time
<cit.>. Furthermore, fundamental scaling relationships
<cit.> were observed to be fulfilled: e.g.  the averaged
avalanche size scales as ⟨ S⟩∼ T^γ and the set
of exponents obey γ=(α-1)/(τ-1) <cit.>.

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)
<cit.>. 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
<cit.>.  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/propagation dynamics above
their respective upper critical dimensions
<cit.>.

Thus, it was conjectured that neuronal systems might operate close to
the edge of marginal propagation of neural (electro-chemical) activity
<cit.>, opening the door to exciting theoretical
perspectives and some debate (see <cit.> for a recent review).
However, as extensively discussed in the literature, diverse
  generative processes for the emergence of power-laws exist
  <cit.>, 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)
  <cit.> or even be unrelated to critical
  behavior <cit.>.

  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 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 timeseries [For instance, since the same
    avalanche could e.g. be split in two by the effect of a higher
    threshold, it would introduce correlations between different
    avalanches, altering the measurement of exponents
    <cit.>.]  <cit.>.





§ 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 <cit.>.  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.



  §.§.§ 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.<ref>– generated by
a Wiener Process, i.e. by a continuous-time unbiased random walk (RW)
defined by the following Langevin equation <cit.>:

    ẋ(t)=ση(t),

where η(t) is a Gaussian white noise with zero mean and unit
variance, and σ is the noise amplitude.  For such a time series
(which can be thought as describing the time course of the activity of
some arbitrary system) the duration T of an avalanche is the amount
of time for which x stays above a given threshold, i.e. an avalanche
begins/ends when the activity signal crosses the threshold from
below/above; the avalanche size S is the area covered between the
walk trajectory and the threshold reference line (see Fig. <ref>).
Observe that similarly, given the symmetry of the process, one could
also define avalanches as excursions below threshold.

The probability distribution of avalanche durations T can be
straightforwardly identified with the first-return time statistics of
random walks (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=3/2 are well-known for
random walks (see Fig. <ref> and Table I); pedagogical 
derivations of these results, as well as a comparison with the
branching process class can be found in
e.g. <cit.>.
Importantly, these results for the random walk 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 <cit.>:

    ẋ(t)=-ax(t)+ση(t),

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. <cit.> for a detailed derivation).  Thus, avalanches
intended as excursions above a given threshold in a process with a
well-defined steady-state value, have power-law-distributed sizes and
durations, with the exponents of the RW class (as in table I and as in
Fig.1), but only up to an upper-cut-off scale controlled by 1/a,
such that it goes to infinity, i.e. it disappears when a vanishes.

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 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
  debated whether this type of behavior –describable in terms of
  random-walk excursions above a threshold– can be called
  “avalanching”. Actually, for most of the examples of interest in
  physics, as discussed in the first paragraph of the Introduction,
  this does not constitute an adequate description as it does not
  include any absorbing state.





  §.§.§ On the definition of avalanche size

In many other circumstances time series exhibit asymmetric 
excursions around a mean value and/or may 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 a so-defined
avalanches <cit.>:

(A) Following the random walk analogy, as done above, for a given
avalanche, one can define its size S as the area in between the
time-series curve and the threshold (θ) reference line.

(B) Alternatively, one can define the avalanche size Σ as the
overall integral of the time series during the avalanche,
i.e. above the reference line x=0 (see
e.g. <cit.> but there are other works making
this choice).





The difference between the two criteria to define avalanche sizes
  is sketched in Figure <ref>.  Σ, the total integral
of the activity is equal to Σ=s^*+S, where S is 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.<ref> using S, i.e. criterion A, one reproduces the
expected theoretical results for all three avalanche exponents. On the
other hand, as illustrated in Figure <ref>, the statistics
of avalanche sizes, as determined employing Σ 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 observation of,
τ≈ 3/2 could be (wrongly) taken as evidence of branching
process-like scaling <cit.>.  The fact that
there is something suspicious with the definition (ii) 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. “tent like”) shape of avalanches <cit.>.




  §.§.§ Scaling arguments


The correction s^* for a given avalanche (such that
Σ= S+ s^*) is nothing but s^*=θ T, where T is 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
  cut-off.  Thus, Σ has a correction s^* with respect to S
that scales as the avalanche duration: P(s^*) ∼s^*^-3/2.
Assuming that the probability to observe a given size S
  conditioned to a given avalanche duration T, 
P(S|T), is a peaked function around its mean value (as
usually occurs for avalanches <cit.>) and using the fact
that ⟨ S ⟩∼ T^γ, with γ=3/2, then

    Σ(T)=S(T)+s^*(T)= c̃ T^3/2+θ T,

from where it follows that

    dΣ=(cT^1/2+θ) dT

Thus, we can readily write (using the implicit function theorem):

    P(Σ(T))=P(T)dT/dΣ= NT^-3/2/cT^1/2+θ.

From this, in the limit of vanishing threshold θ in Eq.(<ref>), one has 

    P(Σ)≈ N/cT^-2≈
        N/c[(Σ/c)^2/3]^-2∼Σ^-4/3,

which is the correct result for the avalanche size distribution of an
Ornstein-Uhlenbeck process. On the other hand, for larger values of
the threshold θ and relatively small values of T (and, thus,
also typically small values of Σ) one has

    P(Σ)≈ N/θT^-3/2≈
        N/θ[Σ/θ]^-3/2∼Σ^-3/2,

in agreement with the numerical observation above. In other words, the
additional contribution s^* dominates the scaling behavior of the
avalanche size Σ when θ is 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
Σ– of the avalanche size exponent (τ=4/3) for any value
of θ but this requires going to larger and larger avalanche
sizes as θ is chosen larger and larger. In particular, Figure
<ref> illustrates 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 cut-off to
scaling may exist (controlled e.g. by 1/a in 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 cut-off, so that the true asymptotic behavior can be
unobservable with criterion B if large values of θ are 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
random-walk-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
miss-classification of scaling behavior in existing works analyzing
neuronal avalanches (see e.g. <cit.>).










§ 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 timeseries 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: (i) as the integrated activity S over a
given threshold and (ii) integrating the total activity signal in
between two threshold crossings, as illustrated in
Fig. <ref>.  We have shown 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 θ are
considered, then –for relatively small avalanches– one observes the
exponent value τ≈ 3/2 which could lead to the erroneous
interpretation that an effective un-biased 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 as 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 <cit.>)
obtain τ≈ 3/2 –compatible with the scaling of a critical
branching process–, a result that our analysis reveal that 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 γ as well as
  the scaling relations between them, in order to avoid possible
  errors and misleading interpretations and properly identify the type
  of scaling behavior.






  We acknowledge the Spanish Ministry and Agencia Estatal de
  investigación (AEI) through grant FIS2017-84256-P (FEDER funds)
  for financial support.  This study has been partially financed by
  the Consejera de Conocimiento, Investigacin y Universidad, Junta
  de Andaluca and European Regional Development Fund (ERDF),
  ref. SOMM17/6105/UGR as well as to TEACH IN PARMA for support to
  M.A. M. We thank V. Buendía for very useful comments and a
  critical reading of the manuscript.








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@bib@innerbibempty

[Altshuler and Johansen(2004)]Altshuler
  author author E. Altshuler and author T. Johansen, @noop  journal journal
  Reviews of Modern Physics volume 76, pages 471 (year 2004)NoStop
[Alessandro et al.(1990)Alessandro, Beatrice, Bertotti, and  Montorsi]ABBM
  author author B. Alessandro, author C. Beatrice, author G. Bertotti,
   and author A. Montorsi,  @noop  journal journal Journal of applied
  physics volume 68, pages 2901
  (year 1990)NoStop
[Cizeau et al.(1997)Cizeau,
  Zapperi, Durin, and Stanley]Zapperi
  author author P. Cizeau, author S. Zapperi,
  author G. Durin,  and author H. E. Stanley, @noop 
  journal journal Physical Review Letters  volume 79, pages 4669 (year
  1997)NoStop
[Wang and Dai(2013)]Wang2013
  author author F. Wang and author Z. Dai,  @noop  journal journal Nat. Phys.  volume 9, pages 465 (year
  2013)NoStop
[Bak et al.(2002)Bak,
  Christensen, Danon, and Scanlon]Bak2002
  author author P. Bak, author K. Christensen,
  author L. Danon,  and author T. Scanlon, @noop 
  journal journal Phys. Rev. Lett. volume 88, pages 178501 (year
  2002)NoStop
[Peters et al.(2001)Peters,
  Hertlein, and Christensen]Peters
  author author O. Peters, author C. Hertlein,   and author K. Christensen,  @noop  journal journal Phys. Rev. Lett.  volume 88, pages 018701 (year 2001)NoStop
[Beggs and Plenz(2003)]BP2003
  author author J. M. Beggs and author D. Plenz,  @noop  journal journal J. Neurosci.  volume 23, pages 11167 (year
  2003)NoStop
[Roli et al.(2015)Roli,
  Villani, Filisetti, and Serra]Serra-review
  author author A. Roli, author M. Villani,
  author A. Filisetti,  and  author R. Serra, @noop 
  journal journal J. Systems Sciences and
  Complexity , pages 1 (year 2015)NoStop
[Kinney et al.(2005)Kinney,
  Crucitti, Albert, and Latora]powergrids
  author author R. Kinney, author P. Crucitti,
  author R. Albert,  and author V. Latora, @noop 
  journal journal The European Physical Journal
  B-Condensed Matter and Complex Systems volume 46,  pages 101 (year 2005)NoStop
[Bak(2013)]Bak
  author author P. Bak, @noop  title How nature works: the
  science of self-organized criticality (publisher Springer
  Science & Business Media, year 2013)NoStop
[Bak et al.(1987)Bak,
  Tang, and Wiesenfeld]BTW
  author author P. Bak, author C. Tang,  and  author K. Wiesenfeld,  @noop  journal journal Physical review
  letters volume 59, pages 381
  (year 1987)NoStop
[Jensen(1998)]Jensen
  author author H. J. Jensen, @noop  title Self-organized
  criticality: emergent complex behavior in physical and biological systems  (publisher Cambridge university press, year
  1998)NoStop
[Rubin et al.(2014)Rubin,
  Pruessner, and Pavliotis]Pruessner
  author author K. J. Rubin, author G. Pruessner,   and author G. A. Pavliotis,  @noop  journal journal J. Phys. A  volume 47, pages 195001 (year 2014)NoStop
[Dickman et al.(2000)Dickman, Muñoz, Vespignani, and  Zapperi]BJP
  author author R. Dickman, author M. A. Muñoz, author A. Vespignani,  and author S. Zapperi, @noop  journal journal Braz.
  J. Phys. volume 30, pages 27
  (year 2000)NoStop
[Mora and Bialek(2011)]Mora-Bialek
  author author T. Mora and author W. Bialek,  @noop  journal journal J. Stat. Phys.  volume 144, pages 268 (year
  2011)NoStop
[Muñoz(2018)]MAM-rmp
  author author M. A. Muñoz, @noop  journal journal
  Reviews of Modern Physics volume 90, pages 031001 (year 2018)NoStop
[Plenz and Thiagarajan(2007)]Plenz2007
  author author D. Plenz and author T. C. Thiagarajan, @noop  journal journal
  Trends Neurosci. volume 30, pages
  101 (year 2007)NoStop
[Beggs(2008)]Beggs2008
  author author J. M. Beggs, @noop  journal journal Philos.
  Trans. A Math. Phys. Eng. Sci. volume 366,  pages 329 (year 2008)NoStop
[Lombardi et al.(2012)Lombardi, Herrmann, Perrone-Capano,
  Plenz, and De
  Arcangelis]Arcangelis2012
  author author F. Lombardi, author H. Herrmann,
  author C. Perrone-Capano,
  author D. Plenz,  and author L. De Arcangelis, @noop 
  journal journal Phys. Rev. Lett. volume 108, pages 228703 (year
  2012)NoStop
[Petermann et al.(2009)Petermann, Thiagarajan, Lebedev,
  Nicolelis, Chialvo, and Plenz]Petermann2009
  author author T. Petermann, author T. C. Thiagarajan, author M. A. Lebedev, author M. A. Nicolelis, author D. R. Chialvo,  and author D. Plenz, @noop  journal journal Proc.
  Natl. Acad. Sci. USA volume 106, pages
  15921 (year 2009)NoStop
[Hahn et al.(2010)Hahn,
  Petermann, Havenith, Yu,
  Singer, Plenz, and Nikolić]Hahn2010
  author author G. Hahn, author T. Petermann,
  author M. N. Havenith, author S. Yu, author
  W. Singer, author D. Plenz,  and author D. Nikolić, @noop  journal
  journal J. Neurophysiol. volume 104,  pages 3312 (year 2010)NoStop
[Palva et al.(2013)Palva,
  Zhigalov, Hirvonen, Korhonen,
  Linkenkaer-Hansen, and Palva]Palva2012
  author author J. M. Palva, author A. Zhigalov,
  author J. Hirvonen, author O. Korhonen, author
  K. Linkenkaer-Hansen,  and author S. Palva, @noop 
  journal journal Proc. Natl. Acad. Sci. USA  volume 110, pages 3585 (year
  2013)NoStop
[Bellay et al.(2015)Bellay,
  Klaus, Seshadri, and Plenz]Plenz2015-pyramidal
  author author T. Bellay, author A. Klaus,
  author S. Seshadri,  and  author D. Plenz, @noop 
  journal journal Elife volume 4, pages e07224 (year
  2015)NoStop
[Muñoz et al.(1999)Muñoz, Dickman, Vespignani, and  Zapperi]Avalanches
  author author M. A. Muñoz, author R. Dickman,
  author A. Vespignani,  and  author S. Zapperi, @noop
   journal journal Phys. Rev. E volume 59, pages 6175 (year
  1999)NoStop
[Friedman et al.(2012)Friedman, Ito, Brinkman, Shimono, DeVille, Dahmen, Beggs, and Butler]Beggs2012
  author author N. Friedman, author S. Ito,
  author B. A. Brinkman, author M. Shimono, author
  R. L. DeVille, author
  K. A. Dahmen, author
  J. M. Beggs,  and author
  T. C. Butler, @noop  journal journal Phys. Rev. Lett. volume 108, pages 208102 (year
  2012)NoStop
[Watson and Galton(1875)]Watson
  author author H. W. Watson and author F. Galton, @noop  journal journal J.
  Roy. Anthropol. Inst. volume 4, pages
  138 (year 1875)NoStop
[Liggett(2004)]Liggett
  author author T. Liggett, @noop  title Interacting Particle
  Systems, Classics in Mathematics (publisher Springer,  address New York, year 2004)NoStop
[Zapperi et al.(1995)Zapperi, Lauritsen, and Stanley]SOBP
  author author S. Zapperi, author K. B. Lauritsen,  and author H. E. Stanley, @noop  journal journal Phys.
  Rev. Lett. volume 75, pages 4071
  (year 1995)NoStop
[Redner(2001)]Redner
  author author S. Redner, @noop  title A guide to
  first-passage processes (publisher Cambridge University
  Press, address Cambridge, year 2001)NoStop
[Karrer et al.(2014)Karrer,
  Newman, and Zdeborová]Newman
  author author B. Karrer, author M. E. Newman,
   and author L. Zdeborová,  @noop  journal journal Physical Review
  Letters volume 113, pages 208702
  (year 2014)NoStop
[Binney et al.(1993)Binney,
  Dowrick, Fisher, and Newman]Binney
  author author J. Binney, author N. Dowrick,
  author A. Fisher,  and author M. Newman, @noop  title The Theory of Critical Phenomena (publisher Oxford University Press, address Oxford,  year 1993)NoStop
[Dobrinevski et al.(2015)Dobrinevski, Le Doussal, and Wiese]Wiese
  author author A. Dobrinevski, author P. Le Doussal,  and author K. J. Wiese, @noop  journal journal EPL
  (Europhysics Letters) volume 108, pages
  66002 (year 2015)NoStop
[Garcia-Millan et al.(2015)Garcia-Millan, Font-Clos, and Corral]Corral
  author author R. Garcia-Millan, author F. Font-Clos,  and author Á. Corral, @noop  journal journal Phys.
  Rev. E volume 91, pages 042122
  (year 2015)NoStop
[Bonachela et al.(2010)Bonachela, De Franciscis, Torres, and  Munoz]JABO2
  author author J. A. Bonachela, author S. De
  Franciscis, author J. J. Torres,  and author M. A. Munoz, @noop  journal journal J.
  Stat. Mech. Theory Exp. volume 2010, pages P02015 (year 2010)NoStop
[Chialvo(2004)]Chialvo2004
  author author D. R. Chialvo, @noop  journal journal
  Physica A volume 340, pages 756
  (year 2004)NoStop
[Beggs and Timme(2012)]Critics
  author author J. Beggs and author N. Timme,  @noop  journal journal Front. Physiol.  volume 3, pages 163 (year
  2012)NoStop
[Newman(2005)]Newman-powerlaws
  author author M. E. J.  Newman, @noop  journal journal Contemp. Phys. volume 46, pages 323 (year 2005)NoStop
[Mitzenmacher(2004)]Mitzenmacher2004
  author author M. Mitzenmacher, @noop  journal journal
  Internet Math. volume 1, pages 226
  (year 2004)NoStop
[Martinello et al.(2017)Martinello, Hidalgo, Maritan, di Santo, Plenz, and Muñoz]neutral-neural
  author author M. Martinello, author J. Hidalgo,
  author A. Maritan, author S. di Santo, author
  D. Plenz,  and author
  M. A. Muñoz, @noop  journal journal Phys. Rev. X volume
  7, pages 041071 (year 2017)NoStop
[di Santo et al.(2018)di Santo, Villegas, Burioni, and  Muñoz]PNAS-LG
  author author S. di Santo, author P. Villegas,
  author R. Burioni,  and  author M. A. Muñoz,  @noop  journal journal Proc. Nat. Acad.
  Sci. volume 115, pages E1356
  (year 2018)NoStop
[di Santo et al.(2016)di Santo, Burioni, Vezzani, and  Muñoz]SOB
  author author S. di Santo, author R. Burioni,
  author A. Vezzani,  and  author M. A. Muñoz,  @noop  journal journal Phys. Rev. Lett.  volume 116, pages 240601 (year 2016)NoStop
[Touboul and Destexhe(2017)]Touboul
  author author J. Touboul and author A. Destexhe, 10.1103/PhysRevE.95.012413 journal journal Phys. Rev. E volume
  95, pages 012413 (year 2017)NoStop
[Priesemann and Shriki(2018)]Viola-Shriki
  author author V. Priesemann and author O. Shriki, @noop  journal journal PLoS
  computational biology volume 14, pages
  e1006081 (year 2018)NoStop
[Note1()]Note1
  note For instance, since the same avalanche could e.g. be split
  in two by the effect of a higher threshold, it would introduce correlations
  between different avalanches, altering the measurement of exponents <cit.>.Stop
[Laurson et al.(2009)Laurson, Illa, and Alava]Perils1
  author author L. Laurson, author X. Illa,   and author M. J. Alava,  @noop  journal journal J. Stat. Mech.
  Theory Exp. volume 2009, pages P01019
  (year 2009)NoStop
[Font-Clos et al.(2015)Font-Clos, Pruessner, Moloney, and  Deluca]Perils2
  author author F. Font-Clos, author G. Pruessner, author N. R. Moloney,  and author A. Deluca, @noop  journal journal New J.
  Phys. volume 17, pages 043066
  (year 2015)NoStop
[di Santo et al.(2017)di Santo, Villegas, Burioni, and  Muñoz]Branching
  author author S. di Santo, author P. Villegas,
  author R. Burioni,  and  author M. Muñoz, @noop
   journal journal Phys. Rev. E volume 95, pages 032115 (year
  2017)NoStop
[Gardiner(2004)]Gardiner
  author author C. W. Gardiner, @noop  title Handbook of
  stochastic methods: for physics, chemistry and the natural sciences; 3rd
  ed., Springer Series in Synergetics (publisher
  Springer, address Berlin, year
  2004)NoStop
[Artime et al.(2018)Artime,
  Khalil, Toral, and San Miguel]Artime
  author author O. Artime, author N. Khalil,
  author R. Toral,  and author M. San Miguel, @noop 
  journal journal Physical Review E volume 98, pages 042143 (year
  2018)NoStop
[Dalla Porta and Copelli(2018)]Dalla2018
  author author L. Dalla Porta and author M. Copelli, @noop  journal journal
  bioRxiv , pages 423921 (year
  2018)NoStop
[Poil et al.(2012)Poil,
  Hardstone, Mansvelder, and Linkenkaer-Hansen]Linken2012
  author author S.-S. Poil, author R. Hardstone,
  author H. D. Mansvelder,  and  author K. Linkenkaer-Hansen,  @noop  journal journal J. Neurosci.  volume 32, pages 9817 (year
  2012)NoStop
[Larremore et al.(2014)Larremore, Shew, Ott, Sorrentino, and Restrepo]Larremore2014
  author author D. B. Larremore, author W. L. Shew,
  author E. Ott, author
  F. Sorrentino,  and author
  J. G. Restrepo, @noop  journal journal Phys. Rev. Lett. volume 112, pages 138103 (year
  2014)NoStop
[Baldassarri et al.(2003)Baldassarri, Colaiori, and Castellano]Colaiori
  author author A. Baldassarri, author F. Colaiori,  and author C. Castellano, @noop  journal journal
  Physical review letters volume 90, pages 060601 (year 2003)NoStop
[Papanikolaou et al.(2011)Papanikolaou, Bohn, Sommer, Durin, Zapperi, and Sethna]shape
  author author S. Papanikolaou, author F. Bohn,
  author R. L. Sommer, author G. Durin, author
  S. Zapperi,  and author
  J. P. Sethna, @noop  journal journal Nature Physics volume
  7, pages 316 (year 2011)NoStop
[Laurson et al.(2013)Laurson, Illa, Santucci, Tallakstad, Måløy, and Alava]Laurson
  author author L. Laurson, author X. Illa,
  author S. Santucci, author K. T. Tallakstad, author K. J. Måløy,  and  author M. J. Alava, @noop
   journal journal Nature communications  volume 4, pages 2927 (year
  2013)NoStop






