# What is critical about criticality: In praise of the correlation function

Henrik Jeldtoft Jensen  Centre for Complexity Science and Department of
Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ, UK
and
Institute of Innovative Research, Tokyo Institute of Technology, 4259,
Nagatsuta-cho, Yokohama 226-8502, Japan
<EMAIL_ADDRESS>
###### Abstract

We present a brief review of power laws and correlation functions as measures
of criticality and the relation between them. By comparing phenomenology from
rain, brain and the forest fire model we discuss the relevant features of
self-organisation to the vicinity about a critical state. We conclude that
organisation to a region of extended correlations and approximate power laws
may be behaviour of interest shared between the three considered systems.

  * 17 June 2021

Keywords: Rain, Brain, Forest Fire, Finite size criticality

##  1 Introduction

The critical state and its vicinity have repeatedly attracted great interest
probably starting from the study of the physics of second order phase
transitions through out the 20th century which culminated in the
renormalisation group theory[1]. The notions such as “Edge of Chaos”[2] and
and “Self-Organised Criticality” (SOC)[3] from the 1980ties coming from
different starting points, dynamical systems theory and high-dimensional non-
equilibrium statistical mechanics, respectively, were inspired by the peculiar
nature of the hyper sensitive critical state and sought mechanisms that would
explain the seemingly omnipresence of this remarkable state.

Although the reference to edge of chaos and criticality sometimes is use
synonymously in a somewhat loose descriptive way[4, 5], both concepts have
precise mathematical definitions and only occur if the control parameters of
the system are tuned to certain values.

A dynamical system is chaotic if it has a positive Lyapunov export[6]. Chaotic
behaviour can occur even when only a few parameters, usually denoted as
degrees of freedom, are involved. Incase the time evolution can be captured by
an iterative map, as for example the population size of a species with non-
over lapping generations, e.g. annual insects, chaos can occur even for a
single degree of freedom, $x_{n}$. The most famous is probably the logistic
map[7] given by the equation

$x_{n+1}=rx_{n}(1-x_{n}),$ (1)

which evolves chaotically for the coefficient $r$ larger than
$r_{\infty}\simeq 3.57$.

In contrast the critical state of equilibrium statistical mechanics can
strictly speaking only occur in the limit of infinitely many degrees of
freedom. The hallmark of the critical state is the long range correlations[1].
The prototypic system supporting a critical state is the Ising model. In its
basic version, the model consist of spin variables $S_{i}\pm 1,i=1,2,...,N$
with $i$ labelling the sites on a $d$ dimensional hypercube of linear
extension $L$, so $N=L^{d}$. The variance coefficient

$C(r_{i,j})=\langle S_{i}S_{j}\rangle-\langle S_{i}\rangle\langle
S_{j}\rangle,$ (2)

where $r_{ij}$ is the distance between site $i$ and site $j$, is taken as a
measure of the interdependence between different locations. Although this
object is denoted the variance in statistics, in the statistical mechanics
literature it has always been denoted the correlations function. We will
therefore stick to this tradition. The dependence on the separation $r=r_{ij}$
is found to be described by the following mathematical form

$C(r)\propto r^{-a}e^{-r/\xi}.$ (3)

Here $\xi$ is called the correlation length and is a finite length scale
except in the critical state. The critical state is reached by tuning the
temperature $T$ to a specific value denoted the critical temperature $T_{c}$.
As $T$ approaches $T_{c}$ the correlation length goes to infinity
$\xi(T)\rightarrow\infty$.

We notice that for $T\neq T_{c}$, the correlation function will decay
algebraically, i.e. as a power law $C(r)\sim r^{-a}$ for length scales short
compared to $\xi$, i.e. $r\ll\xi$ and exponentially, $C(r)\sim e^{-r/\xi}$ for
large separations $r\gg\xi$.

Only exactly in the critical state $T=T_{c}$ will the correlations decay
slowly as a power law $C(r)\sim r^{-a}$ no matter how large the separation $r$
and and since $\xi\rightarrow\infty$ no single characteristic scale
distinguish the spatial and temporal organisation. We say the state is scale
invariant because it looks and behaves in the same manner at all length and
time scales except for a magnification factor. As a consequence spatial
structures are expected to take the form of fractals. And probability
distribution, such as those describing the response to external perturbations,
are expected to follow power laws.

The suggestion of SOC[3, 8, 9] is that in very many systems out of
equilibrium, the dynamics, when slowly driven, will self-organise to a state
critical in the sense just described. The philosophy of the original paper[3]
was that since the critical state is scale invariant, criticality can be
studied by investigating the probability distributions describing the abrupt
releases of pent-up strain as a consequence of gradually added stress. If such
distributions follow power laws, we expect to be dealing with a critical
state, even if we are unable to check directly that the correlation function
in Eq. (2) does indeed depend on separation algebraically. Although this is a
sensible starting point, it may not always be correct.

Distributions of event sizes, we will follow the tradition and use the generic
term avalanches, can follow power laws even if the individual relaxation
events generating the avalanche are entirely independent. The uncorrelated
branching process and the independent random walk are representative examples.

Figure 1: A realisation of a branching process. Each node produce $k$
offspring $k=0,1,2,...$ with the same independent probabilities $p_{k}$. Time
is counted in generations $t$ and the evolution of the tree, or avalanche, is
described by $n(t)$, the number of nodes (see top panel) in generation $t$.
The duration $T$ of the avalanche is from generation $t=0$ up to the last
generation for which $n(t)>0$, so in the figure $T=6$.

Let us briefly recall the situation. An uncorrelated branching process
consists of “nodes” producing offspring. One may schematically think of a node
as one neurone that fires and the offspring as the nodes receiving charges
from the first node. Fig. 1 indicates the process. Every node produces $k$
offspring with the same independent probabilities $p_{k},k=1,2,...$. The size
of the generated tree, $S$, i.e. the total number of nodes produced by the
root node, follows a distribution of the form[10]

$P(S)\sim s^{-3/2}e^{-S/S_{0}(\mu)}\;\mbox{for $S\gg 1$},$ (4)

where the scale $S_{0}$ is determined by the average branching ration

$\mu=\sum_{k=0}^{\infty}kp_{k}.$ (5)

When the branching ration $\mu$ becomes equal to one, the process is said to
become critical. As $\mu=1$ is approach the scale $S_{0}$ diverges according
to $S_{0}\sim(1-\mu)^{-2}$. This means, see Eq. (4), that the avalanche size
distribution for the critical branching process is described by a power law
with exponent of 3/2.

The duration of the avalanche is given by the total number of generations $T$
until no new descendants of the root node are produced, see Fig. 1. When the
branching ration is brought to the critical value $\mu=1$ the distribution of
durations also becomes a power law with $P(T)\sim T^{-2}$

How can uncorrelated events produce scale free power laws. Because the the
variable $n(t)$ denoting the number of nodes in generation number $t$ after
the initiation by the root will be correlated. Though the offspring production
of any node is independent of any other node, the size of generation $t+1$ is
clearly not independent of the size $n(t)$ of the previous generation. If
there are many nodes in generation $t$, i.e. $n(t)$ large, the number of nodes
in generation $t+1$ is more likely also to be large, than if $n(t)$ is small.
But this dependence is not very profound.

The situation for a random walk signal $f(t)$ is similar. If
$f(t+1)=f(t)+\Delta_{t}$ where $\Delta_{t}$ are random numbers drawn
independently from the same distribution, the changes in $f(t+1)-f(t)$ are
independent random numbers but the signal $f(t)$ is nevertheless characterised
by by power laws. For example, consider the distribution of first return times
$T$. The first return time is the time $T$ before $f(t+T)$ for the first time
returns to the value $f(t)$ assumed at time $t$. This distribution follows a
power law $T^{-3/2}$.

After having recapitulated that uncorrelated stochastic processes can produce
power law avalanche distributions, the question is how relevant this is for
the discussion about whether the brain exhibit critical behaviour and more
specifically how relevant, and in what way, SOC is to brain dynamics. Let us
first recall that the exponent 3/2 and 2 indeed have been reported for brain
neuronal activity avalanches observed at very different scales. It is well
known that at the level of individual neurones firing, Beggs and Plenz [11]
found the exponent 3/2 and 2 and the 3/2 exponent was also observed by Chialvo
and collaborators for the distribution of avalanches sizes detected by
fMRI[12].

We should however keep in mind that accurate determination of the exponents is
difficult and alternatives to power law behaviour has been suggested, see e.g.
[13]. Our focus here will be slightly different, namely how approximate power
law behaviour may still be interesting in terms of long range correlations.

Even within the framework of SOC, the accurate values of these exponents and
how they depend on the details of the brain activation has been the focus of
detailed discussion, see e.g. [14] and references therein. Here we just want
to stress that although the individual neuronal avalanches may exhibit
exponents characteristic of an uncorrelated branching process with branching
ratio equal to one, the waiting time distribution between the avalanches show
that the initiation of the avalanches are correlated. It is well know that if
neuronal avalanches were released with constant probability per time unit,
i.e. as a Poisson process, the time between avalanches would be exponentially
distributed. In contrast analysis shows that the waiting time is power law
distributed[15, 14]. These indicators of correlated brain activity are
supplemented by the correlations in both space and time extracted from fMRI
reported in [16] where power law decay as function of spatial separation were
observed together with 1/f temporal power spectra indicating logarithmic decay
of correlations in time. So indeed it is still relevant to understand better
the aspects of the establishment through self-organisation of a correlated
critical state.

Before we in the next section turn to this discussion, we will finish our
brief overview by pointing to another indication of the relevance of the
avalanche dynamics of SOC. In a series of work de Arcangelis and collaborators
have developed SOC inspired models[17], which have been developed to allow
analysis of the importance of inhibition[15, 14], modular structure[18], the
role of synaptic plasticity[19] and even address how avalanche dynamics may be
able to support pattern recognition in a neuronal network[20]. This bulk of
work certainly suggest that even beyond the question about criticality, SOC
inspired modelling and analysis is highly relevant.

##  2 Burst dynamics and correlations

Let us now turn to the question concerning in what sense self-organisation to
criticality occurs and what different systems and models may share in this
respect, which will lead us to suggesting that even if no perfect tuning to a
critical state in the sense of equilibrium statistical mechanics happens when
judged from the scaling of the event distributions, interesting long range
correlations may still play an important role and may also be operationally
accessible.

We will elaborate on the observation put forward by Chialvo and collaborators
that the spatiotemporal statistics of the burst of activities in the brain as
monitored by fMRI[21, 12] exhibits remarkable commonalties with the observed
activity of precipitation in the atmosphere[22]. In turn the statistics of the
brain activity and precipitation both behave very similar[23] to the activity
of the Forest Fire Model (FFM) of SOC[24].

We need to describe the FFM briefly. The dynamics of the model can be thought
of as a schematic representation of spreading across a population of some
activity. The model considers a hyper-cubic lattice of dimension $d$, here we
only discuss results for $d=2$ and use a square lattice of linear size $L$
containing $N=L^{2}$ sites. Each lattice site can be in three states: empty
$E$, contain a tree $T$ or contain a tree on fire $F$. The basic dynamics
consists of the following steps. Choose an empty site and plant a tree with
probability $p$: $E\rightarrow T$. Choose one of the sites occupied by a tree
at random and ignited it, i.e. turn it into a fire site, with probability $f$.
A tree next to a site on fire becomes a fire site in the next time step:
$T\rightarrow F$. A fire site becomes and empty site in the next time step:
$F\rightarrow E$. The model has been studied extensively, for a recent
simulation study see e.g. [25] and references therein.

The FFM is studied in the limit of $f\ll p$, i.e. very rare external ignition
of fires. In this limit the salient aspects of the dynamics consists in the
formation of connected clusters of tree sites. Since external ignition happens
very rarely, a connected cluster is considered to burn down instantaneously on
the time scale of the input of external energy represented by the external
ignition. Simulations have shown that the distribution of sizes of burned-down
clusters follows approximate power laws. But the detailed nature of of
criticality and scale invariance in the model has been found to be unusual[26,
27]. It has for instance been suggested by Grassberger[26] that the FFM model
for large enough values of the ration $\theta=p/f$, and hence systems sizes
big enough to accomodate the huge tree clusters generated, may exhibit the
scaling known from percolation. This may be the case but the analysis of [28]
indicates that this scaling behaviour will not be observed until
$\theta>10^{40}$, which to avoid that the simulations are influenced by non-
wanted effects due to the finite size, will need totally unreachable system
sizes, namely linear size of order $L\sim 10^{40}$[25].

We conclude from this that the true asymptotic (system size going to infinity)
behaviour is beyond reach at present, but hurry to say that the models
behaviour for system sizes that can be studied by simulations appear indeed to
be instructive as a way to gain insights on how SOC is relevant to real
systems.

Let us briefly sketch the situation. Peters and Neelin[22] observed that when
the amount of precipitation is plotted agains the humidity of the atmosphere,
a sharp onset is seen in a narrow range of humidities. Fig. 2 shows a
depiction of the behaviour relevant to precipitation, brain activity and
activity in the FFM model.

Figure 2: Generic sketch of a broad region of onset of behaviour reminiscent
of the criticality familiar from equilibrium phase transitions. The three
curves indicate the following: The solid the order parameter, the long dashed
line the resident time and the dotted line the variance (in equilibrium
systems equivalent to the susceptibility). In a critical equilibrium phase
transition (depicted in the insert), such as the Ising model and magnetic
systems, the order parameter is strictly zero on one side of the transition
for then to become non-zero at a sharp value, the critical value, see
$\rho_{c}$ and the susceptibility diverges at $\rho_{c}$. The residence time
does not have an equivalent in equilibrium systems because the control
parameter $\rho$ is fixed externally. But only one value of $\rho$ is relates
to each equilibrium state.

The figure summarises the behaviour presented in four figures in the
literature. Firstly, Fig. 1 in [22] that plot the amount of precipitation and
its variance as function of the amount of vapour in the atmosphere. The time
the atmosphere is found to spend at a certain vapour level is presented in
Fig. 3 in [22]. This time is called the residence time. Secondly, Fig. 3E in
[12] that shows the normalised size of the largest size of the cluster of
active fMRI voxels (above a certain threshold) as function of the total number
of active voxels, its variance and the residence time. Finally, a similar
analysis was presented in Fig. 1 of [23] that contains the largest tree
cluster, its variance and the residence time plotted against the number of
trees (cnsidered to be the sites able to generate activity).

In all three cases it was observed that the so-called order parameter, i.e the
parameter that distinguish where the system is relative to the critical state,
pickups from very low value over a relatively narrow region of values of the
control parameter. The order parameter for the three considered cases are:
amount of precipitation, size of connected cluster of active voxels and size
of burnt down clusters. The control parameter for the three cases are: amount
of vapour in the atmosphere, number of active voxels and number of sites
occupied by trees. In all three cases approximative power laws for the event
distribution (the size of precipitation events, the size of the largest
cluster of active voxels, the size of the largest burnt down cluster of tress,
respectively) are observed at some value of the control parameter at about the
middle of the region where the order parameter picks up and where the variance
peaks. This means that in all three cases we observe a spread out version of
what would appear as a sharp transition in critical equilibrium phase
transitions, see insert in Fig. 2.

We want to stress two important points. First that these three entirely
different systems share remarkable similarities of how they organise
themselves in a way so they are most frequently found in a region exhibiting
aspects of an underlying nearby critical transition. The other point we want
to emphasise is that despite both the brain and the atmosphere being very
large systems, they share behaviour with the finite size properties of the
FFM. This suggests that in contrast to equilibrium critical phenomena, for
which we know it is crucial to understand the asymptotic behaviour in the
limit of diverging system size, in the case of manifestations of the avalanche
dynamics associated with SOC, the finite system size behaviour is of
particular interest. This implies on the other hand that rather than self-
organisation to one sharply defined critical point, the self-organisation of
relevance to applications consists of a tendency for the dynamics to bring
itself to a certain region of near critical behaviour.

The question then remains what makes this region near some kind of criticality
interesting. We therefore return to correlations and recall that the power
laws and diverging sensitivity (susceptibility) of the equilibrium critical
state all are a consequence of long range algebraic correlations. Is there any
evidence that the region, where our three systems exhibit a peak in the
residence time, is associated with correlations behaving as described by Eq.
(3) with a correlation length $\xi$ large enough to make the slow power law
decay $r^{-a}$ able to essentially determine the behaviour.

We do not know yet the answer for the atmosphere. The fMRI study of the BOLD
signal correlations in [16] suggests this is the case for the brain and recent
studies of the FFM confirm that despite of the puzzling scaling observed for
the event distributions, spatial correlations are long range[28]. In this
approach the correlation length is considered to be a stochastic variable. For
each instantaneous configuration of the considered system, the correlation
function in Eq. (2) is fitted to the functional form in Eq. (3) in order to
determine the correlation length $\xi$ describing this specific configuration.
This approach differs from the usual way the correlation length is determined
in studies of equilibrium system. Simulations usually first average the
correlation function in Eq. (2) over all the generated configurations and then
afterwards extract the correlation length from a fit to this averaged
function. The new method, which focus on the correlation length as a
characteristic of each individual configuration, was shown in [28] to
reproduce the known equilibrium behaviour for the two prototype models: the
Ising model and the 2d-XY model. And more interesting, the method produces new
understanding of correlations in the FFM.

For the FFM model, the variable $S_{i}$ is taken to be $S_{i}=1$ for tree
sites and $S_{i}=0$ for empty sites. Fire sites are ignored since fires remove
connected clusters in a single step on the time scale given by $p$ the tree
growth probability. As growth and fire dynamics is simulated, the
distribution, $P(\xi)$, of the correlation lengths for instantaneous
configurations is sampled. Simulations found that for system sizes up to
$L=3000$, the distribution $P(\xi)$ is well described by finite system size
scaling meaning that $P(\xi)$ for different values of $L$ can be collapsed to
one “universal” function when the activity per site given by $\theta/L^{2}$ is
kept fixed as $L$ is varied. The dependence on system size indicates that the
average correlation length extracted from the distribution $P(\xi)$ depends on
system size as $\langle\xi\rangle\propto L^{1.1}$. This suggests that as $L$
increases correlations will become of significant long range when the drive
given by the fire probability $f$ is well separated from the growth
probability $p$, i.e. when the external drive is slow.

##  3 Summary and Discussion

We have notice that the exact tuning to a critical point first anticipated by
SOC seems not to happen in rain, brain and the Forest Fire Model. This does
not leave the framework of SOC irrelevant. On the contrary self-organisation
does occur though to a region near the rise of an order parameter. Moreover,
studies of the FFM model suggest that although power laws of event
distributions may only be approximate the region where these three systems
spend most time is nevertheless associated with very extended spatial
correlations.

The similar behaviour shared between so different systems indicates that the
basic ingredients of dynamics driven by load, spreading and relaxation will
tend to organise towards configurations poised near some kind of onset of
(correlated) percolation. The buildup of spatially extended structures, which
then abruptly collapse through release of precipitation, neuronal firing or
fires respectively in the three cases discussed, keeps turning “over critical”
configuration into under “under critical” ones.

Simulations of the FFM show that for parameter regimes manageable the region
of preferentially visited configurations, i.e. the region around the peak in
the residence time distribution, broadens as the separation between the two
time scales given by loading (tree growth) and triggering of release of energy
(externally induced fires), i.e. $\theta=p/f$ increases. The width of the peak
of the residence time in the FFM behaves approximately like
$\sigma\sim\theta^{0.1}$, see [25]. So driving the model towards more extended
power laws and longer correlation lengths also broadens the range of visits to
both under and over critical configurations.

The conclusion is that the load and release dynamics does not manage to fine
tune to a critical point but to a region of extended correlations.

This has been suggested before, see for example [29, 30, 13], but here we
emphasise that though strict criticality is not achieved correlations are
sufficiently extensive to make load and relaxation dynamics in the three
completely different systems rain, brain and FFM share remarkable
similarities.

It is often suggested that the reason the brain operates at or near a critical
point is the hypersensitivity of this state, indicated by the divergence of
the susceptibility. This seems certainly very reasonable, but there may even
be reasons why the brain does not sit exactly in a critical state. As
suggested by Lizier and collaborators[31] operating in the region across the
critical point may have a computational advantages for the brain in terms of
combining high data storage capability in the sub-critical region with
increased information transfer in the super-critical region. So after all,
lack of exact tuning to criticality may in the end be seen as a bonus.

##  4 Acknowledgments

It is a great pleasure to acknowledge stimulating interaction spanning many
years with Dante Chialvo and Lucilla de Arcangelis and the recent very
fruitful collaboration with Lorenzo Palmieri. I am grateful to Hardik Rajpal
for pointing me to the work by Lizier and collaborators.

##  5 References

## References

  * [1] Nigel Goldenfeld.  Lectures on Phase Transitions and the Renormalization Group.  CRRC Press, Taylor & Francis, 1992.
  * [2] Norman H Packard.  Adaptation Toward the Edge of Chaos.  University of Illinois at Urbana-Champaign, Center for Complex Systems Research, 1988.
  * [3] Per Bak, Chao Tang, and Kurt Wiesenfeld.  Self-Organized Criticality: An Explanation of 1/f Noise.  Phys. Rev. Lett., 59(4):381–384, 1987.
  * [4] Collin Barras.  Mind Maths; Your brain teeters on the edge of chaos.  NewScientist - https://institutions.newscientist.com/article/mg21729032-100-mind-maths-your-brain-teeters-on-the-edge-of-chaos/, (2903), 2013.
  * [5] Washington University in St.  Brain tunes itself to criticality, maximizing information processing: Criticality is a hallmark of normally functioning neural networks in the intact brain.  ScienceDaily - www.sciencedaily.com/releases/2019/10/191007113314.htm, 2019.
  * [6] Edward Ott.  Ott, E. Chaos in Dynamical Systems (Cambridge Univ. Press, 1993). Cambridge University Press, 1993.
  * [7] R. M May.  Simple Mathematical Models with Very Complicated Dynamics.  Nature, 261:459, 1976.
  * [8] Henrik Jeldtoft Jensen.  Self-Organized Criticality. Emergent Complex Behvavior in Physical and Biological Systems.  Cambridge University Press, 1998.
  * [9] Gunnar Pruessner.  Self-Organsed Criticality. Theory, Models and Charaterisaion.  Cambridge University Press, 2012.
  * [10] T.E. Harris.  The Theory of Branching Processes.  Springer, 1963.
  * [11] John M. Beggs and Diermar Plenz.  Neuronal Avalanches in Neocortical Circuits.  J. Neuroscience, 23(35):11167–1117, 2003.
  * [12] Enzo Tagliazucchi, Pablo Balenzuela, Daniel Fraiman, and Dante R. Chialvo.  Criticality in large-scale brain fMRI dynamics unveiled by a novel point process analysis.  frontiers in Physiology, 3:15, 2012.
  * [13] J. Wilting and V. Priesemann.  25 Years of Criticality in Neuroscience — Established Results, Open Controversies, Novel Concepts.  Current Opinion in Neurobiology, 58:105–111, 2019.
  * [14] F. Lombardi, H. J. Herrmann, and L. De Arcangelis.  Balance of excitation and inhibition determines 1/f power spectrum in neuronal networks.  Chaos, 047402:1–7, 2017.
  * [15] F. Lombardi, H. J. Herrmann, C. Perrone-Capano, D. Plenz, and L. De Arcangelis.  Balance between excitation and inhibition controls the temporal organization of neuronal avalanches.  Physical Review Letters, 108(22):1–5, 2012.
  * [16] Paul Expert, Renaud Lambiotte, Dante R. Chialvo, Kim Christensen, Henrik Jeldtoft Jensen, David J. Sharp, and Federico Turkheimer.  Self-similar correlation function in brain resting-state functional magnetic resonance imaging.  Journal of the Royal Society Interface, 8(57):472–479, 2011.
  * [17] Lucilla De Arcangelis, Carla Perrone-Capano, and Hans J. Herrmann.  Self-organized criticality model for brain plasticity.  Physical Review Letters, 96(2):1–4, 2006.
  * [18] R. Russo, H. J. Herrmann, and L. De Arcangelis.  Brain modularity controls the critical behavior of spontaneous activity.  Scientific Reports, 4:1–5, 2014.
  * [19] L. Michiels Van Kessenich, L. De Arcangelis, and H. J. Herrmann.  Synaptic plasticity and neuronal refractory time cause scaling behaviour of neuronal avalanches.  Scientific Reports, 6(August):1–7, 2016.
  * [20] L. Michiels Van Kessenich, D. Berger, L. De Arcangelis, and H. J. Herrmann.  Pattern recognition with neuronal avalanche dynamics.  Physical Review E, 99(1):1–6, 2019.
  * [21] Dante R. Chialvo.  Emergent complex neural dynamics.  Nature Physics, 6(10):744–750, 2010.
  * [22] Ole Peters and J. David Neelin.  Critical phenomena in atmospheric precipitation.  Nature Physics, 2(6):393–396, 2006.
  * [23] L. Palmieri and H. J. Jensen.  The emergence of weak criticality in SOC systems.  Epl, 123(2), 2018.
  * [24] B. Drossel and F. Schwabl.  Self-organized critical forest-fire model.  Physical Review Letters, 69(11):1629–1632, 1992.
  * [25] Lorenzo Palmieri and Henrik Jeldtoft Jensen.  The Forest Fire Model: The Subtleties of Criticality and Scale Invariance.  Frontiers in Physics, 8(September), 2020.
  * [26] Peter Grassberger.  Critical behaviour of the Drossel-Schwabl forest fire model.  New Journal of Physics, 4, 2002.
  * [27] Gunnar Pruessner and Henrik Jeldtoft Jensen.  Broken scaling in the forest-fire model.  Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 65(5):8, 2002.
  * [28] Lorenzo Palmieri and H.J. Jensen.  Investigating critical systems via the distribution of correlation lengths.  Phys Rev Res., 013199, 2020.
  * [29] Juan A. Bonachela and Miguel A. Muñoz.  Self-organization without conservation: True or just apparent scale-invariance?  Journal of Statistical Mechanics: Theory and Experiment, 2009(9), 2009.
  * [30] Juan A. Bonachela, Sebastiano De Franciscis, Joaquín J. Torres, and Miguel A. Mũoz.  Self-organization without conservation: Are neuronal avalanches generically critical?  Journal of Statistical Mechanics: Theory and Experiment, 2010(2), 2010.
  * [31] Mike Li, Yinuo Han, Matthew J. Aburn, Michael Breakspear, Russell A. Poldrack, James M. Shine, and Joseph T. Lizier.  Transitions in information processing dynamics at the whole-brain network level are driven by alterations in neural gain.  PLoS computational biology, 15(10):e1006957, 2019.