###### Abstract

Collective behaviors exhibited by animal groups, such as fish schools, bird
flocks, or insect swarms are fascinating examples of self-organization in
biology. Concepts and methods from statistical physics have been used to argue
theoretically about the potential consequences of collective effects in such
living systems. In particular, it has been proposed that such collective
systems should operate close to a phase transition, specifically a
(pseudo-)critical point, in order to optimize their capability for collective
computation. In this chapter, we will first review relevant phase transitions
exhibited by animal collectives, pointing out the difficulties of applying
concepts from statistical physics to biological systems. Then we will discuss
the current state of research on the “criticality hypothesis”, including
methods for how to measure distance from criticality and specific functional
consequences for animal groups operating near a phase transition. We will
highlight the emerging view that de-emphasizes the optimality of being exactly
at a critical point and instead explores the potential benefits of living
systems being able to tune to an optimal distance from criticality. We will
close by laying out future challenges for studying collective behavior at the
interface of physics and biology.

##  Chapter 0  Phase Transitions and Criticality in the Collective Behavior of
Animals - Self-organization and biological function

###### Contents

  1. 0 Phase Transitions and Criticality in Collective Behavior
    1. 1 Introduction
    2. 2  Phase transitions in collective behavior
      1. 1 The applicability of phase transition concepts to biology
      2. 2 Flocking
      3. 3 Collective decision making
      4. 4 Behavioral contagion
      5. 5 Synchronization
    3. 3  Quantifying criticality in biological collectives
    4. 4  Consequences of phase transition phenomena on biological function
    5. 5  Mechanisms of self-organization towards criticality
    6. 6  Future challenges & concluding remarks

###  1 Introduction

Collective behavior exhibited by large animal aggregations such as swarms of
insects, schools of fish, and flocks of birds are ubiquitous and fascinating
examples of biological self-organization. Over the past decades it has
attracted scientists from a wide range of scientific disciplines far beyond
biology. Theoretical physicists in particular investigate the analogy between
large animal collectives and systems studied in statistical physics, such as
fluids or magnets, where local interactions between a large number of rather
simple components lead to emergence of novel properties at the macroscopic
level, which can be difficult to trace back to properties of the individual
components[1, 2, 3].

In the past century, since the introduction of the famous Ising model[4, 5,
6]. statistical physics has developed a powerful methodology to study self-
organization in complex systems and the emergence of order, through the theory
of phase transitions and critical phenomena[7, 8]. Whereas most of the
research was devoted to understanding systems at thermal equilibrium, where an
encompassing body of theory confirmed by experiments has been firmly
established, we have also witnessed decades of research into self-organization
and pattern formation in non-equilibrium systems [8, 9], relevant to
biological systems in general, and collective animal behavior in particular.
Whereas many questions regarding non-equilibrium phase transitions remain
open, it is beyond doubt that physics provides a diverse and powerful tool set
that can further our understanding of the complexity of the living world.

However, a question which physics is not well suited to address in biology is
the “Why?”, the question on the ultimate causes and the biological function of
a particular self-organized behavior. This issue must be viewed and addressed
through the prism of evolutionary theory and behavioral biology, and it
demonstrates the need for a truly interdisciplinary exchange between physics
and biology in understanding complex living systems.

An intriguing property of living systems that distinguishes them from most (or
all) inanimate matter is their ability to react adaptively to changing
environments. In general, this capacity for enacting functional adaptive
behavior relies on distributed processing of information at various levels,
characterized by the collective dynamics of a large number of interacting
components or agents constituting the complex biological system. Examples
range in scale from regulatory interactions among proteins in cells[10] to
neurons interacting in brains[11, 12] to the group behavior of animals and
humans [13].

The parameter space in which such collective biological systems operate is
vast, both due to the complexity of single individuals and to the large number
of individuals that can constitute a functional group. The important question
emerges whether particular parameter combinations, or parameter regions, are
particularly suited for their biological function by rendering their
collective behavior in some sense near optimal. In this context, the so-called
_criticality hypothesis_ has been proposed. It suggests that complex living
systems processing information in a distributed way should operate at or close
to a critical point that separates qualitatively different aggregate
behavior.111 To be more precise, a finite-size biological system would be
hypothesized to be close to pseudo-critical point: strictly speaking phase
transitions and criticality are only properly defined in the thermodynamic
limit, in which the number of components $N\rightarrow\infty$. At such
critical points, statistical physics predicts maximal susceptibility, i.e.
maximal sensitivity, of the collective dynamics at the macroscopic level to
small differences in an external input, as well as fast transmission of
information across arbitrarily large scales[14].

One challenge for the criticality hypothesis lies in explaining how
distributed systems are able to regulate their behavior to stay in the
critical region of parameter space. Research on self-organized criticality, a
concept introduced by Bak et al in [15, 16], suggests one set of mechanisms
for how complex systems might tune towards critical points in a self-organized
way without external control, which assumes a time-scale separation between
the fast (relaxation) dynamics of the system and a slow driving of the system
towards the critical point. In this context, it was also suggested that (self-
organized) critical dynamics can explain the overabundance of power-law
distributions in empirical data[15, 17]. Self-organized criticality
experienced significant scientific interest in the 1990s, with the
corresponding research focusing mostly on idealized mathematical models[18,
19]. Nevertheless, the concept remained controversial due to many questions
remaining open and, more importantly, its unclear relevance to real-world
phenomena.

After moving out of the scientific focus towards the end of the 20th century,
the criticality hypothesis received renewed attention largely based on new
experimental observations in the field of neuroscience, following the report
of critical neuronal avalanches by Beggs & Plenz in 2003 [20]. Since then, a
broad range of theoretical and experimental work has been published on the
criticality hypothesis in different biological systems, including neuronal
systems[11, 21, 22, 23], gene-regulatory networks[24, 25], and collective
dynamics of animal groups[26, 27, 28, 29]. This more recent research is
characterized by 1) being more directly connected to experimental
observations, 2) suggesting empirically motivated mechanisms for self-tuning
towards the critical point, and 3) questioning the sole focus on benefits of
criticality, instead emphasizing the importance of adaptively managing
competing trade-offs, potentially by actively tuning the distance to critical
transitions depending on environmental context.

A side note on _self-organization_ : Throughout this chapter we will use this
term when referring to the behavior of animal groups. It is a term widely used
in the context of complex systems, yet its exact meaning and relevance in the
context of biological systems is far from obvious. In physics, self-
organization typically refers to spatio-temporal structure formation in
systems consisting of rather simple, homogeneous components such as atoms or
molecules. Yet animal groups are composed of individuals with a complex
behavioral repertoire, and may exhibit high levels of heterogeneity and
hierarchical organization. Interestingly, even clonal fish exhibit significant
behavioral individuality despite near-identical rearing conditions[30]. If we
consider an extreme example of a single dominant individual — a “leader” —
persistently determining the behavior of an entire group, one would not view
this as self-organized collective behavior. However, this extreme hypothetical
case is rarely encountered in nature. Even for strongly hierarchical societies
of eusocial insects, such as bees[31] or ants[32, 33], the colony level
behavior is far from being determined by the queen alone or a particular caste
within the colony. Furthermore, it has been shown that despite strong rank
hierarchies in baboon troops, the actual movement initiation decisions are
very much driven by rank-independent consensus mechanisms[34]. Here, we will
use the term _self-organization_ in a general sense to refer to the emergence
of macroscopic (group-wide) collective behaviors from interactions between
individuals, which lead to formation of spatio-temporal dynamics and patterns
that cannot be attributed solely to the behavior of isolated individuals or to
environmental factors.

In this chapter, we will refrain from a general discussion of self-organized
criticality in biology, but instead focus specifically on the role of
criticality in the collective behavior of animals, which has received
comparatively little attention. However, when useful we will touch upon
selected results in other fields, referring to results in neuroscience
relevant to distributed information processing, and comparing animal groups to
other biological systems with respect to the criticality hypothesis. We refer
readers interested in the broader perspective to some excellent recent review
articles on the criticality hypothesis [14] in particular in the field of
neuroscience [11, 21, 23].

In the following, we will first discuss in section 2 the applicability of the
theory of phase transitions to the collective behavior of animals, and review
the most relevant types of phase transitions in this context. We will continue
in with a discussion of methods for quantifying criticality in biological
collectives (section 3) and ways that criticality can impact biological
function (section 4), before turning our attention in section 5 to potential
mechanisms that may enable animal collectives to tune themselves in a self-
organized manner towards or away from critical points. Finally, we will close
this chapter with a section on open questions and future challenges regarding
the criticality hypothesis in animal groups.

###  2  Phase transitions in collective behavior

####  1 The applicability of phase transition concepts to biology

Biological systems, and animal groups in particular, share some basic
properties with classical statistical physics systems in which the concept of
a phase transition was first developed. Yet there are also important
differences, and thus the application of the phase transition concept to
collective animal behavior needs to be critically assessed.

First, a crucial difference of biological systems in comparison to systems
classically studied in statistical physics is their far-from-equilibrium
nature. The theory of phase transitions under non-equilibrium conditions is a
very active field of research of modern statistical physics [35, 9, 36]. While
many questions remain open, there is no fundamental reason to believe that the
corresponding theoretical concepts do not extend to living systems, including
animal groups, which is further supported by a large body of literature on
phase transitions in biology over the past decades [37, 38, 39, 40].

Second, an important difference is the sheer size of the systems in terms of
the number of constituting units. Whereas a macroscopic volume of matter
typically contains $N\sim 10^{23}$ particles, a typical animal aggregation
consists of $N\sim 10-10^{3}$ individuals. In rare cases, significantly larger
collectives with $\sim 10^{6}$ individuals are observed, typically in the
context of large migration movements in animals such as pelagic fish (e.g.
sardines) [41] or desert locusts [38]. However, phase transitions in
statistical physics are rigorously defined only in the thermodynamic limit of
extremely large, or more precisely infinite, systems. The mathematical
abstraction of $N\to\infty$, while a very good approximation for the bulk
behavior of “classical” physical matter, is certainly questionable for
describing most biological systems and animal groups in particular. This
critique, while correct, does not refute the significance of phase transitions
and criticality in the description of finite-sized biological systems. The
conjectured benefits of (quasi-)criticality for collective animal behavior do
not rely on the assumption of the thermodynamic limit. Various aspects of
collective information processing, such as correlation lengths, information
transmission, and susceptibility to inputs, still become maximal at quasi-
critical points in finite-sized systems [12, 27, 14, 29].

However, the theoretical concept of universality is intrinsically linked to
the concept of thermodynamic limit through the application of renormalization
methods. Universality predicts the emergence of scaling laws and critical
exponents that depend only on fundamental properties of the system such as
dimensionality and symmetry, and not on microscopic details. This simplified
scaling behavior may only be observed above a critical system size.
Unfortunately, due to the complexity of social behavior of animals, there are
massive gaps in our knowledge on the nature of social interactions and
fluctuations. Therefore even rough estimates of corresponding critical system
sizes, above which universal behavior in the statistical physics sense could
be observed, appear currently impossible to be established. Thus, it is
imperative to be extremely cautious when interpreting empirically observed
scaling laws in small to mesoscale animal groups in the context of
universality classes.

For these reasons, it is our opinion that categorizing biological phase
transitions into distinct universality classes is less practical than in
classic systems from physics, and may indeed be impossible. Still, the concept
that a single relevant combination of parameters dominates collective behavior
near a transition, and that this might allow for immense simplification of
effective models of particular systems, remains viable.

Due to the relatively small system sizes, boundary conditions must be expected
to play a non-negligible — if not a dominating — role for self-organized
collective behavior[42]. For many statistical physics theories, this would be
considered as rather problematic. However, from a biological point of view
this is likely an important or even a defining feature of animal collectives
[43]. If we consider distributed sensing of environmental cues and collective
processing of this information as the core function of animal aggregates, for
example in the context of predator detection or food search, then it becomes
obvious that boundaries are of fundamental importance; e.g. in visual
perception, most of the environmental information will be perceived by
individuals at the edge of the group.

####  2 Flocking

In the context of animal groups, arguably the most easily observed phase
transition is the emergence of orientational order due to spontaneous symmetry
breaking [38]. This flocking transition separates a disordered state, with
individuals moving in random directions with a vanishing center of mass speed,
from an ordered flocking state, with a non-vanishing average momentum of the
entire system. The phenomenon of flocking is strikingly visible in groups of
birds such as starlings [44].

The first theoretical treatment of flocking as a phase transition was reported
by Vicsek and co-authors in 1995 in their seminal work on self-propelled
particles moving with constant speed $v_{0}$ and interacting with a
ferromagnetic (or polar) alignment interaction[37]. It immediately received a
lot of attention as the reported emergence of long-range orientational order
in this non-equilibrium extension of the classical XY-model appeared to
violate the Mermin–Wagner theorem222In equilibrium systems, the Mermin–Wagner
theorem predicts that no long-range orientational order is possible in two
dimensions, and the original Vicsek model was formulated in 2D. [45, 46]. Very
soon after, Toner and Tu were able to show that the non-equilibrium nature of
the model for a non-zero self-propulsion speed makes the decisive difference
[47, 48, 49] why the Mermin–Wagner theorem does not apply. Following these
initial publications, the nature of the transition in the Vicsek model[50, 51,
52, 53], and more generally in models of self-propelled particles with
spatially local alignment interactions, both polar and nematic, has been
intensely researched [54, 55, 56, 57, 58, 59]. Whereas originally the
transition was believed to be continuous[37, 52], later systematic numerical
simulations as well as theoretical analyses have shown that the homogeneous
ordered state is unstable in the vicinity of the critical point with respect
to longitudinal density modulations [50, 60, 61, 62, 53]. This leads to
formation of large-scale, high-density bands moving through a disordered, low
density gas-like “background”. The emergence of these spatial heterogeneities
eventually changes the nature of the phase transitions to a discontinuous one.
However, the discontinuous nature of the transition is often masked by strong
finite size effects and can only be reliably observed at very large system
sizes and/or high self-propulsion speeds [63]. The fundamental mechanism
behind the density instability is the presence of a density-order coupling,
where, on average, regions with higher density are also more ordered [64].

There exists a large variety of flocking models of self-propelled particles,
some even lacking explicit alignment interactions[39, 65, 66, 67]. However, as
long as the interactions are short-ranged and result in effective alignment
while the system exhibits fluid-like lack of positional order, the transition
towards an ordered state will resemble the one observed in Vicsek model,
including the above mentioned density-order coupling, and can be assumed to
fall into the same universality class, if it can be defined. There exists also
a large class of systems which consider self-propelled particle systems with
attractive and repulsive forces exhibiting potentially different types of
flocking transitions[50, 68, 69].

Based on a detailed analysis of experimental data obtained from 3D tracking of
starling flocks, it has been suggested that interactions between pairs of
individuals are governed by topological distance rather than the metric
distance[44]. Such interactions in corresponding topological flocking models
can be also long-ranged, as a focal individual pays attention to others if
they are within a set of nearest neighbors, independently of their Euclidean
distance[70, 71]. This metric-free nature of the interaction was assumed to
disable the density-order coupling, thus eliminating the density instability
and resulting in a continuous flocking transition[70]. However, recent
research into flocking models with distance-independent $k$-nearest-neighbor
interactions has shown the formation of bands due to a weak yet non-vanishing
density-order coupling [72, 71], which can be further enhanced by the presence
of spatial heterogeneities[71].

Most flocking models assume movement of individuals with a constant speed.
This simplifying assumption not only reduces the model complexity but offers a
direct analogy to the fixed spin amplitude in closely related statistical
physics models like Ising, Potts or XY models. However, animals moving in
groups typically exhibit variable speed that can be modulated by social
interaction, and thus has to be considered as an additional degree of freedom
[73, 74, 75]. Scale-free speed correlations observed in bird flocks can only
be explained by variable speed models at criticality[76]. It has been also
shown that variable speed may dramatically alter and expand the self-organized
collective behaviors and lead to new types of order-disorder transitions [73,
74].

Last but not least, recently, based on empirical observations of highly
polarized, collective turning behavior in bird flocks, so-called inertial spin
models featuring non-dissipative couplings have been proposed[77, 78]. In
contrast to the dissipative Vicsek model, the inertial spin model has been
reported to correctly reproduce the dynamical correlations of velocities and
non-exponential relaxation dynamics [79].

####  3 Collective decision making

Collective decision making is another prominent example of collective animal
behavior, where symmetry-breaking phase transitions play an important role.
Examples include social insects choosing a new nest site[80], fish schools or
baboon troops choosing where to forage [81, 34], and social groups coming to
consensus about power hierarchies[82, 83].

Collective decision making in biology can take place across a variety of
spatial and temporal scales, but the fundamental decision dynamics can often
be well captured by simple, non-spatial models[84, 85]. Most collective
decision models consider binary decision tasks, which can be modelled by
Ising-type models [86, 87, 88], while multi-choice decisions can in principle
be described with a Potts model [89]. However, the majority of collective
decision making models considers some sort of threshold or quorum
interactions, where an agent instantaneously updates its decision state based
on the absolute number, or relative ratio, of its neighbors already committed
to a certain choice. Thus instead of formulating the model in terms of a
Hamiltonian as a starting point, these models are typically formulated in
terms of behavioral algorithms in discrete time. An encompassing review of a
variety of such models is given in [90]. Another complementary approach of
modelling collective decisions is rooted in dynamical and stochastic system
theory and involves formulation of (stochastic) differential equations or the
evolution of continuous decision variables of interacting agents[91, 92].

In general, a collective decision is defined as the commitment of a majority
of a group to a single option, starting from an initially undecided collective
state. Thus it directly corresponds to a breaking of symmetry. A major
difference from flocking literature is that models of collective decision
making typically assume either well-mixed or all-to-all interaction between
agents[84, 85], or a static interaction network ranging from lattices [86, 87]
to scale-free[93] and small-world networks [94]. Notable exceptions are works
combining explicit spatial dynamics, such as flocking, with collective
decision making [95, 81], which allow for study of the interplay of spatial
self-organization of the group structure and the decision dynamics. Here, a
fascinating example of spatially-explicit collective behavior is the
cooperative cargo transport by ants [96, 40].

Interestingly, despite the obvious analogy to phase transitions, the
discussion of criticality in the context of collective decision making has
received little attention (see e.g.[96, 97] ). This may be related to the
strong interdisciplinary nature of the field, as well as the rather small
group sizes studied. Furthermore, modeling of collective decision making is
often rooted in the theory of dynamical systems and stochastic processes, with
transitions between states often discussed in the context of bifurcation
theory. While there are important conceptual differences between bifurcations
and phase transitions[98]333The main difference is that phase transitions rely
on a thermodynamic limit in which the number of components goes to infinity,
while bifurcation theories rely on a steady-state limit in which time goes to
infinity., the functional benefits of criticality, such as maximum sensitivity
to external perturbations, are also a feature of bifurcations in dynamical
systems[99]. Finally, rather than arguing for a single optimal point in
parameter space as in the criticality hypothesis, the collective decision
making literature focuses on functional trade-offs, such as the ubiquitous
speed-accuracy trade-off. Interestingly, speed and accuracy of collective
decisions are strongly modulated by the distance from a critical point, as we
have recently shown by investigating collective decision making on networks
[97].

####  4 Behavioral contagion

Another commonly observed pattern in collective behavior is the spread of
behavior or information through a group, resulting in behavioral cascades that
can quickly encompass the entire collective [100, 101, 29]. Examples include
startles spreading through a fish school [102] and conflict spreading through
a macaque society [103].

In contrast to flocking and collective decision making, this type of
collective dynamics typically does not correspond to a spontaneous symmetry-
breaking transition but rather to percolation [104], and is directly related
to the non-equilibrium phase transition studied in epidemic spreading.
Depending on a control parameter, e.g. coupling strength, a behavior initially
adopted by a few individuals may stay localized to a small neighborhood, or
spread quickly in an avalanche-like manner through the entire group, similar
to a pathogen. One distinguishes two fundamental types of spreading processes:
simple and complex contagions. In a simple contagion, the “infection” of a
focal individual results from a superposition of independent pair-wise
contacts with active or infected neighbors. In contrast, in complex contagion
the infection probability depends the state of the entire neighborhood, i.e.
on higher-order interactions, and cannot be decomposed into binary
interactions. Disease spreading represents a classic example of simple
contagion, whereas spreading of behavior is in general assumed to be a complex
contagion processes [105, 106]. In particular, various threshold models
proposed for social contagion represent examples of complex contagions [105,
107].

In the well-mixed case, simple contagion processes exhibit a continuous phase
transition between a phase of vanishing infection load to a phase with a
finite fraction of the system being infected. Complex contagion, on the other
hand, may exhibit different types of transitions depending on model details,
even at the level of mean field analysis [104, 100, 101, 108]. In 2004, Dodds
& Watts proposed a generalized model of contagion [100, 109]. The model is
formulated in discrete time and the coupling between individuals is determined
by the probability $p$ of sending an activation signal ( “infection dose”) to
a neighbor, which by definition is bounded between $0$ and $1$. Depending on
the model parameters, in particular the distribution of thresholds, Dodds &
Watts distinguished three fundamental types of transitions: (I) a continuous
onset of an infected state above a critical interaction probability $p>p_{c}$
for dose-response behavior resembling simple contagion, (II) a discontinuous
one with a bounded coexistence region, and (III) a discontinuous one with a
coexistence region that extends over the entire possible infection probability
range. We note that, in order to re-formulate the model in continuous time,
one has to replace the probability $p$ by a probability rate $\rho$, which is
unbounded, as the probability of infection interaction over a short time
interval $dt$ is given by $p_{dt}=\rho dt$ [102]. Thus, it is unclear whether
the two discontinuous transition types (II and III) are indeed fundamentally
different or just a consequence of the model formulation.

Recently, Böttcher et al[101] have analyzed analytically the mean-field
behavior of a related contagion model formulated in continuous time and
confirmed the existence of both continuous and discontinuous transitions.

####  5 Synchronization

When individuals in a group engage in periodic behaviors, interactions can
cause their behavior to become synchronized. One human example is the
synchronization of clapping in an audience, and in biology synchronization is
observed in the collective activity of neurons, in the behavior of heart
cells, and in the flashing of certain species of fireflies [110].

One classic model for synchronization is the Kuramoto model [111, 112], in
which oscillators influence one another via pairwise coupling that depends on
the difference between their phases. Given variation in the natural
frequencies for interacting oscillators, a continuous phase transition
controlled by the interaction strength separates unsynchronized activity from
a collective state in which a fraction of the oscillators oscillate at a
single consensus frequency [111]. Other model variations, for instance
oscillators that interact only through momentary pulses [113] also show
similar phenomenology of synchronization.

The language of statistical physics is useful for describing the
synchronization of large groups of oscillators, particularly using the concept
of an order parameter. For phase synchronization, the relevant order parameter
is the amplitude of the mean field, treating each oscillator’s state as a unit
vector in the complex plane.

###  3  Quantifying criticality in biological collectives

If we suspect that a living system exists near a collective transition, how
can we test this and make the statement more precise? Typical methods from
statistical physics can be useful, though applying to animal groups often
requires modification due to the importance of finite size and nonequilibrium
dynamics.

One generic strategy for continuous phase transitions is inspired directly by
statistical physics: Define a collective order parameter and look for peaks in
the sensitivity of the order parameter to external perturbations (also known
as susceptibility in the language of magnetic systems). A peak in sensitivity
corresponding to a collective instability becomes a convenient way to define
an effective critical point in a finite system [114, 103, 29]. This approach
is often a natural one because the sensitivity of global behavior to changes
at the individual scale is commonly connected to hypothesized functions of
criticality. More generally, order parameters can be defined in terms of a
Fisher information measure that acts as a generalized sensitivity [115].
Finally, in spatial cases, the correlation length can similarly be measured
and compared to the peak expected at criticality.

Phase transitions in animal groups often involve cascades of activity, where
the relevant transition is the percolation-like transition between cascades
that quickly die out and those that spread through the entire system. In this
case, more specific measures of distance from criticality can be used. First,
a measure of local amplification: the degree to which changes to individuals
spread to change the state of other individuals. In epidemiology, this
corresponds to the reproduction number $R_{0}$, measuring the number of other
individuals infected directly due to one individual becoming sick [116]. In an
infinite system, an amplification value of 1 identifies the critical point. In
finite systems, this corresponds only to local instability, but can still be
useful as a rough indicator for when collective effects will be maximized [29,
103]. Second, a global measure can be constructed from cascade size
distributions, which are expected to decay like power laws at the transition.
Measuring the deviation from a power law has been used as an indicator of
criticality (for instance, in neural avalanches [117]).

Detecting and characterizing discontinuous transitions has generally been less
emphasized in the literature. Yet discontinuous transitions do occur in models
of collective behavior. For instance, in a model of fish schooling with
resource detection, evolutionary stable states occur near discontinuous
transitions between cohesive and dispersed collective states [118]. Such
transitions can be detected by measuring bistability in the collective order
parameter in the vicinity of the transition, or hysteresis in the presence of
time-dependent driving.

Beyond merely demonstrating that a system is “near” a transition, a main goal
for these measures of criticality is to help reason about how system
parameters are tuned. In particular, near a continuous phase transition,
changes to collective behavior are dominated by changes in the distance from
the transition, with changes to other dimensions in parameter space having
relatively small contributions. This can simplify the question of how
individual-scale parameters must be tuned to achieve functional collective
behavior.

###  4  Consequences of phase transition phenomena on biological function

Figure 1: Biological function near a continuous phase transition. (A) A
simplistic view recognizes that collective sensitivity is enhanced only near
the critical point — hence the idea that being tuned near criticality can be
functionally advantageous. (B) Sufficiently near the transition, collective
effects related to the transition are graded. Functional consequences, which
can be advantageous or harmful depending on context, can be traded off against
one another.

The most basic view of the relationship between collective transitions and
biological function is summed up by the fact that, when a system is far away
from a transition, its collective behavior is boring (Fig. 1A). For instance,
when interactions are very weak, then the system’s responsiveness is
unenhanced by the group, simply given by the sum of the responsiveness of each
component. When interactions are very strong, the system is likewise
unresponsive because it is difficult to modify the system’s prevailing order
(or all order may be lost in the case of a chaotic system). It is only near a
collective transition that information exchange among individual components
can hope to have an appreciable effect on the macroscopic behavior. This idea
has been explored in a number of theoretical treatments that identify
parameter regions of marginal stability — the “edge of chaos” — as defining
dynamics that are capable of nontrivial information exchange and computation
[119, 120].

Beyond this basic assessment, there are a number of distinct phenomena that
happen near collective transitions that can be functionally relevant. Yet
these collective phenomena are not necessarily beneficial. This fact, along
with the typical “blurriness” of biological transitions arising from
finiteness and heterogeneity, motivates zooming in near collective transitions
to study how varying slightly away from maximal sensitivity affects biological
function (Fig. 1B). Going beyond the simple statement that “criticality is
best”, this more nuanced perspective often leads to trade-offs that can be
managed by the system maintaining a distance from the transition or even
actively changing this distance[29, 121].

Continuous transitions (“critical points”) in particular have been the subject
of much speculation regarding impacts on biological function[26, 14]. An oft-
cited advantage of criticality is maximal susceptibility of the aggregate
state. That is, at a critical point, changes to behavior of individual animals
in a group are amplified to have maximal control over the group’s behavior
(Fig. 2) [122, 27, 96]. This can clearly be beneficial when individuals have
relevant and accurate information to share, but can equally well be
detrimental, as when individual information is noisy or individuals’ motives
are not aligned. For this reason, being near but not directly at a collective
transition has been hypothesized to be important for maintaining (and perhaps
actively tuning) a particular tradeoff between robustness and sensitivity.
This is thought to be the case in fish schools responding to a threatening
stimulus, where the optimal spreading of startle behavior can depend on the
current threat of predation [29]. The fastest spread of information happens at
the transition point, but this can be suboptimal if false alarms are much more
more likely (and therefore on average more costly) in an environment with low
predation risk. Such tuning between robustness and sensitivity has also been
hypothesized to be important for understanding conflict spreading in macaque
societies. Here, individuals have both noisy information and competing
interests regarding the collective social hierarchy, but can nonetheless
achieve beneficial consensus [83] perhaps by actively controlling the distance
from a transition point where conflict outbreaks are maximally sensitive to
the behavior of individuals [103].

Figure 2: Maximal susceptibility at criticality in a behavioral contagion
model as studied by Poel et al.[29]: The sensitivity, defined via the
difference in the average relative cascade sizes triggered by different
numbers of initially activated individuals, $m$ vs $m+1$ (left) and $1$ vs $m$
(right), is shown as a function of the median nearest neighbor distance (NND).
The NND is a control parameter determining the coupling between individuals
and the probability of an activation to spread. This susceptibility measure
becomes maximal in the vicinity of the quasi-critical point, which separates a
phase of only small-scale, local cascades (high NND, weak coupling) from a
phase of global activation (low NND, strong coupling). In functional terms,
the system is best at discriminating different numbers of initially active
agents at criticality. The vertical dashed lines indicates an analytical
approximation of the critical point. Figure taken from Supplementary
Information of Poel et al.[29]

A second potential advantage of a continuous transition is the dynamical
phenomenon of critical slowing down. In neuroscience, critical slowing is
hypothesized to be a mechanism through which long timescales can be created
using components with short memories [82]. Similarly, in decision-making in
animal groups, slow timescales may be important for integrating over noisy
information [97]. In situations with small signal-to-noise ratio, this
phenomenology also implies a trade-off between speed and accuracy as the
distance from the transition varies, since slower dynamics allows for more
time to integrate noisy information. The speed–accuracy trade-off in
collective decisions is thought to be important for many types of animal
groups, and has been studied extensively in social insects [123, 124].

Collective transitions in general can display the phenomenon of spontaneous
symmetry breaking, in which aggregate states select a particular global order
even when multiple possible orderings are equally likely. In an animal group,
coming to a single group consensus about a decision may be important even if
all possible decisions are equally favored, in which case spontaneous symmetry
breaking could be advantageous. On the other hand, a system that remains at a
continuous transition would have large, slow fluctuations that explore the
alternate global symmetries, and moving beyond the critical point could be
required to force long-term commitment or consensus on a particular collective
decision [125, 97].

While much attention has focused on the theoretically rich area of continuous
transitions, discontinuous transitions can be just as important in
biology.444Note that, given the typical blurriness and nonequilibrium nature
of biological transitions, it may be difficult to resolve whether a given
transition is continuous or discontinuous — in our view, this categorization
is not always important in analyzing a particular transition’s functional
consequences. In particular, hysteresis can create a sustained switch-like
response. This can be advantageous in collective decision-making when
consensus or strong commitment is important. For instance, a bee swarm
committing to a particular nest site may use strong hysteresis to avoid
splitting its individuals between two similar sites. On the other hand,
hysteresis can be costly when it leads to being trapped in a suboptimal
collective state.

Finally, while synchronization is observed across many animal groups, its
functional relevance is still debated. Generally speaking, it is likely that
multiple costs and benefits of coordination must be weighed against those of
acting alone. In fireflies, not all species synchronize, and in those that do,
they tend to synchronize only at high density, suggesting that synchronization
is either difficult or not always beneficial. The selective advantage is not
fully understood but is likely to involve both female choice of mate and
avoidance of predators [126].

###  5  Mechanisms of self-organization towards criticality

A crucial open question with respect to the criticality hypothesis in animal
collectives is that of the actual mechanisms that enable animal groups to tune
towards critical points in a fully distributed manner. Here, the main
challenge is that phase transitions, and therefore critical points, are
macroscopic properties of the system at the level of the entire group, while
the adaptation of behavior can be only performed by individuals. First, in the
common case that single individuals can only perceive their local
neighborhood, it is unclear how they can know at all the global state of the
group, and in consequence how they are able adapt their behavior to tune the
entire collective towards or away from a phase transition. Second, it has also
to be assumed that at any time only a fraction of individuals may adapt its
behavior, which in general will increase the heterogeneity of the group with
unclear consequences on the ability of the system to tune the macroscopic
state, or potentially even detrimental effects on collective behavior by
affecting the spatial structure of the group up to a loss of cohesion.

Regarding the first point of individuals having access only to local
information, a potential solution has been proposed in the field of
neuroscience, as a mechanism for self-tuning of neuronal networks towards
criticality based only on local node dynamics [127]. The basic idea is that
local averages over time provide some information about the global state of
the network. The network can evolve towards a global critical point when
individual neurons that are too silent or too active over time decrease or
increase their response thresholds, respectively. This mechanism relies
implicitly on two important properties of the network dynamics: (1)
ergodicity, i.e. the assumption that the temporal average of local dynamics
corresponds to an ensemble average over the entire network, and (2) time-scale
separation between the fast firing dynamics of individual neurons and the slow
adaptation dynamics of individual nodes changing their thresholds or synaptic
weights. Both assumptions are likely violated in animal groups, leading to
open questions about the applicability of the mechanism to self-tuning towards
criticality in animal collectives.

In general, while the benefits of criticality for collective information
processing have been addressed in various studies, the actual mechanisms on
how animal collectives could tune themselves towards critical points have been
rarely investigated [128, 129, 130, 28]. Here, different adaptive mechanisms
are potentially thinkable.

First, for uncovering the proximate mechanisms of self-organization towards
criticality, it is important to identify how changes in individual behavior
modulate macroscopic group properties, and how these may control the distance
to criticality at the collective level. A fundamental variable at the
macroscopic level is $N$, the number of individuals in the collective. In
finite-sized systems the location of the maximum of the susceptibility is a
function of $N$ due to finite size scaling, and it has been argued that in
midge swarms it may be the decisive parameter tuning the system in a self-
organized way towards criticality[128].

For a fixed group size, group density plays an important role. Assuming that
influence of a neighbor on a focal individual is distance dependent, a change
in density will directly impact the average strength of social interactions in
a group. In fact, it has been recently shown that density modulation is the
prime mechanism for tuning the collective escape responses in fish due to
changes in perceived risk of predation [102]. Surprisingly, adaptations of
individual response thresholds appear to play a negligible role in this
context [102]. However, in other ecological contexts or different animal
species, adaptation of individual sensitivity to social cues (analogously to
synaptic plasticity in neurons), while keeping the typical nearest neighbor
distance unchanged, is a reasonable and biologically viable individual-level
mechanism for tuning the distance to criticality.

Finally, individuals can exhibit high levels of heterogeneity in their
behavioral traits, and may even assume very different roles in a collective (“
_division of labor_ ”). Therefore changes in the composition of the group at
similar group size and density may provide another indirect mechanism for
self-organization toward or away from a critical point. Here, however, one
should note that strong heterogeneity could lead to fundamental changes in the
phase diagram of the system, for example by facilitating the emergence of
Griffith phases, extended regions in parameter space of (quasi)-critical
behavior [131].

Over longer time scales, evolutionary adaptations[129, 28] can tune the
behavioral parameters of individuals towards critical points of the collective
behavior. This would first require that being critical is on average
beneficial across all ecological contexts encountered by animal collectives.
Secondly, the critical point — if beneficial for collective computations —
represents a group level optimum, while evolutionary selection is assumed to
act predominantly at the level of individuals. In general, there is no
guarantee that group optima correspond also to evolutionary stable states with
respect to individual-level adaptation. On the contrary, so-called social
dilemmas, situations where evolutionarily stable individual strategies differ
from group-level optima are rather the rule than the exception[28]. Multi-
level selection [132] could provide a mechanism for evolutionary adaptation
towards group-level optima. However, recent mathematical analysis demonstrated
that as soon as individual-level selection plays a non-negligible role — which
can be safely assumed for most, if not all, animal systems — group-level
optima in general differ from evolutionarily stable strategies [133].

Interestingly, Hidalgo et al[129] have shown that agents modelled as boolean
genetic networks do evolutionarily adapt toward phase transitions if exposed
to heterogeneous, variable environments. Further, they have also shown that
interacting groups of such agents, with environmental input coming from other
members of the group, also self-tune to criticality. However, a recent study
by Klamser & Romanczuk[28] questions the generality of this result with
respect to collective behavior of animal groups. It considers a spatially-
explicit model of prey agents performing cohesive, collective motion subject
to attacks by a predator agent, and investigates both whether being close to
the symmetry-breaking flocking transition is optimal for collective predator
response and whether individual-level evolution provides a viable mechanism
for these systems to tune towards criticality. While being at criticality
indeed appears to be beneficial for the entire collective in terms of optimal
information propagation and minimal predation risk for individuals in the
group, it turns out that the critical point is not an evolutionarily stable
state. On the contrary, the phase transition turns out to be maximally
evolutionarily unstable, as that is where the collective dynamics is most
sensitive to heterogeneity in individual behavior. This results in individuals
with different behavioral parameters having the largest possible differences
in their ability to avoid predation, and therefore leads to steepest fitness
gradients accelerating evolution of individual behavior away from the critical
point deep into the ordered flocking state[28].

A final self-tuning mechanism for animal groups toward critical points could
be some sort of (reinforcement) learning on much shorter time scales than
evolutionary adaptations. However, a similar question as discussed above is
relevant here as well: How is the system able to tune to a global collective
state despite predominantly individual-level learning and access to only local
information? In the context of multi-agent reinforcement learning, further
questions may arise about the convergence of the learning process in a multi-
agent setting [134].

###  6  Future challenges & concluding remarks

The emergence of novel computational techniques for tracking individuals in
large animal collectives[135, 136] provides the ability to quantify their
collective behavior at unprecedented scale, as well as temporal and spatial
resolution. This enables tests of previously theoretical predictions and
theory-driven hypotheses, including the criticality hypothesis. And indeed,
through a combination of experimental and theoretical work, grounded in
statistical physics, it is now possible to quantify whether animal collectives
operate close to critical points [128, 29]. Undoubtedly, depending on the
particular system and behavior of interest, this task may be technically very
challenging, even if the corresponding methodological approaches are known. In
particular, the inverse problem of inferring models of collective behavior
from data is known to be difficult [137], with potential pitfalls for spurious
detection of criticality [138].

For future research, it appears that the fundamental open questions to be
addressed, both from theoretical and empirical point of view, are about the
actual mechanisms on how animal collectives can adapt their distance to
criticality in a fully distributed manner, and with only local information
available to individuals. Here the evolutionary perspective and the existence
of potential social dilemmata between group level optima (criticality) and
individual-level evolutionary stable behavioral strategies cannot be ignored.

As laid out in this chapter, phase transition theory and the concept of
criticality are highly relevant for understanding and quantifying the
interplay of self-organization and function of collective behavior in biology.
The “criticality hypothesis”, stating that complex biological systems should
operate in the vicinity of phase transitions in order to maximize their
collective computation capabilities, has the potential to provide a unifying
principle potentially relevant to a wide range of different biological
systems. It sets up a theory-driven conceptual framework which can inspire,
and even structure, entire research agendas at the interface of empirical and
theoretical research on collective information processing systems in
biophysics and biology. Even in cases where research fails to demonstrate
criticality in collective behavior, or even appears to disprove it, exploring
the reasons for such “negative” results enhances our understanding of
collective behavior and generates new research questions and hypotheses [28,
29]. This also very much aligns with recent calls from within biology for more
novel ideas grounded in theory [139]. However, investigating criticality in
biological systems should not be done in epistemic isolation from the physics
perspective. On the contrary, we argue that connecting corresponding research
to the large body of theories in biology should be a priority, starting from
the most fundamental one, the evolutionary theory, to more specific ones such
as the many-eyes hypothesis in collective anti-predator behavior [140] or
theories on the role of ecological factors[141, 142, 143]. A truly
bidirectional exchange between physics and biology opens new avenues of
research with enormous potential for better understanding the fundamental
processes of life.

#### Acknowledgments

P.R. was funded by the Deutsche Forschungsgemeinschaft (DFG) (German Research
Foundation), grant RO47766/2-1 and acknowledges funding by the DFG under
Germany’s Excellence Strategy–EXC 2002/1 “Science of Intelligence,” project
390523135. B.C.D. was supported by a fellowship at the Wissenschaftskolleg zu
Berlin and by the ASU-SFI Center for Biosocial Complex Systems.

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## Index

  * behavioral contagion §4
  * bird flocks §1
  * collective behavior §1, §1
  * collective decision making §3, §3
  * critical point §4
  * critical slowing down §4
  * criticality hypothesis §1, §1
  * fish schools §1
  * flocking §2, §2
  * insect swarms §1
  * Kuramoto model §5
  * Mermin–Wegner theorem §2
  * phase transitions §1, §2
  * self-organization §1, §1
  * self-organized criticality §1, §1
  * spontaneous symmetry breaking §4
  * synchronization §5
  * universality §1
  * Vicsek model §2