# Semantic information in a model of resource gathering agents

Damian R Sowinski<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Rochester  Jonathan Carroll-Nellenback  Department of
Physics and Astronomy, University of Rochester  Robert N Markwick  Department
of Physics and Astronomy, University of Rochester  Jordi Piñero  Complex
Systems Lab, University Pompeu Fabra  Marcelo Gleiser  Department of Physics
and Astronomy, Dartmouth College  Artemy Kolchinsky  Universal Biology
Institute, The University of Tokyo, Japan  Gourab Ghoshal
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Rochester  Adam Frank<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Rochester

###### Abstract

We explore the application of a new theory of _Semantic Information_ to the
well-motivated problem of a resource foraging agent. Semantic information is
defined as the subset of correlations, measured via the transfer entropy,
between agent $A$ and environment $E$ that is necessary for the agent to
maintain its viability $V$. Viability, in turn, is endogenously defined as
opposed to the use of exogenous quantities like utility functions. In our
model, the forager’s movements are determined by its ability to measure, via a
sensor, the presence of an individual unit of resource, while the viability
function is its expected lifetime. Through counterfactual
interventions—scrambling the correlations between agent and environment via
noising the sensor—we demonstrate the presence of a critical value of the
noise parameter, $\eta_{c}$, above which the forager’s expected lifetime is
dramatically reduced. On the other hand, for $\eta<\eta_{c}$ there is little-
to-no effect on its ability to survive. We refer to this boundary as the
semantic threshold, quantifying the subset of agent-environment correlations
that the agent actually needs to maintain its desired state of staying alive.
Each bit of information affects the agent’s ability to persist both above and
below the semantic threshold. Modeling the viability curve and its semantic
threshold via forager/environment parameters, we show how the correlations are
instantiated. Our work provides a useful model for studies of established
agents in terms of semantic information. It also shows that such semantic
thresholds may prove useful for understanding the role information plays in
allowing systems to become autonomous agents.

##  I Introduction

Questions about the role of information in the physics of life extend as far
back as Schrödinger’s seminal 1944 work “What is Life” Schrodinger [1944].
Four years later, Shannon published his seminal work on information theory
Shannon [1948], shortly followed by the discovery that DNA serves as a code
for living organisms Watson and Crick [1953]. These developments have led to a
deep interest in the relationship between information, physics, and biology
Hopfield [1994]. Since then, the applications of information theory to biology
have grown exponentially Godfrey-Smith [2007]; Mattingly  _et al._ [2021];
Küppers [1990]; Adami [2004]; Uda [2020]; Gohari  _et al._ [2016]; Rhee  _et
al._ [2012]; Tkačik and Bialek [2016]; Donaldson-Matasci  _et al._ [2010];
Rivoire and Leibler [2011]; Hazen  _et al._ [2007], allowing researchers to
unpack the ways organisms store and process data about the environment and
their own internal states Sowinski  _et al._ [2022]; Oh  _et al._ [2020];
Bowen  _et al._ [2022].

One difficulty with applications of Shannon’s information theory to biological
systems is its “syntactic” nature. That is, the kinds of extant measures
employed in information theory capture statistical correlations between
systems without any consideration of the relevance or meaning of those
correlations. Living systems, however, act as agents for whom information is
intrinsically meaningful in the most basic sense; that is, whether it can be
useful for its self-production and self-maintenance Schlosser [1998]; Mossio
_et al._ [2009]. Life, as a driven, non-linear, and far-from-equilibrium
system, is always in a precarious position and must gather information about
the state of the environment and its internal state to endure Fang and Wang
[2020]. Some of this information will be useful for this purpose and some will
be irrelevant. In this setting, relevance and meaning can be considered
synonymous.

Unlike the well-developed field of syntactic information theory, there exists
no widely accepted or applied formal theory of semantic information (some
attempts at developing such a theory include Refs. D Polani [2001]; Thompson
and Stapleton [2009]; Nehaniv  _et al._ [2003]; Barham [1996]; Deacon [2007];
Corning [2007]; Gleiser and Sowinski [2018a]). A goal of a mathematical theory
would be to provide an operational definition useful for characterizing
nonlinear far-from-equilibrium systems which can be identified as agents
(e.g., organisms or robots). Recently, Kolchinsky and Wolpert Kolchinsky and
Wolpert [2018] (henceforth KW18) developed an explicit formalism for semantic
information based on the use of counterfactuals and a notion of viability.
Their formulation uses the state spaces and probability distributions for an
agent $A$ and its environment $E$ to characterize the mutual information
between the two, while the persistence of $A$ (its ability to maintain a
desired state) is measured through a viability function $V$. The concept of
meaning here is thus taken in the most basic sense of being related to an
agent’s continued existence. By running intervened versions of the system
dynamics in which some fraction of the mutual information between agent and
environment is scrambled, a formal working definition of the semantic
information was characterized in terms of the response of the viability
function to such interventions. Importantly, the viability is determined by
the inherent coupled dynamics of the system and the environment Kolchinsky and
Wolpert [2018]; Rovelli [2018], rather than through exogenous utility, cost,
error, or loss functions (as is sometimes done when studying the value of
information in statistics or in engineering applications Gould [1974];
Stratonovich [2020]; Shannon  _et al._ [1959]; Hazen  _et al._ [2007];
Sowinski  _et al._ [2021]).

There are several classes of models that describe specific attributes of
living systems such as synchronization Acebrón  _et al._ [2005]; García-Selfa
_et al._ [2021], pattern formation Cooper  _et al._ [2013]; Mimar  _et al._
[2019], competition between species for resources COHEN [1977], stability of
ecosystems Rooney  _et al._ [2006], simple models of metabolism Xavier  _et
al._ [2020] as well as synthetic cells Blain and Szostak [2014]. Each of these
models are settings where one can test the theoretical framework of semantic
information. Indeed, there has been a recent study on semantic information in
the context of synthetic cells Ruzzante  _et al._ [2023]. Our approach here,
is three-fold: First, we apply the semantic information framework to the well-
motivated problem of resource foraging agents. Second, we develop a detailed
mathematical and numerical implementation of the original KW18 formalism to
explore what simplifying approximations lead to a clear connection between
semantic information and the viability of the foraging agents. Finally, we
demonstrate the efficacy of our approach in uncovering new insights on the
general features of agent/environment dynamics.

Section II focuses on a class of models that jointly address the interrelation
between exploration and resource-consumption. In such _forager models_
Bénichou and Redner [2014]; Bhat  _et al._ [2017]; Bénichou  _et al._ [2016];
Bhat and Redner [2022], an agent navigates its environment (using various
exploration strategies) in search of resources (food) that it then consumes to
maintain its desired internal state (staying alive). The resources are either
at fixed locations along the spatial extent of the environment or are
replenished at a constant rate in random locations. These dynamics are
described in detail in section II.1. In such a setting, a natural choice for a
viability function is the agent’s lifetime, while the environment is simply
the field of resources. The correlations between the agent and its
environment, which are contingent on the agent’s sensorial capability, carry
meaningful information about the location of the resources. These correlations
are then scrambled by tuning the fidelity of the sensor and then measuring its
effect on the viability function, the quantitative details of which make up
section II.2.

We demonstrate in section III the existence of a plateau in the viability,
capturing the subset of correlations that has no effect on the agent’s ability
to stay alive. Below this threshold the lifetime of the agent monotonically
decays with increased scrambling of the mutual information. We introduce the
concept of _viability-per-bit_ that captures the degree to which each bit of
the agent’s information on its environment is relevant. The quantity peaks at
the boundary separating the plateau from the decaying region indicating the
existence of a _semantic threshold_. Information above the threshold has
little-to-no effect on the agent’s viability, while below it, each bit becomes
crucial for the agent to stay alive. We show that our results are agnostic to
the exploration strategy (that is a random walker or ballistic foraging) or
the choice of the particular viability function, as long as it reflects the
general ability of the agent to remain alive. We end in section IV with a
discussion of the implications of our findings and possible future directions.
Three appendices clarify a few technical aspects of our model and its
approximations.

##  II Foraging in a Replenishing Environment

###  II.1 Definition of the model

Consider a system $A\times E$, defined to be a foraging agent, $A$, exploring
an environment, $E$. The agent has a fixed metabolic rate $\mu$ powered by an
ever-dwindling fuel reserve, and a sensor allowing it to detect resources
within some finite circular range $R$. Moving at constant speed $v$, the agent
changes its direction $\hat{{\boldsymbol{n}}}$ only in response to detected
resources. Any resource that falls within the collection radius $r\leq R$ of
the agent can be harvested, its energy content added to the fuel supply of the
agent. We note that this ballistic variant of the forager model is different
from the ones considered for instance in Bénichou  _et al._ [2016]; Bhat and
Redner [2022] where the forager conducts a random walk. In such a formulation,
only when the forager detects a resource does it move toward it ballistically,
after which it resumes its random walk exploration strategy. As we will show
later, the qualitative nature of the results are agnostic to whether the
forager is ballistic or diffusive.

The state of the agent is described by the tuple
$a=(s,\tau,\hat{{\boldsymbol{n}}},{\boldsymbol{x}})$. $s\in[0,S]$ is the
agent’s stored fuel supply; when $s=0$ the agent is no longer functional,
otherwise it is considered alive. The parameter $\tau\in\\{0,1\\}$ is Boolean
and describes if the agent has locked on to a target. The agent will not
change its direction of motion—indicated by the unit vector
$\hat{{\boldsymbol{n}}}$—if it has a target. Consuming a resource resets
$\tau\mapsto 0$. Finally, ${\boldsymbol{x}}\in[0,L]^{2}$ is the position of
the agent in the environment, defined as a square of side length $L\gg R$.

The state of the environment is specified by the set of locations of a
fluctuating number of resources,
$e=\\{{\boldsymbol{y}}_{1},{\boldsymbol{y}}_{2},\dots,{\boldsymbol{y}}_{N}\\}$
with ${\boldsymbol{y}}_{n}\in[0,L]^{2},\ \forall n$. This model differs from
extant models in that resources are renewable—there is a source of energy flux
per unit area into the environment, $\Gamma$, resulting in the growth of fixed
energy $\epsilon$ resources across it, which, in turn, decay at a rate
$\gamma$. In the absence of an agent, the two processes lead to an equilibrium
average resource density $n_{\text{eq}}=\Gamma/\epsilon\gamma$ with
$\mathcal{O}(L^{-1}\sqrt{\Gamma/\epsilon\gamma})$ fluctuations, so that the
average distance between resources is
$\ell_{\\!Re}=\sqrt{\epsilon\gamma/\Gamma}$ (See Appendix A). Large
inhomogeneities in the resource density, such as those which are created by a
foraging agent, are repopulated at a rate $\gamma$. We work in the small back-
reaction regime where forager changes to the equilibrium spacing are on the
order $\delta\ell_{\\!Re}/\ell_{\\!Re}\sim\mathcal{O}(R^{2}/L^{2})\ll 1$, so
that the agent is effectively uncorrelated with the environmental degrees of
freedom on distances larger than some $\mathcal{O}(1)$ multiple of $R$. Table
1 summarizes the parameters that define the model.

Figure 1: An overview of the forager model. The agent moves continuously on a plane at constant speed in a direction $\hat{{\boldsymbol{n}}}$, sensing at a distance $R$ and collecting resources within a radius $r$. Resources are continuously produced by the environment. Agent movement is powered by a metabolism that drains the agents energy reserves; collecting resources replenishes them. When a resource comes within sensor range, the agent targets it (red circle $\tau$) and orients towards it. The agent then moves until the resource falls within collection range and is consumed. Table 1: Model Parameters _Parameter_ | _Symbol_ | _Units_
---|---|---
Collection Radius | $r$ | L
Detection Radius | $R$ | L
Speed | $v$ | LT-1
Metabolic Rate | $\mu$ | ML2T-3
Maximum Stored Energy | $S$ | ML2T-2
Resource Energy | $\epsilon$ | ML2T-2
Resource Decay Rate | $\gamma$ | T-1
Energy Influx | $\Gamma$ | MT-3

As the agent moves through the environment its collection diameter $2r$ acts
as a cross-section, sweeping out an area for harvest proportional to its
speed. Interesting regimes of the model happen when the speed falls within the
following bounds

$v_{\star}<v<v^{\star}\ \ \text{where}\ \
v_{\star}=\frac{\mu\ell_{\\!Re}^{2}}{2rS}=\frac{\epsilon}{S}v^{\star}.$ (1)

For $v<v_{\star}$, resources are spread so thin that the foraging area on a
full tank has an expected harvest less than $1$, leading to agents with rather
short lives. For $v>v^{\star}$, the harvest area fueled by a single resource
has an expected harvest greater than $1$, the agent is coupled to an
environment of abundance, and is effectively immortal. In between, the agent’s
probability of survival decays exponentially at long lifetimes, while still
forming significant correlations with the environment. This is the regime
interesting for exploration. In a strong back-reaction setting these bounds
are dynamic: agent harvesting increases the average spacing between resources
thereby increasing both $v_{\star}$ and $v^{\star}$. Again, we will work in
the small back-reaction limit, where the forager doesn’t affect the
environment. Fig. 1 provides a schematic of the agent-centric dynamics.

###  II.2 The viability function and interventions

While there are a number of ways of quantifying the viability of the agent, an
obvious choice is the expected lifetime $V=\mathds{E}[T]$; here $\mathds{E}$
is an expectation value over an ensemble of agents, and $T$ is the time at
which the agent first reaches the dead state ($s=0$) (as we will later show,
our results are robust to other variants of the function). This is easily
measured by running an ensemble of agents as described in Fig. 1, and
characterizing the distribution of lifetimes. The viability of the agent
depends crucially on it correctly setting its trajectory
$\hat{{\boldsymbol{n}}}$ when a resource is targeted, which amounts to having
a properly working sensor. A broken sensor results in an agent moving in
directions that do not correlate with the location of a resource, increasing
the chance of starvation and thus death.

To escape the vagaries of chance, the agent needs a sensor that correlates
target direction with resource location. Let $\rho_{\\!AE}(a,e)$ be the joint
distribution over states of the agent and environment. We use a standard
notation where the subscript identifies the state space that the arguments are
drawn from. Translational symmetry allows us to eliminate ${\boldsymbol{x}}$
from $a$, and imposes a natural partial ordering, via a permutation of the
resource indices $\pi$, of the environment state $e\mapsto
e^{\prime}=({\boldsymbol{y}}^{\prime}_{1},{\boldsymbol{y}}^{\prime}_{2},\dots,{\boldsymbol{y}}^{\prime}_{N})$,
where the effective forager-resource displacement is
${\boldsymbol{y}}_{\pi(n)}^{\prime}={\boldsymbol{y}}_{n}-{\boldsymbol{x}}$
such that if $\pi(n)<\pi(m)$ then
$|{\boldsymbol{y}}_{\pi(n)}^{\prime}|\leq|{\boldsymbol{y}}_{\pi(m)}^{\prime}|$.
Henceforth we work in this agent-centric coordinate system and drop the
primes. To hone in on the specific correlation between sensor and environment,
we next marginalize over all but the closest resource,
${\boldsymbol{y}}={\boldsymbol{y}}_{1}$, and condition on the agent being
alive, leading to the simpler joint distribution
$\rho_{AE}(\hat{{\boldsymbol{n}}},\tau,{\boldsymbol{y}}|s>0)$. Rather than
computing the correlation between all the agent and environmental degrees of
freedom, we now have the simpler task of finding the correlations between the
subsets $\\{\hat{{\boldsymbol{n}}},\tau\\}$ and $\\{{\boldsymbol{y}}\\}$.

Denote the living agent distribution
$\rho_{A}=\rho_{A}(\hat{{\boldsymbol{n}}},\tau)$ and the environment
distribution $\rho_{E}=\rho_{E}({\boldsymbol{y}})$. Correlations between agent
and environment prevent a factorization of the joint distribution so that
$\rho_{AE}\neq\rho_{A}\rho_{E}$. (The distribution factorizes in the case of a
non-functional agent, given that the lack of an interaction pathway rapidly
suppresses correlations between the agent and environment.) While alive,
interactions transfer information between the two, generating the correlations
which are then responsible for prolonging the agent’s viability. Consequently,
we examine the flow of information between the environment and the agent
during a detection event.

Consider a resource detection and locking event during the time interval
$\Delta t$, when a resource is detected at a distance
$r<|{\boldsymbol{y}}|\leq R$. The agent’s orientation evolves from
$\hat{{\boldsymbol{n}}}\mapsto\hat{{\boldsymbol{n}}}^{\prime}$ while its
target degree of freedom flips from $\tau=0\rightarrow\tau^{\prime}=1$. For an
agent with a perfect sensor, the transition probability for this detection
event, conditioned on the environment state, $e$, is

$\displaystyle\rho_{A\rightarrow
A^{\prime}|e}(\hat{{\boldsymbol{n}}}^{\prime},\tau^{\prime}|\hat{{\boldsymbol{n}}},0,{\boldsymbol{y}})=\delta_{\tau^{\prime}}^{1}\delta\left(\theta(\hat{{\boldsymbol{n}}}^{\prime},{\boldsymbol{y}})\right),$
(2)

where $\theta({\boldsymbol{a}},{\boldsymbol{b}})$ is the angle between vectors
${\boldsymbol{a}}$ and ${\boldsymbol{b}}$, $\delta^{1}_{\tau^{\prime}}$ is the
Kronecker delta, and $\delta(\theta)$ is the Dirac delta function. By the
rotational symmetry of the environment, we also have that $\rho_{A\rightarrow
A^{\prime}}(\hat{{\boldsymbol{n}}}^{\prime},\tau^{\prime}|\hat{{\boldsymbol{n}}},0)=\delta_{\tau^{\prime}}^{1}(2\pi)^{\mathchar
45\relax 1}$, which is not equal to Eq. 2 indicating that it is likely that
$\rho_{AE}\neq\rho_{A}\rho_{E}$.

To investigate the extent these correlations are necessary to insure the
agent’s longevity, we can compromise the agent’s ability to find resources by
adding noise to its sensor. This will impact the agent’s viability function
$V$. We thus add a small amount of noise $\eta$ to the agent’s ideal sensor,
which affects the transition probability for detection defined in Eq. 2 as
$\rho_{A\rightarrow A^{\prime}|e}\equiv\rho_{A\rightarrow
A^{\prime}|e}^{\eta=0}\mapsto\rho_{A\rightarrow A^{\prime}|e}^{\eta}$, with
$\rho_{A\rightarrow A^{\prime}|e}^{1}=\rho_{A\rightarrow A^{\prime}}$. We then
measure its effect on the agent’s viability as $\eta$ is increased. Adding
noise changes the Dirac delta of Eq.2 into a uniform distribution of width
$2\pi\eta$, $\delta\mapsto\delta_{\eta}$, where

$\displaystyle\delta_{\eta}(\theta)=\begin{cases}\dfrac{1}{2\pi\eta}&\theta\in[\mathchar
45\relax\pi\eta,\pi\eta]\\\ 0&\text{otherwise}\end{cases}.$

As the noise parameter $\eta\rightarrow 0$, the perfect sensor is recovered,
while $\eta\rightarrow 1$ gives the uniform distribution.

Figure 2: (Top) The distribution of lifetimes extracted from the simulations.
The dark curve is for a noiseless (ideal) agent sensor ($\eta=0$). The nearly
indistinguishable distribution is from near critical scrambling
($\eta\approx\eta_{c}$), while the leftmost distribution is for above critical
scrambling ($\eta>\eta_{c}$). (Bottom) The probability of being alive for an
unscrambled agent (dark line) and two agents whose sensor correlations with
the environment have been scrambled: one has been scrambled up to just across
the semantic threshhold, while the other has been scrambled well beyond it.
Note the exponential decay of the tails.

The information flowing from the environment to the agent during the detection
event is quantified using the transfer entropy Schreiber [2000],

$\displaystyle\mathcal{T}_{E\rightarrow A}^{\eta}$
$\displaystyle=\mathds{E}_{\rho_{AE}}[\log_{2}\frac{\rho_{A\rightarrow
A^{\prime}|e}}{\rho_{A\rightarrow A^{\prime}}}]$
$\displaystyle=\log_{2}\frac{1}{\eta}.$ (3)

The information gathered during detection approaches $0$ as $\eta\rightarrow
1$, preventing the formation of correlations between agent and environment. It
diverges as $\eta\rightarrow 0$, which reflects the infinite precision needed
to specify a direction in space, and is discussed in Gleiser and Sowinski
[2018b].

##  III Results

Returning to the notion of viability, we extend our definition to the noised
sensor channel by defining $V=V_{0}\mapsto V_{\eta}=\mathds{E}_{\eta}[T]$,
where the expectation is now taken over an ensemble of agents with noisy
sensors. We interpret the addition of noise as an intervention that scrambles
the information transfer between an agent with a perfect sensor and its
environment. We extract $V_{\eta}$ by simulating agents with noised sensors
within an equilibrated environment at half of full health and measuring how
long they survive. We do this $10^{4}$ times for each of $200$ values of
$\eta\in[0,1]$, the first $100$ equally spaced in $[0,0.2]$ and the second
hundred in $(0.2,1]$. In the upper panel of Fig. 2 we plot the distribution of
lifetimes for three different values of the noise parameter $\eta$. The solid
black line corresponds to the case of a perfect sensor (no scrambling,
$\eta=0$) and sets the baseline distribution of lifetimes which is peaked near
the expected lifetime (vertical dashed dark red-line) with a broadly decaying
tail. The thinner lines are for larger values of $\eta$ and one can see a much
lower value for $\mathds{E}_{\eta}[T]$ once a critical value of noise,
$\eta_{c}$, is surpassed.

The actual viability $V_{\rm actual}$ is defined as the expected lifetime of
an agent with noiseless sensor. As predicted Kolchinsky and Wolpert [2018],
the viability plateaus near the actual viability $V_{\rm actual}$ even as
noise degrades the information flow during detection. However, once the sensor
efficiency has been sufficiently degraded, the viability begins to drop
dramatically, as seen in the top panel of Figure 3. This rapid decline in
viability is interpreted as a semantic threshold—it is the minimal information
acquired in a sensing event responsible for maintaining agents close to their
actual viability. To better characterize this critical information, we
introduce the differential measure of viability per bit (VpB), as seen in the
lower panel of Figure 3. The peak in VpB illustrates what an agent requires
from its interactions with the environment. Very noisy sensors provide little
VpB, even with increased accuracy. However, once a certain level of accuracy
is achieved, the VpB grows dramatically and the sensor provides the agent with
meaningful information about its environment, resulting in the agent’s
improved survivability. Too much accuracy, however, is wasted, and the VpB
drops again once the sensor goes above and beyond the constraints dictated by
the physical nature of the agent within the environment.

Figure 3: (Top) The viability curve (thick black line) as a function of the
transfer entropy, $\mathcal{T}_{E\rightarrow A}^{\eta}$. The actual viability
$V_{\rm actual}$ is the expected lifetime with no scrambling. The light grey
line is the median, with the shaded regions representing intervals of $10\%$
above and below the median. (Bottom) The viability per bit. The dots are
actual data, and the line is a smoothed and interpolated curve added to better
visualize the semantic region.

It should be pointed out that the results in Figure 3 don’t depend on using
the expected lifetime as a viability function; any percentile of the lifetime
distribution will do. The light grey line shows the median of the
distribution, while the grey regions represent jumps by $\pm 10\%$ from the
median. The semantic content is robust to all of these, which can easily be
understood by considering how scrambling the sensor relates to the foraging
efficacy of the agent. Since the collection radius, $r$, acts as an impact
parameter, and targets are set when they enter within the sensing radius $R$,
there is a critical noise parameter above which the agent will sometimes miss
their target, and thus affect its viability. This critical value,
$\eta_{c}=\sin^{-1}(r/R)/\pi$, points to a semantic threshold of

$\displaystyle\mathcal{T}_{E\rightarrow
A}^{c}=\log_{2}\pi-\log_{2}\sin^{\mathchar 45\relax 1}\frac{r}{R}.$ (4)

(See Appendix B for details.) There is low VpB when the information
transferred during a detection event is well above $\mathcal{T}_{E\rightarrow
A}^{c}$, and high VpB as $\mathcal{T}_{E\rightarrow
A}^{\eta}\rightarrow\mathcal{T}_{E\rightarrow A}^{c}$.

The minute change in the viability plateau also has a geometric origin. Due to
the circular shape of both the resource collection and sensing zones, a sub-
critically scrambled sensor will force agents to travel farther ever so
slightly to collect a targeted resource. Assume an agent targets a resource a
distance $y=|{\boldsymbol{y}}|\leq R$ away, but the sensor is mistaken by an
angle $\theta$ as to the direction of the resource. Rather than having to
travel $y-r$ to collect its target, the agent now has to travel
$y\cos\theta-\sqrt{r^{2}-y^{2}\sin^{2}\theta}$. Given an information transfer
of $\mathcal{T}$ during detection, averaging over all angles we see that
resources a distance $y$ from the agent are effectively a distance
$(1+\lambda)y$ with

$\displaystyle\lambda$
$\displaystyle=\frac{r}{y}\left(\\!1\\!-\\!\frac{2^{\mathcal{T}}}{\pi}E\left(\frac{\pi}{2^{\mathcal{T}}}\Big{|}\frac{y^{2}}{r^{2}}\right)\\!\right)\\!-\\!\left(\\!1\\!-\\!\frac{2^{\mathcal{T}}}{\pi}\sin\\!\frac{\pi}{2^{\mathcal{T}}}\\!\right)$
$\displaystyle\underset{\eta\rightarrow
0}{\sim}\frac{\pi^{2}}{6}\frac{y-r}{r}4^{-\mathcal{T}}$
$\displaystyle\underset{y\rightarrow
r^{+}}{\sim}\frac{y-r}{r}\left(\frac{\tanh^{\mathchar 45\relax
1}\tan\frac{\pi}{2}2^{\mathchar
45\relax\mathcal{T}}}{\frac{\pi}{2}2^{\mathchar
45\relax\mathcal{T}}}-1\right).$ (5)

Here $E(\varphi|k^{2})$ is the indefinite elliptic integral of the second kind
with modulus $k=y/r$. (See Appendix C for details of the derivation.)

The expression for the dilation factor $\lambda$, indicates that sub-
critically scrambled agents can be thought of as unscrambled agents making
their way through a distorted world. As the information acquisition from the
environment is restored ($\mathcal{T}\rightarrow\infty$), the distortion
decays exponentially as expected. Of course, within the extraction radius
there is no dilation as the scrambled agent still knows how far it can reach,
$y=r$, even if it cannot judge properly distances $y>r$. Maximal dilation
$\lambda^{\max}$, occurs at $y=R$ in Eq. (III). This constitutes an upper
bound enabling the determination of a lower bound on the viability of a
scrambled agent, given the actual viability of an unscrambled agent.

We scale down the length scale of the scrambled agent by $(1+\lambda^{\max})$
so that it matches the length scale of the unscrambled agent. Unfortunately,
this means the speeds of the two no longer match; to restore equality we must
scale down the timescale by the same amount. Numerical simulations indicate
that effects of this rescaling does not affect the environmental variables.
Focusing therefore only on the agent variables, the rescaled viability is

$\displaystyle V_{\eta}=\mathds{E}_{\eta}[T]$
$\displaystyle=\mathds{E}_{0}[(1+\lambda^{\max})^{\mathchar 45\relax 1}T]$
$\displaystyle=(1+\lambda^{\max})^{\mathchar 45\relax 1}V_{\text{actual}}.$
(6)

Expanding this expression for $\lambda\ll 1$, above the semantic threshold,
the viability as a function of the transfer entropy is bounded from below by

$\displaystyle\frac{V(\mathcal{T})}{V_{\text{actual}}}\gtrsim$ $\displaystyle\
\ 2-\frac{r}{R}+\frac{2^{\mathcal{T}-\mathcal{T}^{c}}}{\sin^{\mathchar
45\relax 1}\frac{r}{R}}$
$\displaystyle\times\\!\\!\left[\frac{r}{R}E\left(\left.\frac{\sin^{\mathchar
45\relax
1}\frac{r}{R}}{2^{\mathcal{T}-\mathcal{T}^{c}}}\right|\frac{R^{2}}{r^{2}}\right)\\!-\\!\sin\left(\frac{\sin^{\mathchar
45\relax 1}\frac{r}{R}}{2^{\mathcal{T}-\mathcal{T}^{c}}}\right)\right].$ (7)

The expression is plotted for both the ballistic forager described here, as
well as the more common diffusive forager, in the right panels of Figure 4. As
the figure indicates, the agreement with numerical simulations is excellent
until one reaches the semantic threshold.

Figure 4: A numerical analysis of the viability lower bound for both our model
(the Ballistic Forager, BF), and another model popular in the literature (the
Diffusive Forager, DF). The larger inset figure is the same as the top panel
of Fig. 3, with the viability plateau highlighted for each type of forager,
and expanded in the figures on the right. The semantic threshold is the dashed
blue vertical line, while the pink line in both insets is the expression in
Eq. (III). The expression is in good agreement with the simulations, until one
reaches the semantic threshold.

Thus the viability factorizes into a plateau that depends implicitly on both
environment and agent, and a scaling term that depends only on properties of
the agent. Furthermore, below the semantic threshold the viability curve of
the ballistic forager asymptotes to the diffusive forager. This is fairly
straightforward to understand. With an increasingly noisy sensor, the agent is
as likely to move toward a resource as they are to move away, which correponds
to an ordinary random walk.

One can estimate the minimal viability, $V(0)=\mathds{E}_{1}[T]$, by noting
that resources encountered by such an agent follow a Poisson distribution with
rate $2vr/\ell_{Re}^{2}$. The total fuel processed by the agent will therefore
be $s_{0}+\epsilon\mathds{E}_{1}[T]$, where $s_{0}$ is the initial stored
fuel. But this total fuel divided by the metabolic rate is precisely the
expected lifetime; a little algebra reveals

$V(0)=\frac{s_{0}/\mu}{1-v/v^{\star}},$

where $v^{\star}$ is the upper velocity limit defined in Eq. (1). As mentioned
earlier, the agent approaches immortality when its velocity approaches this
limit, even if it has a completely random sensor.

##  IV Discussion and Conclusion

In this work, we explored the application of Semantic Information to the well-
motivated problem of a resource foraging agent. Semantic information is
defined as the subset of correlations—measured here via the transfer entropy
$\mathcal{T}_{E\rightarrow A}^{\eta}$— between agent $A$ and environment $E$
that is necessary for the agent to maintain its viability $V$. Viability, in
turn, is endogenously defined as opposed to the use of exogenous quantities
like utility functions. The semantic information content in a particular
agent-environment system is determined by rerunning the system evolution for
intervened versions of the dynamics. Such interventions involve “scrambling”
the information transfer between the agent and environment in a specified way.
Tracking changes in agent viability for a distribution of intervened system
trajectories allows the semantic information content to be determined.

Applying this procedure to our forager model required finding appropriate
approximations to the fully specified state space of the original KW18
formalism. For realistic systems with many degrees of freedom, the full
specification of the joint probability distribution $\rho_{AE}$ over the full
state space, as required in the initial formalism, may prove computationally
expensive or even intractable. For our model we adopted a phenomenological
perspective which traced over those degrees of freedom not essential to
specification of the viability. This left us with a reduced state space in
which intervened trajectories could be simulated and the subsequent transfer
entropy could be calculated.

In our model, the forager’s movements were determined by its ability to sense
the presence of an individual unit of resource. Once detected (sensing was
limited to within a radius R) the forager moved towards the resource. The
transfer entropy was scrambled by adding noise to the forager’s sensor via a
parameter $\eta$ where $\eta=1$ implied a complete loss of forager’s ability
to sense the direction of the resource. Our results, expressed in terms of a
viability function defined as the expectation value of the forager’s life-
time, clearly showed the effect of adding noise to the sensor. For
$\eta>\eta_{c}$, where $\eta_{c}$ is a critical value set by the sensing
radius, the forager’s expected lifetime was dramatically reduced (Fig. 2). We
refer to this as the semantic threshold.

This result by itself represents an important extension of previous work on
forager dynamics Bénichou and Redner [2014]; Bhat  _et al._ [2017]; Bénichou
_et al._ [2016]; Bhat and Redner [2022]. What is novel in our work was the way
in which casting the problem in terms of semantic information reveals useful
aspects of the model dynamics. The transfer entropy represents correlations
established between the forager (an agent) and the environment via the agent’s
sensor. A blind forager $\eta=1$ would be fully decoupled from the
environment. By tracking how the forager’s viability changes as these
correlations are either increased or decreased, we gain some understanding
about the role they play in the forager’s ability to persist. In particular
the upper panel of Fig. 3 which shows the viability curve $V(\mathcal{T})$,
reveals two essential ways to understand the role of such correlations for the
forager (or for any agent).

Above the semantic threshold we find a plateau of high viability. Moving from
right to left in this region we are removing correlations between agent and
environment, however this does not effect the agents ability to maintain its
existence. Thus the correlations that are being removed are not essential to
the couplings between agent and environment that maintain viability. Below the
semantic threshold each bit of information, affects the agents ability to
persist. Once this threshold is passed we see the viability monotonically
decrease to zero, corresponding to a dead agent. Thus casting the
forager/environment system into the semantic information formalism allows us
to see exactly how much information matters. In addition our ability to model
the shape of the $V(\mathcal{T})$ curve and the location of the semantic
threshold in terms of the forager/environment parameters ($R,r,\mu,\eta$)
allows us to see how the correlations are instantiated. Thus our work may
provide a useful model for others who want to cast studies of established
agents in terms of semantic information in order to better understand the
underlying nature of correlations and information dynamics.

Coming from the left in Fig. 3 leads us to a different perspective which may
prove useful in using semantic information to understand how agents arise in
the first place. Beginning with the low viability region on the left we see
that adding correlations, initially has little effect. The forager dies
quickly and adding an additional bit of correlation with the environment does
not change that outcome. As more bits of information are acquired, the agent’s
viability slowly rises. However, it is only near the semantic threshold that
the slope of the curve accelerates and _viability per bit_ (VpB) peaks. Thus,
it is possible that this threshold may prove useful for understanding the role
information plays in allowing systems to become autonomous agents. Both
hurricanes and cells are non-linear, driven, far-from-equilibrium systems, but
only cells are considered agents. Future work could explore the relationship
between the accumulation of semantic information and the emergence of agent-
like behavior while also considering the thermodynamic cost of that
accumulation.

## Acknowledgements

This project was partly made possible through the support of Grant 62417 from
the John Templeton Foundation. The opinions expressed in this publication are
those of the author(s) and do not necessarily reflect the views of the John
Templeton Foundation. AK thanks Sosuke Ito for support and encouragement. JP
is supported by “María de Maeztú” fellowship MDM-2014-0370-17-2.

##  Appendix A Equilibrium resource density

Consider a large patch of the environment, $A\subset\mathds{R}^{2}$. The
number of resources in the patch is

$\displaystyle N(t)=\int_{A}\\!\mathrm{d}^{2}x\ n({\boldsymbol{x}},t),$ (8)

where $n$ is the resource density. Resource do not move, but they can be
created or destroyed. Creation is due to an energy flux impinging on $A$,
denoted $\Gamma({\boldsymbol{x}},t)$; there exists some mechanism in the
environment that converts this flux into localized resource deposits of energy
$\epsilon$. Destruction is due to another mechanism by which resources decay,
their energy lost as heat, denoted $\gamma({\boldsymbol{x}},t)$. Note that the
former rate is independent of the number of resources, the latter is not. The
rate of change of resources in this large patch is

$\displaystyle\frac{dN}{dt}=\int_{A}\\!\mathrm{d}^{2}x\
\left(\frac{\Gamma({\boldsymbol{x}},t)}{\epsilon}-\gamma({\boldsymbol{x}},t)n({\boldsymbol{x}},t)\right).$
(9)

Taking $\Gamma$ and $\gamma$ both to be static, the resource density satisfies

$\displaystyle\frac{\partial n}{\partial t}$
$\displaystyle=\frac{\Gamma}{\epsilon}-\gamma n$ $\displaystyle\Downarrow$
$\displaystyle n(t)$ $\displaystyle=n_{0}e^{-\gamma t}+n_{eq}(1-e^{-\gamma
t}).$ (10)

where the equilibrium density is $n_{eq}=\Gamma/\epsilon\gamma$. The average
area occupied by a single resource is the reciprocal of this,
$\epsilon\gamma/\Gamma$, so that the average spacing between resources at
equilibrium is

$\displaystyle\ell_{R_{e}}=\sqrt{\frac{\epsilon\gamma}{\Gamma}}.$ (11)

If instead we coarse grain over the positions of resources and care only about
how many their are, then the transition probabilities for a resource being
generated or destroyed in a time $\Delta t\rightarrow 0$ are $p(N\rightarrow
N+1)=\Gamma A\Delta t/\epsilon$ and $p(N\rightarrow N-1)=\gamma N\Delta t$.
The transitions between these coarse grained states define an $M/M/\infty$
queue — a well known stochastic process with a stationary distribution that is
Poisson:

$\displaystyle p(N)=\frac{1}{N!}\left(\frac{\Gamma
A}{\epsilon\gamma}\right)^{N}e^{-\frac{\Gamma A}{\epsilon\gamma}}.$ (12)

The expected number of resources is $\langle N\rangle=\frac{\Gamma
A}{\epsilon\gamma}$, which under the assumption of a homogeneous distribution,
gives an equilibrium number density that matches $n_{eq}$ from the preceding
paragraphs. The variance in the number of resources is $\delta N^{2}=n_{eq}A$,
giving a fluctuation in the number density of $\delta n=\ell_{Re}^{\mathchar
45\relax 1}A^{-1/2}$. Thus when considering large environmnets,
$A\gg\ell_{Re}^{2}$, the relative fluctuations around the equilibrium density
are negligible, $\delta n/n\rightarrow 0$.

##  Appendix B The Semantic Threshold

Consider an agent for which a resource has just entered sensor range, and a
targeting event has occurred. We scramble the information relayed by the event
so that the agent reorients itself with a target that is misaligned with the
resource. The agent begins its journey towards the target, as shown in Fig 5.

$r$$R$AgentTargetResource Figure 5: The geometry of an agent (mis)targeting a
resource at the edge of sensor range.

For an agent moving towards a target any resource that falls within the grey
region will be collected. The maximal misalignment angle between target and
resource is $\pi\eta$. From the diagram once can infer a critical level of
scrambling, $\eta_{c}$, satisfying

$\sin\pi\eta_{c}=\frac{r}{R},$ (13)

below which the agent will always collect the resource. Above it, the agent
will sometimes miss the resource - an event that could potentially result in
starvation. Combining Eq.13 with Eq.II.2 gives the semantic threshold, Eq. 4.

##  Appendix C The Viability Plateau

Let’s examine a case below the critical scrambling strength, so that every
targetted resource is still collected. Once again we consider a scrambled
targeting event, but assume it occurs for a resource a distance $y<R$ away
from the agent. This type of event occurs regularly after the agent has
collected a resource and there is at least one other resource in sensor range.
The agent begins to move towards the target as depicted in Fig. 6.

$y$$y\sin\theta$$r$AgentTargetResource$p_{1}$$p_{2}$$p_{3}$$\theta$ Figure 6:
The geometry describing how far an agent has to travel to collect a
mistargeted resource.

Had the sensor not been scrambled, the agent would traverse a distance $y-r$
to collect the resource. Meanwhile, the scrambled agent needs to travel the
distance from $p_{2}$ to the resource for collection to occur; a little
geometry and trigonometry on the diagram give us this distance as
$y\cos\theta-\sqrt{r^{2}-y^{2}\sin^{2}\theta}$. Since $p_{2}$ to $p_{3}$ is
also a distance $y-r$, denote the remaining distance $\lambda(\theta)y$ where
$\lambda(\theta)$ is referred to as a dilation factor. It will become clear
why in what follows.

A little algebra gets us the angular dependence of the dilation factor,

$\displaystyle\lambda(\theta)=$
$\displaystyle\frac{r}{y}-1+\cos\theta-\sqrt{\frac{r^{2}}{y^{2}}-\sin^{2}\theta}.$
(14)

Of course this factor will be different each time a targeting event occurs,
depending on distance to resource and the particular mismatch angle of the
target. However, for a given $\eta$ we know the distribution of mismatch
angles - it is uniform over $[-\pi\eta,\pi\eta]$ \- so, for fixed $y$, the
expected dilation is

$\displaystyle\mathds{E}_{\eta}[\lambda]=\frac{1}{2\pi\eta}\int_{\mathchar
45\relax\pi\eta}^{\pi\eta}\\!\\!\\!d\theta\ \lambda(\theta).$

Examining the functional form of $\lambda(\theta)$ we see that the first two
terms are trivial and the third term easily integrates to
$\sin\pi\eta/\pi\eta$, while the fourth term requires a little massaging.
After factoring out a $-r/y\pi\eta$ and using the evenness of the integrand,
the remaining factor is massaged into the form

$\displaystyle\int_{0}^{\pi\eta}\\!\\!\\!d\theta\
\sqrt{1-\frac{y^{2}}{r^{2}}\sin^{2}\theta}.$

The integral is not elementary, but turns out to be an incomplete elliptic
integral of the second kind,

$\displaystyle E(\varphi|k^{2})=\int_{0}^{\varphi}\\!\\!\\!d\theta\
\sqrt{1-k^{2}\sin^{2}\theta},$ (15)

with modulus $k$. There’s a slight nuance since normally $0<k^{2}<1$, while in
our case $y>r$ gives a modulus greater than unity. This bound ensures the
integrand remains real for all values of $\varphi$. Fortunately, since we’re
considering cases below critical scrambling, $y\sin\theta<r$ is automatically
satisfied and there’s no need to worry about which branch we’re on. With this
minutiae out of the way, we write the expected dilation for fixed $y$

$\displaystyle\mathds{E}_{\eta}[\lambda]=\frac{r}{y}-1+\frac{\sin\pi\eta}{\pi\eta}-\frac{r}{y}\frac{E(\pi\eta|\frac{y^{2}}{r^{2}})}{\pi\eta},$
(16)

which when combined with Eq. II.2 and 4, yields the first line of Eq. III. To
simplify the notation, we simply denote this as $\lambda$ in the main text.

## References

  * Schrodinger [1944] E. Schrodinger,  _What is life_ (Cambridge: Cambridge University Press, 1944).
  * Shannon [1948] C. E. Shannon, The Bell system technical journal 27, 379 (1948).
  * Watson and Crick [1953] J. D. Watson and F. H. Crick, Nature 171, 737 (1953).
  * Hopfield [1994] J. Hopfield, Journal of Theoretical Biology 171, 53 (1994).
  * Godfrey-Smith [2007] P. Godfrey-Smith, “Information in biology,” in  _The Cambridge Companion to the Philosophy of Biology_, Cambridge Companions to Philosophy, edited by D. L. Hull and M. Ruse (Cambridge University Press, Cambridge, 2007) pp. 103–119.
  * Mattingly  _et al._ [2021] H. Mattingly, K. Kamino, B. Machta, and T. Emonet, Nature physics 17, 1426 (2021).
  * Küppers [1990] B.-O. Küppers,  _Information and the Origin of Life_ (Mit Press, 1990).
  * Adami [2004] C. Adami, Physics of Life Reviews 1, 3 (2004).
  * Uda [2020] S. Uda, Biophysical reviews 12, 377 (2020).
  * Gohari  _et al._ [2016] A. Gohari, M. Mirmohseni, and M. Nasiri-Kenari, IEEE Transactions on Molecular, Biological and Multi-Scale Communications 2, 120 (2016).
  * Rhee  _et al._ [2012] A. Rhee, R. Cheong, and A. Levchenko, Physical biology 9, 045011 (2012).
  * Tkačik and Bialek [2016] G. Tkačik and W. Bialek, Annual Review of Condensed Matter Physics 7, 89 (2016).
  * Donaldson-Matasci  _et al._ [2010] M. C. Donaldson-Matasci, C. T. Bergstrom, and M. Lachmann, Oikos 119, 219 (2010).
  * Rivoire and Leibler [2011] O. Rivoire and S. Leibler, Journal of Statistical Physics 142, 1124 (2011).
  * Hazen  _et al._ [2007] R. M. Hazen, P. L. Griffin, J. M. Carothers, and J. W. Szostak, Proceedings of the National Academy of Sciences 104, 8574 (2007).
  * Sowinski  _et al._ [2022] D. R. Sowinski, J. Carroll-Nellenback, J. DeSilva, A. Frank, G. Ghoshal, and M. Gleiser, Entropy 24, 1378 (2022).
  * Oh  _et al._ [2020] S. Oh, E. F. Bowen, A. Rodriguez, D. Sowinski, E. Childers, A. Brown, L. Ray, and R. Granger, arXiv preprint arXiv:2011.00088 (2020).
  * Bowen  _et al._ [2022] E. F. Bowen, A. M. Rodriguez, D. R. Sowinski, and R. Granger, Journal of Vision 22, 4 (2022).
  * Schlosser [1998] G. Schlosser, Synthese 116, 303 (1998).
  * Mossio  _et al._ [2009] M. Mossio, C. Saborido, and A. Moreno, The British Journal for the Philosophy of Science 60, 813 (2009).
  * Fang and Wang [2020] X. Fang and J. Wang, Annual Review of Biophysics 49, 227 (2020).
  * D Polani [2001] J. K. D Polani, T Martinez,  _An information-theoretic approach for the quantification of relevance_ (Springer, Berlin, Germany, 2001).
  * Thompson and Stapleton [2009] E. Thompson and M. Stapleton, Topoi 28, 23 (2009).
  * Nehaniv  _et al._ [2003] C. Nehaniv, D. Polani, and K. Dautenhahn,  _Meaningful Information, Sensor Evolution, and the Temporal Horizon of Embodied Organisms_ (2003).
  * Barham [1996] J. Barham, Biosystems 38, 235 (1996).
  * Deacon [2007] T. W. Deacon, Cognitive Semiotics 1, 123 (2007).
  * Corning [2007] P. A. Corning, Systems Research and Behavioral Science: The Official Journal of the International Federation for Systems Research 24, 297 (2007).
  * Gleiser and Sowinski [2018a] M. Gleiser and D. Sowinski, The Map and the Territory: Exploring the Foundations of Science, Thought and Reality , 141 (2018a).
  * Kolchinsky and Wolpert [2018] A. Kolchinsky and D. H. Wolpert, Interface focus 8, 20180041 (2018).
  * Rovelli [2018] C. Rovelli, in  _Wandering Towards a Goal_, The Frontiers Collection (Springer, Cham, 2018) pp. 17–27.
  * Gould [1974] J. P. Gould, Journal of Economic Theory 8, 64 (1974).
  * Stratonovich [2020] R. L. Stratonovich,  _Theory of Information and its Value_, edited by R. V. Belavkin, P. M. Pardalos, and J. C. Principe (Springer International Publishing, Cham, 2020).
  * Shannon  _et al._ [1959] C. E. Shannon _et al._ , IRE Nat. Conv. Rec 4, 1 (1959).
  * Sowinski  _et al._ [2021] D. R. Sowinski, M. D. McGarry, E. E. Van Houten, S. Gordon-Wylie, J. B. Weaver, and K. D. Paulsen, Frontiers in physics 8, 617582 (2021).
  * Acebrón  _et al._ [2005] J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, Reviews of Modern Physics 77, 137 (2005).
  * García-Selfa  _et al._ [2021] D. García-Selfa, G. Ghoshal, C. Bick, J. Pérez-Mercader, and A. P. Muñuzuri, Chaos, Solitons & Fractals 145, 110809 (2021).
  * Cooper  _et al._ [2013] F. Cooper, G. Ghoshal, A. Pawling, and J. Pérez-Mercader, Physical Review Letters 111, 044101 (2013).
  * Mimar  _et al._ [2019] S. Mimar, M. M. Juane, J. Park, A. P. Muñuzuri, and G. Ghoshal, Physical Review E 99, 062303 (2019).
  * COHEN [1977] J. E. COHEN, Nature 270, 165 (1977).
  * Rooney  _et al._ [2006] N. Rooney, K. McCann, G. Gellner, and J. C. Moore, Nature 442, 265 (2006).
  * Xavier  _et al._ [2020] J. C. Xavier, W. Hordijk, S. Kauffman, M. Steel, and W. F. Martin, Proceedings of the Royal Society B: Biological Sciences 287, 20192377 (2020).
  * Blain and Szostak [2014] J. C. Blain and J. W. Szostak, Annual Review of Biochemistry 83, 615 (2014), pMID: 24606140, https://doi.org/10.1146/annurev-biochem-080411-124036 .
  * Ruzzante  _et al._ [2023] B. Ruzzante, L. Del Moro, M. Magarini, and P. Stano, IEEE Transactions on Molecular, Biological and Multi-Scale Communications (2023).
  * Bénichou and Redner [2014] O. Bénichou and S. Redner, Physical review letters 113, 238101 (2014).
  * Bhat  _et al._ [2017] U. Bhat, S. Redner, and O. Bénichou, Physical Review E 95, 062119 (2017).
  * Bénichou  _et al._ [2016] O. Bénichou, M. Chupeau, and S. Redner, Journal of Physics A: Mathematical and Theoretical 49, 394003 (2016).
  * Bhat and Redner [2022] U. Bhat and S. Redner, Journal of Statistical Mechanics: Theory and Experiment 2022, 033402 (2022).
  * Schreiber [2000] T. Schreiber, Phys. Rev. Lett. 85, 461 (2000).
  * Gleiser and Sowinski [2018b] M. Gleiser and D. Sowinski, Physical Review D 98, 056026 (2018b).