# Decomposing the local arrow of time in interacting systems

Christopher W. Lynn  Initiative for the Theoretical Sciences, The Graduate
Center, City University of New York, New York, NY 10016, USA  Joseph Henry
Laboratories of Physics and Lewis–Sigler Institute for Integrative Genomics,
Princeton University, Princeton, NJ 08544, USA  Caroline M. Holmes  Joseph
Henry Laboratories of Physics and Lewis–Sigler Institute for Integrative
Genomics, Princeton University, Princeton, NJ 08544, USA  William Bialek
Initiative for the Theoretical Sciences, The Graduate Center, City University
of New York, New York, NY 10016, USA  Joseph Henry Laboratories of Physics and
Lewis–Sigler Institute for Integrative Genomics, Princeton University,
Princeton, NJ 08544, USA  David J. Schwab  Initiative for the Theoretical
Sciences, The Graduate Center, City University of New York, New York, NY
10016, USA

###### Abstract

We show that the evidence for a local arrow of time, which is equivalent to
the entropy production in thermodynamic systems, can be decomposed. In a
system with many degrees of freedom, there is a term that arises from the
irreversible dynamics of the individual variables, and then a series of
non–negative terms contributed by correlations among pairs, triplets, and
higher–order combinations of variables. We illustrate this decomposition on
simple models of noisy logical computations, and then apply it to the analysis
of patterns of neural activity in the retina as it responds to complex dynamic
visual scenes. We find that neural activity breaks detailed balance even when
the visual inputs do not, and that this irreversibility arises primarily from
interactions between pairs of neurons.

A system held in steady state, away from thermal equilibrium, must
continuously dissipate heat to the surrounding bath, causing an increase in
entropy. Such a system also violates detailed balance, so that the time
reversed trajectories must be measurably less probable than the true
trajectories; observation of the system trajectory thus provides evidence for
the arrow of time. An important result of modern non–equilibrium statistical
mechanics is that the rate at which evidence—in the precise,
information–theoretic sense—accumulates for the arrow of time is equal to the
rate at which entropy is produced in the bath Seifert (2005); Kawai  _et al._
(2007). This idea has been used to search for signatures of irreversibility in
experimental data, notably on a wide range of living systems, across scales
from single cells Battle  _et al._ (2016); Gnesotto  _et al._ (2018); Tan  _et
al._ (2021) to global brain dynamics Lynn  _et al._ (2021); Sanz Perl  _et
al._ (2021).

In almost all the systems where we want to study entropy production and the
arrow of time, there are many interacting degrees of freedom. If we try to
estimate the entropy, then we know that treating each variable independently
results in an over–estimate, and that as we take account of correlations among
pairs, triplets, and larger groups of variables we generate a monotonically
decreasing hierarchy of bounds Schneidman  _et al._ (2003). Here we show that
the opposite is true of the entropy production, or the evidence for the arrow
of time: we can decompose this measure of irreversibility into a series of
non–negative terms, corresponding to successive orders of correlation or
interaction among the variables in the system. While correlations always
decrease the entropy, we find that correlations always increase the evidence
for irreversibility. This leads to a new way of analyzing the origins of
irreversibility in interacting systems, which we apply to the neural
representation of visual inputs.

Evidence for the arrow of time arises because the probability of observing a
trajectory and its time reverse are different, consistently. The proper
information–theoretic measure of this difference is the Kullback–Leibler (KL)
divergence Cover and Thomas (2012). If we write trajectories schematically as
$x(t)$, with $0<t<T$, and the corresponding time reversed trajectories as
$\tilde{x}(t)$, then the evidence for the arrow of time is

$E\equiv
D_{KL}\left(P[x(t)]\,||\,P[\tilde{x}(t)]\right)=\sum_{x(t)}P[x(t)]\log\left(\frac{P[x(t)]}{P[\tilde{x}(t)]}\right).$
(1)

For large $T$ in steady–state systems, under mild assumptions, this evidence
grows linearly with time, so it is natural to define a (global) rate

$\dot{I}_{\rm global}\equiv\lim_{T\rightarrow\infty}\frac{E}{T}.$ (2)

For a wide range of systems that can be described as making transitions
coupled to a heat bath, this rate is equal to the rate $\dot{S}$ of entropy
production that results from heat dissipation to the bath Seifert (2005);
Kawai  _et al._ (2007). To simplify the discussion, it is convenient to think
of time advancing in discrete steps, and to consider observing just one
transition, or two steps of the dynamics, at a time. This “local arrow of
time” or “local irreversibility” can be written

$\dot{I}=\sum_{x,x^{\prime}}P(x\rightarrow
x^{\prime})\log\left[\frac{P(x\rightarrow x^{\prime})}{P(x^{\prime}\rightarrow
x)}\right],$ (3)

where

$P(x\rightarrow x^{\prime})=\text{Prob}(x_{t}=x,\,x_{t+1}=x^{\prime}).$ (4)

One can view this not as the Markovian approximation to the global
irreversibility, but instead as the exact evidence for the arrow of time
contained in individual transitions. Equation (3) makes very explicit that
irreversibility is the breaking of detailed balance.

We are interested in systems where the state $x$ encompasses many interacting
variables, $x\equiv\\{x_{\rm i}\\}$, with ${\rm i}=1,\,2,\,\cdots,\,N$. If we
can measure dynamics with sufficient temporal resolution, then no two
variables can change state at the same time. Then instead of considering the
full distribution $P(x\rightarrow x^{\prime})$ we can focus on the $N$
individual distributions $P_{\rm i}(x_{\rm i}\rightarrow x_{\rm
i}^{\prime},\,x_{-{\rm i}})$, each of which describes a single variable
transitioning from $x_{\rm i}$ to $x_{\rm i}^{\prime}$ and the rest of the
system remaining in the same state, denoted $x_{-{\rm i}}$. Such dynamics are
referred to as multipartite, and exhibit a number of useful properties
Horowitz and Esposito (2014); Horowitz (2015); Wolpert (2020). Chief among
these properties is the fact that the local irreversibility simplifies to a
sum over the irreversibilities associated with the individual elements:

$\dot{I}=\sum_{{\rm i}=1}^{N}\dot{I}_{\rm i},$ (5)

where

$\dot{I}_{\rm i}=\sum_{x_{-{\rm i}}}\sum_{x_{\rm i},x_{\rm i}^{\prime}}P_{\rm
i}(x_{\rm i}\rightarrow x_{\rm i}^{\prime},\,x_{-{\rm
i}})\log\left[\frac{P_{\rm i}(x_{\rm i}\rightarrow x_{\rm
i}^{\prime},\,x_{-{\rm i}})}{P_{\rm i}(x_{\rm i}^{\prime}\rightarrow x_{\rm
i},\,x_{-{\rm i}})}\right].$ (6)

If the different variables in the system are independent of one another, then
the dynamics are fully defined by the marginal probabilities $P_{\rm i}(x_{\rm
i}\rightarrow x_{\rm i}^{\prime})=\sum_{x_{-{\rm i}}}P_{\rm i}(x_{\rm
i}\rightarrow x_{\rm i}^{\prime},\,x_{-{\rm i}})$, leading to an
irreversibility

$\dot{I}^{\text{ind}}=\sum_{{\rm i}=1}^{N}\dot{I}^{\text{ind}}_{\rm i},$ (7)

where

$\dot{I}_{\rm i}^{\rm ind}=\sum_{x_{\rm i},x_{\rm i}^{\prime}}P_{\rm i}(x_{\rm
i}\rightarrow x_{\rm i}^{\prime})\log\left[\frac{P_{\rm i}(x_{\rm
i}\rightarrow x_{\rm i}^{\prime})}{P_{\rm i}(x_{\rm i}^{\prime}\rightarrow
x_{\rm i})}\right].$ (8)

Even if the variables are not independent we can always define this
“independent irreversibility”. The difference between this and the true
irreversibility is the result of interactions,

$\dot{I}^{\rm int}\equiv\dot{I}-\dot{I}^{\rm ind}=\sum_{{\rm
i}=1}^{N}\dot{I}^{\text{int}}_{\rm i},$ (9)

where

$\displaystyle\dot{I}^{\text{int}}_{\rm i}$ $\displaystyle=\dot{I}_{\rm
i}-\dot{I}^{\text{ind}}_{\rm i}$ (10) $\displaystyle=\sum_{x_{-{\rm
i}}}\sum_{x_{\rm i},x_{\rm i}^{\prime}}P_{\rm i}(x_{\rm i}\rightarrow x_{\rm
i}^{\prime},\,x_{-{\rm i}})\log\left[\frac{P_{\rm i}(x_{-{\rm i}}\,|\,x_{\rm
i}\rightarrow x_{\rm i}^{\prime})}{P_{\rm i}(x_{-{\rm
i}}\,|\,x_{i}^{\prime}\rightarrow x_{\rm i})}\right]$ (11)
$\displaystyle=\sum_{x_{\rm i},x_{\rm i}^{\prime}}P_{\rm i}(x_{\rm
i}\rightarrow x_{\rm i}^{\prime})D_{KL}^{\rm i},$ (12) $D_{KL}^{\rm
i}=D_{KL}\left[P_{\rm i}(x_{-{\rm i}}\,|\,x_{\rm i}\rightarrow x_{\rm
i}^{\prime})\,||\,P_{\rm i}(x_{-{\rm i}}\,|\,x_{\rm i}^{\prime}\rightarrow
x_{\rm i})\right],$ (13)

and $P_{\rm i}(x_{-{\rm i}}\,|\,x_{\rm i}\rightarrow x_{\rm
i}^{\prime})=P_{\rm i}(x_{\rm i}\rightarrow x_{\rm i}^{\prime},\,x_{-{\rm
i}})/P_{\rm i}(x_{\rm i}\rightarrow x_{\rm i}^{\prime})$. Since each
$D_{KL}^{\rm i}\geq 0$, this demonstrates that $\dot{I}^{\text{int}}_{\rm
i}\geq 0$, so that interactions can only increase the irreversibility of a
system.

Equation (11) admits a simple interpretation: interactions contribute to the
arrow of time if the observation of $x_{\rm i}\rightarrow x_{\rm i}^{\prime}$
as opposed to $x_{\rm i}^{\prime}\rightarrow x_{\rm i}$ points toward
different states $x_{-{\rm i}}$ of the rest of the system. Thus, if $\rm i$’s
forward- and reverse-time dynamics contain the same information about the rest
of the system, then interactions do not contribute to $\rm i$’s local
irreversibility ($\dot{I}^{\text{int}}_{\rm i}=0$), and violations of detailed
balance can only arise from independent dynamics ($\dot{I}_{\rm
i}=\dot{I}^{\text{ind}}_{\rm i}$).

Together, Eqs. (7-13) establish our first main result: that the local arrow of
time can be split into two non–negative components,

$\dot{I}=\dot{I}^{\text{ind}}+\dot{I}^{\text{int}},$ (14)

where $\dot{I}^{\text{ind}}$ reflects the local irreversibility of the
individual elements and $\dot{I}^{\text{int}}$ reflects the local
irreversibility due to interactions among the elements. Notice that for binary
or Ising variables in steady state, we must have $P_{\rm i}(x_{\rm
i}\rightarrow x_{\rm i}^{\prime})=P_{\rm i}(x_{\rm i}^{\prime}\rightarrow
x_{\rm i})$ (such that $\dot{I}^{\text{ind}}=0$), and so the local arrow of
time necessarily arises from interactions ($\dot{I}=\dot{I}^{\text{int}}$).
Additionally, we note that decomposition in Eq. (14) requires multipartite
dynamics; if multiple elements can change at once, then $\dot{I}^{\text{int}}$
is ill–defined (see Ref. Lynn  _et al._ ).

When we say that interactions contribute to irreversibility, we have the
intuition that these contributions can be further decomposed into interactions
among pairs, triplets, etc.. Saying that we know only about interactions among
pairs, for example, is equivalent to saying that we know all the marginal
distributions

$P_{\rm i}(x_{\rm i}\rightarrow x_{\rm i}^{\prime},\,x_{\rm
j})=\sum_{x_{-\\{{\rm i},{\rm j}\\}}}P_{\rm i}(x_{\rm i}\rightarrow x_{\rm
i}^{\prime},\,x_{-{\rm i}}),$ (15)

where the sum runs over the states of all elements other than $\rm i$ and $\rm
j$. We can then ask: what is the minimal irreversibility, or the weakest arrow
of time, implied by these pairwise dynamics? To answer this question, we can
search over all possible distributions $P_{\rm i}(x_{\rm i}\rightarrow x_{\rm
i}^{\prime},x_{-{\rm i}})$ that are consistent with the marginals in Eq. (15).
Among these hypothetical systems, one will achieve a minimum of the local
irreversibility in Eq. (5), thus defining the minimum irreversibility
consistent with the observed pairwise dynamics, denoted $\dot{I}^{(2)}$. This
generalizes to higher orders, and since knowledge of $k^{\rm th}$–order
dynamics includes all information about dynamics of lower orders
$k^{\prime}<k$, the result is a series of nested bounds,

$0\leq\dot{I}^{(1)}\leq\dot{I}^{(2)}\leq\cdots\leq\dot{I}^{(N-1)}\leq\dot{I}^{(N)}=\dot{I},$
(16)

where we separately verify that $\dot{I}^{(1)}=\dot{I}^{\rm ind}$ Lynn  _et
al._ .

The hierarchy of bounds in Eq. (16) is analogous to the hierarchy of bounds on
the entropy itself Schneidman  _et al._ (2003), but with inequalities
reversed. When subjected to linear constraints on the underlying probability
distributions—such as constraints on marginal distributions—entropy has a
maximum while mutual information or KL divergence have a minimum Cover and
Thomas (2012). Colloquially, “telling you more” about the distribution
increases the evidence for the local arrow of time. Note that computing each
bound $\dot{I}^{(k)}$ requires finding a probability distribution that
minimizes the irreversibility $\dot{I}$ subject to constraints on the $k^{\rm
th}$–order dynamics Lynn  _et al._ ; because $\dot{I}=\dot{S}$ in
thermodynamic contexts, this is equivalent to asking for dynamics that
minimize entropy production. Minimizing entropy production is an idea that has
been widely explored Schnakenberg (1976); Wolpert (2020); Skinner and Dunkel
(2021a, b); Still (2020), since the foundational work of Onsager and Prigogine
Onsager (1931); Prigogine (1945). Importantly we are not claiming that real
systems minimize their entropy production, but rather are asking for
hypothetical systems that have minimal entropy production consistent with a
series of increasingly detailed constraints Skinner and Dunkel (2021a, b).

As a final interpretative step, it is natural to compare $\dot{I}^{(k)}$ with
$\dot{I}^{(k-1)}$. If $\dot{I}^{(k)}=\dot{I}^{(k-1)}$, then the
$k^{\text{th}}$–order dynamics are redundant in the sense that their
irreversibility is entirely determined by lower–order correlations; by
contrast, if $\dot{I}^{(k)}>\dot{I}^{(k-1)}$, then the $k^{\text{th}}$–order
dynamics contain new information about the arrow of time. In this way, we can
think about the difference between $\dot{I}^{(k)}$ and $\dot{I}^{(k-1)}$ as
the contribution of interactions of order $k$ to the local arrow of time,
$\dot{I}^{(k)}_{\text{int}}=\dot{I}^{(k)}-\dot{I}^{(k-1)}\geq 0$. This yields

$\displaystyle\dot{I}=\begin{array}[]{c}\\\ \vspace{-4pt}\hfil\\\
\underbrace{\begin{array}[]{c}\dot{I}^{(1)}_{\text{int}}\\\
\vspace{-9pt}\hfil\\\
\end{array}}_{\text{\normalsize$\dot{I}^{\text{ind}}$}}\end{array}+\begin{array}[]{c}\\\
\vspace{-4pt}\hfil\\\
\underbrace{\begin{array}[]{c}\dot{I}^{(2)}_{\text{int}}+\dot{I}^{(3)}_{\text{int}}+\cdots+\dot{I}^{(N)}_{\text{int}}\\\
\vspace{-9pt}\hfil\\\
\end{array}}_{\text{\normalsize$\dot{I}^{\text{int}}$}}\end{array},$ (27)

which is our central result: the local arrow of time can be decomposed into
non–negative contributions from individual elements in the system,
interactions between pairs of elements, interactions among triplets, and so
on.

To illustrate this decomposition, we consider a minimal system of three binary
variables $x,y,z$. At each moment in time, $z$ is a noisy logical function of
$x$ and $y$, and from one time step to the next the variables $x$ and $y$ flip
between their two states with probability $p_{\text{flip}}$, as in Fig. 1(a).
These dynamics are Markovian, so the local arrow of time is also the true
global arrow of time. Since the variables are steady–state and binary, we have
$\dot{I}^{\rm ind}=0$, and since there are only three variables, the possible
contributions are $\dot{I}_{\rm int}^{(2)}$ and $\dot{I}_{\rm int}^{(3)}$.

Figure 1: Decomposing the irreversibility of logical functions. (a) System of
three binary variables $x$, $y$, and $z$, where $z$ performs a noisy logical
function on the inputs $x$ and $y$. At each point in time, one of the
variables is updated at random. With probability $p_{\text{flip}}$, the inputs
$x$ and $y$ change value, and with probability $p_{\text{error}}$, the output
$z$ fails to perform the specified function. (b-d) Pairwise and triplet
contributions to the local arrow of time for different logical functions.
Across all functions, irreversibility decreases with $p_{\text{error}}$ and
increases with $p_{\text{flip}}$. (b) When $z$ copies either $x$ or $y$, the
triplet irreversibility vanishes and all irreversibility arises from pairwise
dynamics. (c) For AND and OR, irreversibility is driven by both pairwise and
triplet dynamics. (d) For XOR, the pairwise irreversibility vanishes and all
irreversibility arises from triplet dynamics.

To begin, consider the simplest function, where $z$ copies either $x$ or $y$
while ignoring the other input [Fig. 1(b)]. As $p_{\text{error}}$ increases
(that is, as the accuracy of the function decreases), we find that the
irreversibility $\dot{I}$ decreases, until at $p_{\text{error}}=1/2$ (when the
output $z$ completely decouples from the inputs $x$ and $y$) the system
becomes reversible ($\dot{I}=0$). Additionally, the irreversibility vanishes
if the inputs $x$ and $y$ are static ($p_{\text{flip}}=0$) and grows as the
inputs become more dynamic ($p_{\text{flip}}$ increases). Notably, for all
values of $p_{\text{flip}}$ and $p_{\text{error}}$, we find that
$\dot{I}^{(3)}_{\text{int}}=0$, and thus the irreversibility of the system
arises entirely from pairwise dynamics, $\dot{I}=\dot{I}^{(2)}_{\text{int}}$.

For comparison, consider the AND, OR, and XOR functions [Figs. 1(c) and (d)].
As before, the irreversibility increases as the functions become more accurate
and as the inputs become more dynamic. In contrast to the copy functions,
however, the irreversibilities of AND and OR (which are identical) arise in
nearly equal amounts from pairwise and triplet interactions [Fig. 1(c)].
Indeed, for both AND and OR, the output $z$ tends to increase with each of the
inputs independently (yielding pairwise irreversibility), yet the full
dynamics are not defined until all three variables are taken into account
(yielding triplet irreversibility). The XOR function is the classic example of
an irreducibly combinatorial interaction, so that the behavior of the system
only becomes apparent once all three variables are observed simultaneously. As
such, the pairwise dynamics are completely reversible, and all of the
irreversibility is driven by triplet dynamics, so that
$\dot{I}=\dot{I}^{(3)}_{\text{int}}$, as shown in Fig. 1(d). Together, the
results in Fig. 1 demonstrate how the local arrow of time can emerge from
different orders of interactions among the many variables, even in relatively
simple cases.

Figure 2: Local irreversibility in groups of neurons. (a-c) A group of three
neurons in the salamander retina responding to a natural movie. (a) Points
mark the times of action potentials in each of three neurons. To define neural
states, we slide a window of width $\Delta t=20$ ms forward in time. When a
cell generates an action potential inside the window we assign $x_{\rm i}=1$,
and when it is silent we assign $x_{\rm i}=0$. Transitions occur when a spike
enters the front or exits the back of the window, as shown. (b) The
distribution $P(x\rightarrow x^{\prime})$ over pairs of the eight possible
states. Black entries indicate disallowed transitions. (c) Fluxes of
probability $P(x\rightarrow x^{\prime})-P(x^{\prime}\rightarrow x)$ between
neural states, demonstrating that this group of neurons breaks detailed
balance. (d) Distributions of local irreversibilities $\dot{I}$ over 5-cell
groups responding either to a natural movie (blue) or a Brownian stimulus
(red), which is designed to be time-reversal invariant. Out of 100 random
5-cell groups, we only plot those ($\sim 2/3$) with significant local
irreversibility.

Our perception of the arrow of time is driven by patterns of activity in
networks of neurons. In particular, our experience of the visual world is
constructed from the activity of retinal ganglion cells in the optic nerve,
carrying information from eye to brain. Experimental developments have made it
possible to record, simultaneously, from larger numbers of these cells, to the
point that we can monitor all of the information that the brain receives about
a small patch of the visual world Marre  _et al._ (2012). Here we focus on one
such experiment, monitoring the activity of fifty–three neurons in the
salamander retina as it responds to movies with different statistical
structure Palmer  _et al._ (2015). Neurons have a naturally binary response,
generating an action potential (spike; $x_{\rm i}=1$) or remaining silent
($x_{\rm i}=0$) in each small window of time ($\Delta t=20\,{\rm ms}$), as in
Fig. 2. As we slide the window forward in time, the time resolution of
detecting action potentials is high enough that the dynamics are multipartite,
so that a spike from only one neuron will enter the front of the window
($x_{\rm i}=0\rightarrow 1$) or exit the back ($x_{\rm i}=1\rightarrow 0$) at
a time, as in Fig. 2(a).

Figure 2(b) shows the probabilities $P(x\rightarrow x^{\prime})$ for a group
of three neurons as they respond to a naturalistic movie, and Fig. 2(c) shows
the corresponding fluxes of probability between states. Qualitatively we see
that there are loops of flux, a hallmark feature of broken detailed balance in
steady–state systems Battle  _et al._ (2016); Gnesotto  _et al._ (2018);
Schnakenberg (1976); Li  _et al._ (2019). In fact, we have checked that the
neurons are in steady state Lynn  _et al._ , such that the observed
irreversibility cannot be attributed to non–stationarity. As with other
information–theoretic quantities, estimating the local irreversibility from
data is challenging, and prone to systematic errors due to finite data; we
find that these can be controlled using the strategy of Ref. Strong  _et al._
(1998) if we restrict our attention to groups of no more than five cells.
These and other technical details will be addressed in a longer paper Lynn
_et al._ . When we choose groups of five cells at random from the fifty–three
in the experimental population, more than two–thirds exhibit values of
$\dot{I}$ that are significantly different from zero, with the distribution
shown in Fig. 2(d).

Movies taken from natural settings, which provide the visual stimuli for the
experiments analyzed here, obviously break time–reversal invariance. But one
can construct movies for which the time–reverse is equally likely, for example
a single bar moving along the trajectory generated by equilibrium Brownian
motion of a damped harmonic oscillator, as was done (for different reasons) in
the experiments of Ref. Palmer  _et al._ (2015). Among the same groups of five
neurons, we again find that roughly two–thirds exhibit nonzero local
irreversibility, but perhaps surprisingly, these groups are more irreversible
during the Brownian stimulus than the natural movie [Fig. 2(d)]. This
difference in local irreversibility holds for all group sizes from $N=2$ to
$N=5$. Together, these results demonstrate that retinal neurons break detailed
balance in a way that does not simply reflect the irreversibility of the
stimulus. In fact, the neuronal dynamics can define an arrow of time, even
when the stimulus does not.

The decomposition in Eq. (LABEL:eq_dec) gives us the opportunity to ask how
the local arrow of time in neural responses is distributed across dynamics of
different order. We recall that the interaction irreversibilities
$\dot{I}^{(k)}_{\text{int}}$ fundamentally quantify the local arrow of time in
$k^{\text{th}}$–order dynamics, and need not be driven by direct connections
between neurons themselves. As noted above, the irreversibility that we
observe is not the result of non–stationarity; more rigorously we find that in
all the data we consider here, distributions really are in steady state within
experimental error Lynn  _et al._ . But for steady–state binary systems, the
individual neurons cannot establish a local arrow of time ($\dot{I}^{\rm
ind}=0$); note that extended sequences of transitions from a single cell could
generate irreversibility, but we focus here only on the local term. Thus, any
irreversibility necessarily arises from statistical dependencies between two
or more neurons at a time. For the same groups of $N=5$ cells in Fig. 2(d)
responding to the natural movie, we find that pairwise dynamics account for
much more of the local irreversibility than complex higher–order dynamics
[Fig. 3(a)]. In fact, for both the natural and (time–reversal invariant)
Brownian movies, pairwise dynamics contribute $66-74\%$ of the local
irreversibility, more than any of the higher order terms [Fig. 3(b)].

Figure 3: Decomposing local irreversibility in neuronal activity. (a)
Distributions of interaction irreversibilities $\dot{I}^{(k)}_{\text{int}}$ of
different orders $k$ for random groups of five neurons responding to the
natural movie. (b) Interaction irreversibility $\dot{I}^{(k)}_{\text{int}}$,
normalized by the full local irreversibility $\dot{I}$, as a function of the
order $k$, averaged over the same 5-cell groups.

To summarize, we have shown how evidence for the local arrow of time
accumulates from the behavior of individual degrees of freedom and their
interactions. Progressively higher–order dynamics each make a non–negative
contribution, adding to the local irreversibility. As a practical matter, this
decomposition allows us to (lower) bound the local irreversibility through
measurements of low–order correlations in many–body systems, in much the same
way that the maximum entropy method allows us to (upper) bound the entropy
itself. We have focused here on the magnitude and decomposition of the local
arrow of time, but it would be interesting to explore the hierarchy of
minimally irreversible models that we construct along the way, especially as
models for living systems. Perhaps these will be as successful in describing
dynamics as the maximum entropy models have been in describing distributions
of states at single moments in time Schneidman  _et al._ (2006); Bialek  _et
al._ (2012); Meshulam  _et al._ (2017); Lynn  _et al._ (2019); Russ  _et al._
(2020). It would be interesting to understand the relationship of the
minimally irreversible models to maximum entropy models for trajectories,
sometimes called maximum caliber Pressé  _et al._ (2013); Cavagna  _et al._
(2014).

As a first step we have used our decomposition to analyze the responses of
small groups of neurons in the retina as they encode complex visual inputs. It
is not surprising that this initial neural representation of the visual world
defines an arrow of time, although it is reassuring that this can be
quantified, reliably. It is perhaps surprising that irreversibility is
stronger in response to inputs that obey detailed balance, raising questions
about how our internal perception of the arrow of time becomes aligned with
the external world. Despite these large differences in the strength of the
local arrow of time in response to different inputs, the way in which large-
scale irreversibility is built out of fine-scale dynamics is constant, with
the dominant role played by correlations among pairs of neurons. This relative
simplicity holds out promise for simplified models of the neural dynamics,
similar to pairwise maximum entropy models in the study of steady–state
distributions. Generally, the emergence of irreversibility from pairwise
dynamics opens the door for future investigations into whether, and how, the
physical connections between neurons combine to produce a collective arrow of
time.

###### Acknowledgements.

We thank SE Palmer for helpful discussions and for guiding us through the data
of Ref. Palmer  _et al._ (2015). This work was supported in part by the
National Science Foundation, through the Center for the Physics of Biological
Function (PHY–1734030) and a Graduate Research Fellowship (CMH); by the
National Institutes of Health through the BRAIN initiative (R01EB026943); by
the James S McDonnell Foundation through a Postdoctoral Fellowship Award
(CWL); by the Simons Foundation; and by a Sloan Research Fellowship (DJS).

Citation diversity statement.—Recent work in several fields of science
Mitchell  _et al._ (2013); Dion  _et al._ (2018); Caplar  _et al._ (2017);
Dworkin  _et al._ (2020); Bertolero  _et al._ (2020), and physics in
particular Teich  _et al._ (2021), has identified citation bias negatively
impacting women and other minorities. Here we sought to proactively consider
choosing references that reflect the diversity of the field in thought, form
of contribution, gender, and other factors. Excluding (including) self-
citations to the current authors, our references contain $19\%$ ($25\%$) women
lead authors and $31\%$ ($30\%$) women senior authors.

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