intellecton/papers/project_paper_2_neuroscience/references/Friston2013.txt

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Cite this article: Friston K. 2013 Life as we
know it. J R Soc Interface 10: 20130475.
http://dx.doi.org/10.1098/rsif.2013.0475

Received: 27 May 2013
Accepted: 12 June 2013

Subject Areas:
biomathematics

Keywords:
autopoiesis, self-organization, active inference,
free energy, ergodicity, random attractor

Author for correspondence:
Karl Friston
e-mail: k.friston@ucl.ac.uk

Life as we know it

Karl Friston

The Wellcome Trust Centre for Neuroimaging, Institute of Neurology, Queen Square, London WC1N 3BG, UK

This paper presents a heuristic proof (and simulations of a primordial soup)
suggesting that life—or biological self-organization—is an inevitable and
emergent property of any (ergodic) random dynamical system that possesses
a Markov blanket. This conclusion is based on the following arguments: if
the coupling among an ensemble of dynamical systems is mediated by
short-range forces, then the states of remote systems must be conditionally
independent. These independencies induce a Markov blanket that separates
internal and external states in a statistical sense. The existence of a Markov
blanket means that internal states will appear to minimize a free energy
functional of the states of their Markov blanket. Crucially, this is the same
quantity that is optimized in Bayesian inference. Therefore, the internal
states (and their blanket) will appear to engage in active Bayesian inference.
In other words, they will appear to model—and act on—their world to pre-
serve their functional and structural integrity, leading to homoeostasis and a
simple form of autopoiesis.

1. Introduction

How can the events in space and time which take place within the spatial boundary of
a living organism be accounted for by physics and chemistry?
Erwin Schro¨dinger [1, p. 2]

The emergence of life—or biological self-organization—is an intriguing issue
that has been addressed in many guises in the biological and physical sciences
[1–5]. This paper suggests that biological self-organization is not as remarkable
as one might think—and is (almost) inevitable, given local interactions between
the states of coupled dynamical systems. In brief, the events that ‘take place
within the spatial boundary of a living organism’ [1] may arise from the very
existence of a boundary or blanket, which itself is inevitable in a physically
lawful world.
The treatment offered in this paper is rather abstract and restricts itself
to some basic observations about how coupled dynamical systems organize
themselves over time. We will only consider behaviour over the timescale
of the dynamics themselves—and try to interpret this behaviour in relation to
the sorts of processes that unfold over seconds to hours, e.g. cellular proces-
ses. Clearly, a full account of the emergence of life would have to address
multiple (evolutionary, developmental and functional) timescales and the
emergence of DNA, ribosomes and the complex cellular networks common
to most forms of life. This paper focuses on a simple but fundamental aspect
of self-organization—using abstract representations of dynamical processes—
that may provide a metaphor for behaviour with different timescales and
biological substrates.
Most treatments of self-organization in theoretical biology have addressed
the peculiar resistance of biological systems to the dispersive effects of fluctu-
ations in their environment by appealing to statistical thermodynamics and
information theory [1,3,5–10]. Recent formulations try to explain adaptive be-
haviour in terms of minimizing an upper (free energy) bound on the surprise
(negative log-likelihood) of sensory samples [11,12]. This minimization usefully
connects the imperative for biological systems to maintain their sensory states
within physiological bounds, with an intuitive understanding of adaptive
behaviour in terms of active inference about the causes of those states [13].

& 2013 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution
License http://creativecommons.org/licenses/by/3.0/, which permits unrestricted use, provided the original
author and source are credited.

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Under
ergodic
assumptions,
the
long-term
average
of surprise is entropy. This means that minimizing free
energy—through selectively sampling sensory input—places
an upper bound on the entropy or dispersion of sensory
states. This enables biological systems to resist the second law
of thermodynamics—or more exactly the fluctuation theorem
that applies to open systems far from equilibrium [14,15].
However, because negative surprise is also Bayesian model
evidence, systems that minimize free energy also maximize a
lower bound on the evidence for an implicit model of how
their sensory samples were generated. In statistics and machine
learning, this is known as approximate Bayesian inference and
provides a normative theory for the Bayesian brain hypothesis
[16–20]. In short, biological systems act on the world to place
an upper bound on the dispersion of their sensed states,
while using those sensations to infer external states of the
world. This inference makes the free energy bound a better
approximation to the surprise that action is trying to minimize
[21]. The resulting active inference is closely related to formu-
lations in embodied cognition and artificial intelligence; for
example, the use of predictive information [22–24] and earlier
homeokinetic formulations [25].
The ensuing (variational) free energy principle has been
applied widely in neurobiology and has been generalized
to other biological systems at a more theoretical level [11].
The motivation for minimizing free energy has hitherto used
the following sort of argument: systems that do not mini-
mize free energy cannot exist, because the entropy of their
sensory states would not be bounded and would increase
indefinitely—by the fluctuation theorem [15]. Therefore, bio-
logical systems must minimize free energy. This paper
resolves the somewhat tautological aspect of this argument
by turning it around to suggest: any system that exists will
appear to minimize free energy and therefore engage in
active inference. Furthermore, this apparently inferential or
mindful behaviour is (almost) inevitable. This may sound
like a rather definitive assertion but is surprisingly easy to
verify. In what follows, we will consider a heuristic proof
based on random dynamical systems and then see that bio-
logical self-organization emerges naturally, using a synthetic
primordial soup. This proof of principle rests on four attributes
of—or tests for—self-organization that may themselves have
interesting implications.

2. Heuristic proof

We start with the following lemma: any ergodic random dynami-
cal system that possesses a Markov blanket will appear to actively
maintain its structural and dynamical integrity. We will associate
this behaviour with the self-organization of living organisms.
There are two key concepts here—ergodicity and a Markov
blanket. Here, ergodicity means that the time average of any
measurable function of the system converges (almost surely)
over a sufficient amount of time [26,27]. This means that one
can interpret the average amount of time a state is occupied
as the probability of the system being in that state when
observed at random. We will refer to this probability measure
as the ergodic density.
A Markov blanket is a set of states that separates two
other sets in a statistical sense. The term Markov blanket was
introduced in the context of Bayesian networks or graphs
[28] and refers to the children of a set (the set of states that

are influenced), its parents (the set of states that influence it)
and the parents of its children. The notion of influence or
dependency is central to a Markov blanket and its existence
implies that any state is—or is not—coupled to another. For
example, the system could comprise an ensemble of subsys-
tems, each occupying its own position in a Euclidean space.
If the coupling among subsystems is mediated by short-range
forces, then distant subsystems cannot influence each other.
The existence of a Markov blanket implies that its states
(e.g. motion in Euclidean space) do not affect their coupling or
independence. In other words, the interdependencies among
states comprising the Markov blanket change slowly with
respect to the states per se. For example, the surface of a cell
may constitute a Markov blanket separating intracellular and
extracellular states. On the other hand, a candle flame cannot
possess a Markov blanket, because any pattern of molecular
interactions is destroyed almost instantaneously by the flux
of gas molecules from its surface.
The existence of a Markov blanket induces a partition of
states into internal states and external states that are hidden
(insulated) from the internal (insular) states by the Markov
blanket. In other words, the external states can only be seen
vicariously by the internal states, through the Markov blanket.
Furthermore, the Markov blanket can itself be partitioned into
two sets that are, and are not, children of external states. We
will refer to these as a surface or sensory states and active
states, respectively. Put simply, the existence of a Markov blan-
ket S � A implies a partition of states into external, sensory,
active and internal states: x [ X ¼ C � S � A � L. Exter-
nal states cause sensory states that influence—but are not
influenced by—internal states, while internal states cause
active states that influence—but are not influenced by—
external states (table 1). Crucially, the dependencies induced
by Markov blankets create a circular causality that is reminis-
cent of the action–perception cycle (figure 1). The circular
causality here means that external states cause changes in
internal states, via sensory states, while the internal states
couple back to the external states through active states—such
that internal and external states cause each other in a reciprocal

Table 1. Definitions of the tuple ðV; C; S; A; L; p; qÞ underlying active
inference.

a sample space V or non-empty set from which random fluctuations
or outcomes v [ V are drawn

external states C : C � A � V ! R states of the world that
cause sensory states and depend on action

sensory states S : C � A � V ! R the agent’s sensations that
constitute a probabilistic mapping from action and external states

action states A : S � L � V ! R an agent’s action that depends

on its sensory and internal states
internal states L : L � S � V ! R the states of the agent that

cause action and depend on sensory states
ergodic density pðc; s; a; ljmÞ a probability density function over

external c [ C, sensory s [ S, active a [ A and internal states

l [ L for a system denoted by m
variational density q(cjl) an arbitrary probability density function

over external states that is parametrized by internal states

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fashion. This circular causality may be a fundamental and ubi-
quitous causal architecture for self-organization.
Equipped with this partition, we can now consider the
behaviour of any random dynamical system m described by
some stochastic differential equations:

_x ¼ f ðxÞ þ v

and
f ðxÞ ¼

fcðc; s; aÞ
fsðc; s; aÞ
faðs; a; lÞ
flðs; a; lÞ

2

664

3

775:

9
>
>
>
>
=

>
>
>
>
;

ð2:1Þ

Here, f(x) is the flow of system states that is subject to random
fluctuations denoted by v. The second equality formalizes
the dependencies implied by the Markov blanket. Because
the system is ergodic it will, after a sufficient amount of
time, converge to an invariant set of states called a pullback
or random global attractor. The attractor is random because
it itself is a random set [29,30]. The associated ergodic den-
sity p(xjm) is the solution to the Fokker–Planck equation
(a.k.a. the Kolmogorov forward equation) [31] describing
the evolution of the probability density over states

_p(xjm) ¼ r � G rp � r � ð fpÞ:
ð2:2Þ

Here, the diffusion tensor G is the half the covariance (ampli-
tude) of the random fluctuations. Equation (2.2) shows that
the ergodic density depends upon flow, which can always be
expressed in terms of curl and divergence-free components.

This is the Helmholtz decomposition (a.k.a. the fundamen-
tal theorem of vector calculus) and can be formulated in
terms of an antisymmetric matrix R(x) ¼ 2R(x)T and a scalar
potential G(x) we will call Gibbs energy [32],

f ¼ �ðG þ RÞ � rG:
ð2:3Þ

Using this standard form [33], it is straightforward to show
that p(xjm) ¼ exp(2G(x)) is the equilibrium solution to the
Fokker–Planck equation [12]:

pðxjmÞ ¼ expð�GðxÞÞ ) rp ¼ �prG ) _p ¼ 0:
ð2:4Þ

This means that we can express the flow in terms of the
ergodic density

f ¼ðG þ RÞ � r ln pðxjmÞ;

flðs; a; lÞ ¼ðG þ RÞ � rl ln pðc; s; a; ljmÞ

and
faðs; a; lÞ ¼ðG þ RÞ � ra ln pðc; s; a; ljmÞ:

9
>
>
=

>
>
;
ð2:5Þ

Although we have just followed a sequence of standard
results, there is something quite remarkable and curious
about this flow: the flow of internal and active states is essen-
tially a (circuitous) gradient ascent on the (log) ergodic
density. The gradient ascent is circuitous because it contains
divergence-free (solenoidal) components that circulate on
the isocontours of the ergodic density—like walking up a
winding mountain path. This ascent will make it look as if
internal (and active) states are flowing towards regions of

active states

E[a]µ–—aF (s,a,l)

E[l]µ–—lF (s,a,l)

external states
internal states

sensory states

.

.

.

external states
internal states

y ŒY 

s = fs(y,s,a) + w

y = fy(y,s,a) + w

s ŒS
a ΠA
l ŒL

Figure 1. Markov blankets and the free energy principle. These schematics illustrate the partition of states into internal states and hidden or external states that are
separated by a Markov blanket—comprising sensory and active states. The upper panel shows this partition as it would be applied to action and perception in the
brain; where—in accord with the free energy principle—active and internal states minimize a free energy functional of sensory states. The ensuing self-organization
of internal states then corresponds to perception, while action couples brain states back to external states. The lower panel shows exactly the same dependencies but
rearranged so that the internal states can the associated with the intracellular states of a cell, while the sensory states become the surface states or cell membrane
overlying active states (e.g. the actin filaments of the cytoskeleton). See table 1 for a definition of variables.

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state space that are most frequently occupied despite the
fact their flow is not a function of external states. In other
words, their flow does not depend upon external states
(see the right-hand side equation (2.5)) and yet it ascends
gradients that depend on the external states (see the right-
hand side of equation (2.5)). In short, the internal and
active states behave as if they know where they are in the
space of external states—states that are hidden behind the
Markov blanket.

We can finesse this apparent paradox by noting that the
flow is the expected motion through any point averaged
over time. By the ergodic theorem, this is also the flow aver-
aged over the external states, which does not depend on the
external state at any particular time: more formally, for any
point v[V ¼ S � A � L in the space of the internal states
and their Markov blanket, equations (2.1) and (2.5) tell us
that flow through this point is the average flow under the
posterior density over the external states:

flðvÞ ¼ Et½_lðtÞ � ½xðtÞ [ v�� ¼
ð

C
pðcjvÞ � ðG þ RÞ � rl ln pðc; vjmÞdc;

faðvÞ ¼ Et½_aðtÞ � ½xðtÞ [ v�� ¼
ð

C
pðcjvÞ � ðG þ RÞ � ra ln pðc; vjmÞdc;

)

flðvÞ ¼ ðG þ RÞ � rl ln pðvjmÞ;
and
faðvÞ ¼ ðG þ RÞ � ra ln pðvjmÞ:

9
>
>
>
>
>
>
>
>
>
>
>
=

>
>
>
>
>
>
>
>
>
>
>
;

ð2:6Þ

The Iverson bracket [x(t) [ v] returns a value of one
when the trajectory passes through the point in question
and zero otherwise—and the first expectation is taken over
time. Here, we have used the fact that the integral of a deri-
vative of a density is the derivative of its integral—and
both are zero.
Equation (2.6) is quite revealing—it shows that the flow of
internal and active states performs a circuitous gradient
ascent on the marginal ergodic density over internal states
and their Markov blanket. Crucially, this marginal density
depends on the posterior density over external states. This
means that the internal states will appear to respond to
sensory fluctuations based on posterior beliefs about under-
lying fluctuations in external states. We can formalize this
notion by associating these beliefs with a probability density
over external states q(cjl) that is encoded (parametrized) by
internal states.

Lemma 2.1 Free energy. Forany Gibbs energy G(c, s, a, l) ¼ 2ln
p(c, s, a, l), there is a free energy F(s, a, l) that describes the flow of
internal and active states:

flðs; a; lÞ ¼ � ðG þ RÞ � rlF;

faðs; a; lÞ ¼ � ðG þ RÞ � raF

and
Fðs; a; lÞ ¼ �
ð

c
qðcjlÞ ln pðc; s; a; ljmÞ
qðcjlÞ
dc

¼ Eq½Gðc; s; a; lÞ� � H½qðcjmÞ�:

9
>
>
>
>
>
>
>
=

>
>
>
>
>
>
>
;

ð2:7Þ

Here, free energy is a functional of an arbitrary (variational) density
q(cjl) that is parametrized by internal states. The last equality just
shows that free energy can be expressed as the expected Gibbs
energy minus the entropy of the variational density.

Proof. Using Bayes rule, we can rearrange the expression for
free energy in terms of a Kullback–Leibler divergence [34]:

Fðs;a;lÞ ¼ �lnpðs;a;ljmÞ þ DKL½qðcjlÞjjpðcjs;a;lÞ�;
)
flðs;a;lÞ ¼ ðG þ RÞ � rl lnpðs;a;ljmÞ � ðG þ RÞ � rlDKL
and faðs;a;lÞ ¼ ðG þ RÞ � ra lnpðs;a;ljmÞ � ðG þ RÞ � raDKL:

9
>
>
=

>
>
;

ð2:8Þ

However, equation (2.6) requires the gradients of the
divergence to be zero, which means the divergence must be
minimized with respect to internal states. This means that
the variational and posterior densities must be equal:

qðcjlÞ ¼ pðcjs; a; lÞ ) DKL ¼ 0 )
ðG þ RÞ � rlDKL ¼ 0;
ðG þ RÞ � raDKL ¼ 0:

�

In other words, the flow of internal and active states
minimizes free energy, rendering the variational density
equivalent to the posterior density over external states.

Remarks 2.2. Put simply, this proof says that if one inter-
prets internal states as parametrizing a variational density
encoding Bayesian beliefs about external states, then the
dynamics of internal and active states can be described as a
gradient descent on a variational free energy function of
internal states and their Markov blanket. Variational free
energy was introduced by Feynman [35] to solve difficult
integration problems in path integral formulations of quan-
tum physics. This is also the free energy bound that is used
extensively in approximate Bayesian inference (e.g. variational
Bayes) [34,36,37]. The expression for free energy in equation
(2.8) discloses its Bayesian interpretation: the first term is
the negative log evidence or marginal likelihood of the internal
states and their Markov blanket. The second term is a relative
entropy or Kullback–Leibler divergence [38] between the vari-
ational density and the posterior density over external states.
Because (by Gibbs inequality) this divergence cannot be less
than zero, the internal flow will appear to have minimized
the divergence between the variational and posterior density.
In other words, the internal states will appear to have solved
the problem of Bayesian inference by encoding posterior
beliefs about hidden (external) states, under a generative
model provided by the Gibbs energy. This is known as
approximate Bayesian inference—with exact Bayesian inference
when the forms of the variational and posterior densities are
identical. In short, the internal states will appear to engage in
some form of Bayesian inference: but what about action?
Because the divergence in equation (2.8) can never be less
than zero, free energy is an upper bound on the negative log

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evidence. Now, because the system is ergodic we have

Fðs; a; lÞ � � ln pðs; a; ljmÞ )
Et½Fðs; a; lÞ� � Et½� ln pðs; a; ljmÞ� ¼ H½ pðs; a; ljmÞ�:

�
ð2:9Þ

This meansthat action will (on average) appear to minimize free
energy and thereby place an upper bound on the entropy of the
internal states and their Markov blanket. If we associate these
states v ¼ fs, a, lg with biological systems, then action places
an upper bound on their dispersion (entropy) and will appear
to conserve their structural and dynamical integrity. Together
with the Bayesian modelling perspective, this is exactly consist-
ent with the good regulator theorem (every good regulator is a
model of its environment) and related treatments of self-organ-
ization [2,5,12,39,40]. Furthermore, we have shown elsewhere
[11,41] that free energy minimization is consistent with infor-
mation-theoretic formulations of sensory processing and
behaviour [23,42,43]. Equation (2.7) also shows that minimizing
free energy entails maximizing the entropy of the variational
density (the final term in the last equality)—in accord with the
maximum entropy principle [44]. Finally, because we have
cast this treatment in terms of random dynamical systems,
there is an easy connection to dynamical formulations that
predominate in the neurosciences [40,45–47].
The above arguments can be summarized with the
following attributes of biological self-organization:

— biological systems are ergodic [26]: in the sense that the aver-
age of any measure of their states converges over a
sufficient period of time. This includes the occupancy of
state space and guarantees the existence of an invariant
ergodic density over functional and structural states;
— they are equipped with a Markov blanket [28]: the existence of a
Markov blanket necessarily implies a partition of states into
internal states, their Markov blanket (sensory and active
states) and external or hidden states. Internal states and
their Markov blanket (biological states) constitute a biological
system that responds to hidden states in the environment;
— they exhibit active inference [11]: the partition of states implied
by the Markov blanket endows internal states with the
apparent capacity to represent hidden states probabilisti-
cally, so that they appear to infer the hidden causes of
their sensory states (by minimizing a free energy bound
on log Bayesian evidence). By the circular causality induced
by the Markov blanket, sensory states depend on active
states, rendering inference active or embodied; and
— they are autopoietic [4]: because active states change—but
are not changed by—hidden states (figure 1), they will
appear to place an upper (free energy) bound on the dis-
persion (entropy) of biological states. This homoeostasis is
informed by internal states, which means that active states
will appear to maintain the structural and functional
integrity of biological states.

When expressed like this, these criteria appear perfectly
sensible but are they useful in the setting of real biophysical
systems? The premise of this paper is that these criteria apply
to (almost) all ergodic systems encountered in the real world.
The argument here is that biological behaviour rests on the
existence of a Markov blanket—and that a Markov blanket is
(almost) inevitable in coupled dynamical systems with short-
range interactions. In other words, if the coupling between
dynamical systems can be neglected—when they are separated
by large distances—the intervening systems will necessarily

form a Markov blanket. For example, if we consider short-
range electrochemical and nuclear forces, then a cell membrane
forms a Markov blanket for internal intracellular states
(figure 1). If this argument is correct, then it should be possible
to show the emergence of biological self-organization in any
arbitrary ensemble of coupled subsystems with short-range
interactions. The final section uses simulations to provide a
proof of principle, using the four criteria above to identify
and verify the emergence of lifelike behaviour.

3. Proof of principle

In this section, we simulate a primordial soup to illustrate the
emergence of biological self-organization. This soup comprises
an ensemble of dynamical subsystems—each with its own
structural and functional states—that are coupled through
short-range interactions. These simulations are similar to (hun-
dreds of) simulations used to characterize pattern formation in
dissipative systems; for example, Turing instabilities [48]: the
theory of dissipative structures considers far-from-equilibrium
systems, such as turbulence and convection in fluid dynamics
(e.g. Be´nard cells), percolation and reaction–diffusion systems
such as the Belousov–Zhabotinsky reaction [49]. Self-assembly
is another important example from chemistry that has biologi-
cal connotations (e.g. for pre-biotic formation of proteins). The
simulations here are distinguished by solving stochastic differ-
ential equations for both structural and functional states. In
other words, we consider states from classical mechanics that
determine physical motion—and functional states that could
describe electrochemical states. Importantly, the functional
states of any system affect the functional and structural states
of another. The agenda here is not to explore the repertoire of
patterns and self-organization these ensembles exhibit—but
rather take an arbitrary example and show that, buried
within it, there is a clear and discernible anatomy that satisfies
the criteria for life.
3.1. The primordial soup
To simulate a primordial soup, we use an ensemble of
elemental subsystems with (heuristically speaking) Newto-
nian and electrochemical dynamics f~p;~qg [ X:

_~p ¼ fpð~p;~qÞ þ v

and
_~q ¼ fqð~p;~qÞ þ v

)

ð3:1Þ

Here, ~pðtÞ ¼ ð p; p0; p00; . . .Þ are generalized coordinates of motion
describing position, velocity, acceleration—and so on—of the
subsystems, while ~qðtÞ correspond to electrochemical states
(such as concentrations or electromagnetic states). One can
think of these generalized states as describing the physical and
electrochemical state of large macromolecules. Crucially, these
states are coupled within and between the subsystems compris-
ing an ensemble. The electrochemical dynamics were chosen
to have a Lorenz attractor: for the ith system with its own rate
parameter k(i):

_qðiÞ ¼ kðiÞ �

10ðqðiÞ
2 � qðiÞ
1 Þ

ð32 þ �qð jÞ
1 Þ � qðiÞ
1 � qðiÞ
2 � x3qðiÞ
1
qðiÞ
1 qðiÞ
2 � 8
3qðiÞ
3

2

664

3

775 þ kðiÞ � �qðiÞ þ v;

�qðiÞ ¼ P
j qð jÞ � Aij;

Aij ¼ ½jDijj , 1�

and
Dij ¼ pð jÞ � pðiÞ:

9
>
>
>
>
>
>
>
>
>
>
>
=

>
>
>
>
>
>
>
>
>
>
>
;

ð3:2Þ

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Changes in electrochemical states are coupled through
the local average �qðiÞof the states of subsystems that lie within
a distance of one. This means that A can be regarded as an
(unweighted) adjacency matrix that encodes the dependencies
among the functional (electrochemical) states of the ensemble.
The local average enters the equations of motion both linearly
and nonlinearly to provide an opportunity for generalized syn-
chronization [50]. The nonlinear coupling effectively renders

the Rayleigh parameter of the flow 32 þ �qð jÞ
1
state-dependent.
The Lorenz form for these dynamics is a somewhat
arbitrary choice but provides a ubiquitous model of electrody-
namics, lasers and chemical reactions [51]. The rate parameter
kðiÞ ¼ 1
32ð1 � expð�4 � UÞÞ was specific to each subsystem,
where U [ (0, 1) was selected from a uniform distribution.
This introduces heterogeneity in the rate of electrochemical
dynamics, with a large number of fast subsystems—with a
rate constant of nearly one—and a small number of slower sub-
systems. To augment this heterogeneity, we randomly selected
a third of the subsystems and prevented them from (electro-
chemically) influencing others, by setting the appropriate
column of the adjacency matrix to zero. We refer to these as
functionally closed systems.
In a similar way, the classical (Newtonian) motion of each
subsystem depends upon the functional status of its neighbours:

_pðiÞ ¼ p0ðiÞ þ v;

_p0ðiÞ ¼ 1
32 � wðiÞ � 1
4 � p0ðiÞ �
1
1024 pðiÞ þ v;

wðiÞ ¼
X

j

Dij
jDijj �

wðiÞ
f
jDijj �
1

jDijj2

0

@

1

A � Aij

and
wðiÞ
f
¼ 8 � expð2 � jqð jÞ
3 � qðiÞ
3 jÞ � 2:

9
>
>
>
>
>
>
>
>
>
>
=

>
>
>
>
>
>
>
>
>
>
;

ð3:3Þ

This motion rests on forces w(i) exerted by other subsys-
tems that comprise a strong repulsive force (with an inverse
square law) and a weaker attractive force that depends on
their electrochemical states. This force was chosen so that
systems with coherent (third) states are attracted to each
other but repel otherwise. The remaining two terms in the
expression for acceleration (second equality) model viscosity
that depends upon velocity and an exogenous force that
attracts all locations to the origin—as if they were moving
in a simple (quadratic) potential energy well. This ensures
the synthetic soup falls to the bottom of the well and enables
local interactions.
Note that the ensemble system is dissipative at two levels:
first, the classical motion includes dissipative friction or vis-
cosity. Second, the functional dynamics are dissipative in
the sense that they are not divergence-free. We will now
assess the criteria for biological self-organization within this
coupled random dynamical ensemble.

3.2. Ergodicity
In the examples used below, 128 subsystems were integrated
using Euler’s (forward) method with step sizes of 1/512 s
and initial conditions sampled from the normal distribution.
Random fluctuations were sampled from the unit normal
distribution. By adjusting the parameters in the above equa-
tions of motion, one can produce a repertoire of plausible
and interesting behaviours (the code for these simulations
and the figures in this paper are available as part of
the SPM academic freeware). These behaviours range from

gas-like behaviour (where subsystems occasionally get close
enough to interact) to a cauldron of activity, when sub-
systems are forced together at the bottom of the potential
well. In this regime, subsystems get sufficiently close for the
inverse square law to blow them apart—reminiscent of sub-
atomic particle collisions in nuclear physics. With particular
parameter values, these sporadic and critical events can
render the dynamics non-ergodic, with unpredictable high
amplitude fluctuations that do not settle down. In other
regimes, a more crystalline structure emerges with muted
interactions and low structural (configurational) entropy.
However, for most values of the parameters, ergodic be-
haviour emerges as the ensemble approaches its random
global attractor (usually after about 1000 s): generally, subsys-
tems repel each other initially (much like illustrations of the
big bang) and then fall back towards the centre, finding
each other as they coalesce. Local interactions then mediate
a reorganization, in which subsystems are passed around
(sometimes to the periphery) until neighbours gently jostle
with each other. In terms of the dynamics, transient synchro-
nization can be seen as waves of dynamical bursting (due to
the nonlinear coupling in equation (3.2)). In brief, the motion
and electrochemical dynamics look very much like a restless
soup (not unlike solar flares on the surface of the sun, figure
2)—but does it have any self-organization beyond this?

3.3. The Markov blanket
Because the structural and functional dependencies share
the same adjacency matrix—which depends upon position—
one can use the adjacency matrix to identify the principal
Markov blanket by appealing to spectral graph theory:
the Markov blanket of any subset of states encoded by a
binary vector with elements xi [ f0, 1g is given by [B . x] [
f0, 1g, where the Markov blanket matrix B ¼ A þ AT þ ATA
encodes children, parents and parents of children. This
follows because the ith column of the adjacency matrix
encodes the directed connections from the ith state to all its
children.
The
principal
eigenvector of
the
(symmetric)
Markov
blanket
matrix
will—by
the
Perron–Frobenius
theorem—contain positive values. These values reflect the
degree to which each state belongs to the cluster that is most
interconnected (cf., spectral clustering). In what follows, the
internal states were defined as belonging to subsystems with
the k ¼ 8 largest values. Having defined the internal states,
the Markov blanket can be recovered from the Markov blanket
matrix using [B . x] and divided into sensory and active
states—depending upon whether they are influenced by the
hidden states or not.
Given the internal states and their Markov blanket, we can
now follow their assembly and visualize any structural or func-
tional characteristics. Figure 3 shows the adjacency matrix used
to identify the Markov blanket. This adjacency matrix has
non-zero entries if two subsystems were coupled over the last
256 s of a 2048 s simulation. In other words, it accommoda-
tes the fact that the adjacency matrix is itself an ergodic
process—due to the random fluctuations. Figure 3b shows
the location of subsystems with internal states (blue) and
their Markov blanket—in terms of sensory (magenta) and
active (red) locations. A clear structure can be seen here,
where the internal subsystems are (unsurprisingly) close
together and enshrouded by the Markov blanket. Interestingly,
the active subsystems support the sensory subsystems that are

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exposed to hidden environmental states. This is reminiscent of
a biological cell with a cytoskeleton that supports some sensory
epithelia or receptors within its membrane.
Figure
3c
highlights
functionally
closed
subsystems
(filled circles) that have been rusticated to the periphery of
the system. Recall that these subsystems cannot influence or
engage other subsystems and are therefore expelled to the
outer limits of the soup. Heuristically, they cannot invade
the system and establish a reciprocal and synchronous exchange
with other subsystems. Interestingly, no simulation ever pro-
duced a functionally closed internal state. Figure 3d shows the
slow subsystems that are distributed between internal and

external states—which may say something interesting about
the generalized synchrony that underlies self-organization.

3.4. Active inference
If the internal states encode a probability density over the
hidden or external states, then it should be possible to predict
external states from internal states. In other words, if internal
events represent external events, they should exhibit a signifi-
cant statistical dependency. To establish this dependency, we
examined the functional (electrochemical) status of internal
subsystems to see whether they could predict structural

–8
–6
–4
–2
0
2
4
6
8
–8

–6

–4

–2

0

2

4

6

8
(i)
(ii)
(a)

(b)

(c)

position

ensemble
synchronization

50
100
150
200
250
300
350
400
450
500

–30

–20

–10

0

10

20

30

dynamics

–30

–20

–10

0

10

20

30

200
400
600
800
1000
1200
1400
1600
1800
2000

time

motion

position

Figure 2. Ensemble dynamics. (a) The position of (128) subsystems comprising an ensemble after 2048 s. a(i) The dynamical status (three blue dots per subsystem)
of each subsystem centred on its location (larger cyan dots). a(ii) The same information, where the relative values of the three dynamical states of each subsystem
are colour-coded (using a softmax function of the three functional states and a RGB mapping). This illustrates the synchronization of dynamical states within each
subsystem and the dispersion of the phases of the Lorenzian dynamics over subsystems. (b,c) The evolution of functional and structural states as a function of time,
respectively. The (electrochemical) dynamics of the internal (blue) and external (cyan) states are shown for the 512 s. One can see initial (chaotic) transients that
resolve fairly quickly, with itinerant behaviour as they approach their attracting set. (c) The position of internal (blue) and external (cyan) subsystems over the entire
simulation period illustrate critical events (circled) that occur every few hundred seconds, especially at the beginning of the simulation. These events generally reflect
a pair of particles (subsystems) being expelled from the ensemble to the periphery, when they become sufficiently close to engage short-range repulsive forces.
These simulations integrated the stochastic differential equations in the main text using a forward Euler method with 1/512 s time steps and random fluctuations of
unit variance.

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events (movement) in the external milieu. This is not unlike
the approach taken in brain mapping that searches for statisti-
cal dependencies between, say, motion in the visual field and
neuronal activity [52].
To test for statistical dependencies, the principal patterns
of activity among the internal (functional) states were sum-
marized using singular value decomposition and temporal
embedding (figure 4). A classical canonical variates analysis
was then used to assess the significance of a simple linear
mapping between expression of these patterns and the move-
ment of each external subsystem. Figure 4a illustrates these
internal dynamics, while figure 4c shows the Newtonian
motion of the external subsystem that was best predicted.
The agreement between the actual (dotted line) and predic-
ted (solid line) motion is self-evident, particularly around
the negative excursion at 300 s. The internal dynamics that
predict this event appear to emerge in their fluctuations
before the event itself (figure 4)—as would be anticipated if
internal events are modelling external events. Interestingly,
the subsystem best predicted was the furthest away from
the internal states (magenta circle in figure 4d).
This example illustrates how internal states infer or
register distant events in a way that is not dissimilar to

the perception of auditory events through sound waves—or
the way that fish sense movement in their environment.
Figure 4d also shows the subsystems whose motion could be
predicted reliably. This predictability is the most significant
at the periphery of the ensemble, where the ensemble has
the greatest latitude for movement. These movements are
coupled to the internal states—via the Markov blanket—
through generalized synchrony. Generalized synchrony refers
to the synchronization of chaotic dynamics, usually in skew-
product (master-slave) systems [53,54]. However, in our
set-up there is no master–slave relationship but a circular
causality induced by the Markov blanket. Generalized syn-
chrony was famously observed by Huygens in his studies of
pendulum clocks—that synchronized themselves through the
imperceptible motion of beams from which they were sus-
pended [55]. This nicely illustrates the ‘action at a distance’
caused by chaotically synchronized waves of motion. Circular
causality begs the question of whether internal states predict
external causes of their sensory states or actively cause them
through action. Exactly the same sorts of questions apply
to perception [56,57]: for example, are visually evoked neur-
onal responses caused by external events or by our (saccadic
eye) movements?

element
20
40
60
80
100 120

20

(a)
(b)

(c)
(d)

40

60

80

100

120

–8 –6 –4 –2
0
2
4
6
8
–8

–6

–4

–2

0

2

4

6

8

–8

–6

–4

–2

0

2

4

6

8

position

–8 –6 –4 –2
0
2
4
6
8
position
–8 –6 –4 –2
0
2
4
6
8
position

–8

–6

–4

–2

0

2

4

6

8

hidden states

sensory states

active states

internal states

Figure 3. Emergence of the Markov blanket. (a) The adjacency matrix that indicates a conditional dependency (spatial proximity) on at least one occasion over the
last 256 s of the simulation. The adjacency matrix has been reordered to show the partition of hidden (cyan), sensory (magenta), active (red) and internal (blue)
subsystems, whose positions are shown in (b)—using the same format as in the previous figure. Note the absence of direct connections (edges) between external or
hidden and internal subsystem states. The circled area illustrates coupling between active and hidden states that are not reciprocated (there are no edges between
hidden and active states). The spatial self-organization in the upper left panel is self evident; where the internal states have arranged themselves in a small loop
structure with a little cilium, protected by the active states that support the surface or sensory states. When viewed as a movie, the entire ensemble pulsates in a
chaotic but structured fashion, with the most marked motion in the periphery. (c,d) Highlights those subsystems that cannot influence others (closed subsystems (c))
and those that have slower dynamics (slow subsystems (d)). The remarkable thing here is that all the closed subsystems have been rusticated to the periphery—
where they provide a locus for vigorous dynamics and motion. Contrast this with the deployment of slow subsystems that are found throughout the hidden, sensory,
active and internal partition.

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3.5. Autopoiesis and structural integrity
The previous section applied a simple sort of brain mapping
to establish the statistical dependencies between external
and internal states—and their functional correlates. The
final simulations also appeal to procedures in the biological
sciences—in particular neuropsychology to examine the
effects of lesions. To test for autopoietic maintenance of struc-
tural and functional integrity, the sensory, active and internal
subsystems were selectively lesioned—by rendering them
functionally closed—in other words, by preventing them
from influencing their neighbours. This is a relatively mild
lesion, in the sense that they remain physically coupled
with intact dynamics that respond to neighbouring elements.
Because active states depend only on sensory and internal
states one would expect to see a loss of structural integrity
not only with lesions to action but also to sensory and internal
states that are an integral part of active inference.

Figure 5 illustrates the effects of these interventions by fol-
lowing the evolution of the internal states and their Markov
blanket over 512 s. Figure 5a shows the conservation of struc-
tural (and implicitly functional) integrity in terms of spatial
configuration over time. Contrast this with the remaining
three panels that show structural disintegration as the integ-
rity of the Markov blanket is lost and internal elements are
extruded into the environment.

4. Conclusion

Clearly, there are many issues that need to be qualified and
unpacked under this formulation. Perhaps the most prescient
is its focus on boundaries or Markov blankets. This contrasts
with other treatments that consider the capacity of living
organisms to reproduce by passing genetic material to their

time

modes

100
200
300
400
500

time

external states

position

position

100
200
300
0

10

20

30

40

50

c2

frequency

–0.4

–0.3

–0.2

–0.1

0

100
200
300
400
500
–8

–6

–4

–2

0

4

6

8

–5
0
5

2

5

(a)
(b)

(c)
(d)

10

15

20

25

30

Figure 4. Self-organized perception. This figure illustrates the Bayesian perspective on self-organized dynamics. (a) The first (principal) 32 eigenvariates of the
internal (functional) states as a function of time over the last 512 s of the simulations reported in the previous figures. These eigenvariates were obtained by a
singular value decomposition of the timeseries over all internal functional states (lagged between plus and minus 16 s). These represent a summary of internal
dynamics that are distributed over internal subsystems. The eigenvariates were then used to predict the (two-dimensional) motion of each external subsystem using
a standard canonical variates analysis. The (classical) significance of this prediction was assessed using Wilks’ lambda (following a standard transformation to the x2

statistic). The actual (dotted line) and predicted (solid line) position for the most significant external subsystem is shown in (c)—in terms of canonical variates (best
linear mixture of position in two dimensions). The agreement is self-evident and is largely subtended by negative excursions, notably at 300 s. The fluctuations in
internal states are visible in (a) and provide a linear mixture that correlates with the external fluctuation (highlighted with a white arrow). The location of the
external subsystem that was best predicted is shown by the magenta circle on (d). Remarkably, this is the subsystem that is the furthest away from the internal
states and is one of the subsystems that participates in the exchanges a closed subsystem in the previous figure. (c) Also shows the significance with which the
motion of the remaining external states could be predicted (with the intensity of the cyan being proportional to the x2 statistic above). Interestingly, the motion
that is predicted with the greatest significance is restricted to the periphery of the ensemble, where the external subsystems have the greatest latitude for move-
ment. To ensure this inferential coupling was not a chance phenomenon, we repeated the analysis after flipping the external states in time. This destroys any
statistical coupling between the internal and external states but preserves the correlation structure of fluctuations within either subset. The distribution of the
ensuing x2 statistics (over 82 external elements) is shown in (b) for the true (black) and null (white) analyses. Crucially, five of the subsystems in the true analysis
exceeded the largest statistic in the null analysis. The largest value of the null distribution provides protection against false positives at a level of 1/82. The
probability of obtaining five x2 values above this threshold by chance is vanishingly small p ¼ 0.00052.

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offspring [1]. In this context, it is not difficult to imagine
extending the simulations above to include slow (e.g. diur-
nal) exogenous fluctuations—that cause formally similar
Markov blankets to dissipate and reform in a cyclical fashion.
The key question would be whether the internal states of a
system in one cycle induce—or code for—the formation of
a similar system in the next.
The central role of Markov blankets speak to an important
question: is there a unique Markov blanket for any given
system? Our simulations focused on the principal Markov
blanket—as defined by spectral graph theory. However, a
system can have a multitude of partitions and Markov blan-
kets. This means that there are many partitions that—at some
spatial and temporal scale—could show lifelike behaviour.
For example, the Markov blanket of an animal encloses
the Markov blankets of its organs, which enclose Markov
blankets of cells, which enclose Markov blankets of nuclei
and so on. Formally, every Markov blanket induces active
(Bayesian) inference and there are probably an uncountable
number of Markov blankets in the universe. Does this mean
there is lifelike behaviour everywhere or is there something

special about the Markov blankets of systems we consider
to be alive?
Although speculative, the answer probably lies in the stat-
istics of the Markov blanket. The Markov blanket comprises a
subset of states, which have a marginal ergodic density. The
entropy of this marginal density reflects the dispersion or
invariance properties of the Markov blanket, suggesting
that there is a unique Markov blanket that has the smal-
lest entropy. One might conjecture that minimum entropy
Markov blankets characterize biological systems. This conjec-
ture is sensible in the sense that the physical configuration
and dynamical states that constitute the Markov blanket
of an organism—or organelle—change slowly in relation to
the external and internal states it separates. Indeed, the
physical configuration must be relatively constant to avoid
destroying anti-edges (the absence of an edge or coupling)
in the adjacency matrix that defines the Markov blanket.
This perspective suggests that there may be ways of charac-
terizing the statistics (e.g. entropy) of Markov blankets that
may quantify how lifelike they appear. Note from equation
(2.9) that systems (will appear to) place an upper bound on

–8 –6
–4 –2
0
2
4
6
8
–8

–6

–4

–2

0

2

4

6

8
(a)
(b)

(c)
(d)

position

–8 –6 –4 –2
0
2
4
6
8

position

–8

–6

–4

–2

0

2

4

6

8

–8 –6
–4 –2
0
2
4
6
8

position

–8 –6 –4 –2
0
2
4
6
8

position

simulated lesions

Figure 5. Autopoiesis and oscillator death. These results show the trajectory of the subsystems for 512 s after the last time point characterized in the previous
figures. (a) The trajectories under the normal state of affairs; showing a preserved and quasicrystalline arrangement of the internal states (blue) and the Markov
blanket (active states in red and sensory states in magenta). Contrast this formal self-organization with the decay and dispersion that ensues when the internal
states and Markov blankets are synthetically lesioned (b,c,d). In all simulations, a subset of states was lesioned by simply rendering their subsystems closed—in
other words, although the Newtonian interactions were preserved, they were unable to affect the functional states of neighbouring subsystems. (b) The effect of this
relatively subtle lesion on active states—that are rapidly expelled from the interior of the ensemble, allowing sensory states to invade and disrupt the internal
states. A similar phenomenon is seen when the sensory states were lesioned (c)—as they drift out into the external system. There is a catastrophic loss of structural
integrity when the internal states themselves cannot affect each other, with a rapid migration of internal states through and beyond their Markov blanket (d). These
simulations illustrate the effective death of biological self-organization that is a well-known phenomenon in dynamical systems theory—known as oscillator death:
see [58]. In our setting, they are a testament to autopoiesis or self-creation—in the sense that self-organized dynamics are necessary to maintain structural or
configurational integrity.

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the entropy of the Markov blanket (and internal states).
This means that the marginal ergodic entropy measures the
success of this apparent endeavour.
However, minimum entropy is clearly not the whole story,
in the sense that biological systems act on their environment—
unlike a petrified stone with low entropy. In the language of
random attractors, the (internal and Markov blanket) states of
a system have an attracting set that is space filling but has a
small measure or entropy—where the measure or volume
upper bounds the entropy [11]. Put simply, biological systems
move around in their state space but revisit a limited number
of states. This space filling aspect of attracting sets may rest
on the divergence-free or solenoidal flow (equation (2.3)) that
we have largely ignored in this paper but may hold the key
for characterizing life forms.
Clearly, the simulations in this paper are a long way off
accounting for the emergence of biological structures such as
complex cells. The examples presented above are provided
as proof of principle and are as simple as possible. An interest-
ing challenge now will be to simulate the emergence of
multicellular structures using more realistic models with a
greater (and empirically grounded) heterogeneity and formal
structure. Having said this, there is a remarkable similarity
between the structures that emerge from our simulations and
the structure of viruses. Furthermore, the appearance of little
cilia (figure 3) are very reminiscent of primary cilia, which
typically serve as sensory organelles and play a key role in
evolutionary theory [59].
A related issue is the nature of the dynamical (molecular
or cellular) constituents of the ensembles considered above.
Nothing in this treatment suggests a special role for carbon-
based life or, more generally, the necessary conditions for
life to emerge. The contribution of this work is to note
that if systems are ergodic and possess a Markov blanket,
they will—almost surely—show lifelike behaviour. However,
this does not address the conditions that are necessary for the
emergence of ergodic Markov blankets. There may be useful
constraints implied by the existence of a Markov blanket
(whose constituency has to change more slowly than the
states of its constituents). For example, the spatial range of
electrochemical forces, temperature and molecular chemistry
may determine whether the physical motion of molecules
(that determine the integrity of the Markov blanket) is
large or small in relation to fluctuations in electrochemical
states (that do not). However, these questions are beyond
the scope of this paper and may be better addressed in
computational chemistry and theoretical biology.

This touches on another key issue, namely that of evolu-
tion. In this treatment, we have assumed biological systems
are ergodic. Clearly, this is a simplification, in that real
systems are only locally ergodic. The implication here is
that self-organized systems cannot endure indefinitely and
are only ergodic over a particular (somatic) timescale,
which raises the question of evolutionary timescales: is evol-
ution itself the slow and delicate unwinding of a trajectory
through a vast state space—as the universe settles on its
global random attractor? The intimation here is that adap-
tation and evolution may be as inevitable as the simple sort
of self-organization considered in this paper. In other
words, the very existence of biological systems necessarily
implies they will adapt and evolve. This is meant in the
sense that any system with a random dynamical attractor
will appear to minimize its variational free energy and can
be interpreted as engaging in active inference—acting upon
its external milieu to maintain an internal homoeostasis.
However, the ensuing homoeostasis is as illusory as the free
energy minimization upon which it rests. Does the same
apply to adaptation and evolution?
Adaptation on a somatic timescale has been interpreted
as optimizing the parameters of a generative model (encoded
by slowly changing internal states like synaptic connection
strengths in the brain) such that they minimize free energy. It
is fairly easy to show that this leads to Hebbian or associative
plasticity of the sort that underlies learning and memory [21].
Similarly, at even longer timescales, evolution can be cast in
terms of free energy minimization—by analogy with Bayesian
model selection based on variational free energy [60]. Indeed,
free energy functionals have been invoked to describe natural
selection [61]. However, if the minimization of free energy is
just a corollary of descent onto a global random attractor,
does this mean that adaptation and evolution are just ways of
describing the same thing? The answer to this may not be
straightforward, especially if we consider the following possi-
bility: if self-organization has an inferential aspect, what
would happen if systems believed their attracting sets had
low entropy. If one pursues this in a neuroscience setting, one
arrives at a compelling explanation for the way we adaptively
sample our environments—to minimize uncertainty about the
causes of sensory inputs [62]. In short, this paper has only con-
sidered inference as emergent property of self-organization—
not the nature of implicit (prior) beliefs that underlie inference.

Acknowledgements. I would like to thank two anonymous reviewers for
their detailed and thoughtful help in presenting these ideas. The
Wellcome Trust funded this work.

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