Sentient Self-Organization: Minimal dynamics and circular causality

Biswa SenguptaDept. of Bioengineering, Imperial College London

Dept. of Engineering, University of CambridgeCortexica Vision Systems Ltd.

b.sengupta@imperial.ac.uk

Karl FristonWellcome Trust Centre for Neuroimaging

University College Londonk.friston@ucl.ac.uk

Abstract

Theoretical arguments and empirical evidencein neuroscience suggests that organisms repre-sent or model their environment by minimizinga variational free-energy bound on the surpriseassociated with sensory signals from the envi-ronment. In this paper, we study phase transi-tions in coupled dissipative dynamical systems(complex Ginzburg-Landau equations) under avariety of coupling conditions to model the ex-change of a system (agent) with its environment.We show that arbitrary coupling between sensorysignals and the internal state of a system – orthose between its action and external (environ-mental) states – do not guarantee synchronousdynamics between external and internal states:the spatial structure and the temporal dynamicsof sensory signals and action (that comprise thesystem’s Markov blanket) have to be pruned toproduce synchrony. This synchrony is necessaryfor an agent to infer environmental states – a pre-requisite for survival. Therefore, such sentientdynamics, relies primarily on approximate syn-chronization between the agent and its niche.

1 Introduction

In any living system, many physical and chemical reac-tions (along with their respective transport processes) inter-act, leading to fluctuations and instabilities. This promotesthe emergence of new traits that co-exist with other (con-served) traits. In other words, biological systems evolvedue to their innate ability to sustain a dialogue with theirenvironment [12]. Interactions of a biological system withits external milieu are far from being reversible, isothermalor in equilibrium [15, 21, 24]. These interactions entail ma-terial and energy exchange with the environment, renderingorganisms thermodynamically open.

This simple observation calls for a formal treatment of

nonequilibrium steady-state (NESS) that accounts for bi-ological self-organisation and its peculiar resistance to thesecond law or, more precisely, the fluctuation theorem thatgeneralizes the second law to nonequilibrium states [26].We address this challenge using a principle of least (vari-ational) action or free energy in three steps: this paperrepresents the first step by establishing a model systemthat possesses the dynamical repertoire necessary to sustainnonequilibrium steady-state. Crucially, this model systemhas a well-defined thermodynamic free energy or Lyapunovfunction that we will use to characterize its evolution andconvergence to nonequilibrium steady states. Our particu-lar focus here is on equipping a system with a Markov blan-ket that separates internal states from external states. Weillustrate the emergence of synchronization between inter-nal and external states that is mediated through the Markovblanket; thereby modelling an elementary form of percep-tion and action. In brief, we will see the dynamical cou-pling of states separated by a Markov blanket depends on aparticular sparsity and pruning of conditional dependenciesthat may characterise biological systems, as well as artifi-cial systems that mimic them.

In the second step, we will treat internal states as the pa-rameters or sufficient statistics of a probability distributionover external states, where this distribution defines varia-tional free energy. This enables us to write the dynamicsof the internal states as a variational free energy functional.We hope to show that thermodynamic and variational freeenergy change in the same way following external pertur-bations and share a common minimum at non-equilibriumsteady-state. In the final step, we will show how internalstates can enslave external states to minimize (variationaland thermodynamic) free energy at nonequilibrium steady-state. In effect, this rests on interpreting internal states asencoding probabilistic beliefs about the external states (i.e.perception) and acting on the external states – through theMarkov blanket – to make those beliefs come true (i.e.,action). Our ultimate objective is to show that one, andonly one, belief is sufficient to account for biological self-organisation; namely, the belief that the external states min-imize (Shannon) entropy production.

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2 Methods

In this section, we set up the basic form of the systemswe will used to model self organisation and nonequilibriumsteady-state. We start with coupled dynamics based uponthe Ginzburg-Landau equation and then consider the min-imal requirements for distinguishing between two sets ofstates; for example, states that are internal and external tothe system of interest. This involves the introduction of aMarkov blanket that provides a separation between internaland external states in terms of dependencies.

2.1 Dissipative systems – the Ginzburg-Landauequation

The (complex) Ginzburg-Landau equation (CGLE) de-scribes the dynamics of a system on a manifold in a metricspace that is undergoing a Hopf bifurcation from a station-ary to an oscillatory state,

λ(X, t) = λ+ (1 + iα)∇2λ− (1 + iβ) |λ|2λ (1)

The order parameter λ(X, t) is complex, such thatλ(X, t) = ρeiυ . Here, ρ(X, t) and υ(X, t) are space andtime-dependent amplitude and phase respectively. α andβ measure linear and non-linear dispersion (defining therelation between frequency of the waves and the wavenum-ber), respectively. The real GL equation can be obtainedfrom Eqn. 1 by setting α = β = 0. The evolution of thismacroscopic state can also be written in terms of a thermo-dynamic free energy function F ,

λ(X, t) = − δFδλ†

F =

∫ ∣∣∣∣ ∂λ∂X∣∣∣∣2 − η|λ|2 +

1

2|λ|4dX (2)

Another limiting case of Eqn. 1 is the conserved nonlinearSchrodinger equation that is obtained by setting α = β =∞,

iλ(X, t) = ∇2λ− |λ|2λ (3)

Yet another limiting case of Eqn. 1 is the Kuramoto-Sivashinsky (KS) equation. As the dynamics becomes un-stable, any amplitude perturbation is enslaved by the phaseϕ leading to the KS equation,

ϕ+∇4ϕ+∇2ϕ+ ϕ∇ϕ = 0 (4)Flows described by such partial differential equations(PDEs; Eqns. 1-4) occur in an infinite dimensional space.This is because if one writes sets of ordinary differentialequations (ODEs) to describe such flows, an infinite set ofODEs are required to represent the dynamics of a singlePDE. In practice, as dissipation gets larger, the asymptoticbehaviour becomes confined to a finite dimensional inertialmanifold.

2.2 Weak (Galerkin) formulation of PDEs

Using a Lagrange or Hermite finite-element basis, we cantransform the strong form of the PDEs (Eqn. 1) into avariational (weak) form by multiplying by a test functionon a Sobolev space W1. We can then use the divergencetheorem and discretise in time using the Crank-Nicolsonmethod [23]. Analytically (and numerically), this enablesone to represent Eqn. 1 as a system of coupled systemswith periodic boundaries, s.t. λ(x, y, t) = λr(x, y, t) +iλc(x, y, t) where {x, y} ⊂ {X}

λr(x, y, t) = λr +∇2λr − |λ|2λr − α∇2λc + β|λ|2λcλc(x, y, t) = λc +∇2λc − |λ|2λc + α∇2λr − β|λ|2λr

(5)

Although we describe the dynamical behaviour in C2 (rdenotes the real part and c denotes the complex part ofthe field), the finite-element framework is well equippedfor higher dimensions. For example, in C3 one parcellatesthe domain into 6-faced elements with nodes at the ver-tices and use quadratic Lagrange interpolation. For sim-plicity, we discuss the CGLE on a 2D domain. In our sim-ulations, Eqn. 1 was solved iteratively for 400 time-steps,using the generalized minimal residual method (GMRES)with incomplete LU (ILU) pre-conditioning [23]. Cubic-Hermite finite elements were used for increased precision.The mesh-size (L) was set to 512 discretisation units. Peri-odic boundary conditions were imposed on opposing ends.

2.3 Positional coupling of PDEs

Two or more PDEs (CGLE or reaction-diffusion-advection) are coupled using (positional) coupling betweenthe real and complex valued fields as described below. Thetwo CGLEs (ψ and λ) in Figure 2 are reciprocally coupledas,

ψ(x, y, t) = ψ + (1 + iα)∇2ψ − (1 + iβ) |ψ|2ψ + d(λ− ψ)

λ(x, y, t) = λ+ (1 + iα)∇2λ− (1 + iβ) |λ|2λ+ d(ψ − λ)

(6)

Again – for numerical simulations – the coupled system offour PDEs (two each for the real and complex parts) wasintegrated for 400 time-steps, using the generalized mini-mal residual method (GMRES) with incomplete LU (ILU)pre-conditioning. d is the coupling coefficient.

Equation 6 couples the modes of two systems and may thusappear sufficient to consider self organisation in terms ofthe coupling between a subsystem of interest (e.g., λ) andanother; say its heat bath, econiche or external milieu (e.g.,ψ). However, this setup is not sufficient to distinguish thesystem of interest from the system in which it is immersed.This is because there is no unique way of separating the in-ternal states (e.g., λ) from external states (e.g., ψ). This fol-lows because Eqn. 6 does not admit a partition of states inwhich internal states do not depend on external states, whenconditioned upon a third set of states known as a Markovblanket. To introduce a Markov blanket, we need to con-sider a partition into internal, external and Markov blanketstates; where the Markov blanket states can be further parti-tioned into active and sensory states. Crucially, the minimaldependency structure requires that internal states cannot di-rectly influence sensory states and external states cannot di-rectly influence active states (see below). This ensures thatthe dynamics of internal states can be separated from ex-ternal states but remain coupled through sensory and activestates. Assigning a partition of the Markov blankets intoactive and sensory states appeals to the notion of an action-perception cycle; in which external states influence internalstates through sensory states (i.e., perception), while inter-

nal states influence of external states through active states(i.e., action).

To create the sparse coupling necessary to support a dy-namics of self organisation (i.e., the action-perception cy-cle in Figure 3) additional PDEs governing the sensoryand action fields are needed. Here, we chose complexvalued reaction-diffusion-advection equations of the gen-eral form, s(X, t) = f (s,∇Xs,∆Xs) and a(X, t) =f2 (a,∇Xa,∆Xa). In this treatment, we use the reactionterm (s) to represent the positional coupling. Sensation andaction were represented using PDEs for two reasons: (i) inbiology, both sensory receptors and motor effectors are sys-tems that are spatially coupled in addition to their tempo-ral dynamics, (ii) reaction-diffusion-advection underwritesmost computational models in biology – from telegraphequations (advection set to zero) describing pulse propa-gation in nerve cells to modelling advection and diffusionof oxygen in the vasculature.

Eight coupled complex-valued PDEs then provide a novelform for an (generalised) action-perception cycle; with thefield ψ representing the external environment, S is the sen-sory field, λ are the internal states of a system (e.g., yournervous system) and a represents action in space-time. Thescalar fields d1...4 represent coupling kernels between pairsof fields, while the scalars α and β with their respectivesub-scripts represent the parameters of the CGLEs for ex-ternal states and internal states. The subscripts r and c de-note the real and complex parts of the PDEs,

ψr = ψr + β3(ψ2c + ψ2

r

)ψc −

(ψ2c + ψ2

r

)ψr − α1∆ψc + ∆ψr + d1 (ar − ψr)

ψc = ψc − β3(ψ2c + ψ2

r

)ψr −

(ψ2c + ψ2

r

)ψc + α1∆ψr + ∆ψc + d1 (ac − ψc)

sr = ∆sr +∇sr + d3 (ψr − sr)sc = ∆sc +∇sc + d3 (ψc − sc)

λr = λr + β4(λ2c + λ2r

)λc −

(λ2c + λ2r

)λr − α2∆λc + ∆λr + d2 (sr − λr)

λc = λc − β4(λ2c + λ2r

)λr −

(λ2c + λ2r

)λc + α2∆λr + ∆λc + d2 (sc − λc)

ar = ∆ar +∇ar + d4 (λr − ar)ac = ∆ac +∇ac + d4 (λc − ac)

(7)

Crucially, note that the flow of sensory states does not de-pend upon internal states and the flow of active states doesnot depend upon external states. This form therefore ad-mits a Markov blanket and a suitable model of exchangebetween internal and external states. The coupling kernelsd allow for a spatially limited coupling with the Markovblanket that allows us to adjust both the strength and ex-

tent to which the Markov blanket states mediate betweeninternal and external states. Having established the form ofa system we will study, we now consider how to quantifycoupling across the Markov blanket.

A CB

ED

Figure 1: A 2D and a 3D (complex) Ginzburg-Landau sys-tem. In A-B and D-E the x- and y- co-ordinates of theabsolute value (

√λ2r + λ2c) of the field are plotted whilst

in C the z co-ordinate is plotted in addition. Solutionsat the last time-step (T = 400 units) are displayed for:(A) α = +8, β = −8, (B) α = −1, β = +2, (C)α = +1, β = +0.5, (D) α = 0, β = 0 (real GL) and(E) α = +100, β = +100 (imaginary GL). Generalizedminimal residual method (GMRES) with incomplete LU(ILU) pre-conditioning was used to integrate the PDEs us-ing cubic-Hermite as the finite elements. Periodic bound-ary conditions were imposed with grid-size set at 512×512.

3 Results

In this section, we will use the Ginzburg-Landau equationto illustrate the basic dynamics afforded by this systemand illustrate generalised synchrony, starting with identi-cal synchronisation and then considering generalised syn-chrony. Having established the basic phenomenology, wewill then look at generalised synchronisation when insert-ing a Markov blanket between internal and external states;i.e., in the context of self organisation through action andperception.

3.1 The basic phenomenology of theGinzburg-Landau equation

Following Hohenberg and Halperin [16], we study a singletype of dissipative model – Model A, representing the com-plex Ginzburg-Landau equation (CGLE). It is the normalform for dynamics, where fixed points lose their stabilityvia a Hopf bifurcation, leading to a stable limit cycle oscil-lation. In fluid mechanics, this is known as the Newell-Whitehead equation, after the authors who derived it tostudy Benard convection [20]. For the CGLE, bifurcationdiagrams depicting the transitions between different phaseshave been charted for one and two spatial dimensions [9, 6].These studies have shown that for the Benjamin-Feir (BF)stable region of parameter space – defined by 1+α·β > 0 –one can always produce a plane wave that is linearly stable.

We used our simulation framework to study the CGLE intwo- and three-dimensions. As is known for the CGLE, avariety of coherent structures exist in 2D [2] depending onthe values of α and β in Eqn. 1. These coherent structuresrange from defect turbulence (|λ| = 0) in Figure 1A tospiral-waves in Figure 1B and scroll-waves in Figure 1C.Spiral waves are waves that rotate without changing theirshape. After they destabilize in a Hopf bifurcation, they bi-furcate into a drifting spiral wave. In three-dimensions theCGLE exhibits the analogue of a 2D spiral wave in the formof a scroll wave [13]. We simulated the CGLE on a toroid(Figure 1C) and observed turbulence of scroll rings aroundthe toroid (Figure 1C). Unlike spiral waves, scroll waveshave more degrees of freedom – enabling them to flowin space with varying shapes. As noted by Aranson andKramer [2], the 3D-CGLE also displays vortices that canbe highly unstable for the range of parameters over whichtheir 2D analogue is completely stable. Setting, α = 0,β = 0 gives us the real Ginzburg-Landau equation (Fig-ure 1D), representative of coherent brain states of the sortmeasured in electrophysiology. Similarly, as α and β at-tain larger values, the CGLE equation becomes conserva-tive (i.e., Hamiltonian) and we can approximate the epony-mous nonlinear Schrodinger equation (Figure 1E).

In summary, this single equation is capable of exhibiting awide variety of spatiotemporal structures (cf. deep recur-rent networks); including deterministic chaos: this is theprimary reason that we have selected the CGLE to repre-sent an interacting agent and its environment. Furthermore,this equation can be reduced to a neural field equation (pri-vate communication, Jack Cowan) that allows us to placewhat follows in a neural context; in other words, it canrepresent millions of neurons interacting with each other.Our intention here is not to meticulously describe the bi-furcation structure of this equation but to use it as a modelsystem to study the self-organisation of systems and theirapproximate milieu that display similar bifurcations and –most importantly – are dissipative.

3.2 Coupling and synchrony in spatially extendedsystems

To simulate self-organisation, it is necessary to distinguishbetween the system per se and its (proximate) environ-ment. Thermodynamically speaking, the environment actsas a heath bath for the agent wherein the agent accrues en-ergy for its survival from the environment. In order to ex-change information it is vital that the spatiotemporal dy-namics of both parties are coordinated; i.e., they are syn-chronized. We will start with the simplest form of gener-alised synchrony, known as identical synchronization be-cause the states or fields of the system and its environmentbecome identical over time. In other words, the motionin the joint space of the system and its environment areconstrained to an attracting set on a hyperplane, known as

||Re(e-h)||1||Im(e-h)||1

Erro

r E

rror

||Re(e-h)||1||Im(e-h)||1

A CB

ED F

Figure 2: Two 2D CGLE reciprocally coupled using posi-tional interactions i.e., d(λ2,1 − λ1,2) with d = 1, whereλ1 and λ2 describe the individual CGLEs. The x- and y-axis represent the absolute value of the field. (A,B) TwoCGLE with identical parameters – α = −1,β = +2. (C)The difference between the real and the imaginary partsof fields shown in A and B. (D,E) Two CGLE with non-identical parameters – (D) has α = +2,β = −6 and (E)has α = −1,β = −9. (F) Same as C but between non-identical fields.

the synchronization manifold. When the environment andthe agent have identical chaotic spatiotemporal dynamics(i.e., the parameters of the equations above are the same),it takes less than 20 time-units of simulation for two recip-rocally coupled CGLE to reduce real and complex errors tozero. This is shown in Figure 2A-C, where the synchronisa-tion error corresponds to the difference between the statesor distance from the identity synchronisation manifold (asquantified by the surface integral of the difference betweenthe respective real and the imaginary parts of the equation).

More often than not, coupled systems have different pa-rameters; i.e., they have different spatiotemporal dynamics.Generally these systems develop some sort of (generalised)synchrony as time unfolds. Indeed, in two different but re-ciprocally coupled CGLEs, both the real and the imaginaryparts approach zero with time, yet they are never identi-cal. This means there exists a smooth invertible mappingΦ : ψ → λ between the fields representing the environment(ψ) and the system or agent (λ). Such a form of synchronyis known as generalized synchronization [17, 3]. This isuseful, as it tells us that the agent can predict the dynamicsof the environment: for every point on ψ there is anotherpoint on λ that lies on the synchronization manifold.

3.3 The Markov blanket

Statistically, a reciprocally coupled dynamical system re-quires the existence of a Markov blanket to emulate self-organization (Figure 3) [28]. In other words, to distinguishinternal (agent) and external (environmental) states statis-tically, we have to consider the Markov blanket that sep-arates them. The Markov blanket is defined in terms ofdependencies that can be modelled in terms of the equa-

environment internal

sensory

action

ψ = ψr + i ψc λ = λr + i λc

s = sr + i sc

a = ar + i ac

Figure 3: The Action-Perception cycle. Two complex-valued Ginzburg-Landau equations representing the envi-ronment and the agent’s internal states are coupled via spa-tiotemporal states that constitute its Markov blanket; i.e.,the sensory and action fields. Here, we represent sensationand action as arbitrary complex-valued reaction-diffusion-advection equations, to model the spatiotemporal dynamicsof sensation and action. For details please see Eqn. 7.

tions of motion or flow underlying the dynamics in ques-tion. The Markov blanket of any subset of states is definedas its children, its parents and the children of the parents[22]. There are many ways in which one could model apartition between internal and external states. Here, weelected to do this by coupling two non-identical systemsof the sort described in the previous section through addi-tional states that have the necessary dependencies to form aMarkov blanket. The systems described in Figure 2 do notpossess a Markov blanket i.e., they complete knowledge ofone another and cannot be separated in a meaningful way.To introduce a Markov blanket we turn to Eqn. 7, wherethe Markov blanket is partitioned into states that are not in-fluenced by internal states (sensory states) and states thatare not influenced by external states (active states). Heuris-tically, this means that the internal states (e.g., agent) andexternal states (e.g., its environment) have vicarious knowl-edge of one another – it is only through the sensory states(e.g., sensors for sensory receptors or cameras/LIDAR forartificial agents) that the agent acquires knowledge of itsenvironment. It uses this partially observed knowledge toelaborate an action, mediated through active states (e.g.,motor effectors for biological agents or controllers for arti-ficial agents). Note that this partition is symmetrical withrespect to the dependency structure – the fundamental dis-tinction between internal and external states will becomeapparent in subsequent papers. Note also that the presenceof generalised synchrony implies the existence of an at-tracting set and therefore weakly mixing ergodicity. Thismeans that if generalised synchrony can be maintainedthrough a Markov blanket, the system self organises in afundamental (ergodic) sense through the existence of an at-tracting set. This is why we consider generalised synchrony

to be such an important characteristic of self organisation.

A crucial aspect of considering sensory and active com-ponents of the Markov blanket is that the internal and ex-ternal states are reciprocally coupled. In other words, wedo not consider a skew-product (master-slave) system – buttwo systems that influence and are influenced by each other(circular causality). Our interest here is in the way that theysynchronise and how this synchronisation reflects the emer-gence of a non-equilibrium steady state. This nonequilib-rium steady-state is synonymous with the attracting set orsynchronisation manifold that underwrites generalised syn-chrony. The reciprocal coupling across the Markov blanketmeans that there is a circular causality in play. In otherwords, in terms of the action perception cycle, nonequi-librium steady-state rests on both the agent (i.e., internalstates) acting on the environment (i.e., external states) andthe environment acting on the agent.

For the CGLE model, two non-identical systems (on two-dimensional manifolds) were coupled using auxiliary statesthat constitute the Markov blanket as follows – complex-valued sensory and action states were introduced to playthe role of sensation (via sensory receptors) and motor ac-tion (via effector organs), respectively. The sensory stateis governed by arbitrary reaction-diffusion-advection dy-namics (see Methods) but depends on the agent’s environ-ment, while active states depend on the internal states ofthe agent. Both sensory and the action fields are endowedwith a spatial structure due to the fact that sensory receptorsand motor effectors have spatiotemporal (and not just tem-poral) dynamics. Let us motivate this construction – mostcreatures do not have a single photo-receptor to detect lightbut an array of receptors that have a unique spatial and tem-poral architecture [19]. Similarly, motor effectors such asjoints require co-ordination in space as well as time to en-able locomotion [1]. Such a coupled system then formsan explicit way of perturbing the environment based on theinformation accrued by the agent.

Coupling the CGLE equations through a Markov blanket(represented as reaction-diffusion equations as per Figure3) with low coupling strength results in an exponential di-vergence between the external and the internal states (Fig-ure 4A-B). This is generally expected when deterministicchaos in one field is unable to attract the dynamics of thereciprocally connected field onto the synchronization man-ifold. This is because the phase space keeps expanding. Inthis scenario, no synchronisation between the environmentand the agent is seen (Figure 4A-B). Trivially, increasingthe coupling coefficient from 0.003 to 0.3 remedies the ex-ponential divergence; such that error between the real andthe imaginary parts of the internal and external fields re-main bounded (Figure 4C) and, most importantly, constant(Figure 4D).

In fact, one can obtain identical synchronization by altering

||Re(e-h)||1

||Im(e-h)||1

Time

||Re(e-h)||1

||Im(e-h)||1

Time

A B

C D

Error

Error

Re(e-h)2+Im

(e-h)2

Re(e-h)2+Im

(e-h)2

Figure 4: (A) The error between the real and imaginaryparts of the environment and the internal states. (B) Theerror magnitude between the fields in (A). (C) Same as A –but with the coupling coefficient set at d = 0.3 instead ofd = 0.003. (D) Same as B with d = 0.3 coupling coeffi-cient.

the dynamics of the sensory and the action fields, therebychanging how they (the Markov blanket) are influenced bythe environment and the internal states. Specifically, wereplaced the sensory (equivalently for action) fields in Eqn.7 by

sr = ∆sr +∇sr + d3ψr

sc = d3ψc (8)

This effectively suppresses the decay of the sensory and ac-tive states and simplifies the spatial memory of their imag-inary part. In effect, this renders sensory and active states amore direct reflection of external and internal states respec-tively.

Using such a dynamics for the sensory and action fieldsmean that the divergence between the environment and in-ternal state fields tends to zero over time (Figure 5) – acharacteristic of a specific form of generalized synchrony(identity mapping) under weak coupling. In other words,generalised synchrony rests upon particular conditions onthe coupling under this generic model of self organisation.

This concludes our illustration of non-equilibrium steady-state dynamics in dynamical systems that are coupledthrough a Markov blanket.

4 Discussion

The complex-valued Ginzburg-Landau equation (CGLE)offers a numerical framework to pose and study the inter-action of an agent with its environment (for example, toanalyze reinforcement learning systems). Quite naturally,both parties display rich behaviour in the form of phase

Environment �eld Sensation �eld Action �eldInternal �eldA DCB

||Re(e-h)||1||Im(e-h)||1

E

Erro

r

Time

Figure 5: Coupled CGLE as per the scheme in Figure 3but with Eqn. 7 replaced by Eqn. 8 i.e., when the dynam-ics of the Markov blanket is simplified to suppress theirmemory. A-D represent the external, sensory, internal andaction fields respectively. Coupling between fields was setto d = 0.003. (E) The error between the real and imaginaryparts of the external and the internal fields.

transitions (bifurcations). These phase transitions rangefrom spiral wave formation in two dimensions to the incep-tion of scroll waves in three dimensions. We have shownhow coupling of two dissipative systems lead to particularforms of synchronization. However, a direct mapping be-tween the external states of the environment and the inter-nal states of the agent renders them formally inseparable;requiring us to introduce the notion of Markov blankets.Here, we have used two intermediary reaction-diffusion-advection equations; namely, sensory and action fields.Such a setup supports a generalised action-perception cyclein a space-time dependent formalism. Our numerical sim-ulations of biological self-organization via the dynamics ofthe action (motor) – perception (sensory) fields highlightthe fact that synchronization between the environment andan agent can only occur with specific forms of coupling,through the Markov blanket. This is important because itsuggests the emergence of self-organisation is non-trivial;requiring the evolution of particular mappings (receptorsand appendages) that guarantee synchronization betweenthe agent and its ecological niche. In the absence of syn-chronization, the agent is simply unequipped to infer andlearn about its local niche and therefore cannot adapt itselfto sustain itself under burgeoning uncertainties.

By altering two parameters (α and β) the CGLE equationcan exhibit phase turbulence, defect turbulence, spatiotem-poral intermittency and bichaos, many of which are com-mon in systems operating out of equilibrium [25, 29, 10].These recurring instabilities require a continuous supply ofmatter or energy. They cease to exist as soon as the sup-ply of energy is removed. For infinite dimensional sys-tems, such as PDEs, and invariably for biological organ-isms, self-organization can therefore be understood as the

convergence of the dynamics under certain initial condi-tions to solutions that are stable and synchronized underexternal perturbations.

Our numerical experiments speak to the hypothesis thatpredictability is intertwined with generalized synchroniza-tion; i.e., there can be no reduction in uncertainty about theenvironment encoded by the internal states of the agent un-less the dynamics of both the agent and environment havesettled onto a synchronization manifold. The intuition fol-lows from the fact that in coupled chaotic systems the in-ternal state-space can be exponentially expanding; any in-ferential mechanism that reduces uncertainty works on thepremise that the system being inferred has certain proper-ties (say being smooth, continuous, existence of Markovpartition, etc.). This means by constantly changing sensa-tions and generating actions the internal states of the agentought to squeeze the internal states to a synchronizationmanifold. Squeezing is used in the study of dissipative dy-namics to portray the evolution of a trajectory to an iner-tial manifold. In short, our working hypothesis is infer-ence (instantiated as some form of energy minimization)enables the agent to squeeze dissipative states into a syn-chronization manifold. We will show in forthcoming pa-pers that it is the existence of such a manifold that enablesone to minimize the prediction error (using MCMC or vari-ational Bayes) in order to prescribe the most probable in-ternal states that mimic the environment.

The construction that we have adopted to study self-organization assumes that the spatial extent of the sensoryfield is equivalent to the environmental field – or a restrictedversion of it; i.e., they are mapped to one another in a one-to-one fashion. It would be an interesting exercise to studyhow different spatial forms (kernels) of coupling betweenthe environment and the sensations or between the inter-nal states of the agent and the action give rise to a zoo ofspatiotemporal pattern formation. The practical motivationbehind the simple coupling kernels considered in this paperis numerical – using identically sized finite-element meshwith simple coupling bounds the stiffness of the underly-ing differential equations, enabling us to iteratively solvethe coupled system of eight partial differential equations(i.e., action-perception cycle) in a reasonable amount oftime. However, our sensory receptors only have access torestricted and nonlinear samples from the environment (abottleneck) and it will be important to study the effect ofpunctuated coupling kernels.

We may have modelled the action-perception cycle inpurely phenomenological terms yet the principles manifestat this level percolate to lower levels (under a change ofgauge; see [28]). For example, chemical reactions in bi-ological systems do not take place homogeneously over asurface but via propagation of waves of oxidation that travelacross the reaction surface. The non-linearity in chemicalkinetics, the diffusion of chemical species – and lastly, the

external perturbations that force the reaction to occur in far-from-equilibrium conditions, are the major ingredients forbiotic self-organization. Equally, it is important to realizethat it is not a single instability but a hierarchy of insta-bilities that enables a biological system to spontaneouslyproduce increasingly complex modes. This happens as thebiological organism dissipates more and more energy – af-ter instability – creating conditions favourable for more in-stability.

It has been argued that the thermodynamic (Helmholtz)free-energy and variational free-energy share the same min-imum at thermodynamic equilibrium [27]. Yet a direct con-nection between non-equilibrium entropy production andShannon entropy has been missing in the context of bio-logical self-organization and evolution. Our formalism al-lows one to develop intuitions about how the two quantitiescould be linked. Given that a thermodynamic free-energyfunctional can be written for the complex valued Ginzburg-Landau equation [8] and variational inference methodologycan be applied to formulate a free-energy functional overthe environmental states; both could be compared qualita-tively using analysis as well as numerical simulations. Thisputs us in a unique position to compare thermodynamic andinformational energies for the first time.

It is well-known that under the chaotic hypothesis ([14]; formore general fluctuation theorems see [30, 18, 7, 11, 26]),the evolution of internal states will invariably converge to anon-equilibrium steady state (one of them being the Sinai-Ruelle-Bowen (SRB) measure [30, 5]). Biological systemsand artificial decision making systems are no different.Crucially, under Hamiltonian Anosov flows, the chaotic hy-pothesis reduces to the ergodic hypothesis of equilibriumthermodynamics [4] and converges to a Boltzmann-Gibbsmeasure. In our action-perception cycle (Figure 3) the sit-uation is a little more involved, as the PDEs describing theinternal states indirectly (via the sensory forcing) embedthe dynamics of the reservoir (the action field). We arguethat sensations drive the internal states of the agent to anon-equilibrium steady state (with a valid SRB measure).This enables the internal states to cool the information re-ceived from the sensorium wherein the extra heat (entropy)produced is returned as action, the thermal reservoir in thismicrocosm.

In summary, we have taken the first step in formulatinga framework where the interplay between thermodynamicand variational energies could be studied. The second stepis to compare and contrast these energies numerically andanalytically under a variety of parameter combinations andcoupling kernels, esp. those kernels that impose a bottle-neck.

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