Metaphorising and bayesian inference according to biology of cognition and enaction

The most accepted paradigm of brain function sustaining mathematical thinking is the “Brain as Computer” metaphor. Since the early 1970s, however, a new viewpoint based on the Biology of Cognition, created by Humberto Maturana and Francisco Varela, has slowly positioned itself as a challenger to the computer metaphor. Here, we interpret the notion of metaphorising in mathematics in the context of Biology of Cognition. Specifically, we introduce the fundamental concept of Structural Coupling, which is the mechanism by which living systems create the “objects” populating their niche by exploiting correlations in the never-ending Perception ⟲ Action loop. Furthermore, we show that how a living system decides which action to perform is the outcome of a Bayesian Inference mechanism; therefore, randomness is fundamental to living systems and not just a consequence of the mere lack of sufficient information to compute the next action. Additionally, in Bayesian inference, the underlying conditional probability distribution P(action|perception) changes by the very execution of every Perception ⟲ Action loop, performing a biased random walk according to a Hebbian-like rule. With this rich set of concepts derived from cutting-edge biology, we show that Biology of Cognition is a good fit to understand mathematical metaphorisation.


Introduction
The most used metaphor to understand brain function is "The Brain is a Computer".This metaphor can be traced to the early years after the Second World War, where influential books (Cybernetics by Wiener [1]) and influential papers [2] introduced the idea of computing in neuroscience.The power of this metaphor, with its seductive ideas of Codes, Programs, Optimization, Decision Theory, Memory and Information made it the mainstream framework by the early 1970s.According to this widespread and often implicit metaphor, the brain is a machine surrounded by a sensory apparatus that almost perfectly transduces -like a one-to-one correspondence-the properties of external objects into internal data.Data, once produced, is interpreted as an (almost) perfect representation of the external object [3].
Following the Brain-Computer metaphor, once transduction is completed, the internal representation is manipulated using self-evident formal rules that reflect the dynamics of the objective world surrounding the organism.The sequence of rules is implemented by a Turing Automaton that a) recognises the external object, b) analyses the context of the object and Sequence of interpretation of an object by a brain acting as an information processing device.
the circumstances where it is found, and c) designs an appropriate response (Figure 1).This is why the period 1970-2000 is rich in neuroscience papers that try to interpret the visual system as a simple telephoto system [4,5] or a cooperative system where high-level image descriptors (i.e., a cat) are built from simpler visual elements (contrast, borders, edges, occlusion) [6].These ideas reached wide acceptance as experimental research seemed to indicate, mainly in the visual and auditory systems, that the inner workings were similar to electronic devices built by humans (radars, sonars, correlators, linear predictors, etc.).Even in the motor system, a kind of population coding was found that, while different from normal circuits, was interpreted as an evolutionary adaptation of ideas of multivariate linear predictions [7].Although the use of the computer metaphor, plus the idea of Information, plus the many experimental results of the period 1960-1990 were formidable, some obvious cracks appeared in that conceptual edifice.Namely, the neuroanatomy of the vertebrate's brain is very different from von Neumann's computers.And perhaps the main difference is the non-hierarchical connectivity of the brain.Indeed, in modern electronics, the cabling follows a simple logic: massive connections go from sensors to the processing area (usually a CPU/RAM-memory assembly) and from there, a few cables go to the effectors (motors) and only a few cables project back (from the CPU/RAMmemory assembly to sensors or from the effectors to CPU/RAM-memory).Brain connectivity is radically different as feedback connections are massive and ubiquitous (for example, in the mammalian visual system, for every visual axon connecting the lateral geniculate nucleus to the visual cortex two axons from the cortex return to the geniculate [8]).
But the fundamental reason why the computer metaphor has lost its previous pre-eminence, uncontested during the 70-80s, is that it has not helped to advance, in the sense of producing deep ideas, in neuroscience.Furthermore, the computer metaphor did clash, from its very beginnings, with cherished ideas in theoretical biology, primarily the notion that living systems construct their Umwelt (the world as it is experienced) by the sequence of actions they perform.The Umweltian viewpoint has been stated since the seminal works of Jakob von Uexküll in 1920s [9] and since its inception, it has steadily grown in acceptance.
The point of view derived from Biology of Cognition [10] is firmly anchored in the Umwelt lineage and it can be considered its most technically precise formulation as it uses ideas obtained from current neuroscience.From the outset, Biology of Cognition posits that the nervous system is not a computer or a machine processing information but rather it is an infinite spiral of P erception ⟲ Action elementary acts where the main problem is deciding the next action.Thus, at its core, every living system is an unending stream of recursive Perception⟲Action loops and not a device that interprets, using the immutable laws of logic, an objective external world [11,12].
In our formulation, the Perception⟲Action stream is recursive because at any given moment t, the perception at time t, denoted P erception(t), discerned by a living system, is the direct result of the action previously performed, that is, the action at time (t−1), denoted Action(t−1), and in turn triggers Action(t + 1).(Figure 2). 1or example, if you perceive a lion (P erception(t)) and decide to approach it slowly (Action(t + 1)), your subsequent perception (P erception(t + 2)) is different than if you decide to run away (alternative Action(t + 1)).Here Action(t − 1) might have been "scanning the landscape with your binoculars". of the organism with respect to its medium.Thus, the actual objective of living systems is not to understand "reality" but rather to modulate their perception-action spiral to maintain their internal (living) organisation invariant.According to this view, the main formal problem that living systems must solve is not to implement a "capture → recognize → analyse → decide" stream, but rather to create a Perception-Action spiral where the notion of formal truth is replaced by a notion of coherence between the living system and its circumstances.

The basic mechanism of Structural Coupling/Enaction
Structural Coupling, as first proposed by Maturana [11] and later refined by Varela [12] as Enaction, is the phenomenon by which living systems, as they undergo the never-ending perception⟲action spiral, create the "objects"2 they encounter, thus creating their ecological niche (Figure 3)."Objects" are an unavoidable side effect of the Perception⟲Action loop when embodied by an autopoietic system.The basic idea (see Figure 4) is that when an organism finds itself in recurrent interaction with its medium, the relationship that emerges from the perception-action loop is initially undefined (at t = t 0 ).But, as living systems are self-fabricating mechanisms with myriads of regulatory mechanisms, the re-occurrence of the interactions begins to change the structure of the living system as well as the interface with its medium (at t = t 1 ).If the history of mutual interactions is dense enough, in the sense of being recurrent, then the structure of the living system and of the interface become perfectly complementary (at t = t 2 ) and, for an observer not privy to the history of previous interactions, it appears that the living system has "adapted" to a preexistent object.This misinterpretation is understandable because the observer is only witnessing the final stage of the process (t 2 ), not seeing the most important aspect of the phenomenon of Structural Coupling: that all objects are co-constructed relationships between the organism and its medium.

All Objects are always relational.
Our world is populated by a wide variety of objects that appear to us as given objective entities.An object is, however, defined by the list of behaviours that we can have towards it.For example, Initially (t = t 0 ), the relation is undefined (grey blob), but as the recurrent interaction is maintained (t = t 1 ), the inner structure of the living system changes, as well as the interface with its medium.In well-developed interactions (t = t 2 ), the organism's structure has undergone a significant change along with the interface, and these changes are perfectly complementary.External Observers (upper right corner) could misinterpret the overall situation if they only focus on stage t = t 2 and not on the overall process; they might describe the situation as an adaptation of the organism to a pre-existent external object.Thus if the observer only considers the situation at t = t 2 he or she will miss the crucial fact that the object (the tree-like contour) started as an un-structured blob.Structural Coupling refers to the complete process of internal changes in the organism and changes in the interface organism/medium.
the object "cat" is the list of actions: "to feed the cat", "to play with the cat", "to comb the cat", "to brag about the cat",...This is also true for more abstract objects such as vector spaces and linear transformations in mathematics.Thus, we posit that every object is a list of possible behaviours we can engage vis à vis this entity.Every object, either real or formal is defined relationally by our behaviours.This is true for us (Homo sapiens) as well as for bacteria [13].

The problem of Living System: What to do next?
As explained, a living system is a machine that relates perception to action.This relation is mediated by a metabolic organisation whose internal logic is implemented by a self-fabricating system -a concept that still eludes modern science.Indeed, as it has been explained in the last 30 years [10,14], the "secret" of living systems is not that they behave like computers (they do not) or that they have genetic material (like DNA / RNA macromolecules, although they do).Their secret is that they fabricate the vast majority of their own components (Autopoiesis), and regulate the speed and nature of their internal process of transformations (i.e. the living system controls the kinetics of all biochemical reactions by producing and modulating all the catalysts -enzymes-in charge of facilitating the myriad of internal processes) [10,15,16].
As a consequence of the complex network of interlocking processes that constitute metabolism, when the sensory lamina (sensorium) of an organism encounters an "object" (y) in its milieu, a series of internal signals (Sy) are activated within metabolism.Crucially, the relation between "objects" and internal signals is not a simple one-to-one correspondence.Each encounter triggers a complex spectrum of internal signals, not all of which have the same temporal course.Some signals occur as soon as the object is encountered, while others are only slowly recruited.Thus, in the enactive view of the organism, the capture (or transduction) process is far more complex than a simple functional relation between an object and some limited set of internal variables [16,17,18].From the point of view of the organism, a fundamental complexity arises as it must decide which course of action it must implement using as input variable the set of internal signals (S) that are not in direct one-to-one correspondence with the "external world".The organism, from this point of view, should not be considered as a simple computer but as a self-fabricating Bayesian inference machine entangled with the world through Structural Coupling.At all times, it must answer the following question: What is the probability (denoted P (A x |S y )) that I need to implement action A x given that my internal signals are S y ? Figure 5.When a living system encounters a given object in its niche, many internal signals arise inside the metabolic network (arrow connecting the Sensorium to the internal mechanism defining the next action).These signals modulate the mechanism defining the next action that the living system must perform (arrow arriving to the Effectorium).The organism does not detect the "external reality out there" -a non existing entity-it only sees the internal changes as manifested by the spectrum of internal signals.Some of these internals signals however could be generated by the metabolic network bypassing the Sensorium.
In a compact form, we then have a Conditional Probability Matrix (denoted by CPM), associated with the organism, relating the set of signals (S y ) to the set of possible subsequent actions (Ax) 3 , i.e.: CP M = (P (A x |S y )) x,y In this view of the organism (as a self-fabricating Bayesian inference machine entangled with the world through structural coupling), randomness is an intrinsic property of the world of the organism and not just simple ignorance of the factors controlling a perfectly deterministic situation (Fig 4 and Fig 5).
We can interpret the conditional probability matrix CP M as a stochastic mapping4 , call it Φ, which sends a signal vector S y (triggered by the perception process) to the random action vector: Φ(S y ) = x Notice that our stochastic mapping Φ admits a sort of "dual", or "mirroring analogue" stochastic mapping Φ * , going from actions to signals, defined by the "dual" conditional probability matrix: CP M * = (P (S y |A x )) y,x whose (y, x)-entry can be interpreted as the probability that the action vector A x triggers the signal vector S y .In some sense, the matrix CP M * can be seen as the "CPM of the medium of the organism" if we become aware that actions of the organism trigger inner signals of the medium and the other way around.It follows that by combining the stochastic mapping Φ, together with its dual Φ * , we have a random walk in perception-action space.Indeed, Φ sends signals (and more generally random signals) to random actions, then Φ * sends random actions back to random signals, which are sent to random actions by Φ * and so on...We then have a random walk in the space of signals and actions, given by the iterated application of the stochastic mappings Φ, Φ * , Φ, Φ * , • • • To be more precise about this random walk, which intertwines a random walk on signals with another on actions, we need to take into account the temporal unfolding of the "object construction process" of our living being, via the infinite spiral of perception ⟲ action loops of Figure 2. We address this in the following subsections.where the action performed must be consistent with the state of the surrounding medium.This viewpoint is valid for all living systems from bacteria to kittens and Homo Sapiens.The big difference is the size of the Conditional Probability Matrix (CPM).In bacteria, the medium can be catalogued by just a few circumstances (place with low food, place with high temperature, place with many bacteria, etc.), whereas a cat's medium has thousands of circumstances (master's home, it is August!, morning nap, mean dog running down the street, another nap, etc).Any Homo sapiens' medium, thanks to the amazing plastic abilities of neurones, and specially language, could reach a billion of circumstances (Marilyn Monroe, Alexander Grothendieck, Prime Number, Darwin, Password, Electron, Enzyme, Goldbach Conjecture, Gödel's theorem, red colour, Motherland, credit card, debt, inflation, UNIX, grades, tristesse, The Beatles, CyberBullying, etc.).Furthermore, as internal signals are also products of the normal action of self-fabrication, it happens that internal states (i.e.those not triggered by specific encounters between organism and niche) also produce configurations of signals that are treated by the organism as being triggered by specific encounters.In other words, because of the many regulatory mechanisms as well as the complexity of the self-fabricating network, internal states are treated as external events because the only relevant fact is the variation in the spectrum of internal signals.In this sense, we agree with Seth's claim that we "hallucinate reality" [19].

Plasticity of the Bayesian Inference mechanism behind Enaction/Structural Coupling
As mentioned above, the main problem of an organism is to decide its next action after internally detecting a given signal S. Thus, we can think that the decision mechanism uses the Conditional Probability Matrix P (Action|Signal) as the starting point to decide on the final action to implement.This is the core of Structural Coupling [20] and similar theories like the Free Energy Principle [21,22].To explain the following we start from a very simple case (see Figure 6).Suppose that an organism can implement 3 different actions (A, B, C) and detect 4 different signal configurations (S 1 , S 2 , S 3 , S 4 ).Once an action has been implemented (imagine C) because a given signal configuration (S 2 ) was detected, the corresponding conditional probability P (C|S 2 ) must be updated to reflect the increased strength (or effectiveness) of the association S 2 → C.
Thus, we can write a simple evolution equation: where ∆P t represents the change in the probability P (C|S 2 ) t and can take many forms, such as somewhat akin to the Hebbian mechanism of synaptic plasticity [23].
If action C was effective, from the point of view of the organism, ∆P t would be positive and, in that case, P (C|S 2 ) t+1 > P (C|S 2 ) t at the expense of one or more other action/signal couples, for which we would have had P (action|signal) t+1 < P (action|signal) t .
Other rules for the variation of P (C|S 2 ) t can be proposed, but overall the important point is that the Conditional Probability Matrix (CP M ) t at time t dynamically changes moment by moment according to the specific trajectory of (• • • action|internal signal, . . . ) undergone by the organism.
In other words, the transition probabilities of the random walk arising from the iterated application of the stochastic mappings Φ, Φ * , Φ, Φ * , • • • (or the corresponding conditional probability matrices CP M, CP M * , CP M, CP M * , • • • if you prefer) are not stationary.
Indeed, if a signal vector S y arises in the metabolism of our living being at time t, the ensuing action A x at time t+1 is probabilistically afforded by applying the stochastic mapping Φ at time t (or equivalently, the conditional probability matrix CP M at time t).We should then write, Therefore, the temporal unfolding of our perception ⟲ action random walk, starting, say, at an initial signal vector S y (t 0 ), would be given by the iterated application of Φ t 0 , Φ * t 0 +1 , Φ t 0 +2 , Φ * t 0 +3 , • • • to S y (t 0 ).Of course, we could see this random walk as the intertwining, like two threads of the same cord, of the "pure perception" random walk given by Φ * t 0 +1 • Φ t 0 , Φ * t 0 +3 • Φ t 0 +2 , • • •5 and the "pure action" random walk given by Φ Notice now that updating Φ t and Φ * t (or equivalently, (CP M ) t and (CP M ) * t ) in turn enacts a random walk in the space of conditional probability matrices.This random walk "hovers" above the above random walks in perception space and in action space, given by the suitable iterated application of our changing stochastic mappings.So, in fact, the latter are random walks with randomly varying transition probabilities, "second-order" random walks of sorts!

Metaphors as homologous paths in CP M space
In our view, Aristotle's classical description of metaphor [24] is just the tip of the iceberg of metaphorising as a biological cognitive process.
Indeed, the crucial role of metaphorising in mathematics and mathematics education has been increasingly acknowledged only in the last decades [25,26,27,28,29,30,31,32,33].We are especially interested in the poietic role of "hallucinatory metaphors ": while Aristotle described metaphor as giving a thing a name that belongs to something else, we say that metaphorising is looking at something and seeing something else, i.e., "hallucinating", in the sense of Seth [19], for whom "reality" is a "controlled hallucination" of ours.Then, metaphorising enables us to fathom, grasp, or construct new concepts by taking advantage of our previous experiences.For example, when we look at the symmetric random walk of a frog on a row of stones in a pond and we wonder where it will be after a given number of jumps, we may "hallucinate", seeing the frog repeatedly splitting evenly into pieces instead of jumping.Or, engaging in an animal-friendlier hallucination, we may see one litre of fruit juice being shared IOP Publishing doi:10.1088/1757-899X/1292/1/0120129 equally among next neighbours in a row of drinkers, starting with one holding the whole litre.This allows us to construct the abstract notion of probability.Indeed, the probability of finding the frog on a given stone after n jumps can be understood as the amount of fruit juice a specific drinker has after n sharings.Initially, the problems of finding the frog and the distribution of juice after a given number of steps appear unrelated, but metaphorically they can be regarded as the "same".Metaphor can then be seen as the use of previous experience to explain current experience, as suggested by Tall's felicitous pun: "a metaphor is a met-afore" [30].
On the other hand, we can fathom the arising of metaphorising as a biological cognitive process, from our enactivist viewpoint, as follows.The interaction profile between an organism and its Umwelt is not a random collection of perceptions followed by certain actions, because the encounters between an organism and its medium follow regularities created by its ontogeny and phylogeny.In other words, the local history of interactions organism/medium contains strong correlations that are reflected in that the time series of internal signals (..., S t−2 , S t−1 , S t , S t+1 , ..) contains internal correlations reflecting the regularities in the interactions between the organism and its Umwelt.Thus, the random walk is not fully random as any effective (P erception, Action) trajectory contains local regularities or correlations.
By way of example, let us consider two sequences of interactions, i.e. sequences of perception/action) I 1 and I 2 that contain some correlations in common; for example, they consist of petting a cat (I 1 ) and petting a dog (I 2 ), see Figure 7.In this case, random trajectories in CP M space induced by interactions I 1 and I 2 are themselves correlated.Thus, in some sense, each trajectory could be thought of as a metaphor of the other.This metaphorising could be embodied in several ways, from the simple equality of changes in CP M (i.e., for a given P (A i |S k ) the overall change in one interaction is proportional to the one in the other) to more complex forms where internal correlations in CP M are preserved in I 1 and I 2 . .Metaphors and CP M s.Two interactions, sequences of perception ⟲ action (upperrow: petting a cat, lower-row: petting a dog), transform the initial state of their corresponding CP M according to different, but correlated paths.In the end, the states of both CP M are different (compare the corresponding matrices), but subtle correlations exist between both matrices (the two black square patterns are not identical but "homologous").In this case, we would say that each random walk is a metaphor of the other.
We remark that the same applies to more "abstract" circumstances, e. g., in the context of the learning of mathematics, when instead of two pets to fondle, we have two problems to solve.For example, the classical ruin problem for two players, in probability, and the symmetric random walk of our frog now along a row of six stones, say, in a tropical river, numbered 0 to 5, with the caveat that the end stones 0 and 5 are, in fact, alligator heads.Then, initial fortunes of 2 coins and 3 coins of the players, say, correspond to the frog starting from stone number 2. The learners wonder about the unfolding of the game or the fate of the frog.Each problem appears as a metaphor of the other, and both can in turn be metaphorised as the draining down of water through a suitable vertical network of ducts.See [34] for a detailed account of this metaphorisation.
Our first-level introduction of the basic concepts of Enaction/Biology of Cognition uses two main ideas: a) Bayesian inference triggered by internal signals and b) plasticity of the Bayesian mechanism according to a Hebbian-like rule.Notice that our understanding of mathematical thinking based on these premises will very different from the standard viewpoint based on applying perfect logic to a disembodied universe.
We summarise our take-homes below.First, the Enactive viewpoint considers generation of action (rather than analysis) as the central property of a nervous system.
Second, the mechanism generating actions is intrinsically probabilistic, not a mere trick to hide our ignorance about the value of some systemic variables.
Third, the input to our mechanism is not a simple transduction of the external world.The input is an interplay of the confrontation of the nervous system (or organism) against its medium mediated by its own metabolism.This confrontation is always embedded in the task of selffabrication.In some sense the input could be represented by a filtering of the medium by the metabolic network.Thus, the input to the inference mechanism contains many elements of the internal rules produced by the self-fabricating metabolism and is not in a one to one correspondence with the "external world", thus it is more complex than a blurred image of the external world.
Fourth, most actions can be triggered by purely internal events without any input from an external, pre-existing, reality.
Fifth, through the never-ending interplay between the mechanisms of Bayesian inference and Hebbian plasticity, a dynamics akin to a random walk arises.Thus, if we consider simultaneously the random walk generated by the aforementioned mechanism in the space of Perceptions and the associated random walk in the space of Actions, in some sense each random walk is a metaphor of the other.Although both walks exhibit a significant random component, they are strongly correlated to each other.Indeed, the corresponding sequences of perceptions and actions both converge to (the construction of) the same object in the Umwelt of the organism.
Sixth, internal metabolic states are handled by the organism as bona-fide "external object".The identification between internal states and objects (which are defined by recurrent perceptually guided actions) is crucial, as it is the mechanism we use to create, recursively, all sorts of abstract mathematical objects (number, function, Fermat last theorem, etc.).We then present the following diagrammatic description of the processes involved: (Internal-State t → Action t+1 → Internal-State t+2 ), but since internal states are encapsulated as Objects, the above sequence could be written as: making clear that Objects are continuously created by the very fact that the organism handles internal states as external objects.Finally, we can synthetically rewrite this sequence as follows.
Action(Objects) = Objects analogously to the standard functional notation f (x) = y, meaning that actions applied to objects give objects.
But since Objects ←→ Actions, i.e.Actions and Objects are identified with each other, we can write the overall process as a system of self-referential equations [35] to foster the understanding of the intertwining between cognition and action [36,37]" Action(Action) = Action, P erception(P erception) = P erception The latter being entailed by the former because of the intertwining of perception and action, embodied in the recurring perception ⟲ action loops.
Seventh, instead of considering the correlated random walks in perception space and in action space sequentially, as intertwined alternating random walks, we can fathom them simultaneously as a single random walk whose states are ordered pairs (perception, action), so that (perception(t), action(t + 1)) goes to (perception(t + 2), action(t + 3)), etc.This generates an overarching random walk in the Cartesian product of Perception space and Action space, whose transition matrix at time t is a D × D square block diagonal matrix, whose blocks are CP M * t+1 • CP M t and CP M t+2 • CP M * t+1 , with D = dim(P erception) + dim(Action).The "2dimensional" trajectories of this random walk provide a visualisation of our complex sequences of perception ⟲ action.Metaphorisation appears then as similarity of these trajectories, as is the case of Pet a Cat and Pet a Dog in the toy example of Figure 7 (where D = 4 + 5 = 9).
In summary, we have presented the backbone of a theory explaining how living systems construct the objects that populate their niche.The main ideas are that organisms are selffabricating systems and that they live inside a never-ending spiral relating perception to action.Furthermore, we claim that, in order to select the action to be executed among a behavioral repertoire each organism performs Bayesian inference on dynamic probabilities.This viewpoint has the advantage of explaining how a circumstance (an object or a mathematical result) can be viewed as the metaphor of another object.We see mathematical thinking then as centred on the world the organisms create and not on immutable rules of logic.Our perspective opens the door to build an organismic Artificial Intelligence based on actions rather than simple predictions based on oceanic amount of data.In a nutshell we present a novel manner to interpret mathematical work based on the premise that mathematics is the product of concrete self-fabricating living systems entangled with their Umwelt through Structural Coupling.
Figure 1.Sequence of interpretation of an object by a brain acting as an information processing device.

Figure 3 .Figure 4 .
Figure 3. Every Object (from teapots and cats to Schrödinger's equation) are constructed through the perception⟲action spiral as the result of a random walk in the Cartesian product space P erception × Action.

4. 1 .
A further problem for Living Systems: Updating of the Conditional Probability Matrix P (Actions|Signals) Any living system must solve the problem: Internal Signals (S) → Performed Action (A).

Figure 6 .
Figure 6.The Conditional Probability Matrix (CPM) of a Living System.Inside the metabolic network of any organism, from bacteria to Homo sapiens, there exists a mechanism whose operation is akin to a Bayesian probability model driven by a Conditional Probability Matrix (CP M ), whose entries answer questions of the type: Which is the probability of executing action B, given that the internal signals are S 2 .Notably, the matrix CP M is not static, it changes moment by moment according to the particular Perception ⟲ Action spiral Figure 7. Metaphors and CP M s.Two interactions, sequences of perception ⟲ action (upperrow: petting a cat, lower-row: petting a dog), transform the initial state of their corresponding CP M according to different, but correlated paths.In the end, the states of both CP M are different (compare the corresponding matrices), but subtle correlations exist between both matrices (the two black square patterns are not identical but "homologous").In this case, we would say that each random walk is a metaphor of the other.