Rodrick Wallace1
(1)
New York State Psychiatric Institute, New York, NY, USA
Summary
The Data Rate Theorem that establishes a formal linkage between control theory and information theory carries deep implications for the study of embodied cognition and its dysfunctions across human, machine, and composite man–machine cockpit entities. The stabilization of such cognition is a dynamic process deeply intertwined with it, constituting, in a sense, the riverbanks directing the flow of a stream of generalized consciousness. The fundamental role of culture in human life has long been understood. Here we argue that, not only is culture, in the words of the evolutionary anthropologist Robert Boyd, “as much a part of human biology as the enamel on our teeth,” but that it is an important part of the disorders of embodied cognition that different communities socially construct as mental illness. A shift in perspective on such illness is badly needed, from naive geneticism and a simplistic “biomarker” focus to a broad cognitive theory that recognizes the central roles of culture, socioeconomic structure, their histories, and their dynamics, in human development. Unfortunately, stabilizing analogs to cultural influence are unavailable to high-order real-time machine and cockpit systems that, increasingly, will be tasked with control of critical enterprises ranging from communication, power, and travel networks to nuclear and chemical reactors, organized deadly force, and so on (Wallace, Information Theory Models of Instabilities in Critical Systems. World Scientific Series in Information Studies. World Scientific, Singapore, 2017).
6.1 Introduction
Varela et al. (1991), in their pioneering study The Embodied Mind: Cognitive Science and Human Experience, asserted that the world is portrayed and determined by mutual interaction between the physiology of an organism, its sensorimotor circuitry, and the environment. The essential point, in their view, being the inherent structural coupling of brain–body-world. Lively debate has followed and continues (e.g., Clark 1998; Wilson 2002; Wilson and Golonka 2013). As Wilson and Golonka put it,
The most exciting hypothesis in cognitive science right now is the theory that cognition is embodied…[T]he implications of embodiment are…radical…If cognition can span the brain, body, and environment, then the “states of mind” of disembodied cognitive science won’t exist to be modified. Cognition will instead be an extended system assembled from a broad variety of resources…[that] can span brain, body, and environment…
[We] treat perception-action problems and language problems as the same kind of thing…Linguistic information is a task resource in exactly the same way as perceptual information…Our behavior emerges from a pool of potential task resources that include the body, the environment and…the brain…
In the same year as the work by Varela et al. (1991), Brooks (1991) explored analogous ideas in robotics.
Barrett and Henzi (2005) summarize matters as follows:
…[C]ognition is “situated” and “distributed”. Cognition is not limited by the “skin and skull” of the individual…but uses resources and materials in the environment…The dynamic social interactions of primates…can be investigated as cognitive processes in themselves…A distributed approach…considers all cognitive processes to emerge from the interactions between individuals, and between individuals and the world.
We will explore how this picture, of the recruitment of disparate resources into shifting, temporary coalitions in real time to address challenges and opportunities, represents a significant extension of the Baars (1988) model of consciousness, as described in Chap. 1
It is possible to study cognition and embodiment through the Data Rate Theorem that formally relates control theory and information theory, and to include as well the needed regulation and stabilization mechanisms in a unitary construct that must interpenetrate in a similar manner.
Natural cognitive systems operate at all scales and levels of organization of biological process (e.g., Wallace 2012, 2014a). The failure of low level biological cognition in humans is often expressed through early onset of the intractable chronic diseases of senescence (e.g., Wallace and Wallace 2010, 2013; Wallace 2015). Failure of high-order cognition in humans has been the subject of intensive scientific study for over 200 years, with little if any consensus. As Johnson-Laird et al. (2006) put it,
Current knowledge about psychological illnesses is comparable to the medical understanding of epidemics in the early 19th Century. Physicians realized that cholera, for example, was a specific disease, which killed about a third of the people whom it infected. What they disagreed about was the cause, the pathology, and the communication of the disease. Similarly, most medical professionals these days realize that psychological illnesses occur…but they disagree about their cause and pathology.
As the media chatter surrounding the release of the latest official US nosology of mental disorders—the so-called DSM-V—indicates, this may be something an understatement. Indeed, the entire enterprise of the Diagnostic and Statistical Manual of Mental Disorders has been characterized as “prescientific” (e.g., Gilbert 2001). Atmanspacher (2006), for example, argues that formal theory of high-level cognition is itself at a point like that of physics 400 years ago, with the basic entities and the relations between them yet to be determined. Further complications arise via the overwhelming influence of culture on both mental process and its dysfunction (e.g., Heine 2001; Kleinman and Cohen 1997), something to which we will return. Previous chapters provide further exploration.
The stabilization and regulation of high-order cognition may be as complex as such cognition itself.
Some simplification, however, is possible. Cognition can be described in terms of a sophisticated real-time feedback between interior and exterior, necessarily constrained, as Dretske (1994) has noted, by certain asymptotic limit theorems of probability,
Communication theory can be interpreted as telling one something important about the conditions that are needed for the transmission of information as ordinarily understood, about what it takes for the transmission of semantic information. This has tempted people…to exploit [information theory] in semantic and cognitive studies…
…Unless there is a statistically reliable channel of communication between [a source and a receiver]…no signal can carry semantic information…[thus] the channel over which the [semantic] signal arrives [must satisfy] the appropriate statistical constraints of information theory.
Recent intersection of that theory with the formalisms of real-time feedback systems—control theory—may provide insight into matters of embodied cognition and the parallel synergistic problem of embodied regulation and control. Here, we extend that work and apply the resulting conceptual model toward formally characterizing the unitary structural coupling of brain–body-world, a coupling in which, for humans, culture is a central mode. In the process, we will explore dynamic statistical models that can be fitted to data.
We will find, after some considerable development, results recognizably similar to those of Verduzco-Flores (2012). They, using a detailed neural network model, observed that
[Certain network] changes result in a set of dynamics which may be associated with cognitive symptoms associated with different neuropathologies, particularly epilepsy, schizophrenia, and obsessive compulsive disorders…[S]ymptoms in these disorders may arise from similar or the same general mechanisms…[and] these pathological dynamics may form a set of overlapping states within the normal network function..[related] to observed associations between different pathologies.
Apparently, these are robust observations across a variety of cognitive systems.
6.2 The Data Rate Theorem
The Data Rate Theorem, a generalization of the Bode integral theorem for linear control systems (e.g., Yu and Mehta 2010; Kitano 2007; Csete and Doyle 2002), describes the stability of linear feedback control under data rate constraints (e.g., Mitter 2001; Tatikonda and Mitter 2004; Sahai 2004; Sahai and Mitter 2006; Minero et al. 2009; Nair et al. 2007; You and Xie 2013). Given a noise-free data link between a discrete linear plant and its controller, unstable modes can be stabilized only if the feedback data rate 
is greater than the rate of “topological information” generated by the unstable system. For the simplest incarnation, if the linear matrix equation of the plant is of the form xt+1 = A xt + ⋯ , where xt is the n-dimensional state vector at time t, then the necessary condition for stabilizability is
![$$\displaystyle{ \mathcal{H}>\log [\vert det\mathbf{A}^{u}\vert ] }$$](computational-psychiatry.files/image264.png){kind=small}
(6.1)
where det is the determinant and Au is the decoupled unstable component of A, i.e., the part having eigenvalues ≥ 1.
There is, then, a critical positive data rate below which there does not exist any quantization and control scheme able to stabilize an unstable (linear) feedback system.
This result and its variations are as fundamental as the Shannon Coding and Source Coding Theorems, and the Rate Distortion Theorem (Cover and Thomas 2006; Ash 1990; Khinchin 1957).
It is possible to entertain and extend these considerations, using methods from cognitive theory to explore brain–body-world dynamics that inherently take place under data rate constraints.
The essential analytic tool will be something much like Pettini’s (2007) “topological hypothesis”—a version of Landau’s spontaneous symmetry breaking insight for physical systems (Landau and Lifshitz 2007)—which infers that punctuated events often involve a change in the topology of an underlying configuration space, and the observed singularities in the measures of interest can be interpreted as a “shadow” of major topological change happening at a more basic level.
The tool for the study of such topological changes is Morse Theory (Pettini 2007; Matsumoto 2002), summarized in the Mathematical Appendix, and it is possible to construct a relevant Morse Function as a “representation” of the underlying theory.
The first step is a recapitulation of an approach to cognition using the asymptotic limit theorems of information theory (Wallace 2000, 2005a,b, 2007, 2012, 2014a).
6.3 Cognition as an Information Source
Atlan and Cohen (1998) argue that the essence of cognition involves comparison of a perceived signal with an internal, learned or inherited picture of the world, and then choice of one response from a much larger repertoire of possible responses. That is, cognitive pattern recognition-and-response proceeds by an algorithmic combination of an incoming external sensory signal with an internal ongoing activity—incorporating the internalized picture of the world—and triggering an appropriate action based on a decision that the pattern of sensory activity requires a response.
Incoming sensory input is thus mixed in an unspecified but systematic manner with internal ongoing activity to create a path of combined signals x = (a 0, a1, …, a n , …). Each a k thus represents some functional composition of the internal and the external. An application of this perspective to a standard neural network is given in Wallace (2005a, p. 34).
This path is fed into a highly nonlinear, but otherwise similarly unspecified, decision function, h, generating an output h(x) that is an element of one of two disjoint sets B0 and B 1 of possible system responses. Let
Assume a graded response, supposing that if
the pattern is not recognized, and if
the pattern is recognized, and some action bj , k + 1 ≤ j ≤ m takes place.
Interest focuses on paths x triggering pattern recognition-and-response: given a fixed initial state a0, examine all possible subsequent paths x beginning with a0 and leading to the event h(x) ∈ B 1. Thus h(a0, …, a j ) ∈ B 0 for all 0 ≤ j < m, but h(a0, …, a m ) ∈ B 1.
For each positive integer n, take N(n) as the number of high probability paths of length n that begin with some particular a0 and lead to the condition h(x) ∈ B 1. Call such paths “meaningful,” assuming that N(n) will be considerably less than the number of all possible paths of length n leading from a0 to the condition h(x) ∈ B 1.
Identification of the “alphabet” of the states aj , B k may depend on the proper system coarse-graining in the sense of symbolic dynamics (e.g., Beck and Schlogl 1995).
Combining algorithm, the form of the function h, and the details of grammar and syntax are all unspecified in this model. The assumption permitting inference on necessary conditions constrained by the asymptotic limit theorems of information theory is that the finite limit
![$$\displaystyle{H \equiv \lim _{n\rightarrow \infty }\frac{\log [N(n)]} {n} }$$](computational-psychiatry.files/image268.png)
both exists and is independent of the path x. Again, N(n) is the number of high probability paths of length n.
Call such a pattern recognition-and-response cognitive process ergodic. Not all cognitive processes are likely to be ergodic, implying that H, if it indeed exists at all, is path dependent, although extension to nearly ergodic processes, in a certain sense, seems possible (e.g., Wallace 2005a, pp. 31–32).
Invoking the Shannon–McMillan Theorem (Cover and Thomas 2006; Khinchin 1957), it becomes possible to define an adiabatically, piecewise stationary, ergodic information source X associated with stochastic variates X jhaving joint and conditional probabilities P(a0, …, a n ) and P(an | a 0, …, an−1) such that appropriate joint and conditional Shannon uncertainties satisfy the classic relations
![$$\displaystyle\begin{array}{rcl} H[\mathbf{X}]& =& \lim _{n\rightarrow \infty }\frac{\log [N(n)]} {n} \\ & =& \lim _{n\rightarrow \infty }H(X_{n}\vert X_{0},\ldots,X_{n-1}) \\ & =& \lim _{n\rightarrow \infty }\frac{H(X_{0},\ldots,X_{n})} {n} {}\end{array}$$](computational-psychiatry.files/image269.png)
(6.2)
This information source is defined as dual to the underlying ergodic cognitive process, in the sense of Wallace (2005a, 2007).
“Adiabatic” means that, when the information source is properly parameterized, within continuous “pieces,” changes in parameter values take place slowly enough so that the information source remains as close to stationary and ergodic as needed to make the fundamental limit theorems work. “Stationary” means that probabilities do not change in time, and “ergodic” that cross-sectional means converge to long-time averages. Between pieces it is necessary to invoke phase change formalism, a “biological” renormalization that generalizes Wilson’s (1971) approach to physical phase transition (Wallace 2005a).
Shannon uncertainties H(…) are cross-sectional law-of-large-numbers sums of the form − ∑k P k log[P k ], where the Pk constitute a probability distribution. See Cover and Thomas (2006), Ash (1990), or Khinchin (1957) for the standard details.
For cognitive systems, an equivalence class algebra can be constructed by choosing different origin points a0, and defining the equivalence of two states am , a n by the existence of high probability meaningful paths connecting them to the same origin point. Disjoint partition by equivalence class, analogous to orbit equivalence classes for a dynamical system, defines the vertices of a network of cognitive dual languages that interact to actually constitute the system of interest. Each vertex then represents a different information source dual to a cognitive process. This is not a representation of a network of interacting physical systems as such, in the sense of network systems biology (e.g., Arrell and Terzic 2010). It is an abstract set of languages dual to the set of cognitive processes of interest, that may become linked into higher order structures.
Topology has become an object of algebraic study, the so-called algebraic topology, via the fundamental underlying symmetries of geometric spaces. Rotations, mirror transformations, simple (“affine”) displacements, and the like uniquely characterize topological spaces, and the networks inherent to cognitive phenomena having dual information sources also have complex underlying symmetries: characterization via equivalence classes defines a groupoid, an extension of the idea of a symmetry group, as summarized by Brown (1987) and Weinstein (1996). Linkages across this set of languages occur via the groupoid generalization of Landau’s spontaneous symmetry breaking arguments that will be used below (Landau and Lifshitz 2007; Pettini 2007). See the Mathematical Appendix for a brief summary of basic material on groupoids.
Recognize, however, that we are not constrained in this approach to the Atlan–Cohen model of cognition that, through the comparison with an internal picture of the world, invokes representation. The essential inference is that a broad class of cognitive phenomena—with and without representation—can be associated with a dual information source. That is, cognition inevitably involves choice, choice reduces uncertainty, and this implies the existence of an information source.
Again, extension to nonergodic information sources can be done using the methods of Wallace (2005a, Sect. 3.1).
6.4 Environment as an Information Source
Multifactorial cognitive and behavioral systems interact with, affect, and are affected by embedding environments that “remember” such interaction by various mechanisms. It is possible to reexpress environmental dynamics in terms of a grammar and syntax that represent the output of an information source—another generalized language.
Some examples:
1. 1.
2. 2.
3. 3.
Suppose it possible to coarse-grain the generalized “ecosystem” at time t, in the sense of symbolic dynamics (e.g., Beck and Schlogl 1995) according to some appropriate partition of the phase space in which each division A jrepresent a particular range of numbers of each possible fundamental actor in the generalized ecosystem, along with associated larger system parameters. What is of particular interest is the set of longitudinal paths, system statements, in a sense, of the form x(n) = A0, A 1, …, An defined in terms of some natural time unit of the system. Thus n corresponds to an again appropriate characteristic time unit T, so that t = T, 2T, …, nT.
Again, the central interest is in serial correlations along paths.
Let N(n) be the number of possible paths of length n that are consistent with the underlying grammar and syntax of the appropriately coarse-grained embedding ecosystem, in a large sense. As above, the fundamental assumptions are that—for this chosen coarse-graining—N(n), the number of possible grammatical paths is much smaller than the total number of paths possible, and that, in the limit of (relatively) large n, H = lim n → ∞ log[N(n)]∕n both exists and is independent of path.
These conditions represent a parallel with parametric statistics systems for which the assumptions are not true will require specialized approaches.
Nonetheless, not all possible ecosystem coarse-grainings are likely to work, and different such divisions, even when appropriate, might well lead to different descriptive quasi-languages for the ecosystem of interest. Thus, empirical identification of relevant coarse-grainings for which this theory will work may represent a difficult scientific problem.
Given an appropriately chosen coarse-graining, define joint and conditional probabilities for different ecosystem paths, having the form P(A0, A 1, …, An ), P(A n | A0, …, A n−1), such that appropriate joint and conditional Shannon uncertainties can be defined on them that satisfy Eq. (6.2).
Taking the definitions of Shannon uncertainties as above, and arguing backwards from the latter two parts of equation (6.2), it is indeed possible to recover the first, and divide the set of all possible ecosystem temporal paths into two subsets, one very small, containing the grammatically correct, and hence highly probable paths, that we will call “meaningful,” and a much larger set of vanishingly low probability (Khinchin 1957).
6.5 Body Dynamics and Culture as Information Sources
Body movement is inherently constrained by evolutionary Bauplan: snakes do not brachiate, humans cannot (easily) scratch their ears with their hind legs, fish do not breathe air, nor mammals water. This is so evident that one simply does not think about it. Nonetheless, teaching a human to walk and talk, a bird to fly, or a lion to hunt, in spite of evolution, are arduous enterprises that take considerable attention from parents or even larger social groupings. Given the basic bodyplan of head and four limbs, or two feet and wings, or of a limbless spine, the essential point is that not all motions are possible. Bauplan imposes limits on dynamics. That is, if we coarse-grain motions, perhaps using some form of the standard methods for choreography transcription appropriate to the organism (or mechanism) under study, we see immediately that not all “statements” possible using the dance symbols have the same probability. That is, there will inevitably be a grammar and syntax to observed body-based behaviors imposed by evolutionary or explicit design bauplan. Sequences of symbols, say of length n, representing observed motions can be segregated into two sets, the first, and vastly larger, consisting of meaningless sequences (like humans scratching their ears with their feet) that have vanishingly small probability as n → ∞. The other set, consistent with underlying bauplan grammar and syntax, can be viewed as the output of an information source, in precisely the manner of the previous two sections, in first approximation following the relations of equation (6.2).
In precisely the same manner as evolutionary Bauplan constrains possible sequences of motions into high and low probability sets, so too learned culture (and its associated patterns of social interaction) contextually constrain possible behaviors, spoken language, body postures, and many other phenotypes. That is, different cultures impose different probability structures, in a large sense, on essential matters of living and of the life course trajectory. Even sleep is widely discordant across cultural boundaries. Birth, marriage, death, social conflict, economic exchange, and so on are all strongly patterned by culture, in the context of historical trajectory and social segmentation. Some discussion of these matters in the context of mental disorder can be found in Kleinman and Good (1985), Desjarlais et al. (1995), and the references therein. Boyd and Richerson (2005) provide a more comprehensive introduction.
More generally, as Durham (1991) argues, genes and culture are two distinct but interacting systems of heritage in human populations. Information of both kinds has potential or actual influence over behaviors, creating a real and unambiguous symmetry between genes and phenotypes on the one hand, and culture and phenotypes on the other. Genes and culture are best represented as two parallel tracks of hereditary influence on phenotypes, acting, of course, on markedly different timescales. Human species’ identity rests, in no small part, on its unique evolved capabilities for social mediation and cultural transmission, creating, again, high and low probability sets of real-time behavioral sequences.
6.6 Interacting Information Sources
Given a set of information sources that are linked to solve a problem, in the sense of Wilson and Golonka (2013), the “no free lunch” theorem (English 1996; Wolpert and MacReady 1995, 1997) extends a network theory-based theory (e.g., Arrell and Terzic 2010). Wolpert and Macready show there exists no generally superior computational function optimizer. That is, there is no “free lunch” in the sense that an optimizer pays for superior performance on some functions with inferior performance on others gains and losses balance precisely, and all optimizers have identical average performance. In sum, an optimizer has to pay for its superiority on one subset of functions with inferiority on the complementary subset.
This result is well known using another description. Shannon (1959) recognized a powerful duality between the properties of an information source with a distortion measure and those of a channel. This duality is enhanced if we consider channels in which there is a cost associated with the different letters. Solving this problem corresponds to finding a source that is right for the channel and the desired cost. Evaluating the Rate Distortion Function for a source corresponds to finding a channel that is just right for the source and allowed distortion level.
Yet another approach to the same result is through the “tuning theorem” of the Mathematical Appendix which inverts the Shannon Coding Theorem by noting that, formally, one can view the channel as “transmitted” by the signal. Then a dual channel capacity can be defined in terms of the channel probability distribution that maximizes information transmission assuming a fixed message probability distribution.
From the no free lunch argument , Shannon’s insight, or the “Tuning Theorem” , it becomes clear that different challenges facing any cognitive system, distributed collection of them, or interacting set of other information sources, that constitute an organism or automaton, must be met by different arrangements of cooperating modules represented as information sources.
It is possible to make a very abstract picture of this phenomenon based on the network of linkages between the information sources dual to the individual “unconscious” cognitive modules (UCM), and those of related information sources with which they interact. That is, a shifting, task-mapped, network of information sources is continually reexpressed: given two distinct problems classes confronting the organism or automaton, there must be two different wirings of the information sources, including those dual to the available UCM, with the network graph edges measured by the amount of information crosstalk between sets of nodes representing the different sources.
Thus fully embodied systems, in the sense of Wilson and Golonka (2013), involve interaction between very general sets of information sources assembled into a “task-specific device” in the sense of Bingham (1988) that is necessarily highly tunable. This mechanism represents a broad evolutionary generalization of the “shifting spotlight” characterizing the global neuronal workspace model of consciousness (Wallace 2005a). We will return to this point in more detail below.
The mutual information measure of crosstalk is not inherently fixed, but can continuously vary in magnitude. This suggests a parameterized renormalization: the modular network structure linked by crosstalk has a topology depending on the degree of interaction of interest.
Define an interaction parameter ω, a real positive number, and look at geometric structures defined in terms of linkages set to zero if mutual information is less than, and “renormalized” to unity if greater than, ω. Any given ωwill define a regime of giant components of network elements linked by mutual information greater than or equal to it.
Now invert the argument: a given topology for the giant component will, in turn, define some critical value, ωC , so that network elements interacting by mutual information less than that value will be unable to participate, i.e., will be locked out and not be consciously or otherwise perceived. See Chap. 1 for details. Thus ω is a tunable, syntactically dependent, detection limit that depends critically on the instantaneous topology of the giant component of linked information sources defining the analog to a global broadcast of consciousness. That topology is the basic tunable syntactic filter across the underlying modular structure, and variation in ω is only one aspect of more general topological properties that can be described in terms of index theorems, where far more general analytic constraints can become closely linked to the topological structure and dynamics of underlying networks, and, in fact, can stand in place of them (Atiyah and Singer 1963; Hazewinkel 2002).
6.7 Simple Regulation
Continuing the formal theory, information sources are often not independent, but are correlated, so that a joint information source—representing, for example, the interaction between brain, body, and the environment—can be defined having the properties
(6.3)
with equality only for isolated, independent information streams.
This is the information chain rule (Cover and Thomas 2006), and has implications for free energy consumption in regulation and control of embodied cognitive processes. Feynman (2000) describes how information and free energy have an inherent duality, defining information precisely as the free energy needed to erase a message. The argument is quite direct, and it is easy to design an idealized machine that turns the information within a message directly into usable work—free energy. Information is a form of free energy and the construction and transmission of information within living things—the physical instantiation of information—consumes considerable free energy, with inevitable—and massive—losses via the second law of thermodynamics.
Suppose an intensity of available free energy is associated with each defined joint and individual information source H(X, Y ), H(X), H(Y ), e.g., rates MX, Y , MX ,M Y .
Although information is a form of free energy, there is necessarily a massive entropic loss in its actual expression, so that the probability distribution of a source uncertainty H might be written in Gibbs form as
![$$\displaystyle{ P[H] \approx \frac{\exp [-H/\kappa M]} {\int \exp [-H/\kappa M]dH} }$$](computational-psychiatry.files/image271.png)
(6.4)
assuming κ is very small.
To first order, then,
![$$\displaystyle{ \hat{H}\equiv \int HP[H]dH \approx \kappa M }$$](computational-psychiatry.files/image272.png)
(6.5)
and, using Eq. (6.3),
(6.6)
Thus, as a consequence of the information chain rule, allowing crosstalk consumes a lower rate of free energy than isolating information sources. That is, in general, it takes more free energy—higher total cost—to isolate a set of cognitive phenomena and an embedding environment than it does to allow them to engage in crosstalk.
Hence, at the free energy expense of supporting two information sources—X and Y together—it is possible to catalyze a set of joint paths defined by their joint information source. In consequence, given a cognitive module (or set of them) having an associated information source H(…), an external information source Y —the embedding environment—can catalyze the joint paths associated with the joint information source H(…, Y ) so that a particular chosen developmental or behavioral pathway—in a large sense—has the lowest relative free energy.
At the expense of larger global free information expenditure—maintaining two (or more) information sources with their often considerable entropic losses instead of one—the system can feed, in a sense, the generalized physiology of a Maxwell’s Demon, doing work so that environmental signals can direct system cognitive response, thus locally reducing uncertainty at the expense of larger global entropy production.
Given a cognitive biological system characterized by an information source X, in the context of—for humans—an explicitly, slowly changing, cultural “environmental” information source Y, we will be particularly interested in the joint source uncertainty defined as H(X, Y ), and next examine some details of how such a mutually embedded system might operate in real time, focusing on the role of rapidly changing feedback information, via the Data Rate Theorem.
6.8 Extending the Data Rate Theorem
The homology between the information source uncertainty dual to a cognitive process and the free energy density of a physical system arises, in part, from the formal similarity between their definitions in the asymptotic limit. Information source uncertainty can be defined as in the first part of equation (6.2). This is quite analogous to the free energy density of a physical system in terms of the thermodynamic limit of infinite volume (e.g., Wilson 1971; Wallace 2005a). Feynman (2000) provides a series of physical examples, based on Bennett’s (1988) work, where this homology is an identity, at least for very simple systems. Bennett argues, in terms of idealized irreducibly elementary computing machines, that the information contained in a message can be viewed as the work saved by not needing to recompute what has been transmitted.
It is possible to model a cognitive system interacting with an embedding environment using an extension of the language-of-cognition approach above. Recall that cognitive processes can be formally associated with information sources, and how a formal equivalence class algebra can be constructed for a complicated cognitive system by choosing different origin points in a particular abstract “space” and defining the equivalence of two states by the existence of a high probability meaningful path connecting each of them to some defined origin point within that space.
Recall that disjoint partition by equivalence class is analogous to orbit equivalence relations for dynamical systems, and defines the vertices of a network of cognitive dual languages available to the system: each vertex represents a different information source dual to a cognitive process. The structure creates a large groupoid, with each orbit corresponding to a transitive groupoid whose disjoint union is the full groupoid, and each subgroupoid associated with its own dual information source. Larger groupoids will, in general, have “richer” dual information sources than smaller.
We can now begin to examine the relation between system cognition and the feedback of information from the rapidly changing real-time (as opposed to a slow-time cultural or other) environment, , in the sense of equation (6.1).
With each subgroupoid Gi of the (large) cognitive groupoid we can associate a joint information source uncertainty 
, where X is the dual information source of the cognitive phenomenon of interest, and Y that of the embedding environmental context—largely defined, for humans, in terms of culture and path-dependent historical trajectory.
Real-time dynamic responses of a cognitive system can now be represented by high probability paths connecting “initial” multivariate states to “final” configurations, across a great variety of beginning and end points. This creates a similar variety of groupoid classifications and associated dual cognitive processes in which the equivalence of two states is defined by linkages to the same beginning and end states. Thus, it becomes possible to construct a “groupoid free energy” driven by the quality of rapidly changing, real-time information coming from the embedding ecosystem, represented by the information rate , taken as a temperature analog.
For humans in particular, is an embedding context for the underlying cognitive processes of interest, here the tunable, shifting, global broadcasts of consciousness as embedded in, and regulated by, culture. The argument-by-abduction from physical theory is, then, that constitutes a kind of thermal bath for the processes of culturally channeled cognition. Thus we can, in analogy with the standard approach from physics (Pettini 2007; Landau and Lifshitz 2007), construct a Morse Function by writing a pseudoprobability for the jointly defined information sources 
having source uncertainty 
as
![$$\displaystyle{ P[H_{G_{i}}] = \frac{\exp [-H_{G_{i}}/\kappa \mathcal{H})]} {\sum _{j}\exp [-H_{G_{j}}/\kappa \mathcal{H}]} }$$](computational-psychiatry.files/image277.png)
(6.7)
where κ is an appropriate dimensionless constant characteristic of the particular system. The sum is over all possible subgroupiods of the largest available symmetry groupoid. Again, compound sources, formed by the (tunable, shifting) union of underlying transitive groupoids, being more complex, will have higher free-energy-density equivalents than those of the base transitive groupoids.
A possible Morse Function for invocation of Pettini’s topological hypothesis or Landau’s spontaneous symmetry breaking is then a “groupoid free energy” F defined by
![$$\displaystyle{ \exp [-F/\kappa \mathcal{H}] \equiv \sum _{j}\exp [-H_{G_{j}}/\kappa \mathcal{H}] }$$](computational-psychiatry.files/image278.png)
(6.8)
It is possible, using the free energy-analog F, to apply Landau’s spontaneous symmetry breaking arguments, and Pettini’s topological hypothesis, to the groupoid associated with the set of dual information sources.
Many other Morse Functions might be constructed here, for example, based on representations of the cognitive groupoid(s). The resulting qualitative picture would not be significantly different. We will return to this argument below.
Again, Landau’s and Pettini’s insights regarding phase transitions in physical systems were that certain critical phenomena take place in the context of a significant alteration in symmetry, with one phase being far more symmetric than the other (Landau and Lifshitz 2007; Pettini 2007). A symmetry is lost in the transition—spontaneous symmetry breaking. The greatest possible set of symmetries in a physical system is that of the Hamiltonian describing its energy states. Usually states accessible at lower temperatures will lack the symmetries available at higher temperatures, so that the lower temperature phase is less symmetric: The randomization of higher temperatures ensures that higher symmetry/energy states will then be accessible to the system. The shift between symmetries is highly punctuated in the temperature index.
The essential point is that decline in the richness of the cultural control signal , or in the ability of that signal to influence response, as indexed by κ, can lead to punctuated decline in the complexity of cognitive process, according to this model.
This permits a Landau-analog phase transition analysis in which the quality of incoming information from the embedding regulatory system serves to raise or lower the possible richness of cognitive response to patterns of challenge. If 
is relatively large—a rich and varied regulatory cultural environment—then there are many possible cognitive responses. If, however, noise or simple constraint limit the magnitude of 
, then behavior collapses in a highly punctuated manner to a kind of ground state in which only limited responses are possible, represented by a simplified cognitive groupoid structure.
Certain details of such information phase transitions can be calculated using “biological” renormalization methods (Wallace 2005a, Sect. 4.2) analogous to, but much different from, those used in the determination of physical phase transition universality classes (Wilson 1971).
These results represent a significant generalization of the Data Rate Theorem, as expressed in Eq. (6.1).
6.9 Another Picture
Here we use the rich vocabulary associated with the stability of stochastic differential equations to model, from another perspective, phase transitions in the composite system of “brain/body/environment” (e.g., Horsthemeke and Lefever 2006; Van den Broeck et al. 1994, 1997).
Define a “symmetry entropy” based on the Morse Function F of equation (6.8) over a set of structural parameters Q = [Q 1, …, Qn ] (that may include and other information source uncertainties) as the Legendre transform
(6.9)
The dynamics of such a system will be driven, at least in first approximation, by Onsager-like nonequilibrium thermodynamics relations having the standard form (de Groot and Mazur 1984):
(6.10)
where the 
are appropriate empirical parameters and t is the time. A biological system involving the transmission of information may not have local time reversibility: in English, for example, the string “eht” has a much lower probability than “the.” Without microreversibility, 
.
Since, however, biological systems are quintessentially noisy, a more fitting approach is through a set of stochastic differential equations having the form
(6.11)
where the 
and σi, jare appropriate functions, and different kinds of “noise” dBj will have particular kinds of quadratic variation affecting dynamics (Protter 1990).
Several important dynamics become evident:
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
In particular, setting the expectation of equation (6.11) to zero generates an index theorem (Hazewinkel 2002) in the sense of Atiyah and Singer (1963), that is, an expression that relates analytic results, the solutions of the equations, to underlying topological structure, the eigenmodes of a complicated geometric operator whose groupoid spectrum represents symmetries of the possible changes that must take place for a global workspace to become activated.
6.10 Large Deviations and Epileptiform Disorders
As Champagnat et al. (2006) describe, shifts between the quasi-steady states of a coevolutionary system like that of equation (6.11) can be addressed by the large deviations formalism . The dynamics of drift away from trajectories predicted by the canonical equations can be investigated by considering the asymptotic of the probability of “rare events” for the sample paths of the diffusion.
“Rare events” are the diffusion paths drifting far away from the direct solutions of the canonical equation. The probability of such rare events is governed by a large deviation principle, driven by a “rate function” 
that can be expressed in terms of the parameters of the diffusion.
This result can be used to study long-time behavior of the diffusion process when there are multiple attractive singularities. Under proper conditions, the most likely path followed by the diffusion when exiting a basin of attraction is the one minimizing the rate function over all the appropriate trajectories.
An essential fact of large deviations theory is that the rate function almost always has the canonical form
(6.12)
for some probability distribution (Dembo and Zeitouni 1998).
The argument relates to Eq. (6.11), now seen as subject to large deviations that can themselves be described as the output of an information source (or sources), say LD , defining , driving Qj -parameters that can trigger punctuated shifts between quasi-steady state topological modes of interacting cognitive submodules.
It should be clear that both internal and feedback signals, and independent, externally imposed perturbations associated with the source uncertainty , can cause such transitions in a highly punctuated manner. Some impacts may, in such a coevolutionary system, be highly pathological over a developmental trajectory, necessitating higher order regulatory system counterinterventions over a subsequent trajectory.
Similar ideas now pervade systems biology (Kitano 2004). See Chap. 1, Fig. 1.4
An information source defining a large deviations rate function in Eq. (6.12) can also represent input from “unexpected or unexplained internal dynamics” (UUID) unrelated to external perturbation. Such UUID will always be possible in sufficiently large cognitive systems, since crosstalk between cognitive submodules is inevitable, and any critical value can be exceeded if the structure is large enough or is driven hard enough. This suggests that, as Nunney (1999) describes for cancer, large-scale cognitive systems must be embedded in powerful regulatory structures over the life course. Wallace (2005b), in fact, examines a “cancer model” of regulatory failure for mental dysfunction.
Wallace (2000) uses a large deviations argument to examine epileptiform disorders. Martinerie et al. (1998) and Elger and Lenertz (1998) find a simplified “grammar” and “syntax” characterize brain dynamic pathways to epileptic seizure. As Martinerie et al. put it,
The view of chronic focal epilepsy now is that abnormally discharging neurons act as pacemakers to recruit and entrain other normal neurons by loss of inhibition and synchronization into a critical mass. Thus preictal changes should be detectable during the stages of recruitment…Nonlinear indicators may undergo consistent changes around seizure onset…We demonstrated that in most cases…seizure onset could be anticipated well in advance [using nonlinear analytic methods] and that all subjects seemed to share a similar “route” towards seizure.
Wallace (2000) looks at such a phase transition in the dual source uncertainty of a spatial array of resonators itself as a large fluctuational event, having a pattern of optimal/meaningful paths defined by a (pathological) information source L D . Failure of regulation then permits entrainment of normal neurons by abnormally discharging pacemakers, producing a seizure.
6.11 Ground State Collapse 1: Anxiety/Depression Analogs
There is, however, more to the influence of large deviations. The arguments leading to Eqs. (6.7) and (6.8) could be reexpressed using a joint information source
(6.13)
providing a more complete picture of large-scale cognitive dynamics in the presence of embedding regulatory systems, or of sporadic external “therapeutic” interventions. However, the joint information source of equation (6.13) now represents a de-facto distributed cognition involving interpenetration between both the underlying embodied cognitive process and its regulatory machinery.
That is, we can now define a composite Morse Function of embodied cognition-and-regulation, 
, as
![$$\displaystyle{ \exp [-\mathcal{F}/\omega (\mathcal{H},\mu )] \equiv \sum _{i}\exp [-H(X_{G_{i}},Y,L_{D})/\omega (\mathcal{H},\mu )] }$$](computational-psychiatry.files/image289.png)
(6.14)
where 
is a monotonic increasing function of both the control data rate and of the “richness” of the internal cognitive function defined by an internal—strictly cognitive—network coupling parameter μ, a more limited version of the argument in Sect. 6.6. Typical examples might include 
, ![$$\omega _{0}[\mathcal{H}\mu ]^{\gamma }$$](computational-psychiatry.files/image292.png){kind=small}
, γ > 0, ![$$\omega _{1}\log [\omega _{2}\mathcal{H}\mu + 1]$$](computational-psychiatry.files/image293.png){kind=small}
, and so on.
More generally, in Eq. (6.14) can be replaced by the norm
for appropriately chosen representations Γ of the underlying cognitive-defined groupoid, in the sense of Bos (2007) and Buneci (2003). That is, many Morse Functions similarly parameterized by the monotonic functions 
are possible, with the underlying topology, in the sense of Pettini, itself subtly parameterized by the information sources Y and LD .
Applying Pettini’s topological hypothesis to the chosen Morse Function, reduction of either or μ, or both, can trigger a “ground state collapse” representing a phase transition to a less (groupoid) symmetric “frozen” state. In higher organisms, which must generally function under real-time constraints, elaborate secondary back-up systems have evolved to take over behavioral control under such conditions. These typically range across basic emotional, as well as hypothalamic–pituitary–adrenal (HPA) and hypothalamic–pituitary–thyroid (HPT) axis, responses (e.g., Wallace 2005a, 2012, 2013; Wallace and Fullilove 2008). Failures of these systems are implicated across a vast range of common, and usually comorbid, mental and physical illnesses (e.g., Wallace 2005a,b; Wallace and Wallace 2010, 2013).
6.12 Ground State Collapse 2: Obsessive Compulsive Disorders
Following Overduin and Furnham (2012), obsessive compulsive disorder (OCD) is a widespread condition with prevalence rates from about 1% (current) and 2–2.5% (lifetime). Subclinical manifestations are frequently present in individuals without OCD, ranging perhaps as high as 25% of the general population. They state that
Individuals with OCD, or with a high risk of developing OCD, suffer from recurrent, unwanted, and intrusive thoughts (obsessions) and engage in repetitive ritualistic behaviors (compulsions), usually aimed to prevent, reduce, or eliminate distress or feared consequences of the obsessions. Relief by rituals is generally temporary and contributes to future ritual engagement…[U]ntreated symptoms often persist or increase over time, causing significant impairment in social, professional, academic, and/or family functioning…
OCD and its subclinical manifestations thus appear widespread in studied culturally Western populations, indeed, perhaps a canonical failure mode (but see Heine 2001 for another possible interpretation).
Adapting the Onsager method of Sect. 6.9 to the Morse Function 
of equation (6.14) leads to a generalized form of equation (6.11), now involving gradients in the extended entropy-analog
(6.15)
Again, setting the expectation the resulting set of stochastic differential equations to zero and solving for stationary sets may provide individual nonequilibrium steady states, “Red Queen” limit cycles—where the system seems to chase its tail in repititive cycles—or even characteristic “strange attractor” sets over which the system engages in pseudorandom excursions.
Most typically, then, the Red Queen behaviors, analogous to computer thrashing, seem to provide a compelling model of OCD in sophisticated cognitive structures that are generally both embodied and distributed.
6.13 Topological Dysfunctions: Autism Spectrum and Schizophreniform Analogs
Recall that the global workspace model of consciousness (Baars 1988; Baars et al. 2013; Wallace 2005a, 2007, 2012) posits a “theater spotlight” involving the recruitment of unconscious cognitive modules of the brain into a temporary, tunable, general broadcast fueled by crosstalk that allows formation of the shifting coalitions needed to address real-time problems facing a higher organism. Similar exaptations of crosstalk between cognitive modules at smaller scales have been recognized in wound healing, the immune system, and so on (Wallace 2012, 2014a). Theories of embodied cognition envision that phenomenon as analogous, that is, as the temporary assembly of interacting modules from brain, body, and environment to address real-time problems facing an organism. This is likewise a dynamic process that sees many available information sources—not limited to those dual to cognitive brain or internal physiological modules—again linked by crosstalk into a tunable real-time phenomenon that is, in effect, a generalized consciousness.
The perspective has particular implications for disorders of brain connectivity like schizophrenia and autism . Figure 6.1 shows a schematic of a “generalized consciousness” involving dynamic patterns of crosstalk between information sources—the Xj —representing brain, body, culture, socioeconomy, and environment, in no particular order, and treated as fundamentally equivalent. The full and dotted lines represent recruitment of these dispersed resources (involving crosstalk at or above some tunable value ω) in two different topological patterns to address two different kinds of problems in real time.
Fig. 6.1
Full and dotted lines represent two different recruitments of brain, body, cultural, socioeconomic, and other environmental information sources to address real-time problems facing an individual (or composite distributed cognition system). Both underlying topology and the crosstalk index 
are dynamically tunable, representing a generalized consciousness. Pathological restrictions on connectivity or topology would be manifest as analogs to autism or schizophrenia, in this model, in addition to the “anxiety/depression” mode of ground state collapse
“Mental disorders,” in a large sense, emerge as a synergistic dysfunction of internal process and regulatory milieu, which above was simply characterized by the interaction between the driving parameters μand . Other forms of dysfunction likely involve characteristic irregularities in topological connections. For example, autism spectrum and schizophreniform disorders are widely viewed as caused by failures in linkage that limit recruitment of unconscious cognitive brain modules (e.g., Wallace 2005b). Thus analogous disorders might arise from similar “topological failures” affecting the real-time recruitment of brain, body, culture, regulatory, and environmental information sources. The central role of culture in human biology means, of course, that, for humans, all such disorders are inherently “culture bound syndromes,” much in the spirit of Kleinman and Cohen (1997) and Heine (2001).
6.14 Discussion and Conclusions
Here, we have made formal use of the Data Rate Theorem in exploring the dynamics of such an embodied cognition, and of a necessarily related embodied regulation. These, according to theory, inevitably involve a synergistic interpenetration among nested sets of actors, represented here as information sources. They may include dual sources to internal cognitive modules, body bauplan, environmental information, language, culture, and so on. Following the arguments of Wallace (2014b), similar considerations apply to machine and man/machine cockpit systems, a matter discussed more fully in Wallace (2017) and addressed in Chaps. 8 and 9.
Two factors determine the possible range of real-time cognitive response, in the simplest version of the model. These are the magnitude of the environmental feedback signal and the inherent structural richness of the underlying cognitive groupoid. If that richness is lacking—if the possibility of internal μ-connections is limited—then even very high levels of may not be adequate to activate appropriate behavioral responses to important real-time feedback signals, following the argument of equation (6.14).
Cognition and regulation must, then, be viewed as interacting gestalt processes, involving not just an atomized individual (or, taking an even more limited “NIMH” perspective, just the brain of that individual), but the individual in a rich context that must include both the body that acts on the environment, and the environment that reacts on body and brain. Huys et al. (2016) make much the same point.
The large deviations analysis suggests that cognitive function also occurs in the context, not only of a powerful environmental embedding, but also of a specific regulatory milieu: there can be no cognition without regulation. The “stream of generalized consciousness” represented by embodied cognition must be contained within regulatory riverbanks.
For humans and some other animal species (e.g., Avital and Jablonka 2000), this view must be expanded by another layer of information sources: as the evolutionary anthropologist Robert Boyd has expressed it, “Culture is as much a part of human biology as the enamel on our teeth.” Thus, for humans, the schematic hierarchy of interacting information sources becomes
Current theory on embodied cognition omits the critical level of cultural modulation, which is, by structure, unavailable to machine and man/machine systems.
We thus significantly extend the criticisms of Bennett and Hacker (2003) who examined the mereological fallacy of a decontextualization that attributes to “the brain” what is the province of the whole individual. Here, the “whole individual” includes essential interactions with embedding environmental and regulatory settings that, for humans, must include cultural heritage and socioeconomic dynamics.
We have explored in particular epileptiform seizures and “ground state collapses” analogous to anxiety/depression and OCD, but more diverse and subtle “topological” failures seem likely. As Johnson-Laird et al. (2006) indicate, surprisingly little is known about such dysfunction in humans. For some time, the study of mental disorders has been strongly dominated by a simplistic paradigm driven largely by the interests of the pharmaceutical industry, which has since abandoned the effort as a dry hole. The story is well known, and parallels the arguments in Chap. 1 of Wallace and Wallace (2013).
A shift in perspective is needed to a comprehensive cognitive theory of mental disorders that recognizes the central roles of culture, socioeconomic structure, and their dynamics in both healthy human development and its pathologies. At present, little such work is actively supported in the USA, for deep cultural, ideological, and political reasons.
In addition, this work can be seen as extending the considerations of Wallace (2014b, 2017) from a simple “ground state collapse” of machines and man/machine cockpits in which “all possible targets are enemies” to spectra of more subtle failures that will afflict real-time systems increasingly being given control of critical human structures and processes. Indeed, considering the collapse of “pharmaceutical industry” paradigms, it seems possible that study of machine and man/machine failures could provide the intellectual and financial horsepower necessary to lift psychiatry from its longstanding and ideologically enforced doldrums.
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