ARTICLE IN PRESS Journal of Theoretical Biology 245 (2007) 378–390 www.elsevier.com/locate/yjtbi Culture and inattentional blindness: A global workspace perspective Rodrick Wallace The New York State Psychiatric Institute, Box 47, 1051 Riverside Dr., New York, NY 10032, USA Received 29 August 2006; accepted 5 October 2006 Available online 12 October 2006 Abstract A recent ‘necessary conditions’ mathematical treatment of Baars’ global workspace consciousness model, analogous to Dretske’s communication theory analysis of high level mental function, is used to explore the effects of embedding cultural heritage on inattentional blindness. Culture should express itself quite distinctly in this basic psychophysical phenomenon across a great variety of sensory modalities because the limited syntactic and grammatical bandpass of the rate distortion manifold characterizing conscious attention must conform to topological constraints generated by cultural context. r 2006 Elsevier Ltd. All rights reserved. Keywords: Bandpass; Cognition; Consciousness; Culture; Directed homotopy; Global workspace; Groupoid; Inattentional blindness; Information theory; Random network; Rate distortion manifold; Topology 1. Introduction Inattentional blindness (IAB) occurs when focus of attention on a single aspect of a complicated perceptual field precludes detection of others, which may be quite strong and normally expected to register on consciousness. Mack (1998) and Simons and Chabris (1999) provide background. The phenomenon was apparently well known in the early part of the 20th century, but its study languished thereafter, seemingly for many of the reasons that consciousness studies fell into disfavor for nearly a century. Simons and Chabris (1999) describe a particularly spectacular example. A videotape was made of a basketball game between teams in white and black jerseys. Experimental subjects who viewed the tape were asked to keep silent mental counts of either the total number of passes made by one or the other of the teams, or separate counts of the number of bounce and areal passes. During the game, a figure in a full gorilla suit appears, faces the camera, beats its breast, and walks off the court. About one half of the experimental subjects completely failed to notice the Gorilla during the experiment. See Simons Tel.: +1 212 928 0631; fax: +1 212 928 2219. E-mail address: wallace@pi.cpmc.columbia.edu. 0022-5193/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2006.10.006 (2000) for an extended discussion, and Wayand et al. (2005) for more recent results. Other case histories, involving an aircraft crew which became fixated on an unexpectedly flashing control panel light during a landing, or a man walking a railroad track while having a cell phone conversation, are less benign. Dehaene and Changeux (2005) recently reported a neural network simulation of Baars’ global workspace model of consciousness in which ignition of a coherent, spontaneous, excited state blocked external sensory processing, an observation they relate to IAB. Here, by contrast, we use a Dretske-style necessary conditions analytic treatment of Baars’ model based on communication theory to address the phenomenon, a perspective which does not suffer the sufficiency indeterminacy inherent to neural network simulations of high level mental phenomena (Krebs, 2005). A particular utility of the approach is that, treating culture as a kind of embedding language for conscious attention, itself expressed in terms of a language-analog, it becomes possible to model cultural influence by using a rate distortion formalism. The necessity for the inclusion of culture lies in the observations of Nisbett et al. (2001), and others, following the tradition of Markus and Kitayama (1991), regarding fundamental differences in perception between test subjects ARTICLE IN PRESS R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 of Southeast Asian and Western cultural heritage across an broad realm of experiments. East Asian perspectives are characterized as holistic and Western as analytic. Nisbett et al. (2001) find: (1) Social organization directs attention to some aspects of the perceptual field at the expense of others. (2) What is attended to influences metaphysics. (3) Metaphysics guides tacit epistemology, that is, beliefs about the nature of the world and causality. (4) Epistemology dictates the development and application of some cognitive processes at the expense of others. (5) Social organization can directly affect the plausibility of metaphysical assumptions, such as whether causality should be regarded as residing in the field vs. in the object. (6) Social organization and social practice can directly influence the development and use of cognitive processes such as dialectical vs. logical ones. Nisbett et al. (2001) conclude that tools of thought embody a culture’s intellectual history, that tools have theories build into them, and that users accept these theories, albeit unknowingly, when they use these tools. Heine (2001) states the underlying case as follows: Cultural psychology does not view culture as a superficial wrapping of the self, of as a framework within which selves interact, but as something that is intrinsic to the self. It assumes that without culture there is no self, only a biological entity deprived of its potential... Cultural psychology maintains that the process of becoming a self is contingent on individuals interacting with and seizing meanings from the cultural environment... More recently Masuda and Nisbett (2006) examined cultural variations in change blindness, a phenomenon related to inattentional blindness, and found striking differences between Western and East Asian subjects: We presented participants with still photos and with animated vignettes having changes in focal object information and contextual information. Compared to Americans, East Asians were more sensitive to contextual changes than to focal object changes. These results suggest that there can be cultural variation in what may seem to be basic perceptual processes. The central focus of this work is how culture can affect basic perceptual process, in essence creating a topological structure for individual consciousness. The central strategy is to invoke a detailed mathematical model of consciousness in humans, which, taking the perspectives of cultural psychology, must necessarily include the influences of embedding culture. 379 2. The formal theory 2.1. The global workspace consciousness model Bernard Baars’ global workspace theory (Baars, 1988, 2005) is rapidly becoming the de facto standard model of consciousness (e.g. Dehaene and Naccache, 2001; Dehaene and Changeux, 2005). The central ideas are as follows (Baars and Franklin, 2003): (1) The brain can be viewed as a collection of distributed specialized networks (processors). (2) Consciousness is associated with a global workspace in the brain—a fleeting memory capacity whose focal contents are widely distributed (broadcast) to many unconscious specialized networks. (3) Conversely, a global workspace can also serve to integrate many competing and cooperating input networks. (4) Some unconscious networks, called contexts, shape conscious contents, for example unconscious parietal maps modulate visual feature cells that underlie the perception of color in the ventral stream. (5) Such contexts work together jointly to constrain conscious events. (6) Motives and emotions can be viewed as goal contexts. (7) Executive functions work as hierarchies of goal contexts. Although this basic approach has been the focus of many researchers for nearly two decades, academic consciousness studies have only recently, under the relentless pressure of a deluge of empirical results from brain imaging experiments, begun digesting the perspective and preparing to move on. To reiterate, currently popular agent-based and artificial neural network (ANN) treatments of cognition, consciousness and other higher-order mental functions, taking Krebs’ (2005) view, are little more than sufficiency arguments, in the same sense that a Fourier series expansion can be empirically fitted to nearly any function over a fixed interval without providing real understanding of the underlying structure. Necessary conditions, as Dretske argues (Dretske, 1981, 1988, 1993, 1994), give considerably more insight. Perhaps the most cogent example is the difference between the Ptolemaic and Copernican models of the solar system: one need not always expand in epicycles, but can seek the central motion. Dretske’s perspective provides such centrality. Keplerian and Newtonian treatments, unfortunately, still lie ahead of us: Atmanspacher (2006) has likened the current state of consciousness theory to that of physics 400 years ago. Wallace (2005a, b) has addressed Baars’ theme from Dretske’s viewpoint, examining the necessary conditions which the asymptotic limit theorems of information theory impose on the global workspace. A central outcome of this ARTICLE IN PRESS 380 R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 work has been the incorporation, in a natural manner, of constraints on individual consciousness, i.e. what Baars calls contexts. Using information theory methods, extended by the obvious homology between information source uncertainty and the free energy density of a physical system, it is possible to formally account for the effects on individual consciousness of parallel physiological modules like the immune system, embedding structures like the local social network, and, most importantly, the all-encompassing cultural heritage which so uniquely marks human biology (e.g. Richerson and Boyd, 2004). This embedding evades the mereological fallacy which fatally bedevils brain-only theories of human consciousness (Bennett and Hacker, 2003). Transfer of phase change approaches from statistical physics to information theory via the same homology generates the punctuated nature of accession to consciousness in a similarly natural manner. The necessary renormalization calculation focuses on a phase transition driven by variation in the average strength of nondisjunctive weak ties (Granovetter, 1973) linking unconscious cognitive submodules. A second-order universality class tuning allows for adaptation of conscious attention via rate distortion manifolds which generalize the idea of a retina. A version of the Baars model (including contexts) emerges as an almost exact parallel to hierarchical regression, based, however, on the Shannon–McMillan rather than the Central Limit Theorem. Wallace (2005b) recently proposed a somewhat different approach, using classic results from random and semirandom network theory (Erdos and Renyi, 1960; Albert and Barabasi, 2002; Newman, 2003) applied to a modular network of cognitive processors. The unconscious modular network structure of the brain is, of course, not random. However, in the spirit of the wag who said ‘‘all mathematical models are wrong, but some are useful’’, the method serves as the foundation of a different, but roughly parallel, treatment of the global workspace to that given in Wallace (2005a), and hence as another basis for a benchmark model against which empirical data can be compared. The first step is to argue for the existence of a network of loosely linked unconscious cognitive modules, and to characterize each of them by the richness of the canonical language—information source—associated with it. This is in some contrast to attempts to explicitly model neural structures themselves using network theory, e.g. the neuropercolation approach of Kozma et al. (2004, 2005), which nonetheless uses many similar mathematical techniques. Here, rather, the central focus is on the necessary conditions imposed by the asymptotic limits of information theory upon any realization of a cognitive process, be it biological wetware, silicon dryware, or some direct or systems-level hybrid. All cognitive processes, in this formulation, are to be associated with a canonical dual information source which will be constrained by the Rate Distortion Theorem, or, in the zero-error limit, the Shannon–McMillan Theorem. It is interactions between nodes in this abstractly defined network which will be of interest here, rather than whatever mechanism or biological system, or mixture of them, actually constitute the underlying cognitive modules. The second step is to examine the conditions under which a giant component (GC) suddenly emerges as a kind of phase transition in a network of such linked cognitive modules, to determine how large that component is, and to define the relation between the size of the component and the richness of the cognitive language associated with it. This level of approximation subsumes both Baars’ ‘fleeting memory capacity’ which acts as an analog to Newell’s blackboard computing model, and the specialized modules which have been recruited by broadcast, into a single object, and is one way to produce the large-scale brain connectivity which is the sine qua non of consciousness, in conformance with a large and growing body of brain imaging studies (e.g. Wallace, 2005b). Implicit, however, is the possibility of there being a number of different mechanisms which achieve such largescale structure. Wallace (2005a), for example, explores phase transitions centering around an inverse temperature analog involving the average strength of weak ties between modules. Intermediate models are possible. The giant component approach, however, seems particularly simple. Empirical comparisons of consciousness, which appears to be a very old evolutionary adaptation, between different animal orders, for example fish, reptiles, birds, and mammals, would likely be particularly illuminating, as different fundamental linking mechanisms may have evolved in each. The third step, following Wallace (2005b), is to use the renormalization parameters to tune the threshold at which the GC comes into being, along with its topological structure, via a second-order iteration which, when coupled with a tunable rate distortion manifold retina-analog, generalizes Newell’s blackboard model to give a highly flexible version of Baars’ ‘fleeting memory capacity’. Wallace (2005a), by contrast, uses ‘universality class tuning’ to direct the phase transitions associated with changing the average strength of weak ties between modules. These are clearly two analytically tractable asymptotic limits in a much larger domain of possible modeling approaches. Although the second level approximations are sufficient to produce large-scale brain connectivity, a basic kind of consciousness which may be characteristic of many animal families, the third level seems required to produce higher mental function. Some second level models may be more amenable to third-order development than others, again a likely matter of empirical study across animal orders. The information theoretic modular network treatment can be enriched by introducing a groupoid formalism which is roughly similar to recent analyses of linked dynamic networks described by differential equations ARTICLE IN PRESS R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 (e.g. Stewart et al., 2003; Stewart, 2004; Weinstein, 1996; Connes, 1994). Internal and external linkages between information sources break the underlying groupoid symmetry, and introduce more structure, the global workspace and the effect of contexts, respectively. The analysis provides a foundation for further mathematical exploration of linked cognitive processes. 2.2. Cognition as language Cognition is not consciousness. Most mental, and many physiological, functions, while cognitive in a formal sense, hardly ever become entrained into the global workspace of consciousness: one seldom is able to consciously regulate immune function, blood pressure, or the details of binocular tracking and bipedal motion, except to decide ‘what shall I look at’, ‘where shall I walk’. Nonetheless, many cognitive processes, conscious or unconscious, appear intimately related to language, broadly speaking. The construction is fairly straightforward (Wallace 2000, 2005a, b). Atlan and Cohen (1998) and Cohen (2000) argue, in the context of immune cognition, that the essence of cognitive function involves comparison of a perceived signal with an internal, learned picture of the world, and then, upon that comparison, choice of one response from a much larger repertoire of possible responses. Cognitive pattern recognition-and-response proceeds by an algorithmic combination of an incoming external sensory signal with an internal ongoing activity—incorporating the learned picture of the world—and triggering an appropriate action based on a decision that the pattern of sensory activity requires a response. More formally, a pattern of sensory input is mixed in an unspecified but systematic algorithmic manner with a pattern of internal ongoing activity to create a path of combined signals x ¼ ða0 ; a1 ; . . . ; an ; . . .Þ. Each ak thus represents some functional composition of internal and external signals. Wallace (2005a) provides two neural network examples. This path is fed into a highly nonlinear, but otherwise similarly unspecified, decision oscillator, h, which generates an output hðxÞ that is an element of one of two disjoint sets B0 and B1 of possible system responses. Let B0  b0 ; . . . ; bk , 381 fixed initial state a0 , examine all possible subsequent paths x beginning with a0 and leading to the event hðxÞ 2 B1 . Thus hða0 ; . . . ; aj Þ 2 B0 for all 0ojom, but hða0 ; . . . ; am Þ 2 B1 . For each positive integer n, let NðnÞ be the number of high probability grammatical and syntactical paths of length n which begin with some particular a0 and lead to the condition hðxÞ 2 B1 . Call such paths ‘meaningful’, assuming, not unreasonably, that NðnÞ will be considerably less than the number of all possible paths of length n leading from a0 to the condition hðxÞ 2 B1 . While combining algorithm, the form of the nonlinear oscillator, and the details of grammar and syntax, are all unspecified in this model, the critical assumption which permits inference on necessary conditions constrained by the asymptotic limit theorems of information theory is that the finite limit H  lim n!1 log½NðnÞ n (1) both exists and is independent of the path x. Define such a pattern recognition-and-response cognitive process as 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 (Wallace, 2005a). Invoking the spirit of the Shannon–McMillan Theorem, it is then possible to define an adiabatically, piecewise stationary, ergodic (APSE) information source X associated with stochastic variates X j having joint and conditional probabilities Pða0 ; . . . ; an Þ and Pðan ja0 ; . . . ; an1 Þ such that appropriate joint and conditional Shannon uncertainties satisfy the classic relations log½NðnÞ n ¼ lim HðX n jX 0 ; . . . ; X n1 Þ H½X ¼ lim n!1 n!1 ¼ lim n!1 HðX 0 ; . . . ; X n Þ . n This information source is defined as dual to the underlying ergodic cognitive process (Wallace, 2005a). Recall that the Shannon uncertainties Hð. . .Þ are crosssectional law-of-large-numbers sums of the form P  k Pk log½Pk , where the Pk constitute a probability distribution. See Khinchin (1957), Ash (1990), or Cover and Thomas (1991) for the standard details. B1  bkþ1 ; . . . ; bm . Assume a graded response, supposing that if hðxÞ 2 B0 , the pattern is not recognized, and if hðxÞ 2 B1 , the pattern is recognized, and some action bj ; k þ 1pjpm takes place. The principal objects of formal interest are paths x which trigger pattern recognition-and-response. That is, given a 2.3. The cognitive modular network symmetry groupoid A formal equivalence class algebra can be constructed by choosing different origin points a0 and defining equivalence by the existence of a high probability meaningful path connecting two points. Disjoint partition by equivalence class, analogous to orbit equivalence classes for dynamical systems, defines the vertices of the proposed network of cognitive dual languages. Each vertex then represents a different information source dual to a cognitive process. ARTICLE IN PRESS 382 R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 This is not a representation of a neural network as such, or of some circuit in silicon. It is, rather, an abstract set of ‘languages’ dual to the cognitive processes instantiated by either biological wetware, mechanical dryware, or their direct or systems-level hybrids. This structure is a groupoid, in the sense of Weinstein (1996). States aj ; ak in a set A are related by the groupoid morphism if and only if there exists a high probability grammatical path connecting them, and tuning across the various possible ways in which that can happen—the different cognitive languages—parametizes the set of equivalence relations and creates the groupoid. This assertion requires some development. Note that not all possible pairs of states ðaj ; ak Þ can be connected by such a morphism, i.e. by a high probability, grammatical and syntactical cognitive path, but those that can define the groupoid element, a morphism g ¼ ðaj ; ak Þ having the natural inverse g1 ¼ ðak ; aj Þ. Given such a pairing, connection by a meaningful path, it is possible to define ‘natural’ end-point maps aðgÞ ¼ aj ; bðgÞ ¼ ak from the set of morphisms G into A, and a formally associative product in the groupoid g1 g2 provided aðg1 g2 Þ ¼ aðg1 Þ; bðg1 g2 Þ ¼ bðg2 Þ, and bðg1 Þ ¼ aðg2 Þ. Then the product is defined, and associative, i.e. ðg1 g2 Þg3 ¼ g1 ðg2 g3 Þ. In addition there are natural left and right identity elements lg ; rg such that lg g ¼ g ¼ grg whose characterization is left as an exercise (Weinstein, 1996). An orbit of the groupoid G over A is an equivalence class for the relation aj Gak if and only if there is a groupoid element g with aðgÞ ¼ aj and bðgÞ ¼ ak . The isotropy group of a 2 X consists of those g in G with aðgÞ ¼ a ¼ bðgÞ. In essence a groupoid is a category in which all morphisms have an inverse, here defined in terms of connection by a meaningful path of an information source dual to a cognitive process. If G is any groupoid over A, the map ða; bÞ : G ! A  A is a morphism from G to the pair groupoid of A. The image of ða; bÞ is the orbit equivalence relation G, and the functional kernel is the union of the isotropy groups. If f : X ! Y is a function, then the kernel of f, kerðf Þ ¼ ½ðx1 ; x2 Þ 2 X  X : f ðx1 Þ ¼ f ðx2 Þ defines an equivalence relation. As Weinstein (1996) points out, the morphism ða; bÞ suggests another way of looking at groupoids. A groupoid over A identifies not only which elements of A are equivalent to one another (isomorphic), but it also parametizes the different ways (isomorphisms) in which two elements can be equivalent, i.e. all possible information sources dual to some cognitive process. Given the information theoretic characterization of cognition presented above, this produces a full modular cognitive network in a highly natural manner. The groupoid approach has become quite popular in the study of networks of coupled dynamical systems which can be defined by differential equation models, e.g. Stewart et al. (2003), Stewart (2004). This work extends the technique to networks of interacting information sources which, in a dual sense, characterize cognitive processes, and cannot at all be described by the usual differential equation models. These latter, it seems, are much the spiritual offspring of 18th century mechanical clock models. Cognitive and conscious processes in humans involve neither computers nor clocks, but remain constrained by the limit theorems of information theory, and these permit scientific inference on necessary conditions. 2.4. Internal forces breaking the symmetry groupoid The symmetry groupoid, as constructed for unconscious cognitive submodules in a kind of information space, is parametized across that space by the possible ways in which states aj ; ak can be equivalent, i.e. connected by a meaningful path of an information source dual to a cognitive process. These are different, and in this approximation, non-interacting unconscious cognitive processes. But symmetry groupoids, like symmetry groups, are made to be broken: by internal cross-talk akin to spin–orbit interactions within a symmetric atom, and by cross-talk with slower, external, information sources, akin to putting a symmetric atom in a powerful magnetic or electric field. As to the first process, suppose that linkages can fleetingly occur between the ordinarily disjoint cognitive modules defined by the network groupoid. In the spirit of Wallace (2005a), this is represented by establishment of a non-zero mutual information measure between them: a cross-talk which breaks the strict groupoid symmetry developed above. Wallace (2005a) describes this structure in terms of fixed magnitude disjunctive strong ties which give the equivalence class partitioning of modules, and non-disjunctive weak ties which link modules across the partition, and parametizes the overall structure by the average strength of the weak ties, to use Granovetter’s (1973) term. By contrast the approach of Wallace (2005b), outlined here, is to simply look at the average number of fixed-strength nondisjunctive links in a random topology. These are obviously two analytically tractable limits of a much more complicated regime. Since nothing is known about how the cross-talk connections can occur, at first assume they are random and construct a random graph in the classic Erdos/Renyi manner. Suppose there are M disjoint cognitive modules— M elements of the equivalence class algebra of languages dual to some cognitive process—which we now take to be the vertices of a possible graph. For M very large, following Savante et al. (1993), when edges (defined by establishment of a fixed-strength mutual information measure between the graph vertices) are added at random to M initially disconnected vertices, a remarkable transition occurs when the number of edges becomes approximately M=2. Erdos and Renyi (1960) studied random graphs with M vertices and ðM=2Þð1 þ mÞ edges ARTICLE IN PRESS R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 as M ! 1, and discovered that such graphs almost surely have the following properties (Molloy and Reed, 1995, 1998; Grimmett and Stacey, 1998; Luczak, 1990; Aiello et al., 2000; Albert and Barabasi, 2002): (1) If mo0, only small trees and unicyclic components are present, where a unicyclic component is a tree with one additional edge; moreover, the size of the largest tree component is ðm  lnð1 þ mÞÞ1 þ Oðlog log nÞ. (2) If m ¼ 0, however, the largest component has size of order M 2=3 . (3) If m40, there is a unique GC whose size is of order M; in fact, the size of this component is asymptotically aM, where m ¼ a1 ½lnð1  aÞ  1, which has an explicit solution for a in terms of the Lambert W-function. Thus, for example, a random graph with approximately M lnð2Þ edges will have a GC containing  M=2 vertices. Such a phase transition initiates a new, collective, cognitive phenomenon: the global workspace of consciousness, emergently defined by a set of cross-talk mutual information measures between interacting unconscious cognitive submodules. The source uncertainty, H, of the language dual to the collective cognitive process, which characterizes the richness of the cognitive language of the workspace, will grow as some monotonic function of the size of the GC, as more and more unconscious processes are incorporated into it. Wallace (2005b) provides details. Others have taken similar network phase transition approaches to assemblies of neurons, e.g. neuropercolation (Kozma et al., 2004, 2005), but their work has not focused explicitly on modular networks of cognitive processes, which may or may not be instantiated by neurons. Restricting analysis to such modular networks finesses much of the underlying conceptual difficulty, and permits use of the asymptotic limit theorems of information theory and the import of techniques from statistical physics, a matter we will discuss later. 2.5. External forces breaking the symmetry groupoid Just as a higher-order information source, associated with the GC of a random or semirandom graph, can be constructed out of the interlinking of unconscious cognitive modules by mutual information, so too external information sources, for example in humans the cognitive immune and other physiological systems, and embedding sociocultural structures, can be represented as slower-acting information sources whose influence on the GC can be felt in a collective mutual information measure. For machines these would be the onion-like ‘structured environment’, to be viewed as among Baars’ contexts (Baars, 1988, 2005; Baars and Franklin, 2003). The collective mutual information measure will, through the Joint Asymptotic Equipartition Theorem which generalizes the Shannon–McMillan Theorem, be the splitting criterion for high and low probability joint paths across the entire system. 383 The tool for this is network information theory (Cover and Thomas, 1991, p. 388). Given three interacting information sources, Y 1 ; Y 2 ; Z, the splitting criterion, taking Z as the ‘external context’, is given by IðY 1 ; Y 2 jZÞ ¼ HðZÞ þ HðY 1 jZÞ þ HðY 2 jZÞ  HðY 1 ; Y 2 ; ZÞ, ð2Þ where Hð::j::Þ and Hð::; ::; ::Þ represent conditional and joint uncertainties (Khinchin, 1957; Ash, 1990; Cover and Thomas, 1991). This generalizes to IðY 1 ; . . . ; Y n jZÞ ¼ HðZÞ þ n X HðY j jZÞ  HðY 1 ; . . . ; Y n ; ZÞ. j¼1 (3) If the global workspace/GC involves a very rapidly shifting, and indeed highly tunable, dual information source X, embedding contextual cognitive modules like the immune system will have a set of significantly slowerresponding sources Y j ; j ¼ 1; . . . ; m, and external social, cultural and other environmental processes will be characterized by even more slowly acting sources Zk ; k ¼ 1; . . . ; n. Mathematical induction on Eq. (3) gives a complicated expression for a mutual information splitting criterion of the general form IðX jY 1 ; . . . ; Y m jZ1 ; . . . ; Z n Þ. (4) This encompasses a fully interpenetrating biopsychosociocultural structure for individual consciousness, one in which Baars’ contexts act as important, but flexible, boundary conditions, defining the underlying topology available to the far more rapidly shifting global workspace (Wallace, 2005a, b). This result does not commit the mereological fallacy which Bennett and Hacker (2003) impute to excessively neurocentric perspectives on consciousness in humans, that is, the mistake of imputing to a part of a system the characteristics which require functional entirety. The underlying concept of this fallacy should extend to machines interacting with their environments, and its baleful influence probably accounts for a significant part of AI’s failure to deliver. See Wallace (2006) for further discussion along these lines. 2.6. Punctuation phenomena As a number of researchers have noted, in one way or another—see Wallace (2005a) for discussion—Eq. (1), log½NðnÞ , n is homologous to the thermodynamic limit in the definition of the free energy density of a physical system. This has the form H  lim n!1 F ðKÞ ¼ lim V !1 log½ZðKÞ , V (5) ARTICLE IN PRESS 384 R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 where F is the free energy density, K the inverse temperature, V the system volume, and ZðKÞ is the partition function defined by the system Hamiltonian. Wallace (2005a) shows at some length how this homology permits the natural transfer of renormalization methods from statistical mechanics to information theory. In the spirit of the Large Deviations Program of applied probability theory, this produces phase transitions and analogs to evolutionary punctuation in systems characterized by piecewise, adiabatically stationary, ergodic information sources. These biological phase changes appear to be ubiquitous in natural systems and can be expected to dominate machine behaviors as well, particularly those which seek to emulate biological paradigms. Wallace (2002) uses these arguments to explore the differences and similarities between evolutionary punctuation in genetic and learning plateaus in neural systems. 2.7. Tuning the GC The random network development above is predicated on there being a variable average number of fixed-strength linkages between components. Clearly, the mutual information measure of cross-talk is not inherently fixed, but can continuously vary in magnitude. This we address by a parametized renormalization. In essence the modular network structure linked by mutual information interactions has a topology depending on the degree of interaction of interest. Suppose we define an interaction parameter o, a real positive number, and look at geometric structures defined in terms of linkages which are zero if mutual information is less than, and ‘renormalized’ to unity if greater than, o. Any given o will define a regime of GCs of network elements linked by mutual information greater than or equal to it. The fundamental conceptual trick is to invert the argument: a given topology for the GC will, in turn, define some critical value, oC , 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 perceived. We hence are assuming that the o is a tunable, syntactically dependent, detection limit, and depends critically on the instantaneous topology of the GC defining the global workspace of consciousness. That topology is, fundamentally, the basic tunable syntactic filter across the underlying modular symmetry groupoid, and variation in o is only one aspect of a much more general topological shift. More detailed analysis is given below in terms of a topological rate distortion manifold. Suppose the GC at some ‘time’ k is characterized by a set of parameters Ok  ok1 ; . . . ; okm . Fixed parameter values define a particular GC having a particular topological structure (Wallace, 2005b). Suppose that, over a sequence of ‘times’ the giant component can be characterized by a (possibly coarse-grained) path xn ¼ O0 ; O1 ; . . . ; On1 having significant serial correlations which, in fact, permit definition of an APSE information source in the sense of Wallace (2005a). Call that information source X. Suppose, again in the manner of Wallace (2005a), that a set of (external or else internal, systemic) signals impinging on consciousness, i.e. the GC, is also highly structured and forms another APSE information source Y which interacts not only with the system of interest globally, but specifically with the tuning parameters of the GC characterized by X. Y is necessarily associated with a set of paths yn . Pair the two sets of paths into a joint path zn  ðxn ; yn Þ, and invoke some inverse coupling parameter, K, between the information sources and their paths. By the arguments of Wallace (2005a) this leads to phase transition punctuation of I½K, the mutual information between X and Y, under either the Joint Asymptotic Equipartition Theorem, or, given a distortion measure, under the Rate Distortion Theorem. I½K is a splitting criterion between high and low probability pairs of paths, and partakes of the homology with free energy density described in Wallace (2005a). Attentional focusing then itself becomes a punctuated event in response to increasing linkage between the organism or device and an external structured signal, or some particular system of internal events. This iterated argument parallels the extension of the General Linear Model into the Hierarchical Linear Model of regression theory. Call this the Hierarchical Cognitive Model (HCM). The HCM version of Baars’ global workspace model stands in some contrast to other current work. Tononi (2004), for example, takes a complexity perspective on consciousness, in which he averages mutual information across all possible bipartitions of the thalamocortical system, and, essentially, demands an infomax clustering solution. Other clustering statistics, however, may serve as well or better, as in generating phylogenetic trees, and the method does not seem to produce conscious punctuation in any natural manner. Dehaene and Changeux (2005) take an explicit Baars global workspace perspective on consciousness, but use an elaborate neural network simulation to generate a phenomenon analogous to IAB. While their model does indeed display the expected punctuated behaviors, as noted above, Krebs (2005) unsparingly labels such constructions with the phrase ‘neurological possibility does not imply neurological plausibility’, suggesting that the method does little more than fit a kind of Fourier series construction to high level mental processes. The approach here attempts a central motion model of consciousness, focusing on modular networks defined by function rather than by structure. 2.8. Cognitive quasi-thermodynamics A fundamental 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 ARTICLE IN PRESS R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 formal similarity between their definitions in the asymptotic limit. Information source uncertainty can be defined as in Eq. (1). This is quite analogous to the free energy density of a physical system, Eq. (5). Feynman (1996) provides a series of physical examples, based on Bennett’s work, where this homology is, in fact, an identity, at least for very simple systems. Bennett argues, in terms of 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. Feynman explores in some detail Bennett’s microscopic machine designed to extract useful work from a transmitted message. The essential argument is that computing, in any form, takes work, the more complicated a cognitive process, measured by its information source uncertainty, the greater its energy consumption, and our ability to provide energy to the brain is limited. IAB, we will argue, emerges as an inevitable thermodynamic limit on processing capacity in a topologically fixed global workspace, i.e. one which has been strongly configured about a particular task. Understanding the time dynamics of cognitive systems away from phase transition critical points requires a phenomenology similar to the Onsager relations of nonequilibrium thermodynamics. If the dual source uncertainty of a cognitive process is parametized by some vector of quantities K  ðK 1 ; . . . ; K m Þ, then, in analogy with nonequilibrium thermodynamics, gradients in the K j of the disorder, defined as S  HðKÞ  m X K j qH=qK j (6) j¼1 become of central interest. Eq. (6) is similar to the definition of entropy in terms of the free energy density of a physical system, as suggested by the homology between free energy density and information source uncertainty described above. Pursuing the homology further, the generalized Onsager relations defining temporal dynamics become X dK j =dt ¼ Lj;i qS=qK i , (7) i where the Lj;i are, in first order, constants reflecting the nature of the underlying cognitive phenomena. The Lmatrix is to be viewed empirically, in the same spirit as the slope and intercept of a regression model, and may have structure far different than familiar from more simple chemical or physical processes. The qS=qK are analogous to thermodynamic forces in a chemical system, and may be subject to override by external physiological driving mechanisms (Wallace, 2005c). Eqs. (6) and (7) can be derived in a simple parameter-free covariant manner which relies on the underlying topology of the information source space implicit to the development. We suppose that different physiological cognitive phenomena have, in the sense of Wallace (2000, 2005a–c, 385 Chapter 3), dual information sources, and are interested in the local properties of the system near a particular reference state. We impose a topology on the system, so that, near a particular ‘language’ A, dual to an underlying cognitive process, there is (in some sense) an open set U of ^ such that A; A^  U. Note that closely similar languages A, it may be necessary to coarse-grain the physiological responses to define these information sources. The problem is to proceed in such a way as to preserve the underlying essential topology, while eliminating ‘high frequency noise’. The formal tools for this can be found, e.g. in Chapter 8 of Burago et al. (2001). Since the information sources dual to the cognitive processes are similar, for all pairs of languages A; A^ in U, it is possible to (1) Create an embedding alphabet which includes all symbols allowed to both of them. (2) Define an information-theoretic distortion measure in that extended, joint alphabet between any high probability (i.e. grammatical and syntactical) paths in A and ^ which we write as dðAx; AxÞ ^ (Cover and Thomas, A, 1991). Note that these languages do not interact, in this approximation. (3) Define a metric on U, for example, R ^ A;A^ dðAx; AxÞ ^ (8) MðA; AÞ ¼ lim R 1 , ^ A;A dðAx; AxÞ using an appropriate integration limit argument over the high probability paths. Note that the integration in the denominator is over different paths within A itself, while in ^ the numerator it is between different paths in A and A. Consideration suggests M is a formal metric, having MðA; BÞX0; MðA; AÞ ¼ 0, MðA; BÞ ¼ MðB; AÞ; MðA; CÞpMðA; BÞ þ MðB; CÞ. Other approaches to constructing a metric on U may be possible. Since H and M are both scalars, a ‘covariant’ derivative can be defined directly as ^ HðAÞ  HðAÞ , ^ ^ A!A MðA; AÞ dH=dM ¼ lim (9) where HðAÞ is the source uncertainty of language A. Suppose the system to be set in some reference configuration A0 . To obtain the unperturbed dynamics of that state, we impose a Legendre transform using this derivative, defining another scalar S  H  M dH=dM. (10) The simplest possible Onsager relation here—again an empirical equation like a regression model—is just dM=dt ¼ L dS=dM, (11) where t is the time and dS=dM represents an analog to the thermodynamic force in a chemical system. This is seen as ARTICLE IN PRESS 386 R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 acting on the reference state A0 . For dS=dMjA0 ¼ 0, d2 S=dM2 jA0 40 ð12Þ the system is quasistable, a Black hole, if you will, and externally imposed physiological forcing mechanisms will be needed to effect a transition to a different state. Conversely, changing the direction of the second condition, so that dS 2 =dM2 jA0 o0, leads to a repulsive peak, a White hole, representing a possibly unattainable realm of states. Explicit parametization of M introduces standard—and quite considerable—notational complications (e.g. Burago et al., 2001; Auslander, 1967): imposing a metric for different cognitive dual languages parametized by K leads to Riemannian, or even Finsler, geometries (Wallace, 2005c), including the usual geodesics. One can apply this formalism to the example of the GC, with the information source uncertainty/channel capacity taken as directly proportional to the component’s size, which increases monotonically with the average number of (renormalized) linkages, a, after the critical point. HðaÞ then rises to some asymptotic limit: the homology between information source uncertainty and free energy density suggests that raising the cognitive capacity of the giant component, making it larger, requires energy. Beyond a certain point, the system just runs out of steam. Altering the topology of the network, no longer focusing on a particular demanding task, would allow detection of crosstalk signals from other submodules, as would the intrusion of a signal above the renormalization limit o. The manner in which the system runs out of steam involves a maxed-out, fixed topology for the GC of consciousness. As argued above, the renormalization parameter o then becomes an information/energy bottleneck. To keep the GC at optimum function in its particular topology, i.e. focused on a particular task involving a necessary set of interacting cognitive submodules, a relatively high limit must be placed on the magnitude of a mutual information signal which can intrude into consciousness. Consciousness is tunable, and signals outside the chosen syntactical/grammatical bandpass are often simply not strong enough to be detected, broadly accounting for the phenomena of IAB. This basic focus mechanism can be modeled in far more detail, leading toward incorporation of the effects of embedding culture which are the central concern of this work. 2.9. Focusing the mind’s eye: the simplest rate distortion manifold The second-order iteration above—analogous to expanding the General Linear Model to the Hierarchical Linear Model—which involved paths in parameter space, can itself be significantly extended. This produces a generalized tunable retina model which can be interpreted as a ‘Rate Distortion manifold’, a concept which further opens the way for import of a vast array of tools from geometry and topology. Suppose, now, that threshold behavior in conscious reaction requires some elaborate system of nonlinear relationships defining a set of renormalization parameters Ok  ok1 ; . . . ; okm . The critical assumption is that there is a tunable zero-order state, and that changes about that state are, in first order, relatively small, although their effects on punctuated process may not be at all small. Thus, given an initial m-dimensional vector Ok , the parameter vector at time k þ 1, Okþ1 , can, in first order, be written as Okþ1  Rkþ1 Ok , (13) where Rtþ1 is an m  m matrix, having m2 components. If the initial parameter vector at time k ¼ 0 is O0 , then at time k, Ok ¼ Rk Rk1 R1 O0 . (14) The interesting correlates of consciousness are, in this development, now represented by an information-theoretic path defined by the sequence of operators Rk , each member having m2 components. The grammar and syntax of the path defined by these operators is associated with a dual information source, in the usual manner. The effect of an information source of external signals, Y, is now seen in terms of more complex joint paths in Y and R-space whose behavior is, again, governed by a mutual information splitting criterion according to the JAEPT. The complex sequence in m2 -dimensional R-space has, by this construction, been projected down onto a parallel path, the smaller set of m-dimensional o-parameter vectors O0 ; . . . ; Ok . If the punctuated tuning of consciousness is now characterized by a ‘higher’ dual information source—an embedding generalized language—so that the paths of the operators Rk are autocorrelated, then the autocorrelated paths in Ok represent output of a parallel information source which is, given Rate Distortion limitations, apparently a grossly simplified, and hence highly distorted, picture of the ‘higher’ conscious process represented by the R-operators, having m as opposed to m  m components. High levels of distortion may not necessarily be the case for such a structure, provided it is properly tuned to the incoming signal. If it is inappropriately tuned, however, then distortion may be extraordinary. Let us examine a single iteration in more detail, assuming now there is a (tunable) zero reference state, R0 , for the sequence of operators Rk , and that Okþ1 ¼ ðR0 þ dRkþ1 ÞOk , where dRk is ‘small’ in some sense compared to R0 . (15) ARTICLE IN PRESS R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 Note that in this analysis the operators Rk are, implicitly, determined by linear regression. We thus can invoke a quasi-diagonalization in terms of R0 . Let Q be the matrix of eigenvectors which Jordan-block-diagonalizes R0 . Then QOkþ1 ¼ ðQR0 Q1 þ QdRkþ1 Q1 ÞQOk . (16) If QOk is an eigenvector of R0 , say Y j with eigenvalue lj , it is possible to rewrite this equation as a generalized spectral expansion Y kþ1 ¼ ðJ þ dJkþ1 ÞY j  lj Y j þ dY kþ1 n X ai Y i . ¼ lj Y j þ ð17Þ i¼1 J is a block-diagonal matrix, dJkþ1  QRkþ1 Q1 , and dY kþ1 has been expanded in terms of a spectrum of the eigenvectors of R0 , with jai j5jlj j; jaiþ1 j5jai j. (18) The point is that, provided R0 has been tuned so that this condition is true, the first few terms in the spectrum of this iteration of the eigenstate will contain most of the essential information about dRkþ1 . This appears quite similar to the detection of color in the retina, where three overlapping non-orthogonal eigenmodes of response are sufficient to characterize a huge plethora of color sensation. Here, if such a tuned spectral expansion is possible, a very small number of observed eigenmodes would suffice to permit identification of a vast range of changes, so that the rate distortion constraints become quite modest. That is, there will not be much distortion in the reduction from paths in R-space to paths in O-space. Inappropriate tuning, however, can produce very marked distortion, even IAB. Reflection suggests that, if consciousness indeed has something like a grammatically and syntactically tunable retina, then appropriately chosen observable correlates of consciousness may, at a particular time and under particular circumstances, actually provide very good local characterization of conscious process. Large-scale global processes are another matter, and inappropriate focus can lead to large errors in this analysis. Note that Rate Distortion manifolds can be quite formally described using standard techniques from topological manifold theory (Glazebrook, 2006). The essential point is that a rate distortion manifold is a topological structure which constrains the ‘stream of consciousness’ much the way a riverbank constrains the flow of the river it contains. This is a fundamental insight, which we pursue further. 2.10. The cultural topology of consciousness The groupoid treatment of modular cognitive networks above defined equivalence classes of states according to whether they could be linked by grammatical/syntactical high probability ‘meaningful’ paths. Next we ask the precisely complementary question regarding paths: for any 387 two particular given states, is there some sense in which we can define equivalence classes across the set of meaningful paths linking them? This is of particular interest to the second-order hierarchical model which, in effect, describes a universality class tuning of the renormalization parameters characterizing the dancing, flowing, tunably punctuated accession to consciousness. A closely similar question is central to recent algebraic geometry approaches to concurrent, i.e. highly parallel, computing (e.g. Pratt, 1991; Goubault and Raussen, 2002; Goubault, 2003), which we adapt. For the moment we restrict the analysis to a GC system characterized by two renormalization parameters, say o1 and o2 , and consider the set of meaningful paths connecting two particular points, say a and b, in the two dimensional o-space plane of Fig. 1. The generalized quasiOnsager arguments surrounding Eqs. (6), (7) and (12) suggests that there may be regions of fatal attraction and strong repulsion, Black holes and White holes, which can either trap or deflect the path of consciousness. Figs. 1a and b show two possible configurations for a Black and a White hole, diagonal and cross-diagonal. If one requires path monotonicity—always increasing or remaining the same—then, following, e.g. Goubault (2003, Figs. 6, 7), there are, intuitively, two direct ways, Fig. 1. Diagonal Black and White holes in the two-dimensional o-plane. Only two direct paths can link points a and b which are continuously deformable into one another without crossing either hole. There are two additional monotonic switchback paths which are not drawn. (b) Crossdiagonal Black and White holes as in (a). Three direct equivalence classes of continuously deformable paths can link a and b. Thus the two spaces are topologically distinct. Here monotonic switchbacks are not possible, although relaxation of that condition can lead to ‘backwards’ switchbacks and intermediate loopings. ARTICLE IN PRESS 388 R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 without switchbacks, that one can get from a to b in the diagonal geometry of Fig. 1a, without crossing a Black or White hole, but there are three in the cross-diagonal structure of Fig. 1b. Elements of each ‘way’ can be transformed into each other by continuous deformation without crossing either the Black or White hole. Fig. 1a has two additional possible monotonic ways, involving over/under switchbacks, which are not drawn. Relaxing the monotonicity requirement generates a plethora of other possibilities, e.g. loopings and backwards switchbacks, whose consideration is left as an exercise. It is not clear under what circumstances such complex paths can be meaningful, a matter for further study. These ways are the equivalence classes defining the topological structure of the two different o-spaces, analogs to the fundamental homotopy groups in spaces which admit of loops (e.g. Lee, 2000). The closed loops needed for classical homotopy theory are impossible for this kind of system because of the ‘flow of time’ defining the output of an information source—one goes from a to b, although, for non-monotonic paths, intermediate looping would seem possible. The theory is thus one of directed homotopy, dihomotopy, and the central question revolves around the continuous deformation of paths in o-space into one another, without crossing Black or White holes. Goubault and Raussen (2002) provide another introduction to the formalism. These ideas can, of course, be applied to lower level cognitive modules as well as to the second-order HCM where they are, perhaps, of more central interest. We propose that empirical study based on fMRI or other data will show how the influence of cultural heritage on developmental process defines quite different consciousness dihomotopies in humans. That is, the topology of blind spots and their associated patterns of perceptual completion in human consciousness will be culturally modulated. It is this developmental cultural topology of consciousness which, acting in concert with the inherent limitations of the rate distortion manifold, generates the pattern of IAB. 3. Discussion and conclusions The simple groupoid defined by underlying cognitive modular structure can be broken by intrusion of (rapid) crosstalk within it, and by the imposition of (slower) crosstalk from without—the embedding culture. The former, if strong enough, can initiate a topologically determined GC global workspace of consciousness, in a punctuated manner, while the latter deforms the underlying topology of the entire system, the directed homotopy limiting what paths can actually be traversed by consciousness, formalizing the torus-and-sphere arguments of Wallace (2005a). Broken symmetry creates richer structure in systems characterized by groupoids, just as it does for those characterized by groups. Conscious attention acts through a Rate Distortion manifold, a kind of retina-like filter for grammatical and syntactical meaningful paths, which affects what can be brought to consciousness in a punctuated manner akin to a phase transition. Signals outside the topologically constrained tunable syntax/ grammar bandpass of this manifold are subject to lessened probability of punctuated conscious detection: generalized IAB. Culture will, according to this model, profoundly affect the phenomenon by imposing additional topological constraints defining the ‘surface’ along which consciousness can (and cannot) glide. Glazebrook (2006) has suggested that, lurking in the background of this basic construction, is what Bak et al. (2006) have called the groupoid atlas, i.e. an extension of topological manifold theory to groupoid mappings. Also lurking is identification and exploration of the natural groupoid convolution algebra which so often marks these structures (e.g. Weinstein, 1996; Connes, 1994). Consideration suggests, in fact, that a path may be meaningful according to the groupoid parametization of all possible dual information sources, and that tuning is done across that parametization via a rate distortion manifold. Baars’ global workspace of consciousness is, in effect, a movable bucket of limited capacity, constrained to a culturally determined surface. If it is already filled up by attention to a particular task, droplets from other tasks will likely overflow, and may not be consciously perceived. On the other hand, the bucket itself can traverse only certain permitted sets of paths, and will not likely capture droplets falling outside the allowed topology. If one is, then, intensely focused on watching a basketball game and counting passes, requiring a very particular fixed (but highly tunable) cognitive topology, a gorilla beating its chest may simply not be a strong enough syntactically/grammatically correct signal to intrude on consciousness. On the other hand, falling off one’s chair, a hotfoot, or a particularly sharp comment from one’s significant other, might prove intrusive enough—above the tunable syntax limit characterized by o—to permit detection in the given topological configuration, or else powerful enough to shift conscious topology altogether, i.e. to retune the operator R0 in the rate distortion manifold argument above. Short of that, there remains a significant probability that signals outside the range of the grammar/ syntax filter of conscious attention will not be meaningful and will simply not be detected: IAB. Implicit, however, are the constraints imposed by embedding cultural heritage, which may further limit the properties of R0 , i.e. hold it to a developmentally determined topology. Clearly the phenomenon should not be restricted to the visual system, but, in one form or another, is likely to be ubiquitous across conscious experience (e.g. Wayand et al., 2005), and display particular cultural characteristics (e.g. Masuda and Nisbett, 2006). The mathematical ecologist Pielou (1977, p. 106) describes the principal utility of mathematical models of complex ecosystem phenomena to be not in answering ARTICLE IN PRESS R. Wallace / Journal of Theoretical Biology 245 (2007) 378–390 questions but in raising them. Models can be used to inspire new field investigations and these are the only source of new knowledge as opposed to new speculation. Extending that perspective slightly, the model we have presented, like a regression analysis, would perhaps provide the most scientific value through its violation, i.e. new science is often found in the residuals: Kepler and Newton extending Copernicus. Variations in the forms of IAB across the various senses and their interactions, should give deeper understanding of consciousness. It further seems probable that the phenomenon must be subject to elaborate regulation: too much distractibility while hunting, like too much fixation on one’s prey while one is, in turn, being hunted, could be rapidly fatal. Here we have attempted to reexpress this trade-off in terms of a syntactical/grammatical version of conventional signal theory, i.e. as a ‘tuned meaningful path’ form of the classic balance between sensitivity and selectivity, as particularly constrained by the directed homotopy imposed by cultural heritage on basic conscious experience. 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