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A formal equivalence class algebra can now be constructed by choosing different origin and end points S0, S∞ and defining equivalence of two states by the existence of a high probability meaningful path connecting them with the same origin and end. Disjoint partition by equivalence class, analogous to orbit 16 equivalence classes for dynamical systems, defines the vertices of the proposed network of cognitive dual languages, much enlarged beyond the spinglass example. We thus envision a network of metanetworks. Each vertex then represents a different equivalence class of information sources dual to a cognitive process. This is an abstract set of metanetwork ‘languages’ dual to the cognitive processes of gene expression and development. This structure generates 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 to the same base and end points, and tuning across the various possible ways in which that can happen – the different cognitive languages – parameterizes the set of equivalence relations and creates the (very large) groupoid. See the mathematical appendix for a summary of standard material on groupoids. There is a hierarchy in groupoid structures. First, there is structure within the system having the same base and end points, as in Ciliberti et al. Second, there is a complicated groupoid structure defined by sets of dual information sources surrounding the variation of base and end points. We do not need to know what that structure is in any detail, but can show that its existence has profound implications. First we examine the simple case, the set of dual information sources associated with a fixed pair of beginning and end states. Taking the serial grammar/syntax model above, we find that not all high probability meaningful paths from S0 to S∞ are the same. They are structured by the uncertainty of the associated dual information source, and that has a homological relation with free energy density. Let us index possible dual information sources connecting base and end points by some set A = ∪α. Argument by abduction from statistical physics is direct: Given metabolic energy density available at a rate M, and an allowed development time τ , let K = 1/κMτ for some appropriate scaling constant κ, so that Mτ is total developmental free energy. This is just a version of the fundamental probability relation from statistical mechanics, as above. The sum in the denominator, the partition function in statistical physics, is a crucial normalizing factor that allows the definition of of P[Hβ] as a probability. A basic requirement, then, is that the sum/integral always converges. K is the inverse product of a scaling factor, a metabolic energy density rate term, and 17 a characteristic development time τ . The developmental energy might be raised to some power, e.g., K = 1/(κ(Mτ ) b ), suggesting the possibility of allometric scaling. Some dual information sources will be ‘richer’/smarter than others, but, conversely, must use more metabolic energy for their completion. While we might simply impose an equivalence class structure based on equal levels of energy/source uncertainty, producing a groupoid, we can do more by now allowing both source and end points to vary, as well as by imposing energylevel equivalence. This produces a far more highly structured groupoid that we now investigate. Equivalence classes define groupoids, by standard mechanisms (e.g., Weinstein, 1996; Brown, 1987; Golubitsky and Stewart, 2017). The basic equivalence classes – here involving both information source uncertainty level and the variation of S0 and S∞, will define transitive groupoids, and higher order systems can be constructed by the union of transitive groupoids, having larger alphabets that allow more complicated statements in the sense of Ash above. Landau’s insight was that second order phase transitions were usually in the context of a significant symmetry change in the physical states of a system, with one phase being far more symmetric than the other. A symmetry is lost in the transition, a phenomenon called spontaneous symmetry breaking, and symmetry changes are inherently punctuated. 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 – in this case limited by available metabolic free energy densities – ensures that higher symmetry/energy states – mixed transitive groupoid structures – will then be accessible to the system. Absent high metabolic free energy rates and densities, however, only the simplest transitive groupoid structures can be manifest. A full treatment from this perspective seems to require invocation of groupoid representations, no small matter (e.g., Buneci, 2017; Bos 2017). Something like Pettini’s (2017) Morse-Theory-based topological hypothesis can now be invoked, i.e., that changes in underlying groupoid structure are a necessary (but not sufficient) consequence of phase changes in FD[K]. Necessity, but not sufficiency, is important, as it, in theory, allows mixed groupoid symmetries. Using this formulation, the mechanisms of epigenetic catalysis are accomplished by allowing the set B1 above to span a distribution of possible ‘final’ states S∞. Then the groupoid arguments merely expand to permit traverse of both initial states and possible final sets, recognizing that there can now be a possible overlap in the latter, and the epigenetic effects are realized through the joint uncertainties H(XDi , Z), so that the epigenetic information source Z serves to direct as well the possible final states of XDi . Again, Scherrer and Jost (2017a, b) use information theory arguments to suggest something similar. ‘Phase change’ and the developmental holonomy groupoid in phenotype space There is a more direct way to look at phase transitions in cognitive, and here culturally-driven, gene expression, adapting the topological perspectives of homotopy and holonomy directly within phenotype space. We begin with ideas of directed homotopy. In conventional topology one constructs equivalence classes of loops that can be continuously transformed into one another on a surface. The prospect of interest is to attempt to collapse such a family of loops to a point while remaining within the surface. If this cannot be done, there is a hole. Here we are concerned, as in figure 1, with sets of one-way developmental trajectories, beginning with an initial phenotype Si , and converging on some final phenotype, here characteristic (highly dynamic) brain phenotypes labeled, respectively, Sn and So. One might view them as, respectively, ‘normal’ and ‘other’, and the developmental pathways as representing convergence on the two different configurations. The filled triangle represents the effect of a composite external epigenetic catalyst – including the effects of culture and culturally-structured social interaction – acting at a critical developmental period represented by the initial phenotype Si . We assume phenotype space to be directly measurable and to have a simple ‘natural’ metric defining the difference between developmental paths. Developmental paths continuously transformable into each other without crossing the filled triangle define equivalence classes characteristic of different information sources dual to cognitive gene expression, as above. Given a metric on phenotype space, and given equivalence classes of developmental trajectories having more than one path each, we can pair one-way developmental trajectories to make loop structures. In figure 1 the solid and dotted lines above and below the filled triangle can be pasted together to make loops characteristic of the different developmental equivalence classes. Although figure 1 is represented as topologically flat, there is no inherent reason for the phenotype manifold itself to be flat. The existence of a metric in phenotype space permits determining the degree of curvature, using standard methods. Figure 2 shows a loop in phenotype space. Using the metric definition it is possible to parallel transport a tangent vector starting at point s around the loop, and to measure the angle between the initial and final vectors, as indicated. The holonomy group is defined as follows (e.g., Helgason, 1962): If s is a point on a manifold M having a metric, then the holonomy group of M is the group of all linear transformations of the tangent space Ms obtained by parallel translation along closed curves starting at s. For figure 1 the phenotype holonomy groupoid is the disjoint union of the different holonomy groups corresponding to the different branches separated by ‘developmental shadows’ induced by epigenetic information sources acting as developmental catalysts. The relation between the phenotype groupoid as defined here and the phase transitions in FD[K] as defined above is an open question, and is a central focus of ongoing work.()，英语论文，英语论文题目 |
