A combination of directed homotopy topological and Morse theoretic methods can significantly extend control and information theories, permitting deeper understanding of ‘developmental' pathologies afflicting a broad spectrum of biological, psychological, socioeconomic, machine, and hybrid processes across different time scales and levels of organization. Such pathologies emerge as phase transitions driven by synergistic forms of environmental insult under stochastic circumstances, causing `comorbid condensations' through groupoid symmetry breaking. The resulting statistical models should be useful for the analysis of experimental and observational data in many fields.
More explicitly, developmental process -- ontology -- is ubiquitous across vast biological, social, economic, and machine realms. Rosen (2012) characterizes this as ‘...anticipatory behavior at all levels of... organization'. Maturana and Varela (1980) see cognition permeating biology. Atlan and Cohen (1998) invoke a ‘cognitive paradigm' for the immune system that generalizes to wound healing, blood pressure regulation, neural dynamics, and so on (Wallace 2012). West-Eberhard (2003; 2005) sees ontology as a matter of ‘choice' at developmental branch points. Traffic flow involves repeated ‘ontological' choices by atomistic vehicles at road junctions, as well as during ordinary passage in heavy traffic (Wallace 2016a Ch.9). Indeed, machine cognition quite generally requires repeated choice of response to environmental cues (Wallace 2016a). A firm responding to market pressures must, at least annually, reconfigure product lines and marketing strategies, also a cognitive process (e.g., Wallace 2015 and references therein). Democratic state actors confronted by changing patterns of threat and affordance must, at least during elections, repeatedly choose among the different patterns of response made available by the contending parties and candidates. Active warfare involves constantly repeated choice at all levels of organization leading up to, and during, combat operations.
All developmental phenomena are, however, subject to patterns of failure and dysfunction. These range from neurodevelopmental disorders such as autism and schizophrenia (Wallace 2016b) to collapse of vehicle flow in traffic jams (Kerner and Klenov 2009), and catastrophes of governance like Brexit, or the US occupation of Iraq. Here, we attempt to extend results from information and control theories to statistical tools useful in understanding developmental failure.
The underlying model of development is that a system begins at some initial ‘phenotype' So confronting a branch point Co leading to two (or more) possible subsequent ‘phenotypes' S1 and S2, where new branch points C1 and C2 will be confronted, and at which choices must again be made, and so on.
Two of the three essential components of this model are intrinsically linked.
The first component is that of directed homotopy, in the sense of Grandis (2009) and Fajstrup et al. (2016). That is, there are equivalence classes of paths leading from ‘phenotype' S_{n} to S_{n+1}, as defined by the branch conditions C_{n}. A group structure -- the so-called ‘fundamental group' -- is imposed on a geometric object by convolution of loops within it that can be reduced without crossing a hole (e.g., Hatcher 2001). An algebraic topology of directed homotopy can be constructed from the composition of paths that constitutes a groupoid (Weinstein 1996), an object in which a product need not be defined between every possible object, here the equivalence classes of possible linear paths. As Weinstein (1996) emphasizes, almost every interesting equivalence relation on a space B arises in a natural way as the orbit equivalence relation of some groupoid G over that space. Instead of dealing directly with the orbit quotient space B/G$as an object in the category of sets and mappings, one should consider instead the groupoid G itself as an object in the category of groupoids and homotopy classes of morphisms. An exactly similar perspective involves use of the homotopy and homology groups of algebraic topology to characterize complicated geometric objects (Hatcher 2001).
The second component is recognition that choice at developmental branch points involves active selection of one possible subsequent path from a larger number that may be available. This is often done, in the sense of Atlan and Cohen (1998), by comparison of ‘sensory' data with an internalized -- learned or inherited -- picture of the world, and upon that comparison, an active choice of response is made from a larger number of those possible. Rosen (2012) invokes `anticipatory models' for such processes. Following the Atlan/Cohen model, choice involves reduction in uncertainty, and reduction in uncertainty implies the existence of an information source that we will call `dual' to the underlying cognitive process. Wallace (2012) provides a somewhat more formal treatment.
What is clear is that the dual information source or sources associated with developmental process must be deeply coupled with the underlying groupoid symmetries characterizing development. As development proceeds, the groupoid symmetry becomes systematically richer.
As Feynman (1996) argues, information is not ‘entropy', rather it can be viewed as a form of free energy. Indeed, Feynman (1996), following Bennett, constructs an idealized machine that turns the information within a message into useful work.
Second, groupoids are almost groups, and it becomes possible to apply Landau's symmetry breaking/making arguments to the dual information sources characterizing developmental process (Pettini 2007). In that theory, phase transitions are recognized in terms of sudden shifts in the underlying group symmetries available to the system at different temperatures. High temperatures, with the greatest available energies, have the greatest possible symmetries. Symmetry breaking occurs in terms of the sudden nonzero value of some `order parameter' like magnetization at a sufficiently low critical temperature.
For a road network, for example, the `order parameter' would be the number of road turnoffs blocked by a traffic jam. The temperature analog is an inverse function of the linear vehicle density (Kerner and Klenov 2009; Wallace 2016a).
The third component of the model looks in detail at the embedding regulatory apparatus that must operate at each branch point to actively choose a path to the desired ‘phenotype'. This requires exploration of the intimate connection between control and information theories represented by the Data Rate Theorem (Nair et al. 2007).
In a sense, the underlying argument is by abduction from recent advances in evolutionary theory: West-Eberhard (2003, 2005) sees development as a key, but often poorly appreciated, element of evolutionary process, in that a new input, whether it comes from a genome, like a mutation or from the external environment, like a temperature change, a pathogen, or a parental opinion, has a developmental effect only if the preexisting phenotype can respond. A novel input causes a reorganization of the phenotype, a `developmental recombination' in which phenotypic traits are expressed in new or distinctive combinations during ontogeny, or undergo correlated quantitative changes in dimensions. Developmental recombination can result in evolutionary divergence at all levels of organization.
Most importantly, perhaps, West-Eberhard characterizes individual development as a series of branching pathways. Each branch point is a developmental decision, a switch point, governed by some regulatory apparatus, and each switch point defines a modular trait. Developmental recombination implies the origin or deletion of a branch and a new or lost modular trait. The novel regulatory response and the novel trait originate simultaneously, and their origins are inseparable events: there cannot be a change in the phenotype without an altered developmental pathway.
Thus, there are strong arguments for the great evolutionary potential of environmentally induced novelties. An environmental factor can affect numerous individuals, whereas a mutation initially can affect only one, a perspective having implications, not only for evolutionary economics, but across a full spectrum of ubiquitous `developmental' phenomena: even traffic streams `evolve' under changing selection pressures, and, indeed, such pressures act at every level of biological, social, or economic organization, as well as across rapidly expanding realms of machine cognition.
That is, just as the Atlan/Cohen ‘cognitive paradigm' for the immune system generalizes across many different systems (Wallace 2012), so too does the West-Eberhard model of development: repeated branching under the control of an embedding regulatory apparatus responding to environmental cues is widely observed. Here, we apply a control theory formalism via the Data Rate Theorem, and using information theory, invoke the dual information source necessarily associated with regulatory cognition. The intent is to examine developmental disorders, in a large sense, over a spectrum that ranges from cellular to socioeconomic and emerging machine levels of organization, and across time scales from those of biological evolution to extremely rapid machine response.
The main focus is on exploring the influence of environmental insult on developmental dysfunction, where insult itself is measured by a projected scalar `tangent space' defined in terms of the invariants of a complicated `fog-of-war matrix' representing interacting environmental factors. The synergism between control and information theories via the Data Rate Theorem, and the extensions using topological and `free energy' Morse Theory methods, provide a new theoretical window into the dynamics of many developmental processes, via the construction of statistical models that, like more familiar regression procedures, can be applied to a broad range of experimental and observational data.
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