The Continuum Between Temperament and Mental Illness as Dynamical Phases and Transitions

The full range of biopsychosocial complexity is mind-boggling, spanning a vast range of spatiotemporal scales with complicated vertical, horizontal, and diagonal feedback interactions between contributing systems. It is unlikely that such complexity can be dealt with by a single model. One approach is to focus on a narrower range of phenomena which involve fewer systems but still cover the range of spatiotemporal scales. The suggestion is to focus on the relationship between temperament in healthy individuals and mental illness, which have been conjectured to lie along a continuum of neurobehavioral regulation involving neurochemical regulatory systems (e.g., monoamine and acetylcholine, opiate receptors, neuropeptides, oxytocin), and cortical regulatory systems (e.g., prefrontal, limbic). Temperament and mental illness are quintessentially dynamical phenomena, and need to be addressed in dynamical terms. A meteorological metaphor suggests similarities between temperament and chronic mental illness and climate, between individual behaviors and weather, and acute mental illness and frontal weather events. The transition from normative temperament to chronic mental illness is analogous to climate change. This leads to the conjecture that temperament and chronic mental illness describe distinct, high level, dynamical phases. This suggests approaching biopsychosocial complexity through the study of dynamical phases, their order and control parameters, and their phase transitions. Unlike transitions in physical systems, these biopsychosocial phase transitions involve information and semiotics. The application of complex adaptive dynamical systems theory has led to a host of markers including geometrical markers (periodicity, intermittency, recurrence, chaos) and analytical markers such as fluctuation spectroscopy, scaling, entropy, recurrence time. Clinically accessible biomarkers, in particular heart rate variability and activity markers have been suggested to distinguish these dynamical phases and to signal the presence of transitional states. A particular formal model of these dynamical phases will be presented based upon the process algebra, which has been used to model information flow in complex systems. In particular it describes the dual influences of energy and information on the dynamics of complex systems. The process algebra model is well-suited for dealing with the particular dynamical features of the continuum, which include transience, contextuality, and emergence. These dynamical phases will be described using the process algebra model and implications for clinical practice will be discussed.

In the Process Algebra model, each process is considered to be a generator of a finite, discrete set of informons. Complex systems often possess state descriptions in the form of wave functions or probability distributions, both of which mathematically form a structure called a Hilbert space. Many types of Hilbert space (and in particular that used in quantum mechanics) have an additional feature which makes them reproducing kernel Hilbert spaces (1). Given a reproducing kernel Hilbert space H(X) with base space X, one can find a discrete subspace Y of X (sampling subspace), and a Hilbert space H(Y) on Y, such that each function in H(Y) can be lifted to a function in H(X) via interpolation. Interpolation means that if ( ) z  is a function in H(X), then for each y Y  there exists an interpolation function ( ) y z  on H(X) such that ( ) ( ) ( ) In general there are usually an infinite number of these sampling subspaces. These interpolations functions are not unique but in the case that the subspace Y has the form of a regular lattice the interpolation functions may be taken to be sinc functions ( sin / x x ) (1). If the subspace has an irregular structure with density matching the Beurling density (2), Fechtinger-Gröchenik interpolation theory may be used instead (1).
In the Process Algebra, the discrete subsets Y are considered to be fundamental, their elements representing the actual occasions of Whitehead's process theory. The elements of H(Y) are considered to be ontological, while the elements of H(X) are derived (emergent) through an (arbitrary) interpolation procedure. The elements of Y are considered to be generated by process, P, and the value ( ) y  assigned to a point y in Y is also generated by P by causally propagating specific information from prior actual occasions to nascent actual occasions by means of a causal propagator, K. The resulting wave function ( ) Interactions between processes are conjectured as being triggered by the generation of informons according to the compatibility between the processes. Compatibility between interacting complex systems is an idea first proposed by Trofimova (3). In the current context it can be thought of as a generalization of the idea of coupling factors. Compatibility Ξ(P,M) is conjectured to be a function of fixed factors such as mass, charge, coupling constants and of the local compatibilities. The probability of an interaction taking place Π(P,M) is in turn a function of the compatibility, Π(P,M) = χ(Ξ(P,M). The precise form of these functions depends upon the particular case. The Born rule (from which probabilities are derived in quantum mechanics) is expected to arise from these interactions and from the compatibility, but a precise derivation is not yet in hand. If one naively applies the Born rule, then probability will be proportional to the Each informon is represented as [n]< n p ; n m : ( ) n  z ; n  >{ n  } where 1) n is a heuristic mathematical label, 2) n p is a structured set of intrinsic properties, The local process strength at an informon n is given as * n n   . The information residing in the informons of the content is utilized by the generating process to create the informon. The intrinsic properties n p are attributed to the generating process P and imparted to each informon generated by P. The extrinsic properties are unique to each informon but are frame dependent. Each informon n is interpreted as a point n m (causal manifold interpretation or embedding) in some causal manifold M (which can be thought of in various ways as space-time or as a state space). Its content set n  causally embeds into M. Each causal tapestry forms a causal antichain in M, and thus represents a discrete sampling of a spacelike hypersurface in M. Each informon n is associated with a local Hilbert space interpretation of the form ( ) ( , ) n n n n f    r r m (which can be thought of as a wave, or a field, or a local probability), the Hilbert space H(M) being that over the causal manifold M. Each causal tapestry I is associated with two different maps: a tapestry realization (or allowing a slight misnomer, a tapestry "wave function") of the form ( ) r . The lattice spacing must be consistent with the Beurling density (2). Maymon and Oppenheim (5) have shown that non-uniform embeddings still provide a highly accurate approximation using sinc interpolation so long as the spatiotemporal density is large enough. A more realistic model requires the use of nonuniform embeddings and more sophisticated interpolation techniques, such as Fechtinger-Gröchenik theory (1).
Processes possess three additional intrinsic characteristics: 1) r, the number of prior informons whose information is incorporated into an informon n. It is also the cardinality of n  , and the number of short rounds needed to form n.
2) N, the number of informons in each generation, and thus the number of rounds and the cardinality of the causal tapestry I.
3) R, the number of informons generated per round. A primitive process has R=1. Otherwise the process is compound. The propagator will be determined by particle and interacting potentials.
An important concept is that of epistemological equivalence. Epistemological equivalence of two processes P and Q means that their global Hilbert space interpretations, ( If two processes are epistemologically equivalent then the specifics of informon generation do not matter in so far as NRQM is concerned. They generate the same emergent wave functions and therefore will yield the same NRQM predictions. This is useful because processes can be modeled heuristically based upon mathematical convenience just so long as they are epistemologically equivalent to any real processes. In particular one can use processes based upon combinatorial games which have particularly valuable characteristics (6)(7)(8). Epistemological equivalence may also possess ontological implications in that it might be impossible on principle for macroscopic observers to be able to access information about this most fundamental level. To use a computer analogy, it is generally inadvisable for a computer program to be able to access and change its own code. Perhaps that is the case for nature as well.
The Process Algebra is a formal language and mathematical structure for describing interactions between processes. Epistemological equivalence allows for the use of many different forms of representation for modeling the actions of processes. A particularly fruitful and accessible representation is based on the concept of combinatorial games.
These are generally two player games which may be competitive or co-operative, partisan or non-partizan, with discrete moves, finite resources, a well defined end point, and perfect information. Combinatorial games have been harnessed for a variety of purposes mathematically ranging from recreational play (6) to the generation of mathematical structures in formal logic (7,8), where there have proven to be a powerful tool. Here, the Process Algebra is used to model the flow of information throughout the generation of informons by processes as well as their different modes of interaction and the effects that such interactions have on information flow.
Processes may influence one another in two different ways. The first (coupling) involves the generation of individual informons, their relative timing as well as the sources of information which enters into their generation. Coupling results in epistemologically equivalent processes, so properties are unaltered. The second (interaction) involves the activation or inactivation of individual processes and the creation of new processes. Epistemological equivalence is broken and properties are altered.
Two processes P 1 ,P 2 may be independent, meaning that the neither constrains the actions of the other in any way. This relationship is denoted simply by the comma ",". Compound processes (R>1) can be formed from primitive processes (R=1) by various coupling operations. A coupling affects timing and information flow. Two processes may generate informons concurrently (products) during each round, or sequentially (sums), with only one process generating informons during a given round. Information from either or both processes may enter into the generation of a given informon (free) or information incorporated into an informon by a process may only come from informons previously generated by that process (exclusive). This leads to four possible operators: 1: Free sequential (free sum): 1 2  P P The operation of concatenation is used to denote processes that act in successive generation cycles. Thus 1 2  P P (or simply 1 2 P P ) indicates that 1 P acts during the first generation cycle, while 2 P acts during the second generation cycle.
Interactions between processes may activate an inactive process or inactivate an active process. In addition, an interaction among processes P 1 ,P 2 ,..,P n may generate a new process, P, which can be described in functional form as F(P 1 ,P 2 ,..,P n ) = P. If Θ(P 1 ,P 2 ,..,P n ) describes a coupling among P 1 ,P 2 ,..,P n then the functional relation may be described using the operation of concatenation, as Θ(P 1 ,P 2 ,..,P n ) P. This can also be placed into a linguistic from as 1 2 ( , , ,

P P 
Independence, sums and products are commutative, associative and distributive operations. Concatenation is non-commutative and non-associative in general. The zero process, O, is the process that does nothing.
An important and special form of interaction is the coupling interaction. Such interactions respect epistemological equivalence and thus are potentially reversible through a subsequent coupling interaction. An example is a rotation to a different eigenbasis as a result of an engagement with a measurement apparatus.
The impact of these different operations is best demonstrated using a process graph. The process graph ( ) P  of a process P is defined as follows: rounds 0 to N are laid out in order. At round 0 one places the informons of the prior causal tapestry. At round k, place each informon n that was generated during round k and draw a directed line from each prior informon in its content set G n to n and label it with the causal distance between the two informons. Note that no lines link informons of the prior causal tapestry to one another or nascent informons to one another since no information passes among them.
, the subgraph of ( ) P  consisting of n and its content set. The process graph is used to determine the causal manifold interpretation of the nascent causal tapestry and the global Hilbert space interpretation. If a process acts on the same prior causal tapestry it may produce a different process graph, thus a different history. The process covering map gathers together the global Hilbert space interpretations of these different process graphs, thus all of the possible histories required for a sum over histories calculation. A configuration space graph and configuration space covering map can be defined for products of processes.
The basic rules for applying these operations in combining processes are the following: 1. The free sum is only used for single systems and combining states which possess identical property sets (pure states).
2. The exclusive sum is used for single systems and combining states which possess distinct property sets (mixed states).
3. The free product is used for multiple systems which possess distinct character (scalar, spinorial, vectorial and so on) (for example coupling a boson and a fermion).
4. The exclusive product is used for multiple systems which possess the same character (for example coupling two bosons or two fermions).