Process Compartments as a Foundation

By Paul S. Prueitt
Behavioral Computational Neuroscience Group
75664.2705@compuserve.com
    IEEE Workshop
      on
        Architectures for Semiotic Modeling
        and Situation Analysis
        in Large Complex Systems

    August 27-29, Monterey, Ca, USA
    Organizers:

      J. Albus, A. Meystel, D. Pospelov, T. Reader


Abstract: A theory of process organization is proposed as foundation to a computationally grounded class of generic mechanisms. This theory takes into account the creation and annihilation of processes localized into coherent phenomena by the cooperative behavior of complex systems. A relationship between imaging and action-perception cycles is delineated. An analogy to human self image is used to understand how these processes act in a natural setting.


What are process compartments?

Section 1

The theory of process compartments describes the necessary organization involved in the expression of attentional behavior. Using this theory, the relationship between imaging and action-perception cycles can be delineated as generic properties available to an intentional system behaving within an ecology.

According to the theory, action-perception cycles generate compartments out of an interaction between system competence and system image in the presence of external stimuli. The theory defines system image as a reflection of temporal non-locality having unstable dynamics, or ecological affordance (Kugler et al, 1990).

The notion of ecological affordance was coined by J.J. Gibson (1979) to refer to structural invariance as perceived within the flow of information through the optic system. Initially this concept was oriented, by Gibson's behavioral training, towards the external world. In the theory of process compartments the notion is extended to system image emergence under a non-local coupling expressed by the two complementary notions of "external affordance" and "internal affordance." These complementary notions are not separable. It is important for historical reasons to view system orientation towards temporal invariance as an intertwined relationship between observation and observed. Ecology control theory emerges from an understanding of the generic principles involved in this relationship (Kugler, in this volume).

In counter distinction to all current forms of machine intelligence, a primitive measurement without memory occurs through a generic mechanism: opponency induced symmetry. This opponency mechanism is available to natural processes but in living systems is responsible for intentional expression through imaging.

The system image produces a non-localized action during the initial period of compartment formation and shapes the emergence through synchronization of sub processes. In highly intentional systems, the duration of emergence is extended to allow self image to interact more strongly during the formation of a symmetric barrier to action. In this way, plans and goals are incorporated into the substance of the resulting compartment.

Observation takes place concurrent with primitive measurement and results in the formation of initial conditions through subfeatural combinatorics. Recognition occurs when subfeatures combinatorically express to produce features, invariant sets and other attractors, of the compartment manifold - and via this expression allows the initialization of boundary conditions (initial conditions) within the compartment.

Section 2

Consider a small or large set of coupled oscillators:

      dj/dt = w + SUM( l G(j)),

(with j the oscillation phase, w the intrinsic (constant) oscillation, l the coupling and G the sin function), having various types of network connections (architectures) and initial conditions.

In some cases the resulting phase locking between oscillators is easily seen to be trajectories in the very simple dissipative system

H(x,dx/dt) = 1/2 m dx/dt 2+ 1/2 k x 2

In this simple case, the intrinsic oscillation of each trajectory can be mapped to closed loops (circles) on the surface of the manifold described by the above equation as a 2n+1 dimensional parabola. When n = 1 this is the unforced, undamped pendulum. The oscillators are merely coupled, but harmonic, pendulums. The coupling occurs at the micro-process level but discordant interaction must be represented as a reshaping of the potential (energy) manifold at the macro level.

The harmonic case is simple. Consider a system of rotators that is fully connected, the connection strengths constant, and the initial conditions evenly distributed around a circle. The oscillators will phase lock into their initial intrinsic oscillations since connectivity averages out over the entire architecture.

Discordant interaction between subsystems is correlative to opponency based symmetry induction and consequent formation of a new compartment. Since many of the open problems in fields related to situational analysis, control theory and medical/psychological/ecological therapy are related to the systemic response to discordant interaction, it is important to have a formal system that produces new compartments. For the purposes of computational analysis, symmetry can be broken in several ways: (1) unevening the distribution of initial conditions for the phases j, (2) unevening the connection strengths l, and (3) varying the intrinsic rotation of individual oscillators.

Section 3

A simulation environment appropriate to the simulation of cross scale phase locking allows us to create conditions whereby the emerging manifold has strange attractors as discussed by Kowalski et al (1989; 1991) and others. As important to semiotics, the creation of the manifold can be observed directly in the presence of symmetry breaking "seeds" derived jointly from system subfeatural competence and system image.

System architectures can be found where complex cross scale phase locking has occurred and thus stable symmetric induction formed. This symmetry can then be broken while opponent based induction measures informational invariance in an incoming data stream. The resulting process compartment has a manifold with invariant sets that reflect both data invariance and subfeatural combinatorics. Initial conditions are created that complete patterns intrinsic to both the data stream and subfeatural memory, and produce an internal representation to be read by a system downstream.



The Process Compartment Hypothesis (PCH):

PCH (statement): Temporal coherence is produced by systems that are stratified into numerous levels and produce compartmentalized energy manifolds.

Section 1

Connectionist models assume that a neural system builds internal representations with geometrical and algebraic isomorphism to temporal spatial invariance in the world. The PCH provides a common foundation for investigations of internal representations of this type, the transformations of these representations, and the interaction between independent systems supporting these representations. It assumes a hierarchical organization; with compartments emerging from substrata and images providing metacompartments. In each case, compartments exist as transient complex subsystems of other compartments.

Definition: A compartmentalized process, or process compartment, is a process that is localized in space and time.

This definition shifts the ambiguity from the term "process compartment" to the two terms "process" and "localization," separating the description of a compartment to a localization process and to evolution in time from initial conditions. It delineates a natural characteristic generic to all process compartments; i.e., that compartments have a formation phase and a stability phase.

The formation phase has not been fully described in the research literature, except as a singularity in the classical formulation of its thermodynamics.

The classical time cone is formed by intersecting two symmetric cones in such a way as to overlap only on a non-empty set of dimension zero and aligning both cones to a single axis of symmetry. Choose a direction for the flow of time and associate each line, that is fully contained in the union of the cones and that contains the single point of intersection, with the trajectory in state space of a set of observables for a particular system. This geometric exercise produces a linear model of the sequential nature of present moments, with a unique relationship between the past and the future. This time cone assumes a Laplacian, fully predicable from one set of initial conditions, world. It does not account for non-locality in space or time.

Section 2

The stability phase is better understood than the formation phase. Here the stability itself enforces a single set of laws and these laws act, more or less, in a classical fashion. The compartment itself has formed from an election of degrees of freedom, or observables. The combinatorial span of these observables define a state space. The logic, or dynamical entailment, of initial conditions is established by the same formative mechanisms that shape the emergence.

The intrinsic observables of a compartment embedded within a biological system will reflect an "internal" representation of its degrees of freedom. This representation is not as simple as that of a system of coupled pendulums. However, as a general property, the sum of energy transfer in and out of a compartment boundary remains almost constant. Moreover, conditions on the total energy and the interchange between potential and kinetic energy is clearly one constraint that is placed on all compartments during its stability phase.

In the most general case; consider a connected surface in a high dimensional Euclidean space and set of initial conditions for a trajectory. The trajectory will describe the evolution of a full set of observables for the compartment. A simple relationship, expressed in equations 1- 4, captures this model.

An individual dissipative system has the internal form:

H(x,dx/dt) = 1/2 m dx/dt2 + V(x) + D(dx/dt) + E(x)
(1)

where the first term on the right hand side is an energy function, the second corresponds to architectural constraints, and the last two terms correspond to flows of energy originating from outside the compartment. The compartment is said to be closed when the dissipative, D(dx/dt), and escapement, E(x), terms sum to zero.

The first term represents the basin of attraction classically associated with a potential whose expression depends on the value of the right hand side as well as kinetics described by V(x). As already noted, an initial condition x is required to fully define the compartment. In many cases the so specified trajectory in state space will move into an invariant set. This invariant set can be: (1) a (zero dimensional) point of equilibrium, i.e., a singularity; (2) a closed limit cycle; (3) a spiral into a singularity; (4) a radial motion to a singularity, or (5) a more exotic trajectory.

Architectural constraints, V(x), reflect the connectivity between regional process centers as in a traditional connectionist layered network and more complex non-layered associative networks (as within a single Grossberg type gated dipole). The term, 1/2 m dx/dt2, is, of course, generalized from the potential energy term that describes the standard model of the undamped pendulum.

A very general theory can now be proposed. In a single compartment, higher order terms, existent at transitions, may be modeled by generalizing equation 1 to:

H(x,dx/dt) = U(dx/dt) + V(x) + p(x,dx/dt)
(2)

where U is a potential space, V is a kinetic space and p is a polynomial in x and dx/dt. x and dx/dt are coordinates generalized from equation location and velocity to systemic degrees of freedom and the associated rates of change. In the early part of the formation phase, the degrees of freedom may simply be a union of all degrees of freedom across all interacting compartments. Duplicate invariances are eliminated through phase locking. The relative importance of individual members of a feature set is ordered through competitive/ cooperative circuits.



Cross scale stability:

Section 1

Dissipative coupled harmonic oscillators have a fast and a slow time scale, reflecting the average duration of events. At the slow time scale we have a simple coupling relationship between oscillation phase:

dj/dt = w + SUM l G(j).
(3)

The entrainment of phase differences produce a constraining manifold having relative minima (basins of attraction).

The components of the vector j are the phases (angle measure) of a set of weakly coupled rotators; w is the intrinsic rotational inertia of the rotators; l is a vector of coupling measures, and G is a non-linear periodic vector transformation that depends on the pairwise differences between rotator phases.

At the fast time scale we have a large number of subsystems, each defined as a modification of equation 2 through shunting and additive functions, Si and Ai :

Hi(x,dx/dt) = U(dx/dt) + (Ai(ji)/Vi(x) +
Si(ji)) Vi(x) + pi(x,dx/dt)
(4)

Si and Ai depend on the "ith" component of the vector j. Depending on the shape of the last two terms on the right hand side, U(dx/dt) is generally parabolic, the dissipative system has multiple invariant sets called basins of attractions.

Cross scale synchronism occurs under the condition that an initial condition in one of the basins produces a limit cycle whose frequency evenly divides the frequency of the variable ji. The limit cycle, viewed as a microprocess, is seen as a simple rotator when viewed from the macro observation frame.

Reasonable assumptions about the functions Ai and Si ; i.e., Ai and Si are finite sums of polynomials, leads to the following conjecture:

Conjecture (The Stability Conjecture): If all subsystems in the form of equation 4 exhibits cross scale synchronism, with a system in the form of equation 3, then the combined system is stable; i.e., the number of oscillators and the number of basins of attraction will remain constant. In this case the system in the form of equation 3 is called the system image of the systems in the form of equation 4.

Section 2

Compartments arise in the process hierarchy that are themselves dissipative systems constrained by micro/macro influences. The situation can become quite complex. To manage this complexity we use the language of a tree structure with multiple branches. The nodes of these branches are compartments.

One branch of a temporally stratified process hierarchy has m levels of compartment ensembles {Ek} where m is an integer dependent on time scale and observation. Each compartment within the ensemble Ek has the form of equation (4) but with time variable tk ; where tk = ak t1 and {ak} is a finite positive monotone decreasing sequence with a1 less than 1. Multiple branches can be established below any compartment.

The resulting computational model relies on the selection of a manifold that is stable only within a finite period; e.g., within the corresponding compartment. Under the stability conjecture, it is assumed that a one-to-one correspondence is established between one basin of attraction in each manifold and the members of a "community" of sub-compartments.

The span of a manifold is measured by the continuum of time,

t e [ ta, tb ].

This representation is degenerate; i. e., one to many, and yet it is deterministic within the compartment. It has non-algorithmic singularities at ta and tb.

Section 3

Two important aspects of the intuition behind the notion of a process compartment are (1) that within the life span of the compartment the evolution rules are fixed, and (2) the period over which the nominated set of rules operate is finite. Fixed evolution rules are reflected in the independence of equation 3 from equation 4. The slow process, once formed from the faster ones, will keep the subfeatures stable, via cross scale synchronization, and thus the slow process has indirectly ensured a stable process environment.

This stability can be challenged by external perturbation. In certain cases, the perturbation will collapse the system image, producing the singularity at tb . The collapse frees up energy and kinematic constraints formerly enslaved by the now collapsed compartment. The singularity at ta allows for a dynamic restructuring as a new synergism is formed through cooperative dynamics. A single distributed representation will emerge during the short critical period beginning the new phenomenon.

During this short period the dissipative system might have a very general internal form, where the right hand side is a polynomial:

H(x,dx/dt) = q(x,dx/dt),
(5)

and the dimension of the observation vector might be considered infinite. At the end of this very short formation period, the dimension of the observation vector becomes finite and in fact is minimized by a correspondence to the relative minima of the emerging ensemble manifold. This representation will then partition into potential, kinetic, dissipative and escapement terms, as in equation 1.



Will this approach lead to a unified theory?

Section 1

One way to a complete the theory of process compartments is to combine the work on coupled oscillators (Hoppensteadt, 1986; Kowalski et al., 1989; 1993) with quantum neurodynamics (Pribram, 1973; 1991; 1993; 1994). However, some central problems remain.

First, compartmentalization can not be fully reduced to the conservation laws of physics - at least not the classical ones. Second, measurement by biological systems involves symmetry induction and what might reasonably be called a non-algorithmic process. Whereas symmetry induction has an adequate literature, the representation of compartment creation, under non-algorithmic measurement, has not been fully explored.

The philosophical and mathematical (logical) foundations to issues concerning the non- reducibility of theoretical biology to classic conservation laws are effectively addressed by Robert Rosen (1985).

Section 2

We know much about the focal characteristic of compartment formation. For example, the process compartment always arises under the influence of an opponency between complementary principles. Simple inhibition and suppression properties widely used in artificial neural network research describe opponency, but are not sufficiently used to induce symmetry. Furthermore, the control of compartment formation during the collapse of symmetry has not been taken fully exploited.

Temporal stratification is also an essential concept, and one that is problematic for formal systems to represent. The problem is not merely a logical one, it is also observational. Process compartments are created and annihilated within levels that are observable only at specific time scales. The creation itself nominates the system degrees of freedom to reflect a set of intrinsic observables. These observables are, by definition, subject to a uncertainty principle that disallows any standard method to measure the observation. Thus, we are forced to think in terms of a system envelop where we see the shape of system evolution but not its laws.

When created, a compartment separates phenomena that once had a direct interface. When a compartment has been annihilated, two phenomena that were separated now interact. Input to a compartment conforms to the new compartment's degrees of freedom. The result is modeled, as in Pribram's neurowave equation, as the action of a transform on a vector. The computation of transform coefficients are presumed to be accomplished by fast adaptation of structural constraints, like dendrite spine shape, protein conformation, or neurochemical agents reflecting neurotransmitter concentration. This must be done with the flexibility required to select from multiple (optional) responses in ambiguous situations (see Pribram 1991, pp 264-265).

Hypothesis: Indeterminacy is not merely a function of observational scale but is in fact a property of the world that is used by an intentional system to express preferred goals and plans.

At a conference hosted by Daniel Levine in 1991 on optimality, I discussed the rationale for regarding response degeneracy (see Edelman, 1987) as complementary to optimality (Prueitt, in press). I expressed the view that the creation of viable options would be complementary to optimality if all but a few selective response potentials are somehow held ineligible for deterministic evolution toward a local basin of attraction. This viewpoint provides the critical introduction of non-computational processes into the theory of process compartments.

Hypothesis: Non-computational events are initiated by some physical mechanism that induces a symmetry between multiple potential trajectories in the phase space of a slower (meta) compartment.

Section 3

System image is instrumental in any full description of the central issues of intelligence and intentionality, and we need a scientific foundation for its discussion. Establishing this foundation may not be far away. As suggested by Sir John Eccles (1993), intentionality could be expressed during brief non-linear restructural transitions of process compartments. Eccles points out that the geometrical arrangement of synaptic boutons supports a femto-second process controlling the release of the transmitter vesicle mediated by Ca+ influxes. The control is provided by a process selected through evolution to conserve neurotransmitter. The process has the effect of creating a homogeneous probability distribution measuring the likelihood that any one of six boutons will carry the gradient field interchanges between the presynaptic and post-synaptic events. Eccles sees this mechanism as the interface in which the mind couples to the brain by changing the probability and timing of synaptic events.

Synaptic events play a role in the formation of network connections at a higher level of organization. At critical locations, even distributions are maintained while the gradient increases to a high level. Symmetry and increased gradients result in a barrier and thus in the formation of a trigger. The trigger is released when self organization at a higher level of organization can be effectively influenced (Eccles, 1993).

Hameroff has identified similar symmetry generating mechanisms in the geometric structures of the microtubulin assembly as well as in the temporal dynamics of microtubulin formation. Microtubulin play an important role in cell mitosis and may control some aspect of the connectivity of neuronal ensembles through the fine alteration of dendritic arboration as well as influence second messenger cascades guiding long term potentiation (Hameroff, 1987).

The notion of human self image, although not well defined within a scientific literature (Bandura, 1978), can be used as metaphor for an higher order "agent" that mediates the formation of compartments and shapes the compartments' evolution. The metaphor suggests that compartments involved in the transformation of stimulus traces are shaped by cross scale interaction like that modeled mathematically, as temporally stratified coupled dissipative systems in section one. Since, according to equation 3 and 4, slower processes are required to conform to the oscillation frequency of the emergent process manifold, the initial biases created during symmetry induction are structurally reflected in a system image. The longer the symmetry is maintained the more completely the system competence is sampled.

Simple combinatorics of system competence express elements of a behavioral repertoire. The longer the symmetry the more completely the consequences of action in the external world is sampled. The result is no more than a simple average (or convolution) of two scales of observation resulting in a new set of observables. The convolution, however, can have a kernel that biases outcome and thus differentially delineates the creation and initial conditions of emergent compartments. This kernel is the system image.

Human self image itself needs not be a non-material mind/body interface as perceived by Eccles (1993), but it could alter probabilities during phase transitions of compartments whose existence is brief when compared to the agent and thus have many of the same properties as envisioned by Eccles. Nothing mystical is ruled out or in. Of course, this type of interface is more complex than a simple cross scale interaction between dissipative systems.

The primary distinction between a coupled dissipative system with system images and a system with self images is that self image is a phenomenon that interacts intentionally in a probability space. This intentionality is in opposition to the local laws of dynamics. The induction of a symmetric barrier to the expression of lawful processes is evidence that space/time non-locality is involved in the expression of human self image.

Summary: I have suggested that temporal coherence is produced by systems that are stratified into numerous levels and that produce compartmentalized energy manifolds (PCH). These process compartments, and not merely networks of neurons, are prime candidates for the proximal causal mechanisms producing behavior. This view is consistent with those expressed in Changeux and Dehaene (1989):

"A given function (including a cognitive one) may be assigned to a given level of organization and, in our view, can in no way be considered to be autonomous. Functions obviously obey the laws of the underlying level but also display, importantly, clear-cut dependence on the higher levels. At any level of the nervous system, multiple feedback loops are known to create reentrant mechanisms and to make possible higher-order regulations between the levels" (pgs. 71-72).

At all levels, in anatomical regions and across time scales, generic mechanisms appear to operate. Complex models of prefrontal cortex interaction (Levine & Prueitt, 1989; Levine et al., 1991) with other cortical systems and with limbic systems, require a formal model of intentional processes (Rosen, 1985; Kugler et al., 1990) that rely on these mechanisms.

To understand compartment emergence, it is necessary to address the issue of boundary formation in nonautonomous transitions between episodes. Of particular interest is the nonstationary response to symmetry breaking that accompanies these transitions.

The theory provides a framework to integrate models of processes operating at different time scales, as well as clarify the natural role for structural constraints in signal production and interpretation within and between levels of organization (Pattee, 1972). Stratified processing within and between transient compartments can then be seen in ecological terms.

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