Chapter 4: Psychological interpretation of system behavior.

(Updated: 24 Oct. 2002, 15 May 2003, 29 April 2006, 27 April 2007)

Contents:

Summary.

Introduction.

1. Definitions from systems theory.

2. Psychological interpretation.

3. Application of the psychological interpretation.

Acknowledgement.

References.

Summary.

This Chapter discusses a psychological interpretation of system behavior, with an emphasis on cases where this system is the brain. States of the system are psychologically interpreted as data and/or operations on these data. Attractors are generalized to regions of attraction, which are equated with concepts. Hamiltonian dynamics are compared to automatic processing. The distribution of states over the phase space, that is obtained when the system is observed at discrete time intervals, is formalized by a fuzzy hypergraph. As a general structure, the fuzzy hypergraph describes how regions of attraction convey meaning to the states they contain.

By viewing the creation of a region of attraction as fundamental, the psychological interpretation lends more importance to some relatively obscure psychological phenomena, such as the early recognition of abstract properties and semantic satiation. In addition, it suggests that there is a continuum between data and operation, and between episodic and semantic memory.

Introduction.

The definition of psychological terms can be clarified or fine-tuned by testing a model based on these terms. For example, neural network models have been developed for a range of cognitive processes in learning, memory and concept formation (examples in Arbib 1995, Parks and Long 1998). The opposite approach is to observe the behavior of a dynamical system in general, without making assumptions on the nature of the system, and examine to what extent psychological terms are applicable. Such an approach clarifies the definition of these terms, by comparing them with terms in the description of system behavior. In this Chapter, a psychological interpretation will be given to attractors in terms of concept formation and the attribution of meaning to input.

1. Definitions from systems theory.

Dynamical systems.

For the purpose of this discussion, dynamical systems can be defined as a set of differential equations of variables x (vector): dx_i/dt = f_i(x). These equations are called the system equations. (The system equations generate a vector field over the configuration space; a corresponding set of equations describes the change in covariant coordinates, Choquet-Bruhat et al. 1977, p253.) A trajectory of the system is the curve starting at an initial set of values, x(t₀), such that dx/dt is the tangent vector at every point (integral curve, Choquet-Bruhat et al. 1977, p141). Higher-order time derivatives can be rewritten as first-order derivatives by using auxiliary variables, for example: d²x/dt² = - k.x can be rewritten by using the auxiliary variable y: dx/dt = y, dy/dt = -k.x. The space with dimensions x, dx/dt etc. is the phase space (Takahashi et al. 1972). [The formal definition includes the covariant coordinates (Choquet-Bruhat et al. 1977, p253), however, for systems up to second-order the covariant coordinates can be identified with the time derivatives (Dekker 1998).] The set of values in phase space at a given time is the state of the system: x(t) (cf. Choquet-Bruhat et al. 1977, p253).

Absorbing set, attracting set, attractor.

An absorbing set is a bounded, invariant set A (in IR^p), such that for any bounded set B, for all xÎB, there is a time T > 0, with x(t)ÎA for all t ³ T (Stuart and Humphries 1996, p81).

An attracting set is a closed invariant set A, with a neighborhood U of A, such that for all xÎU there is a time T > 0, with x(t)ÎA for all t ³ T (Tufillaro et al. 1992, p196, Stuart and Humphries 1996, p71).

An attractor is an attracting set that contains a dense orbit; it is topologically transitive (any part of the set can be reached from any other part over time) (Wiggins 1990, p45, Stuart and Humphries 1996, p71). Attractors can therefore be fixed points, limit cycles, tori etc. Here, the term 'attractor' is used mainly for fixed point attractors; other attractors are called higher-dimensional attractors.

Repelling sets and repellors can be defined analogously, by reversing the time criteria: t £ T.

Dissipative system.

A dissipative (dynamical) system is a system with an absorbing set (Stuart and Humphries 1996, p81).

Gradient system.

For gradient systems, the trajectories follow the gradient of a potential function V:

dx(t)/dt = grad V(x(t)).

The function V is everywhere semipositive and V(x) → ¥ for ôxô → ¥. In addition, grad V is only zero for fixed points (i.e points where dx/dt = 0) (Stuart and Humphries 1996, p85). These conditions imply that for a gradient system every trajectory leads to a fixed point (i.e. the system has only fixed point attractors, Stuart and Humphries 1996, p86). If the set of fixed points is bounded, the gradient system is dissipative (Stuart and Humphries 1996, p187).

In physics, dissipation refers to loss of energy. The trajectory of a dissipative system cuts across constant energy surfaces.

In the extreme, the trajectory follows the steepest descent in an energy landscape and the dynamics are equivalent to those of a gradient system. (It will be assumed here that gradient systems fulfill the condition of having a bounded set of attractors, so that all gradient systems can be considered extreme versions of dissipative systems.) The opposite extreme is that of systems without energy loss (zero dissipation). If a system has a constant energy, it is called Hamiltonian.

Following these definitions, dissipative systems can be considered intermediate or mixed forms between Hamiltonian and gradient systems. The tangent vector to the trajectory of a dissipative system can be resolved into two parts: one aligned with the Hamiltonian part and one aligned with the gradient part of the trajectory (cf. Wiggins 1990, p603).

Technically, the vector field generated by the system equations can be resolved into a curl-free and a divergence-free part (= gradient and Hamiltonian components). The resolution of a vector field in three dimensions is discussed by Arfken (1970, p66-70). For a vector field A in n dimensions,

Div(n) A = Σ ¶ A_i /¶ x_i

and

curl(n) A = 1/2 . ε(n) : Rot A,

where ε(n) is the Levi‑Civita tensor in n dimensions,

Rot A_ij = ¶ A_i /¶ x_j ‑ ¶ A_j /¶ x_i

and : denotes double contraction. The linearization of the system equations, given by the matrix M:

M_ij = ¶ f_i /¶ x_j,

can be written as the sum of a symmetric and an anti-symmetric part:

M_{S ij} + M_{A ij} = 1/2 (¶ f_i /¶ x_j + ¶ f_j /¶ x_i) + 1/2 (¶ f_i /¶ x_j ‑ ¶ f_j /¶ x_i).

By applying the definitions given above, it can be seen that the first part is curl-free and the second part divergence-free. (Although any vector field can be resolved in this way, the resolution is not unique if no additional restrictions are introduced, since a curl‑free + divergence-free component can be added to either part and subtracted from the other. However, components that are both curl-free and divergence-free can be ignored in the remainder of this discussion.) By calculating these parts at a point, the local gradient and Hamiltonian components can be found.

Region of attraction.

A region of attraction is defined as a region in which there is a net influx. The Gauss theorem (or divergence theorem) can be used to assess the net influx into a region, if the vector field v generated by the system equations is known (Arfken 1970, p48).

ò Div v dvol < 0

vol

( There are cases where the Gauss theorem applies only with exception of a point (e.g. Arfken, 1970, p68‑69). For a more extensive discussion of the Gauss theorem and the conditions under which it holds, see Kellog (1953, p84‑121). )

Regions of repulsion can be defined analogously as regions that have a net efflux.

Regions of attraction/repulsion and their inclusion relations in wider regions can be formalized with a spatial fuzzy hypergraph. A fuzzy hypergraph is an ordered pair <X,S> , where X is a set of elements and S is a family of fuzzy sets containing these elements. The union of all S_i is X (È_i S_i = X) and no S_i is empty ("_i S_i ¹ Æ). In the present discussion, S is formed by a set of regions and the set X is the set of all states contained in these regions. (X is finite when it is the output of an observation process of finite duration at discrete time intervals.) The characteristic function value of a state in an attracting region is set equal to the net influx into a region around that state. This ensures that the characteristic function can be used at several levels of resolution. In addition, this characteristic function can be defined for nested regions (see below).

The fuzzy hypergraph provides a qualitative description of the gradient component of the vector field. Fuzzy hypergraphs can be considered the lowest degree of structure where structure can still be recognized.

The distinction between levels of nested regions is the result of repeatedly applying an algorithm of region growing for successively lower influx/efflux values. This procedure can be sketched as follows. Starting with the smallest spatial unit under consideration and with the highest influx/efflux values, spatial units are merged as determined by a standard algorithm for region growing (Schoenmakers 1995). This provides the maximal extent of a region at the lowest inclusion level. Next, the process of region growing is repeated on a wider spatial scale, with lower influx/efflux values. The characteristic function of one region within another is the influx/efflux in the contained region.

Remark. When an unknown system is observed, this nesting of regions will show itself in subclustering of observed states. However, not all subclustering is the result of nested regions of attraction. For example, subclustering of observed states can be the result of subclustering of attractors that do not necessarily have nested regions of attraction. In addition, if not all variables of the system are observed, the subclustering can be the result of projection. The detection of these projection effects is only possible by including an additional variable in the observations (Chapter 3).

2. Psychological interpretation.

States.

The state of a dynamical system includes both configuration variables and their time derivatives. Therefore, the psychological interpretation of a state may differ between a configuration that has non-zero time derivatives and a configuration that remains unchanged over a finite time interval.

If parts of the system are directly coupled to input and other parts process this input, these parts span dimensions that can be distinguished as to their psychological interpretation: (1) input data, such as episodes that are experienced or instances of a concept that are encountered, and (2) operations on these data, resulting in internal representations of the input. However, the states themselves combine the activities of various parts of the system, so that the distinction between operation and data becomes blurred. This holds to an even greater extent for states along a trajectory (where different stages of an operation would have to be distinguished) and for systems that are not organized into separate parts. In this general case, one state can correspond to the execution of several operations simultaneously and the distinction between data and operations on these data disappears. If a state can be both data and operation, states are possible that are recursive in an algorithmic sense.

Regions of attraction.

Regions of attraction in the phase space can be equated to concepts (cf. Homa et al. 1979), where wider regions correspond to higher degrees of abstraction. A concept therefore spatially envelops instances, or less abstract concepts. It follows that repellors should correspond to 'negative concepts'. An example of a repellor in psychology is a state in which two perceptions of an ambiguous figure are simultaneously present.

Fuzzy hypergraphs can be used as a general description of the structure of memory. They include more specific descriptions: (fuzzy) sets, networks and spatial models (for a brief discussion of these models, see: Dekker 1998).

Recognition guided by regions of attraction.

In neural network models, recognition of an input is described as reaching a steady state, i.e. the trajectory of the system that responds to an input approaches a fixed point attractor for this input (e.g. Hopfield 1982, Hopfield et al. 1983, Amari 1983, Cohen and Grossberg 1983, Hogg and Hubermann 1984, Huberman and Hogg 1984, see also Hirsch 1989).

This description would imply that the system ends in a steady state after a single input. However, in the case of the brain, convergent dynamics are rarely observed (Hirsch 1989). A steady state can be avoided if higher-dimensional attractors (e.g. limit cycles) are included. However, the system would still be stuck in a single attractor if there is no escape mechanism. Therefore, in order to be more realistic, the description of recognition has to be expanded in several ways, to include (1) higher-dimensional attractors and regions of attraction, (2) dynamic recognition (along the trajectory) and (3) mechanisms to escape from attractors. This broader form of classifying an input will be denoted by the term interpretation.

As to higher-dimensional attractors and regions of attraction, it is postulated here that interpretation follows the local gradient component of the vector field.

Regions of attraction are interpreted in as far as the local gradient component affects the trajectory. The distinction between nesting-levels of regions of attraction, given by the algorithm discussed in Section 1, is a discrete approximation; interpretation varies continuously between levels and does not make use of an explicit distinction between levels.

The global interpretation of a higher-dimensional attractor is equivalent to that of a point attractor producing the same region of net influx. (In other words: the equivalent point attractor at a level of coarseness, where the maximal resolution is the size of the attractor).

The postulate formulated above implies that motion in a direction for which the gradient component does not change is not interpreted. In other words, trajectories along a constant energy level should have a constant interpretation. Therefore, locally, a higher-dimensional attractor receives the interpretation that is averaged over the attractor. For example, all points on a limit cycle have the same (averaged) interpretation.

The qualitative information provided by the stable or unstable directions for point attractors is provided by the negative and positive Lyapunov exponents for higher-dimensional attractors (Wiggins 1990, p603-608). The directions in which the Lyapunov exponent equals zero do not add qualitative information. The twisting and folding of trajectories (Wiggins 1990, p607) is therefore not used for interpretation.

As to dynamic recognition, it can be noted that interpretation takes place not only in the final state, but also along the way, as the trajectory is affected by local regions of attraction. Interpretation becomes more available as the gradient component along the trajectory changes more slowly (in a qualitative sense). [Note that this is the generalization of the description for neural networks: an input can be recognized if the system ends in a steady state.]

Since abstract concepts spatially envelop less abstract concepts, the earliest properties encountered along the trajectory are the most abstract ones, and these can be recognized first. This phenomenon was observed in an experiment in which subjects had to recognize words, that were presented tachistoscopically. The type of errors in selecting the presented word from several alternatives suggests that with shortest presentation times subjects derive information about the meaning of the word, and with longer presentation times, more specific aspects (pronunciation, physical form) can be read out (Turvey, 1974). [ In contrast, other studies, such as EEG event-related potential experiments, suggest that semantic information is available relative late. This apparent contradiction can be resolved by noting that in these studies another concept or category is presented first. The results may therefore indicate the effect of leaving such a concept, where specific information is lost first. ]

As to escape mechanisms, several possibilities can be discussed.

The first escape mechanism is that provided by new input. A new input can set the system on a new trajectory (Hirsch 1989). Note that a steady state can involve active searching for a new input, since states can represent operations as well as information. This steady state is left as soon as this input is found in the environment. (A variant of this mechanism is possible for neural networks, where the attractor is given by a local minimum in an error function. In this case, noise, added to the dynamics in a time‑dependent or output‑dependent fashion, can enable the network to escape the attractor, e.g. Burton and Mpitsos 1992.)

Second, the system can escape from an attractor if the attractor is transient, e.g. by slow feedback from a subsystem. [This mechanism should be slower than the active transients that are the result of inhibitory feedback in the system dynamics after excitation by a stimulus (Ermentrout 1995) or the excitation cycles in the model used by Hjelmfelt and Ross (1994), otherwise the extreme positions of a pendulum would also count as 'transient attractors'. On the other hand, if the dynamics of this mechanism are too slow, it is not useful as an escape mechanism.] Escape is already possible when the attractor is transient for even a single variable. For example, consider a constant in a linear system. For different values of this constant, the type of attractor and the extent of attraction can vary (Takahashi et al. 1972, p94-98). Similar examples can be constructed for different values of a variable and a local linearization of a nonlinear system (involving the remaining variables). More dramatic examples can be seen when the variable acts as a structural parameter within a nonlinear system: the number, dimensionality and nature of attractors can change for different values of such a parameter (Guckenheimer and Holmes 1983, p145-153). In functional terms, the variable in the local linearization example acts to blur or focus an attractor involving the remaining variables. This separation of variables does not have to hold globally; local linearizations at other locations in the phase space select other variables (and/or different numbers of variables) to act as focusing or blurring parameters.

As a third escape mechanism, local attraction can be changed through updating of the system parameters, both through input-linked mechanisms or endogenous mechanisms (metadynamics). However, this process has to be slower than the system dynamics, since otherwise the attractor could not exist. In most cases, therefore, updating will be too slow to be useful as an escape mechanism. (Changing local attraction will be further discussed below under concept formation.)

The Hamiltonian component can be defined as automatic processing, in the sense of having no change in semantic content. This definition overlaps with those in psychology of automatic processes, as (1) having minimal attentional demands, (2) being involuntary and (3) requiring no awareness (Posner 1969, Shiffrin and Schneider 1977, for a short overview, see: Tzelgov et al. 1997). However, there are examples where the two do not match. For example, it is conceivable that even when attention is given to the processing of an input, this attention does not alter the semantic classification of this input. Furthermore, the Hamiltonian component does not have to be equated with a lack of awareness. (The degree of awareness is better described in terms of focusing or blurring, see above.) Interestingly, the psychological definitions are not without exceptions either, since (1) automatic processing is sensitive to resource limitation, (2) can be controlled to some extent and (3) is not completely unconscious (for further discussion, see: Tzelgov et al. 1997). This suggests that the psychological definitions can be fine-tuned in such a way to extend the overlap with Hamiltonian dynamics.

Concept formation: creating regions of attraction.

Input-linked updating of the system parameters can result in the formation of regions of attraction or change the nature of an attractor (examples are given by Takahashi et al. 1972). Similarly, regions of repulsion can be formed, and the nature of a repellor can be changed. (In the remainder of this discussion, only attractors will be mentioned, but analogous statements can be made for repellors.)

In the case of the brain, the overall architecture is already formed at birth and certain parts of the brain are linked to a given sensory modality. However, within the overall architecture, there is considerable flexibility. (Even fixed connections have some plasticity later in life, but this occurs only under extreme conditions, for example: receptive areas in the cortex can be reorganized after amputation or sensory deprivation, for reviews see: Kaas 1991, Chen et al. 2002) To the extent that parts of the brain are linked in a predetermined fashion, dimensions of the phase space are predetermined. If the strength of connections is also fixed, then the location relative to these dimensions is not free: the attribution of meaning may proceed according to absolute location in these cases. However, such cases will be rare; imprinting in birds forms a possible example (cf. Long et al. 2001). To the extent that connections can be modified, dimensions of the phase space are not predetermined. This flexibility (in addition to a flexibility of location) ensures that beyond the immediate registration of input semantic processing is free. Therefore, meaning is derived from relative location - by the regions of attraction and their inclusion relations, in whatever way they have been constructed during the processing of previous input.

The relative position is not directly available to the brain itself; it depends on the value(s) of structural parameters or slower changing variables, which can be read out directly. This would suggest that meaning depends on an absolute location after all. However, the psychological significance of the value of structural parameters can only be determined from their effect on the vector field for the remaining variables.

Illustrations of the mechanisms outlined above are provided by the self-organizing maps in the neural network literature: Kohonen networks show how local interactions, in unsupervised learning, lead to a distribution of responsive elements in the network for a given collection of inputs (for overview, see Ritter 1995). Lateral interactions in the output layer of the Amari-network lead to the formation of topographic map for the inputs starting from a rough map established in early development (Amari 1983). Emphasizing such a pre-existing organization, Toulouse et al. (1986) discuss learning in spin-glass models as a selection of the appropriate pattern of connections from a set of pre-existing possible patterns.

In addition to the formation of regions of attraction, updating can also result in a reduction of attraction. A special case is 'unlearning': updating after a set of random inputs, which reduces the number of spurious attractors for neural networks or spin-glass models. The effect of this unlearning is an increased basin of attraction for the remaining attractors and - in total -a greater storage capacity (Van Hemmen 1990, Hopfield et al. 1983).

If interpretation is linked to the gradient component of the vector field and concepts are equated to regions of attraction, the precision of a concept should be linked to the steepness of the gradient. Concept formation can therefore be extended to concept specification: repeated processing of information can lead to more specific concepts where previously only vague ideas existed (cf. Mackintosh 1974, p493, Homa et al. 1979). However, the formation of a small, strongly attracting region can also dominate the original wider region. This new region can therefore not serve (higher-level) interpretation, as seen in a phenomenon as semantic satiation (Smith 1984, Severance and Washburn 1907).

In the opposite direction, unlearning can be extended to concept blurring, for example when borderline examples of a previously formed concept are encountered.

The description of concept formation suggests that the distinction between episodic and semantic memory (Tulving 1972) has to be replaced by a continuum. Without attraction, the projections of states onto phase space will have a uniform distribution and memory recalled by revisiting a phase space region can be considered episodic (cf. Landauer 1975). The more the distribution of states differs from a uniform one, the more memory is semantic. Such intermediary forms between episodic and semantic memory can be found in experimental studies (e.g. Nahinsky 1992). Finally, where attribution of meaning is important, the basic role of memory is generalization rather than memorizing.

Remark 1. The discussion above deals with examplar-based generalization (unsupervised learning). The opposite process of rule-based discrimination (supervised learning) is not discussed here.

Remark 2. The state of the brain is determined by its intrinsic variables; the brain itself has no way of finding whether their values are appropriate other than through further interaction with the environment. Seen in combination with the environment, i.e. the perspective of an outside observer, the psychological content of a state of the brain may be different from that used by the brain itself. The reconstruction of the latter by an outside observer is not always straightforward. It is comparable to separating out the effects of an input u to a nonlinear system dx/dt = f(x,u). It would be simple to replace the input by intrinsic variables that are affected by this input. However, for the external perspective the part of the input that is ignored by the system also matters. Therefore, in general, it will be necessary to derive sufficient knowledge of the function f.

3. Application of the psychological interpretation.

The psychological interpretation can be used to direct the investigation of system behavior. For example, as discussed in Section 2, states with the same values for the configuration variables may differ in their psychological contents if the values of the covariant variables are different. Therefore, a non-stationary state may be different from a state that is temporarily stationary. A possible illustration is the finding that perception of a visual stimulus is blocked by a masking stimulus presented shortly thereafter, but not when this second stimulus is a uniform dark field (backward masking, Estes 1978, Ganz 1978). A direct test would involve measuring time derivatives of neurochemical variables. Although most techniques available at present do not have sufficient temporal resolution, further development will make this possible. Further experiments can involve presentation of a stimulus in different phases of a cyclical process, or manipulating the rate of change directly through psychopharmaca.

As another example, constancy in the set of concepts suggests that there is a resistance to change (parameters to describe resistance to change are discussed in Dekker 1998). If this resistance also holds for the formation of new concepts, it can be predicted that if a new concept is formed, it will be formed with as little change as possible. This prediction can be tested experimentally and will be discussed in the next Chapter.

Acknowledgement.

The author gratefully acknowledges helpful suggestions by Dr. B. Kappen who commented on a previous draft of this manuscript.

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