Eliasmith, C. (1996). The third contender: A critical examination of the dynamicist theory of cognition. Journal of Philosophical Psychology. Vol. 9 No. 4 pp. 441-463.
[Note: Page breaks in the publication are marked in this document as italicized bold e.g.  ]
The third contender: a critical examination of the dynamicist theory of cognition*
Philosophy-Neuroscience-Psychology Program, Department of Philosophy, Washington University in St. Louis, Campus Box 1073, One Brookings Drive, St. Louis, MO 63130-4899, email@example.com
In a recent series of publications, dynamicist researchers have proposed a new conception of cognitive functioning. This conception is intended to replace the currently dominant theories of connectionism and symbolicism. The dynamicist approach to cognitive modeling employs concepts developed in the mathematical field of dynamical systems theory. They claim that cognitive models should be embedded, low-dimensional, complex, described by coupled differential equations, and non-representational. In this paper I begin with a short description of the dynamicist project and its role as a cognitive theory. Subsequently, I determine the theoretical commitments of dynamicists, critically examine those commitments and discuss current examples of dynamicist models. In conclusion, I determine dynamicism's relation to symbolicism and connectionism and find that the dynamicist goal to establish a new paradigm has yet to be realized.
Since the emergence of connectionism in the 1980s, connectionism and symbolicism have been the two main paradigms of cognitive science (Bechtel & Abrahamsen 1991). However, in recent years, a new approach to the study of cognition has issued a challenge to their dominance; that new approach is called dynamicism. There have been a series of papers and books (Globus 1992; Robertson, et al. 1993; Thelen and Smith 1994; van Gelder 1995; van Gelder and Port 1995) that have advanced the claim that cognition is not best understood as symbolic manipulation or connectionist processing, but rather as complex, dynamical interactions of a cognizer with its environment. Dynamicists have criticized both symbolicism and connectionism and have decided to dismiss these theories of cognition and instead wish to propose a "radical departure from current cognitive theory," one in which "there are no structures" and "there are no rules" (Thelen and Smith 1994, p. xix, italics added).
Dynamicism arose because many powerful criticisms which the symbolicist and connectionist paradigms leveled at one another remained unanswered (Bechtel &  Abrahamsen 1991; Fodor and McLaughlin 1990; Fodor and Pylyshyn 1988; Smolensky 1988); it seems there must be a better approach to understanding cognition. But, more than this, there are a number of issues which dynamicists feel are inadequately addressed by either alternative approach. Dissatisfaction with the symbolicist computational hypothesis and the connectionist hypothesis, because of their different emphases and conceptions of time, architecture, computation and representation, have led dynamicists to forward their own dynamicist hypothesis.
Symbolicism is most often the approach against which dynamicists rebel (van Gelder and Port 1995). Dynamicists have offered a number of clear, concise reasons for rejecting the symbolicist view of cognition. The symbolicist stance is well exemplified by the work of Newell, Chomsky, Minsky and Anderson (van Gelder & Port 1995, p. 1). However, Newell and Simon (1976) are cited by van Gelder as having best identified the computationalist hypothesis with the following Physical Symbol System Hypothesis (Newell 1990, pp. 75-77; van Gelder & Port 1995, p. 4):
Natural cognitive systems are intelligent in virtue of being physical symbol systems of the right kind.
Similarly, though for more obscure reasons, dynamicists wish to reject the connectionist view of cognition. Churchland and Sejnowski espouse a commitment to the connectionist view with the hypothesis that "emergent properties are high-level effects that depend on lower-level phenomena in some systematic way" (Churchland and Sejnowski 1992, p. 2). As a result, they are committed to a low-level neural-network type of architecture to achieve complex cognitive effects (Churchland and Sejnowski 1992, p. 4). These same commitments are echoed by another connectionist, Smolensky, in his version of the connectionist hypothesis (1988, p. 7)1:
The intuitive processor is a subconceptual connectionist dynamical system that does not admit a complete, formal, and precise conceptual-level description.
Or, to rephrase:
Natural cognitive systems are dynamic neural systems best understood as subconceptual networks.
However, dynamicists wish to reject both of these hypotheses in favor of an explicit commitment to understanding cognition as a dynamical system. Taken at its most literal, the class of dynamical systems includes any systems which change through time. Clearly, such a definition is inadequate, since both connectionist networks and symbolicist algorithms are dynamic in this sense (Guinti (1991) as cited in van Gelder 1993). Thus, dynamicists wish to delineate a specific type of dynamical system that is appropriate to describing cognition. This is exactly van Gelder's contention with his version of the Dynamicist Hypothesis (1995, p. 4): 
Natural cognitive systems are certain kinds of dynamical systems, and are best understood from the perspective of dynamics.
The hypothesis suggests the heavy reliance of dynamicism on an area of mathematics referred to as dynamical systems theory. The concepts of dynamical systems theory are applied by dynamicists to a description of cognition. Mathematical ideas such as state space, attractor, trajectory, and deterministic chaos are used to explain the internal processing which underlies an agent's interactions with the environment. These ideas imply that the dynamicist should employ systems of differential equations to represent an agent's cognitive trajectory through a state space. In other words, cognition is explained as a multi-dimensional space of all possible thoughts and behaviors that is traversed by a path of thinking followed by an agent under certain environmental and internal pressures, all of which is captured by sets of differential equations (van Gelder and Port 1995). Dynamicists believe that they have identified what should be the reigning paradigm in cognitive science, and have a mandate to prove that the dynamicist conception of cognition is the correct one to the exclusion of symbolicism and connectionism.
Through their discussion of the dynamicist hypothesis, dynamicists identify those "certain kinds" of dynamical systems which are suitable to describing cognition. Specifically, they are: "state-determined systems whose behavior is governed by differential equations... Dynamical systems in this strict sense always have variables that are evolving continuously and simultaneously and which at any point in time are mutually determining each other's evolution" (van Gelder and Port 1995, p. 5) -- in other words, systems governed by coupled nonlinear differential equations. Thus the dynamicist hypothesis has determined that a dynamicist model must have a number of component behaviors, they must be: deterministic; generally complex; described with respect to the independent variable of time; of low dimensionality; and intimately linked (van Gelder 1995; van Gelder and Port 1995). Before discussing what each of these component behaviors mean to the dynamicist view of cognition, we need to briefly examine the motivation behind the dynamicist project -- dynamical systems theory.
The branch of mathematics called dynamical systems theory describes the natural world with essentially geometrical concepts. Concepts commonly employed by dynamicists include: state space, path or trajectory, topology, and attractor. The state space of a system is simply the space defined by the set of all possible states that the system could ever pass through. A trajectory plots a particular succession of states through the state space and is commonly equated with the behavior of the system. The topology of the state space describes the "attractive" properties of all points of the state space. Finally, an attractor is a point or path in the state space towards  which the trajectory will tend when in the neighborhood of that attractor. Employing these concepts, dynamicists can attempt to predict the behavior of a cognitive system if they are given the set of governing equations (which will define the state space, topology and attractors) and a state on the trajectory. The fact that dynamical systems theory employs a novel set of metaphors for thinking about cognition is paramount. Black's emphatic contention that science must start with metaphor underlines the importance of addressing new metaphors like those used by dynamicists (Black 1962). These metaphors may provide us with a perspective on cognition that is instrumental in understanding some of the problems of cognitive science.
The practical and theoretical advantages of dynamical systems theory descriptions of cognition are multitude. The most obvious advantage is that dynamical systems theory is a proven empirical theory. Thus, the differential equations used in formulating a description of a cognitive system can be analyzed and (often) solved using known techniques. One result of having chosen this mathematical basis for a description of cognition is that dynamicists are bound to a deterministic view of cognition (see section 2.0; Bogartz 1994, pp. 303-4).
As well, the disposition of dynamical descriptions to exhibit complex and chaotic behavior is generally considered by dynamicists as an advantage. Dynamicists convincingly argue that human behavior, the target of their dynamical description, is quite complex and in some instances chaotic (van Gelder 1995; Thelen and Smith 1994).
Dynamical systems theory was designed to describe continuous temporal behaviors, thus the dynamicist commitment to this theory provides for a natural account for behavioral continuity. Though the question of whether or not all intelligent behavior is continuous or discrete is a matter of great debate among psychologists (Miller 1988; Molenaar 1990), dynamical systems models possess the ability to describe both. So, relying on the assumption that behavior is "pervaded by both continuities and discrete transitions" (van Gelder and Port 1995, p. 14) as seems reasonable (Churchland and Sejnowski 1992; Egeth and Dagenbach 1991; Luck and Hillyard 1990; Schweicker and Boggs 1984), dynamicism is in a very strong position to provide good cognitive models based on its theoretical commitments.
Fundamentally, dynamicists believe that the other approaches to cognition "leave time out of the picture" (van Gelder and Port 1995, p. 2). They view the brain as continually changing as it intersects with information from its environment. There are no representations, rather there are "state-space evolution[s] in certain kinds of non-computational dynamical systems" (van Gelder and Port 1995, p. 1). The temporal nature of cognition does not rely on "clock ticks" or on the completion of a particular task, rather it is captured by a continual evolution of interacting system  parts which are always reacting to, and interacting with the environment and each other. These temporal properties can be captured with relatively simple sets of differential equations.
In order to avoid the difficult analyses of high-dimensional dynamical systems, dynamicists have claimed that accurate descriptions of cognition are achievable with low-dimensional descriptions. The aim of dynamicists is to "provide a low-dimensional model that provides a scientifically tractable description of the same qualitative dynamics as is exhibited by the high-dimensional system (the brain)" (van Gelder and Port 1995, p. 28).
The dimension of a dynamical systems model is simply equal to the number of parameters in the system of equations describing a model's behavior. Thus, a low dimensional model has few parameters and a high dimensional model has many parameters. The dimensionality of a system refers to the size of its state space. Therefore, each axis in the state space corresponds to the set of values a particular parameter can have.
The low dimensionality of dynamicist systems is a feature which contrasts the dynamicist approach with that of the connectionists. By noting that certain dynamical systems can capture very complex behavior with low dimensional descriptions, dynamicists have insisted that complex cognitive behavior should be modeled via this property. Thus, dynamicists avoid the difficult analyses of high dimensional systems, necessary for understanding connectionist systems. However, it also makes the choice of equations and variables very difficult (see section 3.3).
The linked, or coupled, nature of a system of equations implies that changes to one component (most often reflected by changes in a system variable) have an immediate effect on other parts of the system. Thus, there is no representation passing between components of such a system, rather the system is linked via the inclusion of the same parameter in multiple equations. The ability of such systems of equations to model "cognitive" behaviors has prompted theorists, like van Gelder, to insist that the systems being modeled similarly have no need of representation (van Gelder and Port 1995; van Gelder 1995). In a way, "coupling" thus replaces the idea of "representation passing" for dynamicists.
Dynamicist systems also have a special relation with their environment in that they are not easily distinguishable from their surroundings: "In this vision, the cognitive system is not just the encapsulated brain; rather, since the nervous system, body, and environment are all constantly changing and simultaneously influencing each other, the true cognitive system is a single unified system embracing all three" (van  Gelder 1995, p. 373). Since the environment is also a dynamical system, and since it is affecting the cognitive system and the cognitive system is affecting it, the environment and cognitive system are strongly coupled. Such embeddedness of the cognitive system makes a precise distinction between the system and the system's environment very difficult -- in other words, the system boundaries are obscure. But this fact, dynamicists claim, is not only a good reflection of how things really are, it is a unique strength of the dynamicist approach (van Gelder and Port 1995, p. 25). Coupling amongst not only the equations describing a cognizing system, but also between those describing the environment and those describing the system results in complex "total system" behaviors.
The power of dynamical systems theory to provide useful descriptions of natural phenomena has been demonstrated through its application to many non-cognitive phenomena on various scales, ranging from microscopic fluid turbulence and cell behavior, to macroscopic weather patterns and ecosystems. Still, the questions remains: Why should we apply these tools to a cognitive system? Why should we accept the claim that "cognitive phenomena, like so many other kinds of phenomena in the natural world, are the evolution over time of a self-contained system governed by differential equations" (van Gelder and Port 1995, p. 6)?
A dynamicist advances this claim because of the embeddedness and obvious temporal nature of cognitive systems (van Gelder and Port 1995, p. 9). The omnipresence of embedded, temporal cognitive systems lead van Gelder and Port to conclude that dynamical descriptions of cognition are not only necessary, but also sufficient for an understanding of mind: "...whenever confronted with the problem of explaining how a natural cognitive system might interact with another system which is essentially temporal, one finds that the relevant aspect of the cognitive system itself must be given a dynamical account" (van Gelder and Port 1995, p. 24, italics added). This strong commitment to a particular form of modeling has resulted in the dynamicists claiming to posit a new "paradigm for the study of cognition" (van Gelder and Port 1995, p. 29) -- not, notably, an extension to either of connectionism or symbolicism, but a new paradigm. Thus, the dynamicists are insisting that there is an inherent value in understanding cognition as dynamical instead of connectionist or symbolicist (van Gelder and Port 1995).
One of the greatest strengths of the mathematics of dynamical systems theory is its inherent ability to effectively model complex temporal behavior. It is a unanimous judgement among the paradigms that the temporal features of natural cognizers must be adequately accounted for in a good cognitive model (Newell 1990; Churchland and Sejnowski 1992; van Gelder and Port 1995). Not only do dynamicists address the temporal aspect of cognition, they make this aspect the most important. The reasons for espousing this theoretical commitment are obvious: we humans exist in time; we act in time; and we cognize in time -- real time.  Therefore, dynamical systems theory, which has been applied successfully in other fields to predict complex temporal behaviors, should be applied to the complex temporal behavior of cognitive agents. Whether or not we choose to subscribe to the dynamicist commitment to a particular type of dynamical model, they convincingly argue that we cannot remove temporal considerations from our models of cognition -- natural cognition is indeed inherently temporal in nature.
Dynamicists have often pointed to their temporal commitment as the most important (van Gelder and Port 1995, p. 14). Unfortunately, it is not clear that dynamicists have a monopoly on good temporal cognitive models. In particular, connectionists have provided numerous convincing models of sensorimotor coordination, sensorimotor integration and rythmic behaviors (such as swimming) in which they "embrace time" (Churchland and Sejnowski 1992, p. 337). If dynamicists do not have this monopoly, it will be difficult to argue convincingly that dynamicism should properly be considered a new paradigm.
The intuitive appeal of a dynamical systems theory description of many systems' behaviors is quite difficult to resist. It simply makes sense to think of the behavior of cognitive systems in terms of an "attraction" to a certain state (e.g. some people seem to be disposed to being happy). However, can such metaphorical descriptions of complex systems actually provide us with new insights, integrate previously unrelated facts, or in some other way lead to a deeper understanding of these systems? In other words, can dynamical descriptions be more than metaphorical in nature?
In order to answer this question in the affirmative, we must be able to show the potential for new predictions and explanations. The dynamicist analogy between cognition and dynamical systems theory (see section 2.0) is compelling, but is it predictive and explanatory? We cannot allow ourselves to accept new concepts and theories which do not deepen our understanding of the system being modeled: "[even though] dynamical concepts and theory are seductive, we may mistake translation for explanation" (Robertson, et al. 1993, p. 119).
Philosopher of science Mary Hesse has noted that theoretical models often rely on this sort of analogy to the already familiar (1988, p. 356):
[Theoretical models] provide explanation in terms of something already familiar and intelligible. This is true of all attempts to reduce relatively obscure phenomena to more familiar mechanisms or to picturable non mechanical systems...Basically, the theoretical model exploits some other system (such as a mechanism or a familiar mathematical or empirical theory from another domain) that is already well known and understood in order to explain the less well-established system under investigation.
Clearly, this tack is the one that dynamicists have taken. They are attempting to address the obscure and poorly understood phenomena of cognition in terms of the  more familiar mathematical theory of dynamical systems, which has been successfully applied to complex mechanical and general mathematical systems.
However, simply providing an analogy is not enough (Robertson, et al. 1993, p. 119):
There is a danger, however, in this new fascination with dynamic theory. The danger lies in the temptation to naively adopt a new terminology or set of metaphors, to merely redescribe the phenomena we have been studying for so long, and then conclude that we have explained them.
In science it is necessary to provide a model. A model is no longer simply a resemblance, but rather a precise description of the properties of the system being modeled. The more important properties of the source that are exactly demonstrated by the model, the better the model. So, to differentiate between model and analogy in science, one can determine if the mapping of these important properties is explicit, leaving no room for interpretation; if so, one is dealing with a model. In this sense, a model can be thought of as "a kind of controlled metaphor" (Beardsley 1972, p. 287)2. Thus, where an analogy may consist of the statement: "The atom is like the solar system" -- leaving room for the listener to fill in the details, and possibly to infer wrongly that the orbits of electrons and planets are similar -- a model would consist of a picture, physical prototype, or mathematical description in which each element of the source would be explicitly represented by some particular aspect of the model. In other words, a model presents a precisely constrained analogy.
An excellent example of the mistake of considering analogical application of dynamical systems theory and concepts to be a valid model can be found in Abraham, et al. (1994). Specifically, their dynamical descriptions of behavior apply the concepts of dynamical systems theory to a Jungian analysis of human behavior. However, applying these concepts in such a metaphorical manner simply seems to relate the phenomena in a new way. There is no rigor added to their model simply because the chosen metaphor is mathematical. They have provided a metaphor, not a model. Barton duly notes that in the paper describing one such dynamical model of a Jungian hypothesis, Abraham et al. "imply a level of measurement precision we don't have in clinical psychology" (Barton 1994, p. 12).
Often, clinical psychologists applying dynamics to their field ignore the differences between their field and the rigorous ones from which dynamical systems theory arose: "One way that the distinction between fields is set aside is when authors use rigorous terminology from nonlinear dynamics to refer to psychological variables that are multidimensional and difficult to quantify" (Barton 1994, p. 12). For example, some psychologists have equated the dynamical concept of chaos with overwhelming anxiety, others with creativity, and still others with destructiveness (Barton 1994). These diverse applications of the concept of chaos are clearly more metaphorical than rigorous, and bear little resemblance to the definitions used in precise dynamical systems theory models.
For this reason, there is no real explanation provided by such psychological applications of dynamical systems theory to the phenomenology or intentionality of cognition. These supposed "models" are simply metaphorical descriptions, they  advance no new insights in clinical psychology. They do not reveal any details about what is being described (i.e. cognition). There are no consistent and explicit mappings between dynamical systems theory and human behavior. We have clearly not been presented with anything resembling Beardsley's desirable "controlled metaphor" (Beardsley 1972,p . 287).
From the standpoint of cognitive models, there is not a lot of value in such descriptions. We cannot generate a rigorous explanatory model, nor produce computational simulations from metaphor, so we are not able to discover if the models are predictive. This is a serious failure for any scientific model (Cartwright 1991; Hesse 1972; Koertge 1992; Le Poidevin 1991). Of course, it is possible to haphazardly generate a model which produces data that seems appropriate, but since we have no explicit map between the concepts of clinical psychology and those of dynamical systems theory, the data is meaningful only in its mathematical context, not in a cognitive one.
Even in the most rigorous of dynamical models, such as the Skarda and Freeman (1987) model of the rabbit olfactory bulb, extending dynamical systems theory concepts beyond the metaphorical still proves difficult: "Given this broad picture of the dynamics of this neural system we can sketch a metaphorical picture of its multiple stable states in terms of a phase portrait" (Skarda and Freeman 1987, p. 166). Despite the application of nonlinear differential equations in their model, when it comes time to show how the model relates to cognition, a metaphorical description is employed.
The concepts of dynamical systems theory provide an interesting method of thinking about cognitive systems, but they have not yet been shown to be successfully transferable to rigorous definitions of human behavior or cognition. The "haziness" of clinical psychology does not allow for quantification of mechanisms in dynamical systems theory terms. Furthermore, even some physiological processes do not seem to lend themselves to precise quantitative dynamicist descriptions that are able to provide the predictive or explanative powers expected of good models (c.f. van Geert, 1996).
In providing any dynamical systems theoretic model, one must provide a set of differential equations. These equations consist of constants and parameters (or variables). In a simple equation describing the motion of a pendulum, an example of a parameter would be the current arm angle, which changes as the pendulum swings, whereas a constant is exemplified by the gravitational field which remains constant no matter the position of the pendulum.
One reason that it has been so difficult for dynamicists to provide good cognitive models is that they have been unable to meet the challenge of identifying and quantifying the parameters sufficiently for a dynamical model. It is extremely difficult, if not impossible, to simply examine a complex cognitive system and select which behaviors are appropriately mapped to parameters to be used in a dynamical model (Robertson, et al. 1993, p. 142): 
The central dilemma faced by any experimentalist hoping to apply dynamic systems theory is ignorance, in particular, ignorance of the state variables...
This is a common problem in investigating complex natural nonlinear systems: "Not only are investigators rarely able to completely characterize all the variables that affect a complex system, but they must isolate a system well enough to cut through what Morrison (1991) called a 'sea of noise'" (Barton 1994, p. 10). Dynamicists must realize that the natural systems they wish to model (i.e. cognitive systems) are among the most complex systems known.
When it is difficult to define or even distinguish individual cognitive behaviors (and thus parameters), and similarly challenging to find the signal of interest in ambient noise, it is common practice for dynamicists to define collective parameters (also referred to as order parameters) (Thelen and Smith 1994). A collective parameter is one which accounts for macroscopic behaviors of the system. In other words, many behaviors which could be identified with unique system parameters are "collected" into a group, and the overall behavior of that group is represented by a single system parameter; the collective parameter. Consequently, assigning a meaning or particular interpretation to a collective variable becomes very difficult to justify. Barton (1994) attributes this practice to a confusion of techniques between levels of analysis. Clearly, if the meaning of a parameter cannot be determined, it becomes next to impossible to test a model, or to verify hypotheses derived from observing the behavior of the model. In other words, having identified a parameter which controls a macroscopic behavior of an equation describing a system, does not mean that the parameter can be interpreted in the context of the system being described3. Thus, it is difficult to determine if such a parameter provides an explanation of the mechanisms at work in the system or what, precisely, the relation between the parameter and the original system is.
However, dynamicists have a mandate to "provide a low-dimensional model that provides a scientifically tractable description of the same qualitative dynamics as is exhibited by the high-dimensional system (the brain)" (van Gelder and Port, 1995, p. 28). The only feasible way to generate low-dimensional models of admittedly high-dimensional systems is to use collective parameters. Thus, dynamicists must reconsider their criteria of accepting only low-dimensional models as being valid models of cognition.
By adopting a purely dynamicist approach and thus necessitating the use of collective parameters, it becomes impossible to identify the underlying mechanisms that affect behavior. In contrast, connectionism provides a reasonably simple unit (the neuron or node) to which behavior can ultimately be referred. Similarly, symbolicism provides fundamental symbols to which we can appeal. In both of these instances, understanding global behavior is achieved through small steps, modeling progressively more complex behavior and allowing a "backtrace" when necessary to explain a behavior. With dynamical equations, on the other hand, no such progression can be made. The model is general to such an extent as to lose its ability to explain from where the behaviors it is producing are coming. 
Perhaps the best solution to the difficulties involved in using collective parameters is simply to not use them. Unfortunately, this solution does not help to avoid an important new problem, one which is a consideration for dynamicist models that used collective parameters as well; the problem of systems boundaries.
Dynamicists claim that through their critique of the current state of cognitive science, they are challenging a conceptual framework which has been applied to the problem of cognition since the time of Descartes. Rather than a Cartesian distinction between the cognizer and its environment, dynamicists hold that "the human agent as essentially embedded in, and skillfully coping with, a changing world" (van Gelder and Port 1995, p. 39) (see section 2.0). Thus, dynamicists feel that it is unnatural to distinguish a cognitive system from its environment.
To begin, let us assume that we are attempting to construct a dynamicist model with a number of parameters, let us say n of them. Thus, we will need an n-tuple that we can use (we hope) to completely characterize the behavior of the system we are modeling. Of course, these n parameters are contained in coupled, nonlinear, differential equations. As an example let us assume we are to model a human cognizer; let us think for a moment about the complexity of the model we are attempting to construct.
The human brain contains approximately one trillion connections (Pinel 1993). Furthermore, the number of parameters affecting this system seems almost infinite. Remember, we must account for not only the cognitive system itself, but all environmental factors as there are no discernible system boundaries on the cognitive system. The environment, which must be coupled to our human cognizer, consists not only of other provably chaotic systems like weather, ocean currents, and species populations but billions of other brains (let alone the artificial systems we interact with every day, or the planets, moons, stars, etc.). Having put ourselves in the place of the dynamicist it seems we have the impossible task of characterizing a nearly limitless system. Thus we will, for argument sake, assume the number of parameters is large, but also finite. This same result could be arrived at by using collective parameters (see section 3.3).
So, we now have a large, finite n-tuple of parameters for the equations describing our complex system. Is such a coupled nonlinear differential equation description of human behavior of any value? Using even the most advanced numerical methods, and the most powerful computers, such a problem would probably be unsolvable. So, let us assume infinite computing power. Let us thus be guaranteed that we can solve our system of equations. However, before we can solve our system, we must ask: What can we use for initial conditions?
Is it feasible for us to be able to measure n starting conditions for our model, with any kind of precision? Unlikely, but let us assume once more, that we have n initial conditions of sufficient accuracy. What kind of answer can we expect? Of course, we will get an n dimensional trajectory through an n dimensional state space. How can we possibly interpret such output? At this point, it seems that an interpretation of such a trajectory becomes, if not impossible, meaningless. There is  absolutely no way to either 'un-collect' parameters, or to find out exactly what it means for the system to move through the trillion dimensional (to be conservative) state space.
As our attempt to construct a dynamicist model progresses, it becomes more and more difficult to continue justifying further assumptions. However, such assumptions are necessary in light of the dynamicist hypothesis and its commitments to a "certain type" of dynamical model which is both low-dimensional and completely embedded in its environment. Perhaps these commitments should be reexamined.
An important distinction between dynamicism and either symbolicism or connectionism is the dynamicists' unique view of representation; to be a truly dynamicist model, there should be no representation. In contrast, symbolicist models are fundamentally dependent on symbolic representations, so clearly they are inadequate. Similarly, connectionists represent concepts (via either distributed representation or local symbolic representation) in their simplified networks. But dynamicists decry the use of representation in cognitive models (Globus 1992; Thelen and Smith 1994; van Gelder 1993, 1995).
In a criticism of connectionism, Globus concludes: "It is the processing of representations that qualifies simplified nets as computational (i.e. symbolic). In realistic nets, however, it is not the representations that are changed; it is the self-organizing process that changes via chemical modulation. Indeed, it no longer makes sense to talk of 'representations'" (Globus 1992, p. 302). Similarly, van Gelder insists: "it is the concept of representation which is insufficiently sophisticated" (van Gelder, 1993, p. 6) for understanding cognition. Again, Thelen and Smith pronounce: "We are not building representations at all!" (Thelen and Smith 1994, p. 338). However, it is never mentioned what it would "make sense" to talk of, or what would be "sophisticated" enough, or what dynamicists are "building". Notably, the dynamicist assertion that representation is not necessary to adequately explain cognition is strongly reminiscent of the unsuccessful behaviorist project.
In the late 1950s there was extensive debate over the behaviorist contention that representation had no place in understanding cognition. One of the best known refutations of this position was given by Chomsky in his 1959 review of B. F. Skinner's book Verbal Behavior. Subsequently, behaviorism fell out of favor as it was further shown that the behaviorist approach was inadequate for explicating even basic animal learning (Thagard 1992, p. 231). The reasons for the behaviorist failure was its fundamental rejection of representation in natural cognizers.
Dynamicists have advanced a similar rejection of representation as important to cognition. Consequently, they fall prey to the same criticism that was forwarded over three decades ago. Furthermore, the early work of researchers like Johnson-Laird, Miller, Simon and Newell firmly established a general commitment to representa-tion in cognitive science inquiries (Thagard 1992, p. 233). There have been no alternatives offered by dynamicists which would fundamentally disturb this commitment.
Thus, it is not easy to convincingly deny that representation plays an important role in cognition. It seems obvious that human cognizers use representation in their dealings with the world around them. For example, people seem to have the ability to rotate and examine objects in their head. It seems they are manipulating a representation (Kosslyn 1980, 1994). More striking perhaps is the abundant use of auditory and visual symbols by human cognizers everyday to communicate with one another. Exactly where these ever-present communicative representations arise in the dynamicist approach is uncertain. It will evidently be a significant challenge, if not an impossibility, for dynamicists to give a full account of human cognition, without naturally accounting for the representational aspects of thinking. Though dynamicists can remind us of the impressive behaviors exhibited by Brooks' (1991) dynamical robots, it is improbable that the insect-like reactions of these sorts of systems will scale to the complex interactions of mammalian cognition.
To better understand the outcome of the theoretical difficulties discussed in the previous sections, we will now examine three examples that dynamicists have cited as being good dynamicist models. These models are not only considered to be examples of application of the dynamicist hypothesis, but are considered by dynamicists to be exemplars of their project.
Though a number of dynamicist models have been proposed by clinical psychologists, many have not been cited as paradigmatic. Because of the difficulties involved in developing convincing, non-metaphorical models of psychological phenomena, even dynamicist proponents tend to shy away from praising these abundant models.
Physiological psychologists, in contrast, have developed far more precise models. Robertson et al. (1990), outlined a model for CM (cyclicity in spontaneous motor activity in the human neonate) using a dynamical approach. It seems that such quantifiable physiological behavior should lend itself more readily to a non-metaphorical dynamical description than perhaps clinical psychology would, allowing the psychophysiologist to avoid the poor conceptual mappings of clinical psychologists.
Indeed, Robertson et al. gathered reams of data on the cyclic motor activity apparent in human children. Because of the availability of this empirical data, this dynamicist CM model is one of the few able to begin to breach the metaphor/model boundary (Thelen and Smith 1994, p. 72) which proves impenetrable to many (see section 3.2). However, it is another matter to be able to understand and interpret the data in a manner which sheds some light on the mechanisms behind this behavior. 
Robertson et al., after "filtering" the observedstate space, obtained a dynamicist model with desirably few degrees of freedom which seemed to be able to model the stochastic process of CM. However, upon further investigation, the only conclusions that could be drawn were: "We clearly know very little about the biological substrate of CM" (Robertson, et al. 1993, p. 147). In the end, there is no completed dynamicist model presented, though various versions of the model which do not work are discounted. So, Robertson et al. have employed dynamicist models to constrain the solution, but not to provide new insights. In their closing remarks, they note (Robertson, et al. 1993, p. 47):
We are therefore a long way from the goal of building a dynamical model of CM in which the state variables and parameters have a clear correspondence with psychobiological and environmental factors.
In other words, a truly dynamicist model is still a future consideration.
The olfactory bulb model by Skarda and Freeman is one of the few well-developed models that dynamicists claim as their own. Many authors, including van Gelder, Globus, Barton, and Newman have cited this work as strong evidence for the value of dynamical systems modeling of cognition. Upon closer examination however, it becomes clear that this model is subject to important theoretical difficulties. Furthermore, it is not even evident that this dynamicist exemplar is indeed a truly dynamicist model.
In Skarda and Freeman's (1987) article How brains make chaos in order to make sense of the world, a dynamical model for the olfactory bulb in rabbits was outlined and tested to some degree. They advanced a detailed model of the neural processing underlying the ability to smell. This model relies on a complex dynamical system description which may alternate between chaotic activity and more orderly trajectories, corresponding to a learning phase or a specific scent respectively. They hypothesized that chaotic neural activity serves as an essential ground state for the neural perceptual apparatus. They concluded that there is evidence of the existence of important sensory information in the spatial dimension of electroencephalogram (EEG) activity and thus there is a need for new physiological metaphors and techniques of analysis.
Skarda and Freeman observed that their model generated output that was statistically indistinguishable from the background EEGs of resting animals. This output was achieved by setting a number of feedback gains and distributed delays "in accordance with our understanding of the anatomy and physiology of the larger system" (Skarda and Freeman 1987, p. 166) in the set of differential equations that had been chosen to model the olfactory bulb. Notably, the behavior of the system can be greatly affected by the choice of certain parameters, especially if the system is potentially chaotic (Abraham and Shaw 1992). It is thus uncertain whether the given model is providing an accurate picture of the behavior, or whether it has been molded by a clever choice of system parameters into behaving similarly to the system being modeled. 
Even assuming that the model is not subject to this objection, a further criticism can be directed at its predictive or correlative properties. Although the model accounts quite well for a number of observed properties, "it does not correspond with the actual EEG patterns in the olfactory lobe" (Barton 1994, p. 10). The consequences of this inaccuracy seem quite severe. For, if both the model, and what is being modeled are indeed chaotic systems (i.e. very sensitive to initial conditions), but they are not the same chaotic system, and if there are any inaccuracies in their initial conditions4, then the divergence of the state spaces of the model and the real system will be enormous within a short time frame. Consequently, the model will not be robust and will be difficult to use in a predictive role.
Finally, the authors themselves see their paper and model as showing that "the brain may indeed use computational mechanisms like those found in connectionist models" (Skarda and Freeman 1987, p. 161). Furthermore, they realized that: "Our model supports the line of research pursued by proponents of connectionist or parallel distributed processing (PDP) models in cognitive science" (Skarda and Freeman 1987, p. 170). Dynamicists, however, wish to rest their cognitive paradigm on the shoulders of this model. Ironically, the model is simply not a dynamicist model; the architecture is very much like a connectionist network, only with the slightly less typical addition of inhibition and far more complex transfer functions at each node. These facts make it rather curious that it is touted as a paradigmatically important dynamical systems model. The model's similarities with connectionism make it quite difficult to accept the assertion that this type of dynamical model is the seed of a new paradigm in cognitive modeling.
The model which has been touted by van Gelder (1995) as an exemplar of the dynamicist hypothesis is the Motivational Oscillatory Theory (MOT) modeling framework by James Townsend (1992; see also Busemeyer and Townsend 1993). In this case, unlike the Skarda and Freeman model, MOT does indeed provide dynamicist models, though simplified versions. However, it is also evident that the model provided falls victim to the theoretical criticisms already advanced (see sections 3.2-5).
The most evident difficulty in the MOT model relates to the correct choice of systems parameters (see sections 3.3 and 3.4). Admittedly, for dynamicist models, "changing a parameter of [the] dynamical system changes its total dynamics" (van Gelder 1995,p. 357). Thus, it is extremely important to be able to correctly select these parameters. However, the MOT model does not seem to have any reliable way of doing so (Townsend 1992, pp. 221-2):
A closely allied difficulty [i.e. allied to the difficulty of setting initial conditions] -- in fact, one that interacts with setting the initial conditions -- is that of selecting appropriate parameter values. Unlike physics, where initial conditions and parameter values are usually prescribed by the situation, usually in psychology, the form of the functions is hypothesized in a "reasonable way". However, we often have little idea as to the "best" numbers to assign, especially for parameters. 
The devastating result of this difficulty is that the model needs new descriptions for each task. In other words, it becomes impossible to apply the model more than once without having to rethink its system parameters. It seems that this points to the likelihood of the model being molded by a clever choice of parameters, not to the ability of the model to predict the trajectory of a class of behaviors.
Furthermore, this model is an admittedly simple one (Townsend 1992, p. 219), which makes it rather disconcerting that it is necessary to fix the system manually when it is not behaving correctly (Townsend 1992, p. 223). This presents a great limitation because the expected complexity and dimension of a truly dynamicist model is immense (see section 3.4). Such manual fixing and redescription of each task would surely be impossible in a full-scale model.
The admissions that: "[MOT] appears very simple indeed but is nevertheless nonlinear, and at this time, we do not have a complete handle on its behavior" (Townsend 1992, p. 220) does not bode well for the dynamicist project. If it is not possible to have a handle on the behavior of the simplest of models, and the dynamicist hypothesis calls for massively complex models, what chance do they have of ever achieving the goal of a truly dynamicist model?
Dynamicists tend to be quite succinct in presenting their opinion of what the relation between dynamicism and the other two approaches should be, and are not shy about their project to replace current cognitive approaches: "we propose here a radical departure from current cognitive theory" (Thelen and Smith 1994, p. xix). The dynamicist project to supersede both connectionism and symbolicism has given them reason to assess critically the theoretical commitments of both paradigms. Dynamicists have effectively distinguished themselves from the symbolicist approach and, in doing so, have provided various persuasive critical arguments (Globus 1992; Thelen and Smith 1994; van Gelder 1995; van Gelder and Port 1995). However, dynamicists are not nearly as successful in their attempts to differentiate themselves from connectionists. When they manage to do so, they encounter their greatest theoretical challenges; e.g. providing a non-representational account of cognition. For this reason, deciding the place of dynamicism in the space of cognitive theories, reduces to deciding its relation to connectionism. If dynamicism does not include connectionism in its class of acceptable cognitive models, or if it is not a distinct cognitive approach, there is no basis for accepting the dynamicist hypothesis as defining a new paradigm.
Critics may claim that a dynamical systems approach to cognition is simply not new -- as early as 1970, Simon and Newell were discussing the dynamical aspects of cognition (Simon and Newell 1970, p. 273). In 1991, Giunti showed that the symbolicist Turing Machine is a dynamical system (van Gelder, 1993), so it could be concluded that there is nothing to gain from introducing a separate dynamicist paradigm for studying cognition. However, Turing Machines and connectionist networks have also been shown to be computationally equivalent yet these approaches are vastly disparate in their methods, strengths, and philosophical commit-ments (Fodor and Pylyshyn 1988, p. 10). Similarly, though Turing Machines are dynamical in the strictest mathematical sense, they are nonetheless serial and discrete. Hence, symbolicist models do not behave in the same ideally coupled, dynamical and continuous manner as dynamicist systems are expected to. Dynamicist systems can behave either continuously or discretely, whereas Turing Machines are necessarily discrete. Furthermore, they are not linked in the same way to their environment, and the types of processing and behavior exhibited is qualitatively different. For these reasons, dynamicists believe their approach will give rise to fundamentally superior models of cognition. Biological evidence and the symbolicists' practical difficulties lend support to many of the dynamicists criticisms (Newell 1990; Churchland and Sejnowski 1992; van Gelder and Port 1995).
However, Smolensky's (1988) claim that connectionism presents a dynamical systems approach to modeling cognition can not be similarly dismissed. Connectionist nets are inherently coupled, nonlinear, parallel dynamical systems. These systems are self-organizing and evolve based on continuously varying input from their environment. Still, dynamicists claim that connectionist networks are limited in ways that a truly dynamical description is not.
However, differentiating between connectionist networks and dynamical systems models is no easy task; connectionists often assert that a connectionist network "is a dynamical system" (Bechtel and Abrahamsen 1991; c.f. Churchland 1992). Frequently, dynamicists themselves admit that connectionist networks are indeed "continuous nonlinear dynamical systems" (van Gelder and Port 1995, p. 26). Smolensky outlined the many ways in which a connectionist network is a dynamical system -- he encapsulated the essence of dynamical systems in their relation to cognition and connectionism (Smolensky 1988). Churchland and Sejnowski have gone further, discussing limit cycles, state spaces, and many other dynamical properties of nervous systems and have included purely dynamical analyses in their connectionist discussions of natural cognitive systems (Churchland and Sejnowski 1992, p. 396).
The relationship between connectionism and dynamicism is undeniably more intimate than that between either of these approaches and symbolicism. Nevertheless, dynamicists wish to subordinate connectionism to their cognitive approach (van Gelder and Port 1995, p. 27). Dynamicists fundamentally reject the connectionist commitment to computationalism, representationalism, and high dimensional dynamical descriptions.
Critiques of connectionism from dynamicists do not seem to present any sort of united front. Some dynamicists note the lack of realism in some networks (Globus 1992). Others reject connectionism not because of a "failure in principle" but because of "a failure of spirit" (Thelen and Smith 1994, p. 41). Still others reject connectionism as being high-dimensional and too committed to symbolicist ideas: ideas like representation (see section 3.5).
The lack of realism in networks is often due to the limitations of current computational power. Networks as complex as those found in the human brain are infeasible to simulate on present-day computers. The complexity of real networks does not represent a qualitatively distinct functioning, rather just the end-goal of  current connectionist models. Thus, claims consonant with: "simplified silicon nets can be thought of as computing but biologically realistic nets are non computational" (Globus 1992, p. 300) are severely misleading. The chemical modulation of neurotransmitter synthesis, release, transport, etc. is simply a more complicated process, not a qualitatively different method of functioning. As Globus (1992) later admits, connectionist networks "severely stretch" the concept of computation in the direction of dynamical systems theory. Currently, most, if not all, types of dynamical behavior have been exhibited by various connectionist networks; including chaos, catastrophe, phase change, oscillation, attraction, etc. (Churchland, 1992; Meade and Fernandez 1994). Thus, it is a very safe assumption that the distinctions between real and simplified networks which Globus advances are ones which will, with time and improved processing power, become obsolete.
The claim that connectionism is simply a "failure in spirit" does nothing to advance the dynamicist cause, it simply reminds us where (perhaps) connectionist modeling should be headed. The final two criticisms of connectionism as being high-dimensional and representational have been addressed in sections 3.3-5. It is not clear from this discussion that either of these properties is a hindrance to connectionism. Rather, denying them provides great theoretical difficulty for dynamicism. What is clear, however, is that dynamicism does not include connectionism in its class of acceptable cognitive models. Maybe, then, dynamicism and connectionism are completely distinct cognitive approaches.
Van Gelder urges us to accept that connectionist networks are too limited a way to think about dynamical systems. He claims that "many if not most connectionists do not customarily conceptualize cognitive activity as state-space evolution in dynamical systems, and few apply dynamical systems concepts and techniques in studying their networks" (van Gelder, 1993, p. 21). However, there are a great number of influential connectionists, including the Churchlands, Pollack, Meade, Traub, Hopfield, Smolensky and many others who have addressed connectionist networks in exactly this manner.
There does not seem to be any lack of examples of the application of dynamical systems descriptions to networks (Churchland and Sejnowski 1992; Pollack 1990; Smolensky 1988). In one instance, Kauffman (1993) discusses massively parallel Boolean networks in terms of order, complexity, chaos, attractors, etc. In fact it seems the only viable way to discuss such large (i.e. 100 000 unit) networks is by appealing to the overall dynamics of the system and thoroughly apply dynamical systems concepts, descriptions and analysis (Kauffman 1993, p. 210).
Van Gelder insists that dynamical descriptions of connectionist networks is where connectionists should be headed; many connectionists would no doubt concur. However, he goes on to conclude that connectionism "is little more than an ill-fated attempt to find a half-way house between the two worldviews [i.e. dynamicism and symbolicism]" (van Gelder and Prot 1995, p. 27). Rather, it seems connectionism may be the only viable solution to a unified cognitive theory, since cognition seems to be neither solely representational/symbolic nor nonrepresentational/dynamical. Connectionism is able to naturally incorporate both dynamical and representational commitments into one theory. In any case, all that van Gelder has  really accomplished is to cast a dynamical systems theory description of cognition into the role of a normative goal for connectionism -- he has not provided a basis for claiming to have identified a new paradigm.
The fundamental disagreement between connectionists and dynamicists seems to be whether or not connectionist networks are satisfactory for describing the class of dynamical systems which describes human cognition. By claiming that connectionist networks are "too narrow" in scope, van Gelder wishes to increase the generality of the dynamicist hypothesis, excluding high-dimensional, neuron-based connectionist networks. However, connectionist networks naturally exhibit both high-level and low-level dynamical behaviors, providing room for van Gelder's desired generality while not sacrificing a unit to which behavior can be referred. In other words, the mechanism of cognition remains comprehensible in connectionist networks and does not fall prey to the difficulties involved with collective parameters (see section 3.3). The fact that connectionist networks are amenable to high-level dynamical descriptions makes it hardly surprising that differentiating between connectionist networks and dynamical systems is no easy task. Frequently, dynamicists realize: "indeed, neural networks, which are themselves typically continuous nonlinear dynamical systems, constitute an excellent medium for dynamical modeling" (van Gelder and Port 1995, p. 26). Furthermore, in Smolensky's paper On the Proper Treatment of Connectionism, he has outlined some of the many ways in which a connectionist network is a dynamical system (1988, p. 6):
The state of the intuitive processor at any moment is precisely defined by a vector of numerical values (one for each unit). The dynamics of the intuitive processor are governed by a differential equation. The numerical parameters in this equation constitute the processor's program or knowledge. In learning systems, these parameters change according to another differential equation.
As Smolensky explicitly noted, a connectionist network represents the state of the system at any particular time by the activity of all units in the network. These units are naturally interpretable as axes of a state space. Their behaviors can be effectively described at a general level in dynamical systems theory terms. Such systems are nonlinear, differentially describable, self-organizing and dynamical as they trace a path through their high order state space. The behavior of these networks is exactly describable by the state space and the system's trajectory; as in any typical dynamical system. In other words, the tools provided by dynamical systems theory are directly applicable to the description of the behavior of connectionist networks. Examples of strange attractors, chaos, catastrophe, etc. are all found in connectionist networks, and such concepts have been used to analyze these networks. These qualities lend such systems all the desirable traits of dynamicism (e.g. natural temporal behavior, amenability to general descriptions) but they remain connectionist and thus representational, computational, and high-dimensional.
So, dynamicism does not include connectionism in its class of models, as some of their theoretical commitments are incompatible. Neither, however, is dynamicism a distinct cognitive approach as important aspects (and the least controversial) of the  dynamicist hypothesis are naturally addressed by connectionist networks. Thus, the dynamicist hypothesis has not provided a foundation on which to build a new paradigm. What it has provided, however, are reasons to intensify a particular type of connectionist modeling; one which uses the tools of dynamical systems theory to understand the functioning of connectionist networks.
It is undeniable that brains are dynamical systems. Cognizers are situated agents, exhibiting complex temporal behaviors. The dynamicist description emphasizes our ongoing, real-time interaction with the world. For these reasons, it seems that dynamical systems theory has far greater appeal for describing some aspects of cognition than classical computationalism.
However, by restricting dynamicist descriptions to low dimensional systems of differential equations which must rely on collective parameters, the dynamicist has created serious problems in attempting to apply these models to cognition. In general it seems that dynamicists have a difficult time keeping arbitrariness from permeating their models. There are no clear ways of justifying parameter settings, choosing equations, interpreting data, or creating system boundaries. The dynamicist emphasis on collective parameter models makes interpretation of a system's behavior confounding; there is no evident method for determining the 'meaning' of a particular parameter in a model.
Similarly, though dynamicists present interesting instances when it seems representation may be inappropriate (e.g. motor control, habitual behavior, etc.) it is difficult to understand how dynamicists intend to explain the ubiquitous use of representation by human cognizers while maintaining a complete rejection of representation. This project failed with the behaviorists, and it is not clear why it should succeed now.
It is difficult to accept that dynamical models can effectively stand as their own class of cognitive models. The difficulties which arise at the proposed level of generality seem insurmountable, no matter the resources available. They seem to offer exciting new ways of understanding these systems and of thinking intuitively about human behavior. However, as a rigorous descriptive model of either, the purely dynamical description falls disappointingly short. At most, dynamicists offer new metaphors and interesting discussion, but shaky models. However, at the very least they offer a compelling normative direction for cognitive science (c.f. Aslin, 1993).
Despite the power and intuitive appeal of dynamical systems theory, the dynamicist interpretation of how this field of mathematics should be applied to cognitive modeling is neither trivial nor obviously preferable to connectionism and symbolicism, as dynamicists would have us believe. However, dynamical systems theory can contribute invaluably to the description, discussion and analysis of cognitive models. Possibly more cognitive scientists should realize "Our brains are dynamical, not incidentally or in passing, but essentially, inevitably, and to their very core" (Churchland and Sejnowski 1992,p. 187). 
* Special thanks to Cameron Shelley, Paul Thagard and Jim Van Evra for helpful comments on earlier drafts of this paper.
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