Neural spaces: a general framework for the understanding of cognition?

A commentary on the article by R. N. Shepard

Shimon Edelman
232 Uris Hall, Department of Psychology
Cornell University
Ithaca, NY 14853-7601, USA

December 2000


A view is put forward, according to which various aspects of the structure of the world as internalized by the brain take the form of ``neural spaces,'' a concrete counterpart for Shepard's ``abstract'' ones. Neural spaces may help us understand better both the representational substrate of cognition and the processes that operate on it.

Shepard's meta-theory of representation, illustrated in the target article on three examples (object motion, color constancy, and stimulus generalization), can be given the following general formulation: the existence of an invariant law of representation in a given domain is predicated on the possibility of finding an ``abstract space'' appropriate for its formulation. The generality of this meta-theory stems from the observation that any sufficiently well-understood physical domain will have a quantitative description space associated with it. In the account of perceived motion, this is the constraint manifold in what is called in mechanics (and robotics) the configuration space. In color vision, it is the low-dimensional linear space that can be related through principal component analysis to the characteristics of natural illumination and surface reflectances. In stimulus learning and generalization, it is the ``probabilistic landscape'' with respect to which Marr (1970) formulated his Fundamental Hypothesis1 and over which Shepard's (1987) ``consequential regions'' are defined.

A central thesis of the target article is that evolutionary pressure can cause certain physical characteristics of the world to become internalized by the representational system. I propose that the internalized structure takes the form of neural spaces, whose topology and, to some extent, metrics, reflect the layout of the represented ``abstract spaces.''2

The utility of geometric formalisms in theorizing about neural representation stems from the straightforward interpretation of patterns of activities defined over ensembles of neurons as points in a multidimensional space [Gallistel, 1990, Churchland and Sejnowski, 1992, Mumford, 1994]. Four of the issues stemming from the neural space (NS) approach to representation that I raise here are: (1) its viability in the light of experimental data, (2) the explanatory benefits, if any, that it confers on a theory of the brain that adopts it, (3) the operational conclusions from the adoption of the NS theoretical stance, and (4) the main theoretical and experimental challenges it faces.


The relatively few psychophysical studies designed specifically to investigate the plausibility of attributing geometric structure to neural representation spaces did yield supporting evidence. For example, the parametric structure built into a set of visual stimuli can be retrieved (using multidimensional scaling) from the perceived similarity relationships among them [Shepard and Cermak, 1973, Cortese and Dyre, 1996, Cutzu and Edelman, 1996]. Psychophysical evidence by itself cannot, however, be brought to bear on the neurobiological reality of a neural space. To determine whether or not the geometry of a postulated neural space is causally linked to behavior, one needs to examine the neural activities directly. Unfortunately, neurophysiological equivalents of the psychophysical data just mentioned are very scarce. The best known direct functional interpretation of neuronal ensemble response was given in connection with mental rotation in motor control [Georgopoulos et al., 1988]. More recently, an fMRI study of visual object representation that used multidimensional scaling to visualize the layout of the voxel activation space yielded a low-dimensional map that could be interpreted in terms of similarities among the stimuli [Edelman et al., 1999].

Potential benefits.

The representation space metaphor has been invoked as an explanatory device in many different areas of cognition, from visual categorization [Edelman, 1999] to semantics [Landauer and Dumais, 1997]. In vision, this move has been used, traditionally, to ground similarity and generalization. When a transduction mechanism connecting the neural space to the external world is specified, the geometric metaphor also provides a framework for the treatment of veridicality of representations [Edelman, 1999], in a manner compatible with Shepard's idea of second-order isomorphism between representations and their referents [Shepard and Chipman, 1970]. Conceptual spaces seem to offer a promising unified framework for the understanding of other aspects of cognition as well [Gärdenfors, 2000].

Operational conclusions.

Adopting the NS idea as a working hypothesis leads to some unorthodox and potentially fruitful approaches to familiar issues in cognition. One of these issues, raised in the target article, is what branch of mathematics will emerge as the most relevant to the understanding of cognition in the near future. Shepard mentions in this context group theory; the work of Tenenbaum and Griffiths suggests that Bayesian methods will be useful. If the spatial hypothesis is viable, cognitive scientists may also have to take up Riemannian and algebraic geometry. Another issue to consider is the basic nature of the information processing in the brain. Assuming that the representations harbored by the brain are intrinsically space-like, the model of computation best suited for the understanding of cognition may be based on continuous mappings [MacLennan, 1999], rather than on symbol manipulation. Finally, one may inquire as to the form of the laws of cognition that can be expected to arise most naturally from the NS hypothesis. The law of generalization proposed by Shepard (1987) is an important first step towards an answer to this question.


The two most serious challenges for the NS framework both stem from varieties of holism, albeit rather different ones. First, representing an entire object or event by a point in a neural space precludes the possibility of acting on, or even becoming aware of, its structure [Hummel, 2000]. Second, the treatment of an object by the cognitive system frequently depends on the context within which the particular problem at hand is situated, and therefore, potentially, on any of the totality of the representations that exist in the system; this observation is used by [Fodor, 2000] to argue for some very severe limitations on the scope of ``computational psychology.'' It appears to me that both these problems can be addressed within the NS framework. Specifically, adopting a configuration space approach, in which the global representation space approximates the Cartesian product of spaces that code object fragments [Edelman and Intrator, 2000], may do away with the unwanted holism in the representation of individual objects. Furthermore, sharing the representation of an object among several neural spaces may support its context-sensitive treatment (as long as the spaces intersect transversally, they can be kept distinct in places away from the intersection). This would allow the system to make the kind of non-compositional, holistic inferences which, as Fodor rightly notes, abound in human cognition.


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Cortese and Dyre, 1996
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Cutzu and Edelman, 1996
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Faithful representation of similarities among three-dimensional shapes in human vision.
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Edelman, 1999
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Edelman et al., 1999
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Edelman and Intrator, 2000
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Fodor, 2000
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Gallistel, 1990
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Gärdenfors, 2000
Gärdenfors, P. (2000).
Conceptual spaces: the geometry of thought.
MIT Press, Cambridge, MA.

Georgopoulos et al., 1988
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Hummel, 2000
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Where view-based theories of human object recognition break down: the role of structure in human shape perception.
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Landauer and Dumais, 1997
Landauer, T. K. and Dumais, S. T. (1997).
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MacLennan, 1999
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Marr, 1970
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Shepard and Cermak, 1973
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Perceptual-cognitive explorations of a toroidal set of free-form stimuli.
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Shepard and Chipman, 1970
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Tversky, 1977
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... Hypothesis1
``Where instances of a particular collection of intrinsic properties (i.e., properties already diagnosed from sensory information) tend to be grouped such that if some are present, most are, then other useful properties are likely to exist which generalize over such instances. Further, properties often are grouped in this way.'' [Marr, 1970].
... spaces.''2
Arguments against this idea based on the observation that perceived similarities can be asymmetrical [Tversky, 1977] is effectively countered, for example, by adopting the Bayesian interpretation of Shepard's approach proposed by Tenenbaum and Griffiths (this volume). Sticking with the physical space metaphor, one can imagine a foliated, curved neural space, riddled with wormholes (corresponding to arbitrary associations between otherwise unrelated objects or concepts).

Last modified on Mon Nov 27 10:48:42 2000