tal representations are defined and interpreted in the low-
dimensional P-spaces, not in the high-dimensional patterns
of activity of the attractor network dynamics. Higher-level
complex features are created by combination of lower-level
features (mechanisms of attention should play a role here).
Alternatively they may be formed by a neural realization of
multidimensional scaling procedure [12]. Preservation of
similarities is the only requirement for the dimensionality
reduction. Mental representations - objects in P-spaces - are
formed slowly transferring the knowledge about categoriza-
tion from the attractor networks to simpler feedforward net-
works.
Geometrical characterization of P-spaces and of the land-
scapes created by the probability density functions defined
on these spaces and obtained as an approximation to neu-
rodynamics lead to an alternative model of mind. Such P-
spaces offer an arena for mind events that is acceptable to
psychology and understandable from neurobiological point
of view.
III. Encoding categories in feature spaces.
The model presented in the previous section may be ap-
plied to categorization in psychology. Although the exam-
plar theory of categorization is usually presented as an alter-
native to the prototype theory [13] neurodynamics lies at the
basis of both theories. Is it possible to distinguish between
categorization based on prototypes and exemplars? In the
first case basins of attractors should be large and the cor-
responding objects in P-spaces should be large and fuzzy.
A prototype is not simply a point with average features for
a given set of examples, but a complex fuzzy object in the
P-space. If categorization is based on exemplars there are
point-like attractors corresponding to these exemplars and
the P-space objects are also point-like. Intermediate cases
are also possible, going from set of points representing ex-
emplars, to a fuzzy object containing all the exemplars. Al-
though representation is different both theories may give
similar behavioral results if processes acting on these rep-
resentations are different. If the neural dynamics is noisy
exemplars become so fuzzy that a prototype is formed. Neu-
ral dynamic models physical processes at the level of brain
matter while dynamic in the P-spaces models a series of suc-
cessive categorizations, or mental events, providing precise
language useful from psychological perspective.
A classic category learning task experiment has been per-
formed by Shepard et.al. [14]. Subjects were tested on six
types of classification problems of increasing complexity. 8
distinct objects had two kinds of shape, two colors and two
sizes. In type I problems only a single feature was relevant,
for example category A included all squared-shaped objects
and category B all triangle shaped objects. In type II prob-
lems two features were relevant for categorization, for ex-
ample shape and color, but not size of the objects. The logic
behind category assignment could be AND, OR, XOR. Type
II-VI problems involve all three features with various logic
behind the assignment.
Since the details of neurodynamics are not important to
understand such categorization experiments, it should be
sufficient to investigate canonical form of simplified neu-
rodynamics. One may claim that any neural dynamics re-
sponsible for categorization in problems with two relevant
features is in principle reducible to one of the simplified dy-
namical systems defined in the 3-dimensional psychologi-
cal spaces (two features plus the third dimension labeling
categories). Parameters defining such simplified dynamics
should allow to reproduce observed behavior. Prototype dy-
namics for all logical functions used in categorization exper-
iments has been found. For example, Type II problems are
solved by the following prototype dynamical system:
V(x,y,z) = Zxyz + - (x2 + y2 + Z2)
∂V ■> / 2 , 2 , 2\
ж = -—— = -Zyz-Ix + у +z)x
OX v '
∂V / 2 , 2 , 2\
У = ~~ду = ~3xz ~ ∖x +y +z)y
∂V / 2 . 2 . 2\
г = -— = -Zxy - [х + у +z)z
This system has 5 attractors (0,0,0), (-1,-1,-1), (1,1,-1); (-
1,1,1), (1,-1,1); the first attractor is of the saddle point type
and defines a separatrix for the basins of the other four. Such
dynamical system may be realized by different neural net-
works. In this example, as well as in the remaining five
types of classification problems of Shepard et.al. [14], it is
easy to follow the path from neural dynamics to the behavior
of experimental subjects during classification task. Starting
from examples of patterns serving as point attractors itis al-
ways possible to construct a formal dynamics and realize it
in the form of a set of frequency locking nonlinear oscilla-
tors [15].
Although polynomial form of canonical dynamical sys-
tem is for the XOR case very simple and has only one saddle
point for other useful functions it is more complex. Mod-
eling point attractors using functions G{Xi, si} localized
around the K attractors, leads to the following equations:
к
V(X) = ∑WiG(Xi,si)
z = l
Y. - -ɪ
l ∂Xi
This form allows us to model the potential by changing
the positions and fuzziness (controlled by si parameters)
of the attractors and their relative weights Wi. Functions
G(Xj, Sj) may either be Gaussianor, if neural plausibilityis
required, a sum of combination of pairs of sigmoidal func-
tions ∑j(σ(Xj- + Sj∙) - σ(Xj∙ - si)) filtered through another
sigmoid. Using this form of the potential one may create
basins of attractors with desired properties and set up the pa-
rameters of these functions to account for experimental data.
People learn relative frequencies (base rates) of cate-
gories and use this knowledge for classification. The in-
verse base rate effect [16] shows that in some cases predic-
tions contrary to the base rates are made. This effect may be