Reduction and Elimination in Philosophy and the Sciences

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From Topology to Logic. The Neural Reduction of Compositional Representation — Markus Werning
Figure 1: Cortical realizations of attributes. a) Fragment (ca. 4mm 2 )
of the neural feature map for the attribute orientation of cat V1
(adapted from Crair et. al., 1997). The arrows indicate the polar
topology of the orientation values represented within each hypercolumn.
Hypercolumns are arranged in a retinotopic topology. b)
Color band (ca. 1 mm 2 ) from the thin stripes of macaque V2
(adapted from Xiao et. al., 2003). The values of the attribute color
are arranged in a topology that follows the similarity of hue as defined
by the Commission Internationale de l’Eclairages (xycromaticity).
The topology among the various color bands of V2 is
retinotopic.
The fact that values of different attributes may be
instantiated by the same object, but are processed in
distinct regions of cortex poses the problem of how this
information is integrated in an object-specific way: the
binding problem. How can it be that the color and the
orientation of an object are represented in distinct regions
of cortex, but still are part of the representation of one and
the same object? A prominent and experimentally well
supported solution postulates oscillatory neural
synchronization as a mechanism of binding: Clusters of
neurons that are indicative for different attribute values
sometimes show synchronous oscillatory activity, but only
when the values indicated are instantiated by the same
object in the perceptual field; otherwise they are firing
asynchronously. Synchronous oscillation, thus, might be
regarded to fulfill the task of binding together various
representations of attibute values to form the
representation of an object with these values (Singer,
1999, for review). Using oscillatory networks as biologically
motivated models, it can be demonstrated how the
topological organization of information in the cortex, by
mechanisms of synchronization, may yield a logically
structured semantics of concepts (Maye & Werning, 2004).
Oscillation functions play the role of object concepts.
Clusters of feature sensitive neurons play the role of
attributive concepts – or predicates.
Oscillatory Networks
From Gestalt psychology the principles governing object
concepts are well known. According to some of the Gestalt
principles, spatially proximal elements with similar attribute
values (e.g., similar color/similar orientation) are likely to
be perceived as one object or, in other words, represented
by one and the same object concept. The Gestalt principles
are implemented in oscillatory networks by the following
mechanism: Oscillators that select input from proximal
stimulus elements with like attribute values tend to synchronize,
while oscillators that select input from proximal
stimulus elements with unlike values (e.g., red and green
for color or horizontal and vertical for orientiation) tend to
desynchronize. As a consequence, oscillators selective for
proximal stimulus elements with like values tend to exhibit
synchronous oscillation functions when stimulated simultaneously.
The oscillation in question can be regarded as
one object concept. In contrast, inputs that contain proximal
elements with unlike values tend to cause antisynchronous
oscillations, i.e., different object concepts.
In our model (Fig. 2) a single oscillator – marked as
a cubicle – renders the statistical electrical discharge behavior
of 100 to 200 biological cells and codes for an attibute
value (z-coordinate) for a stimulus in the relevant region
of the receptive field (x,y-coordinates). Differential
equations describe the dynamics of the i-th oscillator as
the temporal evolution of a variable xi(t). The oscillators for
an attribute are arranged on a three-dimensional grid forming
a module. Two dimensions represent the spatial domain,
while the attribute values are encoded by the third
dimension. Thus the twofold topology of biological feature
maps is reflected in the network architecture. Spatially
close oscillators that represent similar values synchronize.
The desynchronizing connections establish a phase lag
between different groups of synchronously oscillating clusters.
Modules for different attributes can be combined by
establishing synchronizing connections between oscillators
of different modules in case they code for the same stimulus
region
Stimulated oscillatory networks (e.g., by stimulus of
Fig. 3a), characteristically, show object-specific patterns of
synchronized and de-synchronized oscillators within and
across modules. Oscillators that represent attributes of the
same object synchronize, while oscillators that represent
attributes of different objects de-synchronize. We observe
that for each represented object a certain oscillation
spreads through the network. The oscillation pertains only
to oscillators that represent the attributes of the object in
question.
Semantic Interpretation
An oscillation function x(t) of an oscillator is its excitatory
activity as a function of time during a time window [0,T].
Mathematically speaking, activity functions can be conceived
of as vectors in the Hilbert space L2[0,T] of functions
that are square-integrable in the interval [0,T]. Thus,
a precise measure of synchrony can be established and a
powerful algebraic framework for the semantic interpretation
of the network is provided. The degree of synchrony
between two oscillations lies between −1 and +1 and can
be defined as the their normalized inner product
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