7 The representational power of state spaces
Discussion so far has been concentrated on the impressive
computational power of coordinate transformations of state spaces,
and on the possible neural implementation of such activity. But it
is important to appreciate fully the equally powerful representational
capacity of neural state spaces. A global circumstance comprised of n
distinct varables can be economically represented by a single point in
an abstract n-dimensional state space. And such a state-space point
can be neurally implemented, in the simplest case, by a specific
distribution of n spiking frequencies in a system of only n distinct
fibres. Moreover, a state-space representation embodies the metrical
relations between distinct possible positions within it, and thus
embodies the representation of _similiarity_ relations between
distinct items thus represented. These claims can be illustrated by
the outputs of several of our sensory organs.
Consider first a simple case: the human gustatory system has a four-
channel output, with each of the four pathways representing the level
of stimulation at one of the four sets of taste sensors in the mouth,
the so-called sweet, sour, salty, and bitter receptors. Any humanly
possible taste sensation, it is therefore conjectured, is a point
somewhere within a four-dimensional gustatory state space. Or more
literally, it is a quadruple of spiking frequencies in the four
proprietary pathways carrying information from the gustatory
receptors for distribution to the rest of the brain. [ wish I had
a GC/MS combo for input instead, also takes care of the olfaction
-- 'gene]
A sweet taste, for example, is coded as a specific set of activity
levels across the four pathways (a high value on one, and a low
value on each on the other three). Other combinations of activity
levels yield all of the taste sensations possible for humans. The
qualitative or subjective similiarity of sensations emerges, both
theoretically and experimentally, as the proximity of the relevant
codings within gustatory state-space.
Such a coding system also gives us an enormous range of
discrimination at a very low price. Just to illustrate the point,
suppose our discrimination of distinct positions (activity levels)
along each of the four axes of gustatory state space is limited
to just ten positions. This gives us an overall four-dimensional
space with fully 10^4 discriminable points. This state-space
approach to gustatory sensations appears in the neuroscience
literature as the _across-fibre pattern_ theory [...].
An account of this same general kind may old four our olfactory
system, which has six or more distinct types of receptor. A six-
dimensional space, at 10-unit axial discrimination, will permit
the discrimination of 10^6 odors. And if we imagine only a seven-
dimensional olfactory space, with only three times the human
axial discrimination, which space a dog almost certainly possesses,
then we are contemplating a state space with 30^7, or 22 billion,
discriminable positions! Given this, the canines' ability to
distinguish, by smell, any one of the 3.5 billion people on the
planet no longer presents itself as a mystery.
Consider the human 'module' for facial recognition. We apparently
have one, since the specific ability to recognize faces can be
destroyed by specific right parietal lesions. Here it is plausible
to suggest an internal state-space representation of perhaps
twenty dimensions, each coding some salient facial feature such
as nose length, facial width, and so on. (Police 'Identikits'
attempt to exploit such a system, with some success.) Even if
discrimination along each axis were limited to only five distinct
positions such a high-dimensional space would still have an enormous
volume (=5^20 positions), and it would permit the discrimination
and recognition of billions of distinct faces. It would also
embody similiarity relations, so that close relatives could be
successfully grouped, and so that the same person could be
reidentified in photos taken at different ages. Consider two photos
of the young and the old Einstein. What makes them similiar?
They occupy approximate positions in one's facial state space.
Finally, let us turn to a motor example, and let us consider
one's "body image": one's continuously updated sense of one's
overall bodily configuration in space. That configuration is
constituted by the simultaneous position and tension of several
hundreds of muscles, and one monitors it all quite successfully,
to judge from the smooth coordination of most of one's movements.
How does one do it? With a high-dimensional state space,
according to the theories of Pellionicz and Llinas, who ascribe
to the cerebellum the job of computing appropriate transformations
among high-dimensional codings of actual and intended motor
circumstances, codings lodged in the input parallel fibers and
the output Purkinje axons.
[Figure 11: a) Step cycle: feline hind leg. b) Skeletal state
space [Roughly, you see an eadweard muybridge of a stylized
cat's hind limb during one full stride cycle, and an according
scimitar-shaped closed trajectory loop in the joint-angle
space spanned by the hip, knee, ankle angles]]
Some of the possibilities here can be evoked by a very simple
example. Consider a highly complex and critically orchestrated
periodic motion, such as occurs in feline locomotor activity
(Figure 11a). Consider now a three-dimensional joint-angle
motor space for the cat's hind limb, a space in which every
possible configuration of that limb is represented by a point,
and every possible movement is represented by a continuous
path. The graceful step cycle of the galloping cat will be
very economically represented by a closed loop in that joint-
angle state space (figure 11b). If the relevant loop is
specified or "marked" in some way, then the awesome task of
coordinated locomotion reduces to a clear-cut tracking problem:
make your motor state-space position follow the path of that
loop. [ now why must I constantly think of robotics when I
read this passage? Inaudible gliding of position feedback
sensors, hum of motors, digitally homeostased by the dedicated
circuitry containing and executing the binary pattern, the
control automaton network -- 'gene]
Once we have taken the step beyond the coginitiv significance
of points in two-dimensional state space to the cognitive
significance of lines and closed loops in n-dimensional state
spaces, it seems possible that we will also find cognitive
significance in surfaces, and hypersurfaces, and intersection
of hypersurfaces, and so forth. What we have opening before us
is a "geometrical," as opposed to a narrowly syntactic, conception
of cognitive activity.
8 Concluding remarks
We have seen how a representation scheme of this kind can account,
in a biologically realistic fashion, for a number of important
features of motor control, sensory discrimination, and sensorimotor
coordination. But has it the resources to account for the so-called
higher cognitive activities, as represented by language use, for
example, and by our propositional knowledge of the world in general?
Conceivably, yes. One might try to find, for example, a way of
representing "anglophone linguistic hyperspace" so that all
grammatical sentences turn out to reside on a proprietary
hypersurface within that hyperspace, with the logical relations
between them reflected as spatial relations of some kind. I do
not know how to do this, of course, but it holds out the
possibility of a alternative to, or potential reduction of,
the familiar Chomskyan picture.
As for the "set of beliefs" that is commonly supposed to constitute
a person's knowledge, it may be that a geometrical representation
of sentences will allow us to solve the severe problem of "tacit
belief" [...]. Just as a hologram does not "contain" a large
number of distinct three-dimensional images, curiously arranged
so as to present a smoothly changing picture of a real object
as the hologram is viewed from different positions, so may humans
not "contain" a large number of distinct beliefs, curiously arranged
so as collectively to present a coherent account of the world.
Perhaps the truth is rather that, in both cases, a specific image
or belief is just an arbitrary projection or "slice" of a deeper set
of data structures, and the collective coherence of such sample
slices is a simple consequence of the manner in which the global
information is stored at the deeper level. It is not a consequence
of, for example, the busywork of some fussy inductive machine
applying inductive rules for the acceptance or rejection of discrete
slices taken singly. Which means that, to understand learning, we
may have to understand the forces that dictate directly the evolution
of the global data structures at the deeper level. The learning
algorithm of Rumelhart, Hinton, and Williams comes to mind again
here, for in a network of that sort no symbols of any kind are being
manipulated. Rather, in the course of a training session a function
is progressively approximated, and the information the network
acquires is stored in nothing more contentful than a distributed
set of synapse-like weights.
These highly speculative remarks illustrate one direction of
research suggested by the theory outlined in this paper: just what
are the abstract representational and computational capacities
of a system of state spaces interacting of coordinate transformations?
Can we use a system of state spaces to articulate models for the
"higher" forms of cognitive activity? The theory also begs research
in the opposite direction, towards the neurophysiology of the brain.
Given that the brain is definitely not a "general purpose" machine
in the way that a digital computer is, it may often turn out that,
once we are primed to see them, the brain's localized computational
tactics can simply be read off its microstructure. There is point,
therefore, to studying that microstructure [...].
Taken jointly, the prodigious representational and computational
capacities of a system of state spaces interacting by coordinate
transformations suggest a powerful and highly general means of
understanding the cognitive activities of the nervous system,
especially since the physical mechanisms appropriate to implement
such a system are widespread throughout the brain.
[ The "[...]"'s mostly denote copious references to the literature.
Should I get several persons' demand for the bibliography, I shall
supplement -- 'gene ]