From: Eugene Leitl (Eugene.Leitl@lrz.uni-muenchen.de)
Date: Sun Feb 02 1997 - 09:31:44 MST
subject: Paul M. Churchland, cot'd (III)
[ Figure 7 legend:
a) Projections drawing of Nissl-stained cross-section of cat superiour
colliculus illustrating laminar organization. Dots correspond to
collicular neurons. From Kanasecki and Sprague (1974). Reprinted
with permission.
b) Retinotopic map: a metrically deformed topographic map of the
visual hemifield, in rectangular coordinates, on the superficial
layer of the right colliculus of the cat. M = medial; L = lateral;
A = anterior; P = posterior. Adapted from Schiller (1984).]
In humans and the higher mammals the superior colliculus is a
visual center secondary to the more important striate cortex
(areas 17 and 18 on the Brodmann map) located at the rear of the
cerebral hemispheres, but in lower animals such as the frog and
snake, which lack any significant cortex, the superior colliculus
(or optic tectum, as it is called in them) is their principal
visual center. It is an important center even for mammals, however,
and it works roughly as follows.
The top-most layer of the superior colliculus (hereafter, SC)
receives projections directly from the retina, and it constitutes
a metrically-deformed topographic map of the retinal surface (figure
7b) [...]. Vertical elements connect this layer to the deepest layer
of the SC. These vertical connections appear to consist of a chain
of two or three short interneurons descending stepwise through two
intervening layers [...], of which more later. Also, the dendrites
of some of the deep-layer neurons appear to ascend directly into
the visual layer, to make synaptic connections with visual cells
[...]. The neurons of the deepest layer project their output axons
via two distinct nervous pathways, one of which leads eventually to
the pair of extra-ocular muscles responsible for horizontal eye
movements [...].
Intriguingly, this underlying motor layer also embodies a topographic
map, a map of state space that represents changes in the contractile
position of the ocular muscles, the muscles that make the eye jump
[...]. Microstimulation by an electrode at any given point in this
deepest layer causes the eyes to execute a saccade of a size and
direction characterisitic for that point, a saccade which moves
the eye's fovea into the position originally occupied by that retinal
cells which projects to the immediately overlying cell in the
top-most layer of the colliculus [...]. In other words, the relative
metrical deformations in the two maps have placed into correspondence
the appropriate points in the upper and lower maps. (This means
that the "deformation" seen in figure 7b should not of itself
be taken as evidence for the state-space sandwich hypothesis. What
is crucial is the deformation of the maps relative to each other.)
Finally, any sufficiently strong retinally produced stimulation
in that top-most visual map is conveyed downwards to the motor map
by the appropriate vertical elements, where it produces a saccade
of just the size and direction appropriate for the foveation of the
external stimulus that provoked it. The SC thus appears to be an
instance of both the structural and the functional pattern displayed
in figure 6. It foveates on changing or moving visual targets by
essentially the same means whereby the schematic cortex of the crab
reaches out for triangulated objects.
A word of caution is in order here, since the account just offered
does not do justice to the full complexity of the superior
colliculus. In mammals, especially the higher mammals, the SC is
a tightly integrated part of a larger modulating system that
includes inputs from the visual cortex and the frontal eye fields,
and output to the neck muscles. The functional properties of the
entire system are more varied and more subtle than the preceding
suggests, and the job of sorting them out is still underway [...].
The preceding is submitted as an account of the central or more
primitive functions of the SC, at best.
With these examples in mind -- the crab's "cortex," and the
superior colliculus -- it is appropriate to focus on the many
other topographically-organized multilayered cortical areas
scattered throughout the brain, and ask what coordinate
transformation they might be effecting. Here it is very important
to appreciate that the topographic maps we seek to decode need
not be, and generally will not be, maps of something anatomically
obvious, such as the surface of the retina, or the surface of
the skin. More often they will be maps of some abstract state
space, whose dimensional significance is likely to be opaque
to the casual observer, though of great functional importance to
the brain. Two pretty examples of such abstract maps are the
map of the echo delays in the bat's auditory cortex, and the
map of binaural disparities in the owl's inferior colliculus
[...].
All of this suggests that the brain may boast many more topographic
maps that have so far been identified, or even suspected.
Certainly the brain has a teeming abundance of topographically
organized areas, and recent work has expanded the number of
known sensory-related maps considerably [...]. All of this further
suggests that we will make better progress in trying to understand
the significance of the many topographically-organized cortical
areas when we approach them as maps of abstract but functionally
relevant state spaces.
5 Cortex with more than two layers
While we are discussing the biological reality of the laminar
mechanism proposed, consider the objection that our modl cortex
has only two layers, whereas the typical human cortex has six
layers, and, counting fine subdivisions, perhaps eight or nine
in some areas. What are they for?
There is no difficulty in perceiving a function for such additional
layers. Let us return to again to the superior colliculus, which
illustrates one of many possibilities here. Between the visual and
the motor maps of the SC there are, in some creatures, onee or
two intermediate layers (see again figure 7). These appear to
constitute an auditory map and/or a somatosensory map (a facial or
whisker map), whose function is again to orient the eye's fovea,
this time toward the source of sudden auditory and/or somatosensory
stimulation [...]. Not surprisingly, these intervening maps are
each metrically deformed in such a fashion as to be in rough
coordinate "register" with the motor map, and hence with each other.
Altogether, this elegant three- or four-layer topographic sandwich
constitutes a multimodal sensorimotor coordinate transformer.
Multilayered structures have further virtues. It is plain that
maps of several distinct modalities, suitably deformed and placed
in collective register within a "club sandwich," provide a most
effective means of cross-modal integration and comparison. In the
SC, for example, this multimodal arrangement is appropriate in the
production of a motor response to the joint receipt of faint but
spatiotemporally coincident auditory and visual stimuli, stimuli
which, in isolation, would have been subthreshold for a motor
response. For example, a faint sound from a certain compass point
may be too faint to prompt the eyes into a foveating saccade, and
a tiny movement from a certain compass point may be similiarly
impotent; but if both the sound and movement come from the same
compass point (and are thus coded in the SC along the same vertical
axis), then their simultaneouos conjunction will indeed be sufficient
to make the motor layer direct the eyes appropriately. This
prediction is strongly corroborated by the recent results of Meredith
and Stein (1095).
Further exploration reveals that multilayered sandwiches can subserve
decidedly sophisticated cognitive functions. In an earlier publication
[...], I have shown how a three-layer state-space sandwich can code,
and project, the path of a _moving_ object in such a fashion as to
position the crab's arm to catch the moving target on the fly.
Evidently, a multilayered cortex can offer considerable advantages.
6 Beyond state-space sandwiches
The examples studied above are uniform in having an input state
space of only two dimensions, and an output state space of only
two dimensions. It is because of this fact that the required
coordinate transformation can be achieved by a contiguous pair
of sheet like maps. But what of cases where the subsystems involved
each have more than two parameters? What of cases where the
coordinate transformations are from an input spac of n to an
output space of m dimensions, where n and m are different, and
both greater than 2? Consider, for example, the problem of
coordinating the joint angles of a limb with three or more
joints, and the problem of coordinating several such limbs with
each other. Or consider the problem of coordinating the even
larger number of muscles that collectively control such limbs.
As soon as one examines the problems routinely faced, and solved,
by real creatures, one appreciates that many of them are far
more complex than can be represented by simple two-dimensions
to two-dimensions transformation.
Perhaps some of these more complex problems might be solved
by dividing them into a set of smaller ones, problems that
can be managed after all by a set of distinct two-dimensional
state-space sandwiches, each addressing some slice or
aspect of the larger problem [...]. The predominance of
laminar cortex in the brain certainly encourages speculation
along these lines. But such solutions, even approximate ones,
cannot in general be guaranteed. The brain badly needs some
mechanism beyond the state-space sandwich if it is routinely
to handle these higher dimensional problems.
Andras Pellionisz and Rodolfo Llinas [...] have already
outlined a mechanism adequate to the task, and have found
impressive evidence of its implementation within the
cerebellum. The cerebellum is the large structure at the
rear of the brain, just underneath the cerebral hemispheres.
Its principal function, divined initially from lesion studies,
is the coordination of complex bodily movements, such as
would be displayed in in preparing a dinner or in playing
basketball. It displays a neural organization quite different
from that of the cerebral hemispheres, an organization whose
significance may be rendered transparent by the Pellionicz/
Llinas account.
To illustrate this more general mechanism for coordinate
transformation, let us consider an input system of four dimensions
whose inputs a, b, c, d, are transformed into the values x, y, z
of a three-dimensional output system. As before, the outputs
and inputs are can each be regarded as points in a suitable
state space. Since they are n-tuples, each can also be regarded
as a vector (whose base lies at the origin of the relevant
state space, and whose arrowhead lies at the point specified
by the n-tuple).
| p_1 q_1 r_1 |
| p_2 q_2 r_2 |
<a,b,c,d> * | p_3 q_3 r_3 | = <x,y,z>
| p_4 q_4 r_4 |
Figure 8
parallel fibre
input
--p1--------q1--------r1------------------< a
--|-p2------|-q2------|-r2----------------< b
--|-|-p3----|-|-q3----|-|-r3--------------< c
--|-|-|-p4--|-|-|-q4--|-|-|-r4------------< d
| | | | | | | | | | | |
| | | | | | | | | | | |
+-+-+-+ +-+-+-+ +-+-+-+
\ / \ / \ / Purkinje cell
\_/ \_/ \_/ bodies
| | |
x y z
Purkinje cell output
Figure 9
A standard mathematical operation for the systematic transformation
of vectors into vectors is matrix multiplication. Here it is the
matrix that embodies or effects the desired coordinate transformation.
To see how this works, consider the matrix of figur 8, which has four
rows and three columns. To multiply the input vector <a,b,c,d> by this
matrix we multiply a times p_1, b times p_2, c times p_3, d times p_4,
and then sum the four results to yield x. We then repeat the process
with the second column to yield y, and again with the third column to
yield z. Thus results the output vectors <x,y,z>. [of course matrixmult
is but a narrow example of a generic mapping, which can be written as
a table -- 'gene ]
This algebraic operation can be physically realized quite simply by
the neural array of figure 9. The parallel input fibres at the right
each send a train of electrochemical "spikes" towards the waiting
dendritic trees. The numbers a, b, c, d represent the amount by which
the momentary spiking frequency of each of the four fibres is above
[ Figure 10 Schematic section: cerebellum is omitted due to ascii art
limitations. You see lots of treelike Purkinje cells (sprouting
synapses on tree branches) reaching into a thick slab of parallel
fibres. Some granular cells (below the Purkinjes) generate some
of the input of the parallel fibres. (Well, a picture _is_ worth
a 1 kWord ]
(positive number) or below (negative number) a certain baseline
spiking frequency. The top-most input fibre, for example, synapses
into each of the three output cells, making a stimulating connection
in each case, one that tends to depolarize the cell body and make
it send a spike down its vertical output axon. The output frequency
of spike emissions for each cell is determined first by the simple
_frequency_ of the input stimulations it receives from all incoming
synaptic connections, and second, by the _weight_ or strength of
each synaptic connection, which is determined by the placement of
the synapses and by their cross-sectional areas. These strength
values are individually represented by the coefficients of the
matrix of figure 8. The neural interconnectivity thus implements
the matrix. Each of the three cells of figure 9 "sums" the
stimulation it receives, and emits an appropriate train of spikes
down its output axon. Those three output frequences differ from
the background or baseline frequencies of the three output cells
by positive or negative amounts, and these amounts correspond to
the output vector <x,y,z>.
Note that with state-space sandwiches, the coding of information
is a matter of the spatial location of neural events. By contrast,
with the matrix-multiplication style of computation under
discussion, input and output variables are coded by sets of spiking
frequencies in the relevant pathways. The former system uses
"spatial coding"; the latter system used "frequency coding." But
both systems are engaged in the coordinate transformation of
state-space positions.
The example of figure 9 concerns a tree-by-four matrix. But it is
evident that neither the mathematical operation nor its physical
realization suffers any dimensional limitations. In principle,
a Pellionisz/Llinas connectivity matrix can effect transformations
on state spaces of a dimensionality into thousands and beyond.
The schematic archetecture of figure 9 corresponds very closely
to the style of microorganization found in the cerebellum
(figure 10). [...]. The horizontal fibres are called parallel fibres,
and they input from the higher motor centres. The bushy vertical
cells are called Purkinje cells, and they output through the
cerebellar nucleus to the motor periphery. In fact, it was from
the observation of the cerebellum's beautifully regular architecture,
and from the attempt to recreate its functional properties by
modelling its large-scale physical connectivity within a computer,
that Pellionisz and Llinas were originally led to the view that
the cerebellum's job is the systematic transformation of vectors
in one neural hyperspace into vectors in another neural hyperspace
[...].
Given that view of the problem, the tensor calculus emerges as
the natural framework with which to address such matters, especially
since we cannot expect the brain to limit itself to Cartesian
coordinates. In the examples discussed so far, variation in
position along any axis of the relevant state space is independent
of variation along any of the other axes, but this independence
will not characterize state spaces with nonorthogonal axes. Indeed,
the generalization of the approach, to include non-Cartesian
hyperspaces, is regarded by Pellionicz and Llinas as one of the
most important features of their account, a feature that is
essential to understanding all but the simplest coordination
problem. I cannot pursue this feature here.
Four final points about the neural matrix of figure 9. First, it
need not be limited to computing linear transformations. The
individual synaptic connections might represent any of a broad
range of functional properties. They do not need be simple
multipliers. In concert then, they are capable of computing
a large variety of non-linear transformations. Second, a neural
matrix will have the same extraordinarily speed displayed by
a state-space sandwich. And third, given a great many components,
such matrices will also display a fierce functional persistence
despite the scattered loss of their cellular components.
Finaally, such systems can be plastic in the transformations they
effect: changes in the weights and/or numbers of the synaptic
connections are all that is required. What is problematic is how
a useful set of weights, i.e. a set that effects a transformation
useful to the organism, gets established in the first place. How
does the matrix 'learn' ot implement the right transformation?
The problem is certainly solvable in principle. The 'back-
propagation' learning algorithm of Rumelhart, Hinton and Williams
(1986) has shown that distributed processors of this general
type can learn with extraordinary efficiency, given a suitable
regime for propagating discovered error back through the
elements of the system. And it may be solvable in fact, since
the cerebellar network does contain a second input system: the
climbing fibres (not shown in figures 9 and 10 for reasons of
clarity). These ascend from below each of the Purkinje cell
and wrap themselves like vines around the branches of its
dendritic tree. They are thus in a position to have a direct
effect on the nature of the synapses made by the parallel
fibres onto the Purkinje dendrites. According to a recent model
by Pellionicz and Llinas (1985), the cerebellum can indeed
become 'coordinated' by essentially this method.
These brief remarks do not do justice to the very extensive work
of Pellionicz and Llinas, nor have I explored any criticism.
[...]. The reader must turn to the literature for deeper
instruction. The principal lesson of this section is that the
general functional schema being advanced here -- the schema
of representation by state-space position, and computation
by coordinate transformation -- does not encounter
implementational difficulties when te representational and
computational task exceeds the case of two dimensions. On the
contrary, the brain boasts neural machinery that is ideally
suited to cases of very high dimensionality. We have then
at least two known brain mechanisms for performing coordinate
transformations: the state-space sandwich specifically for the
two-dimensional cases, and the neural matrix for cases of
any dimensionality whatsoever.
[ to be continued -- 'gene ]
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