From: Eugene Leitl (Eugene.Leitl@lrz.uni-muenchen.de)
Date: Sat Feb 01 1997 - 16:19:33 MST
[ Notice: I haven't checked the formulas, which appear to be minusculy
garbled, moreover phi and psi angle symbols have been used
inconsistantly. However, these points do not at all affect his
course of argumentation -- 'gene ]
3. Sensorimotor coordination
Let me begin by suggesting that vertically connected laminar structures
are one of evolution's simplest solutions to a crucial type of problem,
one that any sensorimotor system beyond the most rudimentary must
somehow solve. In order to appreciate this type of problem, let us
consider a schematic creature of a deliberately contrived simplicity.
[ sketch of a single-armed crablet omitted ]
# a,b
\
\ # food
| \ * eyes (triangulate food,
| \ \phi angles alpha, beta)
| \__/ x_e, y_e
| \/
| / <-- arm, two segments, each 7 units long
| /
| /
| / \theta
\alpha | / \ \beta
________*__________|/___|_____*______________
0,0
left eye right eye
arm base
Figure 2
Figure 2b is a plan view of a crab-like schematic creature (2a) with
two rotatable eyes and an extendable arm. If this equipment is to be
useful to the crab, the crab must embody some functional relationship
between its eye-angle pairs when an edible object is triangulated, and
its subsequent shoulder and elbow angles, so that the arm can assume a
position that makes contact with the edible target. Crudely, it must be
able to to grasp what it sees, wherever the seen object lies.
We can characterize the required arm/eye relationship as follows. First
of all, let us represent the input (the pair of eye angles) by a point
in a two-dimensional sensory-system coordinate space or state space
(figure 3a). The output (the pair of arm angles) can also be represented
by an appropriate point in a separate two-dimensional motor state space
(figure 3a).
180 180 forearm angle \phi
| right eye angle | |
| \beta | |
| | |
| | |
|...........* | |
| . | |
| . | |
| . | |.....*
| . | | .
| . | | .
| . | | .
|____________________________ 0|_________|__________________
0 180 -90 0 180
left eye angle \alpha upper arm angle \theta
sensory state space motor state space
We now need a function ot take us from any point in the sensory state
space to a suitable point in the motor state space, a function that will
coordinate arm-position with eye-position in the manner described. (I here
sketch the deduction of the relevant functions so that its origin will not
be a mystery, but the reader may leap past the algebra without any loss of
comprehension. The only point to remember is that we are deducing a suitable
function to take us from eye configurations to arm configurations.)
The two eye angles (a,b) determine two lines that intersect at the seen
object. The coordinates (a,b) of that point (in real) space are given by
a = -4(tan \alpha + tan \beta)/(tan \alpha - tan \beta)
b = -8(tan \alpha * tan \beta)/(tan \alpha - tan \beta)
The tip of the arm must make contact with this point. Assuming that both
the forearm and upper arm have a fixed length of seven units, the elbow
will therefore have to lie at the intersection of two circles of radius
seven units: one centered at (a,b), and the other centered at (0,0) where
the upper arm projects from the crab's body. Solving for the relevant
intersection, the real-space elbow coordinates (x_e, y_e) are given by
x_e = ((49-((a^2+b^2)^2/4b^2)*(1-((a^2/b^2)/
((a^2/b^2)+?))))^0.5+(((a/b)*((a^2+b^2)/2b))/
((a^2/b^2)+1)^0.5))/((a^2/b^2)+1)^0.5
y_e = (49-x_e^2)^0.5
The three points in real space, (a,b), (x_e,y_e), (0,0), determine the
position of the arm, whose upper arm and forearm angles (theta, psi) are
finally given by
theta = tan^-1(y_e/x_e)
psi = 180-(\theta-tan^-1((b-y_e)/(a-x_e)))
These are the desired coordinates of the arm in motor state space. The
reader will note that the assembled functions that yield them are rather
tangled ones.
Tangled or not, the crab is drawn on the computer screen, such that its
final arm-position (drawn by the computer as output) is the specified
function of its eye-positions (entered by us as input), it it constitutes
a very effective and well-behaved sensorimotor system, especially if we
write the controlling program as follows.
Let the program hold the crab's arm folded against its chest (at \theta=0,
\psi=180), until some suitable stimulus registers on the fovea of
both eyes. The arm is then moved from its initial state-space position
(0 deg, 180 deg), along a straight line in motor statet space, to its
computed target position in motor state space. This is the state-space
position at which in real space, the tip of the arm contacts the
triangulation point of the eyes. This arrangement produces a superficially
realistic system that reaches unerringly for whatever it sees, anywhere
within reach of its arm (figure 4a-c). [ several multiple-frame computer
renderings of the crab reaching for target omitted ]
The algebraic presentation of the crab's sensorimotor transformation,
as represented in the six equations listed earlier, supplies no intuitive
conception of its overall nature. A geometrical presentation is much more
revealing. Let us therefore consider the projection of the active portion
of the crab's sensory state space (figure 5a) into the orthogonal grid of
its motor state space (figure 5b), as imposed by the function under
discussion. That is to say, for _every_ point in the displayed sensory grid,
we have plotted the corresponding arm-position within the motor grid.
Here we can see at a glance the distortion of the vertical and the
horizontal lines of the sensory space, as projected into motor space. The
topological features of the sensory space are preserved, but its metrical
properties are not. What we see is a systematic transformation of
coordinates. (The heavy scored triangle and rectangle are drawn in solely
to help the reader locate corresponding positions in the deformed and the
undeformed grids. Note also that the left border or \beta-axis of figure
5a shrinks to the lef radial point in figure 5b, and that the top border
of figure 5a shrinks to the right radial point in figure 5b.)
4 Coordinate tranformation: its physical implementation.
The tranformation described above sustains effective and realistic
sensorimotor behaviour. But how could a real nervous system possibly
compute such a complex coordinate tranformation? It is not realistic
to expect it to compute this complex trigonometric function step by
step, as our computer simulation does. Nevertheless, given their
sophisticated sensorimotor coordination, biological systems somehow
must be computing transformations like these, and others more complex
still. How might they do it?
Figure 5 suggests a surprisingly simple means. If we suppose that the
crab contains internal representations of both its sensory state space,
and its motor state space, then the following arrangement will effect
the desired tranformation. Let the crab's sensory state space be
represented by a physical grid of signal-carrying fibres, a grid that
is metrically deformed in real space in just the way displayed in figure
5b. Let its motor state space be represented by a second grid of fibres,
in undeformed orthogonal array. Position the first grid over the second,
and let them be connected by a large number of short vertical fibres,
extending from coordinate intersections in the sensory grid down to the
nearest coordinate in the underlying motor grid, as in figure 6.
Suppose that the fibres of the sensory grid receive input from the eyes'
proprioceptive system, such that the position of each eye simulates a
unique fibre in the upper (deformed) grid. The left eye activates one
fibre from the right radial point, and the right eye activates one from
the left. Joint eye-position will thus be represented by a simultaneous
stimulation at the appropriate coordinate intersection in the upper grid.
Underneath this point in the upper map lies a unique intersection in
the motor grid. Suppose that this intersecting pair of orthogonal motor
fibres, when jointly activated, induces the arm to assume the position that
is appropriate to the specific motor coordinate intersection where this
motor signal originates in the lower map.
Happily, the relative metrical deformations in the maps have placed
in correspondence the appropriate points in the upper and lower maps.
We need now suppose only that the vertical connections between the sensory
grid and the motor grid function as "and-gates" or "threshold switches,"
so that a signal is sent down the vertical connection to the motor grid
exactly if the relevant sensory intersection point is simultaneously
stimulated by both of its intersecting sensory fibres. Such a system will
compute the desired coordinate transformations to a degree of accuracy
limited only by the grain of the two grids, and by the density of their
vertical connections. I call such a system a _state-space sandwich_.
Three points are worth noting immediately about the functional properties
of such an arrangement. First, it will remain partially functional despite
localized damage. A small lesion in either grid will produce only a partial
dyskinesia (two permanent "shadows" of fibre inactivity downstream from
the lesion), for which a shift of bodily position will usually compensate
(by bringing the target's state-space position out of the shadow).
Indeed, if the position of the crab's eyes is coded not by the activation
of a single point in the upper grid, but rather by the activation of a
large area (set of points) centered around the 'correct' point, and if the
arm muscles respond by averaging the distributed output signal from the
now multiply-stimulated lower map, than an appropriate motor response will
be forthcoming even if the sandwich has suffered the scattered loss of a
great many cells. Such a system will be functionally persistent despite
widespread cell damage. The quality of the sensorimotor coordination will
be progressively degraded under cell damage, but a roughly appropriate
motor response will still be forthcoming.
Second, the system will be very, very fast, even with fibres of biological
conduction velocities (10 < \nu < 100 m/s). In a creature the size of a crab,
in which the total conduction path is less than 10 cm, this system will yield
a motor response in well under 10 milliseconds. In the crab-simulation
described earlier, my computer (doing its trigonometry within the software)
takes 20 times that interval to produce a motor response on-screen, and its
conduction velocities are on the order of the speed of light. Evidently,
the massively parallel architecture of the state-space sadwich buys a large
advantage in speed, even with vastly slower components.
And third, the quality of of the crab's coordination will not be uniform
over its field of motor activity, since in the maximally deformed areas
of the sensory grid, small errors in sensory registration produce large
errors in the motor respose (see again figure 5b). Accordingly, the crab
is least well coordinated in the area close between its eyes, and to its
extreme right and left.
All three of these functional properties are biologically realistic. And
the sandwich appears biologically realistic in one further respect: it is
relatively easy to imagine such a system being grown. Distinct layers can
confine distinct chemical gradients, and can thus guide distinct
morphogenetic processes. Accordingly, distinct topographical maps can
appear in closely adjacent layers. But given that the maps are so closely
contiguous, and assuming that they are appropriately deformed, the
problem of connecting them up so as to produce a functional system
becomes a trivial one: the solution is just to grow conductive elements
that are roughly orthogonal to the layers.
Different creatures will have different means of locating objects, and
different motor systems to effect contact with them, but all of them will
face the same problem of coordinating positions in their sensory state space
with positions in their motor state space, and the style of solution here
outlined is evidently quite general in nature. The point to be emphasized
is that a state-space sandwich constitutes a simple and biologically
realistic means for effecting any two-dimensional ot two-dimensional
coordinate transformation, whatever its mathematical complexity, and
whatever features -- external or internal, abstract or concrete -- that
the coordinate axes may represent to the brain. If the transformation can
be graphed at all, a sandwich can compute it. The sensorimotor problem
solved above is merely a transparent example of the general technique at
work.
Switching now from functional to structural considerations, I hope it is
apparent that, beyond the issue of functional realism, the system of
interconnected maps in figure 6 is suggestively similiar to the known
physical structure of typical laminar cortex, including the many topographic
maps distributed across the cerebral surface. In all of these areas, inputs
address a given layer of cells, which layer frequently embodies a metrically
deformed topographic map of something-or-other. And outputs leave the area
from a different layer, with which the first layer enjoys massiver vertical
connections.
I therefore propose the hypothesis that the scattered maps within the
cerebral cortex, and many subcerebral laminar structures as well, are all
engaged in the coordinate transformation of points in one neural state
space into points in another, by the direct interaction of metrically
deformed, vertically connected topographic maps. Their mode of
representation is state-space position; their mode of computation is
coordinate transformation; and both functions are simultaneously
implemented in a state-space sandwich.
I can cite not a single cerebral area for which this functional hypothesis
is known to be true. To decide the issue would require knowing in some detail
both the topical and te metrical character of the topographic maps in each
of the contiguous laminae that constitute a given area of cerebral cortex.
In general, we still lack such information. However, there is a
phylogenetically older and simpler laminar structure located on the
dorsal midbrain whose upper and lower maps have both been decoded, and it
does display both the structural and the functional patter portrayed in
figure 6.
The superior colliculus (figure 7) sustains the familiar reflex whereby the
eye makes an involuntary saccade so as to foveate or to look directly at
any sudded change or movement that registers on the retina away from its
central high-resolution area of fovea. e ahve all had the experience of
being in a darkened movie theater when someone down in the front row left
suddenly ignites a match or lighter to light a cigarette. Every eye in the
house makes a ballistic saccade to fixate this brief stimulus, before
returning to the screen. This reflex is the work of the superiour colliculus.
Appropriately enough, this is sometimes called the 'visual grasp reflex.'
[ to be continued. ciao, 'gene]
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