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Comparison between Karl Pribram's
"Holographic Brain Theory" and more
conventional models of neuronal computation
CHAPTER 3 EUCLIDEAN-BASED GEOMETRIC MODEL CHAPTER 4 HOLONOMIC BRAIN THEORY
One of the problems facing neural science is how to explain
evidence that local lesions in the brain do not selectively impair one
or another memory trace. Note that in a hologram, restrictive damage
does not disrupt the stored information because it has become
distributed. The information has become blurred over the entire extent
of the holographic film, but in a precise fashion that it can be
deblurred by performing the inverse procedure. 2. The more conventional theory that particular features of the
untransformed sensory stimulus is stored in separate places in the
brain. A particular example (in vision) would be that particular
visual cortical cells respond to edges or bar widths in the visual
stimulus. Demonstration #3. Take a pair of binoculars. Just look through one
side focusing at a distant object. Now place your fingers in front of
the lens so that only light coming from in-between your fingers enters
the monocular. You will still see the whole image. If you bring your
fingers together so that the light enters only through tiny slits, the
whole image will still be present, only dimmer (and there will be some
loss of resolution) . If you rotate your hand, exposing the light to
different parts of the lens, the whole image can still be formed. This
is another representation that the light incident at the surface of
the lens at any point is in a holographic form.  Figure 1 The above figure doesn't express the transform-taking aspect of a
lens. The above figure is really more indicative of a pinhole camera.
Figure 4 (shown later in the report) gives a better depiction of what
happens at the lens of the eye. Mathematically (in one implementation), a Fourier transform
converts a function of time f(t) into a function of frequency F(jw)
where the j indicates that it is a complex function of frequency. In
other words, a Fourier transform can convert a signal from the time
domain to the frequency domain. A Fourier transform could also be used
to convert something from a spatial locational domain (the coordinates
in space) to a frequency domain (more about this later). The idea (the mathematics) of the Fourier transform is independent
of what the data sets represent. It will be argued that if the brain
performs a Fourier transform for visual stimuli, then it is possible
that it also performs a Fourier transform for the other senses also
(hearing, taste, smell, touch). The same principle can be shown with optics. Consider, for example,
that a large telescope lens (or mirror) can resolve two distinct
images (for example two stars that only have a small angle separating
them in respect to us). A smaller telescope lens (or mirror) will not
be able to resolve (separate) those two stars. Likewise, small parts
of a hologram, although they have information of the whole, will
suffer some resolution deficit.  Figure 2 Reproduced from Kasper and Feller, The complete book of holograms, 1987, pp 4-5. As seen above in figure 2, the holographic plate records an
interference pattern between the diverged laser light and the
scattered laser light bouncing off the object. The pattern recorded on
the holographic plate is in the holographic domain. All parts of the
holographic plate contain information of the whole. Light bouncing off
each point on the object is distributed (spread out) to every location
on the holographic plate) . Alternatively, the pattern recorded on the
photograph is an image (non-holographic). The image features are
located at particular locations on the photographic plate. Light
scattered off the object (now in the holographic domain) is
transformed to the non-holographic (image) domain by the lens of the
camera (which does an effective inverse Fourier transform) by focusing
the image on the photographic film. For the photograph, there is a
one-to-one mapping between the two-dimensional projection of points on
the object to locations on the photographic plate. Correspondingly,
there is a one-to-all mapping for the holographic plate.  Figure 3 Reproduced from Kasper and Feller, The complete book of holograms, 1987, pp 4-5. In the case of the photograph (see figure 3), light is scattered off the photograph (which is now in the holographic domain) and becomes incident on the eye which does a transformation (focuses) which forms an image on the retina. For the holograph, laser light is shined through the holographic plate (picking up the holographic information from the plate) and becomes incident on the eye which does a transformation (focuses) which forms an image on the retina. The holographic nature of the light incident on the lens is shown in figure 4.  Figure 4 Diagram expressing the holographic nature of light incident on the surface of the lens of the eye. The light scattered from point A is incident at each location of the lens (likewise for the light scattered off of B. The lens functions to transform this holographic domain to an image of A and B at the retina. The discussion so far has just taken us to the image formed at the
retina. The interesting part of the holonomic brain theory is what
happens next. The focal point of the above discussion is that a lens
does an effective (inverse) Fourier transform on the light incident to
it. The Fourier transform (and inverse Fourier transform) consists of
convolution integrals which mathematically smear or de-smear the
information. For continuous functions, the Fourier transform and
inverse Fourier transform are as follows (for transforms between the
time and frequency domain):   The Fourier transform also has meaning between a spatial domain (for instance the position in two dimensional space) and spatial frequency. Mathematically, the two-dimensional spatial Fourier transform is  and the inverse transform is  where x and y are spatial coordinates and a and b are horizontal and vertical frequencies. One realization of the Fourier transform is the principle of
diffraction. If you shine coherent light through one point there will
just appear a large white blob on a screen. If coherent light is
shined through two separated points, though, a diffraction pattern
will appear (see figure 5). The orientation of the (sine-wave) grating
is caused by the relative orientation of the two points.  Figure 5 In each figure pair, coherent light shining through the point(s) on the left would create the (diffraction) pattern seen on the right. The right hand side of each figure pair represents the Fourier transform of the left-hand side of the figure pair. Reproduced from Taylor, Images, 1978, page 27. Mathematically, the diffraction patterns seen are explained by
taking (two-dimensional) Fourier transform of the points. The
right-hand side of each figure pair is the Fourier transform of the
left-hand side (and visa versa). If coherent light (or light from a point source) is shined through
two slits, a diffraction pattern can be demonstrated as seen in figure
6. Note a large blob in the middle and smaller blobs tapering off to
either side. The separation and angular position of the spectral blobs
is dependent on the separation and orientation of the slits.  Figure 6 The left hand side of each pair indicates the geometry of the slits. The right-hand side of each pair shows the optical diffraction pattern. Reproduced from (Taylor, Images, 1978, pages 42-43). Figure 7 shows the mathematical Fourier transform of three
different patterns (square-wave grating, checkerboard, and plaid) into
their respective spatial frequency domains. The spectrum of the
square-wave grating has odd numbered harmonics that taper off in
amplitude to each side. Note that the plaid shape is made up of the
addition of vertically and horizontally oriented square-wave gratings.
Similarly, the spectral representation of the plaid is the
superposition of the spectral representation of the vertical
square-wave grating and what the spectrum would be (not shown) for a
horizontal square-wave grating. Pay particular attention to the
dominate four components to the plaid spectrum (the heaviest four dots
towards the center. Now compare those dominate dots to the
corresponding ones in the spectrum of the checkerboard. Note that
there is a 45° rotation. This can be intuitively understood because
you can perceive rows of white and black squares running at the 45
degree orientation for the case of the checkerboard. This fact will
become very important in the physiological experiment to be described
later in the report.  Figure 7 Stimuli and the corresponding Fourier spectrum. (A) square-ware grating. (B) checkerboard with squares. (C.) checkerboard with rectangles wider than tall. (D) plaid. Reproduced from De Valois et. al., 1979, page 485. Before proceeding, a couple of examples will be shown of the effect
of not using the entire spectral domain in performing the inverse
Fourier transform. This demonstrates the holographic idea of the
"whole" being stored in all the parts. It also shows that
fairly good images can be achieved without taking the full theoretical
transforms. This is important because neural processing isn't infinite
in extent. The brain (because of its finite nature) would only be able
to take a truncated Fourier transform. When an inverse Fourier transform is taken of smaller and smaller
areas of the spectral domain. the "whole" is always
captured, but the resolution deteriorates. See figure 8.  Figure 8 The right-hand side of each figure pair shows the spectrum. The left hand side of each figure pair shows the corresponding image after the inverse transform. Reproduced from (De Valois & De Valois, Spatial vision, 1988, page 17). CHAPTER 3 EUCLIDEAN-BASED GEOMETRIC MODEL The conventional theory is that the main computational event in neurons is the generation of the action potential. The firing of the action potential (for a single cell or a network of cells) indicates the triggering of a particular perception. In the extreme case (the "grandfather cell") the firing of a single cell can trigger a certain memory or perception. More typically, though, it would be the near simultaneous firing of a whole collection of cells in a network that triggers the perception. The perception would then be mediated by the action potential's propagation (through the axon) to other parts of the brain. It would be the integrative emergent response of "the other parts of the brain" (including parallel coupling to other sensory modalities) that yields the sensation of the perception. For visual perception, there is the following information flow
(Kendel, Principles of Neural Science, page 438) Lateral geniculate nucleus: cells also respond to small circular stimuli Primary visual cortex: transforms the concentric receptive field in at least three ways. 1. Visual field decomposed into short line segments of different orientation, through orientation columns. Early discrimination of form and movement. 2. Information about color is processed through "blobs" which lack orientation selectivity 3. Input from the two eyes is combined through the ocular dominance
columns (one of the steps necessary in depth perception. The two different views were (1) that the cortical cells function
as non-linear edge detectors or (2) as linear spatial frequency
filters. These two views each have different predictions about how the
cortical cells would respond to a visual stimulus. By making use of
gratings and checkerboards as the visual stimuli, De Valois et. al.,
1979, were able to distinguish between these two possibilities. Figure 7 (from earlier in the report) shows different patterns
(that can be presented as visual stimuli) and the corresponding
frequency spectrum. Each spectrum here is plotted in polar form where
the distance from center represents the spatial frequency of the
stimulus and the angle (from 0°) represents the phase information or
the orientation of the spatial frequency of the stimulus. The size of
the dots represents amplitude. In rectangular coordinates the spectrum
would be interpreted as frequency components in the vertical and
horizontal directions. For example, a square-wave grating with vertical bars (see figure
7) would manifest a frequency (repeating pattern) in the horizontal
direction. The spectral depiction of this image would be decomposed
into various frequency components all in the horizontal direction.
This would be plotted (in the representation used here) as dots along
the horizontal axis. A spectral plot of a square-wave grating with
horizontal bars would consist of dots along the vertical axis. When an animal is presented with the spatial visual field (the
left-hand side of each figure pair) the question can be asked
"are the cortical cells responding to information in the original
spatial domain or information in the frequency (spectral)
domain?". In what representation is the information getting to
the cortical cells? Can an experiment be devised to distinguish
between these two possibilities? For example, is a particular cortical
cell responding to the presence of a line in the visual field or to
the fundamental Fourier component (at a certain orientation) of the
spectrum? This issue was resolved by comparing the response of the
same cortical cell to different visual fields (De Valois et. al.,
1979). A series of experiments were performed in both cats and monkeys (De
Valois et. al., 1979) to see if the cortical cells responded to
differences in the Fourier spectrums. The first experiment was
designed around the observation that spectral Fourier fundamentals for
the checkerboard were rotated at 45 degrees relative to the Fourier
fundamentals of either the square-wave grating or the plaid (see
figure 7). Vertical square-wave gratings, plaids, and checker-boards
each have vertical edges in the same orientation. Therefor, if a
cortical cell was functioning as an edge detector, the cell should
respond optimally (most number of spikes or action potentials per sec)
to square-wave gratings, plaids, and checkerboards all in the same
orientation. If, however, the cortical cells were responding to the
spectral fundamentals, the cortical cell should respond optimally to a
checkerboard pattern that is rotated 45 degrees relative to either a
square-wave grating or a plaid pattern that was oriented to produce
the optimal response. In both cats and monkeys, the procedure would be as follows. A
micro-electrode would be inserted into a visual cortical cell to
measure the number of action potentials (spikes) per second. The
optimal stimulus parameters were first determined for the cell. The
receptive field was located and the animal was positioned so that the
receptive field was centered on the scope display. Then by
experimenting with different sine-wave gratings, the optimum
orientation and the optimum spatial frequency for the cell was
determined. The optimum temporal frequency was determined by drifting
the best grating pattern across the respective field at different
rates. If the cortical cell was functioning as a true edge detector,
one would expect the square-wave grating, the checkerboard, and the
plaid to all induce maximal spikes/sec in the cell at the same
orientation. The cortical response to the square-wave grating was
determined with various angular rotations. Then the cortical response
(# spikes/sec) was determined from the checkerboard at various angular
rotations. The visual cortical cells responded optimally to the
checkerboard pattern which was rotated 45 degrees relative to the
square-wave grating that was rotated to produce the optimal response
(see figure 9). This was evidence that the visual cortical cell was
responding to the Fourier fundamentals, not as an edge detector.  Figure 9 Reproduced from De Valois et. al., 1979, page 489. In another experiment, checkerboards stimuli of different check dimensions (1/1, 2/1. 0.5/1) were presented (to the animals) for comparison with the square-wave grating visual stimulus. The altered (orthogonal) dimension of the checkerboard checks should not matter if the visual cortical cells are responding to the unaltered edges. If, on the other hand, the visual cortical cells are responding to the fundamental Fourier frequency, the different checkerboard patterns would have to be rotated some to get the maximal spikes/sec from the cells. Note how the Fourier fundamentals (the biggest dots) are at a different angle from the center in comparing figure 7B to 7C). It was indeed found that the different checkerboard patterns had to be rotated an amount that matched exactly to what would be predicted from the mathematics of the Fourier transform (the location of the Fourier fundamentals). When the data was re-plotted with the points rotated according to the mathematically predicted position of the Fourier fundamentals, it was found that a very good match existed. This was further evidence that the visual cortical cell was responding to the angular location of the Fourier fundamentals and not the edge of the squares (or grating) seen in the untransformed pattern. In another experiment, plaid checkerboard patterns with the same
dimensions were presented (with various rotations) as the visual
stimulus to the experimental animals. Again, if the cortical cell was
functioning as an edge detector, it would be predicted that the cell
would respond optimally to the two patters at the same orientation
(when the edges are at the same orientation). It was found, though,
that the cortical cell responded optimally to a checkerboard pattern
that was rotated 45 degrees relative to the orientation of the plaid
pattern (that had been oriented to give an optimal response). In another experiment, the relative check dimensions were changed
for the checkerboard patterns (2/1, 1.1, 0.5/1 ratios). Note from
figure 7 that as one dimension of the check is changed, the distance
(from center) of the Fourier fundamentals changes. It could then be
determined what width (given a certain ratio) produced the best result
(when oriented optimally). If the visual cortical cells were
responding to the check width, then the different height/width ratios
shouldn't influence the cell's response. If, the cell was responding
to the Fourier fundamentals, then it should respond optimally to
different check widths when the height/width ratio changes. It was
found that the cortical cell responded optimally to checkerboard
patterns of different widths and that these widths matched what was
predicted from the Fourier mathematics De Valois et. al., 1979). When
the data was plotted according to the theoretical prediction, the
cortical cell was shown to be responding to the spatial frequency (the
distance from center of the Fourier fundamental) for the various
optimally oriented patterns. This was further evidence that the
cortical cells were responding to the Fourier transform of the
presented visual stimuli. All of these experiments were repeated for multiple visual cortical
cells in both the cat and monkey yielding similar results (data not
shown in this report). The next set of experiments examined whether cortical cells could
be found that were sensitive to higher harmonic components of the
Fourier spectrum. If so, then this would be powerful evidence that
these cortical cells are acting like spatial-frequency filters (and
not as edge and bar detectors). The higher spectral harmonics of the
square-wave grating are at the same orientation as the fundamental
frequency but the higher harmonics of the checkerboard are at other
orientations (see figure 7). If a cortical cell exhibits sufficiently
narrow spatial tuning, it could potentially respond separately to the
fundamental and the third harmonics of patterns. For instance, imagine
a square-wave grating with more narrow bars such that the fundamental
frequency falls on what was the third harmonic for a square-wave
grating with wider bars. A cortical cell sensitive to this spectral
position, would respond to either stimulus (and the stimuli would be
presented at the same orientation). For the checkerboard, the
situation would be a little different. A smaller sized checkerboard
pattern sufficient to produce a Fourier fundamental at the same
frequency location as the third harmonic (that a larger sized
checkerboard pattern would produce) would have to be rotated some for
the optimal response (to get the angle of the fundamental to fall on
where the third harmonic would be for the other pattern). It was demonstrated that a cortical cell (responding to a
square-wave grating of a certain frequency and orientation) would also
respond optimally to a square-wave grating with bar widths three times
the size (which would be one third the spatial frequency) with the
same orientation. The Fourier fundamental of the grating with the more
narrow bars fell on the third harmonic of the grating with the wider
bars. It was also demonstrated that the same cortical cell responding
to a sine-wave grating (optimally at a certain frequency and
orientation) would not respond to a sine-wave grating of 1/3 that
frequency at the same orientation (remember that there are no
harmonics for a sine wave). In order for the cortical cell to optimally respond, a checkerboard
pattern with check size of a certain size had to be rotated relative
to the optimal rotational orientation of a checkerboard with checks
that were three times larger producing the optimal response for the
same cortical cell. This observed rotation matched the theoretical
predicted rotation from the Fourier mathematics (De Valois et. al.,
1979). Similar experiments have been performed with the rat somatosensory
system (Pribram, 1994) where the cortical cells were also found to
respond to spectral information.  Figure 10 An isolated system can itself be divided into a subsystem that is open to energy flow and the subsystem's environment (see figure 11). As such, the whole combined isolated system still obeys the second law of thermodynamics, but it is possible that the subsystem can experience a decrease in entropy at the expense of its environment.  Figure 11 The entropy increase in the "sub-system environment" is
guaranteed (by the second law) to more than offset the entropy
decrease in the subsystem. Also note that the sub-system can only be
maintained away from equilibrium as long as there is usable energy in
its environment. When the environmental entropy is maximized (no
usable energy), the subsystem is guaranteed to itself proceed to
equilibrium. There is a special class of such subsystems (as described above)
where the subsystem's organization comes exclusively from processes
that occur within the sub-system's boundaries. This class of
subsystems was labeled "dissipative structures" by I.
Prigogine, 1984 (who won the Nobel price for his work). Pribram
believes the brain to be such a "dissipative structure". One way of modeling a structure that goes to equilibrium is to
minimize a mathematical expression for the internal energy (which is
the same as maximizing an expression for the entropy). This is called
the lest action principle. This would not be appropriate, though, for
a "dissipative structure" since it is not going towards
equilibrium. "Dissipative structures" self-organize around a
different "least action principle". In the holonomic brain
theory, Pribram has the entropy being minimized (which maximizes the
amount of information possible to store) as the "least action
principle". Thus, the system (the brain) self-organizes such that
more and more information can be stored. In Hopfield networks and the Boltzmann engine (which are computer
models of neural processing), computations proceed in terms of
attaining energy minima. In the holonomic brain theory, computations
proceed in terms of attaining a minimum amount of entropy and therefor
a maximum amount of information. In the Boltzmann formulation the
principle of least action leads to a space-time equilibrium state of
least energy. In the holonomic brain theory, Pribram describes the
principle of least action as leading to maximizing the amount of
information (minimizing the entropy). Independently, (in unrelated work) Schneider and Kay (1994) have
proposed a variation on the second law of thermodynamics which may be
applicable to Pribram's holonomic theory. "The thermodynamic principle which governs systems is that as
they are moved away from equilibrium, they will utilize all avenues
available to counter the applied gradient. As the applied gradients
increase, so does the system's ability to oppose movement from
equilibrium". It would be interesting to see if there is a connection between the
work of Schneider and Kay and Pribram. The holonomic brain theory maintains that the brain is continuously
engaged in correlation processes. This is how we make associations
(how the senses are integrated). There is an obvious computational
advantage for the brain storing sensory information (and perceptions)
in the spectral (or holographic) domain as opposed to the brain
directly storing individual features and characteristics. The holonomic brain theory claims that the act of
"re-membering" or thinking is concurrent with the taking of
the inverse of something like the Fourier transform. The action of the
inverse transform (like in the laser shining on the optical hologram)
allows us to re-experience to some degree a previous perception. This
is what constitutes a memory. The holonomic theory (for the example of vision) summarizes
evidence that the image formed on the retina is transformed to a
holographic (or spectral) domain. The information in this spectral
"holographic" domain is distributed over an area of the
brain (a certain collection of cells) by the polarization of the
various synaptic junctions in the dendritic structures. At this point,
there is no longer a localized image stored in the brain. Correlations
and associations can then be achieved by other parts of the brain
projecting to these same cells. Conscious awareness (and memory) is
the byproduct of the transformation back again from the spectral
holonomic domain back to the "image" domain. Possibly the
most radical part of the holonomic theory is Pribram's claim that a
"receiver" is not necessary to "view" the result
of the transformation (from spectral holographic to
"image"). He claims that the process of transformation is
what we "experience". Memory is a form of re-experiencing or
re-constructing the initial sensory sensation. Conventional neuro-physiology effectively pushes back the line
between observer and what is observed (between subject and object). In
signal processing, there always needs to be an end-user to view the
processed or transformed signal. At best, conventional neuro-science
leaves until later the ultimate explanation of the observer. Who would
bet their grant money (career) on being able to answer this question
in a couple of years? Aspects of Pribram's holonomic brain theory
attempts to address this question. The conventional view is that the brain is a computational device.
There is a growing body of literature, though, that shows that there
are severe limitations to computation (Penrose, 1994; Rosen, 1991;
Kampis, 1991; Pattee, 1995). For instance, Penrose uses a variation of
the "halting problem" to show that the mind cannot be an
algorithmic process. Rosen argues that computation (or simulation) is
an inaccurate representation of the natural causes that are in place
in nature. Kampis shows that the informational content of an
algorithmic process is fixed at the beginning and no "new"
information can be brought forward. Pattee argues that the complete
separation of initial conditions and equations of motion necessary in
a computation may only be a special case in nature. Pattee argues that
systems that can make their own measuring devices can affect what they
see and have "semantic closure". It is possible that the brain transcends computational behavior. If
this is the case, then it will be very interesting to see what aspects
of Pribram's holonomic theory are in collaboration with these
non-computable ideas. 1. The apparent spectral frequency filtering aspect of cortical cells 2. The relationship between Fourier transforms and holograms 3. The fact that selective brain damage doesn't necessarily erase specific memories 4. The computational advantage to performing correlations in the spectral domain. 5. His idea of conscious experience being concurrent with the brain performing these Fourier-like transformations (which simultaneously correlate a perception with other previously stored perceptions). He believes that conscious experience is the act of correlation itself and this correlation occurs in the dendritic structures by the summation of the polarizations (and depolarizations) through the processes in the dendritic networks. 6. The brain is a "dissipative structure" and
self-organizes around a least-action principle of minimizing a certain
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