The important conceptual
point is that
this basin structure deterministically emerges from the coupled
generating processes
(the interconnected nodes). If the connections
between the
nodes are changed, or if the logical relations among the nodes
are
changed, or if anything is changed, then a different basin structure
will
emerge. But given the exact generating process, then the
system
will self-organize into the basin structure that is shown in
Figure 1.
Given that a basin structure has emerged in a system, it is
worth
noting that once in a basin, a system will flow to an attractor and
then cycle in that attractor endlessly unless it is perturbed from the
outside or unless it
perturbs itself (supposing that a system has self-perturbing
capabilities which most complex systems do). The systems in
E42
are self-perturbing but that is a topic not covered here. At
a
minimum, a perturbation consists of
changing the value of one node in one state vector. For
example,
look on
Figure 1
and notice that if you are in Basin 1 and the state vector for some
specific moment of time is {1001}
and you change the first node from 1 to 0, the resulting state vector
will be {0001} which is found in Basin 2. Once in Basin 2,
the system will stay
until perturbed again. Thus systems can shift basins of
attraction through internal or external perturbations.
Now let us turn to
representing attractor cycles as visual forms.
As we noted we can express the binary distinction as 0,1, or
ON, OFF,
or BLACK, WHITE. Let us replace 1's in our state vectors with
black squares and 0's
with white squares. And let us rotate the state vectors from
their easily typed row format to columns. Thus the sequence
of
state vectors mentioned above [
{1001},
{1101},
{1111}, {1011}]
rotated to be columns and with 1's = BLACK and 0's = WHITE, becomes
Panel A in
Figure 2.
Examine Figure 2, Panel A, and note that if you rotate
{1001} and replace 1's with BLACK and 0's with WHITE you will get the
first column of Panel A (with the first node at the top).
Panel
C shows Basin 2 from
Figure 1 and Panels B and D show the
perceptual effects of starting the visualization at different points in
time. Panels E through H show "tiling effects;" that
is, if you let the system run through an attractor cycle multiple times
and then visualize it, the resulting visual form will have Gestalt
characteristics akin to those resulting from tiling a floor
or wall.
------------------------
Insert Figure 2 about here
------------------------
Returning to the larger conceptual
point, flows of difference in a
Boolean system, when the differences are expressed as black and white
squares, self-organize into temporal visual forms generated by
something like an ever-changing line moving through time. These
visualizations
are conceptually parallel to those made by Turing in his
morphogenesis paper; indeed,
the tutorial shows how this type of visualization can produce
camouflage-like zebra stripes. The basic can be illustrated
with
Figure 2.
Dynamic Form
Think of sitting on the ocean beach
watching the waves roll up on the shore and think of the complexity of
the moving patterns as they play across each other making dynamic
patterns within dynamic patterns while sheets of water race from the
surf up to your feet. These patterns playing across patterns
making other patterns is an example of what Bateson called moire
patterns (2002, p. 74)
which he defined as a combination of two patterns producing a
third pattern. As another example think about the flow of a
swiftly
falling rapids on a river. A river runner (boater or kayaker) standing
on the bank looking at the rapids will see standing waves (large and
small), holes (generally to be avoided) and small sub-rivers where the
river's moire current snakes across the surface and hints at ways
a kayak might get through. The boater's success in the rapids depends
critically on
extracting dynamic patterns and sub-patterns from what at first may
seem to be the chaos of the thundering fall of water. This pattern
extraction becomes even more complex once the river runner puts in and
is sucked into the rapids since the boater will be moving in relation
to
river's patterns. How are coherent patterns extracted from such
complexity?
River
rapids or autumn leaves shaking in the wind and partially masking
a herd of
deer moving down a steep mountainside are examples of a kind of form
perception that is of central interest to, at the very least, large
mammals. But these patterns are difficult to capture in a
way that can be easily shared among people. Taking a still photo leaves
most of the moving pattern out of the picture so to speak. And
words often fail when we use them to point to a subtle
aspect of such patterns and we are left with saying, "Look at that cloud
that looks like a camel." There is no "camel" (ding an sich) in the cloud (and indeed
our friend may or may not seen anything in the cloud that resembles a
camel). The camel we see in the cloud is a co-construction of the
dynamic movement of the cloud and the dynamics of our own perceptual processes.
We will use an NK Boolean simulation as a model of Bateson's idea
of double description as a way to demonstrate how a living being may
co-construct dynamic forms through the relationship between two flows
of difference (two descriptions). As a preview, examine Exemplar 3
which demonstrates this approach experientially with wave-like dynamic forms and sets some frames to motivate the
details that follow. If necessary for your operating system/browser combination, get the
Java
plugin.
Recent Apple computers using the Safari browser do not require
the plugin. For PC's we recommend the Firefox browser with the
Java
plugin (
http://java.com/en/).
Kauffman, as noted above, indicated
that his Boolean models were extremely
idealized models of genes. In the same way our Boolean dynamic
systems are not models of neural activity except in the most
idealized way. The simulations are intended to model
Bateson's difference-based epistemology. That said, like Kauffman
did with
genes, we will make a broad parallel to neural activity in order to
have a concrete context from which to generate examples. Suppose
in
this idealized sense, the neurons of the retina can be idealized as
as a system of interconnected Boolean nodes. A Boolean node at a
particular moment in time is
either 0 or 1; the neuron at a particular moment is either not firing
or firing. Assuming that neurons are interconnected then their
on-off patterns of firing can be construed to self-organize into
attractor cycles in some way abstractly related to Boolean systems.
In particular we are interested in framing retinal neurons as
capable transforming differences in the territory
(movement, color, etc.) into a flow of differences in a neural circuit
that can reverberate in attractor cycles like those found in Boolean
models. We therefore model the retina as a net of interconnected
Boolean nodes (again,
see the Map/Territory illustrations) which transform differences in territory into differences within the knower.
Bateson (2000, p. 460) discusses in detail the distinction
between external (may be analogue) and internal flows of difference. Malloy, Jensen
and Song (2005)
offer a more detailed mapping of Bateson's epistemology to Boolean
systems.
As above, we will not describe the mathematical details of the double
Boolean flow of differences that constitute our double description; an
easy and informal tutorial is available (
click here). But we will present the main ideas. Examine
Figure 2 once
again. Notice that Panel A and Panel E show the same
attractor cycle but that Panel A shows that cycle for four moments
of time (iterations) while Panel E shows that cycle across sixteen
iterations. Another way to say this is that Panel A shows the
system passing only once through its attractor cycle while Panel E
shows it passing through the attractor cycle four times (the pattern
repeats four times). This is a fundamental question in
representation: What is the temporal chunk size? If we have a flow
of differences and we want to represent that flow we are required to
ask how long a segment of the flow am I to represent? It is as if
we have a window (or snapshot) into the flow of process and that window
by its nature
must have a width. We can show only one iteration through the
window or four iterations (Panel A) or sixteen interactions (Panel E),
or whatever length we want. But the length of the window into the
flow must be specified. We will call the number of iterations in
the window W. So Panel A is a window of size 4 (W=4) into
the flow of state vectors in Basin 1 and Panel E is window of size
sixteen into the same basin (W=16). Panels C and G in
Figure 2
show W=4 and W=16 for the flow of differences in the attractor cycle of basin 2.
Another way to think of this is that the window is a static
snapshot of system's process and that snapshot can be was wide or narrow as
we want it to be.
System Process versus Representational Process.
Recall that we defined the length, L, of
the attractor cycle as the number of iterations it takes to cycle fully
through the attractor (that is, for the sequence of state vectors to
begin repeating itself); in basins 1 and 2 of the running example
L = 4. Note that L is an
emergent characteristic of
the dynamic
system because the attractor cycles themselves are emergent
characteristics of the
system as argued above. In contrast, W is a characteristic
of a second
system; the second system represents the emergent characteristics
(attractors) of
the original system. The original system could exist without any
such secondary system to represent it; undoubtedly in nature many
systems exist without any intrinsic representation of their process.
W arises
because we want to have a
representational window into the flow of the system's process.
In epistemological theory these are very different kinds of
variables, L
indicating something about the process of the original system itself
and W
indicating something about the processes by which the emergent dynamics
of the
original system are represented. In our Boolean simulations that
distinction
is between mathematical algorithm generating an ongoing systemic
process in a computer's CPU and the representation of that
processes by painting what is going on in the CPU to a monitor.
For purposes of representation to
the screen, the E42 program tracks the systemic process (a series
of
state vectors) for W iterations; then E42 paints those state
vectors to the screen (as a snapshot of black and white squares), then
it tracks the next W iterations and paints another snapshot over the
previous one, and so on. What is seen on the screen is a series
of snapshots presented, rapidly, one after the other so that we can
see the ongoing flow as the system runs in the CPU. Note
that cinematography works by similar principles; movies present
the viewer a series of slightly different still frames, briefly and
rapidly, to create motion. A major difference between
cinematography and our simulations is that movies are recordings
motion. Whereas in our simulations nothing is recorded; they are
real-time representations of process and real-time interactions with
the ongoing process are possible. This is the difference between the
passive nature of old media and the interactive nature of new media.
On a technical note, most
computers calculate Boolean functions faster than monitors can
accurately paint so the
simulation program has a "Delay" control that allows the user to slow
down
the speed of the system (and consequently of the painting).
Fundamental Frequencies. In the general case a system can self-organize in a way that has
sub-cycles within the attractor cycles (circles within circles). The on-off
firing of some nodes fall into patterns that repeat themselves
more frequently than the attractor does as a
whole. In
Figure 2,
Panel D, for example, notice that (counting from the top) the first
node repeats its on-off pattern every two iterations as does the
fourth, bottom, node. But if you look at rows 2 and 3 you will
see that nodes 2 and 3 fire only once every four iterations.
Nodes 2 and 3 are responsible for an attractor cycle length of
L=4. But node 1 (and node 4) fire at a faster frequency so we
will use the notation sub-L to describe the short length of their
faster frequencies. In the case of these two nodes sub-L=2.
Both L and sub-L's emerge as formal characteristics of the system
and we will refer to both as the fundamental frequencies of the system.
Modeling Double Description.
Now we can describe more precisely the nature of the double
description that will produce dynamic form. The first description
is the
flow of differences within the original system (defined rigorously
by a
sequence of state vectors) and
the second description is a parallel flow of snapshots of the
original flow. That is,
we have two flows of process (two descriptions),
systemic and representational; and visual form arises in the
relationship between the two. We will demonstrate that dynamic visual
form is a co-construction of
systemic and representational flows. Let us expand these
idea,emphasizing two points. First, regarding representational
process, there will be a series of snapshots of the system's flow
of state vectors, each snapshot of length W and each occurring rapidly,
in order, one
after the other. As noted, this rapidly occurring series of
snapshots works
like a
movie; one still frame is followed quickly by another.. The
second point is that
we can adjust the length, W, of the snapshots in real time as we go.
Doing so, in mathematical terms,
adjusts the phase relations between the frequency of the system's
attractor cycles and the frequency of the recurring representational
process.
We will show a simple model in which dynamic form emerges from
the the phase relations between
systemic and representational frequencies. Let us map this
Boolean
model to Bateson's epistemology using an idealized notion of the
retina. As Bateson points out,
somebody could (theoretically) go out and put the on-off responses of
individual neurons in the retina onto a piece of paper (2000, p. 460);
to keep up with ongoing
changes in the retinal system across time such a
person would have to draw a series of snapshots and would
consequently end up
with an ordered stack of papers. This Batesonian thought-experiment
maps directly to our series of snapshots.
Apparent Stability and Apparent Motion. We now place these two descriptions in relation to each and define their phase relationship.
Figure 3,
which has time (iterations) on the horizontal axis and two nodes of a hypothetical Boolean system on the
vertical axis, shows what happens when L is not equal to W,
specifically when
L=4 and W=5. To be concrete, let B (black) indicate that a node "fires" and W
(white)
indicate that it does not in
Figure 3. Notice in
Figure 3
that the top node
is cycling BWWW every four iterations (it fires once every four
iterations) while the bottom node is cycling BWBW. For the whole
hypothetical system (of N=2 nodes) L=4, but for node 2 sub-L=2.
The vertical
cross-hatched bars indicate the divisions between a series of three
snapshots (each capturing W = 5 iterations of the system's process).
More formally the system itself is cycling with frequency of 4
iterations and
the representational process is cycling with a frequency of 5
iterations; the processes are out of phase. As a result of this
out-of-phase relationship, for the top node, the position of its
single B (fire) appears twice in the first window and then appears
to move backwards relative to the frames
defined by subsequent two snapshots. This is the same out of
phase relation that
causes a wagon wheel to turn backwards in movie. It is called
apparent motion and is generally described as an illusion. In contrast,
within the framework of this
model apparent motion is not caste as an
illusion but as a process central to form perception. When W = L,
that is
when the frequencies of system and of the representational process are
in
phase, the same snapshot will be painted over itself over and over and
the dynamic pattern appears to stabilize; this is only apparent
stability since
the system processes are cycling as fast as they always do. But
because system process and representational process are in phase the
system's process appears to freeze. Thus such a system can create
the appearance of static objects in a dynamic relational world and
resolves the paradox of how objects seem static in a world of flow.
Moreover
this apparent stability will occur if W is any integer multiple of
L. Also notice that,
excluding integer multiples of 4, if W = 2 or any integer multiple of
2 then node 2 (which has sub-L=2) will freeze even while node 1 keeps
moving.
Thus adjusting phase relations between systemic and
representational process allows the possibility of freezing some parts
of a system while allowing other parts to be perceived dynamically.
You may
confirm for
yourself that if L>W features of the system process (such as
nodes firing) will
apparently move forward rather than backward as they did above when
L<W. We will now make these points experientially.
------------------------
Insert Figure 3 about here
------------------------
Instructions. If
necessary, get the
Java
plugin.
In all the following applets begin by pressing the
Use
Delay radio button and then
adjust the
Delay Slider until your particular computer is painting
windows to the screen at about
25 to 35 frames per second (fps).
Adjusting the delay (between iterations of the system) is crucial
because
different computers paint visualizations on the screen at different
speeds; so the Delay slider allows you to adjust your computer to
paint windows (frames) to
your screen at a known rate; apparent motion research has shown that a particularly useful range is between 25 and 35 fps.
The
fps readout is to the right of the
Stop/Play control bar. The number
of fps is a potent variable and you may set it to any value you want
by adjusting the delay between iterations. Keep the fps below
65 since most monitors cannot paint accurately beyond about 65 fps. To
adjust the delay between each screen-paint, you may either drag the
Delay Slider or, for finer adjustments, single-click on the Delay
Slider Bar either above or below the Slider.
Each applet loads to a default basin;
perceptual experience with the dynamics of the default basin is
discussed on the text accompanying each applet. In the sections
below we do not repeat the details of these discussions but simply note
the major results. Optionally, on each applet, you may click the Perturb
button which (almost always) provokes the system into a different basin
to explore perceptual experiences in other basins. When you
perturb the system
into other attractors the particulars of your observations will change
but the same general conclusions will usually be verifiable; sometimes
the new basin will have characteristics that we are not focusing on
here and other, interesting, observations may apply. Remember,
these are not old media, movie-like set pieces where everything is
known. We can set the system's initial conditions to start in a
particular basin that we find interesting. But when you perturb
you do so pseudo-randomly and there is no way to know where you might
go. Some of these systems have hundreds, even thousands of
basins, many of which have not been observed by users before; some have
only a few basins. When you perturb a system you may find
yourself in a basin that demonstrates something new.
Perceiving
fundamental dynamics.
The first applet demonstrates how the phase relations between the
process flows of the original system and the representational system
can freeze (or cause to move together in a coherent way) formal
characteristics of the original system such as attractor cycles and
attractor sub-cycles. Link to
Exemplar
1
(by clicking); the page that pops up includes detailed
instructions for adjusting phase relations between two flows of Boolean
process. These two flows, in Bateson's terms, are a double
description of the same event. One flow is systemic; it is the
flow across time (iterations) of the Boolean system as it cycles
through an attractor. The other is representational; it captures
a given number (W) of iterations of the system and presents them
visually to the screen. Adjusting the phase relations between these two
descriptions allows you
to "extract" the fundamental frequency
(L) of the systemic attractor cycles as well as systemic sub-cycles
(sub-L). When W is equal to L (or any integer multiple of L) the
whole attractor freezes even though the system is running. When W
is equal to the sub-L (or an integer multiple of sub-L) you the
sub-cycle freezes. Thus you can highlight whole attractors or
sub-cycles of attractors by freezing them or by making them move
coherently together. In summary, adjusting
phase relations between to descriptions allows the extraction of the
fundamental frequencies of the system.
Forms Derived from Relationship. Link
to the
Exemplar
2
applet which will demonstrate how visual forms emerge from the phase
relations between two descriptions (flows of difference) that are not
in any way technical characteristics of the system
per se.
These "derivative forms" only emerge in the relationship between systemic and
representational process. Notice that on the web page that contains the
Exemplar
2 applet
there is, as well as the dynamic representation provided by the applet,
also a static snapshot of the the attractor's ongoing flow. The
snapshot does not,
indeed cannot, show these derivative forms
we are referring to here because they emerge dynamically from the
phase relations between the ongoing systemic and ongoing representational process.
The reader can adjust W to change these phase relations to
experience how
different derivative forms emerge as the phase relations between
systemic
and representational processes change. We will discuss this below.
Ambiguous Motion.
Exemplar
2 (and even more so
Exemplar
1) demonstrates another interesting phenomenon: Ambiguous motion. Set W =
77 and observe the fifth node from the top (lined up with a red hash
mark). If you run a mouse arrow or the tip of a pen back and
forth along the horizontal line of the 5th node you will note that the
motion will change directions. There are many examples of
ambiguous static figures (e.g., Necker Cubes); this appears to be a
more general case in which motion itself changes orientation.
With a little practice users can provoke this
change of direction of movement with their eyes alone. With W =
83,
complex emergent forms with ambiguous motion can be perceived moving
either right to left or left to right; once again, you may require a
horizontally moving pointer to observe this phenomenon. Very triking ambiguous motion is shown in
Exemplar
1
where whole groups of nodes move together to the right or to the left
depending on which way you move the cursor across them.
A Simple Model of Dynamic Form.
Above, we have used E42 to demonstrate how changing phase
relations between the computational flow in a computer's CPU and
the representational flow to a computer's monitor produces interesting
perceptual experiences. We now turn to a more speculative and
risky venture; we propose one possible model that maps the computer
simulation we have just reviewed to a Batesonian difference based
epistemology. While we are not modeling neural activity (neural
network models do that better), let us begin by talking about the
retina in an abstract way as a richly connected network through which differences flow. At
this point we re-emphasize the
map/territory distinction
(Bateson, 2002, p. 27) to make it clear that we do not propose to model any aspect of
the territory itself; rather we are modeling Bateson's proposal that
what gets onto maps from the territory are differences (2000, p. 457)
and that knowledge (2002, chapter 3) emerges in the relationships
among multiple descriptions (multiple flows of difference).
Note that, while we do not
model the territory itself, we do assume that the retina is coupled (i.e.,
entrained) with systemic processes ongoing in the territory (e.g.,
Turvey, 1990, p. 942) and consequently that retinal dynamics have a
useful relationship of dynamics in the territory.
In this spirit then the
retinal image is modeled as a discrete dynamic system
coupled to
the environment. We also propose that form emerges through phase
relations between
at least two streams of differences which we call, after Bateson,
two descriptions. The first description is the retinal image as it
flows toward higher centers. As a second description we propose a
parallel, representational flow. Thus we propose
that dynamic visual form emerges from phase
relations between the flows of the first (retinal) and second
(representational) descriptions. Another assumption is necessary: The
perceptual system must have some mechanism for adjusting the phase
relations between these two descriptions. Such a mechanism would allow
the extraction of different dynamic forms. Presumably one part of
perceptual learning would be learning to adjust these phase relations
in context specific ways that both have utility for a person and
correspond to social conventions in a particular context.
A profound insight (e.g., see Varela,
Thompson and Rosch's concept of
enactment, 1993)
in modern thinking is that knowledge in general and representation in
particular is not a reflection of
(or a photo of, or a tape of) what is "out there" but is more usefully
described as something that emergences in the relationship between the
processes of the knower and the processes of what is known.
Representation is not a passive response to the universe; it is
an
active co-construction, an enactment in Varela, Thompson and Rosch's
terms. Knowledge
is neither here nor there, it is neither in the territory nor on the
map; it is in the relationship
between the two. How do we begin to specify, beyond these
provocative
words, what
such statements might mean? In our proposed model the retina is a
(binary) dynamic system whose characteristics (attractors, etc.) are
entrained with the dynamic system called the ecological context.
Whatever is the appropriate model of the ecology's dynamics might
be, following Bateson, what gets onto maps (retina in this case) from
the territory are differences; thus we model the retina's response as a
Boolean system. The retinal Boolean system already entrained to
the ecology also entrains itself with a proposed representational
Boolean system. To be as explicitly as possible, in our computer
simulations we can specify Boolean process running in the CPU and
representational process painting to the screen. We noted
in
Exemplar
2 and
Exemplar 3
that visual forms emerge that are not formal characteristics of CPU's
Boolean system but are a co-construction of the CPU Boolean system and
the representational process of painting to the screen. Thus we propose
that the dynamics of the retina is Boolean-like process that is
entrained with the ecology around it and additionally with a
representational proceeds such that some visual forms emerge
relationally and are not actually characteristics of the retinal
dynamics per se and therefore not characteristics of the ecological
dynamics. The camel we see in the clouds is a
co-construction of the light patterns reflected to our eyes from the
dynamically systemic flow of turbulence in the cloud
and of our retina as a dynamic system
and
of a representational dynamic system. Certainly there is no camel
in the clouds; much is derived in the relationships among dynamic
systems.
Hierarchies of Knowledge: Differences in Differences
Bateson
(2000), pp. 463f) proposes a mental experiment for the reader: