Suppose that Figures 1 a through d represent frames from a movie. These frames are shown in rapid succession. The dark rectangle (top row) and bottom rectangle (bottom row) will appear to move across the frame from left to right. Timing is important in this effect; when the frames are shown at around 25 frames per second (fps) the effect is maximal.
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| Figure 1. If frames a through d are shown in rapid succession, about 25 frames per second, the objects in the frames will appear to move forward, from left to right and (in 1d) off the frame of reference. | |||||||||||||||||||||||||||
The motion, of course, can be seen to be going backwards (here represented by right to left).
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| Figure 2. If frames a through d are shown in rapid succession, about 25 frames per second, the objects in the frames will appear to enter the blank frame (2a) and move backwards (from right to left). | |||||||||||||||||||||||||||
A Review Basins and Sub-basins in N, K, Boolean Systems
Example:
Two-node dynamic system (Figure 3)
Historical trace
Time: the horizontal axis
| Node 1 |  |
| Node 2 | |
| Iteration # ==> | |
| Figure 3. Historical trace of sixteen iterations of a simple two-node system that has basin length L=4. The repetition of the basin pattern (length 4) is emphasized by alternating shading in the Iteration # row. | |
Basin Length L=4
the pattern of black and white squares repeats every four iterations
Node 1 accounts for L=4
Sub-basin is created by Node 2
Sub-basin Length: SubL=2
Extracting Basins and Sub-basin Patterns
from a Dynamic Universe
How does it work?
What creates the apparent motion/stability?
Phase Relations
between System Dynamics
and Representation:
Apparent Motion and Apparent Stability
Representing Sets of Successive Iterations Simultaneously
in a Single Window
Usual Procedure: historical trace is outputting the state (black or white) of each node on each iteration at the moment that that iteration occurs.
Let the system run for X iterations
Then print the set of X iterations at once in a window
Let the system run for X more iterations
Output those X new iterations on the screen
right over the output of the previous X iterations.
And so on.
The representation will now become a window showing the pattern generated by X iterations. This pattern is over-written as soon as the next X iterations of the system are complete.
Such a window will in effect be like a frame on movie film:
It will show one pattern (of length X iterations) then the next pattern of X iterations then the next pattern of X iterations, and so on. Differences in the positions (relative to the window as frame of reference) of various elements of a pattern on successive windows will generate apparent motion (and apparent stability) effects. Let's examine how that happens.
Backwards Apparent Motion:
Window Size Greater than Basin Length
| Node 1 |  |
| Node 2 | |
| Iteration # ==> | |
| Figure 4. A window of size 5 iterations imposed on sixteen iterations of the the historical trace of a simple two-node system that has basin length L=4. The basin patterns (length = 4) are emphasized by alternating shading in the Iteration # row. | |
Forward Apparent Motion:
Window Size Less than Basin Length
| Node 1 |  |
| Node 2 | |
| Iteration # ==> | |
| Figure 5. A window of size 3 imposed on the historical trace of sixteen iterations of a simple two-node system that has basin length L=4. The basin patterns (length = 4) are emphasized by alternating shading in the Iteration # row. | |
Apparent Stability:
Window Size Equal to Basin Length
| Node 1 |  |
| Node 2 | |
| Iteration # ==> | |
| Figure 6. A window of size 4 imposed on the historical trace of sixteen iterations of a simple two-node system that has basin length L=4. The basin patterns (length = 4) are emphasized by alternating shading in the Iteration # row. | |
More Theoretically:
Window Size adjusts the Phase Relations
between System Dynamics
And
Representation of System Dynamics:
Hypothesis:
The neural circuitry that produces Apparent Motion Illusions
Is capable of adjusting phase relations and therefore extracting dynamic patterns from a dynamic universe.
Extracting General Patterns from a Dynamic Universe
River Rapids.
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 overall current snakes across the surface and hints at ways 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. The boater will be moving in relation to river's patterns. How are coherent patterns perceived?
E42: Simulating A Dynamic Universe
River rapids or wind blown autumn leaves and herds of deer represent the sort of thing that is of central interest; still, they are difficult to capture in a way that can be widely shared among many people. So we will use output from an N, K Boolean system to simulate a dynamic universe from which we will extract various patterns and sub-patterns.
We live in world of complex dynamic patterns whose perception is fundamentally tied to motion. Even that which appears stable is stable only relative to motion.
Physiological Nystagmus. [Note 3 ] Visually static objects in everyday life require the constant movement of our eyes (physiological nystagmus) in order to be seen. Physiological nystagmus is produced by tremors in the eye muscles. These tremors prevent the light from stable objects from falling on the same retinal cells from moment to moment. They do that by shifting the eye less than one degree of visual angle (which is about the distance between two foveal cones) around 10 times a second. Experiments with stabilized images (see Palmer, 1999, p. 521ff) indicate that if an image moves synchronously with these tremors, the perception of that image disappears after a few seconds. Bateson (1979, pp. 90, 91) uses nystagmus as a fundamental example in arguing that difference is the basis of mental process. A static relation between a static object and a static eye would produce no differences and therefore no knowledge; this is in full concordance with the stabilized image results. The perception disappears under such circumstances. Nystagmus assures that the eye receives a continuous flow of differences from environmentally static objects.
Enacting a World, Enacting a Mind. We are proposing that the universe be conceived of as a web of dynamic relations some of which set up a field of dynamic relations on the neural receptors of the retina. This, however, is not a passive receiving, as the word "receptors" taken in its strongest sense might connote. A living being has eye muscles that actively twitch the eyes (nystagmus) so as to create motion relative to the context in which it is enmeshed. Add to this other actions (e.g., saccades and micro-saccades). All these actions, taken in relations to a dynamic universe, co-create the retinal image. That is, the image is not simply received from the world by the eye; the eye (i.e., the being) is an active partner in creating the retinal image. The model we prefer is of dynamically active eyes enacting a receptive field on the retina in relation to dynamic relations ongoing in the universe. Varela, Thompson and Rosch (1993, p. 9) put it this way: "We propose the term enactive to emphasize the growing conviction that cognition is ... the enactment of a world and a mind on the basis of ... actions that a being in the world performs."
What is proposed here is a more general and macro hypothesis than those relating to physiological nystagmus, saccades, and micro-saccades, though those phenomena form a basis for the structure of the model. The proposal is about what happens after the retinal image is enacted. The proposal is that there is a mechanism that can adjust phase relations between the dynamics of the retinal image and and a person's experience. That is, we are proposing that a fundamental mechanism of pattern perception is the ability to create and adjust phase relations. Nystagmus is simply an early example of this process, an example that influences the retinal image itself.
But the mechanism of nystagmus itself requires an explanation of why stable objects look stable. Why do we need such an explanation? Because nystagmus adjusts the phase relations among the firing of neurons in relation to a stationary object. That is, nystagmus creates a periodic motion in the retinal image at a rate of about 10 frames (in our terms) per second. Therefore the fact that objects are seen as stable is not explainable by nystagmus itself since it is creating change not stability. Nystagmus by its nature assures that what see as a stable object is "apparent stability." The fact that such objects are perceived as stable means that somewhere in the system another mechanism has adjusted phase relations so that of the moving retinal image is experienced as "apparently stable." But is is a minor example of apparent stability, macro patterns are also extracted from the retinal image. Let's discuss those cases.
In our Java applets we have given explicit control to users so that they can change the phase relations between the dynamics of a simulated universe and its representation on the screen. We are proposing that in an analogous way phase relation adjustments might occur between the retinal image and experienced representation. We found that adjusting such phase relations produced striking examples of the extraction of patterns. There were a least three different kinds of such pattern extractions--basins, sub-basins, and emergent, dynamic patterns encapsulated in time.
We propose that the neural system has a mechanism that performs adjustments of phase relations as a fundamental process for pattern perception. We do not propose that the neural system has spatially defined "windows" whose size can be adjusted. Rather we propose (that since what our "windows" did was adjust phase relations) that the neural system has mechanism that can and do adjust phase relations so that patterns that could exist on the retinal image actually do come to exist. What do we mean by that? We mean that if we take a static image, see Figure 11, from Exemplar 2, we cannot discern the images we later perceive by dynamically adjusting phase relations (e.g., window size = 75, 77, 83). Those patterns emerge through the action of adjusting phase relations. In a similar way, we propose that a retinal pattern of simulation, supposing we could photograph it, would not show the patterbs that a person experiences. Rather, the patterns that a person perceives are, in part at least, due to processes that shift phase relations between retinal image and experienced representation in way a similar to the way that our window size adjustments changed phase relations. [Note 4]
Computational Level Analysis Revisited. Why does human (and perhaps other) neurology enable the "illusion" of apparent motion? Did evolution teleologically aim for movies and video games? Obviously (to us) not. We propose that the illusion of apparent motion is possible because it is based on neural mechanisms which routinely adjust the kind of phase relations that exist in apparent motion experiments and that are the basis of a broad (yet unspecified) range of dynamic pattern perception phenomena.
What and Why. In David Marr's terms, we want to specify the what and why of our proposed pattern perception mechanism. As we stated above, a being needs to be able extract many distinct dynamic patterns at once. Some of these are spatially coherent, others are spread out spatially. Some are unmoving and stable, others are spatially disconnected flows of movement (like hints of a current though a rapids mostly obscured but appearing here and there in the visual field of the kayaker). How do we form coherent patterns from these fragmented dynamics on the retina?
Computational Constraints. We began this web page with some broad constraints for a mechanism for the extraction of dynamic pattern. For various reasons we would like our proposed mechanism to be able to extract patterns of motion that are not spatially adjacent. We would also like to be able to shift between different patterns of motion. In our example, these spatially separated patterns of motion may be a collective like a herd or a single coherent object, say a deer, that is masked by something irregular like leaves. In short a useful requirement is that a coherent pattern does not have to be coherently adjacent on the retinal image. The experiments with the window size mechanism demonstrate the power of phase relation adjustments in extracting patterns with these general constraints.
A second design constraint we mentioned at the outset is that there be a way of distinguishing dynamic patterns, that there be a way of highlighting one dynamic pattern and then shifting to another dynamic pattern and highlighting it, or, even better, a way of having multiple highlights each pointing at different dynamic patterns in the same scene. Again, the experiments with adjusting phase relations through window size demonstrate the power of a putative neural mechanism that could make analogous adjustments of phase relations.
Summary
Finding the patterns (basins, sub basins, and general pattern) in the world around us is nontrivial and important. We have shown that adjusting phase relations between a visual representation and the dynamics of a system is a way to extract many different patterns within the system's dynamics. We propose that the apparent motion phenomenon presupposes that neurology has a way of of adjusting phase relations between the dynamics of the retinal image and the final representation as experienced. Since that mechanism is in place, we propose it as a candidate for the extraction of complex patterns from complex systems.