Do
not click on or manipulate the figure at the right, it is only an image. The
interactive interface is below.
Pattern Perception in a Dynamic Universe
Exemplar 2 Applet
Emergent Dynamic Forms Derived from Phase Relations
CITATION: Malloy, T.E., Butner, J., & Jensen, G. C. (In Press). The emergence of dynamic form through phase relations in dynamic systems. Nonlinear Dynamics, Psychology, and Life Sciences.
How might the processes which enable apparent motion be a basis for pattern perception in general?
Contents
Instructions
Information about Exemplar 2
Interactive Simulation: Derived Patterns Emerging in Time and Space
Do
not click on or manipulate the figure at the right, it is only an image. The
interactive interface is below.
Adjusting your Computer's speed to your Monitor's speed: Most monitors cannot paint accurately faster than 66 to 77 frames per second (fps).
Moreover, apparent motion effects are most robust when stimuli are presented about 25 frames per second (fps).
Use Delay. Before (or after) you start the system by pressing Play, click the Use Delay radio button (highlighted in yellow) and then set the Delay (between iterations) slider (highlighted in yellow) so that the fps indicator (highlighted in violet) falls somewhere between 25 and 66 fps. In that range you should experience apparent motion effects. If your computer is slow, you may not need to use the Speed control.
Clicking on the Delay Slider Bar. To get finer-grain adjustments, you may click on the slider bar for the Delay Control, and the delay will increment (or decrement) one msec.
Adjusting Window Size. The critical manipulation you will do is adjust the window size. This adjustment sets the phase relations among (1) your perceptual ability to perceive apparent motion/stability, (2) the dynamics of the system, and (3) the representation of those dynamics in a window.
Highlighted in blue on the illustration is the slider bar for adjusting theWindow Size variable. Just drag the slider along the slider bar to increase or decrease the Window Size.
Clicking on the Window Size Slider Bar. To get finer-grain adjustments, you may click on the slider bar for the Window Size Control, and the Window Size will increment (or decrement) one iteration.
Separator Bar. You may want to see as many as 100 iterations, but the output frame (the part of the applet where you see the black and white squares of the historical trace) does not show that many iterations. You may drag the separator bar (highlighed in orange) to increase the size of the output frame. Or, you may click the little left-pointing arrow (highlighted by an orange oval toward the top of the illustration) to expand the output frame to its maximum size and to hide the control frame. If you do, remember you can click on the little right-pointing arrow to return the output frame to its normal size and to show the control frame.
Perturbing the System. Although not particularly related to the theoretical points we are making here, you may want to perturb the system and examine other basins. The perturb button will do this. The slider bar associated with the Perturb button adjusts the number of nodes whose states are reversed when you click the perturb button. Do not perturb the system until you finish the set of experiments outlined below.
Basic Information about the Exemplar 2 Dynamic System
We will repeat the figures from the referring page for convenience here as well as add new material.
Basins and their Attractors. Exemplar 2 has three known basins, one with an Attractor Cycle that has length L=50 (our primary focus) and two with attractors that have L=4. In one sample of 10,000 perturbations, the relative frequency of the basin with L=50 was .6985, the relative frequency of the first basin with a L=4 attractor cycle was .3009, and the relative frequency of the second L=4 basin was .0006.
Sub-basins (Sub-Cycles) . This term refers to a subset of the system's nodes that are cycling at a higher frequency (smaller number of iterations) in that attractor cycle than L (the overall length of the attractor cycle in iterations). The basin with the Attractor of L=50 has no sub-basins. That means that (other than window sizes that are a factor of 50) no row (expressing the dynamics of a single node) ever freezes up. For the two basins that have L=4 attractors, the first, relatively more frequent L=4 attractor cycle, has sub-basins of length SubL=2. The third basin has never been observed (its existence was determined analytically by the TAO Tool).
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| Figure 1. Historical Trace from second exemplar. Figure shows two cycles of an attractor of length L=50. There are no sub-cycles in the standard sense. A rich set of complex patterns, some stable, others dynamic, emerge from adjusting the phase relations between Window Size and the system's dynamics. |
Figure 1 shows 100 iterations from the attractor cycle with length L=50 as a historical trace. It has no sub-basins, but a rich set of patterns, both dynamic and stable, emerge as the phase relations between system dynamics and window size is adjusted.
Some of these patterns are hinted at in the static snapshot shown in Figure 1, but most are not, especially the dynamic ones. The emergence of such regularity that is not tied to components of the system (nodes) poses serious difficulties for the attempts of measure a physical system to find defining characteristics of the experienced percept. The dynamic patterns in many cases have little to do with the static snapshots of the behavior of individual nodes shown in Figure 1.
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| Figure 2. 100 iterations of an attractor with L=4 from Exemplar 2. This basin has sub-basins (attractor sub-cycles) of length SubL=2. |
Patterns Emerging from Phase Relations in Time and Space
Click "Use Delay". This will allow you to adjust speed.
Press Play. First: Adjust the speed to around 25 frames per second using the Delay control.
After you adjust your speed to 25 fps: The output panel of the interface (left) should show what appears to be a pulsing matrix surrounding stable islands of pattern. These islands are mostly black, but some aspects are white. (The pusling matrix a dynamic pattern as well.)
These emergent patterns are not sub-cycles of attractors in the usual sense. That is, there is no single row that is completely static across all iterations. Thus they do not reflect fundamental frequencies of the attractor.
Where and When. Rather, these patterns exist within capsules of time; they exist on some iterations (horizontal axis) but not on others. They exist in an intersection of where (nodes on the vertical axis) and when (certain iterations).
The default Window Size for this applet is 75; so you are looking at patterns that emerge for the phase relations that result from a window size of 75. There are many more patterns of interest for different window sizes.
Window Size = 76. Click on the Window Size slider bar to the right of Slider Control. Window Size should jump up to 76. (Get a Window Size = 76 however you can.)
Now notice that at Window Size = 76 the the islands begin move to the left. Have they changed shape? (You can go back and forth between Window Sizes 76 and 75 to answer that question.)
Forward Flow and Backward Flow. Take a pen and let your eyes follow it as you drag it across the screen back and forth, right and left on top of the dynamic pattern. Change the speed with which you move the pen. (The arrow from the mouse works almost as well as a pen tip.) Notice that as your eyes focus on the tip of the pen to the right you will see a dynamic pattersn moving in that direction (You may have to adjust the speed of movement of the pen tip.) Notice, conversely, that as your eyes focus on the tip of the pen to the left you will see a dynamic pattersn moving in that direction also.
Where do any of these patterns go when you press Stop? (Press Play again.) These patterns exist in the dynamics of the system as it unfolds over time. They can't be found in any static aspect of the system. How is that like the perceptual qualities of a rapids as a kayaker wends through them?
Window Size = 77. Click on the slider bar and move Window Size up one notch to 77. What are the changes?
Ambiguous Motion: Two Directions at Once. With the system running focus your attention on row five counting down from the top of the black and white output. It is lined up with the red hash mark (indicating the fifth row from the top). Start at the left side and run a pen tip (or mouse arrow) to the right along that row fairly rapidly. Do the black squares seem to flow to the right when you do? (You may have to adjust how fast you run your pen pointto the righ; in fact it may take a little practicet.) Now reverse directions. Run your pointer to the left along the fifth row. Do the black squares now seem to move left? It seems easier to see the squares move left than right; but the rightward movement can be learned. What's it mean for our perception of dynamic systems that direction of movment can shift relatively?
Window Size = 83: More Ambiguous Motion . Click the slide bar to adjust Window Size to 83.
Run a pen across the screen both right and left. Notice what patterns emerge as you do so and how they are different for the two directions. After you use a pen for a while, notice how you can get some of these patterns, for both directins, simply by moving your eyes.
Explore. Play with Window Size (which is the number of iterations being painted to the screen at any moment). There are many interesting dynamic patterns which emerge. These patterns never exist across all iterations for any one node; rather, they exist only for some itertions and not for others.
Some patterns are easier to see if you use a very long delay between iterations resulting in a very slow rate of presentation around 4 or 5 fps.
Finding the Fundamental Frequencies of the L=50 Attractor Pattern. You can find the fundamental frequency of the attractor pattern in the usual way by setting the Window Size to a multiple of 50 (50, 100, 150, 200).
Perturb. You can perturb the system and explore the one rather simple and uninteresting basin that you are likely to find, if you wish. Just push the Perturb button a few times.