globals [ inner-nbd ;; the set of inner neighborhood [dx dy] pairs outer-nbd ;; the set of outer neighborhood [dx dy] pairs ] patches-own [ new-color ;; currently, always either white or black ] to calculate-nbds ;; (for this model to work with NetLogo's new plotting features, ;; __clear-all-and-reset-ticks should be replaced with clear-all at ;; the beginning of your setup procedure and reset-ticks at the end ;; of the procedure.) __clear-all-and-reset-ticks set inner-nbd compute-neighborhood R1 set outer-nbd compute-neighborhood R2 setup end to-report compute-neighborhood [R] let neighborhood [] let diameter n-values (2 * R + 1) [? - R] foreach diameter [ let delta-x ? foreach diameter [ if (magnitude delta-x ?) <= R [ set neighborhood fput (list delta-x ?) neighborhood ] ] ] report neighborhood end to-report magnitude [delta-x delta-y] ifelse metric = "L1" [ report abs delta-x + abs delta-y ] [ ifelse metric = "L2" [ report round (sqrt (delta-x ^ 2 + delta-y ^ 2)) ] [ ifelse metric = "L_infinity" [ report max list (abs delta-x) (abs delta-y) ] [ report "unimplemented metric" ] ]] end to setup cp if inner-nbd = 0 or outer-nbd = 0 ;; they didn't CALCULATE_NBDS [ calculate-nbds ] ask patches [ ifelse random-float 100.0 < initial-density [ set pcolor black set new-color black ] [ set pcolor white set new-color white ] ] end to go ask patches [ pick-new-color ] ask patches [ set pcolor new-color ] end to pick-new-color ;; patch procedure let activation sum-within inner-nbd let inhibition (sum-within outer-nbd) - activation let difference h + J1 * activation + J2 * inhibition ifelse difference > 0 [ set new-color black ] [ if difference < 0 [ set new-color white ] ] ;; note that we did not deal with the case that difference = 0. ;; this is because we would then want cells not to change color. end to-report sum-within [nbd] let activity 0 foreach nbd [ ifelse black = [ pcolor ] of patch-at (first ?) (item 1 ?) [ set activity activity + 1 ] [ set activity activity - 1] ] report activity end ;; The following two procedures allow patches to be colored black or white to draw-black if mouse-down? [ask patches [if ((abs (pxcor - mouse-xcor)) < 4) and ((abs (pycor - mouse-ycor)) < 4) [set pcolor black] ]] end to draw-white if mouse-down? [ask patches [if ((abs (pxcor - mouse-xcor)) < 4) and ((abs (pycor - mouse-ycor)) < 4) [set pcolor white] ]] end @#$#@#$#@ GRAPHICS-WINDOW 285 10 539 285 30 30 4.0 1 10 1 1 1 0 1 1 1 -30 30 -30 30 0 0 1 ticks 30.0 BUTTON 185 200 275 233 NIL setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 185 280 275 313 NIL go T 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 5 320 177 353 initial-density initial-density 0 100 50 1 1 % HORIZONTAL BUTTON 185 240 275 273 step go NIL 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 5 65 177 98 R1 R1 0 10 1 1 1 cells HORIZONTAL SLIDER 5 105 178 138 R2 R2 0 10 6 1 1 cells HORIZONTAL BUTTON 95 360 177 393 draw black draw-black T 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 5 360 89 393 draw white draw-white T 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 5 200 177 233 J1 J1 -5 5 1 0.1 1 NIL HORIZONTAL SLIDER 5 240 177 273 J2 J2 -5 5 -0.1 0.1 1 NIL HORIZONTAL SLIDER 5 280 177 313 h h -6 6 0 0.5 1 NIL HORIZONTAL CHOOSER 5 15 175 60 metric metric "L1" "L2" "L_infinity" 1 BUTTON 185 65 275 98 calculate nbds calculate-nbds NIL 1 T OBSERVER NIL NIL NIL NIL 1 TEXTBOX 5 150 280 191 Calculate the neighborhoods before running the simulation. 12 0.0 1 @#$#@#$#@ ## WHAT IS IT? Could it be that a single mechanism underlies such diverse patterns such as the stripes on a zebra, the spots on a leopard, and the blobs on a giraffe? This model is a possible explanation of how the patterns on animals' skin self-organize. If the model is right, then even though the animals may appear to have altogether different patterns, the rules underlying the formation of these patterns are the same and only some of the values (the numbers that the rules work on) are slightly different. Thinking of the formation of fur in terms of rules also helps us understand how offspring of animals may have the same type of pattern, but not the same exact pattern. This is because what they have inherited is the rules and the values rather than a fixed picture. The process by which the rules and values generate a pattern is affected by chance factors, so each individual's pattern is different, but as long as the offspring receive the same rules and values, their own fur will self organize into the same type of pattern as their parents'. ## HOW IT WORKS We model the animal skin by a square array of many melanocytes (pigment cells) that are each in either of two states: colorful ('D' for differentiated) or not-colorful ('U' for undifferentiated). The state of a cell can flip between D and U. The color cells (the D's) have an activity level of +1 and the uncolored cells (the U's) have and activity level of -1. Each cell, whether or not it is colorful, is itself the center of its own neighborhood. For now, suppose the neighborhood is a circle. Say this circular neighborhood has a radius of 6 cells. This means that the cell in the center can be affected by other cells that are as far as 6 cells away from it in any direction. So the activity level of a D cell within this circle will affect this central cell (but a D cell 7 cells away will not directly or immediately affect it). Also, each cell has an inner circle of radius, say, 3 cells. The total activity (+1s and -1s) of the cells within the inner circle each contributes to the activation of the central cell. Between the inner circle and the perimeter of the outer circle we have a ring of cells that are more than 3 cells away from the center but 6 or less cells away from the center. The total activity in this outer ring contributes to the inhibition of the central cell. So at every moment each cell is both activated and inhibited by cells in its circle and the question is: will it ultimately be activated and become colorful or be inhibited and lose its color (or just remain the way it was). The logic is that if the activation power is bigger than the inhibition power then the cell will become colorful and vice versa (and if activation and inhibition are balanced then nothing happens). The idea of "power" is that it's not enough to know total activity of each cell affecting the center cell, but one must multiply each cell by its "power": the total activity of the cells in the inner circle is multiplied by J1 (a positive quantity), giving the activation power; the total activity in the ring of cells is multiplied by J2 (a negative quantity) to give the inhibition power. If the activation exceeds the inhibition in magnitude, the cell becomes activated and D (colored), but if the inhibition is greater in magnitude, the cell becomes U (uncolored). Furthermore, cells may be biased more towards activation or inhibition, so that inhibition has to exceed inhibition be a certain threshold, or vice versa. (A tricky point to notice is that while a certain D-cell is activating a neighboring cell, this same D-cell can be inhibiting a different cell further away.) Here are the rules that summarize what we've been discussing: add up all the activities (+1 or -1) of the cells in the ring and call this number I (inhibitory activity), and add up all the activities of the cells in the iner circle and call this number A (activating activity). Let H be the bias (which makes activation or inhibition more likely, depending on whether it's positive or negative). Then compute J1*A + J2*I + H (recalling that J2 < 0), and: - if it is > 0, set activitiy of the central cell to +1 - if it is < 0, set the activity of the central cell to -1 - if it is = 0, leave the central cell unchanged Note that this computation happens to all cells at the same time. After the first step and once the cells have been set accordingly, the entire business starts over at the next step. Once again, the cells are counted up according to the same rule. The rules have not changed but because some of the D cells are now U and vice versa we might get different counts and because of that -- different results of the "fight" between the activation and inhibition powers. So what you see is that from step to step the individual cells often change from black (representing D or color cells) to white (representing U or no-color cells) and the overall impression is that the configuration of white and black changes as a whole. But these configurations are not random. You will see how these configurations often take form. Understanding how each cell behaves, as we have explained above, can help understanding how these global patterns take form. ## HOW TO USE IT In order that your first experiment will more-or-less match the explanations above, you should choose to set the initial-density slider to 50% (that gives each cell an equal chance of being white or black to start with and so the whole window will be roughly 50% black), set the R1 (inner radius) slider to 1 and the R2 (outer radius) sliders to 6, set inner interaction strength J1 to 1 and outer interacton strength J2 to -0.1 (which makes inner cells activating and outer cells inhibiting), and set H (bias) to 0. Now press SETUP. (In later experiments you are welcome to change those settings in various combinations.) It will converge in very few cycles (less than 20). If you press STEP the model will advance a single step. If you press GO the model will keep stepping indefinitely. The program precomputes the neighborhoods. Because of this, only press CALCULATE NBDS when you change the metric or radius sliders. If you only change the INITIAL-DENSITY, J1, J2, and H sliders or if you'd like to run the model again with the same settings, press RESTART instead of CALCULATE NBDS. The SETUP button doesn't ask the patches to recalculate their neighborhoods. DRAW BLACK and DRAW WHITE allow patches to be colored by hand, while the simulation is running or when it is stopped. ## THINGS TO NOTICE As the model runs, patterns may begin to emerge. Eventually, they stabilize. (Well, sometimes the model will run into an endless flip-flop between two states, but we could call that dynamic stability.) Even when it seems to come to a halt, the model is still running and executing the commands and carrying out the computations, but nothing is changing visibly. This is because for each and every cell the power of activators is equal to that of the inhibitors, so nothing changes. ## THINGS TO TRY Run the model with different INITIAL-DENSITY settings. How, if at all, does the value of the INITIAL-DENSITY affect the emergent pattern? Do you get the same pattern? Do you get a different pattern? Does it take longer? Note how fragile the self organization of the cells is to slight changes in parameters. If you hold all other factors and slightly change just the RATIO, from trial to trial, you will note that for small ratios you will invariably get completely white fur and for high ratios you will invariably get completely black fur (why is that?). For ratios in between it fluctuates. That happens partially because the initial setting of black/white coloration has a random element to it (see the RESTART procedure in the code). To see that even when the state has stabilized, the model is still computing it (called a "stationary state" or "dynamic equilibrium"), try the following. Click DRAW BLACK or DRAW WHITE and change some of cells. Notice how the pattern is almost instantly restored (not always exactly, but to a qualitatively similar pattern). To see the effects of more extensive damage, stop the simulation and use DRAW BLACK and DRAW WHITE to change large areas of cells. Then restart the simulation and see what happens. ## EXTENDING THE MODEL If you find a combination of slider and switch values that consistently give you the fur patterns of a favorite animal, you could create a button, for instance "Zebra," that sets the sliders to those values. That way, if you make several of these, you can go on a virtual safari tour by moving between your favorite animals. One such combination that you could set in a single button could be: Parameters R1 3 R2 6 J1 1 J2 -0.1 H 0 INITIAL-DENSITY 50% You could call this, perhaps, Fish. The model provides three neighborhood shapes, L1, L2, and L_infinity. Add some other neighborhood shapes. In particular, add oval or oblong shapes, and add sliders so that the user can control their width and height. How about adding more colors? What could be the logic here? If you introduced, say, red, you would have to decide on specific conditions under which that color would appear. Also, you'd have to decide how that color influences other cells. ## RELATED MODELS The Fur model, in the Biology section, implements a similar model of pattern formation, using diffusing "morphogens" to implement short-range activation and long-range inhibition. The Voting model, in the Social Science section, is based on simpler rules but generates patterns that are similar in some respects. ## CREDITS AND REFERENCES Developed 2008-09-15 by B. J. MacLennan to be implemented and behave similarly to the activation-inhibition cellular automaton model of pattern formation in Bar Yam (1997, sec. 7.2). Bar Yam, Y. (1997). Dynamics of Complex Systems. Reading, MA: Perseus. http://www.necsi.org/publications/dcs/. Much of this documentation and a few lines of code were borrowed from: Wilensky, U. (2003). NetLogo Fur model. http://ccl.northwestern.edu/netlogo/models/Fur. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. All rights reserved. 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