globals [ changes updates time showing_inhibitor? ] patches-own [ activator inhibitor hold_activator hold_inhibitor new_pcolor old_pcolor ] TO SETUP ;----------------------------------------------------------------------------------------- ca set updates 0 set time 0 ask patches [ ifelse random-float 100.0 < initial_density [set activator 100 set inhibitor 100 set pcolor white set new_pcolor white] [set activator 0 set inhibitor 0 set pcolor black set new_pcolor black] ] reset-ticks end TO DRAW_WHITE ;-------------------------------------------------------------------------------------- if mouse-down? [ask patches [if ((abs (pxcor - mouse-xcor)) < 2) and ((abs (pycor - mouse-ycor)) < 2) [set pcolor white set activator 100 ; creates live white cells where mouse button is depressed set inhibitor 100]] ] end TO DRAW_BLACK ;-------------------------------------------------------------------------------------- if mouse-down? [ask patches [if ((abs (pxcor - mouse-xcor)) < 2) and ((abs (pycor - mouse-ycor)) < 2) [set pcolor black set activator 0 ; creates dead cells where mouse button is depressed set inhibitor 0]] ] end TO DRAW_RANDOM ;-------------------------------------------------------------------------------------- if mouse-down? [ask patches [if ((abs (pxcor - mouse-xcor)) < 2) and ((abs (pycor - mouse-ycor)) < 2) [ifelse random-float 100.0 < initial_density [set activator 100 set inhibitor 100 set pcolor white ] [set activator 0 set inhibitor 0 set pcolor black ] ] ] ] end TO GO ;-------------------------------------------------------------------------------------------- set changes 0 ask-concurrent patches [ ;;ask patches [ repeat x_active ; diffuses activator chemical in x direction [set activator ((activator + ([activator] of patch-at 1 0) + ([activator] of patch-at (-1) 0)) / 3)] repeat y_active ; diffuses activator chemical in y direction [set activator ((activator + ([activator] of patch-at 0 1) + ([activator] of patch-at 0 (-1))) / 3)] repeat x_inhibit ; diffuses inhibitor chemical in x direction [set inhibitor ((inhibitor + ([inhibitor] of patch-at 1 0) + ([inhibitor] of patch-at (-1) 0)) / 3)] repeat y_inhibit ; diffuses inhibitor chemical in y direction [set inhibitor ((inhibitor + ([inhibitor] of patch-at 0 1) + ([inhibitor] of patch-at 0 (-1))) / 3)] set hold_activator activator set hold_inhibitor inhibitor set old_pcolor new_pcolor ifelse (bias + activator - inhibitor + (random (2 * noise)) - noise) > 0 [set activator 100 ; turned on set inhibitor 100] [set activator 0 ; turned off set inhibitor 0] ; set_color ifelse activator = 100 [ set new_pcolor white ] [ set new_pcolor black ] ifelse field_display = "none" [ set pcolor new_pcolor ] [ ifelse field_display = "activator" [ set pcolor scale-color red hold_activator 0 100 ] [ set pcolor scale-color blue hold_inhibitor 0 100 ] ] if new_pcolor != old_pcolor [ set changes changes + 1 ] ] set updates changes set time time + 1 plot updates display end to set-random-seed random-seed read-from-string user-input "Random seed?" end @#$#@#$#@ GRAPHICS-WINDOW 324 21 736 434 -1 -1 4.0 1 10 1 1 1 0 1 1 1 -50 50 -50 50 0 0 1 ticks 30.0 BUTTON 21 20 89 53 set seed set-random-seed NIL 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 21 95 159 128 x_active x_active 0 10 3.0 1 1 NIL HORIZONTAL SLIDER 162 95 304 128 x_inhibit x_inhibit 0 30 5.0 1 1 NIL HORIZONTAL SLIDER 21 133 159 166 y_active y_active 0 10 3.0 1 1 NIL HORIZONTAL SLIDER 162 133 304 166 y_inhibit y_inhibit 0 30 5.0 1 1 NIL HORIZONTAL SLIDER 162 58 304 91 bias bias -10 10 0.0 1 1 NIL HORIZONTAL BUTTON 95 20 158 53 NIL setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 236 21 303 54 NIL go T 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 22 208 109 241 draw black draw_black T 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 13 296 111 341 simulation time time 0 1 11 MONITOR 114 296 206 341 changed cells updates 0 1 11 BUTTON 215 209 304 242 draw white draw_white T 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 54 450 154 495 % changed updates * 100 / (world-width * world-height) 2 1 11 CHOOSER 90 247 228 292 field_display field_display "none" "activator" "inhibitor" 0 BUTTON 161 21 229 54 go once go NIL 1 T OBSERVER NIL NIL NIL NIL 1 PLOT 13 345 309 495 cells changed NIL NIL 0.0 10.0 0.0 100.0 true false "" "" PENS "default" 1.0 0 -16777216 true "" "" SLIDER 22 58 159 91 initial_density initial_density 0 100 50.0 1 1 % HORIZONTAL SLIDER 92 170 231 203 noise noise 0 100 0.0 1 1 NIL HORIZONTAL BUTTON 114 209 210 242 draw random draw_random T 1 T OBSERVER NIL NIL NIL NIL 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 This program simulates vertebrate skin patterns by use of a short-range activator and a long-range inhibitor. Each pigmented cell that is "turned on" (the white cells in the program) produces 100 units of an activating morphogen and 100 units of an inhibiting morphogen. Both chemicals diffuse a specified number of times in the X and Y directions, and then each patch is evaluated. Patches with more activator than inhibitor turn white, and patches with more inhibitor than activator turn black. The process is then repeated; a steady-state pattern is usually quick to emerge. ## HOW TO USE IT The SETUP button randomly colors each patch and deposits the morphogens on patches that are white. The GO button executes the model according to the rules above. The GO ONCE button executes a single cycle of the simulation, sp that the changes are easier to see. The DRAW WHITE and DRAW BLACK buttons allow the user to use the mouse to draw white and black patches, respectively. DRAW RANDOM allows drawing a region that is random black and white. The X_ACTIVE, Y_ACTIVE, X_INHIBIT, and Y_INHIBIT sliders control the diffusion of the activator and inhibitor morphogens in each direction. More specifically, the slider variables control the number of times that each patch shares its contents with its two neighbors (in the given direction), such that each patch is left with an equal share of the patch's original morphogens. INITIAL_DENSITY controls the initial density of activated cells (white patches). The FIELD DISPLAY chooser controls the information displayed. In its default position (NONE) is shows the color (white or black) of the cells. If ACTIVATOR is chosen, then the concentration of the activator morphogen is displayed as a shade of red. If INHIBITOR is chosen, then the concentration of the inhibitor morphogen is displayed as a shade of blue. In either case the concentration is displayed after the morphogen has diffused, but before an activated cell secretes more of the morphogens. ## THINGS TO NOTICE Modification of the absolute and relative sizes of the slider variables can cause a variety of patterns to emerge, resembling everything from zebras to vermiculated rabbit fish. The display shows the number and percentage of cells (patches) that change color on each time step, and plots this as a function of time. Notice how quickly the pattern stabilizes, but some of the cells (typically less than 1%) are always changing, showing that the pattern is not static but rather is a "stationary state." You can see this by changing some of the parameters (diffusion rates and bias), which may destabilize the pattern and cause it to converge to a new stationary state. ## 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? Try very large and very small values (e.g., 99%, 1%). 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 BIAS , from trial to trial, you will note that for large positive numbers you will invariably get completely white fur and for large negative numbers you will invariably get completely black fur (why is that?). For values in between it fluctuates. That happens partially because the initial setting of black/white coloration has a random element to it. Try changing the sliders to have different values in the X and Y directions. Once the pattern has stabilized, use the DRAW WHITE and DRAW BLACK controls to damage the pattern. Then use GO or GO ONCE to see whether the pattern will heal and how quickly. Does healing always restore the original pattern? ## EXTENDING THE MODEL 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 Voting model, in the Social Science section of the NetLogo model library, is based on simpler rules but generates patterns that are similar in some respects. ## CREDITS AND REFERENCES Modified by B. J. MacLennan Sep. 7, 2003 for Java StarLogo 2.0.2, Sep. 15, 2007 for NetLogo 3.2.4, Sep. 12, 2008 for NetLogo 4.0.3, and Jan. 27, 2015 for NetLogo 5.1.0 from original version by William Thies on Scott Camazine's website. Modifications included monitoring of changed cells, field displays, and documentation. Some of the above documentation is from the Fur model in the NetLogo model library (which is programmed differently, however): 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. In other publications, please use: Copyright 2003 Uri Wilensky. All rights reserved. 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