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Controlled cellular automata

The second author is partially supported by the Simons Foundation, award # 359510

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  • Cellular Automata have been successfully used to model evolution of complex systems based on simples rules. In this paper we introduce controlled cellular automata to depict the dynamics of systems with controls that can affect their evolution. Using theory from discrete control systems, we derive results for the control of cellular automata in specific cases. The paper is mostly oriented toward two applications: fire spreading; morphogenesis and tumor growth. In both cases, we illustrate the impact of a control on the evolution of the system. For the fire, the control is assumed to be either firelines or firebreaks to prevent spreading or dumping of water, fire retardant and chemicals (foam) on the fire to neutralize it. In the case of cellular growth, the control describes mechanisms used to regulate growth factors and morphogenic events based on the existence of extracellular matrix structures called fractones. The hypothesis is that fractone distribution may coordinate the timing and location of neural cell proliferation, thereby guiding morphogenesis, at several stages of early brain development.

    Mathematics Subject Classification: Primary: 93C55, 92B05; Secondary: 93C95.


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  • Figure 1.  Left: Configuration $ s^0 $ (classical checkerboard). Right: Configuration $ s^1 $ (alternate checkerboard)

    Figure 2.  Let $ s^0 $ be the configuration with value 0.5 in each cell of the grid. Then we have: $ d(s^0, s^1) = d(s^0, s^2) = d(s^0, s^3)<d(s^0, s^4) $

    Figure 3.  In all the figures, red represents positive values and blue represents negative values. The uncontrolled simulation at 4 different timesteps: (a) 0, (b) 5, (c) 20 and (d) 100. The average value at the four timesteps: (a) 0.010, (b) 0.0079, (c) 0.0041 and (d) 0.00014

    Figure 4.  The pictures show a controlled simulation at the following timesteps: (a) 0, (b) 5, (c) 20, (d) 39, (e) 40, (f) 50, (g) 75, (h) 100 and (i) 150. The control switches on at 20, off at 40, back on at 50 and finally off again at 100. Notice that while the control is off, the sign is constant and the magnitude diminishes at a slow exponential rate. When the control is on, the sign alternates with each timestep and the magnitude increases at a slow exponential rate

    Figure 5.  The average is negative for the odd numbered timesteps for which the control is active. For reference, the distance from the controlled simulation at timestep 150 to a grid of zeros is 632.06

    Figure 6.  Top row using the Von Neumann neighborhood. Timesteps: (a) 1, (b) 50 and (c) 100. Bottom row using the Moore neighborhood. Timesteps: (a) 1, (b) 25 and (c) 50. Note that the Moore neighborhood promotes much faster evolution. For both $ \alpha_1 = 0.1 $, $ \alpha_2 = 0.2 $ and $ \alpha_3 = 0.3 $

    Figure 7.  Top row: $ \alpha_1 = 0.05 $, $ \alpha_2 = 0.1 $ and $ \alpha_3 = 0.15 $. Bottom row: $ \alpha_1 = 0.15 $, $ \alpha_2 = 0.3 $ and $ \alpha_3 = 0.05 $. For both, the three times steps are 1, 50 and 100

    Figure 8.  Timesteps: (a) 1, (b) 125, (c) 175, (d) 200, (e) 210 and (f) 250. The fire is diverted by the obstacles

    Figure 9.  Timesteps: (a) 1, (b) 20, (c) 30, (d) 38, (e) 45 and (f) 100. The fire is diverted by the obstacles

    Figure 10.  Timesteps: (a) 1, (b) 20, (c) 25, (d) 35, (e) 50 and (f) 100. The fire is diverted by the obstacles

    Figure 11.  Timesteps: (a) 1, (b) 15, (c) 50 and (d) 100. A simple, uncontrolled cell growth using the Von Neumann neighborhood

    Figure 12.  Timesteps: (a) 1, (b) 5, (c) 10, (d) 30, (e) 60 and (f) 100. An uncontrolled cell growth using a neighborhood consisting of the three grid units directly above the central unit and the three side-by-side units in the row three rows below the central unit

    Figure 14.  Timesteps: (a) 0, (b) 30, (c) 60 and (d) 100. The interaction between growth of normal cells and tumor cells. The competition between the cell masses plays out at the boundary between the cell masses. The normal cells are in red, the probability of a tumor cell developing is in blue

    Figure 13.  Timesteps: (a) 1, (b) 5, (c) 15, (d) 30, (e) 60 and (f) 100. Fractones are placed along three horizontal lines. One is above the initial cell and consists of fractones that stop cell growth; they are arranged in blocks. One is in line with the initial cell (across the middle of the simulation) and greatly increases cell growth. The final line is below the intial cell and also stops cell growth. The placement of the fractones is clearly visible in (f)

    Table 1.  Eigenvalues corresponding to Example 4 and $ m = 5 $

    $ a $ $ a\pm \sqrt{ed} $ $ a\pm \sqrt{3ed} $
    $ a+\sqrt{bc} $ $ a + \sqrt{bc+de\pm 2\sqrt{bcde}} $ $ a + \sqrt{bc+3de\pm2\sqrt{3bcde}} $
    $ a-\sqrt{bc} $ $ a - \sqrt{bc+de\pm 2\sqrt{bcde}} $ $ a - \sqrt{bc+3de\pm 2\sqrt{3bcde}} $
    $ a+\sqrt{3bc} $ $ a+\sqrt{3bc+3de\pm 6\sqrt{bcde}} $ $ a+\sqrt{3bc+3de+2\sqrt{3bcde}} $
    $ a-\sqrt{3bc} $ $ a-\sqrt{3bc+3de\pm 6\sqrt{bcde}} $ $ a-\sqrt{3bc+3de+2\sqrt{3bcde}} $
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