| Observations |
| Excitable media can support circular and spiral waves | ||
| Spiral formation can be triggered in a variety of ways | ||
| All seem to involve inhomogeneities (broken symmetries): | ||
| in space | ||
| in time | ||
| in activity | ||
| Amplification of random fluctuations | ||
| Circles & spirals are to be expected | ||
| StarLogo Simulation of Streaming Aggregation |
| chemical diffuses | |||
| if cell is refractory (yellow) | |||
| then chemical degrades | |||
| else (itÕs excitable, colored white) | |||
| if chemical > movement threshold then | |||
| take step up chemical gradient | |||
| else if chemical > relay threshold then | |||
| produce more chemical (red) | |||
| become refractory | |||
| else wait | |||
| Demonstration of StarLogo Simulation of Streaming |
| Run SlimeStream.slogo |
| Differentiation & Pattern Formation |
| A central problem in development: How do cells differentiate to fulfill different purposes? | |
| How do complex systems generate spatial & temporal structure? | |
| CAs are natural models of intercellular communication |
| Zebra |
| Vermiculated Rabbit Fish |
| Activation &
Inhibition in Pattern Formation |
| Color patterns typically have a charac-teristic length scale | ||
| Independent of cell size and animal size | ||
| Achieved by: | ||
| short-range activation Þ local uniformity | ||
| long-range inhibition Þ separation | ||
| Emergent Hierarchical Structure |
| Characteristic length scale | |
| Independent of cell size and space size | |
| Structures created at intermediate level |
| Interaction Parameters |
| R1 and R2 are the interaction ranges | |
| J1 and J2 are the interaction strengths |
| CA Activation/Inhibition Model |
| Let states si ë {Ð1, +1} | |
| and h be a bias parameter | |
| and rij be the distance between cells i and j | |
| Then the state update rule is: |
| Example (R1=1, R2=6, J1=1, J2=Ð0.1, h=0) |
| Effect of Bias (h = Ð6, Ð3, Ð1; 1, 3, 6) |
| Effect of Interaction Ranges |
| Demonstration of StarLogo Program for Activation/Inhibition Pattern Formation |
| Run Pattern.slogo |
| Abstract Activation/Inhibition Spaces |
| Consider two axes of cultural preference | ||
| E.g. hair length & interpersonal distance | ||
| Fictitious example! | ||
| Suppose there are no objective reasons for preferences | ||
| Suppose people approve/encourage those with similar preferences | ||
| Suppose people disapprove/discourage those with different preferences | ||
| What is the result? | ||
| Emergent Regions of Acceptable Variation |
| A Key Element of Self-Organization |
| Activation vs. Inhibition | ||
| Cooperation vs. Competition | ||
| Amplification vs. Stabilization | ||
| Growth vs. Limit | ||
| Positive Feedback vs. Negative Feedback | ||
| Positive feedback creates | ||
| Negative feedback shapes | ||
| Additional Bibliography |
| Kessin, R. H. Dictyostelium: Evolution, Cell Biology, and the Development of Multicellularity. Cambridge, 2001. | |
| Gerhardt, M., Schuster, H., & Tyson, J. J. ÒA Cellular Automaton Model of Excitable Media Including Curvature and Dispersion,Ó Science 247 (1990): 1563-6. | |
| Tyson, J. J., & Keener, J. P. ÒSingular Perturbation Theory of Traveling Waves in Excitable Media (A Review),Ó Physica D 32 (1988): 327-61. | |
| Camazine, S., Deneubourg, J.-L., Franks, N. R., Sneyd, J., Theraulaz, G.,& Bonabeau, E. Self-Organization in Biological Systems. Princeton, 2001. | |
| P‡lsson, E., & Cox, E. C. ÒOrigin and Evolution of Circular Waves and Spiral in Dictyostelium discoideum Territories,Ó Proc. Natl. Acad. Sci. USA: 93 (1996): 1151-5. | |
| SolŽ, R., & Goodwin, B. Signs of Life: How Complexity Pervades Biology. Basic Books, 2000. |