Introduction to the Modeling and Analysis of Complex Systems

11.5. EXAMPLES OF BIOLOGICAL CELLULAR AUTOMATA MODELS 201several asynchronous updating mechanisms possible, such as random updating (a randomlyselected cell is updated at each time step), sequential updating (cells are updatedin a predetermined sequential order), state-triggered updating (certain states trigger updatingof nearby cells), etc. It is often argued that synchronous updating in conventionalCA models is too artificial and fragile against slight perturbations in updating orders, andin this sense, the behaviors of asynchronous CA models are deemed more robust and applicableto real-world problems. Moreover, there is a procedure to create asynchronousCA that can robustly emulate the behavior of any synchronous CA [43].11.5 Examples of Biological Cellular Automata ModelsIn this final section, I provide more examples of cellular automata models, with a particularemphasis on biological systems. Nearly all biological phenomena involve somekind of spatial extension, such as excitation patterns on neural or muscular tissue, cellulararrangements in an individual organism’s body, and population distribution at ecologicallevels. If a system has a spatial extension, nonlinear local interactions among itscomponents may cause spontaneous pattern formation, i.e., self-organization of static ordynamic spatial patterns from initially uniform conditions. Such self-organizing dynamicsare quite counter-intuitive, yet they play essential roles in the structure and function ofbiological systems.In each of the following examples, I provide basic ideas of the state-transition rulesand what kind of patterns can arise if some conditions are met. I assume Moore neighborhoodsin these examples (unless noted otherwise), but the shape of the neighborhoodsis not so critically important. Completed Python simulator codes are available fromhttp://sourceforge.net/projects/pycx/files/, but you should try implementing yourown simulator codes first.Turing patterns Animal skin patterns are a beautiful example of pattern formation inbiological systems. To provide a theoretical basis of this intriguing phenomenon, Britishmathematician Alan Turing (who is best known for his fundamental work in theoreticalcomputer science and for his code-breaking work during World War II) developed a familyof models of spatio-temporal dynamics of chemical reaction and diffusion processes[44]. His original model was first written in a set of coupled ordinary differential equationson compartmentalized cellular structures, and then it was extended to partial differentialequations (PDEs) in a continuous space. Later, a much simpler CA version of the samemodel was proposed by David Young [45]. We will discuss Young’s simpler model here.