Self-organizing Systems 1 Self-organizing Systems Introductory article Scott Camazine, Boalsburg, Pennsylvania, USA CONTENTS What is self-organization? Simulation of self-organizing systems Emergent properties in a self-organizing system Self-organization in the neural and cognitive sciences How does a self-organizing system work? Self-organizing systems are physical and biological EMERGENT PROPERTIES IN A systems in which pattern and structure at the global SELF-ORGANIZING SYSTEM level arises solely from interactions among the lower-level components of the system. The rules The term `emergence' refers to a process by which a specifying interactions among the system's com- system of interacting elements acquires qualita- ponents are executed using only local information, tively new pattern and structure that cannot be without reference to the global pattern. understood simply as the superposition of the indi- vidual contributions. Although the term may sug- WHAT IS SELF-ORGANIZATION? gest that something mysteriously or magically materializes within the system, this is not the case. Self-organization is a process whereby pattern at The human mind is generally poor at predicting the the global level of a system emerges solely from properties of systems that consist of multiple com- interactions among the lower-level components of ponents with complex, dynamic interactions. Thus, the system. The rules specifying the interactions even if one has a full knowledge of the system's among the system's components are executed elements and their mode of interaction, the collect- using only local information, without reference to ive properties of a self-organizing system often the global pattern. Examples of self-organization seem to arise unexpectedly. include a wide range of pattern formation pro- cesses in both physical and biological systems: sand grains assembling into rippled dunes, chem- HOW DOES A SELF-ORGANIZING ical reactants forming swirling spiral patterns, the SYSTEM WORK? patterns on sea shells, or fish swimming in coordin- ated schools (Figure 1). `Pattern' is used here in a An example may make this abstract description of broad sense to refer not only to a particular ar- self-organization and emergent properties clearer. rangement of objects in space, but also to structure Striped and mottled patterns are found throughout and organization in time. An example is the re- nature ± on a zebra's coat, on a fish's skin, and in the markable synchronous flashing that sometimes de- ocular dominance columns of the brain (Figure 2). velops among aggregations of thousands of fireflies Experimental and theoretical work suggests that in southeast Asia. In neurobiology, self-organiza- these patterns develop from a few simple rules tion contributes to temporal structure and anatom- that are continually iterated among the components ical organization in systems ranging from central of the system. Suppose, for example, that each pig- pattern generators in simple invertebrates to cogni- ment cell on a zebra's coat could either produce a tion in humans. dark pigment or not, depending on a certain chem- In self-organizing systems, pattern and organiza- ical activation above or below a certain threshold tion develop through interactions internal to the level. Further suppose that the cells in the skin system, that is, without the intervention of external produced both a chemical activator and an antagon- influences, such as a `leader' who directs or over- istic inhibitor (called `morphogens'), which both sees the process. The pattern is an emergent prop- diffused through the skin. The rules regulating the erty of the system itself, rather than a property state of each cell ± either `on' (producing pigment) imposed upon the system by an external supervis- or `off' (not producing pigment) ± depend on the ory influence. relative strengths of the activation and inhibition, 2 Self-organizing Systems (a) (a) (b) (b) 5 mm (c) (c) Figure 2. Striped and mottled patterns found in bio- Figure 1. Examples of self-organized pattern formation logical systems. (a) Alternating stripes on a zebra's coat in physical, chemical, and biological systems. (a) Sand (Equus grevii). (b) Mottled pattern of pigments on the skin dune stripes. (b) Belusov±Zhabotinsky chemical reaction of a vermiculated rabbitfish (Siganus vermiculatus). (image courtesy of Stefan C. MuÈ ller). (c) A cone shell (c) Ocular dominance stripes in the visual cortex of a from Ceylon. macaque monkey. Regions receiving inputs from one eye are shown in black, and regions receiving inputs from the other eye are shown in white. Adapted from: their diffusion rates, the initial distribution of the Hubel DH and Wiesel TN (1977) Functional architecture cells, and their thresholds for pigment production. of the macaque monkey visual cortex. Proceedings of the Royal Society, Series B 198: 1±59. In 1952, Alan Turing first suggested the general scheme for this mechanism of self-organized pat- tern formation. In 1972, A. Gierer and H. Meinhardt developed a model as shown in Figure 3. Their antagonist, the inhibitor. Since the inhibitor dif- system has a series of sites that are the source of fuses rapidly into the surroundings, the result is a a short-range activator, which has two functions: local increase in the activation and a long-range to promote its own productions (autocatalysis), antagonistic effect that restricts the self-enhancing and to cause an increase in the production of an reaction and keeps it localized. Self-organizing Systems 3 SIMULATION OF SELF-ORGANIZING a two-dimensional grid or lattice. Each cell is char- SYSTEMS acterized by its location on the grid and its condi- tion (state). Cells interact with each other according Because of the difficulty of predicting the behavior to a set of simple rules which take into account their of these systems, computer simulations are a useful proximity to neighboring cells, their own state, and means of performing `thought experiments' and for the states of their neighbors. The rules specify the better understanding how these systems work. One transition of the cell from one state to another as the method of modeling these systems is by the use of system evolves over time. nonlinear differential equations. Another method Consider the example of animal coat patterns is to simulate the system by means of cellular presented above. This can be implemented as a automata. cellular automaton model that consists of a set of A cellular automaton is a simulation that is dis- cells laid out on a grid. Each cell is initially assigned crete in time, space, and state. Typically, the a state randomly, `on' or `off'. Each `on' cell is components (cells) of the system are arranged on assumed to produce a specified amount of activator and a specified amount of inhibitor that diffuse at different rates across the grid. In the simulation, each `on' cell is represented as black and each `off' cell is represented as white. At each timestep, the Activator program calculated the net amount of activation at each site on the grid. This is determined as the difference between the sum of all the activation from the cells in the neighborhood and the sum of all the inhibition from those cells in the neighbor- Inhibitor hood. If this total is above a prespecified threshold level, then the cell at that site is assigned the `on' Figure 3. Reaction scheme for pattern formation by auto- state; otherwise, it is assigned the `off' state. In this catalysis and long-range inhibition. The two arrows denote activation, with the activator stimulating both its manner, cells switch from one state to another own production (autocatalysis) and that of the inhibitor. according to a single rule. The program continually The jagged line shows the effect of the inhibitor which iterates the rule, causing a pattern to emerge from provides negative feedback by inhibiting the effect of the the initial random array of `on' and `off' cells, as activator. Adapted from: Meinhardt H (1995) The Algo- shown in Figure 4. For one set of diffusion rules, rithmic Beauty of Sea Shells. Berlin, Germany: Springer. an irregular mottled pattern develops. When the (a) (b) Figure 4. Cellular automaton simulations of pattern formation according to an activation±inhibition model. In each example, the first grid shows the initial random state of the system, the second grid shows an intermediate state, and the third grid shows the final stable pattern. (a) Time sequence showing a striped pattern formation, as in Figures 2(a) and 2(c). (b) Time sequence showing a mottled pattern formation, as in Figure 2(b). 4 Self-organizing Systems conditions are changed slightly, a zebra-stripe pat- information that must be coded within the tern develops. The only differences between the genes. Self-organization is such a mechanism. For two examples shown are that in the zebra-stripe example, the pattern of ocular dominance stripes in pattern the diffusion of the activator and inhibitor the visual cortex of the brain (Figure 2(c)) is a char- is greater in one direction than the other, and that acteristic morphogenic feature of neuroanatomical the relative strengths of the activator and inhibitor organization. The neural inputs from each eye to are different in the two cases. the visual cortex in the back of the brain consist of a series of alternating stripes. This architecture is believed to play an important role in how the SELF-ORGANIZATION IN THE NEURAL brain organizes and interprets visual information AND COGNITIVE SCIENCES received by the retina. This functional architecture To understand the brain is one of the greatest chal- can be seen by injecting radioactive proline into one lenges in biology. The brain of an insect such as eye, and making autoradiographs of sections of the a honey-bee contains relatively few neurons ± cortex. The resulting pattern is reminiscent of the approximately one million.