Self-Organization in Nonlinear Optical Systems

Self-organization refers to the development of spatial and/or temporal patterns during the temporal response of a nonlinear system to a given input. Almost always, self-organization results from symmetry breaking in the system. Many natural phenomena as well as the brain or animal behavioral patterns exhibit self-organization. The convective rolls in a liquid when it is heated beyond the Rayleigh-Bernard instability point are an example of pattern formation in hydrodynamics due to self-organization. Here, the temperature difference is the driving force or input parameter. Below the critical or threshold temperature, one can only observe the random motion of the liquid particles. Besides, in an open container containing the fluid, surface tension can also affect the flow, causing tessellation of the surface and formation of hexagonal cells. As stated above, such spontaneous pattern formation is exactly what is termed self-organization, but there is no agent inside the system that does the organizing. The motion of the whole is no longer the sum of the motion of the parts due to the nonlinear interactions between the parts and the environment. Another example of pattern formation is a ‘‘wave’’ among spectators in a stadium-individual spectators communicate and cluster together in groups to create a nearly synchronized pattern that spreads throughout the stadium. Speaking of which, there is enough evidence that human behavioral patterns are selforganized. The human body, for example, is a complex system comprising about 102 joints, 103 muscles, 103 cell types, and 1014 neurons or neuron connections. The actions of communication, body movement, etc., are the results of self-organization of this complex system pertaining to a certain control or input parameter (stimulus). In a similar fashion, it has been shown that the brain itself is an active, dynamical self-organizing system.

For more on the self-organizing aspects of the brain and human behavior, the readers are referred to Scott Kelso (1995), Kohonen (1984), and Haken (1983).