Basic Concepts of Synergetics II: Formation of Spatio-temporal Patterns
One of the most striking features of self-organizing systems is their ability to form spatio-temporal patterns. Since we shall conceive of the brain as a self-organizing system that produces spatio-temporal patterns of activity, we wish to deal with the mechanisms of their formation from a general point of view. We have already encountered some simple examples, for instance, the famous convection instability in fluid dynamics (Fig. 4.4). Another example is provided by hexagonal or stripe patterns found in certain chemical reactions. These phenomena and the theoretical concepts underlying their explanation have led to models for the formation of patterns on furs, or on the skin of fish (cf. Fig. 4.7). In this chapter we wish to show how the methods of synergetics allow us to derive such patterns and, in a way, to classify them. Since colored patterns can be described in the same way as black and white patterns, we shall elucidate our basic approach by means of black and white patterns, or more precisely speaking, by grey tone patterns, where the tone may change locally and in the course of time. We note that the grey tone may symbolize, for instance, some local activity, for instance in electroencephalograms. The degree of the grey tone will be described by the variable q. Our reference state qo will be homogeneous and time independent and may denote a certain grey level. Then we introduce q as the deviation of the actual grey level from the average one, qo. q will depend on space and time, at least in general. Therefore we shall write q = q(x,t). The order parameter concept will turn out to be extremely useful for studying the individual patterns and for classifying them.
KeywordsSpatial Mode Linear Stability Analysis Stripe Pattern Individual Pattern Stable Fixed Point
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- Haken, H. (1993): Advanced Synergetics, 3rd. ed., Springer, BerlinGoogle Scholar