Local Activity Is the Origin of Complexity

Abstract

Many scientists have struggled to uncover the elusive origin of "complexity", and its many equivalent jargons, such as emergence, self-organization, synergetics, collective behaviors, nonequilibrium phenomena, etc. They have provided some qualitative, but not quantitative, characterizations of numerous fascinating examples from many disciplines. For example, Schrödinger had identified "the exchange of energy" from open systems as a necessary condition for complexity. Prigogine has argued for the need to introduce a new principle of nature which he dubbed "the instability of the homogeneous". Turing had proposed "symmetry breaking" as an origin of morphogenesis. Smale had asked what "axiomatic" properties must a reaction-diffusion system possess to make the Turing interacting system oscillate.The purpose of this paper is to show that all the jargons and issues cited above are mere manifestations of a new fundamental principle called local activity, which is mathematically precise and testable. The local activity theorem provides the quantitative characterization of Prigogine's "instability of the homogeneous" and Smale's quest for an axiomatic principle on Turing instability.Among other things, a mathematical proof is given which shows none of the complexityrelated jargons cited above is possible without local activity. Explicit mathematical criteria are given to identify a relatively small subset of the locally-active parameter region, called the edge of chaos, where most complex phenomena emerge.

Impulsive control of a hysteresis cellular neural network model

“…The answer is given in [9]-the necessary condition for a nonconservative system to exhibit complexity is to have its cell locally active. The theory which will be presented below and which follows [9] offers a constructive analytical method for uncovering local activity.…”

“…The answer is given in [9]-the necessary condition for a nonconservative system to exhibit complexity is to have its cell locally active. The theory which will be presented below and which follows [9] offers a constructive analytical method for uncovering local activity. In particular, for diffusion CNN model, one can determine the domain of the cell parameters in order for the cells to be locally active, and thus potentially capable of exhibiting complexity.…”

“…In particular, for diffusion CNN model, one can determine the domain of the cell parameters in order for the cells to be locally active, and thus potentially capable of exhibiting complexity. This precisely defined parameter domain is called the edge of chaos [9].…”

“…entering the cell connected to node j. In this case, the associated electronic circuit has three terminals with two node-to-datum voltages (u j , v j ) and one terminal current U [5,9]. The importance of the circuit model is not only in the fact that we have a convenient physical implementation, but also in the fact that well-known results from classic circuit theory can be used to justify the cells' local activity [9].…”

“…In this case, the associated electronic circuit has three terminals with two node-to-datum voltages (u j , v j ) and one terminal current U [5,9]. The importance of the circuit model is not only in the fact that we have a convenient physical implementation, but also in the fact that well-known results from classic circuit theory can be used to justify the cells' local activity [9]. In this sense, if there is at least one equilibrium point for which the circuit model of the cell acts like a source of ''small signal'', in a precise sense defined in [9], i.e.…”