Abstract
This chapter is devoted to a brief discussion of systems which exhibit adaptive behaviour, i.e. systems which can modify their activity, if necessary, to bring this activity to a preferred (or optimal) state with respect to particular environmental situations. With this crude characterization of adaptive systems, it is seen that the input-output error-actuated devices discussed in Chapter 8 are adaptive systems, as are the abstract dynamical systems with prescribed cost functional considered in Chapter 10.
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Notes to Chapter 11
The pexceptron was originally described by Frank Rosenblatt in 1958. The original reports occasioned a certain amount of controversy, and much further work has been done since that time. Perhaps the closest approach to a definitive exposition is F. Rosenblatt, Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms, Spartan Press, Washington D.C., 1962.
As should be apparent, many variants of the basic perceptron idea are possible; cf. Rosenblatt op. cit. For instance, each analyser unit may receive inputs from only a random subset of sensory units; the analyser units may be iterated in several layers; the analyser units may be cross-connected, etc.
A particularly lucid exposition is A. Novikoff, ‘On Convergence Proofs for Perceptrons’, Mathematical Theory of Automata Symposium, Brooklyn Polytech., 1962. See also A. Charnes, ‘The Geometry of Convergence of Simple Perceptrons’, J. Math. Anal. Appl., 7 (1963) 475–48 L This last paper is especially interesting because of the utilization of techniques of linear programming (cf. Section 12.3); these programming techniques can indeed be entirely reformulated in terms of perceptron convergence problems.
Referenced in an interesting review paper by J. K. Hawkins, ‘Self-Organizing Systems-A Review and Commentary’, Proc. IRE, 49 (1961) 31–48.
A. S. Householder and H. D. Landahl, Mathematical Biophysics of the Central Nervous System, Principia Press, San Antonio, 1945.
H. D. Landahl, ‘Studies in the Mathematical Biophysics of Discrimination and Conditioning I’, Bull. math. Biophys. 3 (1941) 13–26.
cf. N. Rashevsky, Mathematical Biophysics, loc. cit., Vol. I, especially Chap. XXXII.
H. D. Landahl, ‘Studies in the Mathematical Biophysics of Discrimination and Conditioning II: Errors, Trials and Number of Possible Responses’, Bull. math. Biophys. 3 (1941) 71–77.
The formal equivalence between the neural learning circuits and the differentiation models of Jacob and Monod is implicitly contained in a paper by M. Sugita, ‘Functional Analysis of Chemical Systems in vivo Using a Logical Circuit Equivalent’, /. Theoret. Biol, 1 (1961) 415–30. Sugita’s analysis essentially shows the equivalence between the models of Jacob and Monod, and the discrete neural nets of McCulloch and Pitts (cf. N. Rashevsky, op. cit. Vol. II, pp. 207–29). In view, therefore, of the well-known correspondence between the discrete nets and the continuous neural models of Landahl, one is directly led from a model of differentiation to Landahl’s learning circuits, and conversely.
See for example R. R. Bush and F. Mosteller, Stochastic Models forLearning, Wiley, New York, 1958;orR. R. Bush and S. Estes, (eds.), Studies in Stochastic Learning Theory, Stanford University Press, Stanford, 1959.
As stated in the Preface, stochastic models are not considered in this book, but see the references cited in the preceding note for a full treatment of such models and a more extensive bibliography.
The reader is again referred to the works cited in Note 10.
See H. M. Martinez, ‘Studies in Stochastic Learning Theory IF, Bull math. Biophys., 26 (1964) 63–75.
The mathematical theory of population genetics is concerned with precisely this question. The fundamental variables in that theory are the gene frequencies in a population, and the theory seeks to describe the time rate of change of these frequencies under a variety of conditions. Each individual genotype in the population is associated with a purely formal numerical quantity which is called the ‘fitness’ of the genotype, and which is regarded as a measure of the competitive efficiency of an individual that carries the genotype. Thus the change of gene frequencies in the population determines how the fitness of the population changes in time. It is an essential idea in population genetics that fitness is defined so as to tend towards a maximum ; indeed the rate of increase of fitness is used as a measure of the pressure of selection on the population. Many of these ideas are due to R. A. Fisher, who has analogized the increase of fitness of a population under natural selection with the increase of entropy guaranteed by the Second Law of Thermodynamics : cf. his Genetical Theory of Natural Selection, 2nd revised edn, Dover Publications, New York, 1958, pp. 39–40. Kimura, among others, has sharpened to a considerable extent Fisher’s suggestions on the rate of increase of fitness (cf. M. Kimura, ‘On the Change of Population Fitness by Natural Selection’, Heredity, 12 (1958), 145–67); he has proposed, as a general principle, that in any given time interval natural selection causes the gene frequencies to change within a population in such a manner as to maximize the fitness of the population. Naturally these ideas, and their many important ramifications, cannot be discussed in any detail here, as, among other reasons, they are essentially stochastic in nature. However, the reader’s attention is drawn to the formal and logical similarity between Kimura’s principle and the Principle of Optimal Design, viewed in the light of the discussion of the present section.
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Rosen, R. (1967). Adaptive Systems in Biology. In: Optimality Principles in Biology. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-6419-9_11
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