Hierarchical evolutive systems: a mathematical model for complex systems.

The notion of an evolutive hierarchical system proposed in this paper is a mathematical model for systems, like organisms, with more or less complex objects. This model, based on category theory, retains the following characteristics of natural systems: they have an internal organization consisting of components with interrelations; they maintain their organization in time though their components are changing; their components are divided into several levels corresponding to the increasing complexity of their own organization, and the system may be studied at any of these levels (e.g. molecular, cellular...). The state of the system at a given instant is modeled by a category whose objects are its components, the state transition by a functor, a complex object by the (direct) limit of a pattern of linked objects (which describes its internal organization). The properties of limits in a category make it possible to ‘measure’ the emergence of properties for a complex object with respect to its components, and to reduce the study of a hierarchical system to that of its components of the lowest degree and their links. Categorical constructions describe the formation of a hierarchical evolutive system stepwise, by means of the operations: absorption of external objects, destruction of some components, formation of new complex objects.

[1]  S. Amari,et al.  Competition and Cooperation in Neural Nets , 1982 .

[2]  S. Lane.  Categories for the Working Mathematician , 1971 .

[3]  I. C. Baianu,et al.  Organismic supercategories and qualitative dynamics of systems. , 1971 .

[4]  Louis Leon Thurstone,et al.  The indifference function. , 1931 .

[5]  René Thom,et al.  Modèles mathématiques de la morphogénèse , 1971 .

[6]  Robert Rosen.  Feedforwards and global system failure: a general mechanism for senescence. , 1978, Journal of theoretical biology.

[7]  Mathematical theory of motivation interactions of two individuals. , 1947, The Bulletin of mathematical biophysics.

[8]  R. Rosen,et al.  Role of similarity principles in data extrapolation. , 1983, The American journal of physiology.

[9]  A. Lindenmayer.  Mathematical models for cellular interactions in development. I. Filaments with one-sided inputs. , 1968, Journal of theoretical biology.

[10]  S. Amari,et al.  Mathematical theory on formation of category detecting nerve cells , 1978, Biological Cybernetics.

[11]  I Baianu.  Some algebraic properties of ()-systems , 1973 .

[12]  F. William Lawvere,et al.  Metric spaces, generalized logic, and closed categories , 1973 .

[13]  Gregory Bateson,et al.  La nature et la pensée , 1984 .

[14]  Aristid Lindenmayer,et al.  Mathematical Models for Cellular Interactions in Development , 1968 .

[15]  I. Băianu.  Some algebraic properties of (M, R)-systems , 1973 .

[16]  Categorical system theory and the phenomenological calculus , 1983 .

[17]  P. Auger.  Hierarchically organized populations: interactions between individual, population, and ecosystem levels , 1983 .

[18]  J. Changeux,et al.  A theory of the epigenesis of neuronal networks by selective stabilization of synapses. , 1973, Proceedings of the National Academy of Sciences of the United States of America.

[19]  S. Maclane,et al.  General theory of natural equivalences , 1945 .

[20]  Robert Rosen,et al.  A relational theory of biological systems II , 1958 .

[21]  I Băianu,et al.  Organismic supercategories. II. On multistable systems. , 1970, The Bulletin of mathematical biophysics.

[22]  M. W. Warner.  Representations of (M, R)-systems by categories of automata , 1982 .

[23]  Michael Abib.  Categories of (ℓ, ℛ)-systems , 1966 .

[24]  N. Rashevsky.  Organismic sets: Outline of a general theory of biological and social organisms , 1967 .

[25]  A. H. Louie,et al.  Categorical system theory , 1983 .

[27]  P. Watzlawick,et al.  Pragmatics of human communication , 1975 .

[28]  R. Rosen.  THE REPRESENTATION OF BIOLOGICAL SYSTEMS FROM THE STANDPOINT OF THE THEORY OF CATEGORIES , 1958 .

[29]  J. Changeux.  L'homme neuronal , 1983 .

[30]  N. Rashevsky.  Organismic sets. II. Some general considerations. , 1968, The Bulletin of mathematical biophysics.

[31]  J. Gray.  Fibred and Cofibred Categories , 1966 .

[32]  Robert Rosen,et al.  Pattern Generation in Networks , 1981 .

[33]  F. Linton,et al.  Categories et Structures. , 1968 .

[34]  A. A. Min'ko,et al.  Mathematical modeling of short-term memory , 1981 .

[35]  Charles Ehresmann,et al.  Categories of sketched structures , 1972 .

[36]  F. William Lawvere,et al.  Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body , 1980 .

[37]  J. P. Mcm,et al.  Les problèmes de la vie , 1901 .

[38]  Robert Rosen.  Hierarchical Organization in Automata Theoretic Models of the Central Nervous System , 1969 .

[39]  Robert Rosen,et al.  Chapter 4 – SOME RELATIONAL CELL MODELS: THE METABOLISM-REPAIR SYSTEMS , 1972 .