Galois Perceptron: Cell Assemblies in Cellular Space I
G. Fay, D. V. Takács · Journal of Cybernetics · 1975
(Of Parts I and II) A lattice theoretical model is proposed for the topological structure and the physiological function of the brain tissue. According to the model the brain tissue is isomorphic to the graph of a well defined class of lattices, called Galois-lattice, whose vertices represent the neurons and/or ganglia and edges correspond to axons and/or dendrites. A procedure is described by which one is able to produce this lattice algorithmically once a binary relation is given between the elements of two sets. The algorithm is suitable for computer implementation. As a starting point two sets are considered as of interpretational relevance. First, the set of receptors (of, say, an organism), second, the set of effectors (of the same organism) is thought to be given. Further a binary relation is assumed between receptors and effectors. So, the basic concepts of our theory–quite similarily to that of perceptron theories–are: receptors, effectors, and their connectedness. Now by the algorithm one can select all the closed receptor sets and all the closed effector sets. The concept of a closed set (under a binary relation) is a pure mathematical one. An attempt is made to develop and elaborate its anatomical, physiological, and psychological relevance, meaning, and significance. The ground for doing this is a classic mathematical theorem of the theory of Galois correspondence, stating that the set of all the closed receptor sets, as well as the set of all the closed effector sets, form a lattice each with respect to the set theoretical inclusion relation. These two lattices are dual-isomorphic to each other so that they form essentially the same structure. This algebraic structure is called Galois lattice. Now the closed receptor and effector sets, and in general, the closed sets of neurons are interpreted as the Hebb's cell assemblies. Their more elaborated version, i.e. Milner's theory of the rebounding neurons, calls for some concepts of cellular automata theory. As for the topological structure of a closed set of neurons our studies can be viewed as a theory of reticulation. The concept of a closed set, with respect to a relation, although quite intricate to absorb, it is quite at hand to interpret. A number of disciplines are involves at this point. The theory, especially the isomorphism theorem, offers an excellent possibility for the explication of Spinoza's psychophysical parallelism and reveals some intrinsic relationship with the theory of empirical inferences from von Neumann's quantum logic to Bar Hillel's semantic information theory.