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The purpose of this paper is to consider the following three kinds of cases: first, where a network of stochastic automata consists of a neuron of several types; second, where not only pair‐wise interaction of the neuron occurs, but also that between groups of the neuron; and, third, where neurons are located in d‐dimensional Euclidean space. This is based on neurons of different types having been distinguished by the neurophysiological theory of the retina of vertebrates, where the retina consists of overlapping sheets which are composed of cells receiving light, bipolar cells, ganglion cells, respectively; there are also horizontal cells and amacrine; a functional unit of the central nervous system of vertebrates is composed of the group of neurons which have the form of a network; and the case where neurons are located in d‐dimensional Euclidean space will extend theoretical significance and practical significance of abstract neural automata. The existence of abstract neural automata will be proved in all three cases.

By means of topological conjugate transformation, the previous theory of abstract neural automata (ANA) on d‐dimensional (d≥1) integer lattice is extended to compact Riemannian manifold. This paper points out emphatically that intelligence of ANA is related to the geometrical features. The greater the volume of relative plane, the stronger the intelligence; curved Riemannian manifold X˜ configuration space of ANA are locally flat such that the cognitive process of NAN limits the Gibbs' probability measure for a sufficiently small time i.e. the cognitive process of ANA can determine the solution in a sufficiently small time the problem. This hypothesis was supported by studying the human brain, in particular by studying Einstein's brain.

Considers neurons, neural networks and neural fields from the viewpoint of abstract automata. Introduces Abstract neural automata (ANA) to explain and to provide a mathematical description of neural functions and theory. Surveys some current literature including that concerned with Boltzmann machines and the author’s own view of an associative Boltzmann neural model. Provides definitions and theorems to support the author’s theses on cognition, the human brain and the role of ANA in the understanding of neural networks.

The theory of abstract neural automata has been proposed previously. Believes that in studying change of the structure of abstract neural automata, genetico‐variable structure must be considered. Abstract neural automata are considered to be cognizant machines, machines of thought. The basic effect on cognition and on the thought of heredity, such as genes is considered to be important in this discussion. Following the proof of the theorem of existence of abstract neural automata whose state space of single neuron (stochastic automaton) is Euclidean space E^d, the theorem of ergodicity of evolutionary process of genetico‐variable structure of abstract neural automata is proved.

For a long time, one has carried on an endless debate for the problem of the origin of cognition. Much of the debate surrounding this comes from the problem of infinite regress. In a discussion on theory of the thought of abstract neural automata (ANA), we have considered thought, resulting from the nonuniqueness of ANA, to be an important stage of all evolutionary processes of cognition. If thought is considered to be an evolutionary process, we may ask where is the origin of the evolutionary process? In this paper, using ready‐made theory, merely for the problem of origin of thought, we shall offer views of ourselves, which cannot begin to talk about bringing forth new ideas in theory. The view on the theory of ANA about such a problem is that the thought of ANA results from its nonuniqueness, the nonuniqueness results from its uniqueness, the uniqueness originates from its non‐existence – “empty”, i.e. essential answer having universal sense is that thought of ANA originates from “empty”.

In the view of philosophy, a concept is the highest product of the human brain. Demonstrates abstract neural automata (ANA) – the more perfect of the brain's models have the ability of transition of concept – ability of thought. The transition of the concept of ANA results from non‐uniqueness of its limit Gibbs measure, which is the variability of structure of ANA.

Believes that in the view of philosophy, a concept is the highest form of activity of human brain. This paper demonstrates Abstract Neural Automata and a more perfect brain's models that have the ability of transition of concept‐ability of thought. The transition of the concept of Abstract Neural automata results from the non‐uniqueness of its limit Gibbs measure‐variability of the structure of Abstract Neural Automata.By means of topological conjugate transformation, the previous theory of Abstract Neural Automata on a d‐dimensional (d≥1) integer lattice is extended to the compact Riemannian manifold. We have pointed out emphatically that functions of cognition and thought of Abstract Neural Automata depend crucially on its topological and the Riemannian structure, particularly, on its Riemannian volume of some relative places which are relative learning, memory, cognition and thought. Furthermore, the larger the Riemannian volume, the stronger the intelligent function. In the study of the human brain, and in particular, Einstein's brain, one has discovered such information.