Self-organized topological structures in neural networks for the visual cortex of the brain
Introduction
Most of the cortical areas of the brain possess an expressed layered organization with ordered topological structures. This fundamental feature of the topological structures in the primate neostriatum is called as cell islands and matrices [16] which, as it has been recently established, are present in a less visible form in all mammals. The cortical areas possess the matrix rather than the island neuronal organization. The visible clustering of neurons indicates some boundaries of the mosaic configurations. The human and some primate neurostriatum neurons, in contrast to most other animals, are laid out in clustered formations with a higher or lower cell density [53].
Another aspect of ordered self-organized neuronal structures is bound up with the peristimulus inhibition which is well observed in the somatosensory and visual pathways [41]. This inhibition is a result of the interaction effect of an excitatory stimulus, caused by the surrounding inhibitory action, and is often being referred to as a “Mexican-hat” pattern [44], [45]. Although in the neocortex and small areas of the cortex, the peristimulus inhibition also takes place [14], [21], this fact was successfully used to explain the origin of island formation [33]. Though the authors in [45] confirm that the peristimulus inhibition in sensory pathways, where the lateral inhibitory connections take the same place as in the neocortex and cortex areas, cannot be formed by lateral inhibitory connections as well as in the cortical areas, the works [40], [47] present the experimental results on the primates in which the Mexican-hat shaped profiles apparently exist in the cortical layers 4A and 4B revealed by the lateral inhibition. However, the authors in [44] used the lateral inhibitory connections in their inhibition model to induce the peristimulus inhibition and to “control”, as they told, the competitive model which was designated to model the peristimulus inhibition through a competition mechanism. In this way, they use small hexagon patches for simulation in the cortex [55]. However, these ideas are similar to those which we are going to study in our contribution. They are more coincidental to computational mechanisms than to the physiological sense of the main considered problems.
Apart from the common formation of the stable topological structures in the neocortex and cortex areas of the brain, it is important to consider the appearance of simple stable forms. We devote our direction of research more the self-formation of ordered structures, repetitive mosaics which only partially connected with visual hallucination phenomena. There are very many scientists [3], [8], [11], [23], [30], [46] and others who have investigated epilepsy, migraine, drug-induced hallucinations as well as theoretical aspects of reproduction of such phenomena by mathematical and computational tools. We are not going to investigate the origin of the hallucination phenomenon which has not yet been explained to the end. An important fact is that, as Horowitz and others [23] have experimentally established, the imagery hallucination did not arise in peripheral zones, but in the visual cortex areas and that the cortical enhanced excitatory activity was influenced by disinhibitory actions in the brain [9]. Though and this is not beyond doubt as the experiments [42] have shown. The author of [30] classifies the hallucination patterns, as well as “form constants” into four categories which cover almost completely the set of simplified visual irregularities. We are more interested in repetitive forms such as hexagonal lattices, fretworks, honeycombs which are rather respondent to the spatio-temporal pattern copying mechanism in a cerebral code implementation [6].
Though our investigations based on the works of [5], [8], [11], [48], [49] and a specially on [11], we included some new principal elements such as a strongly nonlinear neuron current–voltage relation, integrate-and-fire neuron action, more correct representation of solutions of bifurcation equations for excitation and inhibition cases, background of stability based on concrete calculations, and final mapping of patterns, as well. Other differences will be noted in the sequel of contribution.
The theoretical principles and neural network models with axonal, dendritic, and synaptic elements and N-shaped current–voltage relations are discussed in Section 2. In Section 3, both the general issues of stability and instability of strong nonlinear evolutionary systems and simplified neural systems with excitatory and inhibitory neuronal populations are considered. The group theory, doubly periodic kernel functions as the bifurcation solutions are presented in Section 4. Section 5 deals with the foundation of hexagonal lattice representation, transformation, reduction of bifurcation equations, and the analysis of stable solutions. The modeling results on the hexagonal lattice and partially on the rectangular and square lattices as well as the computation examples of form constants are widely represented in Section 6. Some mathematical details are presented in Appendix A The neutral stability and solution of stationary equations, Appendix B Transformation on hexagonal lattice.
The paper ends with the discussion on the experimental results of iso-orientation structures in the visual cortex of beings and artificial self-organizing dissipative ordered structures—patterns—obtained by means of the formal methodology.
Section snippets
Precondition of approach
Since the visual cortex consists of numerous different sorts neurons, it is expedient to consider similar groups or populations of neurons. Most of all the excitatory and inhibitory neuronal populations are spread. Though in some local field (their number is about 20,000) of the visual cortex there are about 200 cells which can accomplish distinct operations, the consideration of an individual neuron with a low-dimensional integrate-and-fire approach or a single, uncoupled population of neurons
Basic principles
Many physical as well as biophysical systems characterized by nonlinearities and having a control parameter dependent on its value in the evolution of states might be stable or unstable systems. It is known that the resembling systems in the range of the control parameter, possessing a stable solution which becomes under certain conditions, invariant under a symmetry group, and as the parameter crosses the critical value, new solutions can arise that are invariant only under a subgroup. The
Group theory aspects
The group theory can be easily applied to both , , and the system of two equations (11). In the general case the group is used as the subject of these equations. This group is defined on the continuous Banach space Bs by functions u mapping R2 into Bs [11], [49]. The group is invariant under the group Θ of rigid motions in the plane. Besides, we introduce a group Θ (2) that corresponds to the group of the second order matrices A under the condition that AA−1=I, where I is the identity
Hexagonal lattice representation
We concentrate our attention on the hexagonal lattice as the main formation for the presentation of a copying mechanism of patterns in the cortex of the brain. As to other form constants (square, rhombic) one should refer to [8], [11]. We assume that the basis vectors and satisfy the condition . The angle between two basis vectors is denoted as γ for the symmetry lattice. The hexagonal symmetry group scheme is presented in Fig. 6 and the listing of elements is given in
Solutions of bifurcation equations
The double and triple periodicity, different variants of bifurcation solutions, various possibilities of given angles, conditions of invariant transformations allow us to observe either repeating mosaics or global tunnels, funnels, and spiral form constants.
We present below the possible solutions built on the basis of a hexagonal lattice. According to Table 2 the solutions of (28) will be as follows:
Case (a): , γ=60°, and , i.e., space coordinates. The solution will be
Discussion
A macroscopic theoretical approach to the brain functionality corrobrates that many cells are aggregated in macrogroups or populations which possess, in the general case, multi-objective statements, though some of them are fundamental ones. The iso-orientation domains were detected in the visual cortex a long time ago in the following sequence: (i) neuron orientation selectiveness [1], [20], [24], [56]; (ii) the same orientation if neurons are lying below the point on the cortical surface [25],
Conclusions
A macroscopic theoretical approach to the brain functionality corroborates that many of cells are aggregated in macrogroups which may form iso-orientation domains as stripes or blobs. Stripes of activity are created during early onthogenesis of the visual cortex by patterns of periodic standing waves probably based on the self-organizing dissipative ordered structures. Considering neural network theoretical principles based on an integrate-and-fire two point neurons, the N-shaped
Acknowledgements
The author is grateful to the reviewers for their useful remarks and to Mrs. J. Kazlauskaite for her excellent service in the production of this manuscript.
Algis Garliauskas is a head of Neuroinformatics Laboratory of Mathematics and Informatics Institute, habilitation, doctor of technical science, professor, a member of the Council of Lithuanian Science Society, chairman of its Informatics Department. He graduated from Vilnius Pedagogical University, Faculty of Physics and Mathematics. He received Ph.D. in Research Operation & System Analysis and Habilitation Doctor from the Lithuanian Science Council in 1994. He was involved in research work in
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Algis Garliauskas is a head of Neuroinformatics Laboratory of Mathematics and Informatics Institute, habilitation, doctor of technical science, professor, a member of the Council of Lithuanian Science Society, chairman of its Informatics Department. He graduated from Vilnius Pedagogical University, Faculty of Physics and Mathematics. He received Ph.D. in Research Operation & System Analysis and Habilitation Doctor from the Lithuanian Science Council in 1994. He was involved in research work in the fields of modeling complex systems and neuro-molecular networks. He has published over 160 scientific articles, 5 monographs. In the period of the last ten years the sphere of his scientific activity has been the development of neurocomputers and foundations of neuroinformatics. He deals with the problems of a generalization of ferromagnetic theory to nonlinear neural networks in order to improve the methodology of signal and pattern recognition, to accelerate the learning and classifying procedures based on neuronal nets with nonlinear feedback, chaotic phenomena in neural structures, modeling synapse-dendrite-soma logic functions, and brain-style information systems. In this new areas A.Garliauskas has issued 26 works, among them 11 in foreign referred journals, the monograph, has taken part in more than 30 international conferences. In 1995 he received Foreign Research Award of Canadian Natural Sciences and Engineering Research Council and led a visiting research at Saskatchewan University, Canada, and as a collaborate researcher of RIKEN Brain Science Institute in 1999.