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A Geometric Milieu Inside the Brain

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Abstract

The brain, rather than being homogeneous, displays an almost infinite topological genus, since it is punctured with a high number of “cavities”. We might think to the brain as a sponge equipped with countless, uniformly placed, holes. Here we show how these holes, termed topological vortexes, stand for nesting, non-concentric brain signal cycles resulting from the activity of inhibitory neurons. Such inhibitory spike activity is inversely correlated with its counterpart, i.e., the excitatory spike activity propagating throughout the whole brain tissue. We illustrate how Pascal’s triangles and linear and nonlinear arithmetic octahedrons are capable of describing the three-dimensional random walks generated by the inhbitory/excitatory activity of the nervous tissue. In case of nonlinear 3D paths, the trajectories of excitatory spiking oscillations can be depicted as the operation of filling the numbers of octahedrons in the form of “islands of numbers”: this leads to excitatory neuronal assemblies, spaced out by empty areas of inhibitory neuronal assemblies. These mathematical procedures allow us to describe the topology of a brain of infinite genus, to represent inhibitory neurons in terms of Betti numbers and to highlight how spike diffusion in neural tissues is generated by the random activation of tiny groups of excitatory neurons. Our approach suggests the existence of a strong mathematical background subtending the intricate oscillatory activity of the central nervous system.

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This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Center for Nonlinear Science, University of North Texas, 1155 Union Circle, #311427, Denton, TX, 76203-5017, USA

    Arturo Tozzi

  2. Russian Academy of Sciences, Moscow, Puschino, Russia

    Alexander Yurkin

  3. Department of Electrical and Computer Engineering, University of Manitoba, 75A Chancellor’s Circle, Winnipeg, MB, R3T 5V6, Canada

    James F. Peters

  4. Mathematics Department, Adiyaman University, Adiyaman, Turkey

    James F. Peters

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  1. Arturo Tozzi
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All the Authors had full access to all the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis. All the Authors equally contributed to: study concept and design, acquisition of data, analysis and interpretation of data, drafting of the manuscript, critical revision of the manuscript for important intellectual content, statistical analysis, obtained funding, administrative, technical, and material support, study supervision.

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Correspondence to Arturo Tozzi.

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Tozzi, A., Yurkin, A. & Peters, J.F. A Geometric Milieu Inside the Brain. Found Sci 27, 1477–1488 (2022). https://doi.org/10.1007/s10699-021-09798-x

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