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. (shrink)

Physical and biological measurements might display range values extending towards infinite. The occurrence of infinity in equations, such as the black hole singularities, is a troublesome issue that causes many theories to break down when assessing extreme events. Different methods, such as re-normalization, have been proposed to avoid detrimental infinity. Here a novel technique is proposed, based on geometrical considerations and the Alexander Horned sphere, that permits to undermine infinity in physical and biophysical equations. In this unconventional approach, a continuous (...) monodimensional line becomes an assembly of countless bidimensional lines that superimpose in quantifiable knots and bifurcations. In other words, we may state that Achilles leaves the straight line and overtakes the turtle. (shrink)

Topology has been characterized as an unsuitable mathematical framework for the investigation of geometric-perceptual phenomena. This has been attributed to the highly abstract nature of topology leading to failures in tasks such as making distinctions between geometrical figures (e.g., a cube versus a sphere) in which the human perceptual system succeeds easily. An alternative thesis is proposed on both philosophical and empirical grounds. The present analysis applies the Müller-Lyer (ML) illusion as a method of investigation to examine the suitability of (...) topology in raising questions about geometric-perceptual phenomena and formulating potential answers to these questions. Not only will it be concluded that topology is a suitable mathematical framework in which one can formulate relevant questions and potential answers with respect to perceptual phenomena such as the ML illusion, it will also be discussed that topology will not be replaced by other means of spatial representation consistent with the findings that indicate its contributions to our daily spatial behavior along with other means such as Euclidean geometry. In addition, the analysis will be discussed with respect to two relevant themes: 1) the implications of the proposed view in relation to Husserl’s phenomenology and 2) the effect of developmental stages and education that may change the involvement of topology in such perceptual responses over time. (shrink)

The discrepancy between syntax and semantics is a painstaking issue that hinders a better comprehension of the underlying neuronal processes in the human brain. In order to tackle the issue, we at first describe a striking correlation between Wittgenstein's Tractatus, that assesses the syntactic relationships between language and world, and Perlovsky's joint language-cognitive computational model, that assesses the semantic relationships between emotions and “knowledge instinct”. Once established a correlation between a purely logical approach to the language and computable psychological activities, (...) we aim to find the neural correlates of syntax and semantics in the human brain. Starting from topological arguments, we suggest that the semantic properties of a proposition are processed in higher brain's functional dimensions than the syntactic ones. In a fully reversible process, the syntactic elements embedded in Broca's area project into multiple scattered semantic cortical zones. The presence of higher functional dimensions gives rise to the increase in informational content that takes place in semantic expressions. Therefore, diverse features of human language and cognitive world can be assessed in terms of both the logic armor described by the Tractatus, and the neurocomputational techniques at hand. One of our motivations is to build a neuro-computational framework able to provide a feasible explanation for brain's semantic processing, in preparation for novel computers with nodes built into higher dimensions. (shrink)

The target paper of Schoeller, Perlovsky, and Arseniev is an essential and timely contribution to a current shift of focus in neuroscience aiming to merge neurophysiological, psychological and physical principles in order to build the foundation for the physics of mind. Extending on previous work of Perlovsky et al. and Badre, the authors of the target paper present interesting mathematical models of several basic principles of the physics of mind, such as perception and cognition, concepts and emotions, instincts and learning. (...) Their conceptualization helps to clarify the distinction between conscious and unconscious aspects of mind that is often neglected and further provide a clear description of the mental hierarchy, which extends from physical objects in the physical world to abstract ideas in the mental/subjective realm. While we agree that identification of a few fundamental principles is a first step toward developing the physics of the mind, and we concur with the selection of those principles in the target review paper, we think that the theory of the physics of mind would much benefit from considering also the most basic principles that are common for the physics/matter/brain and the mind/subjectivity/cognition. In this respect, such basic principles as time and space, as well as criticality, self-organization, and emergence seem to be the most interesting. (shrink)