MATHEMATICS, LOGIC, NEURAL NETWORKS

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TOPOLOGY, N-DIMENSIONAL MANIFOLDS

Tozzi A, Peters JF. 2016. A Topological Approach Unveils System Invariances and Broken Symmetries in the Brain. Journal of Neuroscience Research 94 (5): 351–65. doi:10.1002/jnr.23720. A survey of the new variants of the Borsuk-Ulam theorem. Bougin-Yang-type theorems, Hausdorff measure.

Tozzi A, Peters JF, Fingelkurts AA, Fingelkurts AA, Marijuán PC. 2017. Topodynamics of metastable brains. Physics of Life Reviews, 21, 1-20. https://dx.doi.org/10.1016/j.plrev.2017.03.001. Topodynamics of metastable brains: a survey of the applications of the Borsuk-Ulam theorem to nervous nonlinear paths.

Peters JF, Tozzi A. 2016. Region-Based Borsuk-Ulam Theorem. arXiv.1605.02987. A new variant of the Borsuk-Ulam theorem.

Peters JF, Tozzi A. 2016. String-Based Borsuk-Ulam Theorem. arXiv:1606.04031. A new variant of the Borsuk-Ulam theorem.

Tozzi A. 2016. Borsuk-Ulam Theorem Extended to Hyperbolic Spaces. In Computational Proximity. Excursions in the Topology of Digital Images, edited by J F Peters, 169–171. doi:10.1007/978-3-319-30262-1. A new variant of the Borsuk-Ulam theorem.

Peters JF, Alfano R, Smith P, Tozzi A. 2022. Discrete and continuous curved surface antipodal paths. Extensions of the Borsuk-Ulam theorem and application of the Feynman path integral and Wodehouse contour integral. 2022. hal-03698584. A new variant of the Borsuk-Ulam theorem. Featuring Lodato proximity, Strongly near sets, hyperbolic, negative curvature, bounded shapes projection mappings and projections, worldsheets.

Tozzi A, Bormashenko E, Jausovec N. 2021. Topology of EEG wave fronts. Cognit Neurodyn, 15, 887–896 https://doi.org/10.1007/s11571-021-09668-z. The hairy ball theorem and the electric wave fronts in the brain.

Tozzi A, Papo D. 2020. Projective mechanisms subtending real world phenomena wipe away cause effect relationships. Progress in Biophysics and Molecular Biology. 151:1-13. DOI: 10.1016/j.pbiomolbio.2019.12.002. Shape pattern, topological invariance. Ham sandwich theorem, Brower fixed point theorem, Wired friend theorem, Lusternik-Schnirelmann theorem, Hairy ball theorem.

Tozzi A, Peters JF. 2019. The common features of different brain activities. Neurosci Lett, 692 (23): 41-46. https://doi.org/10.1016/j.neulet.2018.10.054. homotopy equivalent shapes, Elldesbrunner-Hearer nerve, Wired-friend theorem, Brouwer fixed-point theorem.

Tozzi A, Peters JF. 2017. The multidimensional world. Lambert Academic Publishing, Saarbrücken, Germany. ISBN-13: 978-3-330-03530-0. General mathematical principles for a multidimensional world. Including Hausdorff measure, Group theory, Lodato proximity, mappings, projections.

Tozzi A, Peters JF, Fingelkurts AA, Fingelkurts AA, Marijuán PC. 2017. Brain projective reality: novel clothes for the emperor. Reply to comments on “Topodynamics of metastable brains” by Tozzi et al. Physics of Life Reviews, 21, 46-55. https://doi.org/10.1016/j.plrev.2017.06.020. Cech complexes, Costa minimal surface, Oloid surface, De Raham cohomology.

Tozzi A, Peters JF. 2019. The Borsuk-Ulam theorem solves the curse of dimensionality: Comment on “the unreasonable effectiveness of small neural ensembles in high-dimensional brain” by Alexander N. Gorban et al. Physics of Life Reviews. 29: 89-92 https://doi.org/10.1016/j.plrev.2019.04.008. The curse of dimensionality: when the spherical dimension increase… the volume decreases!

Tozzi A. 2019. Is shape deformation a topological invariant? (electronic response to: Kawabata K, Higashikawa S, Gong Z, Ashida Y, Ueda M. 2019. Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics. Nature Communications 10: 297). Deformation is not a topological invariant: a critique to topology. bubbles; surface tension.

Peters JF, Ramanna S, Tozzi A, İnan E. 2017. Bold-Independent Computational Entropy Assesses Functional Donut-Like Structures in Brain fMRI Images. Front Hum Neurosci. 2017 Feb 1;11:38. doi: 10.3389/fnhum.2017.00038. eCollection 2017. Introducing 4D maximal nucleus cluster from algebraic topology.

Don AP, Peters JF, Ramanna S, Tozzi A. 2020. Topological View of Flows inside the BOLD Spontaneous Activity of the Human Brain. Front. Comput. Neurosci. DOI: 10.3389/fncom.2020.00034. The occurrence of Betti numbers in neurodata.

Don APH, Peters JF, Ramanna S, Tozzi A. 2021. Quaternionic views of rs-fMRI hierarchical brain activation regions. Discovery of multilevel brain activation region intensities in rs-fMRI video frames. Chaos, Solitons & Fractals. Vol 152, 111351. https://doi.org/10.1016/j.chaos.2021.111351. Further 4D insights in the dynamics of the four-dimensional brain.

Tozzi A, Peters JF, Jausovec N, Don APH, Ramanna S, Legchenkova I, Bormashenko E. 2021. Nervous Activity of the Brain in Five Dimensions. Biophysica; 1(1):38-47. https://doi.org/10.3390/biophysica1010004. How to build a 4D computer through quaternionic networks.

Tozzi A, Peters JF. 2019. Topology of Black Holes’ Horizons. Emerging Science Journal, 3(2):58-63. DOI: 10.28991/esj-2019-01169. t’Hooft’-inspired topological mappings on entangled horizons.

Tozzi A. 2022. Bipolar reasoning in feedback pathways. Biosystems. Volumes 215–216, June 2022, 104652 https://doi.org/10.1016/j.biosystems.2022.104652. Sheaf theory, presheaves, sheaves, globular sets, infinity topoi. Composition of functions, cohomology groups, Lurie, sheaf cohomology, stalk, infinity category.

Tozzi A, Peters J. 2023. Topology of Saliva. Preprints, 2023120139. https://doi.org/10.20944/preprints202312.0139.v1. Topological holes & homogeneity of biological fluids.

GEOMETRY, ALGEBRA

Yurkin A, Peters JF, Tozzi A. 2018. A novel belt model of the atom, compatible with quantum dynamics. Journal of Scientific and Engineering Research, 2018, 5(7):413-419. ISSN: 2394-2630. A geometric model of atom in which a single belt of electrons surrounds the nucleus.

Yurkin A, Tozzi A, Peters JF, Marijuan PC. 2017. Quantifying Energetic Dynamics in Physical and Biological Systems Through a Simple Geometric Tool and Geodetic Curves. Addendum to: Cellular Gauge Symmetry and the Li Organization Principle. Progress in Biophysics and Molecular Biology. https://doi.org/10.1016/j.pbiomolbio.2017.06.007. Geometric curves underlying physical and biological dynamics.

Tozzi A, Yurkin A, Peters JF. 2021. A Geometric Milieu Inside the Brain. Found Sci. https://doi.org/10.1007/s10699-021-09798-x. Geometric models of random walks, including Betti numbers, Infinite topological genus, Pascal triangle, octahedron, island of numbers, Vortex cycles.

Tozzi A, Peters JF. 2022 towards a single parameter for the assessment of EEG oscillations. MedRxiv. doi: https://doi.org/10.1101/2022.02.06.22270548. The Green’s theorem for flows describes EEG traces.

Tozzi, A., Peters, J.F. A Topological Approach to Infinity in Physics and Biophysics. Found Sci 26, 245–255 (2021). https://doi.org/10.1007/s10699-020-09674-0. An Alexander horned sphere to treat both divisible and continuum.

Sengupta B, Tozzi A, Coray GK, Douglas PK, Friston KJ. 2016. Towards a Neuronal Gauge Theory. PLOS Biology 14 (3): e1002400. doi:10.1371/journal.pbio.1002400. A tutorial on differential geometry and dynamics on manifolds (i.e., conjugate gradient-descent algorithm and parallel transport on Riemannian manifolds, Fisher information metric, Levi-Civita connexion, Bayesian models). Featuring: Invariance properties, Data analysis, Inference scheme, Probability distributions, Generative models.

Tozzi A, Peters JF. 2020. A Topological Approach to Infinity in Physics and Biophysics. Found Sci. https://doi.org/10.1007/s10699-020-09674-0. Torus, Connes conjecture, infinity, von Neumann algebras, geometrical concepts of infinity, bounded as well as unbounded geometrical shapes, no point geometry, Alexander horned sphere, pathological object, wild points, Cantor set, positive- and negative-curvature manifold, point-free geometries.

Tozzi A, Peters JF. 2019. Points and lines inside our brains. Cognitive Neurodynamics, 13(5): 417–428. DOI: 10.1007/s11571-019-09539-8. Point-free geometry without lines, point an infinity: straight lines. Parallel transport on straight lines in affine geometry, Jordan curves.

Tozzi A. 2020. Are Borders Inside or Outside? Found Sci. https://doi.org/10.1007/s10699-020-09708-7. The problem of boundaries and borders. Borsuk-Ulam theorem, Borders, Internal and external, decision limit problems, bounded objects with fuzzy edges, infinity; boundary; Jordan curve theorem, four-dimensional torus, stereographic projection of a Clifford torus, Lévy’s zero–one law, projectionism, closeness of sets of points with their neighbourhoods. No points algebra, descriptive nearness, point-free geometries, topological vortices, Betti numbers.

Tozzi A, Peters JF, Jaušovec N. 2018. EEG dynamics on hyperbolic manifolds. Neurosci Lett, 683: 138-143. https://doi.org/10.1016/j.neulet.2018.07.035. Mirzakhani’s geodesics on hyperbolic surfaces.

Tozzi A, Mariniello L. 2022. Unusual Mathematical Approaches Untangle Nervous Dynamics. Biomedicines; 10(10):2581. https://doi.org/10.3390/biomedicines10102581. Knots & braids in your brain. Are There Elliptic Curves in the Brain? Monge’s theorem for the evaluation of depth perception. Lurie’s non-abelian cohomology.

Tozzi A, Peters JF, Jausovec N. 2016. A repetitive modular oscillation underlies human brain electric activity. Neurosci Lett, 653, 234-238. 10.1016/j.neulet.2017.05.051. The unexpected occurrence of J-functions & complex functions in the brain.

Tozzi A. Ramanujan-derived Hypergeometric Functions Describe Hidden Coupled Dynamics in Physical and Biological Systems. viXra:2405.0059. A new math (possibly) subtending quantum entanglement, Turing patterns and gomboc dynamics.

FRACTALS, POWER LAWS

Tozzi A. 2015. How to Turn an Oscillation in a Pink One. Journal of Theoretical Biology 377, 117–18. doi:10.1016/j.jtbi.2015.04.018. Building fractal oscillations from noise.

Tozzi A. 2015. Neural code & power laws. SCTPLS Newsletter, April 7-10. Nervous relationships between information entropies and power laws.

Tozzi A, Peters JF, Cankaya MN. 2018. The informational entropy endowed in cortical oscillations. Cognitive Neurodynamics, 12(5), 501-507. DOI: 10.1007/s11571-018-9491-3. Relationships between information entropies and power laws.

NEURAL NETWORKS

Tozzi A, Peters JF. 2020 Removing uncertainty in neural networks. Cognitive Neurodynamics, 14:339–345. https://doi.org/10.1007/s11571-020-09574-w. When fuzzy holes demarcate objects. How to solve decision limit problems... with holes. Disconnected objects, Uncertainty, Decision limit problems, boundaries, Jordan curve theorem; Markov blanket.

Tozzi A, Fla Tor, Peters JF. 2016. Building a minimum frustration framework for brain functions in long timescales. J Neurosci Res. 94(8): 702–716. Minimum frustration principle-based nervous networks.

Tozzi A, Peters JF. 2018. Multidimensional brain activity dictated by winner-take-all mechanisms. Neuroscience Letters, 678 (21):83-89. https://doi.org/10.1016/j.neulet.2018.05.014. Selfridge’s Pandemonium: the winner takes all also in higher dimensions.

Zhang, Y., Guo, S., Sun, M., Mariniello, L., Tozzi, A., & Zhao, X. (2022). Front Waves of Chemical Reactions and Travelling Waves of Neural Activity. Journal of NeuroPhilosophy, 1(2). https://doi.org/10.5281/zenodo.7254050 Multi-bit binary decoder.

Tozzi A, Peters JF. 2017. The multidimensional world. Lambert Academic Publishing, Saarbrücken, Germany. ISBN-13: 978-3-330-03530-0. Small world networks in multidimensions.

Tozzi A. 2022. Bipolar reasoning in feedback pathways. Biosystems. Volumes 215–216, June 2022, 104652 https://doi.org/10.1016/j.biosystems.2022.104652. Using paraconsistent logics to shape neural networks. Featuring sheaf theory, globular sets, infinity topoi.

Tozzi A. 2022. Ramsey’s Theory Meets the Human Brain Connectome. Neural Process Lett. https://doi.org/10.1007/s11063-022-11099-8. Ramsey theorem: regularities & repeated patterns inside structures equipped with nodes and edges.

Tozzi A. 2023. Mental Recognition of Objects via Ramsey Sentences - How does the Human Brain Recognize Dog? J NeuroPhilos, 2(2). DOI: https://doi.org/10.5281/zenodo.10202881. Using Ramsey sentences in neural networks.

Tozzi A. 2024. Tozzi A. 2024. Flock members experience gas pressures higher than lone individuals. Biosystems, Volume 238, 105192. https://doi.org/10.1016/j.biosystems.2024.105192. K-Nearest Neighbours algorithm to simulate collective animal behavior.

COMPUTER SCIENCE

Tozzi A, Ahmad MZ, Peters JF. 2021. Neural computing in four spatial dimensions. Cognitive Neurodynamics, 15:349-357. https://doi.org/10.1007/s11571-020-09598-2. How to build a 4D computer through the Quantum Hall effect.

Tozzi A, Peters JF, Fingelkurts A, Fingelkurts A, Perlovsky L. 2018. Syntax meets semantics during brain logical computations. ProgrBiophys Mol Biol, 140: 133-141. https://doi.org/10.1016/j.pbiomolbio.2018.05.010. How to build a computer equipped with lower-dimensional syntax and higher-dimensional semantics.

IMAGE PROCESSING

Peters JF, Tozzi A. Ramanna S. 2016. Brain Tissue Tessellation Shows Absence of Canonical Microcircuits. Neuroscience Letters 626: 99–105. doi:10.1016/j.neulet.2016.03.052. Voronoi tessellations in histological brain tissues.

Tozzi A, Peters JF. 2020 Removing uncertainty in neural networks. Cognitive Neurodynamics, 14:339–345. https://doi.org/10.1007/s11571-020-09574-w. Centroids on holes in video frames to solve the demarcation problem. Disconnectedness, Persistent homology, Boundary, Jordan curve, Jordan theorem, Set theory, decision limit problem, topological changes.

Don APH, Peters JF, Ramanna S, Tozzi A. 2021. Quaternionic views of rs-fMRI hierarchical brain activation regions. Discovery of multilevel brain activation region intensities in rs-fMRI video frames. Chaos, Solitons & Fractals. Vol 152, 111351. https://doi.org/10.1016/j.chaos.2021.111351. Further insights in the dynamics of the four-dimensional brain.

SET THEORY

Tozzi A, Peters JF. 2016. Symmetries, Information and Monster Groups before and after the Big Bang. Information 7(4), 73; doi:10.3390/info7040073. The Fisher Griess Monster group and the elliptic module: novel physical insights.

Koczkodaj WW, Magnot J-P, Mazurek J, Peters JF, Rakhshani H, Soltys M, Strzałka D, Szybowski J, Tozzi A. 2017. On normalization of inconsistency indicators in pairwise comparisons. International Journal of Approximate Reasoning.86, 73–79. https://doi.org/10.1016/j.ijar.2017.04.005. Pairwise comparison and the infinity problem: no continuum between three apples and three bananas. Discrete values, Normalization, inconsistency indicators; paradox of infinity

Tozzi A, Peters JF, Jausovec N. 2016. A repetitive modular oscillation underlies human brain electric activity. Neurosci Lett, 653, 234-238. 10.1016/j.neulet.2017.05.051. Modular function, J.function, Compex plane, Monstrous Moonshine.

Tozzi A, Mariniello L. 2022. Unusual Mathematical Approaches Untangle Nervous Dynamics. Biomedicines; 10(10):2581. https://doi.org/10.3390/biomedicines10102581. knot theory, braid groups.

Tozzi A, Ahmad MZ, Peters JF. 2021. Neural computing in four spatial dimensions. Cognitive Neurodynamics, 15:349-357. https://doi.org/10.1007/s11571-020-09598-2. Near set theory and descriptive intersections.

LOGIC

Tozzi A, Peters JF. 2018. What it is like to be “the same”? Progress in Biophysics and Molecular Biology. 133, 30-35. https://doi.org/10.1016/j.pbiomolbio.2017.10.005. “The same” in biology. The principle of identity reloaded: When A=A turns out A≠A.

Tozzi A, Peters JF, Fingelkurts A, Fingelkurts A, Perlovsky L. 2018. Syntax meets semantics during brain logical computations. ProgrBiophys Mol Biol, 140: 133-141. https://doi.org/10.1016/j.pbiomolbio.2018.05.010. A neuroscientific account of lower-dimensional syntax and higher-dimensional semantics says that the Wittgenstein’s Tractatus was right.

Tozzi, A., Peters, J.F. 2021. Information-devoid routes for scale-free neurodynamics. Synthese. 199: 2491-2504. https://doi.org/10.1007/s11229-020-02895-7. Information Entropies Are Unable to Describe Mental Activities: a mathematical option from the Banach–Tarski paradox. Disjointness, ergodic, subflows, information theory, Betti numbers; Fourier analysis, power laws in brain, fractals, Sierpinski triangle.

Tozzi A. 2022. Bipolar reasoning in feedback pathways. Biosystems. Volumes 215–216, June 2022, 104652 https://doi.org/10.1016/j.biosystems.2022.104652. Not just 1 and 0 in logic: paraconsistent logics. Bipolar Logics, William of Ockham, Nicholas Of Autrecourt, Aristotle, the optimistic concept of cosmic goodness