Abstract
Relationships among near set theory, shape maps and recent accounts of the Quantum Hall effect pave the way to neural networks computations performed in higher dimensions. We illustrate the operational procedure to build a real or artificial neural network able to detect, assess and quantify a fourth spatial dimension. We show how, starting from two-dimensional shapes embedded in a 2D topological charge pump, it is feasible to achieve the corresponding four-dimensional shapes, which encompass a larger amount of information. Synthesis of surface shape components, viewed topologically as shape descriptions in the form of feature vectors that vary over time, leads to a 4D view of cerebral activity. This novel, relatively straightforward architecture permits to increase the amount of available qbits in a fixed volume.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmad MZ, Peters JF (2018) Descriptive unions. A fibre bundle characterization of the union of descriptively near sets, arXiv:1811.11129
Borsuk K (1957–1958) Concerning the classification of topological spaces from the standpoint of the theory of retracts. XLVI:177–190
Di Concilio A, Guadagni C, Peters JF, Ramanna S (2018) Descriptive proximities. Properties and interplay between classical proximities and overlap. Math Comput Sci 12(1):91–106. https://doi.org/10.1007/s11786-017.0328-y
Dmochowski JP, Ki JJ, DeGuzman P, Sajda P, Parra LC (2017) Extracting multidimensional stimulus-response correlations using hybrid encoding-decoding of neural activity. Neuroimage. https://doi.org/10.1016/j.neuroimage.2017.05.037
Don A, Peters JF, Ramanna S, Tozzi A (2020) Topological view of flows inside the BOLD spontaneous activity of the human brain. Front Comput Neurosci. https://doi.org/10.3389/fncom.2020.00034
Garcia JO, Ashourvan A, Muldoon SF, Vettel J, Bassett DS (2018) Applications of community detection techniques to brain graphs: algorithmic considerations and implications for neural function. Proc IEEE. https://doi.org/10.1109/jproc.2017.2786710
Hall E (1879) On a new action of the magnet on electric currents. Am J Math 2(3):287–292. https://doi.org/10.2307/2369245
Izhikevich EM (2010) Dynamical systems in neuroscience—the geometry of excitability and bursting. ISBN: 9780262514200
Kraus YE, Ringel Z, Zilberberg O (2013) Four-dimensional quantum Hall effect in a two-dimensional quasicrystal. Phys Rev Lett 111:226401
Lohse M, Schweizer C, Price HM, Zilberberg O, Bloch I (2018) Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553:55–58. https://doi.org/10.1038/nature25000
Matoušek J (2003) Using the Borsuk–Ulam theorem. Lectures on topological methods in combinatorics and geometry. Springer, Berlin
Novoselov KS, Jiang Z, Zhang Y, Morozov SV, Stormer HL et al (2007) Room-temperature quantum Hall effect in graphene. Science 315(5817):1379
Peters JF (2007) Near sets. Special theory about nearness of objects. Fundam Inf 75(1–4):407–433
Peters JF (2014) Topology of digital images. Visual pattern discovery in proximity spaces. In: Intelligent systems reference library. Springer, Berlin, pp 1–342, vol 63. ISBN 978-3-642-53844-5. https://doi.org/10.1007/978-3-642-53845-2
Riley SN, Davies J (2020) A spiking neural network model of spatial and visual mental imagery. Cogn Neurodyn 14:239–251. https://doi.org/10.1007/s11571-019-09566-5
Sen S, Daimi SN, Watanabe K et al (2020) Switch or stay? Automatic classification of internal mental states in bistable perception. Cogn Neurodyn 14:95–113. https://doi.org/10.1007/s11571-019-09548-7
Tafreshi TF, Daliri MR, Ghodousi M (2019) Functional and effective connectivity based features of EEG signals for object recognition. Cogn Neurodyn 13:555–566. https://doi.org/10.1007/s11571-019-09556-7
Tozzi A (2019) The multidimensional brain. Phys Life Rev. https://doi.org/10.1016/j.plrev.2018.12.004(in press)
Tozzi A, Peters JF (2016a) Towards a fourth spatial dimension of brain activity. Cogn Neurodyn 10(3):189–199. https://doi.org/10.1007/s11571-016-9379-z
Tozzi A, Peters JF (2016b) A topological approach unveils system invariances and broken symmetries in the brain. J Neurosci Res 94(5):351–365. https://doi.org/10.1002/jnr.23720
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. Phys Life Rev. https://doi.org/10.1016/j.plrev.2019.04.008
Tozzi A, Peters JF (2020) Removing uncertainty in neural networks. Cogn Neurodyn. https://doi.org/10.1007/s11571-020-09574-w
Tozzi A, Peters JF, Fingelkurts AA, Fingelkurts AA, Marijuán PC (2017) Topodynamics of metastable brains. Phys Life Rev 21:1–20. https://doi.org/10.1016/j.plrev.2017.03.001
Ursino M, Cuppini C, Cappa SF et al (2018) A feature-based neurocomputational model of semantic memory. Cogn Neurodyn 12:525–547. https://doi.org/10.1007/s11571-018-9494-0
Zhang S-C, Hu J (2001) A four-dimensional generalization of the quantum Hall effect. Science 294:823–828
Zilberberg O, Huang S, Guglielmon J, Wang M, Chen KP et al (2018) Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553(7686):59–62. https://doi.org/10.1038/nature25011
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tozzi, A., Ahmad, M.Z. & Peters, J.F. Neural computing in four spatial dimensions. Cogn Neurodyn 15, 349–357 (2021). https://doi.org/10.1007/s11571-020-09598-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11571-020-09598-2