{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T19:09:26Z","timestamp":1778526566522,"version":"3.51.4"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2020,7,25]],"date-time":"2020-07-25T00:00:00Z","timestamp":1595635200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,7,25]],"date-time":"2020-07-25T00:00:00Z","timestamp":1595635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput &amp; Applic"],"published-print":{"date-parts":[[2021,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We address the following problem: given a set of complex images or a large database, the numerical and computational complexity and quality of approximation for neural network may drastically differ from one activation function to another. A general novel methodology, scaled polynomial constant unit activation function \u201cSPOCU,\u201d is introduced and shown to work satisfactorily on a variety of problems. Moreover, we show that SPOCU can overcome already introduced activation functions with good properties, e.g., SELU and ReLU, on generic problems. In order to explain the good properties of SPOCU, we provide several theoretical and practical motivations, including tissue growth model and memristive cellular nonlinear networks. We also provide estimation strategy for SPOCU parameters and its relation to generation of random type of Sierpinski carpet, related to the [<jats:italic>pppq<\/jats:italic>] model. One of the attractive properties of SPOCU is its genuine normalization of the output of layers. We illustrate SPOCU methodology on cancer discrimination, including mammary and prostate cancer and data from Wisconsin Diagnostic Breast Cancer dataset. Moreover, we compared SPOCU with SELU and ReLU on large dataset MNIST, which justifies usefulness of SPOCU by its very good performance.<\/jats:p>","DOI":"10.1007\/s00521-020-05182-1","type":"journal-article","created":{"date-parts":[[2020,7,25]],"date-time":"2020-07-25T18:02:23Z","timestamp":1595700143000},"page":"3385-3401","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":65,"title":["\u201cSPOCU\u201d: scaled polynomial constant unit activation function"],"prefix":"10.1007","volume":"33","author":[{"given":"Jozef","family":"Kise\u013e\u00e1k","sequence":"first","affiliation":[]},{"given":"Ying","family":"Lu","sequence":"additional","affiliation":[]},{"given":"J\u00e1n","family":"\u0160vihra","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Sz\u00e9pe","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2749-5990","authenticated-orcid":false,"given":"Milan","family":"Stehl\u00edk","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,7,25]]},"reference":[{"key":"5182_CR1","doi-asserted-by":"publisher","unstructured":"Achter JD, Webb CT (2006) Pair statistics clarify percolation properties of spatially explicit simulations. Theor Popul Biol, 69 (2): 155 \u2013 164, ISSN 0040-5809. https:\/\/doi.org\/10.1016\/j.tpb.2005.07.003. URL http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0040580905000997","DOI":"10.1016\/j.tpb.2005.07.003"},{"issue":"4","key":"5182_CR2","doi-asserted-by":"publisher","first-page":"390","DOI":"10.1587\/nolta.10.390","volume":"10","author":"M Bucolo","year":"2019","unstructured":"Bucolo M, Buscarino A, Corradino C, Fortuna L, Frasca M (2019) Turing patterns in the simplest mcnn. Nonlinear Theory Appl IEICE 10(4):390\u2013398. https:\/\/doi.org\/10.1587\/nolta.10.390","journal-title":"Nonlinear Theory Appl IEICE"},{"key":"5182_CR3","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1007\/BF00319291","volume-title":"Probab Theory Related Fields","author":"JT Chayes","year":"1988","unstructured":"Chayes JT, Chayes L, Durrett R (1988) Connectivity properties of mandelbrot\u2019s percolation process. Probab Theory Related Fields., pp 307\u2013324. https:\/\/doi.org\/10.1007\/BF00319291 ISSN 1432-2064"},{"issue":"4","key":"5182_CR4","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/BF02551274","volume":"2","author":"G Cybenko","year":"1989","unstructured":"Cybenko G (1989) Approximation by superpositions of a sigmoidal function. Math Control Signals Syst 2(4):303\u2013314. https:\/\/doi.org\/10.1007\/BF02551274 ISSN 0932-4194; 1435-568X\/e","journal-title":"Math Control Signals Syst"},{"issue":"5","key":"5182_CR5","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.1007\/BF01026566","volume":"58","author":"FM Dekking","year":"1990","unstructured":"Dekking FM, Meester RWJ (1990) On the structure of mandelbrot\u2019s percolation process and other random cantor sets. J Stat Phys 58(5):1109\u20131126. https:\/\/doi.org\/10.1007\/BF01026566 ISSN 1572-9613","journal-title":"J Stat Phys"},{"key":"5182_CR6","unstructured":"Falconer K (2013) Fractal geometry: mathematical foundations and applications. Wiley. ISBN 9781118762868. URL https:\/\/books.google.at\/books?id=XJN7AgAAQBAJ"},{"key":"5182_CR7","doi-asserted-by":"publisher","unstructured":"Ghazal GA, Neudecker H (2000) On second-order and fourth-order moments of jointly distributed random matrices: a survey. Linear Algebra Appl, 321 (1): 61 \u2013 93. Eighth special issue on linear algebra and statistics. ISSN 0024-3795. https:\/\/doi.org\/10.1016\/S0024-3795(00)00181-6. URL http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0024379500001816","DOI":"10.1016\/S0024-3795(00)00181-6"},{"issue":"10","key":"5182_CR8","doi-asserted-by":"publisher","first-page":"612","DOI":"10.1109\/81.473568","volume":"42","author":"L Goras","year":"1995","unstructured":"Goras L, Chua LO (1995) Turing patterns in CNNS. II. Equations and behaviors. IEEE Trans Circuits Syst I Fund Theory Appl 42(10):612\u2013626","journal-title":"IEEE Trans Circuits Syst I Fund Theory Appl"},{"key":"5182_CR9","doi-asserted-by":"publisher","unstructured":"Hermann P, Mrkvi\u010dka T, Mattfeldt T, Min\u00e1rov\u00e1 M, Helisov\u00e1 K, Nicolis O, Wartner F, Stehl\u00edk M (2015) Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and quermass-interaction process. Stat Med 34 (18): 2636\u20132661, ISSN 1097-0258. https:\/\/doi.org\/10.1002\/sim.6497. URL http:\/\/dx.doi.org\/10.1002\/sim.6497. sim.6497","DOI":"10.1002\/sim.6497"},{"key":"5182_CR10","doi-asserted-by":"crossref","unstructured":"Kisel\u2019\u00e1k J, Pardasani KR, Adlakha N, Stehl\u00edk M, Agrawal M (2013) On some probabilistic aspects of diffusion models for tissue growth. Open Stat Probab J 5: 14\u201321. ISSN 1876-5270\/e","DOI":"10.2174\/1876527001305010014"},{"key":"5182_CR11","unstructured":"Klambauer G, Unterthiner T, Mayr A, Hochreiter S (2017) Self-normalizing neural networks. CoRR. arxiv:1706.02515"},{"key":"5182_CR12","unstructured":"LeCun Y, Cortes C (2010) MNIST handwritten digit database. URL http:\/\/yann.lecun.com\/exdb\/mnist\/"},{"key":"5182_CR13","doi-asserted-by":"crossref","unstructured":"Liu X, Zhou J, Qian H (2019) Comparison and evaluation of activation functions in term of gradient instability in deep neural networks. In: 2019 Chinese control and decision conference (CCDC), pp 3966\u20133971","DOI":"10.1109\/CCDC.2019.8832578"},{"issue":"2","key":"5182_CR14","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1017\/S0022112074000711","volume":"62","author":"BB Mandelbrot","year":"1974","unstructured":"Mandelbrot BB (1974) Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J Fluid Mech 62(2):331\u2013358. https:\/\/doi.org\/10.1017\/S0022112074000711","journal-title":"J Fluid Mech"},{"issue":"2","key":"5182_CR15","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1080\/07362994.2016.1238766","volume":"35","author":"O Nicolis","year":"2017","unstructured":"Nicolis O, Kise\u013e\u00e1k J, Porro F, Stehl\u00edk M (2017) Multi-fractal cancer risk assessment. Stoch Anal Appl 35(2):237\u2013256","journal-title":"Stoch Anal Appl"},{"key":"5182_CR16","doi-asserted-by":"crossref","unstructured":"Pignon D, Parmiter PJM, Slack JK, Hands MA, Hall TJ, van Daalen M, Shawe-Taylor J (Feb 1996) Sigmoid neural transfer function realized by percolation. Opt Lett 21(3):222\u2013224. 10.1364\/OL.21.000222. http:\/\/ol.osa.org\/abstract.cfm?URI=ol-21-3-222","DOI":"10.1364\/OL.21.000222"},{"key":"5182_CR17","doi-asserted-by":"publisher","unstructured":"Rahaman M, Aldalbahi A, Govindasami P, Khanam NP, Bhandari S, Feng P, Altalhi T (2017) A new insight in determining the percolation threshold of electrical conductivity for extrinsically conducting polymer composites through different sigmoidal models. Polymers, 9 (10), ISSN 2073-4360. https:\/\/doi.org\/10.3390\/polym9100527. URL http:\/\/www.mdpi.com\/2073-4360\/9\/10\/527","DOI":"10.3390\/polym9100527"},{"key":"5182_CR18","doi-asserted-by":"publisher","unstructured":"Roth HR, Farag A, Turkbey EB, Lu L, Liu J, Summers RM. Nih pancreas-ct dataset. https:\/\/doi.org\/10.7937\/K9\/TCIA.2016.tNB1kqBU","DOI":"10.7937\/K9\/TCIA.2016.tNB1kqBU"},{"key":"5182_CR19","doi-asserted-by":"publisher","unstructured":"Shallit J, Stolfi J (1989) Two methods for generating fractals. Comput Gr 13 (2): 185\u2013191. ISSN 0097-8493. https:\/\/doi.org\/10.1016\/0097-8493(89)90060-5. URL http:\/\/www.sciencedirect.com\/science\/article\/pii\/0097849389900605","DOI":"10.1016\/0097-8493(89)90060-5"},{"key":"5182_CR20","unstructured":"Steeb W-H (2011) The nonlinear workbook. Chaos, fractals, cellular automata, genetic algorithms, gene expression programming, support vector machine, wavelets, hidden Markov models, fuzzy logic with C++, Java and SymbolicC++ programs. 5th ed. World Scientific, Hackensack, NJ. ISBN 978-981-4335-77-5\/hbk; 978-981-4335-78-2\/pbk; 978-981-4335-79-9\/ebook"},{"key":"5182_CR21","doi-asserted-by":"publisher","unstructured":"Strelniker YM, Havlin S, Bunde A (2009) Fractals and Percolation. Springer, New York, pp 3847\u20133858. ISBN 978-0-387-30440-3. https:\/\/doi.org\/10.1007\/978-0-387-30440-3_227","DOI":"10.1007\/978-0-387-30440-3_227"},{"issue":"2","key":"5182_CR22","doi-asserted-by":"publisher","first-page":"3453","DOI":"10.1038\/s41467-019-11411-6","volume":"10","author":"W Sun","year":"2019","unstructured":"Sun W, Gao B, Chi M et al (2019) Understanding memristive switching via in situ characterization and device modeling. Nat Commun 10(2):3453","journal-title":"Nat Commun"},{"key":"5182_CR23","unstructured":"Sussillo D, Abbott LF (2014) Random walk initialization for training very deep feedforward networks. Neural Evolutionary Computing. arXiv:1412.6558v3"},{"key":"5182_CR24","doi-asserted-by":"crossref","unstructured":"Wang Y, Li Y, Song Y, Rong X (2020) The influence of the activation function in a convolution neural network model of facial expression recognition. Appl Sci 10 (5). URL https:\/\/www.mdpi.com\/2076-3417\/10\/5\/1897","DOI":"10.3390\/app10051897"},{"key":"5182_CR25","unstructured":"Wolberg WH, Street WN, Mangasarian OL (1992) Breast cancer wisconsin (diagnostic) data set. UCI Mach Learn Repos.http:\/\/archive.ics.uci.edu\/ml\/"},{"key":"5182_CR26","doi-asserted-by":"publisher","unstructured":"Wu H (2009) Global stability analysis of a general class of discontinuous neural networks with linear growth activation functions. Inf Sci 179 (19): 3432 \u2013 3441, ISSN 0020-0255. https:\/\/doi.org\/10.1016\/j.ins.2009.06.006. URL http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0020025509002539","DOI":"10.1016\/j.ins.2009.06.006"},{"key":"5182_CR27","doi-asserted-by":"publisher","unstructured":"Xue D, Zhu Y, Zhu G-X, Yan X (1996) Generalized kronecker product and fractals. https:\/\/doi.org\/10.1117\/12.235499","DOI":"10.1117\/12.235499"},{"key":"5182_CR28","unstructured":"Zhao P (2016) R for deep learning (i). URL https:\/\/github.com\/PatricZhao\/ParallelR\/blob\/master\/ParDNN\/iris_dnn.R"}],"updated-by":[{"DOI":"10.1007\/s00521-020-05412-6","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2020,11,17]],"date-time":"2020-11-17T00:00:00Z","timestamp":1605571200000}}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-020-05182-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00521-020-05182-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-020-05182-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T15:20:34Z","timestamp":1627658434000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00521-020-05182-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,25]]},"references-count":28,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["5182"],"URL":"https:\/\/doi.org\/10.1007\/s00521-020-05182-1","relation":{"correction":[{"id-type":"doi","id":"10.1007\/s00521-020-05412-6","asserted-by":"object"}]},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"value":"0941-0643","type":"print"},{"value":"1433-3058","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,25]]},"assertion":[{"value":"15 February 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 July 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 July 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 November 2020","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s00521-020-05412-6","URL":"https:\/\/doi.org\/10.1007\/s00521-020-05412-6","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"On behalf of all authors, I hereby attest that there are no conflicts of interest regarding financial relationships, intellectual property or any point mentioned under the publishing ethics.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}