{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:39:42Z","timestamp":1760402382380,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,5]],"date-time":"2020-05-05T00:00:00Z","timestamp":1588636800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003741","name":"Instituci\u00f3 Catalana de Recerca i Estudis Avan\u00e7ats","doi-asserted-by":"publisher","award":["academia"],"award-info":[{"award-number":["academia"]}],"id":[{"id":"10.13039\/501100003741","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PGC2018-099443-B-I00"],"award-info":[{"award-number":["PGC2018-099443-B-I00"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study how sensory neurons detect and transmit a weak external stimulus. We use the FitzHugh\u2013Nagumo model to simulate the neuronal activity. We consider a sub-threshold stimulus, i.e., the stimulus is below the threshold needed for triggering action potentials (spikes). However, in the presence of noise the neuron that perceives the stimulus fires a sequence of action potentials (a spike train) that carries the stimulus\u2019 information. To yield light on how the stimulus\u2019 information can be encoded and transmitted, we consider the simplest case of two coupled neurons, such that one neuron (referred to as neuron 1) perceives a subthreshold periodic signal but the second neuron (neuron 2) does not perceive the signal. We show that, for appropriate coupling and noise strengths, both neurons fire spike trains that have symbolic patterns (defined by the temporal structure of the inter-spike intervals), whose frequencies of occurrence depend on the signal\u2019s amplitude and period, and are similar for both neurons. In this way, the signal information encoded in the spike train of neuron 1 propagates to the spike train of neuron 2. Our results suggest that sensory neurons can exploit the presence of neural noise to fire spike trains where the information of a subthreshold stimulus is encoded in over expressed and\/or in less expressed symbolic patterns.<\/jats:p>","DOI":"10.3390\/e22050524","type":"journal-article","created":{"date-parts":[[2020,5,7]],"date-time":"2020-05-07T03:10:38Z","timestamp":1588821038000},"page":"524","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Neuronal Transmission of Subthreshold Periodic Stimuli Via Symbolic Spike Patterns"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7535-3893","authenticated-orcid":false,"given":"Maria","family":"Masoliver","sequence":"first","affiliation":[{"name":"Departament de F\u00edsica, Universitat Polit\u00e8cnica de Catalunya, St. Nebridi 22, 08222 Terrassa, Barcelona, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0768-2019","authenticated-orcid":false,"given":"Cristina","family":"Masoller","sequence":"additional","affiliation":[{"name":"Departament de F\u00edsica, Universitat Polit\u00e8cnica de Catalunya, St. Nebridi 22, 08222 Terrassa, Barcelona, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,5]]},"reference":[{"key":"ref_1","unstructured":"Bialek, W., de Ruyter van Steveninck, R., Rieke, F., and Warland, D. (1999). Spikes: Exploring the Neural Code, MIT Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"828","DOI":"10.1038\/nature02058","article-title":"Independent rate and temporal coding in hippocampal pyramidal cells","volume":"425","author":"Huxter","year":"2003","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1038\/nature06105","article-title":"Temporal precision in the neural code and the timescales of natural vision","volume":"449","author":"Butts","year":"2007","journal-title":"Nature"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1038\/nrn1668","article-title":"Neuronal variability: Noise or part of the signal?","volume":"6","author":"Stein","year":"2005","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1038\/nrn3061","article-title":"The benefits of noise in neural systems: Bridging theory and experiment","volume":"12","author":"McDonnell","year":"2011","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"656","DOI":"10.1103\/PhysRevLett.67.656","article-title":"Time-interval sequences in bistable systems and the noise-induced transmission of information by sensory neurons","volume":"67","author":"Longtin","year":"1991","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/j.physrep.2003.10.015","article-title":"Effects of noise in excitable systems","volume":"392","author":"Lindner","year":"2004","journal-title":"Phys. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"174102","DOI":"10.1103\/PhysRevLett.88.174102","article-title":"Permutation entropy: A natural complexity measure for time series","volume":"88","author":"Bandt","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.3390\/e14081553","article-title":"Permutation entropy and its main biomedical and econophysics applications: A review","volume":"14","author":"Zanin","year":"2012","journal-title":"Entropy"},{"key":"ref_10","first-page":"361","article-title":"Recent progress in symbolic dynamics and permutation entropy","volume":"222","author":"Amigo","year":"2013","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Bandt, C. (2019). Small order patterns in big time series: A practical guide. Entropy, 21.","DOI":"10.3390\/e21060613"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"032218","DOI":"10.1103\/PhysRevE.94.032218","article-title":"Emergence of spike correlations in periodically forced excitable systems","volume":"94","author":"Reinoso","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"8276","DOI":"10.1038\/s41598-018-26618-8","article-title":"Subthreshold signal encoding in coupled FitzHugh-Nagumo neurons","volume":"8","author":"Masoliver","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"105023","DOI":"10.1016\/j.cnsns.2019.105023","article-title":"Neuronal coupling benefits the encoding of weak periodic signals in symbolic spike patterns","volume":"88","author":"Masoliver","year":"2020","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"013123","DOI":"10.1063\/1.5121257","article-title":"Characterizing signal encoding and transmission in class I and class II neurons via ordinal time-series analysis","volume":"30","author":"Estarellas","year":"2020","journal-title":"Chaos"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Budzinski, R.C., Lopes, S.R., and Masoller, C. (2020). Symbolic analysis of bursting dynamical regimes of Rulkov neural networks. Neurocomputing, submitted.","DOI":"10.1016\/j.neucom.2020.05.122"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/S0006-3495(61)86902-6","article-title":"Impulses and physiological states in theoretical models of nerve membrane","volume":"1","author":"FitzHugh","year":"1961","journal-title":"Biophys. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2061","DOI":"10.1109\/JRPROC.1962.288235","article-title":"An active pulse transmission line simulating nerve axon","volume":"50","author":"Nagumo","year":"1962","journal-title":"Proc. IRE"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.compbiomed.2011.03.017","article-title":"Classifying cardiac biosignals using ordinal pattern statistics and symbolic dynamics","volume":"42","author":"Parlitz","year":"2012","journal-title":"Comput. Biol. Med."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1883","DOI":"10.1016\/j.physleta.2017.03.052","article-title":"Permutation entropy based time series analysis: Equalities in the input signal can lead to false conclusions","volume":"381","author":"Zunino","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_21","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Elements of Information Theory, Wiley Sons."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1002\/(SICI)1097-0193(1999)8:4<194::AID-HBM4>3.0.CO;2-C","article-title":"Measuring phase synchrony in brain signals","volume":"8","author":"Lachaux","year":"1999","journal-title":"Hum. Brain Mapp."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.chaos.2018.12.006","article-title":"Ordinal synchronization: Using ordinal patterns to capture interdependencies between time series","volume":"119","author":"Echegoyen","year":"2019","journal-title":"Chaos Sol. Fractals"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/5\/524\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:32:43Z","timestamp":1760362363000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/5\/524"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,5]]},"references-count":23,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["e22050524"],"URL":"https:\/\/doi.org\/10.3390\/e22050524","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,5,5]]}}}