Stochastic Resonance with Colored Noise for Neural Signal Detection
Identifieur interne : 002B51 ( Pmc/Corpus ); précédent : 002B50; suivant : 002B52Stochastic Resonance with Colored Noise for Neural Signal Detection
Auteurs : Fabing Duan ; François Chapeau-Blondeau ; Derek AbbottSource :
- PLoS ONE [ 1932-6203 ] ; 2014.
Abstract
We analyze signal detection with nonlinear test statistics in the presence of colored noise. In the limits of small signal and weak noise correlation, the optimal test statistic and its performance are derived under general conditions, especially concerning the type of noise. We also analyze, for a threshold nonlinearity–a key component of a neural model, the conditions for noise-enhanced performance, establishing that colored noise is superior to white noise for detection. For a parallel array of nonlinear elements, approximating neurons, we demonstrate even broader conditions allowing noise-enhanced detection, via a form of suprathreshold stochastic resonance.
Url:
DOI: 10.1371/journal.pone.0091345
PubMed: 24632853
PubMed Central: 3954722
Links to Exploration step
PMC:3954722Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Stochastic Resonance with Colored Noise for Neural Signal Detection</title><author><name sortKey="Duan, Fabing" sort="Duan, Fabing" uniqKey="Duan F" first="Fabing" last="Duan">Fabing Duan</name><affiliation><nlm:aff id="aff1"><addr-line>Institute of Complexity Science, Qingdao University, Qingdao, P. R. China</addr-line></nlm:aff></affiliation></author><author><name sortKey="Chapeau Blondeau, Francois" sort="Chapeau Blondeau, Francois" uniqKey="Chapeau Blondeau F" first="François" last="Chapeau-Blondeau">François Chapeau-Blondeau</name><affiliation><nlm:aff id="aff2"><addr-line>Laboratoire d'Ingénierie des Systèmes Automatisés, Université d'Angers, Angers, France</addr-line></nlm:aff></affiliation></author><author><name sortKey="Abbott, Derek" sort="Abbott, Derek" uniqKey="Abbott D" first="Derek" last="Abbott">Derek Abbott</name><affiliation><nlm:aff id="aff3"><addr-line>Centre for Biomedical Engineering and School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, Southern Australia, Australia</addr-line></nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">24632853</idno><idno type="pmc">3954722</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3954722</idno><idno type="RBID">PMC:3954722</idno><idno type="doi">10.1371/journal.pone.0091345</idno><date when="2014">2014</date><idno type="wicri:Area/Pmc/Corpus">002B51</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">002B51</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Stochastic Resonance with Colored Noise for Neural Signal Detection</title><author><name sortKey="Duan, Fabing" sort="Duan, Fabing" uniqKey="Duan F" first="Fabing" last="Duan">Fabing Duan</name><affiliation><nlm:aff id="aff1"><addr-line>Institute of Complexity Science, Qingdao University, Qingdao, P. R. China</addr-line></nlm:aff></affiliation></author><author><name sortKey="Chapeau Blondeau, Francois" sort="Chapeau Blondeau, Francois" uniqKey="Chapeau Blondeau F" first="François" last="Chapeau-Blondeau">François Chapeau-Blondeau</name><affiliation><nlm:aff id="aff2"><addr-line>Laboratoire d'Ingénierie des Systèmes Automatisés, Université d'Angers, Angers, France</addr-line></nlm:aff></affiliation></author><author><name sortKey="Abbott, Derek" sort="Abbott, Derek" uniqKey="Abbott D" first="Derek" last="Abbott">Derek Abbott</name><affiliation><nlm:aff id="aff3"><addr-line>Centre for Biomedical Engineering and School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, Southern Australia, Australia</addr-line></nlm:aff></affiliation></author></analytic><series><title level="j">PLoS ONE</title><idno type="eISSN">1932-6203</idno><imprint><date when="2014">2014</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>We analyze signal detection with nonlinear test statistics in the presence of colored noise. In the limits of small signal and weak noise correlation, the optimal test statistic and its performance are derived under general conditions, especially concerning the type of noise. We also analyze, for a threshold nonlinearity–a key component of a neural model, the conditions for noise-enhanced performance, establishing that colored noise is superior to white noise for detection. For a parallel array of nonlinear elements, approximating neurons, we demonstrate even broader conditions allowing noise-enhanced detection, via a form of suprathreshold stochastic resonance.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Benzi, R" uniqKey="Benzi R">R Benzi</name></author><author><name sortKey="Sutera, A" uniqKey="Sutera A">A Sutera</name></author><author><name sortKey="Vulpiani, A" uniqKey="Vulpiani A">A Vulpiani</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Gammaitoni, L" uniqKey="Gammaitoni L">L Gammaitoni</name></author><author><name sortKey="H Nggi, P" uniqKey="H Nggi P">P Hänggi</name></author><author><name sortKey="Jung, P" uniqKey="Jung P">P Jung</name></author><author><name sortKey="Marchesoni, F" uniqKey="Marchesoni F">F Marchesoni</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Wiesenfeld, K" uniqKey="Wiesenfeld K">K 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<front><journal-meta><journal-id journal-id-type="nlm-ta">PLoS One</journal-id><journal-id journal-id-type="iso-abbrev">PLoS ONE</journal-id><journal-id journal-id-type="publisher-id">plos</journal-id><journal-id journal-id-type="pmc">plosone</journal-id><journal-title-group><journal-title>PLoS ONE</journal-title></journal-title-group><issn pub-type="epub">1932-6203</issn><publisher><publisher-name>Public Library of Science</publisher-name><publisher-loc>San Francisco, USA</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">24632853</article-id><article-id pub-id-type="pmc">3954722</article-id><article-id pub-id-type="publisher-id">PONE-D-13-48229</article-id><article-id pub-id-type="doi">10.1371/journal.pone.0091345</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="Discipline-v2"><subject>Biology and Life Sciences</subject><subj-group><subject>Biophysics</subject><subj-group><subject>Biophysical Simulations</subject><subject>Biophysics Theory</subject></subj-group></subj-group><subj-group><subject>Biotechnology</subject><subj-group><subject>Bioengineering</subject></subj-group></subj-group><subj-group><subject>Computational Biology</subject><subj-group><subject>Computational Neuroscience</subject><subj-group><subject>Single Neuron Function</subject></subj-group></subj-group></subj-group></subj-group><subj-group subj-group-type="Discipline-v2"><subject>Engineering and Technology</subject><subj-group><subject>Control Engineering</subject></subj-group><subj-group><subject>Signal Processing</subject><subj-group><subject>Statistical Signal Processing</subject></subj-group></subj-group></subj-group><subj-group subj-group-type="Discipline-v2"><subject>Physical Sciences</subject><subj-group><subject>Mathematics</subject><subj-group><subject>Algebra</subject><subj-group><subject>Linear Algebra</subject></subj-group></subj-group><subj-group><subject>Applied Mathematics</subject></subj-group></subj-group><subj-group><subject>Physics</subject></subj-group></subj-group></article-categories><title-group><article-title>Stochastic Resonance with Colored Noise for Neural Signal Detection</article-title><alt-title alt-title-type="running-head">Stochastic Resonance with Colored Noise</alt-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Duan</surname><given-names>Fabing</given-names></name><xref ref-type="aff" rid="aff1"><sup>1</sup></xref><xref ref-type="corresp" rid="cor1"><sup>*</sup></xref></contrib><contrib contrib-type="author"><name><surname>Chapeau-Blondeau</surname><given-names>François</given-names></name><xref ref-type="aff" rid="aff2"><sup>2</sup></xref></contrib><contrib contrib-type="author"><name><surname>Abbott</surname><given-names>Derek</given-names></name><xref ref-type="aff" rid="aff3"><sup>3</sup></xref></contrib></contrib-group><aff id="aff1"><label>1</label><addr-line>Institute of Complexity Science, Qingdao University, Qingdao, P. R. China</addr-line></aff><aff id="aff2"><label>2</label><addr-line>Laboratoire d'Ingénierie des Systèmes Automatisés, Université d'Angers, Angers, France</addr-line></aff><aff id="aff3"><label>3</label><addr-line>Centre for Biomedical Engineering and School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, Southern Australia, Australia</addr-line></aff><contrib-group><contrib contrib-type="editor"><name><surname>Chacron</surname><given-names>Maurice J.</given-names></name><role>Editor</role><xref ref-type="aff" rid="edit1"></xref></contrib></contrib-group><aff id="edit1"><addr-line>McGill University, Canada</addr-line></aff><author-notes><corresp id="cor1">* E-mail: <email>fabing.duan@gmail.com</email></corresp><fn fn-type="conflict"><p><bold>Competing Interests: </bold>DA is a PLOS ONE Editorial Board member, and the authors here confirm that this does not alter the authors' adherence to all the PLOS ONE policies on sharing data and materials.</p></fn><fn fn-type="con"><p>Conceived and designed the experiments: FCB DA. Performed the experiments: FD. Analyzed the data: FD FCB DA. Contributed reagents/materials/analysis tools: FCB DA. Wrote the paper: FD FCB DA. Proofreading: FCB DA.</p></fn></author-notes><pub-date pub-type="collection"><year>2014</year></pub-date><pub-date pub-type="epub"><day>14</day><month>3</month><year>2014</year></pub-date><volume>9</volume><issue>3</issue><elocation-id>e91345</elocation-id><history><date date-type="received"><day>16</day><month>11</month><year>2013</year></date><date date-type="accepted"><day>10</day><month>2</month><year>2014</year></date></history><permissions><copyright-year>2014</copyright-year><copyright-holder>Duan et al</copyright-holder><license><license-p>This is an open-access article distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution License</ext-link>, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.</license-p></license></permissions><abstract><p>We analyze signal detection with nonlinear test statistics in the presence of colored noise. In the limits of small signal and weak noise correlation, the optimal test statistic and its performance are derived under general conditions, especially concerning the type of noise. We also analyze, for a threshold nonlinearity–a key component of a neural model, the conditions for noise-enhanced performance, establishing that colored noise is superior to white noise for detection. For a parallel array of nonlinear elements, approximating neurons, we demonstrate even broader conditions allowing noise-enhanced detection, via a form of suprathreshold stochastic resonance.</p></abstract><funding-group><funding-statement>This work is sponsored by the NSF of Shandong Province (No. ZR2010FM006). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.</funding-statement></funding-group><counts><page-count count="7"></page-count></counts></article-meta></front><body><sec id="s1"><title>Introduction</title><p>Stochastic resonance has emerged as a significant statistical phenomenon where the presence of noise is beneficial for signal and information processing in both man-made and natural systems <xref rid="pone.0091345-Benzi1" ref-type="bibr">[1]</xref>–<xref rid="pone.0091345-Ward1" ref-type="bibr">[11]</xref>. The excitable FitzHugh–Nagumo (FHN) neuron model has been discussed for exploring the functional role of noise in neural coding of sensory information <xref rid="pone.0091345-Longtin1" ref-type="bibr">[12]</xref>. Following this, the milestone concept of aperiodic stochastic resonance using the FHN neuron model <xref rid="pone.0091345-Collins2" ref-type="bibr">[13]</xref> stimulated a number of interesting investigations in sensory biology <xref rid="pone.0091345-Wiesenfeld1" ref-type="bibr">[3]</xref>, <xref rid="pone.0091345-Lindner1" ref-type="bibr">[7]</xref>, <xref rid="pone.0091345-Bezrukov1" ref-type="bibr">[14]</xref>, <xref rid="pone.0091345-Levin1" ref-type="bibr">[15]</xref> and physiological experiments <xref rid="pone.0091345-Russell1" ref-type="bibr">[6]</xref>, <xref rid="pone.0091345-Moss1" ref-type="bibr">[8]</xref>, <xref rid="pone.0091345-Nozaki1" ref-type="bibr">[16]</xref>–<xref rid="pone.0091345-Kurita1" ref-type="bibr">[20]</xref>. Due to the character of activity in the nervous system, the neuron coding strategy based on stochastic resonance is also found in threshold (level-crossing) <xref rid="pone.0091345-Simonotto1" ref-type="bibr">[21]</xref>–<xref rid="pone.0091345-Patel1" ref-type="bibr">[23]</xref> and threshold-free <xref rid="pone.0091345-Bezrukov2" ref-type="bibr">[24]</xref>–<xref rid="pone.0091345-ChapeauBlondeau3" ref-type="bibr">[28]</xref> neurons. Since there are large numbers of neurons in the nervous system of animals and humans with variations in structure, function and size <xref rid="pone.0091345-Gammaitoni1" ref-type="bibr">[2]</xref>, <xref rid="pone.0091345-Collins1" ref-type="bibr">[4]</xref>, <xref rid="pone.0091345-Lindner1" ref-type="bibr">[7]</xref>–<xref rid="pone.0091345-McDonnell1" ref-type="bibr">[9]</xref>, then the potential exploitation of stochastic resonance in a neuron bundle becomes an interesting open question in neuroscience. In a general summing neural network, Collins <italic>et al</italic>. <xref rid="pone.0091345-Collins1" ref-type="bibr">[4]</xref> reported that the noise intrinsic to each neuron could be used to extend the operating range of the sensitivity of the overall system. This, however, is not a unique case. In the summing multi-threshold network, suprathreshold stochastic resonance discovered by Stocks <xref rid="pone.0091345-Stocks1" ref-type="bibr">[29]</xref> overcomes the restriction of subthreshold signals, and appears to offer a possible explantation of dc adaptation in sensory neurons <xref rid="pone.0091345-McDonnell1" ref-type="bibr">[9]</xref>, <xref rid="pone.0091345-Stocks2" ref-type="bibr">[30]</xref>. One-dimensional coupling <xref rid="pone.0091345-Bulsara1" ref-type="bibr">[31]</xref> and spatio-temporal stochastic resonance <xref rid="pone.0091345-Lindner1" ref-type="bibr">[7]</xref>, <xref rid="pone.0091345-Lindner2" ref-type="bibr">[32]</xref> show that not only an optimal noise intensity but also an optimal coupling strength exists. Recent stochastic resonance research in complex networks <xref rid="pone.0091345-Perc1" ref-type="bibr">[33]</xref>–<xref rid="pone.0091345-Teramae1" ref-type="bibr">[40]</xref> also demonstrates that an interconnected network configuration, as well as the non-zero noise level, can be optimized to achieve the best system performance.</p><p>In many practical situations, the idealization of white noise is never exactly realized <xref rid="pone.0091345-Gammaitoni1" ref-type="bibr">[2]</xref>, <xref rid="pone.0091345-ChapeauBlondeau1" ref-type="bibr">[5]</xref>. Consequently, the effect of colored noise on stochastic resonance has been investigated using the measure of output signal-to-noise ratio of a periodic signal <xref rid="pone.0091345-Gammaitoni1" ref-type="bibr">[2]</xref>, <xref rid="pone.0091345-ChapeauBlondeau1" ref-type="bibr">[5]</xref>, <xref rid="pone.0091345-Nozaki1" ref-type="bibr">[16]</xref>, <xref rid="pone.0091345-Hnggi1" ref-type="bibr">[41]</xref>–<xref rid="pone.0091345-Makra1" ref-type="bibr">[43]</xref>. Although the suppression of stochastic resonance with increasing noise correlation time was demonstrated <xref rid="pone.0091345-Gammaitoni1" ref-type="bibr">[2]</xref>, <xref rid="pone.0091345-ChapeauBlondeau1" ref-type="bibr">[5]</xref>, <xref rid="pone.0091345-Hnggi1" ref-type="bibr">[41]</xref>–<xref rid="pone.0091345-Makra1" ref-type="bibr">[43]</xref>, it is interesting to note that, under certain circumstances, colored noise can be superior to white noise for enhancing the response of a nonlinear system to a weak signal <xref rid="pone.0091345-Nozaki1" ref-type="bibr">[16]</xref>, <xref rid="pone.0091345-Neiman1" ref-type="bibr">[44]</xref>. In the field of signal detection, the employment of noise to enhance signal detectability also becomes a possible option <xref rid="pone.0091345-Inchiosa1" ref-type="bibr">[45]</xref>–<xref rid="pone.0091345-Zeng1" ref-type="bibr">[55]</xref>. However, in most of these studies, the observed noise samples are often assumed to be independent. Colored noise for signal detection <xref rid="pone.0091345-Poor1" ref-type="bibr">[56]</xref>–<xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref> is not adequately investigated in the context of stochastic resonance. In this article, we focus on the weak signal detection problem with the beneficial role of additive colored noise in threshold neurons. Because of the “all-and-none” character of nerve activity <xref rid="pone.0091345-FitzHugh1" ref-type="bibr">[61]</xref>, the problem of threshold-based neural signal detection can be considered as a statistical binary hypothesis test <xref rid="pone.0091345-Lindner1" ref-type="bibr">[7]</xref>, <xref rid="pone.0091345-Patel1" ref-type="bibr">[23]</xref>, <xref rid="pone.0091345-Blanchard1" ref-type="bibr">[27]</xref>. In this situation, explicit expressions for the maximum asymptotic detection efficacy are derived for a given transfer function of neuron model. We prove that colored noise that arises from a moving-average model is superior to white noise in improving the detection efficacy of neurons. It is illustratively shown that, for a single neuron with a signum threshold nonlinearity, the possibility of noise-enhanced detection only holds for non-scaled noise. For scaled noise, the effect of noise-enhanced detection does not occur in a single neuron model. However, when we tune the internal noise components of a parallel array of threshold neurons, it is observed that the constructive role of noise comes into play again in improving the signal detection efficacy, wherein suprathreshold stochastic resonance manifests its potentiality.</p></sec><sec id="s2"><title>Results</title><sec id="s2a"><title>Detection model</title><p>Consider the detection problem formulated as a binary hypothesis test <xref rid="pone.0091345-Patel1" ref-type="bibr">[23]</xref>, <xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref>, <xref rid="pone.0091345-Kay2" ref-type="bibr">[62]</xref><disp-formula id="pone.0091345.e001"><graphic xlink:href="pone.0091345.e001.jpg" position="anchor" orientation="portrait"></graphic><label>(1)</label></disp-formula>Under hypothesis <inline-formula><inline-graphic xlink:href="pone.0091345.e002.jpg"></inline-graphic></inline-formula>, the observation vector <inline-formula><inline-graphic xlink:href="pone.0091345.e003.jpg"></inline-graphic></inline-formula> consists of noise <inline-formula><inline-graphic xlink:href="pone.0091345.e004.jpg"></inline-graphic></inline-formula> only, and under hypothesis <inline-formula><inline-graphic xlink:href="pone.0091345.e005.jpg"></inline-graphic></inline-formula> it consists of noise <inline-formula><inline-graphic xlink:href="pone.0091345.e006.jpg"></inline-graphic></inline-formula> and known signal <inline-formula><inline-graphic xlink:href="pone.0091345.e007.jpg"></inline-graphic></inline-formula> with its strength <inline-formula><inline-graphic xlink:href="pone.0091345.e008.jpg"></inline-graphic></inline-formula>. There exists a finite bound <inline-formula><inline-graphic xlink:href="pone.0091345.e009.jpg"></inline-graphic></inline-formula> such that <inline-formula><inline-graphic xlink:href="pone.0091345.e010.jpg"></inline-graphic></inline-formula>, and the asymptotic average signal power satisfies <inline-formula><inline-graphic xlink:href="pone.0091345.e011.jpg"></inline-graphic></inline-formula><xref rid="pone.0091345-Poor1" ref-type="bibr">[56]</xref>–<xref rid="pone.0091345-Martinez1" ref-type="bibr">[58]</xref>, <xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref>, <xref rid="pone.0091345-Kay2" ref-type="bibr">[62]</xref>. Next, the test statistic <inline-formula><inline-graphic xlink:href="pone.0091345.e012.jpg"></inline-graphic></inline-formula> is compared with a decision threshold <inline-formula><inline-graphic xlink:href="pone.0091345.e013.jpg"></inline-graphic></inline-formula> to decide the hypotheses, as<disp-formula id="pone.0091345.e014"><graphic xlink:href="pone.0091345.e014.jpg" position="anchor" orientation="portrait"></graphic><label>(2)</label></disp-formula>where the coefficient vector <inline-formula><inline-graphic xlink:href="pone.0091345.e015.jpg"></inline-graphic></inline-formula> is associated with the function <inline-formula><inline-graphic xlink:href="pone.0091345.e016.jpg"></inline-graphic></inline-formula> to form <inline-formula><inline-graphic xlink:href="pone.0091345.e017.jpg"></inline-graphic></inline-formula>.</p><p>Assume the <inline-formula><inline-graphic xlink:href="pone.0091345.e018.jpg"></inline-graphic></inline-formula>-dimensional probability distribution <inline-formula><inline-graphic xlink:href="pone.0091345.e019.jpg"></inline-graphic></inline-formula> of noise <inline-formula><inline-graphic xlink:href="pone.0091345.e020.jpg"></inline-graphic></inline-formula> and zero-mean vector of <inline-formula><inline-graphic xlink:href="pone.0091345.e021.jpg"></inline-graphic></inline-formula> (for a shift in mean) <xref rid="pone.0091345-Martinez1" ref-type="bibr">[58]</xref>, <xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref>. Then, for a large sample size <inline-formula><inline-graphic xlink:href="pone.0091345.e022.jpg"></inline-graphic></inline-formula> of observation vector <inline-formula><inline-graphic xlink:href="pone.0091345.e023.jpg"></inline-graphic></inline-formula>, the test statistic <inline-formula><inline-graphic xlink:href="pone.0091345.e024.jpg"></inline-graphic></inline-formula> has zero-mean and asymptotic variance<disp-formula id="pone.0091345.e025"><graphic xlink:href="pone.0091345.e025.jpg" position="anchor" orientation="portrait"></graphic><label>(3)</label></disp-formula>under hypothesis <inline-formula><inline-graphic xlink:href="pone.0091345.e026.jpg"></inline-graphic></inline-formula>. Furthermore, for weak signals (<inline-formula><inline-graphic xlink:href="pone.0091345.e027.jpg"></inline-graphic></inline-formula>) and under hypothesis <inline-formula><inline-graphic xlink:href="pone.0091345.e028.jpg"></inline-graphic></inline-formula>, <inline-formula><inline-graphic xlink:href="pone.0091345.e029.jpg"></inline-graphic></inline-formula> can be expanded to the first-order<disp-formula id="pone.0091345.e030"><graphic xlink:href="pone.0091345.e030.jpg" position="anchor" orientation="portrait"></graphic><label>(4)</label></disp-formula>Then, the characteristics of <inline-formula><inline-graphic xlink:href="pone.0091345.e031.jpg"></inline-graphic></inline-formula> under <inline-formula><inline-graphic xlink:href="pone.0091345.e032.jpg"></inline-graphic></inline-formula>, up to the first-order in <inline-formula><inline-graphic xlink:href="pone.0091345.e033.jpg"></inline-graphic></inline-formula>, can be calculated as<disp-formula id="pone.0091345.e034"><graphic xlink:href="pone.0091345.e034.jpg" position="anchor" orientation="portrait"></graphic><label>(5)</label></disp-formula>Under both hypotheses <inline-formula><inline-graphic xlink:href="pone.0091345.e035.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e036.jpg"></inline-graphic></inline-formula>, the test statistic <inline-formula><inline-graphic xlink:href="pone.0091345.e037.jpg"></inline-graphic></inline-formula>, according to the central limit theorem, converges to a Gaussian distribution. Thus, the binary hypothesis test of Eq. (1) becomes a Gaussian mean-shift detection problem <xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref>, <xref rid="pone.0091345-Kay2" ref-type="bibr">[62]</xref>. Given the false probability, the detection probability is a monotonically increasing function of the detection efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e038.jpg"></inline-graphic></inline-formula><xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref>, <xref rid="pone.0091345-Kay2" ref-type="bibr">[62]</xref> given by<disp-formula id="pone.0091345.e039"><graphic xlink:href="pone.0091345.e039.jpg" position="anchor" orientation="portrait"></graphic><label>(6)</label></disp-formula>where the Cauchy-Schwarz inequality yields<disp-formula id="pone.0091345.e040"><graphic xlink:href="pone.0091345.e040.jpg" position="anchor" orientation="portrait"></graphic><label>(7)</label></disp-formula>with the Fisher information matrix <inline-formula><inline-graphic xlink:href="pone.0091345.e041.jpg"></inline-graphic></inline-formula>. Note that the equality of Eq. (6) is satisfied by the locally optimum nonlinearity<disp-formula id="pone.0091345.e042"><graphic xlink:href="pone.0091345.e042.jpg" position="anchor" orientation="portrait"></graphic><label>(8)</label></disp-formula>for an arbitrary constant <inline-formula><inline-graphic xlink:href="pone.0091345.e043.jpg"></inline-graphic></inline-formula>.</p><p>However, a complete closed-form noise distribution <inline-formula><inline-graphic xlink:href="pone.0091345.e044.jpg"></inline-graphic></inline-formula> may be unavailable, especially in unknown noisy circumstances <xref rid="pone.0091345-Poor1" ref-type="bibr">[56]</xref>–<xref rid="pone.0091345-Martinez1" ref-type="bibr">[58]</xref>, <xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref>, <xref rid="pone.0091345-Kay2" ref-type="bibr">[62]</xref>, which makes the nonlinearity of Eq. (8) difficult or too complex to implement. Thus, there may be compelling reasons for considering the given function <inline-formula><inline-graphic xlink:href="pone.0091345.e045.jpg"></inline-graphic></inline-formula> with an easily implemented feature. In this case, the detection efficacy in Eq. (6) can be maximized as<disp-formula id="pone.0091345.e046"><graphic xlink:href="pone.0091345.e046.jpg" position="anchor" orientation="portrait"></graphic><label>(9)</label></disp-formula>with the Cholesky decomposition of the variance matrix <inline-formula><inline-graphic xlink:href="pone.0091345.e047.jpg"></inline-graphic></inline-formula> and by optimally choosing the coefficient vector <inline-formula><inline-graphic xlink:href="pone.0091345.e048.jpg"></inline-graphic></inline-formula> for an arbitrary constant <inline-formula><inline-graphic xlink:href="pone.0091345.e049.jpg"></inline-graphic></inline-formula>.</p></sec><sec id="s2b"><title>Colored noise</title><p>Consider a useful colored noise model of the first-order moving-average <xref rid="pone.0091345-Poor1" ref-type="bibr">[56]</xref>, <xref rid="pone.0091345-Portnoy1" ref-type="bibr">[59]</xref> as<disp-formula id="pone.0091345.e050"><graphic xlink:href="pone.0091345.e050.jpg" position="anchor" orientation="portrait"></graphic><label>(10)</label></disp-formula>where the correlation coefficients are <inline-formula><inline-graphic xlink:href="pone.0091345.e051.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e052.jpg"></inline-graphic></inline-formula> is an independent identically distributed (i.i.d.) random vector. For small values of <inline-formula><inline-graphic xlink:href="pone.0091345.e053.jpg"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="pone.0091345.e054.jpg"></inline-graphic></inline-formula>), the dependence among noise samples <inline-formula><inline-graphic xlink:href="pone.0091345.e055.jpg"></inline-graphic></inline-formula> will be weak <xref rid="pone.0091345-Poor1" ref-type="bibr">[56]</xref>, <xref rid="pone.0091345-Portnoy1" ref-type="bibr">[59]</xref>. Here, we assume <inline-formula><inline-graphic xlink:href="pone.0091345.e056.jpg"></inline-graphic></inline-formula> have an univariate distribution <inline-formula><inline-graphic xlink:href="pone.0091345.e057.jpg"></inline-graphic></inline-formula> that is symmetric about the origin. We also assume the memoryless nonlinearity <inline-formula><inline-graphic xlink:href="pone.0091345.e058.jpg"></inline-graphic></inline-formula> to be odd symmetric about the origin. Then, up to first order in small correlation coefficients <inline-formula><inline-graphic xlink:href="pone.0091345.e059.jpg"></inline-graphic></inline-formula>, we can expand <inline-formula><inline-graphic xlink:href="pone.0091345.e060.jpg"></inline-graphic></inline-formula> as<disp-formula id="pone.0091345.e061"><graphic xlink:href="pone.0091345.e061.jpg" position="anchor" orientation="portrait"></graphic><label>(11)</label></disp-formula><disp-formula id="pone.0091345.e062"><graphic xlink:href="pone.0091345.e062.jpg" position="anchor" orientation="portrait"></graphic><label>(12)</label></disp-formula>and obtain expectations<disp-formula id="pone.0091345.e063"><graphic xlink:href="pone.0091345.e063.jpg" position="anchor" orientation="portrait"></graphic><label>(13)</label></disp-formula><disp-formula id="pone.0091345.e064"><graphic xlink:href="pone.0091345.e064.jpg" position="anchor" orientation="portrait"></graphic><label>(14)</label></disp-formula><disp-formula id="pone.0091345.e065"><graphic xlink:href="pone.0091345.e065.jpg" position="anchor" orientation="portrait"></graphic><label>(15)</label></disp-formula>Therefore, we have the expectation matrix<disp-formula id="pone.0091345.e066"><graphic xlink:href="pone.0091345.e066.jpg" position="anchor" orientation="portrait"></graphic><label>(16)</label></disp-formula>with the unit matrix <inline-formula><inline-graphic xlink:href="pone.0091345.e067.jpg"></inline-graphic></inline-formula>, and the variance matrix <inline-formula><inline-graphic xlink:href="pone.0091345.e068.jpg"></inline-graphic></inline-formula> has elements<disp-formula id="pone.0091345.e069"><graphic xlink:href="pone.0091345.e069.jpg" position="anchor" orientation="portrait"></graphic><label>(17)</label></disp-formula><disp-formula id="pone.0091345.e070"><graphic xlink:href="pone.0091345.e070.jpg" position="anchor" orientation="portrait"></graphic><label>(18)</label></disp-formula>for <inline-formula><inline-graphic xlink:href="pone.0091345.e071.jpg"></inline-graphic></inline-formula>. Then, based on Eq. (9), the normalized detection efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e072.jpg"></inline-graphic></inline-formula> can be computed as<disp-formula id="pone.0091345.e073"><graphic xlink:href="pone.0091345.e073.jpg" position="anchor" orientation="portrait"></graphic><label>(19)</label></disp-formula>Here, when the equality of Eq. (19) is achieved, the signal <inline-formula><inline-graphic xlink:href="pone.0091345.e074.jpg"></inline-graphic></inline-formula> is the corresponding eigenvector to the minimum eigenvalue <inline-formula><inline-graphic xlink:href="pone.0091345.e075.jpg"></inline-graphic></inline-formula> of the matrix <inline-formula><inline-graphic xlink:href="pone.0091345.e076.jpg"></inline-graphic></inline-formula>. It is known that the eigenvalues of the matrix <inline-formula><inline-graphic xlink:href="pone.0091345.e077.jpg"></inline-graphic></inline-formula> are <xref rid="pone.0091345-Kay2" ref-type="bibr">[62]</xref><disp-formula id="pone.0091345.e078"><graphic xlink:href="pone.0091345.e078.jpg" position="anchor" orientation="portrait"></graphic><label>(20)</label></disp-formula>corresponding to eigenvectors <inline-formula><inline-graphic xlink:href="pone.0091345.e079.jpg"></inline-graphic></inline-formula> for <inline-formula><inline-graphic xlink:href="pone.0091345.e080.jpg"></inline-graphic></inline-formula>. Here, as the nonlinearity <inline-formula><inline-graphic xlink:href="pone.0091345.e081.jpg"></inline-graphic></inline-formula> is assumed to be odd, it is then found that <inline-formula><inline-graphic xlink:href="pone.0091345.e082.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e083.jpg"></inline-graphic></inline-formula>. Therefore, if <inline-formula><inline-graphic xlink:href="pone.0091345.e084.jpg"></inline-graphic></inline-formula> and for a large sample size <inline-formula><inline-graphic xlink:href="pone.0091345.e085.jpg"></inline-graphic></inline-formula>, we take <inline-formula><inline-graphic xlink:href="pone.0091345.e086.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e087.jpg"></inline-graphic></inline-formula>. Otherwise, we choose <inline-formula><inline-graphic xlink:href="pone.0091345.e088.jpg"></inline-graphic></inline-formula>. An illustration of the eigenvector <inline-formula><inline-graphic xlink:href="pone.0091345.e089.jpg"></inline-graphic></inline-formula> is shown in <xref ref-type="fig" rid="pone-0091345-g001">Fig. 1</xref> for <inline-formula><inline-graphic xlink:href="pone.0091345.e090.jpg"></inline-graphic></inline-formula>. In this way, by optimally choosing the input signal (eigenvector) <inline-formula><inline-graphic xlink:href="pone.0091345.e091.jpg"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="pone.0091345.e092.jpg"></inline-graphic></inline-formula>), the maximum efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e093.jpg"></inline-graphic></inline-formula> can be calculated as<disp-formula id="pone.0091345.e094"><graphic xlink:href="pone.0091345.e094.jpg" position="anchor" orientation="portrait"></graphic><label>(21)</label></disp-formula></p><fig id="pone-0091345-g001" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0091345.g001</object-id><label>Figure 1</label><caption><title>Eigenvector <inline-formula><inline-graphic xlink:href="pone.0091345.e095.jpg"></inline-graphic></inline-formula>.</title><p>An illustration of the eigenvector <inline-formula><inline-graphic xlink:href="pone.0091345.e096.jpg"></inline-graphic></inline-formula> of the variance matrix <inline-formula><inline-graphic xlink:href="pone.0091345.e097.jpg"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="pone.0091345.e098.jpg"></inline-graphic></inline-formula>).</p></caption><graphic xlink:href="pone.0091345.g001"></graphic></fig><p>Since, from its definition in Eq. (6), the efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e099.jpg"></inline-graphic></inline-formula> is non-negative, the denominator in Eq. (21) must satisfy<disp-formula id="pone.0091345.e100"><graphic xlink:href="pone.0091345.e100.jpg" position="anchor" orientation="portrait"></graphic><label>(22)</label></disp-formula>In order to validate Eq. (22), we use the Cauchy-Schwarz inequality to yield<disp-formula id="pone.0091345.e101"><graphic xlink:href="pone.0091345.e101.jpg" position="anchor" orientation="portrait"></graphic><label>(23)</label></disp-formula><disp-formula id="pone.0091345.e102"><graphic xlink:href="pone.0091345.e102.jpg" position="anchor" orientation="portrait"></graphic><label>(24)</label></disp-formula>with the Fisher information quantity <inline-formula><inline-graphic xlink:href="pone.0091345.e103.jpg"></inline-graphic></inline-formula> and the variance <inline-formula><inline-graphic xlink:href="pone.0091345.e104.jpg"></inline-graphic></inline-formula> of noise distribution of <inline-formula><inline-graphic xlink:href="pone.0091345.e105.jpg"></inline-graphic></inline-formula><xref rid="pone.0091345-Poor1" ref-type="bibr">[56]</xref>. Thus, we find<disp-formula id="pone.0091345.e106"><graphic xlink:href="pone.0091345.e106.jpg" position="anchor" orientation="portrait"></graphic><label>(25)</label></disp-formula>Substituting Eq. (25) into Eq. (22) and noting<disp-formula id="pone.0091345.e107"><graphic xlink:href="pone.0091345.e107.jpg" position="anchor" orientation="portrait"></graphic><label>(26)</label></disp-formula>we have<disp-formula id="pone.0091345.e108"><graphic xlink:href="pone.0091345.e108.jpg" position="anchor" orientation="portrait"></graphic><label>(27)</label></disp-formula>Since we assume <inline-formula><inline-graphic xlink:href="pone.0091345.e109.jpg"></inline-graphic></inline-formula>, the inequalities of Eqs. (27) and (22) can be satisfied, and the detector efficacy in Eq. (21) will be theoretically analyzed in the following.</p><p>For white noise vector <inline-formula><inline-graphic xlink:href="pone.0091345.e110.jpg"></inline-graphic></inline-formula> with zero correlation coefficients <inline-formula><inline-graphic xlink:href="pone.0091345.e111.jpg"></inline-graphic></inline-formula>, the detection efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e112.jpg"></inline-graphic></inline-formula> in Eq. (21) satisfies<disp-formula id="pone.0091345.e113"><graphic xlink:href="pone.0091345.e113.jpg" position="anchor" orientation="portrait"></graphic><label>(28)</label></disp-formula>Thus, for a given function <inline-formula><inline-graphic xlink:href="pone.0091345.e114.jpg"></inline-graphic></inline-formula>, colored noise is superior to white noise in enhancing the detection efficacy, at a cost of optimally matching the input signal with the eigenvector <inline-formula><inline-graphic xlink:href="pone.0091345.e115.jpg"></inline-graphic></inline-formula> of covariance matrix <inline-formula><inline-graphic xlink:href="pone.0091345.e116.jpg"></inline-graphic></inline-formula>.</p></sec><sec id="s2c"><title>Stochastic resonance in threshold-based neurons</title><p>We will illustratively show the possibilities of noise-enhanced detection in threshold-based neurons. The classical McCulloch-Pitts threshold neuron has the form<disp-formula id="pone.0091345.e117"><graphic xlink:href="pone.0091345.e117.jpg" position="anchor" orientation="portrait"></graphic><label>(29)</label></disp-formula>with the response threshold <inline-formula><inline-graphic xlink:href="pone.0091345.e118.jpg"></inline-graphic></inline-formula>. It is seen that <inline-formula><inline-graphic xlink:href="pone.0091345.e119.jpg"></inline-graphic></inline-formula> can be expressed as a function of <inline-formula><inline-graphic xlink:href="pone.0091345.e120.jpg"></inline-graphic></inline-formula> in terms of the signum (sign) function as <inline-formula><inline-graphic xlink:href="pone.0091345.e121.jpg"></inline-graphic></inline-formula>. Since the constant factor <inline-formula><inline-graphic xlink:href="pone.0091345.e122.jpg"></inline-graphic></inline-formula> does not affect the detection efficacy of the transfer function <inline-formula><inline-graphic xlink:href="pone.0091345.e123.jpg"></inline-graphic></inline-formula>, then we focus on the signum function<disp-formula id="pone.0091345.e124"><graphic xlink:href="pone.0091345.e124.jpg" position="anchor" orientation="portrait"></graphic><label>(30)</label></disp-formula>with response threshold <inline-formula><inline-graphic xlink:href="pone.0091345.e125.jpg"></inline-graphic></inline-formula> in the following parts. Here, the signum function <inline-formula><inline-graphic xlink:href="pone.0091345.e126.jpg"></inline-graphic></inline-formula> is not continuous at <inline-formula><inline-graphic xlink:href="pone.0091345.e127.jpg"></inline-graphic></inline-formula>, but has its derivative <inline-formula><inline-graphic xlink:href="pone.0091345.e128.jpg"></inline-graphic></inline-formula> for any <inline-formula><inline-graphic xlink:href="pone.0091345.e129.jpg"></inline-graphic></inline-formula><xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref>.</p><p>For the colored noise model of Eq. (10), the correlation coefficient <inline-formula><inline-graphic xlink:href="pone.0091345.e130.jpg"></inline-graphic></inline-formula> indicates the noise sequence <inline-formula><inline-graphic xlink:href="pone.0091345.e131.jpg"></inline-graphic></inline-formula> is a causal process that can be physically realized. Here, we assume <inline-formula><inline-graphic xlink:href="pone.0091345.e132.jpg"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="pone.0091345.e133.jpg"></inline-graphic></inline-formula>) and <inline-formula><inline-graphic xlink:href="pone.0091345.e134.jpg"></inline-graphic></inline-formula>, and show the possibility of stochastic resonance in the physically realizable noise environment. First, consider scaled noise <inline-formula><inline-graphic xlink:href="pone.0091345.e135.jpg"></inline-graphic></inline-formula> that has the distribution <inline-formula><inline-graphic xlink:href="pone.0091345.e136.jpg"></inline-graphic></inline-formula><xref rid="pone.0091345-Patel1" ref-type="bibr">[23]</xref>, <xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref>. Here, <inline-formula><inline-graphic xlink:href="pone.0091345.e137.jpg"></inline-graphic></inline-formula> has a standardized distribution <inline-formula><inline-graphic xlink:href="pone.0091345.e138.jpg"></inline-graphic></inline-formula> with unity variance <inline-formula><inline-graphic xlink:href="pone.0091345.e139.jpg"></inline-graphic></inline-formula>. Thus, based on Eq. (21), the absolute moment is<disp-formula id="pone.0091345.e140"><graphic xlink:href="pone.0091345.e140.jpg" position="anchor" orientation="portrait"></graphic><label>(31)</label></disp-formula>where the operator <inline-formula><inline-graphic xlink:href="pone.0091345.e141.jpg"></inline-graphic></inline-formula>. Thus, for the signum function <inline-formula><inline-graphic xlink:href="pone.0091345.e142.jpg"></inline-graphic></inline-formula>, the detection efficacy of Eq. (21) can be expressed as<disp-formula id="pone.0091345.e143"><graphic xlink:href="pone.0091345.e143.jpg" position="anchor" orientation="portrait"></graphic><label>(32)</label></disp-formula>It is seen in Eq. (32) that <inline-formula><inline-graphic xlink:href="pone.0091345.e144.jpg"></inline-graphic></inline-formula> is a monotonically decreasing function of noise variance <inline-formula><inline-graphic xlink:href="pone.0091345.e145.jpg"></inline-graphic></inline-formula>, and no noise-enhanced detection effect will occur in such a single neuron model for scaled noise.</p><p>We further consider non-scaled Gaussian mixture distribution <xref rid="pone.0091345-ChapeauBlondeau1" ref-type="bibr">[5]</xref>, <xref rid="pone.0091345-Zozor1" ref-type="bibr">[47]</xref>, <xref rid="pone.0091345-Kay1" ref-type="bibr">[48]</xref>, <xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref><disp-formula id="pone.0091345.e146"><graphic xlink:href="pone.0091345.e146.jpg" position="anchor" orientation="portrait"></graphic><label>(33)</label></disp-formula>where the variance <inline-formula><inline-graphic xlink:href="pone.0091345.e147.jpg"></inline-graphic></inline-formula> and parameters <inline-formula><inline-graphic xlink:href="pone.0091345.e148.jpg"></inline-graphic></inline-formula>. Then, for the signum function <inline-formula><inline-graphic xlink:href="pone.0091345.e149.jpg"></inline-graphic></inline-formula> in Eq. (30), the detection efficacy of Eq. (21) can be computed as<disp-formula id="pone.0091345.e150"><graphic xlink:href="pone.0091345.e150.jpg" position="anchor" orientation="portrait"></graphic><label>(34)</label></disp-formula>where the error function <inline-formula><inline-graphic xlink:href="pone.0091345.e151.jpg"></inline-graphic></inline-formula>. In <xref ref-type="fig" rid="pone-0091345-g002">Fig. 2</xref>, for the correlation coefficient <inline-formula><inline-graphic xlink:href="pone.0091345.e152.jpg"></inline-graphic></inline-formula> and different values of <inline-formula><inline-graphic xlink:href="pone.0091345.e153.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e154.jpg"></inline-graphic></inline-formula>, we show the detection efficacy of Eq. (34) as a function of noise variance <inline-formula><inline-graphic xlink:href="pone.0091345.e155.jpg"></inline-graphic></inline-formula>. For a given non-zero value <inline-formula><inline-graphic xlink:href="pone.0091345.e156.jpg"></inline-graphic></inline-formula> and as <inline-formula><inline-graphic xlink:href="pone.0091345.e157.jpg"></inline-graphic></inline-formula>, the noise distribution model of Eq. (33) indicates the dichotomous noise <xref rid="pone.0091345-ChapeauBlondeau1" ref-type="bibr">[5]</xref>, <xref rid="pone.0091345-Kay1" ref-type="bibr">[48]</xref>, <xref rid="pone.0091345-Duan1" ref-type="bibr">[53]</xref>. In this situation, as the signal strength <inline-formula><inline-graphic xlink:href="pone.0091345.e158.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e159.jpg"></inline-graphic></inline-formula>, the signum function <inline-formula><inline-graphic xlink:href="pone.0091345.e160.jpg"></inline-graphic></inline-formula> will not change its output whether the signal appears or not. Therefore, the test statistics <inline-formula><inline-graphic xlink:href="pone.0091345.e161.jpg"></inline-graphic></inline-formula> in Eq. (2) will be the same value under hypotheses <inline-formula><inline-graphic xlink:href="pone.0091345.e162.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e163.jpg"></inline-graphic></inline-formula>, and the detection efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e164.jpg"></inline-graphic></inline-formula> in Eq. (34) starts from zero. This explantation can be also validated by Eq. (34) as <inline-formula><inline-graphic xlink:href="pone.0091345.e165.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e166.jpg"></inline-graphic></inline-formula> being fixed, as illustrated in <xref ref-type="fig" rid="pone-0091345-g002">Fig. 2</xref>. However, it is clearly seen that, upon increasing the noise variance <inline-formula><inline-graphic xlink:href="pone.0091345.e167.jpg"></inline-graphic></inline-formula> (actually increasing <inline-formula><inline-graphic xlink:href="pone.0091345.e168.jpg"></inline-graphic></inline-formula>), the noise-enhanced detection effect appears. The smaller the parameter <inline-formula><inline-graphic xlink:href="pone.0091345.e169.jpg"></inline-graphic></inline-formula> is, the more pronounced the resonant peak of <inline-formula><inline-graphic xlink:href="pone.0091345.e170.jpg"></inline-graphic></inline-formula> becomes, as shown in <xref ref-type="fig" rid="pone-0091345-g002">Fig. 2</xref>.</p><fig id="pone-0091345-g002" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0091345.g002</object-id><label>Figure 2</label><caption><title>Stochastic resonance of a single threshold neuron.</title><p>Detection efficacy of <inline-formula><inline-graphic xlink:href="pone.0091345.e171.jpg"></inline-graphic></inline-formula> as a function of noise variance <inline-formula><inline-graphic xlink:href="pone.0091345.e172.jpg"></inline-graphic></inline-formula> for the correlation coefficient <inline-formula><inline-graphic xlink:href="pone.0091345.e173.jpg"></inline-graphic></inline-formula> and different values of <inline-formula><inline-graphic xlink:href="pone.0091345.e174.jpg"></inline-graphic></inline-formula> (red), <inline-formula><inline-graphic xlink:href="pone.0091345.e175.jpg"></inline-graphic></inline-formula> (blue) and <inline-formula><inline-graphic xlink:href="pone.0091345.e176.jpg"></inline-graphic></inline-formula> (green). The resonant peaks of <inline-formula><inline-graphic xlink:href="pone.0091345.e177.jpg"></inline-graphic></inline-formula> are marked by the square (▪), the star (<inline-formula><inline-graphic xlink:href="pone.0091345.e178.jpg"></inline-graphic></inline-formula>) and the down triangle (▾) for <inline-formula><inline-graphic xlink:href="pone.0091345.e179.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e180.jpg"></inline-graphic></inline-formula>, respectively. Here, the transfer function <inline-formula><inline-graphic xlink:href="pone.0091345.e181.jpg"></inline-graphic></inline-formula>, and the noise distribution is Gaussian mixture model of Eq. (33).</p></caption><graphic xlink:href="pone.0091345.g002"></graphic></fig><p>Next, an interesting problem is that, for scaled noise, can we observe the noise-enhanced detection effect in threshold-based neurons? The answer to this question is affirmative. Here, we will resort to the constructive role of internal noise for improving the performance of an array of threshold neurons. Let <inline-formula><inline-graphic xlink:href="pone.0091345.e182.jpg"></inline-graphic></inline-formula> be the vector of <inline-formula><inline-graphic xlink:href="pone.0091345.e183.jpg"></inline-graphic></inline-formula> observation components at the <inline-formula><inline-graphic xlink:href="pone.0091345.e184.jpg"></inline-graphic></inline-formula>-th element of receiving array of <inline-formula><inline-graphic xlink:href="pone.0091345.e185.jpg"></inline-graphic></inline-formula> identical neurons. In this observation model, <inline-formula><inline-graphic xlink:href="pone.0091345.e186.jpg"></inline-graphic></inline-formula> under the hypothesis <inline-formula><inline-graphic xlink:href="pone.0091345.e187.jpg"></inline-graphic></inline-formula>. Here, in each neuron element, the <inline-formula><inline-graphic xlink:href="pone.0091345.e188.jpg"></inline-graphic></inline-formula> noise terms <inline-formula><inline-graphic xlink:href="pone.0091345.e189.jpg"></inline-graphic></inline-formula> are assumed to be mutually independent with the same PDF <inline-formula><inline-graphic xlink:href="pone.0091345.e190.jpg"></inline-graphic></inline-formula> and variance <inline-formula><inline-graphic xlink:href="pone.0091345.e191.jpg"></inline-graphic></inline-formula>. Then, at the observed time <inline-formula><inline-graphic xlink:href="pone.0091345.e192.jpg"></inline-graphic></inline-formula>, the array outputs are collected as <inline-formula><inline-graphic xlink:href="pone.0091345.e193.jpg"></inline-graphic></inline-formula>, and the test statistics can be reconstructed as <inline-formula><inline-graphic xlink:href="pone.0091345.e194.jpg"></inline-graphic></inline-formula> with <inline-formula><inline-graphic xlink:href="pone.0091345.e195.jpg"></inline-graphic></inline-formula>. For the colored noise model of Eq. (10) with <inline-formula><inline-graphic xlink:href="pone.0091345.e196.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e197.jpg"></inline-graphic></inline-formula>, we have<disp-formula id="pone.0091345.e198"><graphic xlink:href="pone.0091345.e198.jpg" position="anchor" orientation="portrait"></graphic><label>(35)</label></disp-formula><disp-formula id="pone.0091345.e199"><graphic xlink:href="pone.0091345.e199.jpg" position="anchor" orientation="portrait"></graphic><label>(36)</label></disp-formula>where the composite noise <inline-formula><inline-graphic xlink:href="pone.0091345.e200.jpg"></inline-graphic></inline-formula> has the convolved distribution <inline-formula><inline-graphic xlink:href="pone.0091345.e201.jpg"></inline-graphic></inline-formula>. Then, we have expectations<disp-formula id="pone.0091345.e202"><graphic xlink:href="pone.0091345.e202.jpg" position="anchor" orientation="portrait"></graphic><label>(37)</label></disp-formula>and<disp-formula id="pone.0091345.e203"><graphic xlink:href="pone.0091345.e203.jpg" position="anchor" orientation="portrait"></graphic><label>(38)</label></disp-formula>with the operator <inline-formula><inline-graphic xlink:href="pone.0091345.e204.jpg"></inline-graphic></inline-formula>. The variance matrix <inline-formula><inline-graphic xlink:href="pone.0091345.e205.jpg"></inline-graphic></inline-formula> has elements<disp-formula id="pone.0091345.e206"><graphic xlink:href="pone.0091345.e206.jpg" position="anchor" orientation="portrait"></graphic><label>(39)</label></disp-formula></p><p><disp-formula id="pone.0091345.e207"><graphic xlink:href="pone.0091345.e207.jpg" position="anchor" orientation="portrait"></graphic><label>(40)</label></disp-formula>Then, based on Eqs. (38), (39) and (40), the maximum efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e208.jpg"></inline-graphic></inline-formula> can be computed by Eq. (21) as<disp-formula id="pone.0091345.e209"><graphic xlink:href="pone.0091345.e209.jpg" position="anchor" orientation="portrait"></graphic><label>(41)</label></disp-formula></p><p>For instance, we assume the initial Gaussian noise components <inline-formula><inline-graphic xlink:href="pone.0091345.e210.jpg"></inline-graphic></inline-formula> have the distribution of <inline-formula><inline-graphic xlink:href="pone.0091345.e211.jpg"></inline-graphic></inline-formula> and the given variance <inline-formula><inline-graphic xlink:href="pone.0091345.e212.jpg"></inline-graphic></inline-formula>. The internal noise components of each neuron is assumed to be the uniform random variable <inline-formula><inline-graphic xlink:href="pone.0091345.e213.jpg"></inline-graphic></inline-formula> with its distribution <inline-formula><inline-graphic xlink:href="pone.0091345.e214.jpg"></inline-graphic></inline-formula> for <inline-formula><inline-graphic xlink:href="pone.0091345.e215.jpg"></inline-graphic></inline-formula> and zero otherwise. The composite random variables <inline-formula><inline-graphic xlink:href="pone.0091345.e216.jpg"></inline-graphic></inline-formula> are distributed by<disp-formula id="pone.0091345.e217"><graphic xlink:href="pone.0091345.e217.jpg" position="anchor" orientation="portrait"></graphic><label>(42)</label></disp-formula>For a given Gaussian noise level <inline-formula><inline-graphic xlink:href="pone.0091345.e218.jpg"></inline-graphic></inline-formula>, it is shown in <xref ref-type="fig" rid="pone-0091345-g003">Figs. 3</xref> (a) and (b) that the maximum detection efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e219.jpg"></inline-graphic></inline-formula> varies as a function of internal uniform noise level <inline-formula><inline-graphic xlink:href="pone.0091345.e220.jpg"></inline-graphic></inline-formula> for different array sizes <inline-formula><inline-graphic xlink:href="pone.0091345.e221.jpg"></inline-graphic></inline-formula> and correlation coefficients <inline-formula><inline-graphic xlink:href="pone.0091345.e222.jpg"></inline-graphic></inline-formula>. It is noted that, at the uniform noise level <inline-formula><inline-graphic xlink:href="pone.0091345.e223.jpg"></inline-graphic></inline-formula>, the detection efficacy in Eq. (41) is just the expression of Eq. (32) for a single neuron. Thus, <inline-formula><inline-graphic xlink:href="pone.0091345.e224.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pone.0091345.e225.jpg"></inline-graphic></inline-formula> for <inline-formula><inline-graphic xlink:href="pone.0091345.e226.jpg"></inline-graphic></inline-formula> (see <xref ref-type="fig" rid="pone-0091345-g003">Fig. 3</xref> (a)) and <inline-formula><inline-graphic xlink:href="pone.0091345.e227.jpg"></inline-graphic></inline-formula> (see <xref ref-type="fig" rid="pone-0091345-g003">Fig. 3</xref> (b)), respectively. By comparing <xref ref-type="fig" rid="pone-0091345-g003">Fig. 3</xref> (a) with <xref ref-type="fig" rid="pone-0091345-g003">Fig. 3</xref> (b), it is seen that the maximum detection efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e228.jpg"></inline-graphic></inline-formula> can be further enhanced for a higher value of <inline-formula><inline-graphic xlink:href="pone.0091345.e229.jpg"></inline-graphic></inline-formula>. For the array size <inline-formula><inline-graphic xlink:href="pone.0091345.e230.jpg"></inline-graphic></inline-formula> and upon increasing uniform noise level <inline-formula><inline-graphic xlink:href="pone.0091345.e231.jpg"></inline-graphic></inline-formula>, it is seen in <xref ref-type="fig" rid="pone-0091345-g003">Fig. 3</xref> that there is no noise-enhanced effect in a single neuron. However, as <inline-formula><inline-graphic xlink:href="pone.0091345.e232.jpg"></inline-graphic></inline-formula>, it is illustrated in <xref ref-type="fig" rid="pone-0091345-g003">Fig. 3</xref> that the internal uniform noise can enhance the detection efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e233.jpg"></inline-graphic></inline-formula>, and the noise-enhanced effect does occur. Moreover, as the array size <inline-formula><inline-graphic xlink:href="pone.0091345.e234.jpg"></inline-graphic></inline-formula> increases, the noise-induced enhancement becomes more visible by adopting an appropriate amount of uniform noise of the neuron array, as shown in <xref ref-type="fig" rid="pone-0091345-g003">Fig. 3</xref>. As the detection problem so far is confined to the weak signal with its strength <inline-formula><inline-graphic xlink:href="pone.0091345.e235.jpg"></inline-graphic></inline-formula> but <inline-formula><inline-graphic xlink:href="pone.0091345.e236.jpg"></inline-graphic></inline-formula>, and the response threshold <inline-formula><inline-graphic xlink:href="pone.0091345.e237.jpg"></inline-graphic></inline-formula> of all neurons is zero, thus <xref ref-type="fig" rid="pone-0091345-g003">Fig. 3</xref> shows the potential capability of suprathreshold stochastic resonance in improving the detection efficacy of a parallel array threshold-based neurons.</p><fig id="pone-0091345-g003" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0091345.g003</object-id><label>Figure 3</label><caption><title>Suprathreshold stochastic resonance in an array of threshold neurons.</title><p>Detection efficacy <inline-formula><inline-graphic xlink:href="pone.0091345.e238.jpg"></inline-graphic></inline-formula> as a function of the internal uniform noise level <inline-formula><inline-graphic xlink:href="pone.0091345.e239.jpg"></inline-graphic></inline-formula> and the neuron array size <inline-formula><inline-graphic xlink:href="pone.0091345.e240.jpg"></inline-graphic></inline-formula>. From the bottom upwards, <inline-formula><inline-graphic xlink:href="pone.0091345.e241.jpg"></inline-graphic></inline-formula>. Here, the initial Gaussian noise level <inline-formula><inline-graphic xlink:href="pone.0091345.e242.jpg"></inline-graphic></inline-formula>, the transfer function <inline-formula><inline-graphic xlink:href="pone.0091345.e243.jpg"></inline-graphic></inline-formula>, the correlation coefficient (a) <inline-formula><inline-graphic xlink:href="pone.0091345.e244.jpg"></inline-graphic></inline-formula> and (b) <inline-formula><inline-graphic xlink:href="pone.0091345.e245.jpg"></inline-graphic></inline-formula>.</p></caption><graphic xlink:href="pone.0091345.g003"></graphic></fig></sec></sec><sec sec-type="methods" id="s3"><title>Methods</title><p>Under the assumption of weak signals, the Taylor expansion of the function is utilized in Eqs. (4), (5), (11) and (12). The Cauchy-Schwarz inequality is used in Eqs. (7), (9), (23), and (24). The maximum of Rayleigh quotients for a symmetric matrix is calculated in Eqs. (19), (21) and (41).</p></sec><sec id="s4"><title>Conclusion</title><p>In this paper, we study the performance enhancement of threshold-based neurons for detecting weak signals in the presence of colored noise. For a given transfer function, we maximize the detection efficacy by optimally choosing the signal waveform. We prove that colored noise is superior to white noise in enhancing the detection efficacy, at a cost of optimally matching the input signal with the eigenvector of the covariance matrix. Furthermore, we illustrate that, for a single threshold neuron, the possibility of noise-enhanced detection cannot occur in scaled noise, but does appear in a non-scaled Gaussian mixture noise model. Furthermore, for scaled noise, we can test a parallel bundle of neurons with the same response threshold, and recover the positive role of internal noise in enhancing the detection efficacy of the neuron array via the mechanism of suprathreshold stochastic resonance. These results demonstrate that the strategy of exploiting stochastic resonance is still interesting in the case of improving the nonlinear system performance by adding more noise to the signal corrupted by colored noise.</p><p>Here, we mainly consider the first-order moving-average noise model of Eq. (10) which is, as we show, amenable to analytical treatment. It is possible to extend the present approach to higher-order moving-average noise models. However, the same analytical treatment maybe no longer feasible. It is also interesting to consider yet other models of colored noise to enhance the detectability of the neuron array. This subject is very promising and currently under study.</p><p>It is noted that the detection efficacy of Eqs. (6) and (9) are established under the assumption of weak signal strength <inline-formula><inline-graphic xlink:href="pone.0091345.e246.jpg"></inline-graphic></inline-formula>. We only consider the first-order Taylor expansion of nonlinearities in Eq. (4), because it makes an analytical treatment possible and the corresponding results are rigorous. In practice, most noise distributions are symmetric and the nonlinear characteristics are odd symmetric about the origin. In this case, we can expand the nonlinearity to the second-order terms. The expectation of the second-order term of Taylor expansion of Eq. (4) vanishes and does not affect the conclusion of this paper. However, for unsymmetrical noise distributions and nonlinearities, the high-order terms of Taylor expansion of Eq. (4) are not exactly zero. For this case, we need to numerically observe the effect of high-order terms on the detector performance. It is interesting to compare the present theoretical results of first-order expansion with the numerical results in the further studies.</p><p>We also note that these equations of Eqs. (4)–(9) are the extension of white noise <xref rid="pone.0091345-Duan2" ref-type="bibr">[54]</xref>, <xref rid="pone.0091345-Poor1" ref-type="bibr">[56]</xref>–<xref rid="pone.0091345-Martinez1" ref-type="bibr">[58]</xref>, <xref rid="pone.0091345-Kassam1" ref-type="bibr">[60]</xref> to the case of colored noise. Then, we consider a model of colored noise allowing for an analytical evaluation of the detection efficacy in Eqs. (6) and (9). The detection efficacy can also be numerically computed to address other models of colored noise, or to explore broader conditions beyond the weak signal limit. As the signal strength <inline-formula><inline-graphic xlink:href="pone.0091345.e247.jpg"></inline-graphic></inline-formula> increases, the Taylor expansion of Eq. (4) and the upper bound of Eq. (6) gradually cease to apply. 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