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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Metastable dynamics in heterogeneous neural fields</title><author><name sortKey="Schwappach, Cordula" sort="Schwappach, Cordula" uniqKey="Schwappach C" first="Cordula" last="Schwappach">Cordula Schwappach</name><affiliation><nlm:aff id="aff1"><institution>Department of German Studies and Linguistics, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></nlm:aff></affiliation><affiliation><nlm:aff id="aff2"><institution>Department of Physics, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></nlm:aff></affiliation></author><author><name sortKey="Hutt, Axel" sort="Hutt, Axel" uniqKey="Hutt A" first="Axel" last="Hutt">Axel Hutt</name><affiliation><nlm:aff id="aff3"><institution>Team Neurosys, Inria</institution><country>Villers-les-Nancy, France</country></nlm:aff></affiliation><affiliation><nlm:aff id="aff4"><institution>Team Neurosys, Centre National de la Recherche Scientifique, UMR nō 7503, Loria</institution><country>Villers-les-Nancy, France</country></nlm:aff></affiliation><affiliation><nlm:aff id="aff5"><institution>Team Neurosys, UMR nō 7503, Loria, Université de Lorraine</institution><country>Villers-les-Nancy, France</country></nlm:aff></affiliation></author><author><name sortKey="Beim Graben, Peter" sort="Beim Graben, Peter" uniqKey="Beim Graben P" first="Peter" last="Beim Graben">Peter Beim Graben</name><affiliation><nlm:aff id="aff1"><institution>Department of German Studies and Linguistics, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></nlm:aff></affiliation><affiliation><nlm:aff id="aff6"><institution>Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">26175671</idno><idno type="pmc">4485166</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4485166</idno><idno type="RBID">PMC:4485166</idno><idno type="doi">10.3389/fnsys.2015.00097</idno><date when="2015">2015</date><idno type="wicri:Area/Pmc/Corpus">000005</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000005</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Metastable dynamics in heterogeneous neural fields</title><author><name sortKey="Schwappach, Cordula" sort="Schwappach, Cordula" uniqKey="Schwappach C" first="Cordula" last="Schwappach">Cordula Schwappach</name><affiliation><nlm:aff id="aff1"><institution>Department of German Studies and Linguistics, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></nlm:aff></affiliation><affiliation><nlm:aff id="aff2"><institution>Department of Physics, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></nlm:aff></affiliation></author><author><name sortKey="Hutt, Axel" sort="Hutt, Axel" uniqKey="Hutt A" first="Axel" last="Hutt">Axel Hutt</name><affiliation><nlm:aff id="aff3"><institution>Team Neurosys, Inria</institution><country>Villers-les-Nancy, France</country></nlm:aff></affiliation><affiliation><nlm:aff id="aff4"><institution>Team Neurosys, Centre National de la Recherche Scientifique, UMR nō 7503, Loria</institution><country>Villers-les-Nancy, France</country></nlm:aff></affiliation><affiliation><nlm:aff id="aff5"><institution>Team Neurosys, UMR nō 7503, Loria, Université de Lorraine</institution><country>Villers-les-Nancy, France</country></nlm:aff></affiliation></author><author><name sortKey="Beim Graben, Peter" sort="Beim Graben, Peter" uniqKey="Beim Graben P" first="Peter" last="Beim Graben">Peter Beim Graben</name><affiliation><nlm:aff id="aff1"><institution>Department of German Studies and Linguistics, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></nlm:aff></affiliation><affiliation><nlm:aff id="aff6"><institution>Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></nlm:aff></affiliation></author></analytic><series><title level="j">Frontiers in Systems Neuroscience</title><idno type="eISSN">1662-5137</idno><imprint><date when="2015">2015</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>We present numerical simulations of metastable states in heterogeneous neural fields that are connected along heteroclinic orbits. Such trajectories are possible representations of transient neural activity as observed, for example, in the electroencephalogram. Based on previous theoretical findings on learning algorithms for neural fields, we directly construct synaptic weight kernels from Lotka-Volterra neural population dynamics without supervised training approaches. We deliver a MATLAB neural field toolbox validated by two examples of one- and two-dimensional neural fields. We demonstrate trial-to-trial variability and distributed representations in our simulations which might therefore be regarded as a proof-of-concept for more advanced neural field models of metastable dynamics in neurophysiological data.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Afraimovich, V S" uniqKey="Afraimovich V">V. S. Afraimovich</name></author><author><name sortKey="Rabinovich, M I" uniqKey="Rabinovich M">M. I. Rabinovich</name></author><author><name sortKey="Varona, P" uniqKey="Varona P">P. Varona</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Afraimovich, V S" uniqKey="Afraimovich V">V. S. Afraimovich</name></author><author><name sortKey="Zhigulin, V P" uniqKey="Zhigulin V">V. P. Zhigulin</name></author><author><name sortKey="Rabinovich, M I" uniqKey="Rabinovich M">M. I. Rabinovich</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Allefeld, C" uniqKey="Allefeld C">C. Allefeld</name></author><author><name sortKey="Atmanspacher, H" uniqKey="Atmanspacher H">H. Atmanspacher</name></author><author><name sortKey="Wackermann, J" uniqKey="Wackermann J">J. Wackermann</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Amari, S I" uniqKey="Amari S">S.-I. Amari</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Barres, V" uniqKey="Barres V">V. Barrès</name></author><author><name sortKey="Simons, A" uniqKey="Simons A">A. Simons</name></author><author><name sortKey="Arbib, M" uniqKey="Arbib M">M. Arbib</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Beim Graben, P" uniqKey="Beim Graben P">P. beim Graben</name></author><author><name sortKey="Hutt, A" uniqKey="Hutt A">A. Hutt</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Beim Graben, P" uniqKey="Beim Graben P">P. beim Graben</name></author><author><name sortKey="Hutt, A" uniqKey="Hutt A">A. Hutt</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Beim Graben, P" uniqKey="Beim Graben P">P. beim Graben</name></author><author><name sortKey="Potthast, R" uniqKey="Potthast R">R. Potthast</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Beim Graben, P" uniqKey="Beim Graben P">P. beim Graben</name></author><author><name sortKey="Potthast, R" uniqKey="Potthast R">R. Potthast</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Connor, C E" uniqKey="Connor C">C. E. Connor</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Coombes, S" uniqKey="Coombes S">S. Coombes</name></author><author><name sortKey="Beim Graben, P" uniqKey="Beim Graben P">P. beim Graben</name></author><author><name sortKey="Potthast, R" uniqKey="Potthast R">R. Potthast</name></author><author><name sortKey="Wright, J" uniqKey="Wright J">J. Wright</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Cowan, J" uniqKey="Cowan J">J. Cowan</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Fukai, T" uniqKey="Fukai T">T. 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Jung</name></author><author><name sortKey="Makeig, S" uniqKey="Makeig S">S. Makeig</name></author><author><name sortKey="Westerfield, M" uniqKey="Westerfield M">M. Westerfield</name></author><author><name sortKey="Townsend, J" uniqKey="Townsend J">J. Townsend</name></author><author><name sortKey="Courchesne, E" uniqKey="Courchesne E">E. Courchesne</name></author><author><name sortKey="Sejnowski, T J" uniqKey="Sejnowski T">T. J. Sejnowski</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Kandel, E R" uniqKey="Kandel E">E. R. Kandel</name></author><author><name sortKey="Schwartz, J H" uniqKey="Schwartz J">J. H. Schwartz</name></author><author><name sortKey="Jessel, T M" uniqKey="Jessel T">T. M. Jessel</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Kiebel, S J" uniqKey="Kiebel S">S. J. Kiebel</name></author><author><name sortKey="Von Kriegstein, K" uniqKey="Von Kriegstein K">K. von Kriegstein</name></author><author><name sortKey="Daunizeau, J" uniqKey="Daunizeau J">J. Daunizeau</name></author><author><name sortKey="Friston, K J" uniqKey="Friston K">K. J. Friston</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lehmann, D" uniqKey="Lehmann D">D. Lehmann</name></author><author><name sortKey="Ozaki, H" uniqKey="Ozaki H">H. Ozaki</name></author><author><name sortKey="Pal, I" uniqKey="Pal I">I. Pal</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lehmann, D" uniqKey="Lehmann D">D. Lehmann</name></author><author><name sortKey="Pascual Marqui, R D" uniqKey="Pascual Marqui R">R. D. Pascual-Marqui</name></author><author><name sortKey="Michel, C" uniqKey="Michel C">C. Michel</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lehmann, D" uniqKey="Lehmann D">D. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Front Syst Neurosci</journal-id><journal-id journal-id-type="iso-abbrev">Front Syst Neurosci</journal-id><journal-id journal-id-type="publisher-id">Front. Syst. Neurosci.</journal-id><journal-title-group><journal-title>Frontiers in Systems Neuroscience</journal-title></journal-title-group><issn pub-type="epub">1662-5137</issn><publisher><publisher-name>Frontiers Media S.A.</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">26175671</article-id><article-id pub-id-type="pmc">4485166</article-id><article-id pub-id-type="doi">10.3389/fnsys.2015.00097</article-id><article-categories><subj-group subj-group-type="heading"><subject>Neuroscience</subject><subj-group><subject>Original Research</subject></subj-group></subj-group></article-categories><title-group><article-title>Metastable dynamics in heterogeneous neural fields</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Schwappach</surname><given-names>Cordula</given-names></name><xref ref-type="aff" rid="aff1"><sup>1</sup></xref><xref ref-type="aff" rid="aff2"><sup>2</sup></xref><uri xlink:type="simple" xlink:href="http://loop.frontiersin.org/people/239293/overview"></uri></contrib><contrib contrib-type="author"><name><surname>Hutt</surname><given-names>Axel</given-names></name><xref ref-type="aff" rid="aff3"><sup>3</sup></xref><xref ref-type="aff" rid="aff4"><sup>4</sup></xref><xref ref-type="aff" rid="aff5"><sup>5</sup></xref><uri xlink:type="simple" xlink:href="http://loop.frontiersin.org/people/5234/overview"></uri></contrib><contrib contrib-type="author"><name><surname>beim Graben</surname><given-names>Peter</given-names></name><xref ref-type="aff" rid="aff1"><sup>1</sup></xref><xref ref-type="aff" rid="aff6"><sup>6</sup></xref><xref ref-type="author-notes" rid="fn001"><sup>*</sup></xref><uri xlink:type="simple" xlink:href="http://loop.frontiersin.org/people/5178/overview"></uri></contrib></contrib-group><aff id="aff1"><sup>1</sup><institution>Department of German Studies and Linguistics, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></aff><aff id="aff2"><sup>2</sup><institution>Department of Physics, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></aff><aff id="aff3"><sup>3</sup><institution>Team Neurosys, Inria</institution><country>Villers-les-Nancy, France</country></aff><aff id="aff4"><sup>4</sup><institution>Team Neurosys, Centre National de la Recherche Scientifique, UMR nō 7503, Loria</institution><country>Villers-les-Nancy, France</country></aff><aff id="aff5"><sup>5</sup><institution>Team Neurosys, UMR nō 7503, Loria, Université de Lorraine</institution><country>Villers-les-Nancy, France</country></aff><aff id="aff6"><sup>6</sup><institution>Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin</institution><country>Berlin, Germany</country></aff><author-notes><fn fn-type="edited-by"><p>Edited by: Emili Balaguer-Ballester, Bournemouth University, UK</p></fn><fn fn-type="edited-by"><p>Reviewed by: Pablo Varona, Universidad Autonoma de Madrid, Spain; Basabdatta Sen Bhattacharya, University of Lincoln, UK</p></fn><corresp id="fn001">*Correspondence: Peter beim Graben, Department of German Studies and Linguistics, Humboldt-Universität zu Berlin, Unter den Linden 6, D–10099 Berlin, Germany <email xlink:type="simple">peter.beim.graben@hu-berlin.de</email></corresp></author-notes><pub-date pub-type="epub"><day>30</day><month>6</month><year>2015</year></pub-date><pub-date pub-type="collection"><year>2015</year></pub-date><volume>9</volume><elocation-id>97</elocation-id><history><date date-type="received"><day>26</day><month>3</month><year>2015</year></date><date date-type="accepted"><day>15</day><month>6</month><year>2015</year></date></history><permissions><copyright-statement>Copyright © 2015 Schwappach, Hutt and beim Graben.</copyright-statement><copyright-year>2015</copyright-year><copyright-holder>Schwappach, Hutt and beim Graben</copyright-holder><license xlink:href="http://creativecommons.org/licenses/by/4.0/"><license-p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</license-p></license></permissions><abstract><p>We present numerical simulations of metastable states in heterogeneous neural fields that are connected along heteroclinic orbits. Such trajectories are possible representations of transient neural activity as observed, for example, in the electroencephalogram. Based on previous theoretical findings on learning algorithms for neural fields, we directly construct synaptic weight kernels from Lotka-Volterra neural population dynamics without supervised training approaches. We deliver a MATLAB neural field toolbox validated by two examples of one- and two-dimensional neural fields. We demonstrate trial-to-trial variability and distributed representations in our simulations which might therefore be regarded as a proof-of-concept for more advanced neural field models of metastable dynamics in neurophysiological data.</p></abstract><kwd-group><kwd>neural fields</kwd><kwd>kernel construction</kwd><kwd>metastability</kwd><kwd>heteroclinic orbits</kwd><kwd>trial-to-trial variability</kwd><kwd>distributed representations</kwd><kwd>sub-networks</kwd><kwd>sparsity</kwd></kwd-group><funding-group><award-group><funding-source id="cn001">Heisenberg Fellowship of the German Research Foundation DFG</funding-source><award-id rid="cn001">GR 3711/1-2</award-id></award-group><award-group><funding-source id="cn002">Bernstein Center for Computational Neuroscience</funding-source></award-group><award-group><funding-source id="cn003">European Research Council</funding-source></award-group><award-group><funding-source id="cn004">European Union's Seventh Framework Programme</funding-source><award-id rid="cn004">257253</award-id></award-group></funding-group><counts><fig-count count="5"></fig-count><table-count count="0"></table-count><equation-count count="13"></equation-count><ref-count count="41"></ref-count><page-count count="8"></page-count><word-count count="4843"></word-count></counts></article-meta></front><body><sec id="s1"><title>1. Introduction</title><p>Metastable states and transient dynamics between metastable states have received increasing interest in the neuroscientific community in recent time. Beginning with Dietrich Lehmann's original idea to identify “atoms of thought” as metastable topographies, so-called <italic>brain microstates</italic>, in spontaneous and event-related electroencephalograms (EEG) (Lehmann et al., <xref rid="B24" ref-type="bibr">1987</xref>; Lehmann, <xref rid="B26" ref-type="bibr">1989</xref>; Lehmann et al., <xref rid="B25" ref-type="bibr">2009</xref>), experimentalists found accumulating evidence that metastability is tentatively an important organization principle in neurodynamical systems. Mazor and Laurent (<xref rid="B28" ref-type="bibr">2005</xref>), e.g., reported metastable states in the locust odor system (cf. Rabinovich et al., <xref rid="B33" ref-type="bibr">2001</xref>, <xref rid="B34" ref-type="bibr">2008a</xref>), while Hudson et al. (<xref rid="B16" ref-type="bibr">2014</xref>) found metastability in the local field potentials of rats recovering from anesthesia. For the analysis of human EEG, several segmentation techniques into metastable states have recently been suggested by Hutt (<xref rid="B19" ref-type="bibr">2004</xref>), Allefeld et al. (<xref rid="B3" ref-type="bibr">2009</xref>), and beim Graben and Hutt (<xref rid="B7" ref-type="bibr">2015</xref>).</p><p>From a theoretical perspective, metastable EEG topographies or components of the event-related potential (ERP) have been identified with saddle-nodes in deterministic low-dimensional systems by Hutt et al. (<xref rid="B18" ref-type="bibr">2000</xref>) and Hutt and Riedel (<xref rid="B17" ref-type="bibr">2003</xref>). Particularly, the discoveries of winnerless competition (Rabinovich et al., <xref rid="B33" ref-type="bibr">2001</xref>; Seliger et al., <xref rid="B39" ref-type="bibr">2003</xref>) and heteroclinic orbits in neural population dynamics (Afraimovich et al., <xref rid="B1" ref-type="bibr">2004a</xref>,<xref rid="B2" ref-type="bibr">b</xref>; Rabinovich et al., <xref rid="B35" ref-type="bibr">2008b</xref>) led to better understanding of metastability and transient behavior in theoretical neuroscience. Winnerless competition is ubiquitous in complex excitation-inhibition networks with strong asymmetries. While symmetric connectivity usually leads to Hopfield-type attractor neural networks (Hopfield, <xref rid="B15" ref-type="bibr">1982</xref>; Hertz et al., <xref rid="B14" ref-type="bibr">1991</xref>) where transient dynamics is only observed for the motion from a basin of attraction toward an asymptotically stable fixed point attractor, winnerless competition between neural Lotka-Volterra populations (Fukai and Tanaka, <xref rid="B13" ref-type="bibr">1997</xref>; Cowan, <xref rid="B12" ref-type="bibr">2014</xref>) allows for hierarchical transient computations, bifurcations, and the resolution of sequential decision problems, as applied for modeling speech processing (Kiebel et al., <xref rid="B23" ref-type="bibr">2009</xref>), bird songs (Yildiz and Kiebel, <xref rid="B41" ref-type="bibr">2011</xref>), syntactic parsing (beim Graben and Potthast, <xref rid="B9" ref-type="bibr">2012</xref>), and, most recently, working memory (Rabinovich et al., <xref rid="B36" ref-type="bibr">2014a</xref>,<xref rid="B37" ref-type="bibr">b</xref>).</p><p>However, these phenomena have been investigated on the rather abstract level of macroscopic neural populations so far, without reference to the mesoscopic and microscopic levels of spatially given nervous tissue and individual neurons. One important approach to characterize the former, nervous tissue at the mesoscopic scale, are <italic>neural fields</italic>, i.e., continuum approximations of infinitely large neural networks (Coombes et al., <xref rid="B11" ref-type="bibr">2014</xref>). In a recent theoretical study, beim Graben and Hutt (<xref rid="B6" ref-type="bibr">2014</xref>) investigated stationary states and heteroclinic dynamics in neural fields with heterogeneous synaptic connectivity. The present work applies this previous study to describe experimentally observed transient neural activity as a proof-of-concept of our theoretical approach. We propose a novel hypothesis on the origin of trial-to-trial variability observed in most experimental data, on episodic cell assembly dynamics and on sparsely sampled neural representations.</p><p>Moreover, we disseminate our software implementation as a MATLAB <italic>neural field toolbox</italic> to facilitate further research on this intriguing field of computational neuroscience.</p></sec><sec id="s2"><title>2. Materials and methods</title><p>In this section we present some of the theoretical findings of beim Graben and Hutt (<xref rid="B6" ref-type="bibr">2014</xref>) and indicate how they have been implemented in our simulations.</p><sec><title>2.1. Theoretical background</title><p>An important representative of neural fields is given through the <italic>Amari equation</italic><disp-formula id="E1"><label>(1)</label><mml:math id="M1"><mml:mrow><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi>Ω</mml:mi></mml:msub><mml:mi>w</mml:mi></mml:mrow></mml:mstyle><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext> </mml:mtext><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mtext> d</mml:mtext><mml:mi>y</mml:mi></mml:mrow></mml:math></disp-formula>describing the evolution of neural activity <italic>u</italic>(<italic>x</italic>, <italic>t</italic>) at site <italic>x</italic> ∈ Ω ⊂ ℝ<sup><italic>d</italic></sup> and time <italic>t</italic> (Amari, <xref rid="B4" ref-type="bibr">1977</xref>). Here, Ω is a <italic>d</italic>-dimensional manifold, representing neural tissue. Moreover, <italic>w</italic>(<italic>x</italic>, <italic>y</italic>) is the synaptic weight kernel, and <italic>f</italic> is a sigmoidal activation function, usually chosen as <italic>f</italic>(<italic>u</italic>) = 1/(1 + exp(−β (<italic>u</italic> − θ))), with gain β > 0, and threshold θ > 0. The time scale of the dynamics, often characterized by a particular time constant is implicitly included in the kernel <italic>w</italic>(<italic>x</italic>, <italic>y</italic>).</p><p>The neural field described by Equation (1) is called homogeneous when the kernel is translation invariant: <italic>w</italic>(<italic>x</italic>, <italic>y</italic>) = <italic>w</italic>(<italic>x</italic> − <italic>y</italic>). If the field is not homogeneous it is called heterogeneous.</p><p>Stationary states, <italic>v</italic>(<italic>x</italic>), of the Amari equation which are obtained from ∂<italic>u</italic>/∂<italic>t</italic> = 0 obey the nonlinear Hammerstein integral equation</p><disp-formula id="E2"><label>(2)</label><mml:math id="M2"><mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi>Ω</mml:mi></mml:msub><mml:mi>w</mml:mi></mml:mrow></mml:mstyle><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mtext> d</mml:mtext><mml:mi>y</mml:mi><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula><p>By choosing a heterogeneous <italic>Pincherle-Goursat kernel</italic> (Veltz and Faugeras, <xref rid="B40" ref-type="bibr">2010</xref>)
<disp-formula id="E3"><label>(3)</label><mml:math id="M3"><mml:mrow><mml:mi>w</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>and carrying out a linear stability analysis, beim Graben and Hutt (<xref rid="B6" ref-type="bibr">2014</xref>) were able to prove that the stationary state <italic>v</italic>(<italic>x</italic>) is either an asymptotically stable fixed point attractor, or a saddle with a one-dimensional unstable manifold, i.e., a metastable state. Since such saddles could be connected along their stable and unstable directions, heterogeneous neural fields may exhibit stable heteroclinic sequences (SHS: Afraimovich et al., <xref rid="B2" ref-type="bibr">2004b</xref>; Rabinovich et al., <xref rid="B35" ref-type="bibr">2008b</xref>).</p><p>Let {<italic>v</italic><sub><italic>k</italic></sub>(<italic>x</italic>)}, 1 ≤ <italic>k</italic> ≤ <italic>n</italic> be such a collection of metastable states which we assume to be linearly independent. Then, this collection possesses a biorthogonal system of adjoints {<italic>v</italic><sup>+</sup><sub><italic>k</italic></sub>(<italic>x</italic>)} obeying</p><disp-formula id="E4"><label>(4)</label><mml:math id="M4"><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi>Ω</mml:mi></mml:msub><mml:mrow><mml:msubsup><mml:mi>v</mml:mi><mml:mi>j</mml:mi><mml:mo>+</mml:mo></mml:msubsup></mml:mrow></mml:mrow></mml:mstyle><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext> d</mml:mtext><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula><p>For the particular case of Lotka-Volterra neural populations, described by activities ξ<sub><italic>k</italic></sub>(<italic>t</italic>),
<disp-formula id="E5"><label>(5)</label><mml:math id="M5"><mml:mrow><mml:mfrac><mml:mrow><mml:mtext>d</mml:mtext><mml:msub><mml:mi>ξ</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mtext>d</mml:mtext><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>ξ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mstyle><mml:msub><mml:mi>ξ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula>with growth rates σ<sub><italic>k</italic></sub> >0, interaction weights ρ<sub><italic>kj</italic></sub> >0 and ρ<sub><italic>kk</italic></sub> = 1 that are tuned according to the algorithm of Afraimovich et al. (<xref rid="B2" ref-type="bibr">2004b</xref>) and Rabinovich et al. (<xref rid="B35" ref-type="bibr">2008b</xref>), the population amplitude
<disp-formula id="E6"><label>(6)</label><mml:math id="M6"><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>ξ</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula>recruits its corresponding metastable state <italic>v</italic><sub><italic>k</italic></sub>(<italic>x</italic>), leading to an order parameter expansion
<disp-formula id="E7"><label>(7)</label><mml:math id="M7"><mml:mrow><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:mstyle><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></disp-formula>of the neural field.</p><p>Under these assumptions, beim Graben and Potthast (<xref rid="B9" ref-type="bibr">2012</xref>) and beim Graben and Hutt (<xref rid="B6" ref-type="bibr">2014</xref>) have explicitly constructed the kernel <italic>w</italic>(<italic>x</italic>, <italic>y</italic>) through a power series expansion of the right-hand-side of the Amari equation (Equation 1),
<disp-formula id="E8"><label>(8)</label><mml:math id="M8"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi>Ω</mml:mi></mml:msub><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mrow></mml:mstyle><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext> d</mml:mtext><mml:mi>y</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext> </mml:mtext><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi>Ω</mml:mi></mml:msub><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi>Ω</mml:mi></mml:msub><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mrow></mml:mstyle><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext> d</mml:mtext><mml:mi>y</mml:mi><mml:mtext> d</mml:mtext><mml:mi>z</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>with Pincherle-Goursat kernels<xref ref-type="fn" rid="fn0001"><sup>1</sup></xref>.</p><disp-formula id="E9"><label>(9)</label><mml:math id="M9"><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mi>k</mml:mi></mml:munder><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mi>v</mml:mi><mml:mi>k</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:math></disp-formula><disp-formula id="E10"><label>(10)</label><mml:math id="M10"><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mstyle><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mi>v</mml:mi><mml:mi>k</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mi>v</mml:mi><mml:mi>j</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula><p>Interestingly, the kernel <italic>w</italic><sub>1</sub>(<italic>x</italic>, <italic>y</italic>) describes a Hebbian synapse between sites <italic>y</italic> and <italic>x</italic> whereas the three-point kernel <italic>w</italic><sub>2</sub>(<italic>x</italic>, <italic>y</italic>, <italic>z</italic>) further generalizes Hebbian learning to interactions between three sites <italic>x</italic>, <italic>y</italic>, <italic>z</italic> of neural tissue.</p></sec><sec><title>2.2. Numerical studies</title><p>For a numerical implementation of the theoretical results above, we have to discretize time and space. Using MATLAB, temporal discretization on the one hand is achieved through the ordinary differential equation solver <monospace>ode15s</monospace> for stiff problems. On the other hand, spatial discretization converts the kernels <italic>w</italic><sub>1</sub> and <italic>w</italic><sub>2</sub> into tensors of rank two and three, respectively. Consequently, the integrals in Equation (8) become contractions over products of tensors and state vectors <italic>u</italic>(<italic>t</italic>). In order to properly deal with tensor algebra, we use the Sandia Tensor Toolbox<xref ref-type="fn" rid="fn0002"><sup>2</sup></xref>. Our neural field toolbox, thus obtained is available as Supplementary Material. We evaluate our implementation in the next subsections by means of two examples.</p><sec><title>2.2.1. One-dimensional neural field</title><p>In our first simulation, we use a <italic>d</italic> = 1 dimensional neural field where we choose <italic>n</italic> = 3 sine functions
<disp-formula id="E11"><label>(11)</label><mml:math id="M11"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>sin</mml:mi><mml:mi>k</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:math></disp-formula>as metastable states on the domain Ω = [0, 2π] discretized with a spatial grid of <italic>N</italic><sub><italic>x</italic></sub> = 100 sites. According to the orthogonality relations
<disp-formula id="E12"><label>(12)</label><mml:math id="M12"><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi>Ω</mml:mi></mml:msub><mml:mrow><mml:mi>sin</mml:mi></mml:mrow></mml:mrow></mml:mstyle><mml:mi>j</mml:mi><mml:mi>x</mml:mi><mml:mi>sin</mml:mi><mml:mi>k</mml:mi><mml:mi>x</mml:mi><mml:mtext>d</mml:mtext><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></disp-formula>we easily obtain the adjoint modes</p><disp-formula id="E13"><label>(13)</label><mml:math id="M13"><mml:mrow><mml:msubsup><mml:mi>v</mml:mi><mml:mi>k</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>π</mml:mi></mml:mfrac><mml:mi>sin</mml:mi><mml:mi>k</mml:mi><mml:mi>x</mml:mi><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula><p>For the temporal dynamics we prepare the stable heteroclinic contour solving (Equation 5) used by beim Graben and Hutt (<xref rid="B7" ref-type="bibr">2015</xref>) with σ<sub>1</sub> = 1, σ<sub>2</sub> = 2, σ<sub>3</sub> = 3. Metastable states <italic>v</italic><sub><italic>k</italic></sub>(<italic>x</italic>) and their population activities ξ<sub><italic>k</italic></sub>(<italic>t</italic>) are shown in Figure <xref ref-type="fig" rid="F1">1</xref>.</p><fig id="F1" position="float"><label>Figure 1</label><caption><p><bold>Prescribed dynamics. (A)</bold> Three sinusoids as spatial patterns. <bold>(B)</bold> Stable heteroclinic contour resulting from winnerless competition in a Lotka-Volterra system (Equation 5). Blue: <italic>k</italic> = 1, green: <italic>k</italic> = 2, red: <italic>k</italic> = 3.</p></caption><graphic xlink:href="fnsys-09-00097-g0001"></graphic></fig><p>We run simulations with one fixed initial condition and also from an ensemble of 60 initial conditions randomly distributed in the vicinity of the first saddle, where we add some small portion of Gaussian observational noise (noise level σ = 0.005) in order to demonstrate trial-to-trial variability and hence event-related phase decoherence (Jung et al., <xref rid="B21" ref-type="bibr">2001</xref>; Makeig et al., <xref rid="B27" ref-type="bibr">2002</xref>).</p></sec><sec><title>2.2.2. Two-dimensional neural field</title><p>For our second demonstration, we assume a spatially distributed response in a neural population to external stimuli triggering a sequence of neural activity patterns. It is well-established that sensory input features (Pasupathy and Connor, <xref rid="B30" ref-type="bibr">2002</xref>) at earlier stages of the object's representation pathway and memory (Rissman and Wagner, <xref rid="B38" ref-type="bibr">2012</xref>) is encoded by distributed cortical neural populations while objects are sparsely coded in later stages of the representation pathway (Connor, <xref rid="B10" ref-type="bibr">2005</xref>). Here we consider a cortical neural population embedded in two-dimensional space involving interleaved patterns. These patterns are <italic>d</italic> = 2 dimensional gray scale bitmap images of the numbers<xref ref-type="fn" rid="fn0003"><sup>3</sup></xref> 1, 2, and 3 (see <bold>Figure 4</bold> in Section 3.2). In the implementation, these bitmaps are downsampled to a 20 × 20 grid and reshaped into vectors with <italic>N</italic><sub><italic>x</italic></sub> = 400 elements. Adjoint patterns are obtained as Moore-Penrose pseudoinverses (Hertz et al., <xref rid="B14" ref-type="bibr">1991</xref>).</p><p>The temporal evolution of these patterns follows the same heteroclinic contour as above. Here, the underlying working assumption is the presence of interacting sub-networks, e.g., reflecting several distributed representations of signal features or of pieces of working memory. The study predicts what one expects to measure in single spatial locations while the neural system encodes information in a spatially distributed population.</p></sec></sec></sec><sec id="s3"><title>3.1. Results</title><p>The results of our simulation studies are presented in this section.</p><sec><title>3.2. One-dimensional neural field</title><p>For the one-dimensional neural field we compare in Figure <xref ref-type="fig" rid="F2">2</xref> the prescribed spatiotemporal dynamics as resulting from the order parameter expansion (Equation 7) with the solution of the Amari (Equation 8).</p><fig id="F2" position="float"><label>Figure 2</label><caption><p><bold>One-dimensional spatiotemporal dynamics. (A)</bold> Prescribed trajectory from order parameter ansatz (Equation 7). <bold>(B)</bold> Solution of the Amari (Equation 8).</p></caption><graphic xlink:href="fnsys-09-00097-g0002"></graphic></fig><p>Figure <xref ref-type="fig" rid="F2">2A</xref> shows the prescribed dynamics on a spatiotemporal grid with time on the x-axis and space on the y-axis. The instantaneous activations are therefore given by vertical slices. Going from left to right, these slices first exhibit one wave crest (in red at the bottom) and one wave trough (in blue at the top), corresponding to metastable state <italic>v</italic><sub>1</sub>(<italic>x</italic>). Around time <italic>t</italic> = 15 the frequency doubles and metastable state <italic>v</italic><sub>2</sub>(<italic>x</italic>) can be observed for approximately seven ticks. The third metastable state met by the trajectory around time <italic>t</italic> = 21 is the mode with tripled frequency. It is only stable for five ticks and evolves thereafter into the first mode again.</p><p>In contrast, Figure <xref ref-type="fig" rid="F2">2B</xref> depicts the numerical solution of the Amari equation (Equation 8). Obviously, no deviation is visible.</p><p>In order to draw neurophysiologically relevant conclusions from our toy model, we consider the metastable states of the heteroclinic contour as “synthetic ERP components” (Barrès et al., <xref rid="B5" ref-type="bibr">2013</xref>) measured with “electrodes” at the particular sampling points. Because ERPs are obtained from averaging spontaneous EEG over ensembles of several trials that are time-locked to the perception or processing of stimuli, we simulate 60 synthetic ERP trials by randomly preparing initial conditions of the Amari equation.</p><p>The results are displayed in Figure <xref ref-type="fig" rid="F3">3</xref> for four “measurement electrodes” at positions 3, 21, 47, and 88. Interestingly, our algorithm exhibited numerical instabilities in five runs which have been marked as “rejected” outliers and excluded from presentation. The resulting 55 trials are shown as colored traces in Figure <xref ref-type="fig" rid="F3">3</xref>. At simulation start all signals are nicely coherent, but later substantial phase dispersions take place (Jung et al., <xref rid="B21" ref-type="bibr">2001</xref>; Makeig et al., <xref rid="B27" ref-type="bibr">2002</xref>).</p><fig id="F3" position="float"><label>Figure 3</label><caption><p><bold>Four selected “recording sites” for neural field simulation with 55 randomly prepared initial conditions (colored traces) and “grand average” (bold black trace). (A)</bold> At position: 3, <bold>(B)</bold> position 21, <bold>(C)</bold> position 47, <bold>(D)</bold> position 88.</p></caption><graphic xlink:href="fnsys-09-00097-g0003"></graphic></fig><p>We also calculated the ERP averages from our simulation shown as bold black traces in Figure <xref ref-type="fig" rid="F3">3</xref>. On the one hand, the averaged ERP is much smoother than the noisy single realizations which justifies averaging in our simulation. However, the averaged ERP significantly decays in the course of time. This is obviously due to the increasing phase decoherence (Jung et al., <xref rid="B21" ref-type="bibr">2001</xref>; Makeig et al., <xref rid="B27" ref-type="bibr">2002</xref>).</p></sec><sec><title>3.2. Two-dimensional neural field</title><p>The numerical simulation of Equation (8) yields a sequence of two-dimensional transient patterns which is shown as a sampled sequence of snapshot maps in Figure <xref ref-type="fig" rid="F4">4</xref>.</p><fig id="F4" position="float"><label>Figure 4</label><caption><p><bold>Two-dimensional spatiotemporal solution of model (Equation 9) considering spatial patterns of the numbers “1,” “2,” and “3” as spatial modes</bold><italic><bold>v</bold></italic><bold><sub>1</sub>(<italic>x</italic>), <italic>v</italic><sub>2</sub>(<italic>x</italic>), and</bold><italic><bold>v</bold></italic><bold><sub>3</sub>(<italic>x</italic>), respectively</bold>. The three color-coded points denote three spatial locations whose temporal evolution is shown in <bold>Figure 5</bold>.</p></caption><graphic xlink:href="fnsys-09-00097-g0004"></graphic></fig><p>According to the different growth rates σ<sub><italic>k</italic></sub> of the populations, pattern “1” stays the longest period of time, pattern “2” is visible for a shorter period of time and pattern “3” can be seen for the shortest period of time. These modes represent interweaved spatial networks reflecting intrinsically stored activity patterns.</p><p>Now assuming that measurement of neural activity takes place at discrete spatial locations (color-coded points in Figure <xref ref-type="fig" rid="F4">4</xref>), one observes different transient dynamics dependent on the spatial location of the measurement point that is shown in Figure <xref ref-type="fig" rid="F5">5</xref>. Considering the red-coded spatial location, one observes strong activity in the time periods when pattern “1” is active, and well-reduced activity in the time windows of active patterns “2” and “3.” Conversely, the activity at the blue-coded location defined in Figure <xref ref-type="fig" rid="F4">4</xref> raises only if pattern “3” is active, otherwise its activity is well-reduced. The green-coded spatial location shows negligible activity in time periods when pattern “1” is active while activity is increased during the emergence of patterns “2” and “3.”</p><fig id="F5" position="float"><label>Figure 5</label><caption><p><bold>The time-dependent activity</bold><italic><bold>u</bold></italic><bold>(</bold><italic><bold>x</bold></italic><sub><italic><bold>l</bold></italic></sub><bold>,</bold><italic><bold>t</bold></italic><bold>) at three spatial locations</bold><italic><bold>x</bold></italic><sub><italic><bold>l</bold></italic></sub><bold>,</bold><italic><bold>l</bold></italic><bold> = 1, 2, 3 defined in Figure <xref ref-type="fig" rid="F4">4</xref></bold>. The upper gray-colored bars denote the emergence time intervals of the corresponding patterns in Figure <xref ref-type="fig" rid="F4">4</xref>. The color codes of the time series correspond to the respective colors of the spatial locations in Figure <xref ref-type="fig" rid="F4">4</xref>.</p></caption><graphic xlink:href="fnsys-09-00097-g0005"></graphic></fig></sec></sec><sec id="s4"><title>4. Discussion</title><p>In this paper, we presented a software implementation (<italic>neural field toolbox</italic>) and numerical simulation results of previously reported theoretical findings on metastable states and heteroclinic dynamics in neural fields (beim Graben and Potthast, <xref rid="B9" ref-type="bibr">2012</xref>; beim Graben and Hutt, <xref rid="B6" ref-type="bibr">2014</xref>). For the particular case of Lotka-Volterra population dynamics and linearly independent spatial modes, the synaptic weight kernel of the Amari neural field equation (Amari, <xref rid="B4" ref-type="bibr">1977</xref>) can be explicitly constructed from the prescribed metastable states and their evolution parameters as Pincherle-Goursat kernels. This is an important finding as our kernel construction method is not a standard training algorithm such as backpropagation (Igel et al., <xref rid="B20" ref-type="bibr">2001</xref>; beim Graben and Potthast, <xref rid="B8" ref-type="bibr">2009</xref>). Yet it implements a straightforward generalization of Hebbian learning algorithms (beim Graben and Potthast, <xref rid="B8" ref-type="bibr">2009</xref>; Potthast and beim Graben, <xref rid="B31" ref-type="bibr">2009</xref>).</p><p>We validated our algorithm by means of two examples, a one-dimensional neural field where metastable states are three sinusoidal excitations over a line, and a two-dimensional example where we have chosen three bitmap images as spatial modes. The temporal dynamics was prescribed as a heteroclinic contour connecting these three patterns in a closed loop. In both simulations, the results were in exact agreement with the prescribed trajectories.</p><p>Furthermore, we examined the issues of trial-to-trial variability and distributed representations. In the first example we created solutions for randomly prepared initial conditions, thereby emulating phase resetting in event-related brain potentials (ERP). We observed increasing phase decoherence in the resulting ERP averages. Our model presents a theoretically satisfying explanation for this ubiquitous experimental finding (Jung et al., <xref rid="B21" ref-type="bibr">2001</xref>; Makeig et al., <xref rid="B27" ref-type="bibr">2002</xref>). Assuming that ERP components are metastable states that are connected along heteroclinic orbits (Hutt and Riedel, <xref rid="B17" ref-type="bibr">2003</xref>; beim Graben and Hutt, <xref rid="B7" ref-type="bibr">2015</xref>), single ERP trials start from randomly distributed initial conditions, sometimes closer and sometimes farther from the respective metastable stable. These initial distances from a metastable state result in acceleration and hence in velocity differences in phase space, eventually leading to dispersion and decoherence. Moreover, such a dependence on initial conditions resembles previous experimental results by Pastalkova et al. (<xref rid="B29" ref-type="bibr">2008</xref>) showing that identical experimental initial conditions in a motor task lead to identical sequences of cell assembly activations, while different initial conditions yield different sequences.</p><p>For the second example we considered the interaction of three two-dimensional populations, cf. Figure <xref ref-type="fig" rid="F4">4</xref>. The transient passage of the system at metastable attractors has been shown experimentally in previous studies, such as in middle-latent auditory evoked potentials (Hutt and Riedel, <xref rid="B17" ref-type="bibr">2003</xref>) or in the population response of olfactory projection neurons to odor stimuli (Mazor and Laurent, <xref rid="B28" ref-type="bibr">2005</xref>). For instance, the study of Mazor and Laurent (<xref rid="B28" ref-type="bibr">2005</xref>) also shows nicely the responses of single neurons in the population revealing different activity in different neurons: some neurons respond to the external stimulus, others remain silent. Such a distinction in response can easily be explained by an insufficient spatial sub-sampling in the measurement and the presence of spatially distributed patterns. However, just spatial sub-sampling does not explain the fully distinct activity of different neurons, such as different episode neurons found in the hippocampus (Pastalkova et al., <xref rid="B29" ref-type="bibr">2008</xref>). Here, different neurons show distinct episodic temporal activities. The equivalent temporal evolution is shown in our simulations in Figure <xref ref-type="fig" rid="F5">5</xref>, where the units at different spatial locations exhibit different temporal sequences of activation that are highly correlated to the presence of the respective pattern representations. This difference results from interacting populations or cell assemblies.</p><p>The latter line of argumentation raises the question whether it may explain previous results on sparse neural representations or even may contribute to the question on the existence of “grandmother cells” (Connor, <xref rid="B10" ref-type="bibr">2005</xref>; Quiroga et al., <xref rid="B32" ref-type="bibr">2005</xref>). At a first glance, the present work assumes the existence of interacting spatially distributed sub-networks and supports their existence by a qualitative comparison to previous experimental results by Mazor and Laurent (<xref rid="B28" ref-type="bibr">2005</xref>) and Pastalkova et al. (<xref rid="B29" ref-type="bibr">2008</xref>). Our assumption of interacting sub-networks does not rule out sparse neural representations since our modeling approach does not stipulate contiguous spatial patterns but also allows for sparse patterns as well.</p><p>Metastable neural field dynamics as an ubiquitous organization principle of the brain is also consistent with findings from neuroanatomy and cognitive neuroscience. Anatomically, neural circuits comprise convergent and divergent pathways between populations (Kandel et al., <xref rid="B22" ref-type="bibr">1991</xref>). Assuming that a particular sub-network gets activated by percolation along a convergent pathway and deactivated along a divergent pathway subsequently entails a saddle-node picture in its phase space description, hence a metastable attractor. In cognitive neuroscience, mental representations are regarded as intermediate results of cognitive computations in discrete time. In order to embed these into continuous physical time, they have to be considered connected through continuous trajectories along their stable and unstable directions, i.e., as metastable states, again (beim Graben and Potthast, <xref rid="B8" ref-type="bibr">2009</xref>, <xref rid="B9" ref-type="bibr">2012</xref>).</p><p>The present study is a first step toward metastability in neural fields. We hope that our work encourages further research on metastability in neural fields to describe transient neural dynamics by interacting populations and contribute to the description of neural information storage, being either distributed or sparse.</p></sec><sec><title>Author contributions</title><p>This study reports results from CS's student internship at Department of German Studies and Linguistics, Humboldt-Universität zu Berlin. CS developed the program code and conducted the numerical simulations. AH included the sub-network study, PbG contributed the study on trial-to-trial variability and compiled the neural field toolbox. All authors wrote the manuscript together.</p></sec><sec><title>Funding</title><p>PbG acknowledges support by a Heisenberg Fellowship of the German Research Foundation DFG (GR 3711/1-2) and of the Bernstein Center for Computational Neuroscience, Berlin, hosting AH as visiting professor during October 2014. AH acknowledges funding from the European Research Council for support under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 257253.</p><sec><title>Conflict of interest statement</title><p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p></sec></sec></body><back><fn-group><fn id="fn0001"><p><sup>1</sup>There was a mistake in our previous reports (beim Graben and Potthast, <xref rid="B9" ref-type="bibr">2012</xref>; beim Graben and Hutt, <xref rid="B6" ref-type="bibr">2014</xref>). Although the kernel construction has been correctly derived, a minus sign was omitted in the final result for kernel <italic>w</italic><sub>2</sub>(<italic>x</italic>, <italic>y</italic>, <italic>z</italic>). This is corrected now.</p></fn><fn id="fn0002"><p><sup>1</sup><ext-link ext-link-type="uri" xlink:href="http://www.sandia.gov/~tgkolda/TensorToolbox/index-2.5.html">http://www.sandia.gov/~tgkolda/TensorToolbox/index-2.5.html</ext-link></p></fn><fn id="fn0003"><p><sup>2</sup>Original images are taken from the webpage <ext-link ext-link-type="uri" xlink:href="http://www.iconarchive.com/tag/number-3">http://www.iconarchive.com/tag/number-3</ext-link> before modifications with respect to color and resolution.</p></fn></fn-group><sec sec-type="supplementary-material" id="s5"><title>Supplementary material</title><p>The Supplementary Material for this article can be found online at: <ext-link ext-link-type="uri" xlink:href="http://journal.frontiersin.org/article/10.3389/fnsys.2015.00097">http://journal.frontiersin.org/article/10.3389/fnsys.2015.00097</ext-link></p><sec><title>Supplemental data</title><p>A GIF animation of the two-dimensional neural field simulation is given as Supplementary Material. Moreover, we deliver a MATLAB software neural field toolbox. This package essentially comprises three routines: <monospace>amarikernels</monospace>, <monospace>amarieq</monospace>, <monospace>iniamari</monospace>, and a main program, <monospace>solveamari</monospace> to be evoked in the the following way: <monospace>amarikernels</monospace> is the training program for the synaptic weights. It takes four arguments: <italic>V</italic>_<italic>patterns</italic>, <italic>sigmarange</italic>, <italic>compbias</italic>, and <italic>contourflag</italic> and returns two kernel tensors <italic>K</italic>1, <italic>K</italic>2. <italic>V</italic>_<italic>patterns</italic> is a matrix whose columns are the metastable states in a spatial discretization, their order corresponds to the desired heteroclinic sequence. <italic>sigmarange</italic> is an interval of Lotka-Volterra grow rates σ characterizing the time scale of the dynamics, while <italic>compbias</italic> denotes the competition bias in the interaction matrix (ρ). The last parameter, <italic>contourflag</italic>, is a binary flag deciding whether the hereroclinic sequence is closed (1) or not (0). A closed hereroclinic sequence is called hereroclinic contour. <monospace>amarieq</monospace> defines the Amari equation (Equation 8) for the ODE solver. It has four input arguments, <italic>t</italic>, <italic>V</italic>, <italic>K</italic>1, <italic>K</italic>2, where <italic>t</italic> is the time span to be simulated, <italic>V</italic> is the actual field activity, and <italic>K</italic>1, <italic>K</italic>2 are the two synaptic weight kernels. <monospace>iniamari</monospace> prepares an initial condition at the surface of the simplex spanned by the metastable states <italic>V</italic>_<italic>patterns</italic>. The other arguments are <italic>lead</italic> and <italic>remain</italic>, denoting the leading direction toward the next saddle and its orthogonal projection on the remaining modes. 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