Neural field simulator: two-dimensional spatio-temporal dynamics involving finite transmission speed
Identifieur interne : 000004 ( Pmc/Corpus ); précédent : 000003; suivant : 000005Neural field simulator: two-dimensional spatio-temporal dynamics involving finite transmission speed
Auteurs : Eric J. Nichols ; Axel HuttSource :
- Frontiers in Neuroinformatics [ 1662-5196 ] ; 2015.
Abstract
Neural Field models (NFM) play an important role in the understanding of neural population dynamics on a mesoscopic spatial and temporal scale. Their numerical simulation is an essential element in the analysis of their spatio-temporal dynamics. The simulation tool described in this work considers scalar spatially homogeneous neural fields taking into account a finite axonal transmission speed and synaptic temporal derivatives of first and second order. A text-based interface offers complete control of field parameters and several approaches are used to accelerate simulations. A graphical output utilizes video hardware acceleration to display running output with reduced computational hindrance compared to simulators that are exclusively software-based. Diverse applications of the tool demonstrate breather oscillations, static and dynamic Turing patterns and activity spreading with finite propagation speed. The simulator is open source to allow tailoring of code and this is presented with an extension use case.
Url:
DOI: 10.3389/fninf.2015.00025
PubMed: 26539105
PubMed Central: 4611063
Links to Exploration step
PMC:4611063Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Neural field simulator: two-dimensional spatio-temporal dynamics involving finite transmission speed</title><author><name sortKey="Nichols, Eric J" sort="Nichols, Eric J" uniqKey="Nichols E" first="Eric J." last="Nichols">Eric J. Nichols</name></author><author><name sortKey="Hutt, Axel" sort="Hutt, Axel" uniqKey="Hutt A" first="Axel" last="Hutt">Axel Hutt</name></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">26539105</idno><idno type="pmc">4611063</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4611063</idno><idno type="RBID">PMC:4611063</idno><idno type="doi">10.3389/fninf.2015.00025</idno><date when="2015">2015</date><idno type="wicri:Area/Pmc/Corpus">000004</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000004</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Neural field simulator: two-dimensional spatio-temporal dynamics involving finite transmission speed</title><author><name sortKey="Nichols, Eric J" sort="Nichols, Eric J" uniqKey="Nichols E" first="Eric J." last="Nichols">Eric J. Nichols</name></author><author><name sortKey="Hutt, Axel" sort="Hutt, Axel" uniqKey="Hutt A" first="Axel" last="Hutt">Axel Hutt</name></author></analytic><series><title level="j">Frontiers in Neuroinformatics</title><idno type="eISSN">1662-5196</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>Neural Field models (NFM) play an important role in the understanding of neural population dynamics on a mesoscopic spatial and temporal scale. Their numerical simulation is an essential element in the analysis of their spatio-temporal dynamics. The simulation tool described in this work considers scalar spatially homogeneous neural fields taking into account a finite axonal transmission speed and synaptic temporal derivatives of first and second order. A text-based interface offers complete control of field parameters and several approaches are used to accelerate simulations. A graphical output utilizes video hardware acceleration to display running output with reduced computational hindrance compared to simulators that are exclusively software-based. Diverse applications of the tool demonstrate breather oscillations, static and dynamic Turing patterns and activity spreading with finite propagation speed. The simulator is open source to allow tailoring of code and this is presented with an extension use case.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Atay, F M" uniqKey="Atay F">F. M. Atay</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. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Front Neuroinform</journal-id><journal-id journal-id-type="iso-abbrev">Front Neuroinform</journal-id><journal-id journal-id-type="publisher-id">Front. Neuroinform.</journal-id><journal-title-group><journal-title>Frontiers in Neuroinformatics</journal-title></journal-title-group><issn pub-type="epub">1662-5196</issn><publisher><publisher-name>Frontiers Media S.A.</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">26539105</article-id><article-id pub-id-type="pmc">4611063</article-id><article-id pub-id-type="doi">10.3389/fninf.2015.00025</article-id><article-categories><subj-group subj-group-type="heading"><subject>Neuroscience</subject><subj-group><subject>Methods</subject></subj-group></subj-group></article-categories><title-group><article-title>Neural field simulator: two-dimensional spatio-temporal dynamics involving finite transmission speed</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Nichols</surname><given-names>Eric J.</given-names></name><xref ref-type="author-notes" rid="fn001"><sup>*</sup></xref><uri xlink:type="simple" xlink:href="http://loop.frontiersin.org/people/147472/overview"></uri></contrib><contrib contrib-type="author"><name><surname>Hutt</surname><given-names>Axel</given-names></name><uri xlink:type="simple" xlink:href="http://loop.frontiersin.org/people/5234/overview"></uri></contrib></contrib-group><aff><institution>Team Neurosys, Loria, Centre National de la Recherche Scientifique, INRIA, UMR no. 7503, Université de Lorraine</institution><country>Nancy, France</country></aff><author-notes><fn fn-type="edited-by"><p>Edited by: Gaute T. Einevoll, Norwegian University of Life Sciences, Norway</p></fn><fn fn-type="edited-by"><p>Reviewed by: Viktor Jirsa, Aix-Marseille University, France; Sid Visser, University of Exeter, UK</p></fn><corresp id="fn001">*Correspondence: Eric J. Nichols <email xlink:type="simple">ericjnichols@gmail.com</email></corresp></author-notes><pub-date pub-type="epub"><day>20</day><month>10</month><year>2015</year></pub-date><pub-date pub-type="collection"><year>2015</year></pub-date><volume>9</volume><elocation-id>25</elocation-id><history><date date-type="received"><day>20</day><month>5</month><year>2015</year></date><date date-type="accepted"><day>02</day><month>10</month><year>2015</year></date></history><permissions><copyright-statement>Copyright © 2015 Nichols and Hutt.</copyright-statement><copyright-year>2015</copyright-year><copyright-holder>Nichols and Hutt</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>Neural Field models (NFM) play an important role in the understanding of neural population dynamics on a mesoscopic spatial and temporal scale. Their numerical simulation is an essential element in the analysis of their spatio-temporal dynamics. The simulation tool described in this work considers scalar spatially homogeneous neural fields taking into account a finite axonal transmission speed and synaptic temporal derivatives of first and second order. A text-based interface offers complete control of field parameters and several approaches are used to accelerate simulations. A graphical output utilizes video hardware acceleration to display running output with reduced computational hindrance compared to simulators that are exclusively software-based. Diverse applications of the tool demonstrate breather oscillations, static and dynamic Turing patterns and activity spreading with finite propagation speed. The simulator is open source to allow tailoring of code and this is presented with an extension use case.</p></abstract><kwd-group><kwd>neural field</kwd><kwd>numerical simulation</kwd><kwd>delay</kwd><kwd>pattern formation</kwd></kwd-group><counts><fig-count count="8"></fig-count><table-count count="1"></table-count><equation-count count="15"></equation-count><ref-count count="66"></ref-count><page-count count="11"></page-count><word-count count="6324"></word-count></counts></article-meta></front><body><sec id="s1"><title>1. Introduction</title><p>The understanding of spatio-temporal electric activity in neural tissue is essential in the study of neurobiological phenomena. To achieve this, mesoscopic-scale models such as neural mass and neural fields (NFM) which describe the dynamics of a large population of neurons reflecting coarse-grained properties of single neurons (Wilson and Cowan, <xref rid="B64" ref-type="bibr">1973</xref>; Deco et al., <xref rid="B16" ref-type="bibr">2008</xref>; Bressloff, <xref rid="B4" ref-type="bibr">2012</xref>; Hutt and Buhry, <xref rid="B27" ref-type="bibr">2014</xref>) play an important role. NFMs serve as a good description of the dynamic source of Local Field Potentials and encephalographic data (Nunez, <xref rid="B39" ref-type="bibr">1974</xref>, <xref rid="B40" ref-type="bibr">2000</xref>; Wright and Kydd, <xref rid="B65" ref-type="bibr">1992</xref>; Wright and Liley, <xref rid="B66" ref-type="bibr">1994</xref>; Jirsa et al., <xref rid="B34" ref-type="bibr">2002</xref>; Nunez and Srinivasan, <xref rid="B41" ref-type="bibr">2006</xref>; Coombes et al., <xref rid="B12" ref-type="bibr">2014</xref>). They allow to consider diverse single neuron features that may tune neural population dynamics, such as somatic (Molaee-Ardekani et al., <xref rid="B38" ref-type="bibr">2007</xref>) and synaptic adaptation (Coombes and Owen, <xref rid="B14" ref-type="bibr">2005</xref>; Kilpatrick and Bressloff, <xref rid="B35" ref-type="bibr">2010</xref>), extra-synaptic receptor dynamics (Hashemi et al., <xref rid="B22" ref-type="bibr">2014</xref>; Hutt and Buhry, <xref rid="B27" ref-type="bibr">2014</xref>) or finite axonal transmission speed (Jirsa and Haken, <xref rid="B33" ref-type="bibr">1996</xref>; Pinto and Ermentrout, <xref rid="B46" ref-type="bibr">2001</xref>; Hutt et al., <xref rid="B26" ref-type="bibr">2003</xref>, <xref rid="B31" ref-type="bibr">2008</xref>; Coombes, <xref rid="B11" ref-type="bibr">2005</xref>; Faye and Faugeras, <xref rid="B19" ref-type="bibr">2010</xref>; Veltz and Faugeras, <xref rid="B59" ref-type="bibr">2011</xref>, <xref rid="B60" ref-type="bibr">2013</xref>). All these applications make NFMs valuable in order to understand spatio-temporal dynamics of neural population activity.</p><p>Mathematical analysis and the numerical integration of NFMs are complementary. The recent years have shown strong attention of research on the mathematical properties of NFMs, whereas the numerical simulation of NFM solutions has been less considered in research. Since NFMs generalize partial differential equations (Coombes et al., <xref rid="B15" ref-type="bibr">2007</xref>; Hutt, <xref rid="B23" ref-type="bibr">2007</xref>) while involving finite transmission delay interactions, they allow to study a large class of pattern forming systems, cf. Hutt (<xref rid="B23" ref-type="bibr">2007</xref>). In recent years, several software tools have been developed to simulate neural network dynamics. Examples for simulators for networks of spiking neurons are <italic>BRIAN</italic> (Stimberg et al., <xref rid="B56" ref-type="bibr">2014</xref>) and <italic>Neuron</italic> (Carnevale and Hines, <xref rid="B10" ref-type="bibr">2006</xref>). <italic>The Virtual Brain</italic> (Sanz Leon et al., <xref rid="B53" ref-type="bibr">2013</xref>) allows to simulate networks of neural mass models to reproduce global brain activity. The simulation platform <italic>DANA</italic> (Rougier and Fix, <xref rid="B50" ref-type="bibr">2012</xref>) simulates a hierarchy of coupled Dynamic Neural Fields which are decentralized, i.e., are updated numerically in time asynchronously (Rougier and Hutt, <xref rid="B51" ref-type="bibr">2011</xref>). These latter software tools are powerful, general and highly adaptive to the framework of their neural network types. However, they do not provide the effective computation for the specific NFM given in Equation (1) which is a stochastic delayed integral-differential equation in two spatial dimensions. The tool presented here fills a gap in the landscape of neural simulator tools which are typically very general and adaptive and, hence, not efficient for NFM. A simulation tool for NFM allows to explore rapidly and in a user-friendly way the solution space of Equation (1) in order to reproduce numerically experimental spatio-temporal dynamics, e.g., to understand neuroimaging data (Friston et al., <xref rid="B21" ref-type="bibr">2014</xref>; Pinotsis and Friston, <xref rid="B43" ref-type="bibr">2014</xref>), retrieve neural sources and lateral connections (Pinotsis et al., <xref rid="B45" ref-type="bibr">2013</xref>), and understand power spectra of electroencephalographic activity (Pinotsis et al., <xref rid="B44" ref-type="bibr">2012</xref>). In addition, the tool promises to allow detection of new numerical solutions, cf. Section 3. The numerical analysis is non-trivial and challenging if NFMs become more complex, e.g., by involving complex dynamical features rendering the model high-dimensional or by considering delayed interactions. The present work considers a two-dimensional spatial embedding of neural populations similar to several previous studies (Laing, <xref rid="B36" ref-type="bibr">2005</xref>; Owen et al., <xref rid="B42" ref-type="bibr">2007</xref>) while taking into account finite axonal transmission speed (Hutt and Rougier, <xref rid="B30" ref-type="bibr">2010</xref>, <xref rid="B29" ref-type="bibr">2014</xref>). By virtue of its modularity, the simulator allows subsequent extensions with additional features, such as extra-synaptic receptor effects or several interacting populations.</p><p>The combination of finite axonal transmission speed and two-dimensional spatial embedding is challenging from a numerical simulation perspective due to the missing convolution structure (Hutt and Rougier, <xref rid="B30" ref-type="bibr">2010</xref>, <xref rid="B29" ref-type="bibr">2014</xref>) leading to long simulation durations. To overcome this problem, a numerical technique has been developed in recent years (Hutt and Rougier, <xref rid="B30" ref-type="bibr">2010</xref>, <xref rid="B29" ref-type="bibr">2014</xref>). Since future research in neural fields will investigate spatio-temporal dynamics involving finite axonal transmission speed, we have developed an open-source simulation toolbox that allows to gain spatio-temporal solutions of NFM models in two spatial dimensions, visualize them and save them, if necessary, as movies. We hope that the tool will provide an essential tool for the computational neuroscience community to advance the research field and the insight into the brain.</p><p>The simulator in this work obeys integral-differential equations of the type</p><disp-formula id="E1"><label>(1)</label><mml:math id="M1"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>η</mml:mi><mml:mfrac><mml:mrow><mml:msup><mml:mo>∂</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mi>γ</mml:mi><mml:mfrac><mml:mo>∂</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><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:mtext> </mml:mtext></mml:mrow></mml:mstyle><mml:mi>K</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>−</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext> </mml:mtext><mml:mi>S</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mo>∥</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>−</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>∥</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:msup><mml:mtext>d</mml:mtext><mml:mn>2</mml:mn></mml:msup><mml:mi>y</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula><p>with a two-dimensional square spatial domain Ω and periodic boundary condition. The mean neuron potential <italic>V</italic> ∈ <inline-formula><mml:math id="M2"><mml:mi mathvariant="-tex-caligraphic">R</mml:mi></mml:math></inline-formula> at location <bold>x</bold> ∈ Ω is evolved by the external stimulus <italic>I</italic>(<italic>x, t</italic>) ∈ <inline-formula><mml:math id="M3"><mml:mi mathvariant="-tex-caligraphic">R</mml:mi></mml:math></inline-formula> and the integral of the synaptic connectivity kernel <italic>K</italic> : <inline-formula><mml:math id="M4"><mml:mi mathvariant="-tex-caligraphic">R</mml:mi></mml:math></inline-formula><sup>2</sup> → <inline-formula><mml:math id="M5"><mml:mi mathvariant="-tex-caligraphic">R</mml:mi></mml:math></inline-formula> and population firing rate <italic>S</italic> ∈ <inline-formula><mml:math id="M6"><mml:mi mathvariant="-tex-caligraphic">R</mml:mi></mml:math></inline-formula> which depend on the distance between spatial locations <bold>x</bold> and <bold>y</bold> with a finite axonal transmission speed <italic>c</italic>. Equation (1) represents the core of most NFM in the sense that most NFMs consider extensions of this equation.</p><p>Motivation for the work arises from a need for a visualization tool that is useful to the largest number of NFM researchers, allows for the tailoring of code and has fast while visually appealing output. The simulator can operate on all major operating systems. Output of data in three dimensions is provided by PyOpenGL which brings the speed and graphical detail of low-level OpenGL to the agile Python language. It is open source, enabling modification of the simulator in any beneficial way.</p></sec><sec id="s2"><title>2. Materials and methods</title><p>The cross-platform simulator is written in Python (version 2.7) and uses the NumPy library in consideration of its speed being close to the computational rate of the platform-dependent C language (Langtangen, <xref rid="B37" ref-type="bibr">2006</xref>). The simulator can be downloaded<xref ref-type="fn" rid="fn0001"><sup>1</sup></xref> in a package along with documentation<xref ref-type="fn" rid="fn0002"><sup>2</sup></xref> describing its installation, running, features, and examples and the code is registered in ModelDB<xref ref-type="fn" rid="fn0003"><sup>3</sup></xref>.</p><p>The following Section 2.1 describes the comprehensive access to field parameters, the subsequent section details the techniques applied to accelerate the simulation and Section 2.3 discusses the 3D visualization.</p><sec><title>2.1. Field parameters</title><p>A textual interface named values.py is provided in the root directory of the simulator code. It allows field values to be changed without knowledge of the inner workings of the simulator. For example, if η in Equation (1) is initialized as a non-zero number, a second order derivative is calculated to solve <italic>V</italic>. Conversely, the interface eliminates the knowledge requirement of the numerical implementation of the derivatives and all other underlying code implementations. The interface has additional benefits of easily modifying variables in one place without searching through the code. This implementation permits changing parameters easily and sharing code amongst others working with similar simulations by the exchange of a single file.</p><p>The most important aspect of a text-based interface from its user experience is its facilitation of novelty by allowing absolute control of all terms of Equation (1). For instance, matrix <italic>I</italic> can be defined in the interface with as many lines of Python code as necessary given the definition ends with an assignment (i.e., <italic>I = …</italic>). Assigning the first parameter in the values.py file, named showData, a value of 3 displays <italic>I</italic> in the simulator, which can be useful when refining its values. Time-varying spatio-temporal input is available in the interface by uncommenting and modifying the body of a function named <italic>updateI</italic> in any manner while maintaining that <italic>I</italic> is returned. Neural field investigations are thereby efficiently implemented with free choice over all the variables accessible through the interface while retaining the full performance.</p></sec><sec><title>2.2. Accelerated simulation</title><p>The simulator is advantageous in its acceleration of spatial and temporal integration. Multiple approaches are used to increase the simulation speed.</p><sec><title>2.2.1. Spatial and temporal integration</title><p>Equation (1) includes a spatial integral with a homogeneous kernel <italic>K</italic>. Please recall that homogeneous kernels just depend on the difference vector <bold>x</bold> − <bold>y</bold> between two spatial locations <bold>x, y</bold> including isotropic kernels, i.e., <italic>K</italic> = <italic>K</italic>(∥<bold>x</bold> − <bold>y</bold>∥), as a specific case. In the absence of the finite transmission delay term, this integral would represent a spatial convolution and would be solvable numerically efficiently by a Fast Fourier Transform (FFT) (Van Loan, <xref rid="B58" ref-type="bibr">1991</xref>). For non-vanishing transmission delay, the convolution structure is less obvious and the FFT is not applicable directly. Nevertheless, it is possible to re-write the spatial integration to utilize a FFT in space (Owen et al., <xref rid="B42" ref-type="bibr">2007</xref>; Hutt and Rougier, <xref rid="B30" ref-type="bibr">2010</xref>, <xref rid="B29" ref-type="bibr">2014</xref>) as</p><disp-formula id="E2"><mml:math id="M7"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi>Ω</mml:mi></mml:msub><mml:mrow><mml:msup><mml:mtext>d</mml:mtext><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mstyle><mml:mi>y</mml:mi><mml:mtext> </mml:mtext><mml:mi>K</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>−</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo stretchy="false">)</mml:mo><mml:mtext> </mml:mtext><mml:mi>S</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mo>∥</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>−</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>∥</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><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:msup><mml:mtext>d</mml:mtext><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mstyle><mml:mi>y</mml:mi><mml:msubsup><mml:mstyle displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mn>0</mml:mn><mml:mrow><mml:msub><mml:mi>τ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext> </mml:mtext><mml:mi>d</mml:mi><mml:mi>τ</mml:mi><mml:mi>L</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>−</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext> </mml:mtext><mml:mi>S</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula><p>with the maximum delay time τ<sub><italic>m</italic></sub> and the spatio-temporal kernel function <italic>L</italic>(<bold>x</bold>, <italic>t</italic>) = <italic>K</italic>(<bold>x</bold>)δ(∥ <bold>x</bold> ∥ ∕<italic>c</italic> − <italic>t</italic>). We observe that the spatial summation represents an integration over delayed spatial rings, which are convolved spatially with the transfer function <italic>S</italic> in Equation (1). Introducing a regular rectangular spatial grid for spatial discretization, finite axonal speed <italic>c</italic> yields rings of width</p><disp-formula id="E3"><label>(2)</label><mml:math id="M8"><mml:mrow><mml:mi>w</mml:mi><mml:mo>=</mml:mo><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mi>c</mml:mi><mml:mo>·</mml:mo><mml:mi>Δ</mml:mi><mml:mi>t</mml:mi><mml:mo>·</mml:mo><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></disp-formula><p>delineated within the field, where <italic>n</italic> and <italic>l</italic> are the number of discretized spatial units and the length of the field, respectively, and Δ<italic>t</italic> is the finite integration time step. The Pythagorean theorem gives the maximum radius of the rings in the field <inline-formula><mml:math id="M9"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>n</mml:mi><mml:mo>∕</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:math></inline-formula> over which the spatial integration is performed, which is applied to obtain the number of rings in a field as</p><disp-formula id="E4"><label>(3)</label><mml:math id="M10"><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>g</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:mo>/</mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:mo>⌋</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mi>c</mml:mi><mml:mi>Δ</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mo>⌋</mml:mo></mml:mrow></mml:math></disp-formula><p>defining the maximum delay to τ<sub><italic>m</italic></sub> = <italic>n</italic><sub><italic>rings</italic></sub>Δ<italic>t</italic>. The spatio-temporal kernel <italic>L</italic> is determined by the spatial kernel <italic>K</italic> and the axonal speed <italic>c</italic> (Hutt and Rougier, <xref rid="B30" ref-type="bibr">2010</xref>, <xref rid="B29" ref-type="bibr">2014</xref>).</p><p>Equation (1) involves distance-dependent delays which represent a specific type of distributed delays (Hutt and Lefebvre, <xref rid="B28" ref-type="bibr">in press</xref>). To this end, it is necessary to initialize the field variable <italic>V</italic> in an initial time interval and the toolbox allows the user to set the initial values arbitrarily. The external input <italic>I</italic> may be deterministic or stochastic and the user may choose it according to her needs, e.g., implementing spatial correlations in stochastic inputs. To integrate the evolution equation in time the user may choose between different integration methods for delay differential equations (Buckwar and Winkler, <xref rid="B7" ref-type="bibr">2006</xref>, <xref rid="B8" ref-type="bibr">2007</xref>). Standard methods discretize time regularly in steps of duration Δ<italic>t</italic> yielding results (Equations 2, 3). In the case of stochastic input, the toolbox includes numerical implementations of the delayed Euler-Maruyama method (Buckwar et al., <xref rid="B6" ref-type="bibr">2008</xref>) and the stochastic version of the Runge-Kutta method for delayed differential equation (Carletti, <xref rid="B9" ref-type="bibr">2006</xref>). For deterministic inputs, the equivalent deterministic methods are available.</p><p>If there is no modification to <italic>K</italic> and <italic>c</italic> during the simulation, then <italic>L</italic> is calculated once only before the start of the simulation while <italic>S</italic>(·) changes over time. The convolution of <italic>L</italic> and <italic>S</italic> is performed using a FFT what greatly increases the speed of the integral convolution compared to conventional integration. This can be understood easily recalling that the two-dimensional FFT needs to sum up <inline-formula><mml:math id="M11"><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mo class="qopname">log</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> terms leading to summands of the total number of <inline-formula><mml:math id="M12"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>F</mml:mi><mml:mi>F</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mo class="qopname">log</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>g</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>. In contrast, conventional integration sums up terms of number <italic>n</italic><sup>4</sup> for each delay time and hence the total number of computation <inline-formula><mml:math id="M13"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>g</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, cf. Appendix I. Hence the FFT implementation speeds up the integration by a factor of</p><disp-formula id="E5"><label>(4)</label><mml:math id="M14"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>p</mml:mi><mml:mi>e</mml:mi><mml:mi>e</mml:mi><mml:mi>d</mml:mi><mml:mi>u</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>F</mml:mi><mml:mi>F</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>log</mml:mi></mml:mrow><mml:mn>2</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula><p>The axonal speed's implementation is described in detail in Hutt and Rougier (<xref rid="B30" ref-type="bibr">2010</xref>). It is important to note that other (conventional) numerical software tools not taking into account the convolution structure have to sum up <italic>N</italic><sub><italic>conv</italic></sub> terms in case of fully connected networks. For instance, this holds true for the simulation tools <italic>BRIAN, Neuron</italic> and <italic>The Virtual Brain</italic> (Sanz Leon et al., <xref rid="B52" ref-type="bibr">2015</xref>) which have to memorize the history and advance the stored activity field <italic>n</italic><sub><italic>rings</italic></sub> times. We also note that the method proposed may be implemented in these simulation tools in the future since they also may consider spatially homogeneous neural fields as specific cases. The FFT-based method presented here computes the network interactions faster than these tools by <italic>f</italic><sub><italic>speedup</italic></sub> given in Equation (4). For instance, for a typical number of spatial grid intervals of <italic>n</italic> = 512 as used in the application showed in Section 3, we obtain the huge speed up factor of <italic>f</italic><sub><italic>speedup</italic></sub> = ≈3236.</p></sec><sec><title>2.2.2. Self-writing code</title><p>The second approach to increase the simulation speed employs self-writing code to reduce the simulator's instruction set. The simulator writes and executes its own code to increase the efficiency of simulations and display only the user defined features. The simulation code is based on interface selections and is self-written by an initialization module at the onset of the program. The interface offers features, such as a second derivative calculation, <italic>I</italic> and <italic>K</italic> updates and added noise, that conditionally run during the simulation and are not performed over time if the user selects to view <italic>V</italic>, <italic>I</italic>, or <italic>K</italic> at <italic>t</italic> = 0. For example, the visual interface offers the choice of viewing the spatial kernel <italic>K</italic> for its design and visualization. Only the code that initializes and displays <italic>K</italic> is written to the executing module if this choice is selected. The self-writing code is also favorable when the full simulation is run with calculations executed unconditionally. The result is very efficient code that is changed with every modification to the interface.</p></sec><sec><title>2.2.3. Implementation on GPUs</title><p>The third approach parallelizes the output calculations on the graphics processing unit (GPU). The displayed matrix is put onto the running system's GPU for hardware acceleration of the visualization. Vertex buffer objects improve visualization throughput by uploading vertices to video device memory where vertex and fragment shaders transform and write neural field data in parallel to the framebuffer for display. The simulator also avoids the CPU to GPU information transfer bottleneck by its storage of data on the video device memory. This is accomplished with the <italic>OpenGL Shading Language</italic> that is used through PyOpenGL to achieve a better visual description of information than is provided in other tools. A background on PyOpenGL and its comparison to other visualization libraries can be found in Rossant and Harris (<xref rid="B49" ref-type="bibr">2013</xref>).</p></sec><sec><title>2.2.4. Optimal visualization rate</title><p>The fourth approach to accelerate simulations is to display field matrices at a rate optimized for continuous visualization perception. Two images are perceived simultaneously when there is an interval of less than 30 ms between them (Wertheimer et al., <xref rid="B63" ref-type="bibr">2012</xref>). The simulator takes advantage of the temporal lag in biological visual perception by stopping the numerical calculations to submit the field data to the GPU once within every 30 ms. This allows for the numerical part of the simulations to continue with fewer stoppages, resulting in faster simulations.</p></sec></sec><sec><title>2.3. 3D visualization</title><p>The open source and cross-platform <italic>show3D</italic> library was written for the Neural Field Simulator to display field information. The library's visualization of neural fields expands two dimensional neural field data into a third dimension to better observe the differences in field locations. This is achieved by raising every value in the 2D spatial plane to a third dimension position [<italic>x</italic>, <italic>y</italic>] ↦ [<italic>z</italic>] relative to other 2D field values.</p><p>Color values are efficiently manipulated with the keyboard keys shown in Appendix II (Table <xref ref-type="table" rid="TA1">A1</xref>). There is a selection of 8 colors, cf. Figure <xref ref-type="fig" rid="F1">1</xref>, available for the background, minimum, middle, and maximum graph values.</p><fig id="F1" position="float"><label>Figure 1</label><caption><p><bold>Selection of colors that can be applied to the background and ranges of the plotted matrix</bold>.</p></caption><graphic xlink:href="fninf-09-00025-g0001"></graphic></fig><p>Intermediate color transformations are encoded in a dictionary containing 8<sup>2</sup> unique 3 element vectors, each representing red, green, and blue colors. The appropriate color transformation vector is uploaded to the GPU where the vector elements represent one mutually inclusive index of [0, 1, <italic>Z</italic>, 1-<italic>Z</italic>] with Z axis locations ∈ <bold>R</bold> ∣ 0 < <italic>Z</italic> < 1. Each location on the Z axis is subsequently colored in parallel by the GPU with the appropriate shade. Graph value colors are interpolated with two choices of ranges: [minimum, maximum] and [minimum, middle], [middle, maximum] graph value colors. Different depths of the graph can be highlighted by raising or lowering the ranges of colors.</p><p>Scrolling the mouse rotates neural fields in the direction of the mouse and the keyboard is used to move fields in various ways, cf. Figure <xref ref-type="fig" rid="F2">2</xref>.</p><fig id="F2" position="float"><label>Figure 2</label><caption><p><bold>Keyboard keys and their corresponding movements</bold>.</p></caption><graphic xlink:href="fninf-09-00025-g0002"></graphic></fig><p>Images and videos of simulations are saved, respectively in .png and .mp4 formats by using the keyboard keys in Appendix II. Visualization parameters are saved by the library after every simulation to reduce graphical adjustments during subsequent simulations of neural fields.</p><p>The show3D graphical visualization library is not limited to neural field data. Every two dimensional NumPy matrix can be displayed in 3 dimensions using the show3D library. Documentation for the show3D library's use, including a tutorial and code API, is online<xref ref-type="fn" rid="fn0004"><sup>4</sup></xref> and packaged with example code along with the library<xref ref-type="fn" rid="fn0005"><sup>5</sup></xref>. However, there is no requirement for the library's separate download for use with the simulator because it is integrated into the Neural Field Simulator.</p></sec></sec><sec id="s3"><title>3. Applications</title><p>The simulator can be used to analyze spatio-temporal neural field dynamics. The simulator's open source code allows modifications and extensions to be added to the code. The subsequent sections describe few of these possible applications.</p><p>Introducing finite axonal transmission speed in neural fields substantially slows numerical computation. However, to omit finite transmission speed is to neglect biological physiology (Idiart and Abbott, <xref rid="B32" ref-type="bibr">1993</xref>). Hutt and Rougier (<xref rid="B30" ref-type="bibr">2010</xref>) have suggested to implement finite axonal transmission speed in a computationally efficient manner that is utilized by the simulator. Numerically, the speed is infinite if <inline-formula><mml:math id="M15"><mml:mi>c</mml:mi><mml:mo>≥</mml:mo><mml:mi>l</mml:mi><mml:mo>∕</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msqrt><mml:mi>Δ</mml:mi><mml:mi>t</mml:mi></mml:math></inline-formula> and there is increasing delay as <italic>c</italic> decreases.</p><sec><title>3.1. Breather</title><p>Breather oscillations have been reported in theory (Folias and Bressloff, <xref rid="B20" ref-type="bibr">2005</xref>; Hutt and Rougier, <xref rid="B30" ref-type="bibr">2010</xref>) and experiments (Wang, <xref rid="B61" ref-type="bibr">2010</xref>). As shown in Hutt and Rougier (<xref rid="B30" ref-type="bibr">2010</xref>), breathers are solutions of Equation (1) for finite axonal transmission speeds. They can be obtained and visualized in the simulator by assuming a temporally constant external input <italic>I</italic> in Equation (1). For a Gaussian-shape input with its apex at the center of the field, one overwrites the <italic>I</italic> variable section in the values.py file as</p><disp-formula id="E6"><mml:math id="M16"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>sigma</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">5</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">65685425</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>I </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">20</mml:mn><mml:mo mathvariant="monospace">*</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>np</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">.</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>exp</mml:mtext></mml:mstyle><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mo mathvariant="monospace">−</mml:mo><mml:mtext>x</mml:mtext><mml:mo mathvariant="monospace">*</mml:mo><mml:mo mathvariant="monospace">*</mml:mo><mml:mn mathvariant="monospace">2</mml:mn><mml:mo mathvariant="monospace">/</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>sigma</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">*</mml:mo><mml:mo mathvariant="monospace">*</mml:mo><mml:mn mathvariant="monospace">2</mml:mn></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">/</mml:mo><mml:mtext> </mml:mtext><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>sigma</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">*</mml:mo><mml:mo mathvariant="monospace">*</mml:mo><mml:mn mathvariant="monospace">2</mml:mn><mml:mo mathvariant="monospace">*</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>np</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">.</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>pi</mml:mtext></mml:mstyle></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow><mml:mtext> </mml:mtext></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula><p>and change the showData variable assignment near the beginning of the values.py file to</p><disp-formula id="E7"><mml:math id="M17"><mml:mrow><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>showData</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">3</mml:mn></mml:mrow></mml:math></disp-formula><p>to show the input <italic>I</italic> in the simulation. In the definition of <italic>I</italic>, the space variable <italic>x</italic> ∈ <inline-formula><mml:math id="M18"><mml:mi mathvariant="-tex-caligraphic">R</mml:mi></mml:math></inline-formula><sup>2</sup> is defined to cover the spatial domain Ω (not shown in the code snippet). A field input similar to Figure <xref ref-type="fig" rid="F3">3A</xref> can be seen when the simulator is run.</p><fig id="F3" position="float"><label>Figure 3</label><caption><p><bold>Breather parameters plotted in the simulator for (A) <italic>I</italic> and (B) <italic>K</italic> in Equation (1)</bold>.</p></caption><graphic xlink:href="fninf-09-00025-g0003"></graphic></fig><p>An inhibitory synaptic connectivity kernel, <italic>K</italic> in Equation (1), can be implemented for the breather and viewed by changing the <italic>showData</italic> and <italic>K</italic> variables in the values.py file to</p><disp-formula id="E8"><mml:math id="M19"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>showData</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">4</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>K</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">−</mml:mo><mml:mn mathvariant="monospace">4</mml:mn><mml:mo mathvariant="monospace">*</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>np</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">.</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>exp</mml:mtext></mml:mstyle><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mo mathvariant="monospace">−</mml:mo><mml:mtext>x</mml:mtext><mml:mo mathvariant="monospace">/</mml:mo><mml:mn mathvariant="monospace">3</mml:mn></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">/</mml:mo><mml:mtext> </mml:mtext><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mn mathvariant="monospace">18</mml:mn><mml:mo mathvariant="monospace">*</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>np</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">.</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>pi</mml:mtext></mml:mstyle></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula><p>and running the simulator. Here, <italic>x</italic> ∈ <inline-formula><mml:math id="M20"><mml:mi mathvariant="-tex-caligraphic">R</mml:mi></mml:math></inline-formula><sup>2</sup>. An inhibitory synaptic kernel similar to the one in Figure <xref ref-type="fig" rid="F3">3B</xref> can be subsequently viewed.</p><p>After overwriting <italic>I</italic> and <italic>K</italic> as noted above, replace the following variables and function in the values.py file with the values below:</p><disp-formula id="E9"><mml:math id="M21"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>showData</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">1</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>endTime</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">−</mml:mo><mml:mn mathvariant="monospace">1</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>dt</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">0</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">002</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>gamma</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">1.0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>eta</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">0</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>c</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">500</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>l</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">30</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>n</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">512</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>V</mml:mtext><mml:mn mathvariant="monospace">0</mml:mn></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>np</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">.</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>zeros</mml:mtext></mml:mstyle><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>n</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">,</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>n</mml:mtext></mml:mstyle></mml:mrow><mml:mo 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mathvariant="monospace">2</mml:mn><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">/</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">32</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">0</mml:mn><mml:mo mathvariant="monospace">+</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>b</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">*</mml:mo><mml:mo mathvariant="monospace">*</mml:mo><mml:mn mathvariant="monospace">2</mml:mn><mml:mo mathvariant="monospace">/</mml:mo><mml:mn mathvariant="monospace">32</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">0</mml:mn></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">/</mml:mo><mml:mtext> </mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext> </mml:mtext><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>np</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">.</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>pi</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">*</mml:mo><mml:mn mathvariant="monospace">32</mml:mn></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow><mml:mo mathvariant="monospace">*</mml:mo><mml:mn mathvariant="monospace">0.1</mml:mn><mml:mo mathvariant="monospace">*</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>np</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">.</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>sqrt</mml:mtext></mml:mstyle><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>dt</mml:mtext></mml:mstyle></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="blue"><mml:mtext>def</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>updateS</mml:mtext></mml:mstyle><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>V</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow><mml:mo mathvariant="monospace">:</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext> </mml:mtext><mml:mstyle mathvariant="monospace" class="text" mathcolor="blue"><mml:mtext>return</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">1</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">0</mml:mn><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">/</mml:mo><mml:mtext> </mml:mtext><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mn mathvariant="monospace">1</mml:mn><mml:mo mathvariant="monospace">+</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>np</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">.</mml:mo><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>exp</mml:mtext></mml:mstyle><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mo mathvariant="monospace">−</mml:mo><mml:mn mathvariant="monospace">10000</mml:mn><mml:mo mathvariant="monospace">*</mml:mo><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>V</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">−</mml:mo><mml:mn mathvariant="monospace">0</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">005</mml:mn></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow><mml:mtext> </mml:mtext></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula><p>Spatially localized breather oscillations are replicated by running the program. Figure <xref ref-type="fig" rid="F4">4</xref> shows two cycles of the oscillations after setting the minimum and maximum z axis values by typing</p><disp-formula id="E10"><mml:math id="M22"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace"><mml:mtext>n </mml:mtext></mml:mstyle><mml:mn mathvariant="monospace">0</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">00</mml:mn><mml:mstyle mathvariant="monospace"><mml:mtext>48 </mml:mtext></mml:mstyle><mml:mrow><mml:mo mathvariant="monospace">[</mml:mo><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mtext>Enter key</mml:mtext></mml:mstyle></mml:mrow><mml:mo mathvariant="monospace">]</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace"><mml:mtext>y </mml:mtext></mml:mstyle><mml:mn mathvariant="monospace">0</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">00</mml:mn><mml:mstyle mathvariant="monospace"><mml:mtext>58 </mml:mtext></mml:mstyle><mml:mrow><mml:mo mathvariant="monospace">[</mml:mo><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mtext>Enter key</mml:mtext></mml:mstyle></mml:mrow><mml:mo mathvariant="monospace">]</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula><p>after running the program.</p><fig id="F4" position="float"><label>Figure 4</label><caption><p><bold>Two cycles of the breather oscillations</bold>.</p></caption><graphic xlink:href="fninf-09-00025-g0004"></graphic></fig></sec><sec><title>3.2. Turing patterns</title><p>Turing patterns (Turing, <xref rid="B57" ref-type="bibr">1952</xref>) have been reported in neural field models (Atay and Hutt, <xref rid="B1" ref-type="bibr">2006</xref>; Elvin et al., <xref rid="B18" ref-type="bibr">2009</xref>; Steyn-Ross et al., <xref rid="B55" ref-type="bibr">2010</xref>). The Neural Field Simulator is able to compute and display noisy neural field activity evolving into Turing patterns.</p><p>Static Turing patterns emerge from noisy initial conditions with the following interface properties:</p><disp-formula id="E11"><mml:math id="M23"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mtext> </mml:mtext><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>showData</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">1</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>endTime</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">10</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" 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mathcolor="red"><mml:mtext>exp</mml:mtext></mml:mstyle><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mo mathvariant="monospace">−</mml:mo><mml:mn mathvariant="monospace">1</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">24</mml:mn><mml:mo mathvariant="monospace">*</mml:mo><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>V</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">−</mml:mo><mml:mn mathvariant="monospace">3</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">0</mml:mn></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula><p>With <italic>c</italic> = 6364 the effective speed is infinite for the given <italic>l</italic> and <italic>dt</italic> properties. Figure <xref ref-type="fig" rid="F5">5</xref> shows the simulation starting with random field potential noise. A pattern begins to emerge at 1 s and evolves into a temporally constant Turing pattern after approximately 5 s.</p><fig id="F5" position="float"><label>Figure 5</label><caption><p><bold>Static Turing pattern emerging over 5 s</bold>.</p></caption><graphic xlink:href="fninf-09-00025-g0005"></graphic></fig><p>Dynamic Turing patterns emerging over time in the simulator with interface values:</p><disp-formula id="E12"><mml:math id="M24"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>showData</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> 1 </mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>endTime</mml:mtext></mml:mstyle><mml:mtext> </mml:mtext><mml:mo mathvariant="monospace">=</mml:mo><mml:mtext> </mml:mtext><mml:mn mathvariant="monospace">40</mml:mn><mml:mtext> 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mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">5</mml:mn><mml:mo mathvariant="monospace">*</mml:mo><mml:mrow><mml:mo mathvariant="monospace">(</mml:mo><mml:mrow><mml:mstyle mathvariant="monospace" class="text" mathcolor="red"><mml:mtext>V</mml:mtext></mml:mstyle><mml:mo mathvariant="monospace">−</mml:mo><mml:mn mathvariant="monospace">3</mml:mn><mml:mo mathvariant="monospace">.</mml:mo><mml:mn mathvariant="monospace">0</mml:mn></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mrow><mml:mo mathvariant="monospace">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula><p>Figure <xref ref-type="fig" rid="F6">6</xref> shows a typical simulation, given random starting field potential noise, with different Turing patterns materializing. Activity patterns form at varying intervals, generally every few seconds, throughout the simulation. The times in Figure <xref ref-type="fig" rid="F6">6</xref> were chosen for clear displays of different (vertical, cone, and horizontal) patterns.</p><fig id="F6" position="float"><label>Figure 6</label><caption><p><bold>Turing patterns in neural field activity forming over time during the same simulation</bold>.</p></caption><graphic xlink:href="fninf-09-00025-g0006"></graphic></fig></sec><sec><title>3.3. Finite spreading speed</title><p>Stimulating a neural population at a single location, as is done in typical physiological experiments applying external stimuli, the neural activity spreads in the population. Since finite transmission speed represents delayed spatial interaction in the population under study, it affects the spreading speed of activity (Hutt, <xref rid="B23" ref-type="bibr">2007</xref>). If the transmission speed is infinite, the activity spreads diffusively involving the instantaneous activation at all spatial locations. Conversely, finite transmission speed delays the activity spread leading to a slowly-moving spreading front (Hutt and Atay, <xref rid="B25" ref-type="bibr">2006</xref>; Hutt, <xref rid="B24" ref-type="bibr">2009</xref>). Figure <xref ref-type="fig" rid="F7">7</xref> shows numerical simulations for large (top row) and small speeds (bottom row), other parameters are identical.</p><fig id="F7" position="float"><label>Figure 7</label><caption><p><bold>Activity spread with large speed <italic>c</italic> (top) and small speed <italic>c</italic> (bottom)</bold>.</p></caption><graphic xlink:href="fninf-09-00025-g0007"></graphic></fig><p>The simulator allows the transmission speed to be examined closely in the visualization window by decreasing the maximum z axis value to be close to the original field value. 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Extensions</title><p>The simulator, being open source, allows the tailoring of code to provide modifications and extensions. An example extension is the addition of a graphical interface to modify parameters and simulate neural fields. Figure <xref ref-type="fig" rid="F8">8</xref> shows an example interface coded with the wxPython<xref ref-type="fn" rid="fn0006"><sup>6</sup></xref> toolkit. Simulations are started, paused, continued, and started anew by clicking a button.</p><fig id="F8" position="float"><label>Figure 8</label><caption><p><bold>A graphical interface for the simulator</bold>.</p></caption><graphic xlink:href="fninf-09-00025-g0008"></graphic></fig><p>Neural field parameters can be modified prior to and during simulations by clicking on the appropriate area of the interface and completing a pop-up dialogue. Running simulations are automatically paused when a mouse hovers above parameter areas of the interface. There is a symbiosis among the show3D library discussed in Section 2.3 and the parameter selection interface. It is possible to view the external input stimulus, kernel, and firing rate in the GLFW window by adding a mouse event and hovering over these sections to automatically view the respective matrices. Viewing these elements while changing their parameters can help to fine-tune field parameters. Moving the mouse away from these areas unpauses paused simulations and the field potential matrix is shown in the GLFW window. Further synergy between the interface and show3D library is implemented with the option to alter graph values from the interface where z axis limits and colors can be selected.</p></sec></sec><sec id="s4"><title>4. Discussion</title><p>The Neural Field Simulator and its dependencies are cross-platform. However, the simulator interacts with graphics hardware using system-specific drivers which can result in problems on some operating systems. The graphical user interface in Section 3.4 is an example of this where the cross platform wxPython toolkit uses OpenGL to draw to the screen. The show3D library likewise uses OpenGL to interact with GPU. The graphical user interface and show3D library function symbiotically on Mac systems via the Apple Graphics Library. Conversely, on other operating systems such as Linux and Windows, unfortunately the separate utilization of OpenGL causes the simulator to crash. To this end, the current version of the simulator is released without the addition of extensions in order to operate properly on every major operating system. Nevertheless, a graphical interface can be a good choice with an appropriate single operating system.</p><p>Apart from the software implementation, in future work some model assumptions can be released. The square geometry can be recast easily to a rectangular geometry, whereas more general geometries (e.g., the impressive implementation work in <italic>The Virtual Brain</italic> Sanz Leon et al., <xref rid="B52" ref-type="bibr">2015</xref>) will take more numerical effort. Periodic boundary conditions guarantee the simple application of the FFT, effective implementations of other boundary conditions like Dirichlet conditions [<italic>V</italic>(<bold>z</bold>, <italic>t</italic>) = const, <bold>z</bold> ∈ ∂Ω] will demand some implementation changes in the spatial integral computation. Such modifications may still retain the fundamental implementation of the FFT. In contrast, rejecting the homogeneity hypothesis of spatial interactions, i.e., <italic>K</italic> = <italic>K</italic>(<bold>x, y</bold>) ≠ <italic>K</italic>(<bold>x</bold> − <bold>y</bold>), abolishes the convolution structure and slows down the numerical simulation, cf. Appendix I.</p><p>Future work will extend the NFM model to multiple equations to render the model even more biologically plausible. In addition, an extension of the implementation to a mixture of constant and space-dependent delays as considered by Veltz and Faugeras (<xref rid="B59" ref-type="bibr">2011</xref>) will be interesting.</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><ack><p>The authors thank Kevin Green for valuable testing of the software under MacOS and Linux and for providing system parameters for Figure <xref ref-type="fig" rid="F6">6</xref>. 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Neural Syst.</source><volume>5</volume>, <fpage>191</fpage>–<lpage>202</lpage>. <pub-id pub-id-type="doi">10.1088/0954-898X_5_2_005</pub-id></mixed-citation></ref></ref-list><app-group><app id="A1"><title>Appendix i</title><sec><title>Heterogeneous neural fields</title><p>Heterogeneous neural fields have attracted increased attention in recent years (Bressloff, <xref rid="B5" ref-type="bibr">2001</xref>; Qubbaj and Jirsa, <xref rid="B47" ref-type="bibr">2007</xref>; Brackley and Turner, <xref rid="B3" ref-type="bibr">2009</xref>; Schmidt et al., <xref rid="B54" ref-type="bibr">2009</xref>; Coombes et al., <xref rid="B13" ref-type="bibr">2012</xref>; beim Graben and Hutt, <xref rid="B2" ref-type="bibr">2014</xref>) since they have been found in biological neural networks such as the prefrontal cortex (Rosenkilde, <xref rid="B48" ref-type="bibr">1979</xref>; Wang et al., <xref rid="B62" ref-type="bibr">2006</xref>) and visual cortex (Demeulemeester et al., <xref rid="B17" ref-type="bibr">1988</xref>). In order to study the neural population activity in such systems, the present implementation could be extended along the following mathematical reasoning. The integral in Equation (1) may be re-written as</p><disp-formula id="E15"><mml:math id="M27"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi>Ω</mml:mi></mml:msub><mml:mrow><mml:msup><mml:mtext>d</mml:mtext><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mstyle><mml:mi>y</mml:mi><mml:mtext> </mml:mtext><mml:mi>K</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo stretchy="false">)</mml:mo><mml:mtext> </mml:mtext><mml:mi>S</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mo>∥</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>−</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>∥</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><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:msup><mml:mtext>d</mml:mtext><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mstyle><mml:mi>y</mml:mi><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>∞</mml:mi></mml:mrow><mml:mi>∞</mml:mi></mml:msubsup><mml:mi>d</mml:mi></mml:mrow></mml:mstyle><mml:mi>τ</mml:mi><mml:mi>K</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo stretchy="false">)</mml:mo><mml:mi>δ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>−</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mo>∥</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>−</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>∥</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:mfrac><mml:mo stretchy="false">)</mml:mo><mml:mtext> </mml:mtext><mml:mi>S</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mi>τ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><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:msup><mml:mtext>d</mml:mtext><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mstyle><mml:mi>y</mml:mi><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mrow><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msubsup><mml:mi>d</mml:mi></mml:mrow></mml:mstyle><mml:mi>T</mml:mi><mml:mi>D</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>−</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext> </mml:mtext><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>x</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mstyle mathvariant="bold" mathsize="normal"><mml:mi>y</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula><p>with <italic>D</italic>(<bold>x</bold>, <italic>t</italic>) = δ(∥ <bold>x</bold> ∥ ∕<italic>c</italic> − <italic>t</italic>), <italic>R</italic>(<bold>x, y</bold>, <italic>t</italic>) = <italic>K</italic>(<bold>x, y</bold>)<italic>S</italic>[<italic>V</italic>(<bold>y</bold>, <italic>t</italic>)]. This integral still has a spatial convolution structure; however, it is not perfect since <italic>R</italic> includes the spatial location <bold>x</bold>. The numerical simulation of the integral consequently involves <inline-formula><mml:math id="M28"><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>g</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> summands and the numerical implementation is slower than in the homogeneous case. The formulation of heterogeneous neural fields is not implemented yet, but it will be part of a future update.</p></sec></app></app-group><app-group><app id="A2"><title>Appendix ii</title><table-wrap id="TA1" position="anchor"><label>Table A1</label><caption><p><bold>Keyboard keys and their actions</bold>.</p></caption><table frame="hsides" rules="groups"><thead><tr><th valign="top" align="left" rowspan="1" colspan="1"><bold>Key</bold></th><th valign="top" align="left" rowspan="1" colspan="1"><bold>Action</bold></th></tr></thead><tbody><tr style="border-top: thin solid #000000;"><td valign="top" align="left" rowspan="1" colspan="1">2</td><td valign="top" align="left" rowspan="1" colspan="1">Interpolate between min and max graph value colors</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">3</td><td valign="top" align="left" rowspan="1" colspan="1">Interpolate between min-mid and mid-max graph value colors</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">↑</td><td valign="top" align="left" rowspan="1" colspan="1">Rotate the field up</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">↓</td><td valign="top" align="left" rowspan="1" colspan="1">Rotate the field down</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">←</td><td valign="top" align="left" rowspan="1" colspan="1">Rotate the field left</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">→</td><td valign="top" align="left" rowspan="1" colspan="1">Rotate the field right</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">a</td><td valign="top" align="left" rowspan="1" colspan="1">Modulate minimum graph value color</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">b</td><td valign="top" align="left" rowspan="1" colspan="1">Modulate background graph color</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">d</td><td valign="top" align="left" rowspan="1" colspan="1">Move the field down</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">e</td><td valign="top" align="left" rowspan="1" colspan="1">Move the field up</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">Esc</td><td valign="top" align="left" rowspan="1" colspan="1">Exit simulation</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">f</td><td valign="top" align="left" rowspan="1" colspan="1">Move the field right</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">g</td><td valign="top" align="left" rowspan="1" colspan="1">Change number of axes lines</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">i</td><td valign="top" align="left" rowspan="1" colspan="1">Save an image</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">j</td><td valign="top" align="left" rowspan="1" colspan="1">Set min and max z axis limits to min and max field values</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">k</td><td valign="top" align="left" rowspan="1" colspan="1">Change color distribution to a higher range</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">l</td><td valign="top" align="left" rowspan="1" colspan="1">Change color distribution to a reduced range</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">m</td><td valign="top" align="left" rowspan="1" colspan="1">Set minimum z axis limit to minimum field value</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">n</td><td valign="top" align="left" rowspan="1" colspan="1">Change minimum z axis limit</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">o</td><td valign="top" align="left" rowspan="1" colspan="1">Equally distribute color range</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">p</td><td valign="top" align="left" rowspan="1" colspan="1">Pause and resume simulation</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1"><italic>pg up</italic></td><td valign="top" align="left" rowspan="1" colspan="1">Zoom in</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1"><italic>pg down</italic></td><td valign="top" align="left" rowspan="1" colspan="1">Zoom out</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">q</td><td valign="top" align="left" rowspan="1" colspan="1">Modulate middle graph value color</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">s</td><td valign="top" align="left" rowspan="1" colspan="1">Move the field left</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">t</td><td valign="top" align="left" rowspan="1" colspan="1">Change axis text size on Mac systems</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">u</td><td valign="top" align="left" rowspan="1" colspan="1">Set maximum z axis limit to maximum field value</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">v</td><td valign="top" align="left" rowspan="1" colspan="1">Begin and end video recording</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">y</td><td valign="top" align="left" rowspan="1" colspan="1">Change maximum z axis limit</td></tr><tr><td valign="top" align="left" rowspan="1" colspan="1">z</td><td valign="top" align="left" rowspan="1" colspan="1">Modulate maximum graph value color</td></tr></tbody></table></table-wrap></app></app-group></back></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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