Synaptic Plasticity in Neural Networks Needs Homeostasis with a Fast Rate Detector
Identifieur interne : 002164 ( Pmc/Curation ); précédent : 002163; suivant : 002165Synaptic Plasticity in Neural Networks Needs Homeostasis with a Fast Rate Detector
Auteurs : Friedemann Zenke [Suisse] ; Guillaume Hennequin [Royaume-Uni] ; Wulfram Gerstner [Suisse]Source :
- PLoS Computational Biology [ 1553-734X ] ; 2013.
Abstract
Hebbian changes of excitatory synapses are driven by and further enhance correlations between pre- and postsynaptic activities. Hence, Hebbian plasticity forms a positive feedback loop that can lead to instability in simulated neural networks. To keep activity at healthy, low levels, plasticity must therefore incorporate homeostatic control mechanisms. We find in numerical simulations of recurrent networks with a realistic triplet-based spike-timing-dependent plasticity rule (triplet STDP) that homeostasis has to detect rate changes on a timescale of seconds to minutes to keep the activity stable. We confirm this result in a generic mean-field formulation of network activity and homeostatic plasticity. Our results strongly suggest the existence of a homeostatic regulatory mechanism that reacts to firing rate changes on the order of seconds to minutes.
Url:
DOI: 10.1371/journal.pcbi.1003330
PubMed: 24244138
PubMed Central: 3828150
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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Synaptic Plasticity in Neural Networks Needs Homeostasis with a Fast Rate Detector</title><author><name sortKey="Zenke, Friedemann" sort="Zenke, Friedemann" uniqKey="Zenke F" first="Friedemann" last="Zenke">Friedemann Zenke</name><affiliation wicri:level="1"><nlm:aff id="aff1"><addr-line>School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, Ecole polytechnique fédérale de Lausanne, Lausanne, Switzerland</addr-line></nlm:aff><country xml:lang="fr">Suisse</country><wicri:regionArea>School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, Ecole polytechnique fédérale de Lausanne, Lausanne</wicri:regionArea></affiliation></author><author><name sortKey="Hennequin, Guillaume" sort="Hennequin, Guillaume" uniqKey="Hennequin G" first="Guillaume" last="Hennequin">Guillaume Hennequin</name><affiliation wicri:level="1"><nlm:aff id="aff2"><addr-line>Computational and Biological Learning Laboratory, Department of Engineering, University of Cambridge, Cambridge, United Kingdom</addr-line></nlm:aff><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>Computational and Biological Learning Laboratory, Department of Engineering, University of Cambridge, Cambridge</wicri:regionArea></affiliation></author><author><name sortKey="Gerstner, Wulfram" sort="Gerstner, Wulfram" uniqKey="Gerstner W" first="Wulfram" last="Gerstner">Wulfram Gerstner</name><affiliation wicri:level="1"><nlm:aff id="aff1"><addr-line>School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, Ecole polytechnique fédérale de Lausanne, Lausanne, Switzerland</addr-line></nlm:aff><country xml:lang="fr">Suisse</country><wicri:regionArea>School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, Ecole polytechnique fédérale de Lausanne, Lausanne</wicri:regionArea></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">24244138</idno><idno type="pmc">3828150</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3828150</idno><idno type="RBID">PMC:3828150</idno><idno type="doi">10.1371/journal.pcbi.1003330</idno><date when="2013">2013</date><idno type="wicri:Area/Pmc/Corpus">002164</idno><idno type="wicri:Area/Pmc/Curation">002164</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Synaptic Plasticity in Neural Networks Needs Homeostasis with a Fast Rate Detector</title><author><name sortKey="Zenke, Friedemann" sort="Zenke, Friedemann" uniqKey="Zenke F" first="Friedemann" last="Zenke">Friedemann Zenke</name><affiliation wicri:level="1"><nlm:aff id="aff1"><addr-line>School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, Ecole polytechnique fédérale de Lausanne, Lausanne, Switzerland</addr-line></nlm:aff><country xml:lang="fr">Suisse</country><wicri:regionArea>School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, Ecole polytechnique fédérale de Lausanne, Lausanne</wicri:regionArea></affiliation></author><author><name sortKey="Hennequin, Guillaume" sort="Hennequin, Guillaume" uniqKey="Hennequin G" first="Guillaume" last="Hennequin">Guillaume Hennequin</name><affiliation wicri:level="1"><nlm:aff id="aff2"><addr-line>Computational and Biological Learning Laboratory, Department of Engineering, University of Cambridge, Cambridge, United Kingdom</addr-line></nlm:aff><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>Computational and Biological Learning Laboratory, Department of Engineering, University of Cambridge, Cambridge</wicri:regionArea></affiliation></author><author><name sortKey="Gerstner, Wulfram" sort="Gerstner, Wulfram" uniqKey="Gerstner W" first="Wulfram" last="Gerstner">Wulfram Gerstner</name><affiliation wicri:level="1"><nlm:aff id="aff1"><addr-line>School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, Ecole polytechnique fédérale de Lausanne, Lausanne, Switzerland</addr-line></nlm:aff><country xml:lang="fr">Suisse</country><wicri:regionArea>School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, Ecole polytechnique fédérale de Lausanne, Lausanne</wicri:regionArea></affiliation></author></analytic><series><title level="j">PLoS Computational Biology</title><idno type="ISSN">1553-734X</idno><idno type="eISSN">1553-7358</idno><imprint><date when="2013">2013</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>Hebbian changes of excitatory synapses are driven by and further enhance correlations between pre- and postsynaptic activities. Hence, Hebbian plasticity forms a positive feedback loop that can lead to instability in simulated neural networks. To keep activity at healthy, low levels, plasticity must therefore incorporate homeostatic control mechanisms. We find in numerical simulations of recurrent networks with a realistic triplet-based spike-timing-dependent plasticity rule (triplet STDP) that homeostasis has to detect rate changes on a timescale of seconds to minutes to keep the activity stable. We confirm this result in a generic mean-field formulation of network activity and homeostatic plasticity. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">PLoS Comput Biol</journal-id><journal-id journal-id-type="iso-abbrev">PLoS Comput. Biol</journal-id><journal-id journal-id-type="publisher-id">plos</journal-id><journal-id journal-id-type="pmc">ploscomp</journal-id><journal-title-group><journal-title>PLoS Computational Biology</journal-title></journal-title-group><issn pub-type="ppub">1553-734X</issn><issn pub-type="epub">1553-7358</issn><publisher><publisher-name>Public Library of Science</publisher-name><publisher-loc>San Francisco, USA</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">24244138</article-id><article-id pub-id-type="pmc">3828150</article-id><article-id pub-id-type="publisher-id">PCOMPBIOL-D-13-00665</article-id><article-id pub-id-type="doi">10.1371/journal.pcbi.1003330</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group></article-categories><title-group><article-title>Synaptic Plasticity in Neural Networks Needs Homeostasis with a Fast Rate Detector</article-title><alt-title alt-title-type="running-head">Stable Hebbian Plasticity Needs Fast Rate Detector</alt-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Zenke</surname><given-names>Friedemann</given-names></name><xref ref-type="aff" rid="aff1"><sup>1</sup></xref><xref ref-type="corresp" rid="cor1"><sup>*</sup></xref></contrib><contrib contrib-type="author"><name><surname>Hennequin</surname><given-names>Guillaume</given-names></name><xref ref-type="aff" rid="aff2"><sup>2</sup></xref></contrib><contrib contrib-type="author"><name><surname>Gerstner</surname><given-names>Wulfram</given-names></name><xref ref-type="aff" rid="aff1"><sup>1</sup></xref></contrib></contrib-group><aff id="aff1"><label>1</label><addr-line>School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, Ecole polytechnique fédérale de Lausanne, Lausanne, Switzerland</addr-line></aff><aff id="aff2"><label>2</label><addr-line>Computational and Biological Learning Laboratory, Department of Engineering, University of Cambridge, Cambridge, United Kingdom</addr-line></aff><contrib-group><contrib contrib-type="editor"><name><surname>Morrison</surname><given-names>Abigail</given-names></name><role>Editor</role><xref ref-type="aff" rid="edit1"></xref></contrib></contrib-group><aff id="edit1"><addr-line>Research Center Jülich, Germany</addr-line></aff><author-notes><corresp id="cor1">* E-mail: <email>friedemann.zenke@epfl.ch</email></corresp><fn fn-type="conflict"><p>The authors have declared that no competing interests exist.</p></fn><fn fn-type="con"><p>Conceived and designed the experiments: FZ GH WG. Performed the experiments: FZ. Analyzed the data: FZ. Contributed reagents/materials/analysis tools: FZ. Wrote the paper: FZ GH WG.</p></fn></author-notes><pub-date pub-type="collection"><month>11</month><year>2013</year></pub-date><pmc-comment> Fake ppub added to accomodate plos workflow change from 03/2008 and 03/2009 </pmc-comment>
<pub-date pub-type="ppub"><month>11</month><year>2013</year></pub-date><pub-date pub-type="epub"><day>14</day><month>11</month><year>2013</year></pub-date><volume>9</volume><issue>11</issue><elocation-id>e1003330</elocation-id><history><date date-type="received"><day>19</day><month>4</month><year>2013</year></date><date date-type="accepted"><day>25</day><month>9</month><year>2013</year></date></history><permissions><copyright-year>2013</copyright-year><copyright-holder>Zenke et al</copyright-holder><license xlink:href="http://creativecommons.org/licenses/by/3.0/"><license-p>This is an open-access article distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/3.0/">Creative Commons Attribution License</ext-link>, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.</license-p></license></permissions><abstract><p>Hebbian changes of excitatory synapses are driven by and further enhance correlations between pre- and postsynaptic activities. Hence, Hebbian plasticity forms a positive feedback loop that can lead to instability in simulated neural networks. To keep activity at healthy, low levels, plasticity must therefore incorporate homeostatic control mechanisms. We find in numerical simulations of recurrent networks with a realistic triplet-based spike-timing-dependent plasticity rule (triplet STDP) that homeostasis has to detect rate changes on a timescale of seconds to minutes to keep the activity stable. We confirm this result in a generic mean-field formulation of network activity and homeostatic plasticity. Our results strongly suggest the existence of a homeostatic regulatory mechanism that reacts to firing rate changes on the order of seconds to minutes.</p></abstract><abstract abstract-type="summary"><title>Author Summary</title><p>Learning and memory in the brain are thought to be mediated through Hebbian plasticity. When a group of neurons is repetitively active together, their connections get strengthened. This can cause co-activation even in the absence of the stimulus that triggered the change. To avoid run-away behavior it is important to prevent neurons from forming excessively strong connections. This is achieved by regulatory homeostatic mechanisms that constrain the overall activity. Here we study the stability of background activity in a recurrent network model with a plausible Hebbian learning rule and homeostasis. We find that the activity in our model is unstable unless homeostasis reacts to rate changes on a timescale of minutes or faster. Since this timescale is incompatible with most known forms of homeostasis, this implies the existence of a previously unknown, rapid homeostatic regulatory mechanism capable of either gating the rate of plasticity, or affecting synaptic efficacies otherwise on a short timescale.</p></abstract><funding-group><funding-statement>FZ was supported by the European Community's Seventh Framework Program under grant agreement no. 237955 (FACETS-ITN) and 269921 (BrainScales). GH was supported by the Swiss National Science Foundation. WG acknowledges funding from the European Research Council (no. 268689). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.</funding-statement></funding-group><counts><page-count count="14"></page-count></counts></article-meta></front><body><sec id="s1"><title>Introduction</title><p>The awake cortex is constantly active, even in the absence of external inputs. This baseline activity, commonly referred to as the “background state”, is characterized by low synchrony at the population level and highly irregular firing of single neurons. While the direct implications of the background state are presently unknown, several neurological disorders such as Parkinson's disease, epilepsy or schizophrenia have been linked to various disruptions thereof <xref ref-type="bibr" rid="pcbi.1003330-Filion1">[1]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Uhlhaas1">[5]</xref>. Theoretically, the background state is currently understood as the asynchronous and irregular (AI) firing regime resulting from a dynamic balance of excitation and inhibition in recurrent neural networks <xref ref-type="bibr" rid="pcbi.1003330-vanVreeswijk1">[6]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Renart1">[9]</xref>. Balanced networks exhibit low activity and small mean pairwise correlations <xref ref-type="bibr" rid="pcbi.1003330-Brunel1">[7]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Renart1">[9]</xref>. However, even small changes in the amount of excitation can disrupt the background state <xref ref-type="bibr" rid="pcbi.1003330-Brunel1">[7]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Kumar1">[10]</xref>. Changes in excitation can arise from Hebbian plasticity of excitatory synapses: Subsets of jointly active neurons form strong connections with each other which is thought to be the neural substrate of memory <xref ref-type="bibr" rid="pcbi.1003330-Hebb1">[11]</xref>. However, Hebbian plasticity has the unwanted side effect of further increasing the excitatory synaptic drive into cells that are already active. The emergent positive feedback loop renders this form of plasticity unstable and makes it hard to reconcile with the stability of the background state <xref ref-type="bibr" rid="pcbi.1003330-Morrison1">[12]</xref>.</p><p>To stabilize neuronal activity, homeostatic control mechanisms have been proposed theoretically <xref ref-type="bibr" rid="pcbi.1003330-vonderMalsburg1">[13]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Clopath1">[19]</xref> and various forms have indeed been found experimentally <xref ref-type="bibr" rid="pcbi.1003330-Abraham1">[20]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Abraham2">[22]</xref>. The term homeostasis comprises any compensatory mechanism that stabilizes neural firing rates in the face of plasticity induced changes. This includes compensatory changes in the overall synaptic drive (e.g. synaptic scaling <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano1">[21]</xref>), the neuronal excitability (intrinsic plasticity <xref ref-type="bibr" rid="pcbi.1003330-Desai1">[23]</xref>) or changes to the plasticity rules themselves (i.e. metaplasticity <xref ref-type="bibr" rid="pcbi.1003330-Abraham1">[20]</xref>). Common to all experimentally found homeostatic mechanisms is their relatively slow response compared to plasticity. While synaptic weights can change on the timescale of seconds to minutes <xref ref-type="bibr" rid="pcbi.1003330-Markram1">[24]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Sjstrm1">[26]</xref>, noticeable changes caused by homeostasis generally take hours or even days <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano1">[21]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano2">[27]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Watt1">[29]</xref>. This is thought to be crucial since it allows neurons to detect their average firing rate by integrating over long times. While fluctuations on short timescales cause Hebbian learning and alter synapses in a specific way to store information, at longer timescales homeostasis causes non-specific changes to maintain stability <xref ref-type="bibr" rid="pcbi.1003330-Desai1">[23]</xref>. The required homeostatic rate detector acts as a low-pass filter and therefore induces a time lag between the rate estimate and the true value of neuronal activity. As a result, homeostatic responses based on this detector become inert to sudden changes. The longer the filter time constant is, the more sluggish the homeostatic response becomes.</p><p>Here we formalize the link between stability of network activity and the timescales involved in homeostasis in the presence of Hebbian plasticity. We first study the stability of the background state during long episodes of ongoing plasticity in direct numerical simulations of large balanced networks with a metaplastic triplet STDP rule <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref> in which the timescale of homeostasis is equal to the one of the rate detector. This allows us to determine the critical timescale beyond which stability is lost. In a second step we reduce the system to a generic two-dimensional mean-field model amenable to analytical considerations. Both the numerical and the analytical approach show that homeostasis has to react to rate changes on a timescale of seconds to minutes. We then show analytically and in simulations that these stability requirements are not specific to metaplastic triplet STDP, but generalize to the case of triplet STDP in conjunction with synaptic scaling.</p><p>In summary we show that the stability of the background state requires the ratio between the timescales of homeostasis and plasticity to be smaller than a critical value <inline-formula><inline-graphic xlink:href="pcbi.1003330.e001.jpg"></inline-graphic></inline-formula> which is determined by the network properties. For realistic network and plasticity parameters this requires the homeostatic timescale to be short, meaning that homeostasis has to react quickly to changes in the neuronal firing rate (on the order of seconds to minutes). Our results suggest that plasticity must either be gated rapidly by a third factor, or be accompanied by a yet unknown homeostatic control mechanism that reacts on a short timescale.</p></sec><sec id="s2"><title>Results</title><p>In the following we first discuss our results obtained from simulating spiking neural networks in the balanced state with a Hebbian learning rule subject to a plausible learning rate. In the beginning we focus on a metaplastic mechanism that regulates the amount of synaptic long term depression (LTD) homeostatically. By systematically varying the time constant of the homeostatic rate detector, we find that stability of the background state requires homeostasis to act on a timescale of minutes. We then strive to understand the underlying mechanism of the instability from a generic mean field model, which we use to analytically confirm the critical time constant found in the spiking network simulations. Finally, to explore the generality of this mean field approach, we apply the analysis to two variations of the triplet learning rule. First, we add a slow weight decay to metaplastic triplet STDP and second we switch from homeostatic metaplasticity to synaptic scaling in combination with triplet STDP. In both cases we confirm analytically and in simulations that a fast rate detector is required to assure stability.</p><sec id="s2a"><title>Simulation results</title><p>To study the stability of the background state in balanced networks with plastic excitatory-to-excitatory (EE) synapses we simulate networks of 25000 randomly connected integrate-and-fire neurons (<xref ref-type="fig" rid="pcbi-1003330-g001">Figure 1 A</xref>). Prior to any synaptic modification by plasticity, we set the network to the balanced state in which membrane potentials exhibit large sub-threshold fluctuations (<xref ref-type="fig" rid="pcbi-1003330-g001">Figure 1 C</xref>), giving rise to irregular activity at low rates (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e002.jpg"></inline-graphic></inline-formula>) and asynchronous firing at the population level (<xref ref-type="fig" rid="pcbi-1003330-g001">Figure 1 D</xref>). In our model more than 90% of the input to each neuron comes from within the network, thus closely resembling conditions found in cortex <xref ref-type="bibr" rid="pcbi.1003330-DeFelipe1">[31]</xref>.</p><fig id="pcbi-1003330-g001" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1003330.g001</object-id><label>Figure 1</label><caption><title>The balanced network model.</title><p>(<bold>A</bold>) Schematic of the network model. Recurrent synapses in the population of excitatory neurons (<bold>*</bold>) are subject to the homeostatic triplet STDP rule. (<bold>B</bold>) Typical magnitude and time course of a single excitatory postsynaptic potential from rest. (<bold>C</bold>) Membrane potential trace of a cell during background activity. (<bold>D</bold>) Histogram of single neuron firing rates (blue) and coefficient of variation (CV ISI, red) across neurons as well as the ISI distribution of all neurons (yellow) of the network during background activity. Arrowheads indicate mean values.</p></caption><graphic xlink:href="pcbi.1003330.g001"></graphic></fig><p>Plasticity of all recurrent EE synapses is modeled as an additive triplet STDP rule (see <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref> and <xref ref-type="sec" rid="s4">Methods</xref>) which accurately describes experimental data from visual cortex <xref ref-type="bibr" rid="pcbi.1003330-Sjstrm1">[26]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>. In this metaplastic triplet STDP rule the amount of LTD is chosen such that LTP and LTD cancel on average, when the pre- and postsynaptic neurons fire with Poisson statistics at rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e003.jpg"></inline-graphic></inline-formula>. Therefore, under the assumption of low spike-spike correlations and irregular firing, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e004.jpg"></inline-graphic></inline-formula> becomes a fixed point of the network dynamics (see <xref ref-type="bibr" rid="pcbi.1003330-Hennequin1">[32]</xref> and <xref ref-type="sec" rid="s4">Methods</xref>). We begin with a fixed learning rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e005.jpg"></inline-graphic></inline-formula>, which is chosen as a compromise between biological plausibility and computational feasibility (<xref ref-type="sec" rid="s4">Methods</xref>). To go towards the fixed point, all neurons constantly estimate their firing rate as the moving average <inline-formula><inline-graphic xlink:href="pcbi.1003330.e006.jpg"></inline-graphic></inline-formula> with exponential decay constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e007.jpg"></inline-graphic></inline-formula>, given by<disp-formula id="pcbi.1003330.e008"><graphic xlink:href="pcbi.1003330.e008"></graphic><label>(1)</label></disp-formula>where <inline-formula><inline-graphic xlink:href="pcbi.1003330.e009.jpg"></inline-graphic></inline-formula> corresponds to the <inline-formula><inline-graphic xlink:href="pcbi.1003330.e010.jpg"></inline-graphic></inline-formula>-th firing time of neuron <inline-formula><inline-graphic xlink:href="pcbi.1003330.e011.jpg"></inline-graphic></inline-formula> (see also <xref ref-type="sec" rid="s4">Methods</xref>, <xref ref-type="disp-formula" rid="pcbi.1003330.e321">Eq. (19)</xref>). If the rate estimate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e012.jpg"></inline-graphic></inline-formula> of the postsynaptic neuron <inline-formula><inline-graphic xlink:href="pcbi.1003330.e013.jpg"></inline-graphic></inline-formula> lies above (below) <inline-formula><inline-graphic xlink:href="pcbi.1003330.e014.jpg"></inline-graphic></inline-formula>, homeostasis increases (decreases) the LTD amplitude. The homeostatic time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e015.jpg"></inline-graphic></inline-formula> is the only free parameter of our model.</p><p>We then explore systematically how a particular choice of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e016.jpg"></inline-graphic></inline-formula> affects the stability of the background state in the network. To allow the moving averages to settle, we run the network for an initial period of duration <inline-formula><inline-graphic xlink:href="pcbi.1003330.e017.jpg"></inline-graphic></inline-formula>, during which synaptic updates are not carried out. After that, plasticity is switched on. To check whether the network dynamics remain stable, simulations are run for 24 h of biological time during which we constantly monitor the evolution of the population firing rate (<xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2 A</xref>). The network is considered unstable if the mean population firing rate either drops to zero or increases above <inline-formula><inline-graphic xlink:href="pcbi.1003330.e018.jpg"></inline-graphic></inline-formula> which happens when run-away potentiation occurs (<xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2 B</xref>). By systematically varying the time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e019.jpg"></inline-graphic></inline-formula> in 1 s steps, we find that for the background state to remain stable (<xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2 C</xref>), <inline-formula><inline-graphic xlink:href="pcbi.1003330.e020.jpg"></inline-graphic></inline-formula> must be shorter than some critical value <inline-formula><inline-graphic xlink:href="pcbi.1003330.e021.jpg"></inline-graphic></inline-formula>. Moreover, we find a sharp transition to instability when <inline-formula><inline-graphic xlink:href="pcbi.1003330.e022.jpg"></inline-graphic></inline-formula> is increased beyond <inline-formula><inline-graphic xlink:href="pcbi.1003330.e023.jpg"></inline-graphic></inline-formula>. For <inline-formula><inline-graphic xlink:href="pcbi.1003330.e024.jpg"></inline-graphic></inline-formula> the network has a tendency to fall silent (<xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2 A</xref>, black line).</p><fig id="pcbi-1003330-g002" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1003330.g002</object-id><label>Figure 2</label><caption><title>Network stability during ongoing synaptic plasticity depends crucially on the homeostatic time constant.</title><p>(<bold>A</bold>) Temporal evolution of the average firing rate in the excitatory population for different homeostatic time constants <inline-formula><inline-graphic xlink:href="pcbi.1003330.e025.jpg"></inline-graphic></inline-formula>. Explosion of firing rate indicated by dashed lines. Curves for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e026.jpg"></inline-graphic></inline-formula> (dark blue), <inline-formula><inline-graphic xlink:href="pcbi.1003330.e027.jpg"></inline-graphic></inline-formula> (light blue), and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e028.jpg"></inline-graphic></inline-formula> (turquoise) overlap on the interval from 2 h to 24 h indicating stability. With <inline-formula><inline-graphic xlink:href="pcbi.1003330.e029.jpg"></inline-graphic></inline-formula> (black) we show one of the cases with very short <inline-formula><inline-graphic xlink:href="pcbi.1003330.e030.jpg"></inline-graphic></inline-formula> where the activity spontaneously dies. (<bold>B</bold>) Spike raster of 200 randomly selected excitatory neurons. The last two seconds are shown before the network activity destabilizes (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e031.jpg"></inline-graphic></inline-formula>). (<bold>C</bold>) For <inline-formula><inline-graphic xlink:href="pcbi.1003330.e032.jpg"></inline-graphic></inline-formula>, the activity stays asynchronous and irregular even after 24 h hours of simulated time. (<bold>D</bold>) Firing statistics in a stable network (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e033.jpg"></inline-graphic></inline-formula>) measured after 24 h of simulated time. Histogram of single neuron firing rates (blue) and coefficient of variation (CV ISI, red) across neurons and the ISI distribution of all neurons (yellow). Arrowheads indicate mean values. Black lines represent the corresponding statistics prior to any synaptic modifications (copied from <xref ref-type="fig" rid="pcbi-1003330-g001">Figure 1</xref>). (<bold>E</bold>) Population firing rate for stable simulation runs at <inline-formula><inline-graphic xlink:href="pcbi.1003330.e034.jpg"></inline-graphic></inline-formula> as a function of the homeostatic time constant. The dashed line indicates the target firing rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e035.jpg"></inline-graphic></inline-formula>. (<bold>F</bold>) Evolution of the synaptic weight distribution during the first 8 hours of synaptic plasticity (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e036.jpg"></inline-graphic></inline-formula>).</p></caption><graphic xlink:href="pcbi.1003330.g002"></graphic></fig><p>During stable simulation runs (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e037.jpg"></inline-graphic></inline-formula>), some synapses grow from their initial value <inline-formula><inline-graphic xlink:href="pcbi.1003330.e038.jpg"></inline-graphic></inline-formula> up to the maximum allowed value <inline-formula><inline-graphic xlink:href="pcbi.1003330.e039.jpg"></inline-graphic></inline-formula>, while the rest of the synapses decay to zero. The resulting bimodal distribution of synaptic efficacies (<xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2 F</xref>) remains stable until the end of the run. This is a known phenomenon for purely additive learning rules <xref ref-type="bibr" rid="pcbi.1003330-Toyoizumi1">[33]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Billings1">[34]</xref> and we will see later that unimodal weight distributions arise by the inclusion of a weight decay or by choosing synaptic scaling as the homeostatic mechanism <xref ref-type="bibr" rid="pcbi.1003330-vanRossum1">[35]</xref>.</p><p>Despite the qualitative change in the weight distribution, the inter-spike-interval (ISI) distribution remains largely unaffected, while the coefficient of variation of the ISI distribution (CV ISI) is shifted to slightly higher values (<xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2 D</xref>). However, we noted that the single-neuron average firing rates, which are widely spread out initially, are at the end clustered slightly above the homeostatic target rate of (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e040.jpg"></inline-graphic></inline-formula>) with a weak dependence on the actual value of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e041.jpg"></inline-graphic></inline-formula> (<xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2 E</xref>). This behavior is characteristic for homeostatic firing rate control in single cells.</p><p>We conclude that metaplastic triplet STDP with a homeostatic mechanism as presented here can lead to stable dynamics in models of balanced networks exhibiting asynchronous irregular background activity. However, the timescale <inline-formula><inline-graphic xlink:href="pcbi.1003330.e042.jpg"></inline-graphic></inline-formula> of the homeostatic mechanism critically determines stability. It has to be on the order of seconds to minutes and therefore comparable to the timescale of plasticity itself (here <inline-formula><inline-graphic xlink:href="pcbi.1003330.e043.jpg"></inline-graphic></inline-formula>). This finding is in contrast to most known homeostatic mechanisms that have experimentally been found to act on effective timescales of hours or days <xref ref-type="bibr" rid="pcbi.1003330-Abraham1">[20]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Watt1">[29]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano4">[36]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano5">[37]</xref>.</p></sec><sec id="s2b"><title>Mean field model</title><p>To understand why the critical time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e044.jpg"></inline-graphic></inline-formula> above which homeostasis cannot control plasticity is so short, we here analyze the stability of the background state in a mean field model. In line with the spiking network model we consider a single population of neurons that fires with the mean population firing rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e045.jpg"></inline-graphic></inline-formula> (<xref ref-type="fig" rid="pcbi-1003330-g003">Figure 3 A</xref>). To find an analytic expression that characterizes the response of the background activity to changes in the recurrent weights <inline-formula><inline-graphic xlink:href="pcbi.1003330.e046.jpg"></inline-graphic></inline-formula> around the initial value <inline-formula><inline-graphic xlink:href="pcbi.1003330.e047.jpg"></inline-graphic></inline-formula>, we begin with a linear neuron model<disp-formula id="pcbi.1003330.e048"><graphic xlink:href="pcbi.1003330.e048"></graphic><label>(2)</label></disp-formula>with the offset <inline-formula><inline-graphic xlink:href="pcbi.1003330.e049.jpg"></inline-graphic></inline-formula> and the slope parameter <inline-formula><inline-graphic xlink:href="pcbi.1003330.e050.jpg"></inline-graphic></inline-formula>. Since we are interested in weight changes around the initial value <inline-formula><inline-graphic xlink:href="pcbi.1003330.e051.jpg"></inline-graphic></inline-formula>, the natural choice for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e052.jpg"></inline-graphic></inline-formula> would be <inline-formula><inline-graphic xlink:href="pcbi.1003330.e053.jpg"></inline-graphic></inline-formula>. However, here we set <inline-formula><inline-graphic xlink:href="pcbi.1003330.e054.jpg"></inline-graphic></inline-formula> to take into account the recurrent feed-back. This choice makes <inline-formula><inline-graphic xlink:href="pcbi.1003330.e055.jpg"></inline-graphic></inline-formula> dimensionless while <inline-formula><inline-graphic xlink:href="pcbi.1003330.e056.jpg"></inline-graphic></inline-formula> is measured in units of Hz. Because weights evolve slowly, while population dynamics are fast we can solve for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e057.jpg"></inline-graphic></inline-formula> and obtain the self-consistent solution<disp-formula id="pcbi.1003330.e058"><graphic xlink:href="pcbi.1003330.e058"></graphic><label>(3)</label></disp-formula></p><fig id="pcbi-1003330-g003" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1003330.g003</object-id><label>Figure 3</label><caption><title>Mean field theory predicts the stability of background activity.</title><p>(<bold>A</bold>) Schematic of the mean field model. Plastic synapses are indicated by <bold>*</bold>. (<bold>B</bold>) Eigenvalues of the Jacobian evaluated at the non-trivial fixed point <inline-formula><inline-graphic xlink:href="pcbi.1003330.e059.jpg"></inline-graphic></inline-formula>. (<bold>C</bold>) Phase portrait for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e060.jpg"></inline-graphic></inline-formula>, a choice where background activity is stable. Nullclines are drawn in black. Arrows indicate the direction of the flow. Two prototypical trajectories starting close to <inline-formula><inline-graphic xlink:href="pcbi.1003330.e061.jpg"></inline-graphic></inline-formula> are shown. Blue line: Typical example of a solution that returns to the stable fixed point. Solutions starting in the shaded area, such as the red line, diverge to infinity. (<bold>D</bold>) The separatrix for four different values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e062.jpg"></inline-graphic></inline-formula>. (<bold>E</bold>) Population firing rate of the spiking network model (simulations: red dots) for different values of weight <inline-formula><inline-graphic xlink:href="pcbi.1003330.e063.jpg"></inline-graphic></inline-formula> for connections from excitatory to excitatory neurons. Black line: Least-square fit of <xref ref-type="disp-formula" rid="pcbi.1003330.e058">Eq. (3)</xref> on the interval <inline-formula><inline-graphic xlink:href="pcbi.1003330.e064.jpg"></inline-graphic></inline-formula> as indicated by the black bar. Extracted parameters are <inline-formula><inline-graphic xlink:href="pcbi.1003330.e065.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e066.jpg"></inline-graphic></inline-formula> (cf. <xref ref-type="disp-formula" rid="pcbi.1003330.e058">Eq. (3)</xref>).</p></caption><graphic xlink:href="pcbi.1003330.g003"></graphic></fig><p>As we will show later, a better qualitative fit to the spiking model can be achieved with this heuristic, which will facilitate finding the right parameters <inline-formula><inline-graphic xlink:href="pcbi.1003330.e067.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e068.jpg"></inline-graphic></inline-formula>.</p><p>To introduce plasticity into the mean field model, we use the corresponding rate-based plasticity rule<disp-formula id="pcbi.1003330.e069"><graphic xlink:href="pcbi.1003330.e069"></graphic><label>(4)</label></disp-formula>which can be directly derived from the triplet STDP rule <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref> and also can be interpreted as a BCM model <xref ref-type="bibr" rid="pcbi.1003330-Bienenstock1">[15]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Gjorgjieva1">[38]</xref>. Here, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e070.jpg"></inline-graphic></inline-formula> is the relative learning rate and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e071.jpg"></inline-graphic></inline-formula> sets the scale of the system. The second equality in <xref ref-type="disp-formula" rid="pcbi.1003330.e069">Eq. (4)</xref> follows because in the recurrent model pre- and postsynaptic rates are the same (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e072.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e073.jpg"></inline-graphic></inline-formula>). The function <inline-formula><inline-graphic xlink:href="pcbi.1003330.e074.jpg"></inline-graphic></inline-formula> scales the strength of LTD relative to LTP just as in the spiking case (cf. <xref ref-type="sec" rid="s4">Methods</xref>, <xref ref-type="disp-formula" rid="pcbi.1003330.e316">Eq. (18)</xref>). In the mean field model, the rate detector <inline-formula><inline-graphic xlink:href="pcbi.1003330.e075.jpg"></inline-graphic></inline-formula> (<xref ref-type="disp-formula" rid="pcbi.1003330.e008">Eq. (1)</xref>) becomes the low pass filtered version of the population firing rate<disp-formula id="pcbi.1003330.e076"><graphic xlink:href="pcbi.1003330.e076"></graphic><label>(5)</label></disp-formula></p><p>To link the network dynamics with synaptic plasticity we take the derivative of <xref ref-type="disp-formula" rid="pcbi.1003330.e058">Eq. (3)</xref>, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e077.jpg"></inline-graphic></inline-formula> and combine it with <xref ref-type="disp-formula" rid="pcbi.1003330.e069">Eq. (4)</xref> to arrive at<disp-formula id="pcbi.1003330.e078"><graphic xlink:href="pcbi.1003330.e078"></graphic><label>(6)</label></disp-formula>which describes the temporal evolution of the mean firing rate as governed by synaptic plasticity. Taken together, <xref ref-type="disp-formula" rid="pcbi.1003330.e076">equations (5)</xref> and <xref ref-type="disp-formula" rid="pcbi.1003330.e078">(6)</xref> define a two-dimensional dynamical system with two fixed points. One lies at <inline-formula><inline-graphic xlink:href="pcbi.1003330.e079.jpg"></inline-graphic></inline-formula> and represents the quiescent network. The remaining non-trivial fixed point is <inline-formula><inline-graphic xlink:href="pcbi.1003330.e080.jpg"></inline-graphic></inline-formula>, which we interpret as the network in its background state.</p><p>Given these choices, we now ask whether this fixed point can be linearly stable (<xref ref-type="sec" rid="s4">Methods</xref>) and find that the stability of the background state requires<disp-formula id="pcbi.1003330.e081"><graphic xlink:href="pcbi.1003330.e081"></graphic><label>(7)</label></disp-formula></p><p>For <inline-formula><inline-graphic xlink:href="pcbi.1003330.e082.jpg"></inline-graphic></inline-formula> infinitesimal excursions from the fixed point diverge, which corresponds to run-away potentiation in this model. We note that <inline-formula><inline-graphic xlink:href="pcbi.1003330.e083.jpg"></inline-graphic></inline-formula> crucially depends on the parameters <inline-formula><inline-graphic xlink:href="pcbi.1003330.e084.jpg"></inline-graphic></inline-formula>, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e085.jpg"></inline-graphic></inline-formula>, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e086.jpg"></inline-graphic></inline-formula>, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e087.jpg"></inline-graphic></inline-formula> and the target rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e088.jpg"></inline-graphic></inline-formula>. However, we can rescale the system to natural units, by expressing firing rates in units of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e089.jpg"></inline-graphic></inline-formula> and time in units of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e090.jpg"></inline-graphic></inline-formula>, and plot the eigenvalues as a function of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e091.jpg"></inline-graphic></inline-formula> (<xref ref-type="fig" rid="pcbi-1003330-g003">Figure 3 B</xref>). The fact that the fixed point of background activity loses stability for too large values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e092.jpg"></inline-graphic></inline-formula> is in good qualitative agreement with what we observe in the spiking model. One should further note that <xref ref-type="disp-formula" rid="pcbi.1003330.e081">Eq. (7)</xref> is independent of the power of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e093.jpg"></inline-graphic></inline-formula> appearing in <inline-formula><inline-graphic xlink:href="pcbi.1003330.e094.jpg"></inline-graphic></inline-formula>, as long as the fixed point of background activity exists (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e095.jpg"></inline-graphic></inline-formula>) and under the condition that at criticality the imaginary parts of the eigenvalues are always non-vanishing (see <xref ref-type="sec" rid="s4">Methods</xref>). This indicates the presence of oscillations which are indeed observed in the spiking network (cf. <xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2 A</xref>, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e096.jpg"></inline-graphic></inline-formula>). The fact that the network falls silent for very small values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e097.jpg"></inline-graphic></inline-formula> (e.g. <inline-formula><inline-graphic xlink:href="pcbi.1003330.e098.jpg"></inline-graphic></inline-formula> in <xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2 A</xref>) is not captured by the mean field model.</p><p>We can make further use of the mean field model to qualitatively understand the behavior of the system far from equilibrium. <xref ref-type="fig" rid="pcbi-1003330-g003">Figure 3 C</xref> shows the phase plane of a network with a stable fixed point (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e099.jpg"></inline-graphic></inline-formula>). When the system is driven away from it, and perturbations are small, the dynamics converge back towards the fixed point. However, when excursions become too large, the network activity diverges (compare <xref ref-type="fig" rid="pcbi-1003330-g003">Figure 3 C</xref>, dotted solution) since the fixed point of background activity is only locally stable. A numerical analysis shows that the basin of attraction is small when <inline-formula><inline-graphic xlink:href="pcbi.1003330.e100.jpg"></inline-graphic></inline-formula> approaches <inline-formula><inline-graphic xlink:href="pcbi.1003330.e101.jpg"></inline-graphic></inline-formula> from below (<xref ref-type="fig" rid="pcbi-1003330-g003">Figure 3 D</xref>). Hence the system is very sensitive to perturbations which easily lead to run-away potentiation. Although we expect the basin of attraction of the mean-field model and the spiking model only to be comparably where <xref ref-type="disp-formula" rid="pcbi.1003330.e058">Eq. (3)</xref> describes the firing rates of the spiking network accurately we can assume that for robust stability <inline-formula><inline-graphic xlink:href="pcbi.1003330.e102.jpg"></inline-graphic></inline-formula> has to be satisfied.</p></sec><sec id="s2c"><title>Model comparison</title><p>To be able to make more quantitative predictions for the spiking network we have to choose values for the parameters on the right hand side of <xref ref-type="disp-formula" rid="pcbi.1003330.e081">Eq. (7)</xref>. These are the effective timescale of plasticity <inline-formula><inline-graphic xlink:href="pcbi.1003330.e103.jpg"></inline-graphic></inline-formula> on the one hand, and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e104.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e105.jpg"></inline-graphic></inline-formula>, which characterize the network dynamics, on the other hand. We will now show that the latter can be determined from the static network model, which is independent of plasticity. Note that the parameters <inline-formula><inline-graphic xlink:href="pcbi.1003330.e106.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e107.jpg"></inline-graphic></inline-formula> in our mean field model are shared with the spiking model which we will use to quantitatively compare the two.</p><p>First, we relate the variables <inline-formula><inline-graphic xlink:href="pcbi.1003330.e108.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e109.jpg"></inline-graphic></inline-formula> to the response of the spiking network when all its EE synapses are modified. Since this is not feasible analytically, we extract the response numerically by systematically varying the EE weights around the initial state with <inline-formula><inline-graphic xlink:href="pcbi.1003330.e110.jpg"></inline-graphic></inline-formula>. While doing so, plasticity is disabled and we record the steady state population rate of the network (<xref ref-type="fig" rid="pcbi-1003330-g003">Figure 3 E</xref>). We then minimize the mean square error for <xref ref-type="disp-formula" rid="pcbi.1003330.e058">Eq. (3)</xref> over a small interval <inline-formula><inline-graphic xlink:href="pcbi.1003330.e111.jpg"></inline-graphic></inline-formula> and determine the following values: <inline-formula><inline-graphic xlink:href="pcbi.1003330.e112.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e113.jpg"></inline-graphic></inline-formula>. For the stability analysis only the derivative of <xref ref-type="disp-formula" rid="pcbi.1003330.e058">Eq. (3)</xref> at <inline-formula><inline-graphic xlink:href="pcbi.1003330.e114.jpg"></inline-graphic></inline-formula> matters. However, it is worth noting that the response of the balanced network is well captured by <xref ref-type="disp-formula" rid="pcbi.1003330.e058">Eq. (3)</xref> over a much wider range than the one used for the fit. This behavior is an expected consequence of the balanced state, which is known to linearize network responses <xref ref-type="bibr" rid="pcbi.1003330-vanVreeswijk1">[6]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-VanVreeswijk1">[39]</xref>. Our approximation by a linear rate model breaks down for higher rates since it does not incorporate refractory effects.</p><p>Second, under the assumption of independent and irregular firing in the background state, the plasticity time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e115.jpg"></inline-graphic></inline-formula> is fully determined by the target rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e116.jpg"></inline-graphic></inline-formula> and known parameters of the triplet STDP model (see <xref ref-type="sec" rid="s4">Methods</xref> and <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>). For <inline-formula><inline-graphic xlink:href="pcbi.1003330.e117.jpg"></inline-graphic></inline-formula> we find <inline-formula><inline-graphic xlink:href="pcbi.1003330.e118.jpg"></inline-graphic></inline-formula>.</p><p>Using these results together with <xref ref-type="disp-formula" rid="pcbi.1003330.e081">Eq. (7)</xref> we predict the critical timescale of homeostasis for different values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e119.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e120.jpg"></inline-graphic></inline-formula> and compare it to the results that we obtain as before from direct simulations of the spiking network. <xref ref-type="fig" rid="pcbi-1003330-g004">Figure 4 A</xref> shows that the dependence of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e121.jpg"></inline-graphic></inline-formula> on the learning rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e122.jpg"></inline-graphic></inline-formula> is remarkably well captured by the mean field model. The fourth power dependence on the background firing rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e123.jpg"></inline-graphic></inline-formula> is described well for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e124.jpg"></inline-graphic></inline-formula> (<xref ref-type="fig" rid="pcbi-1003330-g004">Figure 4 B</xref>), but the theory fails for smaller values, where we start to observe synchronous events in the population activity, which introduce correlations that are not taken into account in the mean field approach. In <xref ref-type="fig" rid="pcbi-1003330-g004">Figure 4 C</xref> we plot the typical lifetimes (i.e. the time when the spiking simulations are stopped, because they either show run-away potentiation or the maximum simulated time <inline-formula><inline-graphic xlink:href="pcbi.1003330.e125.jpg"></inline-graphic></inline-formula> is reached) as a function of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e126.jpg"></inline-graphic></inline-formula>. The figure illustrates nicely that the critical time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e127.jpg"></inline-graphic></inline-formula> coincides with the sharp transition in lifetimes observed in the spiking network.</p><fig id="pcbi-1003330-g004" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1003330.g004</object-id><label>Figure 4</label><caption><title>The mean field predictions agree with results from direct simulation of the spiking network.</title><p>(<bold>A</bold>) Solid line: <inline-formula><inline-graphic xlink:href="pcbi.1003330.e128.jpg"></inline-graphic></inline-formula> as a function of the learning rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e129.jpg"></inline-graphic></inline-formula> (cf. <xref ref-type="disp-formula" rid="pcbi.1003330.e081">Eq. (7)</xref>), with simulation data (red points) for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e130.jpg"></inline-graphic></inline-formula>. The arrow indicates the value used throughout the rest of this figure (the dotted line corresponds to the learning rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e131.jpg"></inline-graphic></inline-formula> as used in <xref ref-type="supplementary-material" rid="pcbi.1003330.s001">Figure S1</xref>). (<bold>B</bold>) Same as before but as a function of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e132.jpg"></inline-graphic></inline-formula> for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e133.jpg"></inline-graphic></inline-formula> fixed. (<bold>C</bold>) Lifetime values for the spiking network (red points) with a scaled step function as predicted by mean field theory (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e134.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e135.jpg"></inline-graphic></inline-formula>). All error bars are smaller than the data points.</p></caption><graphic xlink:href="pcbi.1003330.g004"></graphic></fig><p>When running additional simulations with smaller learning rates (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e136.jpg"></inline-graphic></inline-formula> as opposed to <inline-formula><inline-graphic xlink:href="pcbi.1003330.e137.jpg"></inline-graphic></inline-formula>) we observe that the network destabilizes occasionally for values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e138.jpg"></inline-graphic></inline-formula> smaller than <inline-formula><inline-graphic xlink:href="pcbi.1003330.e139.jpg"></inline-graphic></inline-formula>, but only after 22 h of activity (see <xref ref-type="supplementary-material" rid="pcbi.1003330.s001">Figure S1</xref>). We find, however, that this “late” instability can be avoided by either initializing the EE weights with a weight matrix obtained from a stable run (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e140.jpg"></inline-graphic></inline-formula> at <inline-formula><inline-graphic xlink:href="pcbi.1003330.e141.jpg"></inline-graphic></inline-formula>) or by reducing the maximally allowed synaptic weight (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e142.jpg"></inline-graphic></inline-formula>). Since these changes do not affect the “early” instability (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e143.jpg"></inline-graphic></inline-formula>), the “late” instability seems to have a different origin and might be linked to the spontaneous emergence of structure in the network.</p><p>Here we focus on the “early” instability which is seen in all simulations that do not respect the analytical criterion <inline-formula><inline-graphic xlink:href="pcbi.1003330.e144.jpg"></inline-graphic></inline-formula>, after less than one hour of biological time, and therefore puts a severe stability constraint on <inline-formula><inline-graphic xlink:href="pcbi.1003330.e145.jpg"></inline-graphic></inline-formula>. Moreover the theory is able to quantitatively confirm the timescale <inline-formula><inline-graphic xlink:href="pcbi.1003330.e146.jpg"></inline-graphic></inline-formula> emerging from the spiking network simulations and allows us to see the detailed parameter dependence. In particular for a background rate of 3 Hz and the learning rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e147.jpg"></inline-graphic></inline-formula> we find a critical timescale of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e148.jpg"></inline-graphic></inline-formula> (simulations: <inline-formula><inline-graphic xlink:href="pcbi.1003330.e149.jpg"></inline-graphic></inline-formula>, mean field model: <inline-formula><inline-graphic xlink:href="pcbi.1003330.e150.jpg"></inline-graphic></inline-formula>).</p><p>In summary, our mean field model discussed here makes accurate quantitative predictions about the stability of a large spiking network model with plastic synapses for a given timescale of homeostasis. Furthermore it gives useful insights into parameter dependencies which are computationally costly to obtain from parameter sweeps in simulations of spiking networks. Our theory confirms that metaplastic triplet STDP with biological learning rates has to be matched by a homeostatic mechanism that acts on a timescale of seconds to minutes. In the next sections we will show that the mean field framework described here can be readily extended to other forms of homeostasis.</p></sec><sec id="s2d"><title>Weight decay</title><p>The induction of synaptic plasticity is only a first step towards the formation of long-term memory. In the absence of neuromodulators necessary to consolidate early LTP into late LTP, these modifications have been found to decay away with a time constant of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e151.jpg"></inline-graphic></inline-formula><xref ref-type="bibr" rid="pcbi.1003330-Frey1">[40]</xref>. To study the effect of a slow synaptic decay on the stability of the background state we focus on the early phase of plasticity. In particular we neglect consolidation in the model and introduce a slow decay term<disp-formula id="pcbi.1003330.e152"><graphic xlink:href="pcbi.1003330.e152"></graphic><label>(8)</label></disp-formula>where we already replaced the STDP rule by its equivalent rate based rule (see <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref> and <xref ref-type="sec" rid="s4">Methods</xref>, <xref ref-type="disp-formula" rid="pcbi.1003330.e279">Eq. (17)</xref>), while the effect of the decay term can be written identically in the rate based model and the STDP model. Note that for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e153.jpg"></inline-graphic></inline-formula> we retrieve the model studied in <xref ref-type="fig" rid="pcbi-1003330-g001">Figures 1</xref>–<xref ref-type="fig" rid="pcbi-1003330-g004">4</xref>. Again we determine the critical timescale of homeostasis in numerical simulations of the spiking network by systematically varying <inline-formula><inline-graphic xlink:href="pcbi.1003330.e154.jpg"></inline-graphic></inline-formula> for different values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e155.jpg"></inline-graphic></inline-formula>. We further find that the slow weight decay causes the synaptic weights to stabilize in a unimodal distribution (<xref ref-type="fig" rid="pcbi-1003330-g005">Figure 5 A and B</xref>) which is fundamentally different to what we observed for the decay-free case. However, the critical time constant of homeostasis <inline-formula><inline-graphic xlink:href="pcbi.1003330.e156.jpg"></inline-graphic></inline-formula> is only marginally larger than in the decay-free case (<xref ref-type="fig" rid="pcbi-1003330-g005">Figure 5 C</xref>).</p><fig id="pcbi-1003330-g005" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1003330.g005</object-id><label>Figure 5</label><caption><title>Slow synaptic weight decay renders weight distribution unimodal, but hardly affects global stability.</title><p>(<bold>A</bold>) Evolution of the synaptic weight distribution over 8 h of background activity. (<bold>B</bold>) Synaptic weight distribution at <inline-formula><inline-graphic xlink:href="pcbi.1003330.e157.jpg"></inline-graphic></inline-formula>. (<bold>C</bold>) Predictions for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e158.jpg"></inline-graphic></inline-formula>of mean field theory (solid line) and values obtained from direct simulation (points). (<bold>D</bold>) Final population firing rate as a function of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e159.jpg"></inline-graphic></inline-formula> for values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e160.jpg"></inline-graphic></inline-formula> where the background state is a stable fixed point (dashed line: target rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e161.jpg"></inline-graphic></inline-formula>; error bars: standard deviation over 100 bins of 1 s).</p></caption><graphic xlink:href="pcbi.1003330.g005"></graphic></fig><p>To assess the impact of the decay on the critical timescale, the mean field approach, as it was derived above, can be adapted to take into account the constant synaptic decay (<xref ref-type="sec" rid="s4">Methods</xref>). Provided the decay time constant is sufficiently long, we find the critical time constant to be<disp-formula id="pcbi.1003330.e162"><graphic xlink:href="pcbi.1003330.e162"></graphic><label>(9)</label></disp-formula>which is in good agreement with the results from direct simulations (<xref ref-type="fig" rid="pcbi-1003330-g005">Figure 5 C</xref>). From <xref ref-type="disp-formula" rid="pcbi.1003330.e162">Eq. (9)</xref> we can further confirm that the decay term only causes a small positive shift in the critical time constant as it was also observed in the spiking network. Furthermore, we see that the population firing rate settles to values closer to the actual target rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e163.jpg"></inline-graphic></inline-formula> (<xref ref-type="fig" rid="pcbi-1003330-g005">Figure 5 D</xref>) than this was the case in the decay-free scenario.</p><p>In summary, adding a slow synaptic weight decay to the plasticity model is sufficient to cause substantial change to the steady state weight distribution in the network. Nevertheless this slow process does not affect the need for a rapid homeostatic mechanism.</p></sec><sec id="s2e"><title>Synaptic scaling</title><p>To test whether the previous findings are limited to our particular choice of metaplastic homeostatic mechanism, or whether they are also meaningful in the case of synaptic scaling <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano1">[21]</xref> we now adapt the model by van Rossum et al. <xref ref-type="bibr" rid="pcbi.1003330-vanRossum1">[35]</xref> and combine it with triplet STDP<disp-formula id="pcbi.1003330.e164"><graphic xlink:href="pcbi.1003330.e164"></graphic><label>(10)</label></disp-formula>where the rate of LTD is fixed in the triplet term (cf. <xref ref-type="disp-formula" rid="pcbi.1003330.e279">Eq. (17)</xref>) and synaptic scaling is the only form of homeostasis. One important difference to the previous metaplastic STDP model is the addition of the scaling time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e165.jpg"></inline-graphic></inline-formula> which controls the timescale of synaptic scaling. In the metaplastic model we analyzed above, this time constant is implicit since it is the same as the one of plasticity (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e166.jpg"></inline-graphic></inline-formula>). In contrast to the original model of synaptic scaling (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e167.jpg"></inline-graphic></inline-formula><xref ref-type="bibr" rid="pcbi.1003330-vanRossum1">[35]</xref>) here we choose <inline-formula><inline-graphic xlink:href="pcbi.1003330.e168.jpg"></inline-graphic></inline-formula> to avoid additional unstable fixed points in the phase plane (<xref ref-type="fig" rid="pcbi-1003330-g006">Figure 6 D</xref>).</p><fig id="pcbi-1003330-g006" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1003330.g006</object-id><label>Figure 6</label><caption><title>Triplet STDP with synaptic scaling requires a fast rate detector.</title><p>(<bold>A</bold>) Black line: Eigenvalues of the Jacobian (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e169.jpg"></inline-graphic></inline-formula>) for different values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e170.jpg"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e171.jpg"></inline-graphic></inline-formula>). Gray curve: Values from <xref ref-type="fig" rid="pcbi-1003330-g003">Figure 3 B</xref> for reference. The red line (“sim”) indicates the critical value as obtained from simulating the full spiking network. (<bold>B</bold>) As before, but for different values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e172.jpg"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e173.jpg"></inline-graphic></inline-formula>). (<bold>C</bold>) Lifetimes of the background state in simulated networks of spiking neurons for different values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e174.jpg"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e175.jpg"></inline-graphic></inline-formula>). (<bold>D</bold>) Phase plane with nullclines. <inline-formula><inline-graphic xlink:href="pcbi.1003330.e176.jpg"></inline-graphic></inline-formula>-nullcline in black; <inline-formula><inline-graphic xlink:href="pcbi.1003330.e177.jpg"></inline-graphic></inline-formula>-nullclines: dashed (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e178.jpg"></inline-graphic></inline-formula>), gray (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e179.jpg"></inline-graphic></inline-formula>) and red (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e180.jpg"></inline-graphic></inline-formula>). The latter was used in the rest of the figure. (<bold>E</bold>) Synaptic weight distribution after <inline-formula><inline-graphic xlink:href="pcbi.1003330.e181.jpg"></inline-graphic></inline-formula> of simulation.</p></caption><graphic xlink:href="pcbi.1003330.g006"></graphic></fig><p>Bearing this in mind we move on to linearizing the system around the fixed point of background activity (<xref ref-type="sec" rid="s4">Methods</xref>). We find that for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e182.jpg"></inline-graphic></inline-formula> the eigenvalues of the linearized system qualitatively have the same shape as for the plasticity rule with homeostatically modulated LTD (<xref ref-type="fig" rid="pcbi-1003330-g006">Figure 6 A</xref>). In fact for sensible values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e183.jpg"></inline-graphic></inline-formula>, the stability condition is exactly the same: <inline-formula><inline-graphic xlink:href="pcbi.1003330.e184.jpg"></inline-graphic></inline-formula> (cf. <xref ref-type="disp-formula" rid="pcbi.1003330.e081">Eq. (7)</xref>). However, in the case of synaptic scaling <xref ref-type="disp-formula" rid="pcbi.1003330.e081">Eq. (7)</xref> represents a necessary, but not a sufficient condition for stability. For too large values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e185.jpg"></inline-graphic></inline-formula> stability is lost also in the case of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e186.jpg"></inline-graphic></inline-formula> (<xref ref-type="fig" rid="pcbi-1003330-g006">Figure 6 B</xref>). On the other hand decreasing <inline-formula><inline-graphic xlink:href="pcbi.1003330.e187.jpg"></inline-graphic></inline-formula> indefinitely leads to oscillations without any further effect on stability (see <xref ref-type="sec" rid="s4">Methods</xref> and <xref ref-type="bibr" rid="pcbi.1003330-vanRossum1">[35]</xref>).</p><p>To compare these findings with the equivalent STDP rule we perform numerical simulations with the full spiking network in which we set <inline-formula><inline-graphic xlink:href="pcbi.1003330.e188.jpg"></inline-graphic></inline-formula> and choose <inline-formula><inline-graphic xlink:href="pcbi.1003330.e189.jpg"></inline-graphic></inline-formula> on the order of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e190.jpg"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e191.jpg"></inline-graphic></inline-formula>). By changing <inline-formula><inline-graphic xlink:href="pcbi.1003330.e192.jpg"></inline-graphic></inline-formula> systematically (<xref ref-type="fig" rid="pcbi-1003330-g006">Figure 6 C</xref>) we determine the critical value to be smaller than predicted (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e193.jpg"></inline-graphic></inline-formula>), but within the same order of magnitude (<xref ref-type="fig" rid="pcbi-1003330-g006">Figure 6 A,C</xref>). Conversely when we start with <inline-formula><inline-graphic xlink:href="pcbi.1003330.e194.jpg"></inline-graphic></inline-formula> held fixed, we determine the critical value of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e195.jpg"></inline-graphic></inline-formula> to be on the same order as <inline-formula><inline-graphic xlink:href="pcbi.1003330.e196.jpg"></inline-graphic></inline-formula> (<xref ref-type="fig" rid="pcbi-1003330-g006">Figure 6 B</xref>). At the end of a stable simulation run (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e197.jpg"></inline-graphic></inline-formula>) we find that synaptic weights have formed a unimodal distribution (<xref ref-type="fig" rid="pcbi-1003330-g006">Figure 6 E</xref>), an expected behavior of synaptic scaling <xref ref-type="bibr" rid="pcbi.1003330-vanRossum1">[35]</xref>.</p><p>In summary we have shown here that a fast rate detector is necessary to produce fast homeostatic responses to guarantee stable network dynamics also for the case of synaptic scaling. Although the quantitative agreement between the mean field model and the full spiking simulation is less accurate than in the case of for the metaplastic model above, both models confirm that the rate detector has to act on a timescale of seconds to minutes. Furthermore the time constant of the scaling term <inline-formula><inline-graphic xlink:href="pcbi.1003330.e198.jpg"></inline-graphic></inline-formula> has to be comparable to the time scale of plasticity (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e199.jpg"></inline-graphic></inline-formula>) or stability is compromised, when <inline-formula><inline-graphic xlink:href="pcbi.1003330.e200.jpg"></inline-graphic></inline-formula> is chosen too large (and oscillations occur, when chosen too small).</p></sec></sec><sec id="s3"><title>Discussion</title><p>In this paper we have shown that a realistic additive triplet STDP rule <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref> can sustain a stable background state in balanced networks provided there is a homeostatic mechanism with a fast rate detector that acts on a timescale of seconds to minutes. We confirmed this result in a generic two dimensional mean field model in which the stability of the background state is interpreted as the linear stability of a non-zero fixed point of the system for which the timescale of the homeostatic rate detector <inline-formula><inline-graphic xlink:href="pcbi.1003330.e201.jpg"></inline-graphic></inline-formula> plays the role of a bifurcation parameter. These results are generic, i.e. independent of model details. In particular, we showed that similar results are obtained for triplet STDP with a form of metaplastic homeostasis, where homeostasis was implemented as a modulation of the LTD rate, or alternatively in combination with synaptic scaling. The mean field formalism produces accurate quantitative predictions for metaplastic triplet STDP. Although, in the case of triplet STDP in combination with synaptic scaling, the match of mean field model and direct simulations was less accurate, both support the notion that a fast rate detector is required for stability. For the case of synaptic scaling we found additionally that the homeostatic changes have to be implemented on a timescale comparable to the one of plasticity itself (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e202.jpg"></inline-graphic></inline-formula>), which is fast compared to most homeostatic mechanisms reported in the experimental literature, but consistent with earlier simulation studies that used fast homeostasis <xref ref-type="bibr" rid="pcbi.1003330-vonderMalsburg1">[13]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Miller1">[16]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Clopath1">[19]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-vanRossum1">[35]</xref>.</p><sec id="s3a"><title>Homeostasis and plasticity</title><p>The fact that Hebbian learning has to be opposed by some kind of compensatory mechanism has long been known <xref ref-type="bibr" rid="pcbi.1003330-vonderMalsburg1">[13]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Miller1">[16]</xref> and such mechanisms indeed have been found <xref ref-type="bibr" rid="pcbi.1003330-Abraham1">[20]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano4">[36]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Huang1">[41]</xref>. In the following we will briefly review the different types of homeostasis affecting synaptic weights and how they relate to what was used in the present study.</p><p>Homeostasis can be classified in two main categories. We call models “weight homeostasis” if they try to keep all afferent weights into a cell normalized <xref ref-type="bibr" rid="pcbi.1003330-vonderMalsburg1">[13]</xref>. Such models have been criticized because they are non-local <xref ref-type="bibr" rid="pcbi.1003330-Bienenstock1">[15]</xref>, i.e. they require cell wide spatial averaging over synapses, which can only be achieved in a plausible way if all synaptic weights decay at a global rate modulated by the total afferent synaptic strength <xref ref-type="bibr" rid="pcbi.1003330-Miller1">[16]</xref>. To avoid this, “rate homeostasis” models have been proposed <xref ref-type="bibr" rid="pcbi.1003330-Bienenstock1">[15]</xref> which strive to maintain a certain postsynaptic firing rate. This approach, which we chose in the present study, has more experimental support <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano3">[28]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Watt1">[29]</xref>. In contrast to the spatial filtering as described above, this mechanism requires temporal filtering of the postsynaptic rate over a given time window (represented by <inline-formula><inline-graphic xlink:href="pcbi.1003330.e203.jpg"></inline-graphic></inline-formula> in this study). We can further distinguish between two principal types of homeostasis. A homeostatic mechanism can either act on the synaptic weights directly (e.g. synaptic scaling), or indirectly through metaplasticity <xref ref-type="bibr" rid="pcbi.1003330-Abraham1">[20]</xref>, by changing parameters of the plasticity model over time. The former, direct form of homeostasis allows for synaptic changes even in the absence of activity as it is seen in synaptic scaling experiments <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano1">[21]</xref> on a timescale of days. This is in contrast to theoretical models that apply scaling by algorithmically enforcing weight normalization <xref ref-type="bibr" rid="pcbi.1003330-vonderMalsburg1">[13]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Lazar1">[18]</xref> on the timescale of one or a few simulation time-steps.</p><p>In our study we looked at both approaches. In the metaplastic triplet STDP model homeostasis manifest itself as a shift in the plasticity threshold between LTD and LTP <xref ref-type="bibr" rid="pcbi.1003330-Clopath1">[19]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Clopath2">[42]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-ElBoustani1">[43]</xref>. This is achieved by modulating the rate of LTD induction using the temporal average of the postsynaptic firing rates over a given time window (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e204.jpg"></inline-graphic></inline-formula>). As we have shown, this average has to follow the neuronal spiking activity very rapidly, meaning that plasticity parameters change on a short timescale, which is comparable to the duration of many standard STDP protocols <xref ref-type="bibr" rid="pcbi.1003330-Sjstrm1">[26]</xref>. We therefore predict that if biological circuits rely on such a metaplastic homeostatic mechanism, weight changes are different for cells that are silent prior to a plasticity induction than for cells that have been primed by postsynaptic firing (over an extended period before the induction protocol). In <xref ref-type="fig" rid="pcbi-1003330-g007">Figure 7 A</xref> we demonstrate this idea in the model of metaplastic triplet STDP (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e205.jpg"></inline-graphic></inline-formula>) for a typical LTD induction protocol (75 pairs at 5 Hz with −10 ms spike offset). <xref ref-type="fig" rid="pcbi-1003330-g007">Figure 7 B</xref> shows the relative differences between primed and unprimed experiments in dependence of the length of the priming duration or the priming frequency respectively. Since this plasticity rule implements homeostasis as an activity dependent change of the LTD learning rate, the amount of LTD changes dramatically while LTP is unaffected by priming. However, we expect that the main results of our mean field analysis also hold for cases in which LTP is affected, as long as the net synaptic weight change decreases with the intensity of priming. In either case the functional form of the dependence allows us to draw conclusions on the order of magnitude of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e206.jpg"></inline-graphic></inline-formula> and the exponent of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e207.jpg"></inline-graphic></inline-formula> appearing in <inline-formula><inline-graphic xlink:href="pcbi.1003330.e208.jpg"></inline-graphic></inline-formula> (cf. <xref ref-type="disp-formula" rid="pcbi.1003330.e316">Eq. 18</xref>). Conversely, if homeostasis was exclusively mediated by synaptic scaling, we would expect that it manifests as a heterosynaptic effect. Its impact, however, would likely be smaller than in the case of metaplastic triplet STDP, because synaptic scaling does not have an explicit dependence on the presynaptic firing rate.</p><fig id="pcbi-1003330-g007" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1003330.g007</object-id><label>Figure 7</label><caption><title>Postsynaptic priming affects STDP protocols.</title><p>Simulation of the metaplastic triplet STDP rule <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>. (<bold>A</bold>) Top: Typical protocol for the induction of LTD (75 pairs (post-pre) at 5 Hz with −10 ms spike offset) in the triplet STDP model (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e209.jpg"></inline-graphic></inline-formula>) with a postsynaptic cell which is quiescent prior to the LTD protocol (black) compared to induction after postsynaptic priming (blue). Top, left: Pre- and postsynaptic spikes for priming and. Top, right: LTD induction. Middle: postsynaptic rate estimate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e210.jpg"></inline-graphic></inline-formula> of the postsynaptic cell. Bottom: Weight change <inline-formula><inline-graphic xlink:href="pcbi.1003330.e211.jpg"></inline-graphic></inline-formula> over time. Postsynaptic priming period (duration 100 s): regular firing at <inline-formula><inline-graphic xlink:href="pcbi.1003330.e212.jpg"></inline-graphic></inline-formula> terminated by one second of silence (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e213.jpg"></inline-graphic></inline-formula>) to avoid triplet effects. (<bold>B</bold>) Relative differences in final weight change between quiet (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e214.jpg"></inline-graphic></inline-formula>) and primed protocol (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e215.jpg"></inline-graphic></inline-formula>) at the end of a LTD (gray) plasticity protocol. LTP protocol for reference (hollow, same paring protocol, with reversed timing, +10 ms spike offset). Left: For different durations of the priming period and fixed priming frequency of 3 Hz. Right: Different priming frequencies with fixed priming duration of 60 s. The black line is a RMS fit to LTD data points of: (left) an exponential function; (right) of a quadratic function.</p></caption><graphic xlink:href="pcbi.1003330.g007"></graphic></fig><p>Since stability requires <inline-formula><inline-graphic xlink:href="pcbi.1003330.e216.jpg"></inline-graphic></inline-formula> to be relatively short, it is also worth considering the extreme case where it is on the timescale of a few hundred milliseconds. In that case the learning rule can be interpreted as a quadruplet STDP rule combining a triplet term for LTP (e.g. post-pre-post) with a quadruplet term for LTD (e.g. post-post-post-pre). While such a choice of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e217.jpg"></inline-graphic></inline-formula> would make sense from a stability point of view, this behavior is not seen in experiments <xref ref-type="bibr" rid="pcbi.1003330-Sjstrm1">[26]</xref>.</p></sec><sec id="s3b"><title>Influence of the model design</title><p>The timescales of synaptic plasticity and the time constants behind most homeostatic mechanisms reported in experiments are far apart. While plasticity can cause substantial synaptic changes in less than one minute <xref ref-type="bibr" rid="pcbi.1003330-Markram1">[24]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Sjstrm1">[26]</xref>, homeostatic responses typically differ on the order of several magnitudes (hours or days) <xref ref-type="bibr" rid="pcbi.1003330-Watt1">[29]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano5">[37]</xref>. In this paper we have shown that even if homeostatic changes manifest relatively slowly they have to be controlled by a fast rate detector, else triplet STDP is incompatible with the low background activity observed in cortical circuits. We argue that this statement is likely not to be limited to our particular model, but rather applies to an entire family of existing plasticity models.</p><p>The basic building blocks of our study were a network model and a homeostatic plasticity rule. We used a generic balanced network model <xref ref-type="bibr" rid="pcbi.1003330-Brunel1">[7]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Kumar1">[10]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Compte1">[44]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Vogels2">[46]</xref> to mimic brain-like spiking activity in a recurrent neural network. It is clear that the particular choice of network model does affect our results in a quantitative way and absolute predictions would require a more accurate and detailed network model. Nevertheless, we expect homeostasis to have similar timescale requirements in more detailed models as well. Indeed, as long as a strengthening of the excitatory synapses yields increased firing rates without a major change in the correlations, the qualitative predictions of the mean field model hold. However, our simulations were limited to roughly 1000 recurrent inputs per neuron, which is presumably less than what real cortical neurons receive <xref ref-type="bibr" rid="pcbi.1003330-DeFelipe1">[31]</xref>, so that excitatory run-away could build up even more rapidly in real networks than in our simulations.</p><p>The second building block of our model was the plasticity rule. Here we chose triplet STDP <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref> as a plasticity model that quantitatively captures a large body of experiments <xref ref-type="bibr" rid="pcbi.1003330-Markram1">[24]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Sjstrm1">[26]</xref>. One key feature of this model, which is seen across a range of in-vitro plasticity studies, is the fact that it yields LTP for high postsynaptic firing rates. The emergence of a critical timescale for homeostasis is mainly rooted in this fact and it is largely relaxed for pair-based STDP, be it additive or multiplicative <xref ref-type="bibr" rid="pcbi.1003330-Morrison1">[12]</xref>. However, such models do not capture experimental data as well as triplet STDP.</p><p>With the models we analyzed, namely the metaplastic triplet STDP and triplet STDP with synaptic scaling, we combined a realistic STDP learning rule with two quite different, but commonly used synaptic homeostatic mechanisms <xref ref-type="bibr" rid="pcbi.1003330-Bienenstock1">[15]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Lazar1">[18]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Clopath1">[19]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-vanRossum1">[35]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Gjorgjieva1">[38]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Clopath2">[42]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-ElBoustani1">[43]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Tetzlaff1">[47]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Tetzlaff2">[48]</xref>. The fact that we were able to show in both cases, either using a generic mean field model or numerical simulations of large balanced networks, that a fast rate detector is needed for stability, suggests that these results are quite general. The argument is further strengthened by the fact that existing computational models demonstrating stable background activity in plastic recurrent network models either use a form of multiplicative STDP which can be intrinsically stable <xref ref-type="bibr" rid="pcbi.1003330-Morrison1">[12]</xref>, but has poor memory retention <xref ref-type="bibr" rid="pcbi.1003330-Morrison1">[12]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Billings1">[34]</xref>, or rely on a fast homeostatic mechanism <xref ref-type="bibr" rid="pcbi.1003330-Lazar1">[18]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-ElBoustani1">[43]</xref>. In fact one of the first studies that illustrates stable learning in large recurrent networks combined with long memory retention times <xref ref-type="bibr" rid="pcbi.1003330-ElBoustani1">[43]</xref> is a model of metaplasticity built on top of the triplet model <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>. To describe effects observed in priming experiments <xref ref-type="bibr" rid="pcbi.1003330-Huang1">[41]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Christie1">[49]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Mockett1">[50]</xref>, the authors introduce two floating plasticity thresholds that modulate the rate of LTP and LTD depending on the low-pass filtered neuronal activity. El Boustani and colleagues obtain the time constants behind these filters by fitting their model to experimental data. It is striking, and in agreement with what we report here, that the timescales they find are on the order of 1 s <xref ref-type="bibr" rid="pcbi.1003330-ElBoustani1">[43]</xref>.</p><p>We conclude that current plasticity models that capture experimental data well require homeostasis to be able to react fast in order to maintain a stable background state. Likewise, if there is no rapid homeostatic control, most current plasticity models are probably missing a key ingredient to what makes cortical circuits stable.</p></sec><sec id="s3c"><title>Experimental evidence</title><p>The metaplastic triplet STDP rule we used makes use of an homeostatically modulated rate of LTD and can be mapped to a BCM-like learning rule <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Gjorgjieva1">[38]</xref>. The BCM theory relies on a plasticity rule with a neuron wide sliding threshold <xref ref-type="bibr" rid="pcbi.1003330-Bienenstock1">[15]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Cooper1">[51]</xref>. There seems to be some experimental ground for this idea <xref ref-type="bibr" rid="pcbi.1003330-Wang1">[52]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Hulme1">[53]</xref> and it is intriguing, that the effects reported there are on the order of 30 min or less which points towards a relatively fast mechanism. We should further point out, that the arguments that led us to the critical timescale of homeostasis are not limited to a neuron wide sliding threshold. In fact the mean field equations for a global or local synaptic sliding threshold, or even one based on local dendritic compartments, are identical. Therefore the arguments we put forward also hold for the latter cases, which have experimental support through priming experiments <xref ref-type="bibr" rid="pcbi.1003330-Huang1">[41]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Christie1">[49]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Mockett1">[50]</xref>. Priming experiments highlight changes in the induction of plasticity which depends on the synaptic activity over some 30 min.</p><p>With synaptic scaling we studied another possibility of introducing homeostasis into the triplet STDP model. Homeostatic scaling of synapses has good experimental support <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano1">[21]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Watt1">[29]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Turrigiano5">[37]</xref>. Although it is generally associated with long timescales (order of days), also more rapid forms of scaling are known <xref ref-type="bibr" rid="pcbi.1003330-Sutton1">[54]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Ibata1">[56]</xref> of which some indeed act on the order of minutes <xref ref-type="bibr" rid="pcbi.1003330-Frank1">[57]</xref>. Further modeling is required to test the ability of these rapid forms of homeostasis to guarantee stability in recurrent networks.</p><p>Finally one should note that the critical time scale of the rate detector strongly depends on the firing rates of the background state (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e218.jpg"></inline-graphic></inline-formula>, cf. <xref ref-type="disp-formula" rid="pcbi.1003330.e081">Eq. (7)</xref> and <xref ref-type="sec" rid="s4">Methods</xref>). The low firing rates reported experimentally <xref ref-type="bibr" rid="pcbi.1003330-Burns1">[58]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Barth1">[60]</xref> are therefore potentially necessary to guarantee the stability of the network. Conversely, cells or sub-networks with higher mean firing rates should have lower learning rates in order to be stable.</p></sec><sec id="s3d"><title>Limitations</title><p>Despite the mean field formalism being a drastic simplification of the original spiking model, the results we were able to derive from it were surprisingly accurate in the case of metaplastic triplet STDP and off by a factor of two in the case of triplet STDP with synaptic scaling. In all cases our mean field predictions overestimate the critical timescale obtained from simulations. This discrepancy has multiple potential reasons. First, in the mean field model we completely omit the existence of noise, fluctuations, and correlations. That these factors do play a role follows from the observation that the spiking network does not stabilize at the target rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e219.jpg"></inline-graphic></inline-formula>, but at higher values (cf. <xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2 E</xref>). Although correlations in the AI state are small, they are on average positive <xref ref-type="bibr" rid="pcbi.1003330-Renart1">[9]</xref>. When we estimated <inline-formula><inline-graphic xlink:href="pcbi.1003330.e220.jpg"></inline-graphic></inline-formula> we explicitly ignored correlations and required that LTD and LTP cancel at a firing rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e221.jpg"></inline-graphic></inline-formula>. Adding correlations causes this cancellation to take place at slightly higher rates, which reduces the effective critical time constant. In the rate formulation of the STDP rule we make the simplifying assumption that the synaptic traces are perfect estimates of the postsynaptic firing rates. Indeed it can be shown that fluctuations that are present in the rates, bias the learning rule towards LTP (see <xref ref-type="supplementary-material" rid="pcbi.1003330.s002">Text S1</xref>). Finally, any deviation of the population activity from its target value, initial or spontaneous, can be thought of as perturbations around the fixed point of background activity in the mean field model. This can compromise stability when the basin of attraction is small, as is the case when <inline-formula><inline-graphic xlink:href="pcbi.1003330.e222.jpg"></inline-graphic></inline-formula> is close to criticality (<xref ref-type="fig" rid="pcbi-1003330-g003">Figure 3 D</xref>). Again, such perturbations bias the critical value for the spiking network towards lower values. All the above points concern the simplifications made when going from the spiking model to the mean field model.</p><p>More importantly, the spiking model itself already represents a drastic simplification of the biological reality. For instance, we did not include neuronal firing rate adaptation or synaptic short-term plasticity (STP) in the present model. The timescales involved in firing rate adaptation are typically short (on the order of 100 ms) and their effect therefore negligible at the low firing rates of background activity <xref ref-type="bibr" rid="pcbi.1003330-Benda1">[61]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Brette1">[62]</xref>. While the time constants behind STP can be longer than that, their stabilizing effect is somewhat less clear since they can be facilitating and depressing <xref ref-type="bibr" rid="pcbi.1003330-Markram2">[63]</xref>. Although we do not expect STP to have a strong impact on our main results, it would be an interesting avenue to verify this in future studies.</p><p>All our present studies were limited to spontaneous background activity. In a more realistic scenario we would expect the network to receive external input with spatio-temporal correlations. Such input will generally cause synaptic weights to change, which in the mean field model corresponds to a perturbation of the dynamical network state around the stable fixed point. If the perturbation leaves the system in the basin of attraction of background activity, equilibrium will be restored over time. If, however, the perturbation is strong, or perturbations are in rapid concession and start to pile up, the system loses stability once its dynamical state reaches the separatrix (cf. <xref ref-type="fig" rid="pcbi-1003330-g003">Figure 3 C,D</xref>).</p><p>Another possibility worth mentioning is homeostatic regulation through inhibitory synaptic plasticity (ISP) <xref ref-type="bibr" rid="pcbi.1003330-Lamsa1">[64]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Vogels3">[68]</xref>. Recent theoretical studies <xref ref-type="bibr" rid="pcbi.1003330-Vogels4">[69]</xref>–<xref ref-type="bibr" rid="pcbi.1003330-Srinivasa1">[71]</xref> suggest that ISP could produce an intrinsically stable feed-back system. Although we cannot exclude ISP as an important factor in network homeostasis, we have excluded it in the current study. It is likely that to stabilize Hebbian plasticity at excitatory synapses, ISP has to act on a comparable timescale <xref ref-type="bibr" rid="pcbi.1003330-Sprekeler1">[72]</xref> and it will be interesting to integrate future experimental findings into a similar framework as presented here.</p></sec><sec id="s3e"><title>Conclusion</title><p>In summary, homeostatic mechanisms are necessary to stabilize the background activity in network models subject to Hebbian plasticity. Homeostasis needs to react faster than what is experimentally observed. This raises the important question of how the background activity in the brain can be stable. Our results suggest that the existence of a rapid homeostatic mechanism could be one possible answer. That, however, would require this mechanism to act on the same timescale as most STDP induction protocols. This then raises the question, why it has not been observed so far. Suitable plasticity protocols to detect such a mechanism should be similar to priming experiments <xref ref-type="bibr" rid="pcbi.1003330-Huang1">[41]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Christie1">[49]</xref>, but on the timescale of 1 min (<xref ref-type="fig" rid="pcbi-1003330-g007">Figure 7</xref>). Another possibility would be, that the plasticity rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e223.jpg"></inline-graphic></inline-formula> is not a constant after all, but subject to some neuromodulatory change <xref ref-type="bibr" rid="pcbi.1003330-Pawlak1">[73]</xref>. This could be possible, since it cannot be excluded that conditions in slice preparations, like the ones used to obtain the parameters of triplet STDP <xref ref-type="bibr" rid="pcbi.1003330-Sjstrm1">[26]</xref>, are different from in-vivo conditions. Finally, also fast forms of ISP could play a role in network stability.</p><p>No matter whether through ISP or additional, hitherto unseen excitatory homeostatic effects, a variation of current models of homeostasis and plasticity seem inevitable, to achieve stability in plastic network models whilst making them biologically plausible.</p></sec></sec><sec sec-type="methods" id="s4"><title>Methods</title><p>To study stability in plastic spiking recurrent networks we simulated networks of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e224.jpg"></inline-graphic></inline-formula> integrate-and-fire neurons with conductance-based synapses (<xref ref-type="fig" rid="pcbi-1003330-g001">Figure 1 A</xref>). The size of the network was chosen large enough to allow for an asynchronous irregular (AI) background state with low spiking correlations, but still small enough to enable simulations over long periods of biological time.</p><sec id="s4a"><title>Neuron model</title><p>The networks we study consist of leaky integrate-and-fire neurons with a relative refractory mechanism connected by conductance-based synapses <xref ref-type="bibr" rid="pcbi.1003330-Vogels2">[46]</xref>. The membrane voltage <inline-formula><inline-graphic xlink:href="pcbi.1003330.e225.jpg"></inline-graphic></inline-formula> of neuron <inline-formula><inline-graphic xlink:href="pcbi.1003330.e226.jpg"></inline-graphic></inline-formula> evolves according to<disp-formula id="pcbi.1003330.e227"><graphic xlink:href="pcbi.1003330.e227"></graphic><label>(11)</label></disp-formula>A spike is triggered when <inline-formula><inline-graphic xlink:href="pcbi.1003330.e228.jpg"></inline-graphic></inline-formula> crosses the spiking threshold <inline-formula><inline-graphic xlink:href="pcbi.1003330.e229.jpg"></inline-graphic></inline-formula>. After a spike <inline-formula><inline-graphic xlink:href="pcbi.1003330.e230.jpg"></inline-graphic></inline-formula> is reset to <inline-formula><inline-graphic xlink:href="pcbi.1003330.e231.jpg"></inline-graphic></inline-formula> and the threshold <inline-formula><inline-graphic xlink:href="pcbi.1003330.e232.jpg"></inline-graphic></inline-formula> is increased <inline-formula><inline-graphic xlink:href="pcbi.1003330.e233.jpg"></inline-graphic></inline-formula> to implement refractoriness. In the absence of spikes the threshold relaxes back to its resting value <inline-formula><inline-graphic xlink:href="pcbi.1003330.e234.jpg"></inline-graphic></inline-formula> according to<disp-formula id="pcbi.1003330.e235"><graphic xlink:href="pcbi.1003330.e235"></graphic><label>(12)</label></disp-formula>with <inline-formula><inline-graphic xlink:href="pcbi.1003330.e236.jpg"></inline-graphic></inline-formula> similar to <xref ref-type="bibr" rid="pcbi.1003330-Clopath2">[42]</xref>. Inhibitory neurons were modeled identically except for a shorter membrane time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e237.jpg"></inline-graphic></inline-formula>. All relevant parameters are summarized in <xref ref-type="table" rid="pcbi-1003330-t001">Table 1</xref>.</p><table-wrap id="pcbi-1003330-t001" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1003330.t001</object-id><label>Table 1</label><caption><title>Neuron model and synaptic parameters.</title></caption><alternatives><graphic id="pcbi-1003330-t001-1" xlink:href="pcbi.1003330.t001"></graphic><table frame="hsides" rules="groups"><colgroup span="1"><col align="left" span="1"></col><col align="center" span="1"></col><col align="center" span="1"></col><col align="center" span="1"></col><col align="center" span="1"></col><col align="center" span="1"></col></colgroup><thead><tr><td colspan="2" align="left" rowspan="1">Membrane</td><td colspan="2" align="left" rowspan="1">Threshold</td><td colspan="2" align="left" rowspan="1">Synapse</td></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e238.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">0 mV</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e239.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">5 ms</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e240.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">5 ms</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e241.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">−70 mV</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e242.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">−50 mV</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e243.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">10 ms</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e244.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">−80 mV</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e245.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">100 mV</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e246.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">100 ms</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e247.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">20 ms (10 ms<xref ref-type="table-fn" rid="nt101">*</xref>)</td><td align="left" rowspan="1" colspan="1"></td><td align="left" rowspan="1" colspan="1"></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e248.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">0.5</td></tr></tbody></table></alternatives><table-wrap-foot><fn id="nt101"><label>*) </label><p>only inhibitory neurons.</p></fn></table-wrap-foot></table-wrap><p>The spike train <inline-formula><inline-graphic xlink:href="pcbi.1003330.e249.jpg"></inline-graphic></inline-formula> of neuron <inline-formula><inline-graphic xlink:href="pcbi.1003330.e250.jpg"></inline-graphic></inline-formula> is defined as <inline-formula><inline-graphic xlink:href="pcbi.1003330.e251.jpg"></inline-graphic></inline-formula>, where the sum runs over all <inline-formula><inline-graphic xlink:href="pcbi.1003330.e252.jpg"></inline-graphic></inline-formula> corresponding firing times <inline-formula><inline-graphic xlink:href="pcbi.1003330.e253.jpg"></inline-graphic></inline-formula> of neuron <inline-formula><inline-graphic xlink:href="pcbi.1003330.e254.jpg"></inline-graphic></inline-formula>. It affects the synaptic conductances of downstream neurons as<disp-formula id="pcbi.1003330.e255"><graphic xlink:href="pcbi.1003330.e255"></graphic><label>(13)</label></disp-formula>if the index <inline-formula><inline-graphic xlink:href="pcbi.1003330.e256.jpg"></inline-graphic></inline-formula> corresponds to an inhibitory neuron or<disp-formula id="pcbi.1003330.e257"><graphic xlink:href="pcbi.1003330.e257"></graphic><label>(14)</label></disp-formula><disp-formula id="pcbi.1003330.e258"><graphic xlink:href="pcbi.1003330.e258"></graphic><label>(15)</label></disp-formula>in the case of an excitatory cell. Here <inline-formula><inline-graphic xlink:href="pcbi.1003330.e259.jpg"></inline-graphic></inline-formula> is the weight of the synapse connecting neuron <inline-formula><inline-graphic xlink:href="pcbi.1003330.e260.jpg"></inline-graphic></inline-formula> with <inline-formula><inline-graphic xlink:href="pcbi.1003330.e261.jpg"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e262.jpg"></inline-graphic></inline-formula> if the connection does not exists). Excitatory synapses contain a fast rising AMPA component with exponential decay and a slowly rising NMDA component with its respective exponential decay with time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e263.jpg"></inline-graphic></inline-formula>. For simplicity we implemented the NMDA component as a low pass filtered version of the AMPA conductance (<xref ref-type="disp-formula" rid="pcbi.1003330.e258">Eq. (15)</xref>). The complete excitatory postsynaptic potential (EPSP) is then given by a weighted sum of the AMPA and NMDA conductances<disp-formula id="pcbi.1003330.e264"><graphic xlink:href="pcbi.1003330.e264"></graphic><label>(16)</label></disp-formula>With the chosen parameters (cf. <xref ref-type="table" rid="pcbi-1003330-t001">Table 1</xref>), a typical EPSP has an amplitude of about <inline-formula><inline-graphic xlink:href="pcbi.1003330.e265.jpg"></inline-graphic></inline-formula>, as shown in <xref ref-type="fig" rid="pcbi-1003330-g001">Figure 1 B</xref>. For computational efficiency the voltage dependence of NMDA channels was omitted.</p></sec><sec id="s4b"><title>Network model</title><p>All units (20000 excitatory and 5000 inhibitory units, see <xref ref-type="table" rid="pcbi-1003330-t002">Table 2</xref> for details) are connected randomly with a sparse connectivity of 5%. Additionally each excitatory cell receives external input from a pool of 2500 independent Poisson processes firing at 2 Hz that are connected with 5% probability. The relevant synaptic weight values are summarized in <xref ref-type="table" rid="pcbi-1003330-t002">Table 2</xref>. Due to the high recurrence (on average 1000 out of 1125 connections are from within the network) the mean firing rate and network activity are sensitive to small changes in the recurrent synaptic strength. By appropriate choice of the excitatory weights (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e266.jpg"></inline-graphic></inline-formula>) the network is initially tuned to the balanced state with AI activity at a mean population activity of approximately 3 Hz.</p><table-wrap id="pcbi-1003330-t002" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1003330.t002</object-id><label>Table 2</label><caption><title>Network model parameters.</title></caption><alternatives><graphic id="pcbi-1003330-t002-2" xlink:href="pcbi.1003330.t002"></graphic><table frame="hsides" rules="groups"><colgroup span="1"><col align="left" span="1"></col><col align="center" span="1"></col><col align="center" span="1"></col><col align="center" span="1"></col></colgroup><thead><tr><td colspan="2" align="left" rowspan="1">Neuron groups and connectivity</td><td colspan="2" align="left" rowspan="1">Synaptic weight structure</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural population</td><td align="left" rowspan="1" colspan="1">Size</td><td align="left" rowspan="1" colspan="1">Connection</td><td align="left" rowspan="1" colspan="1">Weight</td></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1">Excitatory (E)</td><td align="left" rowspan="1" colspan="1">20000</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e267.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e268.jpg"></inline-graphic></inline-formula></td></tr><tr><td align="left" rowspan="1" colspan="1">Inhibitory (I)</td><td align="left" rowspan="1" colspan="1">5000</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e269.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e270.jpg"></inline-graphic></inline-formula></td></tr><tr><td align="left" rowspan="1" colspan="1">External Poisson (ext)</td><td align="left" rowspan="1" colspan="1">2500 at 2 Hz</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e271.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e272.jpg"></inline-graphic></inline-formula></td></tr><tr><td align="left" rowspan="1" colspan="1">Network connectivity</td><td align="left" rowspan="1" colspan="1">5%</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e273.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e274.jpg"></inline-graphic></inline-formula></td></tr><tr><td align="left" rowspan="1" colspan="1">Connectivity from ext</td><td align="left" rowspan="1" colspan="1">5%</td><td align="left" rowspan="1" colspan="1">ext Poisson <inline-formula><inline-graphic xlink:href="pcbi.1003330.e275.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e276.jpg"></inline-graphic></inline-formula></td></tr></tbody></table></alternatives></table-wrap></sec><sec id="s4c"><title>Plasticity model</title><p>We model synaptic plasticity after the triplet STDP model of <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>, using the minimal parameter set corresponding to in-vitro visual cortex data <xref ref-type="bibr" rid="pcbi.1003330-Sjstrm1">[26]</xref>. Plasticity only affects the EE recurrent connections. Weight updates <inline-formula><inline-graphic xlink:href="pcbi.1003330.e277.jpg"></inline-graphic></inline-formula> act additively on the matrix elements <inline-formula><inline-graphic xlink:href="pcbi.1003330.e278.jpg"></inline-graphic></inline-formula> and are given by<disp-formula id="pcbi.1003330.e279"><graphic xlink:href="pcbi.1003330.e279"></graphic><label>(17)</label></disp-formula>where <inline-formula><inline-graphic xlink:href="pcbi.1003330.e280.jpg"></inline-graphic></inline-formula> is a small positive number and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e281.jpg"></inline-graphic></inline-formula>, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e282.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e283.jpg"></inline-graphic></inline-formula> are synaptic traces of neuron <inline-formula><inline-graphic xlink:href="pcbi.1003330.e284.jpg"></inline-graphic></inline-formula> defined as <inline-formula><inline-graphic xlink:href="pcbi.1003330.e285.jpg"></inline-graphic></inline-formula> with associated time constants <inline-formula><inline-graphic xlink:href="pcbi.1003330.e286.jpg"></inline-graphic></inline-formula>, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e287.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e288.jpg"></inline-graphic></inline-formula> respectively (see <xref ref-type="table" rid="pcbi-1003330-t003">Table 3</xref> and <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>). Since the original triplet model describes relative synaptic changes, weight updates in <xref ref-type="disp-formula" rid="pcbi.1003330.e279">Eq. (17)</xref> are scaled by the factor <inline-formula><inline-graphic xlink:href="pcbi.1003330.e289.jpg"></inline-graphic></inline-formula>, where <inline-formula><inline-graphic xlink:href="pcbi.1003330.e290.jpg"></inline-graphic></inline-formula> is the initial synaptic weight and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e291.jpg"></inline-graphic></inline-formula> is an additional parameter that can be interpreted as a learning rate, or a conversion factor between the weight scales of the model and the true biological scale. In the model we approximate the biological scale by choosing plausible values for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e292.jpg"></inline-graphic></inline-formula> (cf. <xref ref-type="fig" rid="pcbi-1003330-g001">Figure 1 B</xref>) and therefore expect <inline-formula><inline-graphic xlink:href="pcbi.1003330.e293.jpg"></inline-graphic></inline-formula> to be of the order of one. For a synapse with an initial weight of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e294.jpg"></inline-graphic></inline-formula>, a value of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e295.jpg"></inline-graphic></inline-formula> corresponds to the learning rate that best fits visual cortex data <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>. However, since small values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e296.jpg"></inline-graphic></inline-formula> are computationally expensive we used <inline-formula><inline-graphic xlink:href="pcbi.1003330.e297.jpg"></inline-graphic></inline-formula> in <xref ref-type="fig" rid="pcbi-1003330-g002">Figure 2</xref> to ensure that a stable weight distribution can be observed within a day of simulated biological time (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e298.jpg"></inline-graphic></inline-formula> of computation time). Note that for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e299.jpg"></inline-graphic></inline-formula> we would expect a comparable degree of convergence after 6.25 days of simulated time (roughly four weeks of computation). During ongoing plasticity the allowed weight values are limited to the interval <inline-formula><inline-graphic xlink:href="pcbi.1003330.e300.jpg"></inline-graphic></inline-formula>. Note that to avoid the creation of new synapses, connections that have zero weight initially, remain absent (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e301.jpg"></inline-graphic></inline-formula>) throughout the entire simulation.</p><table-wrap id="pcbi-1003330-t003" orientation="portrait" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1003330.t003</object-id><label>Table 3</label><caption><title>Plasticity model parameters.</title></caption><alternatives><graphic id="pcbi-1003330-t003-3" xlink:href="pcbi.1003330.t003"></graphic><table frame="hsides" rules="groups"><colgroup span="1"><col align="left" span="1"></col><col align="center" span="1"></col><col align="center" span="1"></col></colgroup><tbody><tr><td align="left" rowspan="1" colspan="1">Plasticity window</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e302.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e303.jpg"></inline-graphic></inline-formula></td></tr><tr><td align="left" rowspan="1" colspan="1"></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e304.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">16.8 ms</td></tr><tr><td align="left" rowspan="1" colspan="1"></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e305.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">33.7 ms</td></tr><tr><td align="left" rowspan="1" colspan="1"></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e306.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">114 ms</td></tr><tr><td align="left" rowspan="1" colspan="1">Initial weight</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e307.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">0.16</td></tr><tr><td align="left" rowspan="1" colspan="1">Weight limits</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e308.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">0</td></tr><tr><td align="left" rowspan="1" colspan="1"></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e309.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">1</td></tr><tr><td align="left" rowspan="1" colspan="1">Target firing rate</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e310.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">3 Hz</td></tr><tr><td align="left" rowspan="1" colspan="1">Rel. learning rate</td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e311.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e312.jpg"></inline-graphic></inline-formula><xref ref-type="table-fn" rid="nt102">*</xref></td></tr><tr><td align="left" rowspan="1" colspan="1"></td><td align="left" rowspan="1" colspan="1"><inline-formula><inline-graphic xlink:href="pcbi.1003330.e313.jpg"></inline-graphic></inline-formula></td><td align="left" rowspan="1" colspan="1">1 (<xref ref-type="supplementary-material" rid="pcbi.1003330.s001">Figure S1</xref>)</td></tr></tbody></table></alternatives><table-wrap-foot><fn id="nt102"><label>*) </label><p>As used in <xref ref-type="fig" rid="pcbi-1003330-g002">Figures 2</xref> and <xref ref-type="fig" rid="pcbi-1003330-g004">4 B,C</xref>.</p></fn></table-wrap-foot></table-wrap><p>In simulations with metaplastic triplet STDP the amount of long term synaptic depression (LTD) <inline-formula><inline-graphic xlink:href="pcbi.1003330.e314.jpg"></inline-graphic></inline-formula> is varied homeostatically as a function of the moving average <inline-formula><inline-graphic xlink:href="pcbi.1003330.e315.jpg"></inline-graphic></inline-formula> of the postsynaptic firing rate <xref ref-type="bibr" rid="pcbi.1003330-Bienenstock1">[15]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Clopath1">[19]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>, <xref ref-type="bibr" rid="pcbi.1003330-Gjorgjieva1">[38]</xref> with<disp-formula id="pcbi.1003330.e316"><graphic xlink:href="pcbi.1003330.e316"></graphic><label>(18)</label></disp-formula>This choice of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e317.jpg"></inline-graphic></inline-formula> ensures that for uncorrelated Poisson firing at the rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e318.jpg"></inline-graphic></inline-formula> LTP and LTD cancel on average. The moving average <inline-formula><inline-graphic xlink:href="pcbi.1003330.e319.jpg"></inline-graphic></inline-formula> of the firing rate of neuron <inline-formula><inline-graphic xlink:href="pcbi.1003330.e320.jpg"></inline-graphic></inline-formula> is implemented as a low pass filtered version of its spike train<disp-formula id="pcbi.1003330.e321"><graphic xlink:href="pcbi.1003330.e321"></graphic><label>(19)</label></disp-formula>where <inline-formula><inline-graphic xlink:href="pcbi.1003330.e322.jpg"></inline-graphic></inline-formula> is the timescale which controls of the temporal evolution of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e323.jpg"></inline-graphic></inline-formula> (cf. <xref ref-type="disp-formula" rid="pcbi.1003330.e316">Eq. (18)</xref>).</p><p>In simulations that require an additional slow weight decay of the weights we approximate this exponential decay, to avoid the costly operation of updating all weights after each time step, by periodically (period <inline-formula><inline-graphic xlink:href="pcbi.1003330.e324.jpg"></inline-graphic></inline-formula>) multiplying all weights by the factor <inline-formula><inline-graphic xlink:href="pcbi.1003330.e325.jpg"></inline-graphic></inline-formula>. Finally, simulations of synaptic scaling are performed using a fixed value <inline-formula><inline-graphic xlink:href="pcbi.1003330.e326.jpg"></inline-graphic></inline-formula>. The scaling of the weights is approximated with the same approach as for weight decay. In such cases <inline-formula><inline-graphic xlink:href="pcbi.1003330.e327.jpg"></inline-graphic></inline-formula> is adapted appropriately according to the occurring scaling time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e328.jpg"></inline-graphic></inline-formula>.</p></sec><sec id="s4d"><title>The time constant of plasticity</title><p>We determine the timescale of plasticity in the mean field model by approximating <inline-formula><inline-graphic xlink:href="pcbi.1003330.e329.jpg"></inline-graphic></inline-formula> from the plasticity parameters of the triplet STDP model <xref ref-type="bibr" rid="pcbi.1003330-Pfister1">[30]</xref>. To do so we consider the expectation value of the mean weight update averaged over many spike pairs, and we assume that pre- and postsynaptic firing is uncorrelated with stationary rates <inline-formula><inline-graphic xlink:href="pcbi.1003330.e330.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e331.jpg"></inline-graphic></inline-formula> respectively. The average relative weight change over time then reads<disp-formula id="pcbi.1003330.e332"><graphic xlink:href="pcbi.1003330.e332"></graphic><label>(20)</label></disp-formula><disp-formula id="pcbi.1003330.e333"><graphic xlink:href="pcbi.1003330.e333"></graphic><label>(21)</label></disp-formula><disp-formula id="pcbi.1003330.e334"><graphic xlink:href="pcbi.1003330.e334"></graphic><label>(22)</label></disp-formula>The resulting differential equation can be directly identified with <xref ref-type="disp-formula" rid="pcbi.1003330.e069">Eq. (4)</xref> to obtain the effective time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e335.jpg"></inline-graphic></inline-formula>.</p></sec><sec id="s4e"><title>Numerical simulations</title><p>All differential equations were integrated using forward Euler integration with a 0.1 ms time step. Spiking simulations were written in C++ using Open MPI and the Boost libraries. The sources were compiled using the GNU C compiler. Simulations were run on 5 Linux workstations equipped with Intel(R) Core(TM)2 Duo E8400 CPUs and 24 GB of RAM each. It took approximately four and a half days to simulate one day of biological time.</p><p>Numerical results for the phase plane analysis, such as the position of the separatrix, were obtained by integrating the ODEs of the mean field model numerically using custom-written Python code.</p></sec><sec id="s4f"><title>Derivation of the stability condition in the mean field model</title><p>To analyze the stability of the fixed point of background activity (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e336.jpg"></inline-graphic></inline-formula>) in the case of the metaplastic triplet STDP rule, we consider the Jacobian <inline-formula><inline-graphic xlink:href="pcbi.1003330.e337.jpg"></inline-graphic></inline-formula> of the two dimensional system (cf. <xref ref-type="disp-formula" rid="pcbi.1003330.e076">Eqs. (5)</xref>,<xref ref-type="disp-formula" rid="pcbi.1003330.e078">(6)</xref>) in the general case of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e338.jpg"></inline-graphic></inline-formula> for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e339.jpg"></inline-graphic></inline-formula>.<disp-formula id="pcbi.1003330.e340"><graphic xlink:href="pcbi.1003330.e340"></graphic><label>(23)</label></disp-formula>where we introduced the auxiliary variable <inline-formula><inline-graphic xlink:href="pcbi.1003330.e341.jpg"></inline-graphic></inline-formula>. When evaluated at the fixed point <inline-formula><inline-graphic xlink:href="pcbi.1003330.e342.jpg"></inline-graphic></inline-formula> reduces to<disp-formula id="pcbi.1003330.e343"><graphic xlink:href="pcbi.1003330.e343"></graphic><label>(24)</label></disp-formula>with characteristic polynomial<disp-formula id="pcbi.1003330.e344"><graphic xlink:href="pcbi.1003330.e344"></graphic><label>(25)</label></disp-formula>which determines the eigenvalues to be of the linearized system at the fixed point of background activity<disp-formula id="pcbi.1003330.e345"><graphic xlink:href="pcbi.1003330.e345"></graphic><label>(26)</label></disp-formula></p><p>Stability of the fixed point requires all eigenvalues to have negative real parts (e.g. <xref ref-type="bibr" rid="pcbi.1003330-Strogatz1">[74]</xref>). We now prove that the real part of both eigenvalues is negative if and only if <inline-formula><inline-graphic xlink:href="pcbi.1003330.e346.jpg"></inline-graphic></inline-formula>. The square root in <xref ref-type="disp-formula" rid="pcbi.1003330.e345">Eq. (26)</xref> is either purely imaginary, in which case <inline-formula><inline-graphic xlink:href="pcbi.1003330.e347.jpg"></inline-graphic></inline-formula> follows directly. For the case in which the square root is real we can express the larger of the two eigenvalues as<disp-formula id="pcbi.1003330.e348"><graphic xlink:href="pcbi.1003330.e348"></graphic><label>(27)</label></disp-formula><disp-formula id="pcbi.1003330.e349"><graphic xlink:href="pcbi.1003330.e349"></graphic><label>(28)</label></disp-formula>where we introduced the variable <inline-formula><inline-graphic xlink:href="pcbi.1003330.e350.jpg"></inline-graphic></inline-formula> for the term in the square brackets (<xref ref-type="disp-formula" rid="pcbi.1003330.e348">Eq. (27)</xref>). If <inline-formula><inline-graphic xlink:href="pcbi.1003330.e351.jpg"></inline-graphic></inline-formula> then <inline-formula><inline-graphic xlink:href="pcbi.1003330.e352.jpg"></inline-graphic></inline-formula> and the fixed point is unstable. If, however, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e353.jpg"></inline-graphic></inline-formula> then we know<disp-formula id="pcbi.1003330.e354"><graphic xlink:href="pcbi.1003330.e354"></graphic><label>(29)</label></disp-formula><disp-formula id="pcbi.1003330.e355"><graphic xlink:href="pcbi.1003330.e355"></graphic><label>(30)</label></disp-formula><disp-formula id="pcbi.1003330.e356"><graphic xlink:href="pcbi.1003330.e356"></graphic><label>(31)</label></disp-formula>Here, we used the fact that all occurring constants are positive, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e357.jpg"></inline-graphic></inline-formula> and the argument in the square root is positive as well. Finally we can conclude the fixed point is stable if <inline-formula><inline-graphic xlink:href="pcbi.1003330.e358.jpg"></inline-graphic></inline-formula>. This identifies <inline-formula><inline-graphic xlink:href="pcbi.1003330.e359.jpg"></inline-graphic></inline-formula> as an important limiting case for the stability of the fixed point. It is interesting to note that <inline-formula><inline-graphic xlink:href="pcbi.1003330.e360.jpg"></inline-graphic></inline-formula> is independent of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e361.jpg"></inline-graphic></inline-formula>.</p><sec id="s4f1"><title>Stability condition for weight decay</title><p>If we are to include an additional weight decay in the above model we replace <xref ref-type="disp-formula" rid="pcbi.1003330.e078">Eq. (6)</xref> by<disp-formula id="pcbi.1003330.e362"><graphic xlink:href="pcbi.1003330.e362"></graphic><label>(32)</label></disp-formula>and proceed similarly as before by replacing all occurrences of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e363.jpg"></inline-graphic></inline-formula>. In the decay term we can use the identities <inline-formula><inline-graphic xlink:href="pcbi.1003330.e364.jpg"></inline-graphic></inline-formula> and since <inline-formula><inline-graphic xlink:href="pcbi.1003330.e365.jpg"></inline-graphic></inline-formula> (cf. <xref ref-type="disp-formula" rid="pcbi.1003330.e058">Eq. (3)</xref>) to rewrite<disp-formula id="pcbi.1003330.e366"><graphic xlink:href="pcbi.1003330.e366"></graphic><label>(33)</label></disp-formula>We use this expression together with our results from <xref ref-type="disp-formula" rid="pcbi.1003330.e340">Eq. (23)</xref> and the abbreviation <inline-formula><inline-graphic xlink:href="pcbi.1003330.e367.jpg"></inline-graphic></inline-formula>, to arrive at<disp-formula id="pcbi.1003330.e368"><graphic xlink:href="pcbi.1003330.e368"></graphic><label>(34)</label></disp-formula>which leads to the following Jacobian at the fixed point<disp-formula id="pcbi.1003330.e369"><graphic xlink:href="pcbi.1003330.e369"></graphic><label>(35)</label></disp-formula>The corresponding eigenvalues are given by<disp-formula id="pcbi.1003330.e370"><graphic xlink:href="pcbi.1003330.e370"></graphic><label>(36)</label></disp-formula></p><p>As we have seen earlier the stability is determined by the first term since the square root is purely imaginary around criticality. This leads us to the relaxed stability condition <inline-formula><inline-graphic xlink:href="pcbi.1003330.e371.jpg"></inline-graphic></inline-formula> and therefore with <inline-formula><inline-graphic xlink:href="pcbi.1003330.e372.jpg"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1003330.e373.jpg"></inline-graphic></inline-formula> we get <inline-formula><inline-graphic xlink:href="pcbi.1003330.e374.jpg"></inline-graphic></inline-formula>.</p></sec><sec id="s4f2"><title>Stability condition for synaptic scaling</title><p>Here we will derive the critical time constant <inline-formula><inline-graphic xlink:href="pcbi.1003330.e375.jpg"></inline-graphic></inline-formula> for yet another variation of the triplet rule<disp-formula id="pcbi.1003330.e376"><graphic xlink:href="pcbi.1003330.e376"></graphic><label>(37)</label></disp-formula>which uses synaptic scaling to achieve the target rate <inline-formula><inline-graphic xlink:href="pcbi.1003330.e377.jpg"></inline-graphic></inline-formula> (cf. <xref ref-type="bibr" rid="pcbi.1003330-vanRossum1">[35]</xref>). With the same transformations as before (i.e. <inline-formula><inline-graphic xlink:href="pcbi.1003330.e378.jpg"></inline-graphic></inline-formula>) we can bring <xref ref-type="disp-formula" rid="pcbi.1003330.e376">Eq. (37)</xref> to the form<disp-formula id="pcbi.1003330.e379"><graphic xlink:href="pcbi.1003330.e379"></graphic><label>(38)</label></disp-formula>which taken together with <xref ref-type="disp-formula" rid="pcbi.1003330.e076">Eq. (5)</xref> yields the following Jacobian at the fixed point<disp-formula id="pcbi.1003330.e380"><graphic xlink:href="pcbi.1003330.e380"></graphic><label>(39)</label></disp-formula>with associated eigenvalues<disp-formula id="pcbi.1003330.e381"><graphic xlink:href="pcbi.1003330.e381"></graphic><label>(40)</label></disp-formula></p><p>We can appreciate directly from <xref ref-type="disp-formula" rid="pcbi.1003330.e381">Eq. (40)</xref> that the real part of the largest eigenvalue is lower bounded <inline-formula><inline-graphic xlink:href="pcbi.1003330.e382.jpg"></inline-graphic></inline-formula> and therefore we find that stability requires <inline-formula><inline-graphic xlink:href="pcbi.1003330.e383.jpg"></inline-graphic></inline-formula>, which is the same condition as above for the case of metaplastic triplet STDP. However, in the case of synaptic scaling this stability condition is necessary, but not sufficient. This we can see in <xref ref-type="disp-formula" rid="pcbi.1003330.e381">Eq. (40)</xref> for given <inline-formula><inline-graphic xlink:href="pcbi.1003330.e384.jpg"></inline-graphic></inline-formula>, when <inline-formula><inline-graphic xlink:href="pcbi.1003330.e385.jpg"></inline-graphic></inline-formula> becomes sufficiently small (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e386.jpg"></inline-graphic></inline-formula> sufficiently large) eventually we get <inline-formula><inline-graphic xlink:href="pcbi.1003330.e387.jpg"></inline-graphic></inline-formula>, where the background state loses stability (cf. <xref ref-type="fig" rid="pcbi-1003330-g006">Figure 6 B</xref>). Hence, in addition to <inline-formula><inline-graphic xlink:href="pcbi.1003330.e388.jpg"></inline-graphic></inline-formula> there is also a critical value for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e389.jpg"></inline-graphic></inline-formula> which can be on a comparable scale like <inline-formula><inline-graphic xlink:href="pcbi.1003330.e390.jpg"></inline-graphic></inline-formula>, but not arbitrarily large.</p></sec></sec></sec><sec sec-type="supplementary-material" id="s5"><title>Supporting Information</title><supplementary-material content-type="local-data" id="pcbi.1003330.s001"><label>Figure S1</label><caption><p><bold>Evolution of the population rate for metaplastic triplet STDP model.</bold> (<bold>A</bold>) Temporal evolution of mean population rate for different values of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e391.jpg"></inline-graphic></inline-formula> (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e392.jpg"></inline-graphic></inline-formula>). While the change in stability in the vicinity of <inline-formula><inline-graphic xlink:href="pcbi.1003330.e393.jpg"></inline-graphic></inline-formula> can be understood from the mean field theory, which also predicts the observed oscillations at criticality, the late destabilization of the curve <inline-formula><inline-graphic xlink:href="pcbi.1003330.e394.jpg"></inline-graphic></inline-formula> is not captured by the theory. (<bold>B</bold>) Evolution of mean population rate for <inline-formula><inline-graphic xlink:href="pcbi.1003330.e395.jpg"></inline-graphic></inline-formula>. Black: <inline-formula><inline-graphic xlink:href="pcbi.1003330.e396.jpg"></inline-graphic></inline-formula> and weights are initialized with the weights from a stable run (<inline-formula><inline-graphic xlink:href="pcbi.1003330.e397.jpg"></inline-graphic></inline-formula>, <inline-formula><inline-graphic xlink:href="pcbi.1003330.e398.jpg"></inline-graphic></inline-formula>) at <inline-formula><inline-graphic xlink:href="pcbi.1003330.e399.jpg"></inline-graphic></inline-formula>. Cyan: Same, but with <inline-formula><inline-graphic xlink:href="pcbi.1003330.e400.jpg"></inline-graphic></inline-formula>. Dark blue: <inline-formula><inline-graphic xlink:href="pcbi.1003330.e401.jpg"></inline-graphic></inline-formula>, weight initialization as in (A), but maximally allowed weights limited to <inline-formula><inline-graphic xlink:href="pcbi.1003330.e402.jpg"></inline-graphic></inline-formula>. Light blue: <inline-formula><inline-graphic xlink:href="pcbi.1003330.e403.jpg"></inline-graphic></inline-formula>, network falls silent at <inline-formula><inline-graphic xlink:href="pcbi.1003330.e404.jpg"></inline-graphic></inline-formula>. Purple: <inline-formula><inline-graphic xlink:href="pcbi.1003330.e405.jpg"></inline-graphic></inline-formula>, with <inline-formula><inline-graphic xlink:href="pcbi.1003330.e406.jpg"></inline-graphic></inline-formula> (the learning rate was unchanged), which reduces the initial excursion to low rates.</p><p>(PDF)</p></caption><media xlink:href="pcbi.1003330.s001.pdf"><caption><p>Click here for additional data file.</p></caption></media></supplementary-material><supplementary-material content-type="local-data" id="pcbi.1003330.s002"><label>Text S1</label><caption><p><bold>Rate fluctuations.</bold> Mean field solutions ignore the effect of fluctuations in the postsynaptic firing rate.</p><p>(PDF)</p></caption><media xlink:href="pcbi.1003330.s002.pdf"><caption><p>Click here for additional data file.</p></caption></media></supplementary-material></sec></body><back><ack><p>The authors would like to thank T. P. Vogels and H. 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HfdIndexSelect -h $EXPLOR_AREA/Data/Pmc/Curation/RBID.i -Sk "pubmed:24244138" \
| HfdSelect -Kh $EXPLOR_AREA/Data/Pmc/Curation/biblio.hfd \
| NlmPubMed2Wicri -a HapticV1
| This area was generated with Dilib version V0.6.23. |  |