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| <?xml version="1.0" encoding="UTF-8"?><!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.1d1 20130915//EN" "JATS-archivearticle1.dtd"><article article-type="research-article" dtd-version="1.1d1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><front><journal-meta><journal-id journal-id-type="nlm-ta">elife</journal-id><journal-id journal-id-type="hwp">eLife</journal-id><journal-id journal-id-type="publisher-id">eLife</journal-id><journal-title-group><journal-title>eLife</journal-title></journal-title-group><issn publication-format="electronic">2050-084X</issn><publisher><publisher-name>eLife Sciences Publications, Ltd</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="publisher-id">02370</article-id><article-id pub-id-type="doi">10.7554/eLife.02370</article-id><article-categories><subj-group subj-group-type="display-channel"><subject>Research article</subject></subj-group><subj-group subj-group-type="heading"><subject>Neuroscience</subject></subj-group></article-categories><title-group><article-title>Criticality and degeneracy in injury-induced changes in primary afferent excitability and the implications for neuropathic pain</article-title></title-group><contrib-group><contrib contrib-type="author" id="author-11502"><name><surname>Ratté</surname><given-names>Stéphanie</given-names></name><xref ref-type="aff" rid="aff1"/><xref ref-type="aff" rid="aff2"/><xref ref-type="aff" rid="aff3"/><xref ref-type="fn" rid="con1"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-10799"><name><surname>Zhu</surname><given-names>Yi</given-names></name><xref ref-type="aff" rid="aff3"/><xref ref-type="fn" rid="con3"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-10800"><name><surname>Lee</surname><given-names>Kwan Yeop</given-names></name><xref ref-type="aff" rid="aff1"/><xref ref-type="fn" rid="con4"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" corresp="yes" id="author-5002"><name><surname>Prescott</surname><given-names>Steven A</given-names></name><xref ref-type="aff" rid="aff1"/><xref ref-type="aff" rid="aff2"/><xref ref-type="aff" rid="aff3"/><xref ref-type="corresp" rid="cor1">*</xref><xref ref-type="other" rid="par-1"/><xref ref-type="other" rid="par-2"/><xref ref-type="other" rid="par-3"/><xref ref-type="other" rid="par-4"/><xref ref-type="other" rid="par-5"/><xref ref-type="fn" rid="con2"/><xref ref-type="fn" rid="conf1"/></contrib><aff id="aff1"><institution content-type="dept">Neurosciences and Mental Health</institution>, <institution>The Hospital for Sick Children</institution>, <addr-line><named-content content-type="city">Toronto</named-content></addr-line>, <country>Canada</country></aff><aff id="aff2"><institution content-type="dept">Department of Physiology</institution>, <institution>University of Toronto</institution>, <addr-line><named-content content-type="city">Toronto</named-content></addr-line>, <country>Canada</country></aff><aff id="aff3"><institution content-type="dept">Pittsburgh Center for Pain Research</institution>, <institution>University of Pittsburgh</institution>, <addr-line><named-content content-type="city">Pittsburgh</named-content></addr-line>, <country>United States</country></aff></contrib-group><contrib-group content-type="section"><contrib contrib-type="editor"><name><surname>Calabrese</surname><given-names>Ronald L</given-names></name><role>Reviewing editor</role><aff><institution>Emory University</institution>, <country>United States</country></aff></contrib></contrib-group><author-notes><corresp id="cor1"><label>*</label>For correspondence: <email>steve.prescott@sickkids.ca</email></corresp></author-notes><pub-date date-type="pub" publication-format="electronic"><day>01</day><month>04</month><year>2014</year></pub-date><pub-date pub-type="collection"><year>2014</year></pub-date><volume>3</volume><elocation-id>e02370</elocation-id><history><date date-type="received"><day>22</day><month>01</month><year>2014</year></date><date date-type="accepted"><day>21</day><month>02</month><year>2014</year></date></history><permissions><copyright-statement>© 2014, Ratté et al</copyright-statement><copyright-year>2014</copyright-year><copyright-holder>Ratté et al</copyright-holder><license xlink:href="http://creativecommons.org/licenses/by/3.0/"><license-p>This article is 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 and redistribution provided that the original author and source are credited.</license-p></license></permissions><self-uri content-type="pdf" xlink:href="elife02370.pdf"/><related-article ext-link-type="doi" id="ra1" related-article-type="commentary" xlink:href="10.7554/eLife.02615"/><abstract><object-id pub-id-type="doi">10.7554/eLife.02370.001</object-id><p>Neuropathic pain remains notoriously difficult to treat despite numerous drug targets. Here, we offer a novel explanation for this intractability. Computer simulations predicted that qualitative changes in primary afferent excitability linked to neuropathic pain arise through a switch in spike initiation dynamics when molecular pathologies reach a tipping point (criticality), and that this tipping point can be reached via several different molecular pathologies (degeneracy). We experimentally tested these predictions by pharmacologically blocking native conductances and/or electrophysiologically inserting virtual conductances. Multiple different manipulations successfully reproduced or reversed neuropathic changes in primary afferents from naïve or nerve-injured rats, respectively, thus confirming the predicted criticality and its degenerate basis. Degeneracy means that several different molecular pathologies are individually sufficient to cause hyperexcitability, and because several such pathologies co-occur after nerve injury, that no single pathology is uniquely necessary. Consequently, single-target-drugs can be circumvented by maladaptive plasticity in any one of several ion channels.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02370.001">http://dx.doi.org/10.7554/eLife.02370.001</ext-link></p></abstract><abstract abstract-type="executive-summary"><object-id pub-id-type="doi">10.7554/eLife.02370.002</object-id><title>eLife digest</title><p>Although the pain associated with an injury is unpleasant, it normally serves an important purpose: to make you avoid its source. However, some pain appears to arise from nowhere. Frustratingly, this type of pain, known as neuropathic pain, does not respond to common painkillers and is thus very difficult to treat.</p><p>The neurons that transmit pain and other sensory information do so using electrical signals. In response to a stimulus, ions travel through channels in the membrane of a neuron, which leads to a change in the electrical potential of the membrane. When this change is large enough, a voltage spike is produced: this signal is ultimately transmitted to the brain.</p><p>When certain neurons fire too easily or too often, neuropathic pain can arise. This hyperexcitability can make something painful feel even worse, or it can make things hurt that shouldn’t. To prevent this, extensive research has been devoted to identify drugs that target particular types of ion channels and block them. However, despite the discovery of many promising drugs, those drugs have been frustratingly ineffective in clinical trials.</p><p>Using simulations and experiments, Ratté et al. have examined the behavior of a type of neuron that normally conducts information about touch, but the brain sometimes misinterprets this information as pain. Increasing the flow of ions through the cell membrane in these simulations eventually causes a ‘tipping point’ to be crossed, which triggers a dramatic, discontinuous change in spiking pattern. However, as several different types of ion channels contribute to the current, there are several different ways in which the tipping point can be crossed.</p><p>This ability to produce the same result by multiple means is a common feature of complex systems. Known as degeneracy, it makes systems more robust, as a given result can still be achieved if one particular attempt to achieve this result fails. The work of Ratté et al. helps to explain why drugs that target just one type of ion channel may fail to relieve neuropathic pain: maladaptive changes in any one of several other ion channels may circumvent the therapeutic effect.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02370.002">http://dx.doi.org/10.7554/eLife.02370.002</ext-link></p></abstract><kwd-group kwd-group-type="author-keywords"><title>Author keywords</title><kwd>neuropathic pain</kwd><kwd>degeneracy</kwd><kwd>excitability</kwd><kwd>membrane potential oscillations</kwd><kwd>dynamic clamp</kwd><kwd>bursting</kwd></kwd-group><kwd-group kwd-group-type="research-organism"><title>Research organism</title><kwd>rat</kwd></kwd-group><funding-group><award-group id="par-1"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/100000002</institution-id><institution>National Institutes of Health</institution></institution-wrap></funding-source><award-id>R21 NS074146</award-id><principal-award-recipient><name><surname>Prescott</surname><given-names>Steven A</given-names></name></principal-award-recipient></award-group><award-group id="par-2"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/501100000038</institution-id><institution>Natural Sciences and Engineering Research Council of Canada</institution></institution-wrap></funding-source><principal-award-recipient><name><surname>Prescott</surname><given-names>Steven A</given-names></name></principal-award-recipient></award-group><award-group id="par-3"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/100001447</institution-id><institution>Rita Allen Foundation</institution></institution-wrap></funding-source><principal-award-recipient><name><surname>Prescott</surname><given-names>Steven A</given-names></name></principal-award-recipient></award-group><award-group id="par-4"><funding-source><institution-wrap><institution>Edward Mallinckrodt Jr Foundation</institution></institution-wrap></funding-source><principal-award-recipient><name><surname>Prescott</surname><given-names>Steven A</given-names></name></principal-award-recipient></award-group><award-group id="par-5"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/501100000024</institution-id><institution>Canadian Institutes of Health Research</institution></institution-wrap></funding-source><principal-award-recipient><name><surname>Prescott</surname><given-names>Steven A</given-names></name></principal-award-recipient></award-group><funding-statement>The funder had no role in study design, data collection and interpretation, or the decision to submit the work for publication.</funding-statement></funding-group><custom-meta-group><custom-meta><meta-name>elife-xml-version</meta-name><meta-value>2</meta-value></custom-meta><custom-meta specific-use="meta-only"><meta-name>Author impact statement</meta-name><meta-value>No single molecular change is uniquely necessary to cause neuropathic changes in primary afferent excitability; multiple different changes are sufficient.</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec id="s1" sec-type="intro"><title>Introduction</title><p>Neuropathic pain—pain arising from damage to or dysfunction of the nervous system (<xref ref-type="bibr" rid="bib48">Merskey and Bogduk, 1994</xref>)—is notoriously difficult to treat. Effective treatments have eluded discovery despite intense research and many promising leads (<xref ref-type="bibr" rid="bib63">Woolf, 2010</xref>). Proposed explanations for the lack of clinical translation have focused on preclinical animal models (<xref ref-type="bibr" rid="bib58">Rice et al., 2008</xref>; <xref ref-type="bibr" rid="bib50">Mogil, 2009</xref>) and on clinical trial design (<xref ref-type="bibr" rid="bib12">Dworkin et al., 2011</xref>; <xref ref-type="bibr" rid="bib44">Mao, 2012</xref>). An alternative possibility is that degeneracy within the pain system allows the pathogenic process to circumvent single-target-drugs. If true, the single-target-drug paradigm is bound to fail no matter how good the animal models or clinical trials are, and a new paradigm is needed.</p><p>Degeneracy refers to multiple ‘different’ mechanisms conveying equivalent function (<xref ref-type="bibr" rid="bib13">Edelman and Gally, 2001</xref>); by comparison, redundancy refers to multiple instantiations of the ‘same’ mechanism. Both convey robustness to complex systems (<xref ref-type="bibr" rid="bib35">Kitano, 2004a</xref>). But if a pathogenic process were to hijack degeneracy, the pathological state could itself become robust, or in other words resistant to treatment. Accordingly, degeneracy is recognized as an important factor for cancer and other complex diseases (<xref ref-type="bibr" rid="bib35">Kitano, 2004a</xref>, <xref ref-type="bibr" rid="bib36">2004b</xref>; <xref ref-type="bibr" rid="bib60">Tian et al., 2011</xref>), including epilepsy (<xref ref-type="bibr" rid="bib37">Klassen et al., 2011</xref>) but has yet to inform pain research or analgesic drug development. That said, degeneracy has been recognized in neural systems (<xref ref-type="bibr" rid="bib55">Prinz et al., 2004</xref>) and its existence and functional implications are gaining increasing attention (<xref ref-type="bibr" rid="bib21">Grashow et al., 2009</xref>, <xref ref-type="bibr" rid="bib22">2010</xref>; <xref ref-type="bibr" rid="bib46">Marder, 2011</xref>; <xref ref-type="bibr" rid="bib1">Amendola et al., 2012</xref>; <xref ref-type="bibr" rid="bib69">Zhao and Golowasch, 2012</xref>; <xref ref-type="bibr" rid="bib23">Gutierrez et al., 2013</xref>; <xref ref-type="bibr" rid="bib52">O’Leary et al., 2013</xref>; <xref ref-type="bibr" rid="bib56">Ransdell et al., 2013</xref>).</p><p>A degenerate basis for neuropathic pain is suggested by basic research findings. For example, increased function of the sodium channel Na<sub>v</sub>1.8 is sufficient to produce the hypersensitivity associated with neuropathic pain (<xref ref-type="bibr" rid="bib3">Bierhaus et al., 2012</xref>; <xref ref-type="bibr" rid="bib65">Wu et al., 2012</xref>). Na<sub>v</sub>1.8-targetted manipulations predictably fail in Na<sub>v</sub>1.8 knock-out animals, but those animals can nonetheless develop injury-induced hypersensitivity (<xref ref-type="bibr" rid="bib51">Nassar et al., 2005</xref>). This indicates that injury-induced changes in other ion channels—and indeed many such changes occur (for reviews, see <xref ref-type="bibr" rid="bib4">Campbell and Meyer, 2006</xref>; <xref ref-type="bibr" rid="bib64">Woolf and Ma, 2007</xref>; <xref ref-type="bibr" rid="bib2">Basbaum et al., 2009</xref>; <xref ref-type="bibr" rid="bib45">Marchand et al., 2009</xref>; <xref ref-type="bibr" rid="bib19">Gold and Gebhart, 2010</xref>)—are also sufficient to cause neuropathic pain.</p><p>To directly explore degeneracy in the context of neuropathic pain, we tested whether multiple distinct molecular pathologies are sufficient to produce the cellular hyperexcitability associated with neuropathic pain. Hyperexcitability characterized by quantitative changes such as reduced threshold and qualitative changes such as altered spiking patterns (see below) develops at multiple points along the neuraxis (<xref ref-type="bibr" rid="bib8">Costigan et al., 2009</xref>). This includes primary somatosensory afferents whose spontaneous spiking and exaggerated responsiveness are thought to underlie spontaneous pain and hypersensitivity, respectively (for review, see <xref ref-type="bibr" rid="bib10">Devor, 2005</xref>). Notably, peripheral input helps drive central sensitization (<xref ref-type="bibr" rid="bib9">Devor, 1991</xref>; <xref ref-type="bibr" rid="bib20">Gracely et al., 1992</xref>; <xref ref-type="bibr" rid="bib38">Koltzenburg et al., 1994</xref>), meaning increased peripheral input is amplified rather than attenuated centrally.</p><p>Injury-induced hyperexcitability is not limited to nociceptors; on the contrary, hyperexcitability also develops in myelinated afferents that normally convey innocuous information but that contribute to mechanical allodynia (hypersensitivity) under neuropathic conditions (<xref ref-type="bibr" rid="bib5">Campbell et al., 1988</xref>; <xref ref-type="bibr" rid="bib38">Koltzenburg et al., 1994</xref>; <xref ref-type="bibr" rid="bib11">Devor, 2009</xref>; <xref ref-type="bibr" rid="bib34">King et al., 2011</xref>). Hyperexcitability in myelinated afferents is characterized by a triad of qualitative changes that include repetitive spiking, membrane potential oscillations (MPOs), and bursting (<xref ref-type="bibr" rid="bib42">Liu et al., 2000</xref>; <xref ref-type="bibr" rid="bib26">Herzog et al., 2001</xref>; <xref ref-type="bibr" rid="bib68">Xing et al., 2001</xref>; <xref ref-type="bibr" rid="bib41">Liu et al., 2002</xref>; <xref ref-type="bibr" rid="bib43">Ma and LaMotte, 2007</xref>; <xref ref-type="bibr" rid="bib16">Fan et al., 2011</xref>; <xref ref-type="bibr" rid="bib67">Xie et al., 2011</xref>; <xref ref-type="bibr" rid="bib59">Song et al., 2012</xref>). We refer to this qualitatively altered excitability as ‘neuropathic’. We recently showed through mathematical modeling that all three neuropathic changes arise from a switch in spike initiation dynamics (<xref ref-type="bibr" rid="bib57">Rho and Prescott, 2012</xref>); there is, therefore, a one-to-many mapping between altered spike initiation dynamics and qualitative changes in cellular excitability. On the other hand, we found a many-to-one mapping between parameter values (whose variations represent injury-induced molecular changes) and cellular excitability; moreover, continuous parameter variations led to a discontinuous change in spike initiation dynamics. The many-to-one mapping constitutes degeneracy and the discontinuity, or tipping point, reflects criticality. The results identify spike initiation as a key nonlinear process whose qualitative alteration explains multiple features of cellular hyperexcitability on the basis of several possible molecular pathologies.</p><p>In the current study, we experimentally tested our theory. After identifying the activation properties of currents affecting spike initiation, we used nonlinear dynamical analysis of our mathematical model to define the system’s tipping point. From this, we generated several predictions as to how that tipping point could be crossed. To experimentally test those predictions, we applied different manipulations designed to force neurons in one or the other direction across their tipping point, therein acutely reproducing neuropathic excitability in primary afferent neurons from naive animals or acutely reversing neuropathic excitability in neurons from nerve-injured animals. Manipulations involved decreasing and/or increasing subthreshold currents via pharmacology and/or dynamic clamp, respectively, in identified myelinated primary afferents. All predictions were confirmed, therein supporting our theory that primary afferent hyperexcitability arises through a critical transition whose molecular basis is highly degenerate.</p></sec><sec id="s2" sec-type="results"><title>Results</title><sec id="s2-1"><title>Theory and modeling</title><p>According to our previous theoretical work, spike initiation occurs through a time- and voltage-dependent competition between <italic>net</italic> fast-activating inward current and <italic>net</italic> slower-activating outward current (<xref ref-type="bibr" rid="bib53">Prescott et al., 2008</xref>). Because multiple types of ion channels contribute to each <italic>net</italic> current—currents with similar kinetics sum linearly (<xref ref-type="bibr" rid="bib30">Kepler et al., 1992</xref>)—injury-induced changes in any one of the contributing ion channels can bias the competition (<xref ref-type="bibr" rid="bib57">Rho and Prescott, 2012</xref>). Using a Morris–Lecar model that comprises only two variables whose interaction is sufficient to produce spikes, we adjusted parameters (‘Materials and methods’) to give a spike threshold approximating that observed in the soma of myelinated afferents, which is about −35 mV. The parameters of this base model were thereafter unchanged. Next, we derived the voltage-dependency and kinetics (<xref ref-type="fig" rid="fig1">Figure 1A</xref>) for an additional conductance (<xref ref-type="disp-formula" rid="equ1 equ2 equ3 equ4">Equations 1–4</xref> in ‘Materials and methods’) which, when added to our base model, modulates its spike initiation dynamics. This conductance corresponds to either a sodium or potassium channel depending on its associated reversal potential and, according to our analysis, must be active at subthreshold voltages. To be clear, the parameters for this conductance were chosen to modulate spike initiation dynamics, not to model specific ion channel types that are known to be altered by nerve injury. That said, although our determination of parameters was agnostic to molecular identities, parameter values resemble those of known channel types, as noted in subsequent sections. We then varied the maximal conductance of the additional conductance(s). Based on two-parameter bifurcation analysis, we determined the combinations of sodium conductance <italic>ḡ</italic><sub>Na</sub> and potassium conductance <italic>ḡ</italic><sub>K</sub> required to give repetitive spiking at different stimulus thresholds (<xref ref-type="fig" rid="fig1">Figure 1B</xref>). Somata of myelinated primary afferents normally generate a single spike at stimulus onset, which corresponds to the gray-shaded region, whereas a subset of those neurons spike repetitively after nerve injury (e.g., <xref ref-type="bibr" rid="bib41">Liu et al., 2002</xref>), which corresponds to the colored region.<fig-group><fig id="fig1" position="float"><object-id pub-id-type="doi">10.7554/eLife.02370.003</object-id><label>Figure 1.</label><caption><title>Modeling and theory.</title><p>(<bold>A</bold>) Voltage-dependency and kinetics of subthreshold conductance. In bottom panel, τ = (α+β)<sup>−1</sup> where α and β are defined in <xref ref-type="disp-formula" rid="equ3">Equations 3</xref> and <xref ref-type="disp-formula" rid="equ4">4</xref>. Equivalent activation parameters were used to implement either a sodium or potassium current based on reversal potential. (<bold>B</bold>) Two-parameter bifurcation analysis in which <italic>ḡ</italic><sub>Na</sub> and <italic>ḡ</italic><sub>K</sub> were co-varied to determine the <italic>ḡ</italic><sub>Na</sub>:<italic>ḡ</italic><sub>K</sub> ratio for a given threshold for repetitive spiking. Gray shading shows parameter regime associated with onset-only spiking. Changes in excitability are explained by a switch in spike initiation dynamics (<xref ref-type="fig" rid="fig1s1">Figure 1—figure supplement 1</xref>). (<bold>C</bold>) Predictions (indicated by numbered arrows) of how manipulating <italic>ḡ</italic><sub>Na</sub> and/or <italic>ḡ</italic><sub>K</sub> may force the system across a tipping point, thus reproducing or reversing neuropathic excitability. Position of the tipping point depends on many other parameters including leak conductance (<bold>D</bold>), voltage-dependency of <italic>ḡ</italic><sub>Na</sub> and <italic>ḡ</italic><sub>K</sub> (<bold>E</bold>), and potassium reversal potential (<bold>F</bold>).</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02370.003">http://dx.doi.org/10.7554/eLife.02370.003</ext-link></p></caption><graphic xlink:href="elife02370f001"/></fig><fig id="fig1s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.02370.004</object-id><label>Figure 1—figure supplement 1.</label><caption><title>Spike initiation dynamics differ between normal and neuropathic conditions.</title><p>In bifurcation analysis, a parameter such as stimulus intensity <italic>I</italic><sub>stim</sub> is systematically varied to determine at what value the system qualitatively changes behavior, which is to say it undergoes a bifurcation. We repeated this analysis in a model with the <italic>ḡ</italic><sub>Na</sub>:<italic>ḡ</italic><sub>K</sub> ratio set to give normal excitability (<italic>ḡ</italic><sub>Na</sub> = 2.0 µS/cm<sup>2</sup>; <italic>ḡ</italic><sub>K</sub> = 2.5 µS/cm<sup>2</sup>) or neuropathic excitability (<italic>ḡ</italic><sub>Na</sub> = 2.5 µS/cm<sup>2</sup>; <italic>ḡ</italic><sub>K</sub> = 2.0 µS/cm<sup>2</sup>). Bifurcation diagrams (left) illustrate how the system behaves at different <italic>I</italic><sub>stim</sub>: a stable fixed point corresponds to quiescence whereas a stable limit cycle corresponds to repetitive spiking. A subcritical Hopf bifurcation is evident in the bottom (<italic>Neuropathic</italic>) bifurcation diagram but not in the top (<italic>Normal</italic>) bifurcation diagram, suggesting that repetitive spiking is only possible in the former condition. In both conditions, the <italic>I</italic><sub>stim</sub> range in which a single spike is produced at stimulus onset is indicated with gray shading. Onset-only spiking is achieved through a quasi-separatrix crossing, independent of any bifurcation. Dotted arrows indicate <italic>I</italic><sub>stim</sub> values for sample traces on the right. Membrane potential oscillations (MPOs) occur when the fixed point is in proximity to a Hopf bifurcation; after being perturbed by noise, the system relaxes towards its nearly-unstable fixed point following a loose spiral trajectory, which corresponds to MPOs on the time series. In the absence of a Hopf bifurcation, the fixed point is more stable and MPOs are negligible. Repetitive spiking occurs when <italic>I</italic><sub>stim</sub> exceeds the intensity required for the Hopf bifurcation. Repetitive spiking can also occur in the <italic>I</italic><sub>stim</sub> range between the Hopf bifurcation and where the stable and unstable limit cycles meet (at a saddle-node bifurcation of limit cycles); that region also contains a stable fixed point, meaning that the system is bistable. The neuron will remain in one or the other stable state unless a slow process like adaptation sweeps the system across the bistable region, in which case the neuron will remain quiescent when sweeping rightward toward the Hopf bifurcation, while spiking repetitively when sweeping leftward toward the saddle-node bifurcation of limit cycles, thereby producing bursts as a consequence of hysteresis. Bistability and adaptation are both required for bursting.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02370.004">http://dx.doi.org/10.7554/eLife.02370.004</ext-link></p></caption><graphic xlink:href="elife02370fs001"/></fig></fig-group></p><p>Neuropathic changes in excitability arise from a switch in spike initiation dynamics (<xref ref-type="bibr" rid="bib57">Rho and Prescott, 2012</xref>; <xref ref-type="fig" rid="fig1s1">Figure 1—figure supplement 1</xref>). In brief, the colored region of <xref ref-type="fig" rid="fig1">Figure 1B</xref> corresponds to a parameter regime in which a subcritical Hopf bifurcation occurs when stimulus intensity <italic>I</italic><sub>stim</sub> reaches a critical value <italic>I</italic>*. A Hopf bifurcation represents destabilization of the ‘resting’ state: repetitive spiking occurs when <italic>I</italic><sub>stim</sub> exceeds <italic>I</italic>*, membrane potential oscillations (MPOs) arise in the presence of noise when <italic>I</italic><sub>stim</sub> approaches <italic>I</italic>*, and bursting occurs when adaptation <italic>I</italic><sub>adapt</sub> causes <italic>I</italic><sub>stim</sub>−<italic>I</italic><sub>adapt</sub> to sweep back and forth across <italic>I</italic>*. Outside the colored region of <xref ref-type="fig" rid="fig1">Figure 1B</xref>, spike initiation is effectively limited to a quasi-separatrix crossing; according to this mechanism, the resting state remains stable but a single spike is produced if the system transiently escapes from that state during a stimulus transient. In the absence of a Hopf bifurcation, repetitive spiking, MPOs and bursting are simply not possible.</p><p>The boundary between gray and blue regions in our colored ‘excitability’ diagram (<xref ref-type="fig" rid="fig1">Figure 1B</xref>), thus represents the critical tipping point separating normal and neuropathic parameter regimes. That tipping point is represented by a simple curve in subsequent excitability diagrams. We predict that a cell with normal excitability can be converted to neuropathic excitability (i.e., forced across its tipping point) by a decrease in <italic>ḡ</italic><sub>K</sub>, an increase in <italic>ḡ</italic><sub>Na</sub>, or some combination thereof (arrows 1–3 on <xref ref-type="fig" rid="fig1">Figure 1C</xref>). However, balanced conductance changes may offset one another, resulting in no qualitative alteration of excitability (arrow 4). Contrariwise, the inverse changes are predicted to normalize excitability in a hyperexcitable cell by forcing the system in the opposite direction across its tipping point (arrows 5 and 6). Each arrow corresponds to a prediction tested in the experimental portion of this study.</p><p>Changes in other conductances, in parameters other than maximal conductance (e.g., activation properties), or in parameters not strictly connected to conductance (e.g., reversal potential) can all affect where the tipping point lies (<xref ref-type="fig" rid="fig1">Figure 1D–F</xref>, respectively). It is not feasible to experimentally test all parameter variations and combinations thereof, and hence we focused on predictions outlined in <xref ref-type="fig" rid="fig1">Figure 1C</xref>. But one should bear in mind that the tipping point in a real neuron will depend on numerous parameters and thus correspond to a boundary within a high-dimensional space. The important considerations are (1) that that boundary exists (which implies criticality), and (2) that there are many ways to cross it (which implies degeneracy). In other words, our theory predicts that excitability can change abruptly on the basis of many different molecular changes when and if those molecular changes reach a tipping point.</p></sec><sec id="s2-2"><title>Reproducing neuropathic excitability in neurons from naive animals</title><p>Our next step was to test our predictions experimentally. Unlike typical experiments in which excitability is compared between different neurons before and after nerve injury, and where it is impossible to account for all injury-induced molecular changes and between-cell differences, we compared excitability in the same neuron before and after strictly controlled manipulations dictated by our simulation results. Furthermore, rather than trying to reproduce a complete set of injury-induced molecular pathologies, our manipulations were designed to reproduce one or two of those pathologies at a time. This approach reveals which molecular pathologies, alone or together, are sufficient to reproduce neuropathic excitability without unaccounted for co-variations in other parameters. Furthermore, by applying manipulations acutely, our approach avoids the confounding influence of compensatory changes that might develop over longer time scales.</p><p>We targeted myelinated afferents (somatic diameter ≥30 µm) because they are directly implicated in allodynia and because they have been reported to develop the triad of neuropathic excitability changes described in the ‘Introduction’. We initially targeted myelinated cutaneous afferents retrogradely labeled by DiI injected intradermally into the hindpaw (‘Materials and methods’) because this population comprises low-threshold mechanosensors that may subserve mechanical allodynia; we subsequently tested labeled muscle afferents, as reported at the end of the ‘Results’ section. In keeping with previous studies, neurons harvested from naive, uninjured rats responded to a square pulse of current injection with onset-only spiking, even at high stimulus intensities (<xref ref-type="fig" rid="fig2">Figure 2A</xref>; <italic>n</italic> = 23 cutaneous afferents). Neurons could spike again in response to subsequent increments in stimulus intensity (<xref ref-type="fig" rid="fig2">Figure 2A</xref>, inset), thus ruling out complete sodium channel inactivation as the basis for non-repetitive spiking. This spiking pattern places naive neurons within the gray-shaded region of our excitability diagram.<fig id="fig2" position="float"><object-id pub-id-type="doi">10.7554/eLife.02370.005</object-id><label>Figure 2.</label><caption><title>Reproduction of neuropathic excitability in naive cutaneous neurons.</title><p>(<bold>A</bold>) Medium- to large-diameter (putative myelinated) cutaneous afferents from naive rats respond to depolarization with a single spike at stimulus onset. Neurons can respond to increments in stimulus intensity (inset), which demonstrates that sodium channels are not completely inactivated after the initial spike. (<bold>B</bold>) Prediction 1 was tested by blocking potassium current by application of 1.5–5 mM 4-AP. By repeating stimulation during wash-in and/or wash-out of the drug, we observed repetitive spiking and MPOs (in the order expected based on the change in drug concentration) in response to equivalent stimulation. Gray-shaded windows show enlarged views of voltage traces, with colors corresponding to the power spectra shown at the top. (<bold>C</bold>) Prediction 2 was tested by adding virtual sodium conductance via dynamic clamp. As predicted, MPOs and repetitive spiking developed as virtual sodium conductance was increased, but bursting was absent. Note that a conductance density of 1 nS/pF corresponds to 1 mS/cm<sup>2</sup> assuming a specific membrane capacitance of 1 μF/cm<sup>2</sup>, which means that the conductance densities added by dynamic clamp are the same order of magnitude as those predicted by modeling in <xref ref-type="fig" rid="fig1">Figure 1B</xref>. (<bold>D</bold>) Bursting was achieved in repetitively spiking neurons by introducing an AHP-type adaptation current via dynamic clamp.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02370.005">http://dx.doi.org/10.7554/eLife.02370.005</ext-link></p></caption><graphic xlink:href="elife02370f002"/></fig></p><p>Our first prediction was that decreasing subthreshold potassium conductance would reproduce neuropathic changes in excitability, consistent with injury-induced reduction of K<sub>v</sub>1 channels (<xref ref-type="bibr" rid="bib15">Everill and Kocsis, 1999</xref>; <xref ref-type="bibr" rid="bib29">Ishikawa et al., 1999</xref>; <xref ref-type="bibr" rid="bib32">Kim et al., 2002</xref>; <xref ref-type="bibr" rid="bib24">Hammer et al., 2010</xref>; <xref ref-type="bibr" rid="bib70">Zhao et al., 2013</xref>). To test this, we applied 4-aminopyridine (4-AP) with a maximal concentration between 1.5 mM and 5 mM to decrease the total potassium current activated at subthreshold voltages. As drug concentration increased during wash-in, MPOs developed followed by a switch to repetitive spiking during stimulation; the reverse sequence occurred during washout (<xref ref-type="fig" rid="fig2">Figure 2B</xref>). The broad peak in the power spectrum (<xref ref-type="fig" rid="fig2">Figure 2B</xref>, inset) shows that pharmacologically induced MPOs are consistent with previous descriptions of injury-induced MPOs (<xref ref-type="bibr" rid="bib59">Song et al., 2012</xref>) and with a noise-dependent mechanism (<xref ref-type="bibr" rid="bib57">Rho and Prescott, 2012</xref>). In 6 of 11 neurons tested, MPOs and repetitive spiking co-developed during wash-in of 4-AP, consistent with their common connection with the subcritical Hopf bifurcation. Bursting was not observed (see below). Using 5 nM α-dendrotoxin, which more selectively blocks K<sub>v</sub>1 channels, we obtained equivalent results in 2 of 4 additional neurons (data not shown). Reproduction of neuropathic excitability via potassium channel blockade in a total of 8 of 15 neurons represents a significantly higher conversion rate than expected by chance (i.e., 0 of 23 neurons, as determined from the excitability at the start and end of recordings in each neuron; p<0.001; Fisher’s exact test).</p><p>Our second prediction was that increasing subthreshold sodium conductance would reproduce neuropathic excitability, consistent with injury-induced increase of Na<sub>v</sub>1.3 (<xref ref-type="bibr" rid="bib62">Waxman et al., 1994</xref>; <xref ref-type="bibr" rid="bib31">Kim et al., 2001</xref>; <xref ref-type="bibr" rid="bib18">Fukuoka et al., 2008</xref>; <xref ref-type="bibr" rid="bib28">Huang et al., 2008</xref>). To test this, we added a virtual sodium conductance with the activation properties described in <xref ref-type="fig" rid="fig1">Figure 1A</xref>. As expected, increasing the virtual sodium current produced MPOs and repetitive spiking (<xref ref-type="fig" rid="fig2">Figure 2C</xref>). Power spectral analysis (<xref ref-type="fig" rid="fig2">Figure 2C</xref>, inset) shows that dynamic clamp-induced MPOs are comparable to injury- and pharmacologically-induced MPOs. In 9 of 19 neurons tested, MPOs and repetitive spiking co-developed when sufficient virtual sodium conductance was inserted; this conversion rate is significantly higher than expected by chance (p<0.001; Fisher’s exact test). Of the 10 neurons that did not develop repetitive spiking, five nonetheless developed MPOs. Again, bursting was not observed.</p><p>Before proceeding to test predictions 3 and 4, we sought to explain the absence of bursting. Our modeling indicated that although the subcritical Hopf bifurcation is necessary to create a stimulus range in which the system is bistable (i.e., in which quiescence or repetitive spiking are both possible), bursting also requires a slow process like spike-dependent adaptation to sweep the system back and forth across the bistable region, thus allowing hysteresis to manifest bursting; in other words, bursting requires a subcritical Hopf bifurcation and adaptation (<xref ref-type="bibr" rid="bib57">Rho and Prescott, 2012</xref>). We therefore hypothesized that our repetitively spiking neurons, although capable of bursting, lacked the required adaptation. To test this, we inserted a virtual adaptation current (‘Materials and methods’). As expected, this enabled bursting under conditions associated with repetitive spiking in all three cells tested (<xref ref-type="fig" rid="fig2">Figure 2D</xref>) but had no effect in the same three cells under normal conditions with onset-only spiking (data not shown). These data confirm that bursting would have co-developed with repetitive spiking and MPOs if adaptation currents had been present, and suggest that adaptation currents are upregulated after nerve injury (possibly as a compensatory measure) rather than existing occultly in naive neurons whose spiking is already transient.</p></sec><sec id="s2-3"><title>Additivity and subtractivity of conductance changes</title><p>In a subset of neurons, neither decreasing K<sub>v</sub>1 potassium conductance nor increasing Na<sub>v</sub>1.3-like sodium conductance reproduced neuropathic excitability. Larger manipulations may have succeeded in forcing neurons across their tipping point but technical considerations (e.g., off-target effects at high drug concentrations and recording instability for large virtual conductances) limited the magnitude of manipulations. But as explained in <xref ref-type="fig" rid="fig1">Figure 1</xref>, variation of a single conductance is not the only way to force a neuron across its tipping point; instead, small changes in more than one conductance may combine to produce a net conductance change that is large enough to force the neuron across its tipping point. As an aside, if multiple small changes can recombine in different ways (which is plausible under conditions in which dozens of different ion channels are up- or down-regulated), then degeneracy could exist on the basis of many different combinations being sufficient to produce neuropathic excitability; in other words, no one set of combined changes would be uniquely necessary just as no single change is uniquely necessary. To explore how conductance changes combine, we proceeded with testing predictions 3 and 4.</p><p>Our third prediction was that manipulations tested independently in predictions 1 and 2 could combine to reproduce neuropathic changes in excitability; indeed, in 8 neurons in which neither decreasing potassium conductance nor increasing sodium conductance was sufficient to produce repetitive spiking and MPOs, we tested the two manipulations together and found that the combination was sufficient to produce hyperexcitability in 7 of them (<xref ref-type="fig" rid="fig3">Figure 3A</xref>). We also observed that the minimum virtual sodium conductance needed to reproduce neuropathic excitability was reduced during 4-AP application (<xref ref-type="fig" rid="fig3">Figure 3B</xref>). We quantified the degree to which potassium channel blockade had moved the system towards its tipping point by comparing the minimum virtual sodium conductance needed to force the system across its tipping point without vs with 4-AP. The median (and 25–75 percentile range) virtual sodium conductance was significantly reduced from 0.51 (range 0.47–0.63) nS/pF without 4-AP to 0.23 (range 0.13–0.40) nS/pF with 4-AP (p<0.001; Mann–Whitney U test) (<xref ref-type="fig" rid="fig3">Figure 3C</xref>). Non-parametric statistics were used because of the non-Gaussian distribution of data points.<fig id="fig3" position="float"><object-id pub-id-type="doi">10.7554/eLife.02370.006</object-id><label>Figure 3.</label><caption><title>Additivity and subtractivity of conductance changes.</title><p>(<bold>A</bold>) Example in which neither virtual sodium conductance nor potassium channel blockade reproduced neuropathic excitability, whereas the combination of manipulations did, thus confirming prediction 3, that manipulations can be additive. (<bold>B</bold>) Example in which the minimum virtual sodium conductance required to reproduce neuropathic excitability was reduced when combined with potassium channel blockade. (<bold>C</bold>) Even when 4-AP application did not, on its own, produce repetitive spiking, it nonetheless moved the neuron significantly closer to criticality as evidenced by comparing the minimal virtual sodium conductance needed to produce repetitive spiking without vs with 4-AP (*p<0.001; Mann–Whitney U test). Points represent data from individual neurons and boxes represent median and 25–75 percentile range. Repetitive spiking caused by 4-AP application (<bold>D</bold>) or by virtual sodium conductance (<bold>E</bold>) was reversed by insertion of virtual potassium conductance. Panel <bold>E</bold> confirms prediction 4, that manipulations can be subtractive.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02370.006">http://dx.doi.org/10.7554/eLife.02370.006</ext-link></p></caption><graphic xlink:href="elife02370f003"/></fig></p><p>Our fourth prediction was that different manipulations could offset one another, resulting in no net change in excitability. To test this, we first tested whether inserting a virtual potassium conductance could reverse the hyperexcitability induced by blockade of native potassium conductance by 4-AP. Results confirmed our prediction in 3 of 3 neurons tested (<xref ref-type="fig" rid="fig3">Figure 3D</xref>), thus demonstrating the specificity of the 4-AP effect; in other words, even if channels other than K<sub>v</sub>1 were blocked by 4-AP, the effect of 4-AP on excitability is attributable to blockade of subthreshold potassium current given that reintroducing that type of current reverses the altered excitability. More interesting is the observation that hyperexcitability caused by insertion of a virtual sodium conductance was reversed by insertion of a virtual potassium conductance in 3 of 3 neurons tested (<xref ref-type="fig" rid="fig3">Figure 3E</xref>). These data confirm prediction 4 and demonstrate the subtractivity of different manipulations.</p></sec><sec id="s2-4"><title>Reversing neuropathic excitability in neurons from nerve-injured animals</title><p>In our final two predictions, we sought to move neurons in the opposite direction across their tipping point, which required neurons rendered hyperexcitable by nerve injury (‘Materials and methods’). Surprisingly, only 1 of 9 injured cutaneous afferents exhibited any degree of repetitive spiking defined here as at least three spikes during sustained stimulation, which is not a significant change compared to chance (p=0.28; Fisher’s exact test compared to spontaneous conversion rate of 0 in 23 cutaneous afferents). By comparison, 6 of 9 injured neurons not labeled by intradermal injection of DiI exhibited neuropathic excitability, an example of which is shown in <xref ref-type="fig" rid="fig4">Figure 4A</xref> (p<0.001; Fisher’s exact test compared to 1 in 9 cutaneous). However, because the identity of unlabeled neurons is uncertain and given previous studies showing that muscle afferents are in fact more prone than cutaneous afferents to becoming grossly hyperexcitable (<xref ref-type="bibr" rid="bib49">Michaelis et al., 2000</xref>; <xref ref-type="bibr" rid="bib41">Liu et al., 2002</xref>), we retrogradely labeled muscle afferents. 11 of 20 injured muscle afferents exhibited neuropathic excitability (<xref ref-type="fig" rid="fig4">Figure 4B</xref>), which is a significantly higher proportion than 1 in 9 injured cutaneous afferents (p<0.05; Fisher’s exact test) and the spontaneous conversion rate of 0 in 27 muscle afferents from naive animals (p<0.001; Fisher’s exact test). Neuropathic excitability was reversed by insertion of virtual potassium conductance in 10 of 10 neurons tested or by reduction of sodium conductance via application of 10 µM riluzole in 4 of 4 neurons tested, thus confirming predictions 5 and 6, respectively (<xref ref-type="fig" rid="fig4">Figure 4A,B</xref>). In the injured muscle afferents whose neuropathic excitability was reversed by insertion of virtual potassium conductance, the median conductance (and 25–75 percentile range) needed to cause reversal was only 0.14 (range 0.12–0.15) nS/pF. This suggests that the neuropathic state lies close to the tipping point, which is consistent with our 100% success rate in reversing neuropathic excitability by manipulating either potassium or sodium conductance (see above).<fig id="fig4" position="float"><object-id pub-id-type="doi">10.7554/eLife.02370.007</object-id><label>Figure 4.</label><caption><title>Reversal of neuropathic excitability in neurons from nerve-inured rats.</title><p>(<bold>A</bold>) Typical response of an injured unlabeled primary afferent (top). Repetitive spiking and MPOs were abolished by insertion of a potassium conductance (middle) or by blockade of sodium channels using 10 μM riluzole (bottom), thus confirming predictions 5 and 6, respectively. (<bold>B</bold>) Typical response of an injured muscle afferent whose neuropathic excitability was similarly reversed by increased potassium conductance and reduced sodium conductance. (<bold>C</bold>) Although only 1 of 9 injured cutaneous afferents exhibited repetitive spiking and MPOs, those afferents were nonetheless closer to criticality insofar as they required significantly less virtual sodium conductance than uninjured cutaneous afferents to reach criticality (*p<0.05; Mann–Whitney U test). Points represent data from individual neurons and boxes represent median and 25–75 percentile range.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02370.007">http://dx.doi.org/10.7554/eLife.02370.007</ext-link></p></caption><graphic xlink:href="elife02370f004"/></fig></p><p>The concept of measuring proximity to the tipping point prompted us to ask whether cutaneous afferents, only one of which exhibited grossly abnormal excitability after nerve injury, were nonetheless closer to their tipping point after nerve injury than under control conditions. In the four injured cutaneous afferents to which we added virtual sodium conductance, only 0.30 (range 0.14–0.44) nS/pF of sodium conductance was required to reproduce neuropathic excitability, which is significantly less than the 0.51 (range 0.47–0.63) nS/pF required to reproduce neuropathic excitability in uninjured cutaneous afferents (p<0.05; Mann–Whitney test) (<xref ref-type="fig" rid="fig4">Figure 4C</xref>). Hence, nerve injury leads to qualitatively altered excitability in only a subset of cells (i.e., ones that cross their tipping point) but the remaining cells are nonetheless quantitatively more excitable (i.e., closer to their tipping point).</p></sec><sec id="s2-5"><title>Differential susceptibility of afferent subtypes to developing neuropathic excitability</title><p>Using neurons from naive rats, we subsequently verified that muscle afferents, like cutaneous afferents described in <xref ref-type="fig" rid="fig2">Figure 2</xref>, normally exhibit onset-only spiking (<italic>n</italic> = 27 muscle afferents). Neuropathic excitability was reproduced in 16 of 20 muscle afferents by insertion of virtual sodium conductance (<xref ref-type="fig" rid="fig5">Figure 5A</xref>), which is a significantly greater conversion rate than 9 of 19 cutaneous afferents (p<0.05; Fisher’s exact test). Moreover, we found that the median sodium conductance required to reproduce neuropathic excitability was 0.31 (range 0.25–0.41) nS/pF in muscle afferents compared with 0.51 (range 0.47–0.63) nS/pF in cutaneous afferents, indicating that naive muscle afferents are significantly closer to their tipping point than naive cutaneous afferents (p<0.001; Mann–Whitney test) (<xref ref-type="fig" rid="fig5">Figure 5B</xref>). Because we cannot isolate the density of native Na<sub>V</sub>1.3 cannels, we cannot gauge the relative change that this absolute virtual conductance represents. Furthermore, our measurement represents the distance to tipping point along only one dimension; if criticality exists along a boundary in a high-dimensional space (<xref ref-type="fig" rid="fig1">Figure 1</xref>), then the distance to tipping point may differ significantly along different dimensions. But notably, input resistance did not differ significantly between cutaneous and muscle afferents (p=0.2; unpaired <italic>t</italic> test; 282 ± 47 MΩ in cutaneous afferents vs 377 ± 57 MΩ in muscle afferents). Overall, these data suggest that muscle afferents are more prone to developing neuropathic excitability after nerve injury because they operate closer to their tipping point than do cutaneous afferents.<fig id="fig5" position="float"><object-id pub-id-type="doi">10.7554/eLife.02370.008</object-id><label>Figure 5.</label><caption><title>Reproduction of neuropathic excitability in naive muscle afferents.</title><p>(<bold>A</bold>) As predicted, repetitive spiking developed as virtual sodium conductance was increased. (<bold>B</bold>) Uninjured muscle afferents operate significantly closer to criticality than uninjured cutaneous afferents based on the minimum virtual sodium conductance needed to produce repetitive spiking (***p<0.001; Mann–Whitney U test). In <bold>B</bold> and <bold>D</bold>, points represent data from individual neurons and boxes represent median and 25–75 percentile range. These data suggest that muscle afferents are more prone to develop neuropathic excitability after nerve injury because they start off closer to their tipping point, and not necessarily because injury induces a larger change in membrane conductances. (<bold>C</bold>) Based on the observation that 4-AP and dendrotoxin (DTX) produced repetitive spiking in only 1 of 12 muscle afferents, we measured the DTX-sensitive current. The persistent outward current activated at perithreshold voltages (−50 to −30 mV) and blocked by 10 nM DTX was significantly smaller in muscle afferents than in cutaneous afferents (**p<0.01; two-way ANOVA and post-hoc Student-Newman-Keuls test). (<bold>D</bold>) Although 4-AP produced repetitive spiking in only 1 of 12 muscle afferents, it nonetheless moved neurons closer to criticality as evidenced by the minimal virtual sodium conductance needed to cause hyperexcitability after 4AP application (*p<0.05; Mann–Whitney U test).</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02370.008">http://dx.doi.org/10.7554/eLife.02370.008</ext-link></p></caption><graphic xlink:href="elife02370f005"/></fig></p><p>We also tested whether neuropathic excitability could be reproduced in muscle afferents by reduction of potassium conductance by application or 4-AP or α-dendrotoxin. Contrary to the expectations, 4-AP and dendrotoxin reproduced neuropathic excitability in only 1 of 12 muscle afferents, which is a significantly lower conversion rate than 8 of 15 cutaneous afferents (p<0.05; Fisher’s exact test). We hypothesized that this was either (1) because muscle afferents are further than cutaneous afferents from their tipping point or (2) because K<sub>V</sub>1 expression is lower in muscle afferents. Because hypothesis 1 is inconsistent with data in <xref ref-type="fig" rid="fig5">Figure 5B</xref>, we tested hypothesis 2 by comparing the density of dendrotoxin-sensitive current in muscle and cutaneous afferents measured in voltage clamp after blockade of sodium channels using 1 μM tetrodotoxin; dendrotoxin was used for these experiments because it is a more selective K<sub>V</sub>1 blocker than 4-AP. As predicted, the sustained outward current activated at perithreshold voltages (−50 mV to −30 mV) and blocked by 10 nM dendrotoxin was significantly smaller in muscle afferents than in cutaneous afferents (p<0.01; two-way ANOVA and post-hoc Student-Newman-Keuls test; 1.7 ± 0.6 pA/pF in muscle afferents vs 4.2 ± 0.6 pA/pF in cutaneous afferents after removing variance attributable to voltage) (<xref ref-type="fig" rid="fig5">Figure 5C</xref>). With fewer K<sub>V</sub>1 channels available to be blocked, 4-AP and dendrotoxin naturally have a smaller impact on neuronal excitability. However, amongst muscle afferents in which neuropathic excitability was not reproduced by application of 4-AP and in which virtual sodium conductance was subsequently inserted, significantly less sodium conductance was needed to reproduce neuropathic excitability with 4-AP (0.22, range 0.14–0.27 nS/pF) than without 4-AP (0.31, range 0.25–0.41 nS/pF) (p<0.05; Mann–Whitney U test) (<xref ref-type="fig" rid="fig5">Figure 5D</xref>). This indicates that 4-AP moves muscle afferents toward their tipping point but simply not far enough to reach it, reminiscent of the effects of nerve injury on cutaneous afferents (<xref ref-type="fig" rid="fig4">Figure 4C</xref>). Moreover, although many factors contribute to regulating excitability (‘Introduction’), the relatively low expression of K<sub>v</sub>1 channels in muscle afferents likely contributes to them existing closer to their tipping point, as shown in <xref ref-type="fig" rid="fig5">Figure 5B</xref>.</p></sec></sec><sec id="s3" sec-type="discussion"><title>Discussion</title><p>Using computer simulations and experiments, we have demonstrated that primary afferent hyperexcitability associated with neuropathic pain arises through a switch in spike initiation dynamics that occurs when neurons cross a tipping point. Furthermore, there are many different ways to cross that tipping point. The first observation demonstrates criticality, whereas the latter demonstrates degeneracy. Criticality and degeneracy are both important concepts for understanding how complex systems normally operate and how they fail in disease states, but neither concept has been previously applied to help decipher the pathogenesis of neuropathic pain. We will discuss each concept in turn, and then address their relevance for understanding neuropathic pain.</p><p>Nonlinear dynamical analysis of our mathematical model predicted that qualitative neuropathic changes in primary afferent excitability—repetitive spiking, MPOs, and bursting—co-develop when subthreshold inward and outward currents become sufficiently unbalanced. According to our theory, the tipping point represents a qualitative change in spike initiation dynamics. By pharmacologically decreasing and/or electrophysiologically increasing candidate conductances, we were able to force primary afferent neurons in different directions across their tipping point, thereby reproducing or reversing neuropathic changes in excitability. These data clearly demonstrate criticality and support our attribution of it to a switch in spike initiation dynamics.</p><p>Our theory further predicted, and our experimental data confirmed, that there are multiple ways in which spike initiation dynamics can be altered. To be clear, the tipping point constitutes a unique switch in spike initiation dynamics but there are many different ways to reach that tipping point. Multiple bases for the same outcome constitute degeneracy (<xref ref-type="bibr" rid="bib13">Edelman and Gally, 2001</xref>; <xref ref-type="bibr" rid="bib47">Marder and Taylor, 2011</xref>). Degeneracy prevents emergent network and cellular behaviors from being ascribed to unique ion channel combinations, but therein allows for a disturbance in one type of ion channel to be offset by compensatory changes in other ion channels so that network and cellular function can be robustly maintained (<xref ref-type="bibr" rid="bib17">Flake et al., 2004</xref>; <xref ref-type="bibr" rid="bib27">Howard et al., 2007</xref>; <xref ref-type="bibr" rid="bib46">Marder, 2011</xref>). We focused on manipulating two factors, maximal conductance of subthreshold sodium and potassium conductances, while maintaining all other factors unchanged. Our data clearly demonstrate the degeneracy of neuropathic excitability insofar as distinct manipulations produced equivalent outcomes.</p><p>In contrast to our simplified testing paradigm, nerve injury induces numerous molecular changes within primary afferents as evidenced by gene expression profiling (<xref ref-type="bibr" rid="bib7">Costigan et al., 2002</xref>; <xref ref-type="bibr" rid="bib66">Xiao et al., 2002</xref>; <xref ref-type="bibr" rid="bib61">Valder et al., 2003</xref>; <xref ref-type="bibr" rid="bib24">Hammer et al., 2010</xref>; <xref ref-type="bibr" rid="bib40">LaCroix-Fralish et al., 2011</xref>). Some of those changes increase excitability, others are compensatory, and most are irrelevant for (although nonetheless correlated with) hyperexcitability and instead bear on other aspects of cellular function. That said, alterations in cellular excitability cannot be definitively explained without accounting for all ‘relevant’ changes. This is an insurmountable task if one considers that each neuron experiences different molecular changes and that those changes occur on different molecular backgrounds. However, one can ascertain how close or far a neuron sits from its tipping point based on whether controlled manipulations succeed in forcing the neuron across that point. This requires a theoretical understanding of tipping points (i.e., their nonlinear dynamical basis) to identify telltale signs by which they can be inferred experimentally. It also requires a testing paradigm in which controlled manipulations can be applied, and where unknown changes (differences) do not occur (exist). It is for these reasons that we compared excitability within the same neuron before and after artificial manipulations rather than comparing excitability before and after nerve injury; notably, the latter approach would have necessitated comparison of excitability in separate neurons, each of which is distinct, and is plagued by countless injury-induced changes, not all of which can be simultaneously measured. Moreover, the abruptness of our manipulations precludes compensatory changes from developing. Thus, our approach represents an innovative alternative to experiments using nerve injury models or molecular–genetic techniques in which channel expression is more slowly up- or down-regulated.</p><p>Our results do not reveal a cure for neuropathic pain but, instead, suggest that a paradigm shift is needed in how we approach this task. Criticality and degeneracy may explain features of neuropathic pain that have hitherto eluded explanation. For instance, neuropathic symptoms often develop at long and highly variable delays after the inciting injury or disease. Criticality can explain this insofar as underlying molecular changes can develop occultly, without obvious outward manifestations, until a tipping point is reached, whereupon symptoms develop abruptly because of gross changes in excitability. Furthermore, once near the tipping point, symptoms could wax or wane because of the system moving forward and backward across its tipping point due to subtle fluctuations in underlying factors. From a therapeutic perspective, an obvious goal is to move the system from the ‘abnormal’ side of the tipping point back to the ‘normal’ side. Because of degeneracy, there are many ways to cross the tipping point in the forward direction, and an equally large number of ways to cross back in the reverse direction. This last observation is promising, as is the observation that the neuropathic state seems to lie close to the tipping point. But why then is neuropathic pain so notoriously difficult to treat?</p><p>The answer to the last question hinges on the answer to another question: Why does neuropathic excitability develop in the first place? Degeneracy ought to convey robustness to the regulation of neuronal excitability since effects of pathological molecular changes can be mitigated by compensatory changes in any one of a multitude of other ion channels. But gross changes in excitability do occur, at least in some neurons, and although those changes were easily reversed by our acute manipulations, prolonged therapies in patients and animal models often have only transient effects. One possible explanation is that the homeostatic regulation of excitability is itself deranged, in effect maintaining the system at an incorrect ‘set point’ and hijacking degeneracy to paradoxically maintain hyperexcitability. In that scenario, therapeutically manipulating only one misregulated conductance will fail to sustainably reverse hyperexcitability if a deranged homeostatic control mechanism can fall back on other conductances to achieve its misguided goal. Instead, all misregulated conductances need to be coordinately targeted. If epilepsy has a similarly degenerate molecular basis (<xref ref-type="bibr" rid="bib37">Klassen et al., 2011</xref>), it is no wonder that multi-target drugs tend to work well or that polypharmacy with single-target drugs is beneficial (<xref ref-type="bibr" rid="bib39">Kwan et al., 2001</xref>); indeed, combination pharmacotherapy also offers benefits in the treatment of neuropathic pain (<xref ref-type="bibr" rid="bib6">Chaparro et al., 2012</xref>). But, alternatively, the deranged control mechanism itself could be commandeered or its set point adjusted. The idea that neuropathic pain reflects the maladaptive homeostatic response to injury rather than being a direct consequence of injury has started to gain traction (<xref ref-type="bibr" rid="bib8">Costigan et al., 2009</xref>). Our data emphasize the importance of considering the context in which that plasticity occurs, namely, that it occurs within a system that is both complex and degenerate.</p><p>To conclude, we reproduced and reversed neuropathic excitability in primary afferent neurons using manipulations designed to force the system across the tipping point that separates normal and neuropathic excitability. Existence of that tipping point demonstrates ‘criticality’ and speaks to the importance of the nonlinear interactions that give rise to system complexity. Observation that multiple different manipulations can force the system across that tipping point demonstrates ‘degeneracy’. Criticality and degeneracy are fundamentally important concepts for understanding how complex systems fail, and how to therapeutically intervene to prevent or reverse those failures. Our theory suggests that treatment strategies for neuropathic pain must move away from single-target-drugs and towards a systems-based approach, capitalizing on degeneracy rather than being thwarted by it.</p></sec><sec id="s4" sec-type="materials|methods"><title>Materials and methods</title><sec id="s4-1"><title>Simulations</title><p>Starting with a two-dimensional Morris–Lecar (ML) model, which is sufficient to produce spikes, we added Hodgkin–Huxley (HH) style conductances for the express purpose of modulating spike initiation dynamics in the ML model. For ML equations, see <xref ref-type="bibr" rid="bib57">Rho and Prescott (2012)</xref>. Parameter values were as follows: <italic>C</italic> = 2 μF/cm<sup>2</sup>, <italic>E</italic><sub>Na</sub> = 50 mV, <italic>E</italic><sub>K</sub> = −100 mV, <italic>E</italic><sub>leak</sub> = −70 mV, φ<sub>w</sub> = 0.15, <italic>ḡ</italic><sub>fast</sub> = 20 mS/cm<sup>2</sup>, <italic>ḡ</italic><sub>slow</sub> = 20 mS/cm<sup>2</sup>, <italic>g</italic><sub>leak</sub> = 2 mS/cm<sup>2</sup>, β<sub>m</sub> = −1.2 mV, γ<sub>m</sub> = 14 mV, β<sub>w</sub> = −10 mV, γ<sub>w</sub> = 10 mV, and <italic>I</italic><sub>stim</sub> was varied. HH conductances were modeled according to<disp-formula id="equ1"><label>(1)</label><mml:math id="m1"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mrow><mml:mi>N</mml:mi><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>g</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mi>m</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>V</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>N</mml:mi><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="equ2"><label>(2)</label><mml:math id="m2"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:mi>m</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="equ3"><label>(3)</label><mml:math id="m3"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mfrac><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>V</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>V</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="equ4"><label>(4)</label><mml:math id="m4"><mml:mrow><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>β</mml:mi></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>V</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mi>β</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>β</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where, <italic>V</italic><sub>α,β</sub> = −24 mV, <italic>s</italic><sub>α,β</sub> = −17 mV, and <italic>k</italic><sub>α,β</sub> was between 1 and 1.5 ms<sup>−1</sup>. These parameters were chosen to give activation in the perithreshold voltage range (based on the threshold determined by ML parameters). In the interest of simplicity, the activation variable <italic>m</italic> was not raised to any exponent and nor was inactivation included. The Na<sup>+</sup> and K<sup>+</sup> currents differed only their reversal potential, where <italic>E</italic><sub>Na</sub> = +50 mV and <italic>E</italic><sub>K</sub> = −100 mV. Maximal conductances <italic>ḡ</italic><sub>Na</sub> and/or <italic>ḡ</italic><sub>K</sub> were varied in order to modulate spike initiation dynamics and thereby test our hypotheses. Where indicated, we also included a virtual slow AHP current modeled according to<disp-formula id="equ5"><label>(5)</label><mml:math id="m5"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>H</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>g</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>H</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>V</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="equ6"><label>(6)</label><mml:math id="m6"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mi>z</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>γ</mml:mi><mml:mi>z</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>−</mml:mo><mml:mi>z</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mi>z</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where. β<sub>z</sub> = 0 mV, γ<sub>z</sub> = 5 mV, τ<sub>z</sub> = 300 ms. Setting β<sub>z</sub> to a suprathreshold voltage ensures a spike-dependent form of adaptation (<xref ref-type="bibr" rid="bib54">Prescott and Sejnowski, 2008</xref>). Gaussian noise with standard deviation 0.3 μA/cm<sup>2</sup> and 0 mean was also added, where indicated. Simulations were conducted in XPP and bifurcation analysis was conducted with AUTO using the XPP interface (<xref ref-type="bibr" rid="bib14">Ermentrout, 2002</xref>).</p></sec><sec id="s4-2"><title>Animals</title><p>All experiments were carried out on adult (200–340 g) male Sprague–Dawley rats (Harlan, Indianapolis, IN or Charles River, Montreal) and were approved by the University of Pittsburgh Institutional Animal Care and Use Committee (protocol number 1108600) and by The Hospital for Sick Children Animal Care Committee (protocol number 22919).</p></sec><sec id="s4-3"><title>Neuron labeling</title><p>To retrogradely label cutaneous or muscle afferent cell bodies in the dorsal root ganglion (DRG), animals were anesthetized with isofluorane (4% for induction; 2.5% for maintenance) and DiI (Invitrogen, Carlsbad, CA) dissolved in DMSO (170 mg/ml; Sigma–Aldrich, St. Louis, MO) and diluted 1:10 in 0.9% sterile saline was injected into either the hindpaw skin or gastrocnemius muscle. Specifically, to label cutaneous afferents, 10 μl of DiI solution was injected intradermally in 5 sites (2 μl/site) over an area of ∼5 mm<sup>2</sup> into the glabrous skin of the left hindpaw. To label muscle afferents, 10 μl of DiI solution was slowly injected into five sites of the gastrocnemius muscle of the left leg (2 μl/site); to exclude spurious labeling of cutaneous afferents along the needle track, 10 μl of Fast Blue (1% in sterile saline) was injected intradermally around the intramuscular injection site. Only cells labeled with DiI and not Fast Blue were considered muscle afferents. Neurons were harvested 10–20 days after injections.</p></sec><sec id="s4-4"><title>Nerve injury model</title><p>A subset of animals received spinal nerve ligation (SNL) (<xref ref-type="bibr" rid="bib33">Kim and Chung, 1992</xref>) 2–5 days before terminal experiments. Under isoflurane anesthesia, paraspinal muscles of the lower lumbar and sacral level were separated to access the area around the L6 process. The L6 process was carefully removed and the L5 spinal nerve was tightly ligated with 6-0 silk suture. All nerve-injured animals maintained motor function but developed neuropathic pain as inferred by guarding of the affected paw.</p></sec><sec id="s4-5"><title>Cell dissociation</title><p>To collect DRG neurons, rats were deeply anesthetized by subcutaneous injection of anesthetic cocktail (1 ml/kg of 55 mg/ml ketamine, 5.5 mg/ml xylazine, and 1.1 mg/ml acepromazine) or by isoflurane inhalation (4% for induction; 2.5% for maintenance). DRG were surgically removed (L4 and L5 in naive animals; L5 in nerve-injured animals), enzymatically treated, mechanically dissociated, plated on glass coverslips previously coated by a solution of 0.1 mg/ml poly-D-lysine, and incubated in MEM-BS at 37°C, 3% CO<sub>2</sub>, and 90% humidity for 2 hr. After incubation, coverslips were transferred to a HEPES-buffered L-15 media containing 10% BS and 5 mM glucose and stored at room temperature. Neurons were studied 2–28 hr after harvesting.</p></sec><sec id="s4-6"><title>Electrophysiology</title><p>Coverslips with cultured cells were transferred to a recording chamber constantly perfused with room temperature, oxygenated (95% O<sub>2</sub> and 5% CO<sub>2</sub>) artificial cerebral spinal fluid containing (in mM) 126 NaCl, 2.5 KCl, 2 CaCl<sub>2</sub>, 2 MgCl<sub>2</sub>, 10 D-glucose, 26 NaHCO<sub>3</sub>, 1.25 NaH<sub>2</sub>PO<sub>4</sub>. Using gradient contrast optics, neurons with somatic diameter ≥30 μm were targeted for patching as these give rise to myelinated fibers (<xref ref-type="bibr" rid="bib25">Harper and Lawson, 1985</xref>). Cutaneous or muscle afferents were targeted based on epifluorescent illumination of DiI labeling (and exclusion of Fast Blue labeling in the case of muscle afferents). Cells were recorded in the whole-cell configuration with >70% series resistance compensation using an Axopatch 200B amplifier (Molecular Devices; Palo Alto, CA). Electrodes (2–5 MΩ) were filled with a recording solution containing (in mM) 125 KMeSO<sub>4</sub>, 5 KCl, 10 HEPES, 2 MgCl<sub>2</sub>, 4 ATP, 0.4 GTP, as well as 0.1% Lucifer Yellow; pH was adjusted to 7.2 with KOH and osmolality was between 270 and 290 mosmol/l. The pipette shank was painted with Sylgard to reduce pipette capacitance. Responses were low-pass filtered at 2 KHz, digitized at 20 KHz using a CED 1401 computer interface (Cambridge Electronic Design, Cambridge, UK), and analyzed offline. Membrane potential (after correction for the liquid junction potential of −9 mV) was adjusted to −65 mV through tonic current injection.</p><p>To block native conductances, various drugs were bath applied as reported in the ‘Results’. To add virtual conductances, the dynamic clamp technique was applied using Signal 5 software (Cambridge Electronic Design). Conductances were modeled using the same equations as in computer simulations, as described above. To account for differences in cell size, values of inserted conductances were converted to densities by normalizing by membrane capacitance <italic>C</italic>, which was measured from responses to small (≤50 pA) hyperpolarizing current steps, where <italic>C</italic> = <italic>τ</italic><sub>m</sub>/<italic>R</italic><sub>in</sub>, <italic>τ</italic><sub>m</sub> is the membrane time constant, and <italic>R</italic><sub>in</sub> is input resistance. Notably, given a specific membrane capacitance of 1 μF/cm<sup>2</sup>, a conductance density of 1 mS/cm<sup>2</sup> (as reported for simulations) corresponds to 1 nS/pF (as reported for dynamic clamp experiments).</p></sec></sec></body><back><ack id="ack"><title>Acknowledgements</title><p>This work was supported by NIH grant NS074146, a Natural Sciences and Engineering Research Council of Canada Discovery Grant, and scholar awards from the Rita Allen Foundation and the Edward Mallinckrodt Jr Foundation to SAP. SAP is also a Canadian Institutes of Health Research New Investigator. We thank Nicole Scheff and Michael Gold for providing acutely dissociated DRG neurons for pilot studies, Erica Schwartz for instruction on the spinal nerve ligation procedure, Nadine Simons-Weidenmaier for technical assistance, and Tim Bergel for dynamic clamp modifications. We also thank Yves De Koninck, Jerry Gebhart, and Mike Salter for feedback on the manuscript.</p></ack><sec sec-type="additional-information"><title>Additional information</title><fn-group content-type="competing-interest"><title>Competing interests</title><fn fn-type="conflict" id="conf1"><p>The authors declare that no competing interests exist.</p></fn></fn-group><fn-group content-type="author-contribution"><title>Author contributions</title><fn fn-type="con" id="con1"><p>SR, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article</p></fn><fn fn-type="con" id="con2"><p>SAP, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article</p></fn><fn fn-type="con" id="con3"><p>YZ, Acquisition of data, Analysis and interpretation of data</p></fn><fn fn-type="con" id="con4"><p>KYL, Acquisition of data</p></fn></fn-group><fn-group 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An edited version of the letter sent to the authors after peer review is shown, indicating the substantive concerns or comments; minor concerns are not usually shown. Reviewers have the opportunity to discuss the decision before the letter is sent (see <ext-link ext-link-type="uri" xlink:href="http://elifesciences.org/review-process">review process</ext-link>). Similarly, the author response typically shows only responses to the major concerns raised by the reviewers.</p></boxed-text><p>[Editors’ note: a previous version of this study was rejected after peer review, but the authors submitted for reconsideration. The two decision letters after peer review are shown below.]</p><p>Thank you for choosing to send your work entitled “Injury-induced changes in primary afferent excitability: criticality, degeneracy, and implications for neuropathic pain” for consideration at <italic>eLife</italic>. Your full submission has been evaluated by a Senior editor and 3 peer reviewers, one of whom, Ronald L Calabrese, is a member of our Board of Reviewing Editors, and the decision was reached after discussions between the reviewers. We regret to inform you that your work will not be considered further for publication at this time, although we are willing to consider a thoroughly revised version, which will be treated as an entirely new manuscript. We understand that you may not be interested in this route, and that you may choose to send the present manuscript elsewhere.</p><p>There was considerable enthusiasm for the aims of the work but this enthusiasm was dampened by shortfalls in the data. One reviewer wrote “I would like to reiterate that I feel the study has significant merit and if these issues are addressed appropriately, this could represent a major contribution to the field. Nobody else in the field is really doing this type of work, and the conclusions suggest an entirely new way that we should potentially be looking at the mechanisms of neuropathic pain of peripheral origin.“ Resolving the issues however, would require extensive new experiments. Three primary issues were identified:</p><p>1) Proper identification of afferents particularly deep tissues afferents. More identified hyper-excitable cutaneous afferents are needed as these are the ones that are thought to be involved in the neuropathic pain rodent model. In general larger n's for all experimental groups with appropriate statistical analyses are needed.</p><p>2) Proper controls need to be put in place: e.g., controls for treatments leading to hyper-excitability.</p><p>3) For the dynamic-clamp experiments and the scaling of conductance injection, passive properties need to be measured on each single cell submitted to dynamic-clamp. These measurements are needed so that the need conductance for criticality can be scaled for proper interpretation.</p><p>The reviews of the outside reviewers are attached in their entirety. If these issues are addressed then a resubmission to <italic>eLife</italic> is encouraged.</p><p><italic>Reviewer</italic> <italic>1:</italic></p><p>The authors combine theoretical and experimental studies to address the question of criticality and degeneracy in the mechanisms that convert primary afferent neurons from normal single-spike activity to the repetitive firing characteristic of injury-induced neuropathic pain state. The theoretical analyses based on the dynamical systems approach considering just inward and outward spike conductances indicates that multiple mechanisms can move neuron past a critical point that represents the transition from single-spike to repetitive firing. The critical point is in actuality a critical line that itself is subject to movement as other parameters such a leak conductance are varied. Using dissociated dorsal root ganglion (DRG) neurons from rat - some identified as cutaneous afferents - and a combination of pharmacological blockade of K currents and addition of Na currents with dynamic clamp, they demonstrate criticality and degeneracy in neuron neurons in the transition studied. Most exciting is the finding that when using DRG neurons that show injury-induced neuropathic-like repetitive firing from a rat model that they can rescue normal single-spike firing with pharmacological blockade of subthreshold Na current or addition of K current with dynamic clamp.</p><p>The writing is succinct and the paper is very easy to follow. The figures are simple but elegant and provide the necessary data in an easily digestible form. The work appears carefully done with sufficient attention to detail and adequate numbers of experiments. The work may prove controversial in the pain community because of its radical approach but the data is just too interesting not to put out there. The controversy will come in the interpretation not in the data itself, and it seems to me that <italic>eLife</italic> should support work that pushes a field in new directions.</p><p>Concern:</p><p>1) I was intrigued by the notion that some types of afferents live closer to the criticality edge than others. If I could ask for more experiments it would be to look for more identified cutaneous afferents that show repetitive firing in the injury model and to identify the afferent types that do show repetitive firing with tracers if possible.</p><p><italic>Reviewer</italic> <italic>2:</italic></p><p>In the present article, Ratte and colleagues investigate the criticality and degeneracy of the pathological changes in primary afferent excitability involved in the aetiology of neuropathic pain. Using computer simulation, pharmacological manipulation and dynamic-clamp electrophysiology, the authors show the existence of a tipping point where primary afferent activity qualitatively switches from single-spiking to tonic spiking in response to current injection, therefore demonstrating the existence of criticality in the behavior of these neurons. Then, the authors show that multiple different manipulations can trigger excitability changes similar to those observed in neuropathic pain, therefore demonstrating the degeneracy of the transition between the physiological and the pathological conditions. The authors argue that the existence of multiple equivalent molecular solutions for the transitions between the physiological and pathological states indicates the potential robustness of both of these states and explains, at least in part, the inefficiency of single-target drugs. Therefore, the authors conclude that new drug design strategies, based on the degeneracy principle, may be needed to treat pathologies such as neuropathic pain, in particular the search for drugs that act on multiple-targets.</p><p>Overall, this article is very interesting in that it emphasizes the theory that biological systems use very diverse solutions to solve the same problems, which might be one of the main bases of biological robustness. Moreover, the focus of the current article is to investigate not only physiological robustness but also pathological robustness, especially in regard to drug resistance. The experimental design is elegant (although not as innovative as the authors claim), the experiments were suitably performed, and the article is well written. I have one major concern about some of the conclusions drawn in the present study:</p><p>The authors claim that the dynamic-clamp experiments performed on cutaneous afferents and deep tissue afferents argue in favor of deep tissue afferents being closer to the tipping point. This conclusion arises from the fact that the injected conductance needed to trigger the pathological behavior of deep tissue afferents is lower than that needed to cross the tipping point in cutaneous afferents. However, this conclusion is biased, especially in the absence of control experiments measuring the intrinsic excitability and conductance levels in native neurons. In other words, one could easily imagine that deeper tissue afferents display higher input resistance values than cutaneous afferents, and that their channel densities (in particular sodium and potassium channels) are lower than for cutaneous afferents. If this is the case, what should be compared is not the absolute conductance value that needs to be injected to trigger or reverse the pathological state, but conductance values that are adjusted for the differences in input resistance and channel densities between the two neuronal types. It is known that input resistance and channel densities are highly variable within a same cell type but also highly variable between cell types. Such differences may explain the results presented in <xref ref-type="fig" rid="fig4">Figure 4D</xref> without invoking the differences in distance between the physiological and pathological states in the two cell types analyzed. Although the conclusion drawn by the authors is very interesting and might be true, the absence of control measurements of the native passive and active properties of the two cell types prevents it from being the only interpretation.</p><p><italic>Reviewer</italic> <italic>3:</italic></p><p>This manuscript by Ratté et al describes theoretical and experimental examination of mechanisms of hyperexcitability of sensory neurons from the rat in the context of nerve injury pain. The work is highly unique to the field, and the findings of the study are quite provocative and have the potential to represent a major advance. There are several issues with the study that reduce the potential impact of the work, however.</p><p>1) The entire study is predicated on the reported increase in excitability of sensory neurons from rodents with neuropathic pain. The authors examine ways in which modulation of multiple ion conductances can lead to the generation of this type of hyperexcitability. The problem (and it is a big one) is that the authors' own data fail to demonstrate increased excitability of cutaneous afferents in the context of nerve injury. Thus, only 1/9 cells that innervate the hind paw skin from rats with nerve injury showed repetitive firing - and this is the phenomenon that their theoretical work is meant to be examining. The authors do show that when they examine unlabeled neurons (neurons that do not take up DiI injected into the hind-paw skin), most of these cells (6/9) demonstrate repetitive firing. The major findings were then repeated in these unlabeled neurons, and the findings were consistent. However, the disconnect shines light on the major shortcomings of this study, which are two-fold.</p><p>a) The authors do not demonstrate increased incidence of repetitive firing in injured neurons compared to un-injured neurons, and</p><p>b) The number of cells in each group is extremely small, given the large degree of heterogeneity among firing properties of rodent sensory neurons. Certainly the number of cells in each group needs to be increased.</p><p>2) Many of the comparisons do not have statistical analysis to support the conclusions. Rather, the authors state things like “...was sufficient to produce repetitive spiking and MPOs in 6 of 13 neurons tested with both manipulations. In those 6 neurons, we tested the two manipulations together and found that the combination was sufficient to produce hyperxcitability in 5 neurons.” What does it mean when something happens in 6 of 13 cells? What happened in the other 7 cells? On average, that might suggest that nothing happens. I recognize that there is a high degree of variability, but some statistical analyses are a must here. Or to the point, there are really no “control” experiments – examining cells recorded without the stated manipulation, or using the vehicle in which the drugs were dissolved, for example.</p><p>3) The authors record some neurons that were not labeled via DiI injection into the paw. These neurons are called “deep tissue” neurons. This is not correct. The fact that the neurons do not take up dye injected into the paw skin does not mean they are neurons that project to deeper tissues. These should simply be called “unlabeled” neurons. This brings up a very important point. Either the cutaneous afferents are indeed less excitable after injury than deep neurons, or alternatively, perhaps the DiI is causing changes to the firing properties. The appropriate comparison would be to inject deeper tissues with DiI and record from those neurons to make sure that the observed differences are indeed due to target of innervation as opposed to the presence of the dye. The data provided do <italic>not</italic> demonstrate that “deep tissue afferents are more prone to qualitative changes in excitability....” as the authors claim.</p><p>[Editors’ note: what now follows is the decision letter after the authors submitted for further consideration.]</p><p>Thank you for sending your work entitled “Injury-induced changes in primary afferent excitability: criticality, degeneracy, and implications for neuropathic pain” for consideration at <italic>eLife</italic>. Your article has been favorably evaluated by a Senior editor and 3 reviewers, one of whom, Ronald L Calabrese, is a member of our Board of Reviewing Editors.</p><p>The Reviewing editor and the other reviewers discussed their comments before we reached this decision, and the Reviewing editor has assembled the following comments to help you prepare a revised submission.</p><p>Small concern:</p><p>Please clarify the model a bit more. Is it correct that spiking is generated in the model by the standard M-L equations and that the added HH Na and K currents are strictly subthreshold; i.e., they do not produce spikes? This should be stated explicitly in both Results (at the very beginning of Results) and in Methods. Also lower panel of <xref ref-type="fig" rid="fig1">Figure 1A</xref> is very confusing as it is not mapped onto the model or mentioned in the text.</p></body></sub-article><sub-article article-type="reply" id="SA2"><front-stub><article-id pub-id-type="doi">10.7554/eLife.02370.010</article-id><title-group><article-title>Author response</article-title></title-group></front-stub><body><p>[Editors’ note: the author responses to the first round of peer review follow.]</p><p><italic>1A) Proper identification of afferents particularly deep tissues afferents</italic>.</p><p>We have added new experiments from muscle afferents labeled in naïve and nerve-injured animals. These new data appear in <xref ref-type="fig" rid="fig4 fig5">Figures 4 and 5</xref>.</p><p><italic>1B) More identified hyper-excitable cutaneous afferents are needed as these are the ones that are thought to be involved in the neuropathic pain rodent model</italic>.</p><p>We started under the assumption (and conveyed the impression in the original text) that cutaneous afferents are primarily responsible for allodynia in neuropathic pain conditions, which is why we deliberately targeted them with DiI labeling in our initial experiments. In fact, muscle afferents are more prone to developing hyperexcitability (e.g. Michaelis et al. J Neurosci 2000), which is confirmed by our new data. <xref ref-type="bibr" rid="bib41">Liu et al. (2002)</xref> reported repetitive spiking in only 6 of 33 injured cutaneous neurons, which is comparable to the proportion we observed (1 of 9; p = 1, Fisher’s exact test); by comparison, they observed repetitive spiking in 18 of 21 injured muscle afferents, which is significantly more than the proportion we observed (11 of 20; p < 0.05, Fisher’s exact test) but is still reasonably similar given a variety of methodological differences. In any case, 11 of 20 muscle afferents showing repetitive spiking after nerve injury is a significantly higher proportion than 0 of 27 muscle afferents showing repetitive spiking under control conditions (p < 0.001, Fisher’s exact test). Overall, additional experiments in cutaneous afferents are not warranted given past studies and since our new experiments in muscle afferents now demonstrate that a high proportion of such afferents become grossly hyperexcitable after nerve injury.</p><p>Importantly, mechanical hypersensitivity is tested in rodents by poking the foot with von Frey filaments, which also activates underlying muscle afferents (e.g., Jankowski et al. J Neurophysiol 2013); thus, muscle afferent hyperexcitability very likely contributes to mechanical hypersensitivity in animal models. The difference between punctate stimulation and the light brushing used in clinical studies is a source of controversy (e.g., Mogil Nat Rev Neurosci 2009), but that is beyond the scope of our study. It should also be kept in mind that cutaneous afferents, although not exhibiting gross hyperexcitability, are nonetheless quantitatively more excitable after injury (<xref ref-type="fig" rid="fig4">Figure 4C</xref>) and that muscle afferent hyperexcitability could alter the central processing of cutaneous input.</p><p>Based on the newly added data from muscle afferents from both control and nerve-injured animals, and based on extensive changes to the text, we think this concern has been thoroughly addressed.</p><p><italic>1C In general larger n's for all experimental groups with appropriate statistical analyses are needed</italic>.</p><p>We have more or less doubled the number of cells now reported in the paper. This is largely due to the new data from muscle afferents, but certain experiments were also repeated in cutaneous afferents. We have also included statistical analysis wherever appropriate.</p><p>Although we are not certain it is necessary, we have compared our conversion rates against estimated “chance” conversions rate based on comparing the pattern of excitability at the beginning and end of recording from each cell, without any manipulation. No cell ever spontaneously changed to a neuropathic pattern of excitability over the course of recording. If we insert those 0 conversion rates into a contingency table and compare against the conversion rate after 4-AP application or virtual sodium current insertion, Fisher’s exact tests always shows significance. Thus, we can say that our manipulations are significantly more successful in reproducing neuropathic excitability than expected by chance. We have added this statistical analysis to the revised text, although we worry that it might be seen as rather trivial.</p><p>Given technical limitations and various biological factors, single manipulations sometimes failed to convert the excitability (i.e., we did not achieve a 100 % forward conversion rate, although we did achieve a 100% reversal rate). When we failed to produce neuropathic excitability by a single manipulation like 4-AP or nerve injury, we have now quantitatively shown that the first manipulation moved the system closer to its tipping point as assessed by the reduction in the second manipulation needed for the system to finally cross that tipping point (i.e., the minimum necessary sodium conductance). This shifting is now quantified <xref ref-type="fig" rid="fig3 fig4 fig5">Figures 3-5</xref> in the revised manuscript.</p><p>All of the important comparisons have now been tested statistically and all results were significant. This leads us to believe that the current sample sizes, many of which have indeed been increased since the original submission, are large enough to support our claims. Moreover, the text and figures have been revised to help clearly convey the statistical results.</p><p><italic>2) Proper controls need to be put in place: e.g., controls for treatments leading to hyper-excitability</italic>.</p><p>With respect to our nerve injury model, we chose spinal nerve ligation (SNL) because it has been used for > 20 years and, compared with subtler pain models, is known to consistently cause rapid-onset and robust mechanical hypersensitivity (e.g., Kim et al. Exp Brain Res 1997). Given that the SNL model has been thoroughly validated and that we are using the procedure solely to give us hyperexcitable neurons with which we can test predictions 5 and 6, we consider the sham control to be unnecessary (and therefore unethical since it entails animals experiencing untreated post-operative pain). The fact that we did not observe grossly hyperexcitable cutaneous afferents after nerve injury might suggest that our SNL procedure was ineffective, but this is disproven by three observations: (1) nerve-injured animals exhibited behavioral evidence of pain, (2) the cutaneous afferents were quantitatively hyperexcitable, i.e., closer to their tipping point, and (3) the same procedure caused gross hyperexcitability in a sizeable fraction of muscle afferents. This last point is arguably the most important and is now clearly demonstrated in the revised manuscript. There is, therefore, very clear evidence that the SNL procedure worked.</p><p>We assume this point might also relate to the second major concern expressed by reviewer 3. He/she wrote: <italic>What does it mean when something happens in 6 of 13 cells? What happened in the other 7 cells? On average, that might suggest that nothing happens. I recognize that there is a high degree of variability, but some statistical analyses are a must here. Or to the point, there are really no “control” experiments – examining cells recorded without the stated manipulation, or using the vehicle in which the drugs were dissolved, for example</italic>.</p><p>Most of our key demonstrations rely on within-cell comparisons that were intended to remove effects of cellular heterogeneity, DiI labeling, etc. We routinely turned dynamic clamp on and off to verify its effects, and we washed out drugs after application to demonstrate reversibility. To be very clear, we always examined the cell without the stated manipulation for comparison with the manipulation. As for vehicle controls, the only drug not soluble in water is riluzole; the small amount of DMSO used to dissolve the riluzole has no measurable effect on neuronal excitability.</p><p>As to what happens to cells not converted by 4-AP, we now show in both cutaneous afferents (<xref ref-type="fig" rid="fig3">Figure 3C</xref>) and muscle afferents (<xref ref-type="fig" rid="fig5">Figure 5D</xref>) that 4-AP, if it did not cause repetitive spiking, has nonetheless moved the neuron closer to its tipping point insofar as significantly less virtual sodium current needs to be added to cause conversion, reminiscent of the effects of nerve injury on cutaneous afferents (<xref ref-type="fig" rid="fig4">Figure 4C</xref>). We cannot do the converse experiment since it is impossible to precisely quantify the amount of potassium current blocked in current clamp experiments. Nonetheless, based on an unsuspected difficulty with pharmacologically converting muscle afferents (despite the ease of converting them with virtual sodium current), we conducted voltage clamp experiments to test whether muscle afferents express less dendrotoxin-sensitive potassium current. We found that this was indeed the case (<xref ref-type="fig" rid="fig5">Figure 5C</xref>) With respect to whether DiI or its vehicle may account for differences between labeled and unlabeled neurons, Zhang et al. (J Neurosci Methods 2007) showed that retrograde labeling with up to 5 % DiI (we used 1.7 %) does not affect passive or active membrane properties. In any case, this becomes a moot point since we now compare labeled cutaneous afferents with muscle afferents that are also labeled with DiI.</p><p><italic>3) For the dynamic-clamp experiments and the scaling of conductance injection, passive properties need to be measured on each single cell submitted to dynamic-clamp. These measurements are needed so that the need conductance for criticality can be scaled for proper interpretation</italic>.</p><p>We completely agree and have done this in the revised manuscript. We calculated the membrane capacitance from the measured input resistance and membrane time constant. All dynamic clamp conductance values have been normalized by membrane capacitance. The same normalization was applied to new voltage clamp data.</p><p>Reviewer 1:</p><p><italic>1) I was intrigued by the notion that some types of afferents live closer to the criticality edge than others. If I could ask for more experiments it would be to look for more identified cutaneous afferents that show repetitive firing in the injury model and to identify the afferent types that do show repetitive firing with tracers if possible</italic>.</p><p>We have added extensive new data on labeled muscle afferents. Consistent with past studies, we have confirmed that a significant proportion of muscle afferents are rendered grossly hyperexcitable by nerve injury. This is entirely consistent with our new data showing that uninjured muscle afferents are significantly closer to their tipping point than cutaneous neurons, based on the minimal amount of virtual sodium current needed to produce repetitive spiking. However, we also found that 4-AP application was significantly less likely to produce repetitive spiking in muscle afferents than in cutaneous afferents. This prompted additional experiments which led to a novel finding that muscle afferents have a lower density of dendrotoxin-sensitive potassium current, which may explain why these neurons operate closer to their tipping point. Although these new findings are interesting, especially for those in the pain field, we have tried to retain a clear focus on our initial hypotheses concerning degeneracy and criticality. We hope to have struck a good balance in this revised paper.</p><p>Reviewer 2:</p><p><italic>1) The authors claim that the dynamic-clamp experiments performed on cutaneous afferents and deep tissue afferents argue in favor of deep tissue afferents being closer to the tipping point. This conclusion arises from the fact that the injected conductance needed to trigger the pathological behavior of deep tissue afferents is lower than that needed to cross the tipping point in cutaneous afferents. However, this conclusion is biased, especially in the absence of control experiments measuring the intrinsic excitability and conductance levels in native neurons. In other words, one could easily imagine that deeper tissue afferents display higher input resistance values than cutaneous afferents, and that their channel densities (in particular sodium and potassium channels) are lower than for cutaneous afferents. If this is the case, what should be compared is not the absolute conductance value that needs to be injected to trigger or reverse the pathological state, but conductance values that are adjusted for the differences in input resistance and channel densities between the two neuronal types. It is known that input resistance and channel densities are highly variable within a same cell type but also highly variable between cell types. Such differences may explain the results presented in</italic> <xref ref-type="fig" rid="fig4"><italic>Figure 4D</italic></xref> <italic>without invoking the differences in distance between the physiological and pathological states in the two cell types analyzed. Although the conclusion drawn by the authors is very interesting and might be true, the absence of control measurements of the native passive and active properties of the two cell types prevents it from being the only interpretation</italic>.</p><p>Based on new data in labeled muscle afferents, we now compare the input resistances in muscle and cutaneous afferents and show that they do not significantly differ. We have normalized all dynamic clamp conductance values by capacitance, and so now express those conductance values as densities (i.e., nS/pF). Statistically, uninjured muscle afferents require significantly less virtual sodium conductance than cutaneous afferents to be converted (<xref ref-type="fig" rid="fig5">Figure 5B</xref>). To be clear, these data are based on experiments in naïve animals and show only that muscle afferents are significantly closer to their tipping point based on this one manipulation; indeed, when thinking in terms of a high-dimensional space, a system may be very close to its tipping point in one dimension but very far it in the other dimensions. We think this is a very interesting topic, but dealing with it here will quickly exceed the intended scope of the current paper. These issues are nonetheless mentioned in the revised text.</p><p>Adjusting (or expressing) the virtual conductance value based on native conductance densities is an interesting suggestion but it is technically very challenging. We have now conducted voltage clamp experiments to compare the density of dendrotoxin-sensitive potassium current in muscle and cutaneous afferents. We found the current density to be significantly less in muscle afferents, which is consistent with muscle afferents operating closer to their tipping point, but again, does not exclude other differences. In any case, if the sodium channel mimicked by dynamic clamp was upregulated by nerve injury (and all other conductances remained unchanged) then a certain amount of upregulation could cause gross hyperexcitability amongst muscle afferents whilst failing to cause gross hyperexcitability amongst cutaneous afferents. We express “upregulation” of the virtual channel in absolute terms as there is no virtual channels in the control condition, and because we are not strictly mimicking a native Na<sub>v</sub>1.3 channel and nor could we strictly isolate the density of that channel in order to express the upregulation in relative terms.</p><p>In summary, because we do not know and cannot measure all of the possibly relevant native conductance densities (let alone conduct the dynamic clamp experiment in the same neuron), we opted for a minimalist strategy. This leads to a novel suggestion that proximity to tipping point may be a contributing factor for why certain cell types develop gross hyperexcitability after nerve injury while other cell types do not. We cannot exclude other pre-existing differences or differential injury-induced (pathological or compensatory) changes. Quantifying the contribution of each factor would be a difficult task and the interpretation of such results would be subject to many caveats. Overall, we want to introduce the distance-to-tipping-point idea, as we think it helps quantify certain demonstrations that are directly relevant to our main theses, but we think it wise to keep those demonstrations as simple as possible.</p><p>Reviewer 3:</p><p><italic>1) The entire study is predicated on the reported increase in excitability of sensory neurons from rodents with neuropathic pain. The authors examine ways in which modulation of multiple ion conductances can lead to the generation of this type of hyperexcitability. The problem (and it is a big one) is that the authors' own data fail to demonstrate increased excitability of cutaneous afferents in the context of nerve injury. Thus, only 1/9 cells that innervate the hind paw skin from rats with nerve injury showed repetitive firing – and this is the phenomenon that their theoretical work is meant to be examining. The authors do show that when they examine unlabeled neurons (neurons that do not take up DiI injected into the hind-paw skin), most of these cells (6/9) demonstrate repetitive firing. The major findings were then repeated in these unlabeled neurons, and the findings were consistent. However, the disconnect shines light on the major shortcomings of this study, which are two-fold</italic>.</p><p><italic>a) The authors do not demonstrate increased incidence of repetitive firing in injured neurons compared to un-injured neurons,</italic> <italic>and</italic></p><p><italic>b) The number of cells in each group is extremely small, given the large degree of heterogeneity among firing properties of rodent sensory neurons. Certainly the number of cells in each group needs to be increased</italic>.</p><p>With respect to point (a), we now show that there is indeed a significant increase in the incidence of repetitive spiking among muscle afferents. This is consistent with past publications (see response to consensus report). But we are unclear if the reviewer is questioning whether gross hyperexcitability in either type of afferent actually causes neuropathic pain. This is obviously important, but we think we are on safe ground. For instance, blocking peripheral input relieves pain for the duration of the block (e.g., Gold and Gebhart, Nat Med 2010), which argues that peripheral input is necessary to cause pain although it may not be sufficient since extensive plasticity also develops centrally. As explained in the text, myelinated afferents are believed to subserve the mechanical hypersensitivity (or allodynia) associated with neuropathic pain. A switch from onset-only spiking to repetitive spiking would dramatically amplify signaling. That said, cutaneous mechanosensors are the most likely to be directly involved, and yet neither we nor others observed gross hyperexcitability amongst those neurons. We have shown in this study that those neurons do have a critical point and that they are moved towards that critical point by nerve injury; it is conceivable that a higher proportion of cutaneous afferents cross that critical point under the conditions that exist in vivo. It is also possible that input from grossly hyperexcitable muscle afferents drives central heterosynaptic plasticity or summates with the input from cutaneous afferents (e.g., Nagi et al. J Physiol 2011) and thus contributes to mechanical allodynia in an indirect way.</p><p>The general consensus within the pain research community is that peripheral input is key and, therefore, that primary afferent hyperexcitability is a causal factor in many forms of neuropathic pain. Our study aims to advance understanding of that hyperexcitability, especially regarding criticality and degeneracy. Our new data on muscle afferents will, we hope, dispel the reviewer’s concerns.</p><p>With respect to point (b), please see response to point 1C in the consensus report.</p><p><italic>2) Many of the comparisons do not have statistical analysis to support the conclusions. Rather, the authors state things like “...was sufficient to produce repetitive spiking and MPOs in 6 of 13 neurons tested with both manipulations. In those 6 neurons, we tested the two manipulations together and found that the combination was sufficient to produce hyperxcitability in 5 neurons.” What does it mean when something happens in 6 of 13 cells? What happened in the other 7 cells? On average, that might suggest that nothing happens. I recognize that there is a high degree of variability, but some statistical analyses are a must here. Or to the point, there are really no “control” experiments – examining cells recorded without the stated manipulation, or using the vehicle in which the drugs were dissolved, for example</italic>.</p><p>Please see responses to points 1C and 2 in the consensus report. We believe that our revisions have thoroughly addressed this concern.</p><p><italic>3) The authors record some neurons that were not labeled via DiI injection into the paw. These neurons are called “deep tissue” neurons. This is not correct. The fact that the neurons do not take up dye injected into the paw skin does not mean they are neurons that project to deeper tissues. These should simply be called “unlabeled” neurons. This brings up a very important point. Either the cutaneous afferents are indeed less excitable after injury than deep neurons, or alternatively, perhaps the DiI is causing changes to the firing properties. The appropriate comparison would be to inject deeper tissues with DiI and record from those neurons to make sure that the observed differences are indeed due to target of innervation as opposed to the presence of the dye. The data provided do</italic> not <italic>demonstrate that “deep tissue afferents are more prone to qualitative changes in excitability....” as the authors claim</italic>.</p><p>Although we have left in sample traces from an unlabeled neuron (now using the nomenclature recommended by the reviewer), we have added substantial new data from labeled muscle afferents and now compare identified cutaneous afferents and identified muscle afferents. The muscle afferents were labeled with DiI, and so DiI labeling is not a confounding factor in the comparison. Based on the new analysis, we think we can safely suggest that muscle afferents operate closer to their tipping point, which may contribute to them being more prone to qualitative changes in excitability.</p><p>[Editors’ note: the author responses to the re-review follow.]</p><p><italic>Please clarify the model a bit more. Is it correct that spiking is generated in the model by the standard M-L equations and that the added HH Na and K currents are strictly subthreshold; i.e., they do not produce spikes? This should be stated explicitly in both Results (at the very beginning of Results) and in Methods. Also lower panel of</italic> <xref ref-type="fig" rid="fig1"><italic>Figure 1A</italic></xref> <italic>is very confusing as it is not mapped onto the model or mentioned in the text</italic>.</p><p>We have rewritten the opening paragraph of the Results and relevant parts of the Methods. We are reluctant to refer to HH currents as “strictly” subthreshold since they continue to be active at suprathreshold voltages, but it is true that they are not necessary for spike generation. We hope the revised text is clear in explaining that addition of an inward or outward HH conductance active at voltages near threshold changes how the fast and slow currents interact during the spike initiation process.</p><p>With respect to the lower panel in <xref ref-type="fig" rid="fig1">Figure 1A</xref>, we now explain in the legend that τ = (α+β)<sup>-1</sup>.</p></body></sub-article></article> |