Abstract
Background
Reduction of the transmembrane chloride gradient in spinal lamina I neurons contributes to the cellular hyperexcitability producing allodynia and hyperalgesia after peripheral nerve injury. The resultant decrease in anion reversal potential (i.e. shift in Eanion to less negative potentials) reduces glycine/GABAA receptor-mediated hyperpolarization, but the large increase in membrane conductance caused by inhibitory input can nonetheless shunt concurrent excitatory input. Without knowing the relative contribution of hyperpolarization and shunting to inhibition's modulation of firing rate, it is difficult to predict how much net disinhibition results from reduction of Eanion. We therefore used a biophysically accurate lamina I neuron model to investigate quantitatively how changes in Eanion affect firing rate modulation.
Results
Simulations reveal that even a small reduction of Eanion compromises inhibitory control of firing rate because reduction of Eanion not only decreases glycine/GABAA receptor-mediated hyperpolarization, but can also indirectly compromise the capacity of shunting to reduce spiking. The latter effect occurs because shunting-mediated modulation of firing rate depends on a competition between two biophysical phenomena: shunting reduces depolarization, which translates into reduced spiking, but shunting also shortens the membrane time constant, which translates into faster membrane charging and increased spiking; the latter effect predominates when average depolarization is suprathreshold. Disinhibition therefore occurs as both hyperpolarization- and shunting-mediated modulation of firing rate are subverted by reduction of Eanion. Small reductions may be compensated for by increased glycine/GABAA receptor-mediated input, but the system decompensates (i.e. compensation fails) as reduction of Eanion exceeds a critical value. Hyperexcitability necessarily develops once disinhibition becomes incompensable. Furthermore, compensation by increased glycine/GABAA receptor-mediated input introduces instability into the system, rendering it increasingly prone to abrupt decompensation and even paradoxical excitation.
Conclusion
Reduction of Eanion dramatically compromises the inhibitory control of firing rate and, if compensation fails, is likely to contribute to the allodynia and hyperalgesia associated with neuropathic pain. These data help explain the relative intractability of neuropathic pain and illustrate how it is important to choose therapies not only based on disease mechanism, but based on quantitative understanding of that mechanism.
Background
Neuropathic pain can arise from a multitude of pathophysiologic mechanisms or combinations thereof [1–5]. Reduced inhibition, or disinhibition, of spinal neurons constitutes an important class of these mechanisms [6–8]. Indeed, many features of neuropathic pain syndromes can be reproduced by pharmacologically blocking inhibition in the spinal cord [9–17] or through genetic changes that reduce inhibition [18]. Conversely, increasing inhibition can, in some conditions, reduce neuropathic pain [17,19–22].
Reproducing features of neuropathic pain by reducing inhibition and relieving neuropathic pain by increasing inhibition are important observations but constitute only circumstantial evidence that disinhibition is involved in the pathogenesis of neuropathic pain. More direct evidence comes from studies showing that neuropathy can occlude the effects of pharmacologically reducing inhibition [19] and that inhibition is indeed reduced in animal models of neuropathic pain [23–32]. There is controversy whether reduction of inhibitory transmitters and/or their receptors occurs following neuropathy [33–36], but disinhibition can arise from a broad array of mechanisms.
The recent study by Coull et al. [31] suggests an alternative mechanism to explain disinhibition: reduced expression of the potassium-chloride cotransporter (KCC2) causes reduction of the chloride gradient across the neuronal membrane, which in turn leads to reduction of the anion reversal potential (i.e. Eanion shifts to a less negative membrane potential) [see also [37]]. The change in driving force means that both glycine and GABAA receptor-mediated inputs produce less hyperpolarization and could even paradoxically depolarize the neuron. But even if those inputs become depolarizing on their own, their large conductances mean they may still reduce the depolarization caused by concurrent excitatory input [e.g. [38,39]] – a phenomenon known as shunting.
Without knowing the relative contribution of shunting and hyperpolarization to firing rate modulation, it is difficult to predict how reduction of Eanion will impact inhibitory control of firing rate, especially given the nonlinearity inherent in spike generation [40,41]. Recent work has revealed a good correlation between Eanion and pain threshold [42], but it remains unproven whether reductions in Eanion could compromise inhibition sufficiently to produce the cellular hyperexcitability that may in turn cause the perceptual/behavioral features of neuropathic pain including allodynia (perception of pain in response to normally innocuous stimulation) and hyperalgesia (exaggerated pain perception in response to noxious stimulation).
Allodynia and hyperalgesia are typically thought to arise from hyperexcitability at the cellular and network levels [43]. It is implicit in our analysis that responses in lamina I neurons correlate with pain perception, and that hyper-excitability amongst those neurons could therefore give rise to allodynia and hyperalgesia. Although there is no doubt that lamina I neurons convey information to supraspinal targets [44,45], it has been widely assumed that wide dynamic range cells in lamina V are solely capable of encoding stimulus intensity because lamina I cells are predominantly nociceptive specific (which is mistakenly taken to imply that they lack the capacity to modulate their response magnitude depending on stimulus intensity) and that lamina I cells do not receive low threshold input [for review see [46]]. On the contrary, multiple studies have demonstrated the capacity of lamina I projection neurons to encode stimulus intensity [47–51]; moreover, more recent work shows that the A and C fiber nociceptors that innervate lamina I can encode stimulus intensity [52]. It has also been shown that lamina I neurons receive low threshold inputs via polysynaptic pathways, but transmission through those pathways is normally suppressed by inhibition [53]. The cumulated evidence therefore indicates that lamina I projection neurons can encode nociceptive information relevant for pain perception. It logically follows that hyperexcitability of lamina I neurons may contribute to allodynia and hyperalgesia.
This study was undertaken to investigate quantitatively whether reductions in Eanion could compromise inhibition sufficiently to produce cellular hyperexcitability despite compensatory changes that may develop (e.g. enhanced GABAA components to synaptic events [31]). To this end, we developed a biophysically accurate neuron model for quantitative testing under a variety of conditions. Results demonstrate that even a small reduction of Eanion can cause disinhibition. But whereas compensatory changes may prevent disinhibition resulting from a small reduction of Eanion, reduction of Eanion at magnitudes reported by Coull et al. [31] almost certainly causes incompensable disinhibition. Disinhibition becomes incompensable when compensatory mechanisms fail; the failure of compensation means that the system decompensates and necessarily becomes hyperexcitable. Demonstration that decompensation can occur abruptly, especially in a highly compensated system, may help explain the sometimes unpredictable efficacy of therapeutic interventions and has implications for how to optimize treatment of neuropathic pain.
Results
Impact of Eanion on the inhibitory control of firing rate
A multicompartment model based on available data on lamina I neurons was developed as outlined in the Methods (Fig. 1A). The model neuron was bombarded by synaptic input simulated in a biophysically realistic manner: discrete synaptic events generated inward or outward currents by opening channels in the cell membrane (rather than by simply injecting current) so that synaptic inputs influence the total membrane conductance of the postsynaptic neuron. Simulating synaptic input in this way is crucial for uncovering the role of shunting in firing rate modulation. Three varieties of inhibitory input were tested (Fig. 1B): proportional (to excitation), constant, and feedback. We also tested model neurons with different intrinsic properties (Fig. 1C): basic model containing only fast Na+ and delayed rectifier K+ channels (i.e. Hodgkin-Huxley or HH channels), tonic-spiking, and single-spiking. Tonic- and single-spiking neurons represent the opposite extremes of physiological cell types in lamina I [54]; their responses to somatic injection of constant current are illustrated in Fig. 1C. Using these models, a systematic investigation of the effects of reducing Eanion on firing rate modulation was carried out.
Starting with the basic model neuron, a series of simulations was performed in which the frequency of excitatory synaptic input (fexc), the frequency of inhibitory synaptic input (finh), and Eanion were systematically varied while measuring the frequency of output spiking (fout). We first posited that both fexc and finh increase with increasingly strong stimulation, and that the increase is proportional though not necessarily equal between excitation and inhibition; the ratio of inhibition to excitation (finh/fexc) is reported as α and was tested at multiple values. Figure 2A shows that reduction of Eanion compromises inhibition's capacity to reduce firing rate. The degree to which firing rate modulation is compromised is most readily understood by comparing the fout-fexc curves corresponding to different values of Eanion (colored curves, Fig. 2A) against the fout0-fexc curve with no inhibition (i.e. α = 0; black curve, Fig. 2A). A reduction of Eanion to −55 mV completely incapacitated inhibition, as evidenced by the corresponding fout-fexc curve (light green) lying very close to the no inhibition curve. Reduction of Eanion to −50 or −45 mV caused paradoxical excitation, as evidenced by the corresponding fout-fexc curves (yellow and orange) lying above the no inhibition curve. Even modest reduction of Eanion to −65 or −60 mV caused some disinhibition, as evidenced by the corresponding fout-fexc curves (blue and dark green) lying above the curve for Eanion = −70 mV (purple) but below the no inhibition curve. The transition from modest disinhibition to complete disinhibition to paradoxical excitation as Eanion was shifted from −70 mV to −45 mV occurred regardless of the ratio of inhibition to excitation (compare panels showing different values of α in Fig. 2A). The divergence the fout-fexc curves was, however, exaggerated as α was increased.
Adding feedback inhibition to proportional inhibition with α = 0.5 had an effect (Fig. 2B) similar to increasing the strength of proportional inhibition (i.e. increasing α in Fig. 2A). In other words, feedback inhibition exaggerated the effects of reducing Eanion, which was manifested on the graphs as increased divergence of the fout-fexc curves. Interestingly, the divergence was more exaggerated for large reductions in Eanion (i.e. −45 and −50 mV) compared with small reductions. This is explained by the fact that, when the glycine/GABAA receptor-mediated input becomes paradoxically excitatory, a positive feedback loop is born, whereas the feedback loop is negative under normal conditions. Positive feedback (i.e. feedback excitation) translates into extreme hyperexcitability.
We then tested the effect of constant inhibition. In contrast to the divergent fout-fexc curves seen for proportional inhibition and for feedback inhibition (which is also “proportional” insofar as the feedback neuron's activation is proportional to the output neuron's activity), constant inhibition caused a more parallel shift in the fout-fexc curves (Fig. 2C). The vertical separation of those curves was enhanced by increasing finh.
Using proportional inhibition, we repeated simulations in a tonic-spiking model neuron (Fig. 2D) and in a single-spiking model neuron (Fig. 2E). While the slopes of the fout-fexc curves for these two models were different from those of the basic model (Fig. 2A), the effects of reducing Eanion were very similar: in the tonic-spiking model, reduction of Eanion to −55 mV caused complete disinhibition, the same as in the basic model; the single-spiking model was more resistant, requiring that Eanion be reduced to around −50 mV before disinhibition becomes complete. These data demonstrate that despite the extreme differences in the intrinsic membrane properties of these cell types, firing rate modulation is altered in qualitatively the same way as Eanion becomes reduced.
Data in Figure 2 thus indicate that the degree of reduction in Eanion correlates with the degree to which inhibitory control of spiking is compromised: small reductions (to −65 or −60 mV) cause modest disinhibition, intermediate reductions (to −55 mV) can cause complete disinhibition (i.e. equivalent to completely removing inhibition), and large reductions (to −50 or −45 mV) cause paradoxical excitation. This correlation remains quantitatively true regardless of the circuit connectivity (feedback vs. proportional inhibition), strength of inhibition (magnitude of α or of finh), or intrinsic neuronal properties (tonic- vs. single-spiking). In the more detailed investigation that follows, we focus on proportional inhibition in the basic model but, based on the demonstrations in Figure 2, the results can be reasonably extrapolated to other conditions.
Relative importance of shunting and hyperpolarization for firing rate modulation
The results above demonstrate that reduction of Eanion significantly compromises inhibitory control of firing rate. But although a reduction in Eanion should logically reduce glycine/GABAA receptor-mediated hyperpolarization, a change in Eanion should not reduce glycine/GABAA receptor-mediated shunting. The results, therefore, suggest that shunting plays a relatively minor role in the modulation of firing rate, compared with the dominant role of hyperpolarization. This result was unexpected given previous reports on the importance of shunting [e.g. [55,56]] and therefore required further investigation.
The lack of effect of shunting on firing rate contrasts its known effect on depolarization. To investigate in isolation how shunting modulates depolarization, we modified the basic model by removing HH channels, thereby preventing the model neuron from spiking and allowing us to quantify the underlying depolarization. Figure 3 shows that whereas the depol-fexc curves for Eanion = −45 and −50 mV lay above the curve for no inhibition at low values of fexc, they bent downwards at higher values of fexc so that they eventually lay below the curve for no inhibition (arrow in Fig. 3). This sublinearity in the depol-fexc curves (i.e. downward bend with increasing input) is precisely what one would expect from shunting, where increasingly more excitatory input is shunted through open glycine/GABAA channels as excitation increases. These results thereby demonstrate that glycine/GABAA receptor-mediated shunting remains intact despite reduction of Eanion. As expected, the sublinearity attributable to shunting was absent when there was no inhibition (black depol-fexc curve in Fig. 3 is linear).
But whereas the effects of shunting are clearly evident in the depol-fexc curves of Figure 3, they are absent from the fout-fexc curves of Figure 2 (i.e. the fout-fexc curves with inhibition are only slightly sublinear). If excitatory input drives depolarization, and depolarization drives spiking, then one should look at the relationship between depolarization and firing rate in order to identify why shunting's capacity to modulate firing rate is lost. To do this, we plotted firing rate (fout) against depolarization (Fig. 4A) rather than against fexc (as had been down in Fig. 2). Specifically, we plotted fout generated by a certain value of fexc (based on simulations in the model neuron with HH channels) against depolarization generated by the same value of fexc (based on simulations without HH channels).
Shunting can paradoxically increase firing rate
Figure 4A reveals a rather counterintuitive observation that, for the same amount of depolarization, significantly faster spiking was generated in the presence of synaptic inhibition than in its absence. This was true regardless of the value of Eanion and became more evident as depolarization increased (see below). The most likely explanation for this is modulation of the membrane time constant (τmembrane), since τmembrane becomes shorter as membrane conductance increases (Fig. 4B). To test this, we increased the passive leak conductance in the model neuron to reduce τmembrane by half (open circle on Fig 4B); like inhibitory synaptic input, increasing the passive leak conductance caused fout to increase (dashed curve on Fig. 4A) compared with the original model neuron (solid black curve). Why does shortening τmembrane lead to faster spiking? When the model neuron was shunted and therefore had a short τmembrane, its power spectrum exhibited higher power at all but the lowest frequencies (blue curve in Fig. 4C) compared with the power spectrum for the unshunted neuron with a longer τmembrane (black curve). The most important consequence of this reduced filtering of high frequencies is illustrated in the inset of Figure 4C: reduced filtering allows the membrane to recharge more quickly between spikes so that higher firing rates can be achieved when the neuron is shunted than when it is not shunted.
Therefore, shunting can reduce the depolarization caused by excitatory input (Fig. 3), which reduces firing rate (Fig. 4A), but it also shortens τmembrane (Fig. 4B), which can increase firing rate (Fig. 4C inset). Both effects occur regardless of the value of Eanion, but their relative importance for firing rate modulation depends on other factors. It is important to recall here that the depol-fexc curves with inhibition diverge from the depol-fexc curve without inhibition only when depolarization becomes suprathreshold or, at least, nearly suprathreshold (Fig. 4A). This indicates that that shortening τmembrane is only consequential for firing rate modulation when depolarization is suprathreshold, whereas we know from Figure 3 that shunting reduces both sub- and suprathreshold depolarization.
We have previously demonstrated that, when average depolarization is suprathreshold, spikes are generated deterministically whereas, when average depolarization is subthreshold, spikes are elicited by noisy, suprathreshold voltage fluctuations and are therefore generated probabilistically [41]. The rate of probabilistic spiking is reduced by shunting because, by reducing depolarization, shunting increases the difference between average depolarization and voltage threshold, which in turn reduces the likelihood of voltage fluctuations crossing threshold. If shunting can reduce suprathreshold depolarization so that it becomes subthreshold, deterministic spiking will become probabilistic and its rate will be reduced by shunting according to the above mechanism. If, on the other hand, excitation is sufficiently strong to cause net depolarization that remains suprathreshold despite shunting, spiking will be deterministic. Under those conditions, rather than being limited by the probability of threshold crossing, the interspike interval is limited by the rate of interspike membrane charging which, as demonstrated above, is sensitive to both depolarization and τmembrane. Shunting reduces depolarization but it also shortens τmembrane, where each effect has the opposite impact on the rate of membrane charging. Thus, in regard to firing rate modulation, the dual effects of increased membrane conductance counterbalance each other when spiking is deterministic, whereas the inhibitory effect via reduction of depolarization becomes dominant if spiking is probabilistic. This explanation reconciles the observations in Figures 2 and 3: shunting can reduce depolarization while at the same time having little effect on firing rate.
Since shunting becomes ineffective at modulating firing rate when average depolarization is suprathreshold, one can appreciate that by reducing the hyperpolarization caused by glycine/GABAA receptor-mediated input or, worse yet, causing that input to become depolarizing, a reduction of Eanion can indirectly compromise shunting's capacity to reduce firing rate (Fig. 4D). Thus, both mechanisms through which inhibitory input normally modulates firing rate (i.e. hyperpolarization and shunting) are both susceptible (directly or indirectly) to changes in Eanion. Furthermore, not only are both inhibitory mechanisms compromised by reduction of Eanion, both mechanisms can contribute to paradoxical excitation if the reduction of Eanion is sufficiently large.
Effects of constant vs. intermittent inhibition
We also tested whether the irregularity of inhibitory synaptic input could compromise control of spiking inasmuch as spikes can occur during inhibitory gaps (i.e. between inhibitory synaptic events). This irregularity may be expected to compromise modulation of firing rate by shunting more than it compromises modulation by hyperpolarization since inhibitory gaps are larger in the former case. The difference in size of inhibitory gaps is a direct consequence of the differential time course of inhibitory postsynaptic currents (IPSCs) and inhibitory postsynaptic potentials (IPSPs): IPSPs have a slower time course because of the low pass filtering caused by the membrane time constant, so that whereas slow IPSPs tend to overlap and produce sustained potentials, faster IPSCs (which directly parallel the time course of the synaptic conductance) overlap less and, therefore, shunting is likely to be intermittent (Fig. 5A). Quantitative testing confirmed that gaps between IPSCs close more slowly than gaps between IPSPs as finh increases (data not shown). To test whether inhibitory gaps are functionally significant for firing rate modulation, we replaced the intermittent inhibition associated with irregular synaptic input with constant inhibition equal to the time-averaged synaptic input.
Figure 5B demonstrates that switching from intermittent to constant inhibition had virtually no effect on fout-fexc curves except at the most negative values of Eanion. On initial examination, this suggests that inhibitory gaps are relatively insignificant for firing rate modulation. On closer examination however, the fact that irregular inhibition produces larger gaps in shunting than in hyperpolarization (see above and Fig. 5A) suggests that the irrelevance of switching from intermittent inhibition to constant inhibition may simply reflect the relative impotency of shunting to reduce spiking when spiking is generated in a deterministic manner (see above and Fig. 4). Following from this point, the observation that regularizing inhibition improved the effectiveness of inhibition for Eanion values of −65 and −70 mV (stars in Fig. 5B) is significant, as those Eanion values correspond to the same conditions under which depolarization remained subthreshold at high fexc (Fig. 3). If average depolarization remains subthreshold, then spiking is generated in a probabilistic manner and shunting should retain it capacity to reduce firing rate (see above). In short, Figure 5B shows that regularizing inhibition enhances shunting's capacity to reduce spiking when spiking is probabilistic, but not when spiking is deterministic, consistent with conclusions drawn from Figure 4 regarding the conditions under which shunting can or can not modulate spiking.
Disinhibition in the context of compensatory changes and modulation outside lamina I
Reduction of Eanion does not necessarily occur in isolation. The possibility of compensatory changes complicates the conclusion that disinhibition necessarily follows from reduction of Eanion. Two separate studies suggest that GABAA transmission may increase following neuropathy [31,57]. We therefore investigated whether disinhibition develops if reduction of Eanion is coupled with compensatory increases in GABA/glycine transmission.
Under normal conditions with Eanion = −70 mV, even modest amounts of inhibition (e.g. α = 0.5) can reduce firing rate (Fig. 6A, dotted curve). The decrease in firing rate can be quantified by taking the ratio of the response with inhibition to the response without inhibition (fout/fout0). The fout/fout0 ratio is ~0.6 for α = 0.5, but larger values of α result in greater reduction of that ratio (Fig. 6A, dashed and solid curves). For purposes of illustration, we estimate inhibition conservatively (α = 0.5) and, by extension, consider fout/fout0 > 0.6 to represent disinhibition. If Eanion were to decrease to −60 mV, inhibition with α = 0.5 would be incapable of reducing fout/fout0 to 0.6 (Fig. 6B, dotted curve), meaning disinhibition would occur under those conditions. But if α was increased to 1.2, the fout/fout0 ratio would be returned to 0.6 (Fig. 6B, blue curve). Assuming the network has the capacity to at least double α, this demonstrates that moderate reductions in Eanion cause compensable disinhibition; in other words, the disinhibition that could potentially result from reduction of Eanion can be compensated for by shifting the balance between excitatory and inhibitory input. However, if Eanion were to decrease to −55 mV, even increasing α to 2 could not reduce fout/fout0 to 0.6 (Fig. 6C), demonstrating that disinhibition becomes incompensable if reduction of Eanion becomes large. In fact, an increase in α may not only fail to reduce fout/fout0 sufficiently (as in Fig. 6C), but may even paradoxically increase fout/fout0 as in the case with Eanion = −50 mV (Fig. 6D), resulting in paradoxical excitation. It is interesting to note that paradoxical excitation can occur for values of Eanion below spike threshold, which is approximately −49 mV in the model neuron tested here.
Effects of Eanion on the fout/fout0 ratio are summarized in Figure 6E. fout/fout0 > 0.6 represents disinhibition based on the conservative estimate of α = 0.5 prior to any compensation, while fout/fout0 > 1 represents paradoxical excitation. The relationship between α and the critical value of Eanion beyond which net disinhibition necessarily develops (pink curve on Fig. 6E) shows a ceiling effect wherein increases in α, no matter how large, can not compensate for reductions in Eanion beyond a certain level. For example, assuming α could increase as high as 4, disinhibition would still develop at Eanion ≈ −54 mV (arrowhead in Fig. 6E). Stricter limitation on the increase in α would result in disinhibition occurring for even smaller reductions in Eanion.
In a system capable of compensation, reduction of Eanion does not produce disinhibition until the system decompensates; decompensation occurs when reductions in Eanion outstrip compensatory changes. But although compensatory changes may prevent decompensation from occurring until a large reduction of Eanion has developed, Figure 6F shows that in a system relying on strong compensation (e.g. α = 4 to maintain an fout/fout0 ratio of 0.6), small changes in Eanion may cause large changes in the fout/fout0 ratio; in other words, decompensation occurs abruptly, which reflects instability within the system. This is evident from the steepness of the fout/fout0-Eanion curve where it passes through fout/fout0 = 0.6 (long-dashed curve in Fig. 6F). A system relying on less compensation (e.g. α = 1) may decompensate at a lower value of Eanion, but will do so gradually (dot-dashed curve in Fig. 6F), indicating that the system is more stable. Understanding the abruptness of decompensation may help guide therapeutic intervention insofar as it is preferable to reestablish a robust balance between excitation and inhibition rather than an unstable one (see below).
Pathophysiologic changes associated with neuropathic pain also occur outside lamina I. How do those changes influence activity in lamina I neurons? We have heretofore assumed a constant relationship between the strength of peripheral stimulation and the strength of synaptic input to lamina I. As a postsynaptic mechanism, disinhibition resulting from reduced Eanion within lamina I neurons does not affect that relationship; on the other hand, presynaptic mechanisms including peripheral sensitization and presynaptic inhibition do (Fig. 7A): peripheral sensitization steepens that relationship by increasing the frequency of excitatory input elicited by a given stimulus, while presynaptic inhibition reduces the amplitude of individual synaptic events. The distinction between modulation of event frequency and event amplitude is irrelevant for this discussion, insofar as cumulative synaptic excitation (in Siemens per second) equals amplitude per input (in Siemens per event) multiplied by event frequency (in events per second); in short, peripheral sensitization increases cumulative synaptic excitation whereas presynaptic inhibition decreases it. Following the simple logic outlined in Figure 7B, the relationship between fout and synaptic excitation (which for all intents and purposes is equivalent to the fout-fexc relationship) can be combined with the relationship between synaptic excitation and strength of peripheral stimulation to determine the relationship between fout and strength of peripheral stimulation. Figure 7C illustrates how peripheral sensitization and presynaptic inhibition influence the relationship between fout and peripheral stimulation; note that the relationship between fout and synaptic excitation remains unchanged. If the fout/fout0 ratio is calculated using fout from the test condition and fout0 from the control condition, then the effects of peripheral sensitization and presynaptic inhibition can be visualized as leftward or rightward shifts, respectively, in the relationship between fout/fout0 and Eanion (Fig. 7D). Thus, whereas peripheral sensitization will exacerbate the effects of postsynaptic disinhibition, presynaptic inhibition will mitigate the effects. Beyond that, reduced pre- or postsynaptic inhibition within polysynaptic pathways may uncover low threshold input to lamina I neurons [53], which would also alter the relationship between the strength of peripheral stimulation and synaptic input.
Implications for therapeutic interventions
It seems logical that if a reduction in Eanion has compromised the strength of inhibition, interventions aimed at reestablishing inhibition's strength would be beneficial for correcting disinhibition. Ideally one would target the primary pathophysiologic cause (e.g. KCC2 expression or its direct effects on chloride extrusion from inside the cell, or upstream events including microglial activation and BDNF release [31,42]) but with no clinically available drugs to do this, other targets must be considered. For example, the strength and kinetics of GABAA receptor-mediated input can be modulated by benzodiazepines. We therefore investigated the effects of doubling the strength (w) and τdecay of GABAergic inputs; parameters of glycinergic input were left unchanged. Figure 8A shows that augmenting individual GABAergic inputs mimics the effects of increasing α (compare with Fig. 6F): it delayed decompensation from occurring until higher values of Eanion, but it did so by steepening the curve relating fout/fout0 and Eanion. As explained above, this steepening means that although the intervention may rebalance the system at a normal fout/fout0 ratio, that balance is liable to be disturbed by even small changes in Eanion; in other words, the system becomes increasingly unstable. Increasing GABAergic transmission also risks exacerbating paradoxical excitation if reduction of Eanion is particularly large. Furthermore, although endogenous compensation may be expected to occur specifically in neurons affected by changes in Eanion, a drug would not show the same specificity, suggesting the fout/fout0 ratio would be inadvertently reduced below 0.6 in normal cells (e.g. with Eanion = −70 mV) exposed to the drug at the same time that the fout/fout0 ratio is potentially normalized in affected cells (e.g. with Eanion = −55 mV). Therefore, although a disinhibitory mechanism intuitively suggests that inhibition should be augmented in order to offset the disinhibition, results here suggest that augmenting GABAA receptor-mediated input (or, similarly, glycine receptor-mediated input) may be unwise when disinhibition occurs through reduction of Eanion. In contrast, disinhibition occurring through a different mechanism (e.g. reduced expression or function of GABAA or glycine receptors) could be successfully corrected by augmenting inhibition (see Discussion).
Although increasing inhibition ultimately fails to prevent decompensation if the reduction in Eanion grows too large, and at best reestablishes a normal fout/fout0 ratio that is easily disrupted, interventions that do not target inhibition are likely to be more robust. NMDA receptors are a common pharmacological target whose blockade can reduce neuropathic pain [58]. Figure 8B illustrates that blocking NMDA receptors did not alter the slope of the curve (and thus the stability of the system) and instead caused a uniform reduction in the fout/fout0 ratio, which is similar to the effect of presynaptic changes described in Figure 7. Furthermore, reducing excitatory input did not risk exacerbating paradoxical excitation. The magnitude of effects of NMDA antagonism depends on the contribution of NMDA receptor-mediated excitation relative to AMPA receptor-mediated excitation, which is relatively small for conditions tested here (see Methods) but may increase under neuropathic conditions [59]. In any case, reducing AMPA receptor-mediated excitation had a similar effect (see below). Reducing sodium channel density, which functionally mimics the postsynaptic effects of local anesthetics and many anti-epileptic medications [60], also resulted in modulation very similar to that described above for NMDA antagonism (data not shown). Insertion of a potassium conductance such as might be activated by opioids [61] also had a similar effect (data not shown). All in all, reducing excitatory input and/or reducing intrinsic neuronal excitability reduces the fout/fout0 ratio uniformly across a broad range of Eanion. Combining any of these manipulations with augmentation of inhibition had purely additive effects (Fig. 8C) rather than acting synergistically.
Based on the observation that it may be counterproductive to try to replace inhibition when disinhibition occurs through reduction of Eanion, we explored whether it would be preferable to block inhibitory input altogether and balance the resulting disinhibition with reduction of excitatory input (Fig. 8D). The rationale is that although, on its own, blocking glycine/GABAA receptor-mediated input would increase the fout/fout0 ratio to 1, the appropriate reduction in excitatory input and/or intrinsic excitability could return the ratio to around 0.6 (Fig. 8D, right panel). The benefit of deliberately switching the disinhibitory mechanism (i.e. from reduction of Eanion to blockade of glycine/GABAA receptors) is that blocking glycine/GABAA receptor-mediated input would uniformly adjust the fout/fout0 ratio to 1, thereby trivializing the variability of Eanion and α between affected and unaffected neurons. Moreover, blocking inhibition would prevent inhibitory input from causing paradoxical excitation in the event that reduction of Eanion was particularly large. One significant requirement is that the distribution and kinetics of the drugs affecting inhibitory and excitatory transmission are similar in order to avoid spatial or temporal mismatches between the two effects. Even if this approach may not be feasible in practice, it illustrates that increasing glycine/GABAA receptor-mediated input may be counterproductive in certain neuropathic conditions and that alternative approaches, possibly involving non-intuitive drug combinations, deserve consideration.
Discussion
This modeling study demonstrates that reduction of Eanion in spinal lamina I neurons can result in disinhibition and hyperexcitability. Compensatory changes may successfully prevent disinhibition and maintain a normal input-output relationship, but they ultimately fail (i.e. the system decompensates) if reduction of Eanion exceeds a critical value. Although incompensable disinhibition requires a relatively large reduction in Eanion, the magnitude of that reduction is physiologically plausible and need not be so large as to cause glycine/GABAA receptor-mediated inputs to become paradoxically excitatory. Compensatory mechanisms eventually fail because, although glycine/GABAA receptor-mediated inputs continue to shunt excitatory inputs despite no longer causing hyperpolarization, the increase in membrane conductance that underlies shunting also shortens the membrane time constant which, under certain circumstances, allows for faster spiking. Deciphering this complex interplay between biophysical mechanisms is, ultimately, important for being able to ascribe perceptual changes (e.g. allodynia and hyperalgesia) to pathophysiologic changes at the cellular and molecular level.
Relative importance of shunting and hyperpolarization for firing rate modulation
Glycine and GABAA receptor-mediated inputs are often thought to act via shunting rather than through hyperpolarization given their large conductance and relatively small driving force. Consequently, even depolarizing glycine/GABAA receptor-mediated input can have a net inhibitory effect on spike generation by shunting concurrent excitatory input [38,39]. In the current study, we found that shunting was far less significant than hyperpolarization when it comes to modulation of repetitive spiking. This results from increased membrane conductance having opposing effects: it reduces depolarization by shunting excitatory input (thereby decreasing firing rate) but simultaneously shortens the membrane time constant (thereby increasing firing rate). In terms of firing rate modulation, the first effect predominates when average depolarization is subthreshold and spiking is driven by suprathreshold voltage fluctuations (i.e. probabilistic spiking) but the two effects offset each other when average depolarization is suprathreshold (i.e. deterministic spiking). Shunting therefore becomes ineffective at reducing firing rate when depolarization is suprathreshold, which is especially likely to occur when reduction of Eanion renders glycine/GABAA receptor-mediated input less hyperpolarizing or, worse yet, depolarizing. Kuhn et al. [62] have previously described how firing rate modulation is complicated by the dual effects of membrane conductance, noting that average depolarization was reduced by shunting while voltage fluctuations became larger because of the reduced filtering associated with a shortened membrane time constant. Their result applies to probabilistic spiking whereas the effect that we have described applies to deterministic spiking and, in that sense, is distinct.
A recent modeling study that investigated the effects of changes in Eanion on firing rate modulation in a neocortical pyramidal neuron [63] concluded that, because of shunting, GABA remained inhibitory despite changes in Eanion. That would appear to contradict our conclusion that shunting fails in the face of large, but physiologically plausible (as reported in Coull et al. [31]), reductions in Eanion. However, Morita et al. tested only two values of Eanion. According to our evaluation of the phase-planes shown in Figure 4C of their paper, a Hopf bifurcation would not have prevented repetitive spiking if a slightly lower value of Eanion had been tested. There is therefore no discrepancy between our studies but, simply, a difference in the range of Eanion tested.
Compensable vs. incompensable disinhibition
Given this new understanding of the biophysical mechanisms, we can predict whether the disinhibition caused by reduction of Eanion can account for the hyperexcitability that is presumably responsible for the allodynia and hyperalgesia associated with neuropathic pain. We estimate that decompensation would start occurring at Eanion ≈ −58 mV (see Fig. 6) assuming that, as compensation, the ratio of inhibitory to excitatory input quadruples relative to the ratio under normal conditions (i.e. α increases from 0.5 to 2). Weaker compensation (α increases to only 1) would result in decompensation starting at Eanion ≈ −61 mV, whereas stronger compensation (α increases as high as 4) would prevent decompensation until Eanion ≈ −54 mV. All of those estimates conservatively assume α = 0.5 under normal conditions (see Results).
Variations in compensation may explain why, with acute manipulations including intrathecal application of BDNF and activated microglia, reduction in Eanion to around −62 mV caused almost an equivalent decrease in withdrawal threshold as that associated with reduction of Eanion to −49 mV following peripheral nerve injury [42]: compensation that could have developed in the latter case, may not have had time to develop in the former case; it is also possible, however, that BDNF has other effects [64] that are unaccounted for in this argument. Additionally, we have not taken into account activity-dependent reduction of the chloride gradient [65–68], meaning a much smaller long-term reduction of Eanion (i.e. caused by reduced KCC2 expression) may cause incompensable disinhibition once dynamic, short-term reductions of Eanion (i.e. caused by activity-dependent reduction of the chloride gradient) are taken into account. Activity-dependent potassium accumulation is another fast mechanism that may exacerbate reduction of Eanion leading to incompensable disinhibition [69,70]. In any case, given that Eanion reduces on average to −49 mV following peripheral nerve injury [31], the disinhibition that results is most likely incompensable; indeed, the observation that ~20% of lamina I neurons were paradoxically excited by GABA under those conditions [31] suggests that an even larger fraction were incompensably disinhibited (given that disinhibition requires less reduction in Eanion than paradoxical excitation).
Regardless of the precise value of Eanion at which it occurs, incompensable disinhibition approximates the conditions of pharmacologically reduced inhibition, which previous work has shown to be sufficient to produce allodynia and hyperalgesia [9–17]. It logically follows that incompensable disinhibition resulting from reduced Eanion is sufficient to explain the exaggerated pain perception associated with neuropathic pain. We could not reach that conclusion if disinhibition were compensable (i.e. if compensatory mechanisms could successfully prevent disinhibition) because, in that case, even if the reduction in Eanion could cause allodynia and hyperalgesia, whether or not it did would depend on the success or failure of compensation. Notably, the argument that incompensable disinhibition is sufficient to explain allodynia and hyperalgesia does not exclude other mechanisms from contributing to the aberrant perception; for example, Figure 7 illustrates how peripheral sensitization can exacerbate the hyperexcitability caused by postsynaptic disinhibition. Furthermore, under conditions in which a modest reduction of Eanion occurs, such that some inhibitory capacity remains (e.g. with Eanion <−55 mV), a decrease in GABAergic or glycinergic transmission (either transmitter release or receptor function; see below) would contribute to the disinhibition caused by reduction of Eanion.
Reduction of Eanion vs. other mechanisms of disinhibition
Although there is general consensus that disinhibition occurs following neuropathy, the mechanism underlying disinhibition has been controversial. Studies have reported that the number of inhibitory neurons in the spinal dorsal horn decreases following peripheral nerve injury [24,26–28], but more recent work has argued against this [34–36]. Other studies have reported a reduction of presynaptic GABA, implicating the GABA transporter GAT1 [32] and the GABA synthesizing enzyme GAD65 [30], but again this has been contradicted by the demonstration of no change in synaptosomal glycine or GABA [33]. In fact, Kontinen et al. [57] reported that GABAergic transmission was increased following neuropathy, presumably as a compensatory change, which would be consistent with the increased GABA contribution to background input to lamina I neurons reported by Coull et al. [31]. Disinhibition through reduction of Eanion does not involve reduced glycinergic or GABAergic transmission but instead works at a downstream locus and controls the potency of inhibitory input. Ironically, although the system has built-in redundancy, inasmuch as it uses both glycine and GABA as fast inhibitory neurotransmitters [71–73], the reduction in Eanion subverts both glycine and GABAA receptor-mediated inhibition because of the receptors' common reliance on the transmembrane chloride gradient. This contrasts the mechanism responsible for disinhibition in inflammatory pain, where prostaglandin E2-induced phosphorylation of the glycine receptor decreases glycinergic transmission [74,75] without affecting GABAergic transmission. Under those conditions, increased GABAergic transmission can compensate for decreased glycinergic transmission [76].
In addition to reducing Eanion via decreased KCC2 expression, neuropathy leads to other pathophysiologic changes in lamina I and elsewhere in the pain pathway that undoubtedly contribute to the aberrant perception associated with neuropathic pain [1,3,77–79]. But whereas most other changes perturb a neuron's input-output relationship directly, disinhibition acts indirectly by perturbing a modulatory process. Glycine and GABAA receptor-mediated inhibition are crucial mechanisms for the endogenous modulation of pain [80,81], controlling the inflow of Aδ and C fiber-mediated inputs at the segmental level [32,53,82–85] as well as participating in descending modulation originating from the periaqueductal gray and nucleus raphe magnus [86–89]. These control mechanisms are seriously compromised by reduction of Eanion. Disinhibition therefore equates with modulation of a modulatory process, or meta-modulation, and represents a higher order change that not only perturbs the system, but simultaneously compromises the control mechanisms that would otherwise correct that perturbation. This may contribute to the relative intractability of neuropathic pain compared with other types of pain. The ideal therapy would target the primary pathological change and return Eanion to its normal value; this would not only reestablish a normal input-output relationship within the system, but would also restore the system's full capacity to control the input-output relationship. No such therapy currently exists.
Implications for therapeutics
The mechanism underlying disinhibition does, nonetheless, have significant implications for reestablishing inhibition through therapeutic interventions. Specifically, disinhibition through reduction of Eanion means increasing glycinergic and/or GABAergic transmission may be inconsequential, and potentially even detrimental, depending on the degree of change in Eanion. However, several studies have reported that allodynia was reduced by applying GABAA receptor agonists [17,20,22] or by transplantation of GABA-producing cells into the lumbar subarachnoid space [19]. This could be explained by an increase in presynaptic inhibition, since primary afferent terminals do not express KCC2 and are therefore not prone to disinhibition by reduced KCC2 expression [31]. However, the efficacy of increasing GABAergic transmission is controversial: Stubley et al. [21] reported that transplanted-GABA producing cell could prevent allodynia if transplanted early after nerve injury but could not reverse allodynia once it was established, other studies have reported failure of GABAA agonists to relieve neuropathic pain after ischemic spinal cord injury [90,91]. These inconsistencies may be attributable to variation in the model of neuropathic pain, but that implicitly assumes that the underlying mechanisms are qualitatively different depending on the model. The present study suggests that variation can also be explained by quantitative differences in the degree of Eanion reduction, i.e. whether the system was fully decompensated or whether residual inhibitory capacity remained.
Several authors have advocated the benefits of optimizing the treatment of neuropathic pain by choosing treatments targeted towards mechanisms implicated in the pathogenesis of that pain [92–96]. This study has approached the topic of mechanism-based therapies using quantitative modeling (see Fig. 8). By being quantitative, our results highlight how choosing the optimal therapy might not depend solely on what pathogenic mechanism is involved, but on the degree of change that has occurred, e.g. whether the system would benefit from augmenting inhibition depends on how much Eanion is reduced. This may explain the variable efficacy of treatments amongst patients who are suffering from the same neuropathic pain syndrome, where the underlying mechanism is presumably the same but in whom reduction of Eanion may vary.
Conclusion
Reduction of Eanion can dramatically reduce the inhibitory control of spiking in spinal lamina I neurons, but whether this gives rise to the exaggerated sensitivity characterizing neuropathic pain depends quantitatively on the success or failure of compensatory mechanisms. Understanding the efficacy of therapeutic interventions also depends on this quantitative understanding. These results speak to the importance of using quantitative, biophysically accurate models to bring together the vast panoply of experimental data so that we not only identify mechanisms involved in neuropathic and other types of pain, but so that we understand those mechanisms quantitatively and fully exploit them for clinical benefit.
Methods
All simulations were performed with NEURON simulation software [97] using a compartmental model of a generic spinal lamina I neuron with resting membrane potential (Vrest) = −63 mV, input resistance (Rin) = 470 MΩ, and membrane time constant (τmembrane) = 31 ms, based on average values in Prescott and De Koninck [54] and Coull et al. [31]. Dendrites bifurcated up to fourth order for a total of 60 dendritic compartments (see Fig. 1A). Axial resistivity was 150 Ω·cm. An axon similar to that described by Mainen et al. [98] was attached to the soma. Fast Na+ and delayed rectifier K+ conductances based on Traub and Miles [99] were inserted at 0.1 and 0.01 S/cm2 respectively, in the soma, axon initial segment, and axon nodes. Voltage threshold for spiking was approximately −49 mV. A passive leak conductance was distributed evenly throughout the neuron and was adjusted to produce the passive membrane properties described above. Confirmatory testing (see Fig. 2D and 2E) was also performed on two other model neurons with distinct intrinsic properties (Fig. 1C). The tonic-spiking model is the same as model 2 in Figure 9 of our earlier paper [100]. The single-spiking model was derived from the tonic-spiking model by removing INa, P, ICa, P, ICa, T, and IK, S, and then inserting, at 0.2 mS/cm2 a low-threshold K+ current, IK, LT, modified from the M-type K+ current described in [101]; voltage at half-maximal activation was shifted to −45 mV and kinetics were sped up 100×. For the single-spiking model, all synaptic weights (see below) were tripled in order to overcome the low intrinsic excitability of this cell type.
Synaptic conductances were modeled as a rapid exponential rise in conductance combined with a slower exponential decay in conductance described by τrise and τdecay, respectively. Synaptic current is therefore written as Isyn(t) = w [1-exp(-t/τrise)]exp(-t/τdecay) (Vm-Erev) where t = 0 at the onset of a synaptic event, w is synaptic weight, Vm is membrane potential, Erev is reversal potential, and τrise = 0.5 ms for all synapses. Excitation was mediated by four sets of excitatory synapses distributed throughout the dendrites. Each set consisted of 5 synapses, with 2–3 AMPA synapses and the remainder being NMDA synapses. For AMPA synapses, τdecay = 5 ms and w was set so that AMPA receptor-mediated events had a peak conductance of 333 pS/synapse [100]. For NMDA synapses, τdecay = 25 ms and w was left equal to that for AMPA synapses, such that NMDA-receptor mediated events contributed 14% of the total excitatory current in voltage clamp simulations at −60 mV [102]. The voltage-dependent magnesium block of the NMDA current was modeled after Jahr and Stevens [103,104] [see also [105]]. Reversal potential (Erev) was 0 mV for both AMPA and NMDA receptor-mediated inputs.
Eight sets of 2–3 inhibitory synapses were distributed randomly in the perisomatic region; distributing the inhibitory synapses throughout the dendrites and soma, rather than perisomatically, had no significant effect on the efficacy of inhibition, as confirmed with a separate series of simulations. Half of the inhibitory neuron sets were modeled after glycine synapses with τdecay = 12 ms and while the other half were modeled after GABA synapses with τdecay = 60 ms [31]. Synaptic weight was adjusted to produce glycine receptor-mediated events with a peak conductance of 450 pS/synapse [72]. Synaptic weight was fivefold less for GABA synapses so that, given the fivefold increase in τdecay, GABA receptor-mediated events contributed roughly the same total conductance as glycine receptor-mediated events, as reported by Coull et al. [31]. Mixed glycine/GABAA receptor-mediated inhibition simulates the conditions following neuropathy [31], but switching to full glycinergic inhibition did not significantly affect firing rate modulation; results from mixed inhibition are therefore reported throughout. Reversal potential for inhibitory events (which equates with Eanion) was tested at 5 mV increments between −70 mV and −45 mV, which encompasses the range expected for normal and neuropathic conditions [31].
Simulations were intended to mimic the bombardment of a lamina I neuron with input arriving mono- and polysynaptically from multiple primary afferent fibers, where the spike train in each afferent is assumed to be semi-random (i.e. the interspike interval distribution has a coefficient of variation (CV) between 0 and 1). The summation of those spike trains gives a cumulative spike train whose distribution has a CV approaching 1 (i.e. a Poisson distribution); each set of synapses was therefore driven by an independent Poisson process. For excitatory input, the rate of each Poisson process was multiplied by the number of synaptic sets (i.e. 4) so that fexc specifies the total rate of EPSPs received by the neuron from all presynaptic cells; note that simultaneous activation of multiple excitatory synapses constitutes a single EPSP, which explains why we multiplied by the number of synaptic sets rather than by the total number of synapses in order to calculate the EPSP rate. The EPSP rate reported by Furue et al. [106] using in vivo patch clamp falls within the range we tested. Notably, each set of synapses had a slightly different spatial distribution and a different constitution of AMPA and NMDA synapses so that not all primary afferent activity elicited identical EPSPs in the postsynaptic cell. The rate of IPSPs, reported as finh, was also calculated by multiplying the Poisson process rate by 4 so that fexc and finh are readily comparable. In keeping with the classical view that excitation is driven mainly by input from small diameter fibers while inhibition is driven mainly by input from large diameter fibers [80], where both types of fibers can be driven by the same peripheral stimulus, we posited that fexc and finh are proportional, although not necessarily equivalent; the constant of proportionality is reported as α, where α = finh/fexc. Since activity in large and small diameter fibers is independent, a temporal relation between EPSPs and IPSPs is not likely to exist, except at the stimulus onset because of the differential conduction velocity in differently sized fibers and the minimum number of intervening synapses (monosynaptic excitation vs. disynaptic inhibition). We have not explicitly modeled the stimulus onset and assume, for proportional inhibition, that excitation and inhibition are temporally independent. The variable delays introduced by signal transmission through polysynaptic pathways encourage this temporal independence, and also support our decision to model inputs as Poisson processes.
In one set of simulations, the time-averaged conductance at each value of finh was calculated and subsequently applied as a constant conductance to the soma, thereby replacing the intermittent inhibition produced by irregular synaptic input with constant inhibition. For certain simulations, feedback inhibition was introduced by setting up a simple network (see Fig. 1C) in which the neuron of interest excited another neuron (with the same intrinsic properties as the first) which, in turn, inhibited the neuron of interest. Strength of excitatory synapses was adjusted so that a spike in the presynaptic neuron typically elicited a spike in the postsynaptic neuron, meaning the firing rates of both neurons were roughly equivalent. Based on the delay between spike generation in the output neuron and spike generation in the feedback neuron, and the subsequent synaptic delay, the output neuron experiences an IPSP ~8.5 ms after each spike. Feedback inhibitory synapses were identical to inhibitory synapses described above.
Simulated temperature was 23°C, which is consistent with the kinetics we used here for voltage- and ligand-gated currents. These results can, however, be safely extrapolated to in vivo conditions (i.e. 37°C). The main concern is that kinetics will be faster at warmer temperatures, which, in the case of inhibitory synaptic input, could result in larger inhibitory gaps. Data in Figure 5 argue that such gaps are relatively unimportant, which is consistent with the absence of any appreciable effect when switching between mixed GABA/glycinergic inhibition and full glycinergic input (see above). All Simulations were 20 s long. With such long simulations, standard deviation for firing rate was only ~0.5 Hz across multiple trials; most conditions were therefore tested with a single trial.
Competing interests
The author(s) declare that they have no competing interests.
Authors' contributions
SAP designed the study and performed all simulations. YDK helped conceive the study. All authors contributed to preparation of the final manuscript.
Footnotes
List of abbreviations
Acknowledgements
This research was supported by the Natural Sciences and Engineering Research Council of Canada (YDK) and by the Howard Hughes Medical Institute (TJS). YDK is a senior scholar of the Fonds de la recherche en santé du Québec. SAP was supported by postdoctoral fellowships from the Canadian Institutes of Health Research and from the Human Frontier Science Program.