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Copyright © 2003, The National Academy of Sciences Neuroscience Gain control of firing rate by shunting inhibition: Roles of synaptic noise and dendritic saturation *To whom correspondence should be addressed. E-mail: prescott/at/pharma.mcgill.ca. Communicated by William T. Newsome, Stanford University School of Medicine, Stanford, CA Received June 11, 2002; Accepted December 12, 2002. ![]() | |||||
Abstract Adjusting input–output gain is crucial for information processing by the brain. Gain control of subthreshold depolarization is commonly ascribed to increased membrane conductance caused by shunting inhibition. But contrary to its divisive effect on depolarization, shunting inhibition on its own fails to divisively modulate firing rate, apparently upsetting a critical tenet of neural models that use shunting inhibition to achieve gain control. Using a biophysically realistic neuron model, we show that divisive modulation of firing rate by shunting inhibition requires synaptic noise to smooth the relation between firing rate and somatic depolarization; although necessary, noise alone endows shunting inhibition with only a modest divisive effect on firing rate. In addition to introducing noise, synaptic input is associated with a nonlinear relation between somatic depolarization and excitation because of dendritic saturation; this nonlinearity dramatically enhances divisive modulation of firing rate by shunting inhibition under noisy conditions. Thus, shunting inhibition can act as a mechanism for firing rate gain control, but its modulatory effects (which include both divisive and subtractive components) are fully explained only when both synaptic noise and dendritic saturation are taken into account. | |||||
Gain control refers to modulation of a neuron's responsiveness to input; it is critically important for normal sensory, cognitive, and motor functions (1–5). Gain control is achieved through a divisive process and is observed as a change in the slope of the input–output curve. A longstanding and widely cited candidate mechanism for gain control is shunting inhibition (6, 7) such as that mediated by GABA-gated Cl− conductance, which decreases input resistance with little direct effect on membrane potential (Vm). Whereas hyperpolarization has a subtractive effect on the depolarization caused by concurrent excitation, shunting has a divisive effect (Fig. 1A), thereby modulating gain of synaptic depolarization.
However, neurons communicate with each other through spike trains. One must, therefore, consider how Vm translates into firing rate. Contrary to initial intuition, modeling studies (8–10) supported by in vitro electrophysiological data (11, 12) concluded that shunting inhibition modulates firing rate in a purely subtractive manner. However, electrophysiological studies that analyzed responses in the presence of background synaptic input (13–15) provide circumstantial evidence supporting a role for shunting inhibition in gain control. These dichotomous findings suggest that properties of synaptic input such as its noisiness or its effect on membrane conductance (16, 17), which were neglected in past modeling, may explain the discrepancy regarding the role of shunting inhibition in gain control. Only recently did Chance et al. (18) independently demonstrate that background synaptic noise was key for achieving firing rate gain control by shunting. This study was therefore undertaken to examine whether shunting inhibition can divisively modulate firing rate when a neuron is bombarded by synaptic input comparable to that recorded in the intact brain. The data demonstrate that shunting inhibition can indeed mediate firing rate gain control so long as both synaptic noise and dendritic saturation are taken into account. | |||||
Methods Compartmental Model. A compartmental model was constructed from a layer V neocortical pyramidal neuron described in ref. 19. An axon morphologically identical to that in ref. 20 was attached to the soma. A passive leak conductance reversing at −70 mV was inserted at 0.02 mS·cm−2 in all compartments except axonal nodes, where it was 10-fold higher, giving an input resistance of 59 MΩ and membrane time constant of ≈40 ms. Axial resistivity was set to 200 Ωcm, and membrane capacitance was set to 1 μF· cm−2 except in axon internodes, where it was 0.04 μF·cm−2. Voltage-dependent Na+ and K+ conductances [Hodgkin–Huxley (HH) channels] were inserted at different densities depending on the compartment (Na+, K+ in mS Synaptic Input. Synaptic input was modeled as a nonsaturating conductance with a dual exponential time course of the form [1-exp(-t/τrise)]exp(-t/τdecay). For excitatory input, τrise = 0.2 ms, τdecay = 2 ms, gmax = 1 nS, and Erev = 0 mV; for inhibitory input, τrise = 0.8 ms, τdecay = 8 ms, gmax = 0.5 nS, and Erev = −70 mV (10, 16, 17, 22–24). Two sets of eight inhibitory synapses were positioned randomly in the perisomatic region. One set of 24 excitatory synapses was positioned on the basal dendrites, and another set was positioned on the apical tuft. The number of synapses per set reflects a combination of the number of contacts per presynaptic cell (23, 25) and the coordination of presynaptic activity (17, 26–28) and was adjusted to generate Vm fluctuations of appropriate amplitude (17). Each set was driven by an independent Poisson process. The total number of synapses is estimated to be 24,000 excitatory (26 synapses per 100 μm2 of dendritic surface area; ref. 29) and 4,800 inhibitory (one-fifth the number of excitatory synapses; ref. 17); the 48 excitatory and 16 inhibitory model synapses were therefore driven, respectively, at 500 times and 300 times the event frequency of real synapses, essentially replacing spatial summation with temporal summation. Validity of this assumption was tested in a model cell with a 10-fold increase in the number of sets of synapses and equivalent decrease in frequency of events per synapse; this more complex model reproduced all critical findings reported in Figs. 1 and 2. Data from the more computationally efficient model are reported throughout. Reported frequencies refer to events per synapse after correction for synapse number.
Nonsynaptic Stimulation. Constant current (Iclamp) was injected into the soma to simulate the typical electrophysiological experiment. In some cases, to remove noise but still mimic the conductance-altering properties of synaptic input, fluctuating synaptic conductances were replaced with constant excitatory conductance (gexc) and/or inhibitory conductance (ginh) simulated as either a single-point process in the soma or as multiple-point processes at synaptic sites. Reported values of gexc and ginh refer to total excitatory or inhibitory conductance, based on the time-averaged synaptic conductances scaled by the number of synapses. Contrariwise, to recreate Vm fluctuations caused by synaptic input without the associated conductance change, noisy current was generated by an Ornstein–Uhlenbeck process and injected into the soma. We used a process like that described in ref. 30 where current fluctuates according to dI/dt = −1/τ (I − Iavg) + σ √dt N (0, 1), where N (0, 1) is a number drawn from a Gaussian distribution with average 0 and unit variance with a scaling factor σ, which adjusts noise amplitude. The power spectrum of Vm generated with τ = 5 ms reasonably approximates that generated by synaptic input. All simulations were conducted with neuron 4.3.1 by using a 10-μs time step and simulated temperature of 36°C. Segmentation of compartments was defined by using the d_lamba criterion in neuron (31), such that grid points were spaced no further than 10% of the length constant at 100 Hz. Each datum is based on 10 s of simulated time for output firing rate (fout) values and 1 s for Vm values. | |||||
Results Morphology of the model cell and a representative response to synaptic input are shown in Fig. 1B. Under conditions approximating in vivo background synaptic input [rate of excitatory synaptic events (fexc) = 1.0 Hz/synapse and rate of inhibitory synaptic events (finh) = 5.6 Hz/synapse] (17), our model neuron was depolarized to −60.7 mV, standard deviation of Vm was 2.5 mV, and Rin was reduced by 89%. This is comparable to previously reported values (16, 17) indicating that model parameters were set within reasonable bounds. A critical finding of this study was that shunting inhibition mediates firing rate gain control when the model cell was bombarded by realistically simulated synaptic input. Slope of the relationship between output firing rate and frequency of excitatory input was clearly reduced as the frequency of inhibitory input was increased (Fig. 1 C and D). However, the effect of shunting was not purely divisive; the rightward shift in the fout-fexc curves points to a subtractive component. Equivalent divisive/subtractive modulation was observed when the number of simulated synapses was increased (see Methods) and in two different model neurons tested (data not shown). When spike frequency adaptation was incorporated into the model neuron by addition of an M-type K+ current, the divisive effect was still evident in both the preadapted and adapted firing rates (data not shown). This result is inconsistent with previous modeling studies that concluded shunting inhibition had a purely subtractive effect on fout (8–10). The main difference between models was the manner in which synaptic input was simulated. We therefore tested our model neuron in the absence of synaptic noise by generating excitation with constant somatic Iclamp and shunting with constant somatic ginh. Under these conditions, shunting inhibition had a purely subtractive effect on fout (Fig. 1E), implicating properties of the synaptic stimulation as key factors for achieving division. Neuronal Behavior in the Presence of Synaptic Input. How does synaptic input alter neuronal behavior to allow divisive modulation of fout by shunting inhibition? To first investigate the impact of synaptic input without the added complexity of spike generation, we removed HH channels from all compartments except distal dendrites. This reduced HH condition prevented spiking while still encouraging propagation of distally evoked depolarization into the soma. Fig. 2A shows the noisy fluctuations in somatic Vm produced by random synaptic input. Increased shunting had two effects: reduction of average somatic depolarization caused by excitatory input (Fig. 2B) and reduction of Vm fluctuation amplitude (Fig. 2C). Notably, standard deviation of Vm did not reduce to 0 mV, showing that fluctuations remain even in the presence of strong shunting. The near-horizontal regression lines in Fig. 2C indicate that Vm fluctuation amplitude was relatively unaffected by the strength of excitation, consistent with previously reported data (17, 18, 30). Because of its fluctuations, somatic Vm occasionally crosses spike threshold despite average Vm remaining below threshold (Fig. 2A). Therefore, probability of crossing threshold depends jointly on average Vm (i.e., distance from threshold) and amplitude of Vm fluctuations. The consequences of this for spiking are seen in Fig. 2D after reinsertion of HH channels. Constant Iclamp gives rise to an abrupt increase in fout at rheobase (i.e., minimum constant Iclamp eliciting spikes), where probability of spiking jumps from 0 to 1. By contrast, fout increases more gradually in response to noisy input because the Vm fluctuations allow for probabilistic or noise-driven spiking before average depolarization reaches threshold. This amounts to smoothing the threshold nonlinearity, allowing fout to encode the average underlying Vm even while average Vm remains below threshold (32–35). Given that smoothing allows fout to more closely parallel Vm, divisive modulation of Vm may be predicted to manifest as divisive modulation of fout under noisy conditions. Spiking recorded in vivo is normally highly irregular (36, 37). Fig. 2E shows that spiking caused by Vm fluctuations (which we refer to as probabilistic spiking) is irregular compared with spiking generated by raising average Vm above threshold (27, 38–41). Under in vivo conditions, neurons typically operate in the lower range of fout, in a regime where spiking is driven by noise (42, 43). We therefore paid particular attention to this regime of probabilistic spiking. Role of Synaptic Noise. Having identified noisy fluctuations in Vm as important for determining spiking elicited by synaptic input, an obvious question was whether noise alone is sufficient to allow shunting inhibition to divisively modulate fout. To isolate the influence of noise, Vm fluctuations with amplitude and frequency content similar to that generated by synaptic input were reconstituted in the model neuron by directly injecting noisy current into the soma (Fig. 3A). As with synaptic input (Fig. 2D), Vm fluctuations generated by noisy Iclamp smoothed the threshold nonlinearity in proportion to the intensity of the noise (Fig. 3B).
Under noisy conditions, shunting had some divisive effect on fout (Fig. 3C) in contrast to the purely subtractive effect observed with constant Iclamp (Fig. 1E). However, Fig. 3C shows that the reduction in slope of fout-Iclamp curves was restricted to probabilistic firing (i.e., lower values of fout), with curves becoming parallel at fout > 40 Hz. Furthermore, based on responses with fout < 40 Hz, the reduction in slope was modest compared with that observed under synaptic conditions for the same amount of shunting (Fig. 3D). Thus, the divisive effect of shunting inhibition under synaptic conditions is not fully explained simply by adding noise and, except under certain circumstances (e.g., with precisely balanced excitation and inhibition), the subtractive component predominates (Fig. 6, which is published as supporting information on the PNAS web site, www.pnas.org). What other aspect of synaptic input is responsible for the remaining division? Role of Dendritic Saturation. We have argued that noisy Vm fluctuations allow fout to more accurately encode average Vm by smoothing the threshold nonlinearity. However, the relationship between excitatory input and somatic depolarization depends on the stimulus. Somatic depolarization is linearly related to somatic Iclamp (Fig. 4A Upper), and the relationship becomes only modestly sublinear for excitation by somatic gexc (Fig. 4B Upper). In contrast, somatic depolarization is nonlinearly related to dendritic gexc (Fig. 4C Upper), which is particularly important given that most excitatory synapses occur on dendrites. The strong nonlinearity results from a complex interplay between multiple factors (see Discussion), which we simply refer to as dendritic saturation.
Analysis of the effects of shunting inhibition on somatic Vm-excitation curves (Fig. 4 A Upper–C Upper) allows one to roughly predict the subtractive effect of shunting inhibition on fout for comparison to fout-excitation curves (Fig. 4 A Lower–C Lower). Without noise, the subtractive effect of shunting on fout is predicted by where Vm-excitation curves cross spike threshold. Introducing noise reduces the effective threshold for spiking such that the subtractive effect on fout can be predicted from where Vm-excitation curves enter the region of probabilistic spiking. The subtractive effect is therefore reduced in noisy conditions. Insight into the divisive effect of shunting can be similarly gained by considering alteration in the trajectory of Vm-excitation curves through the region of probabilistic spiking. Acting on the linear Vm-somatic Iclamp curves and near-linear Vm-somatic gexc curves (Fig. 4 A and B, respectively), shunting causes a rather modest change in trajectory compared with the strong change observed when shunting acts on the nonlinear Vm-dendritic gexc curves (Fig. 4C). The relative changes in trajectory for Vm-somatic Iclamp and Vm-dendritic gexc curves are summarized in Fig. 4D and correctly predict the enhanced divisive effect of shunting on fout (compare with Fig. 3D), depending on stimulus conditions. The critical observation is that the nonlinearity caused by dendritic saturation enhances the divisive effect of shunting on fout much more than it enhances the subtractive effect because of the shape of the nonlinearity relative to the region of probabilistic firing. Thus the combined effects of two factors, threshold smoothing by synaptic noise and the nonlinearity caused by dendritic saturation, appear to explain how shunting inhibition modulates fout. We sought to quantitatively test the adequacy of this explanation by reconstituting a family of fout-gexc curves representing different levels of inhibition for comparison to the fout-fexc curves in Fig. 1C. In other words, these fout-gexc curves do not represent results of additional simulations but rather are calculated from earlier simulations aimed at independently measuring the threshold smoothing caused by noise (Fig. 3) and the nonlinearity introduced by dendritic saturation (Fig. 4C). The manner in which the multiple different relationships described earlier in the paper were combined to generate fout-gexc curves directly comparable to fout-fexc curves is explained in Fig. 4E. The calculated fout-gexc curves exhibit a pattern of mixed divisive and subtractive change (Fig. 4F, solid lines) closely resembling that seen in response to synaptic excitation (dashed lines; data from Fig. 1C). By comparison, applying the modest nonlinearity associated with somatic gexc (which circumvents the effects of dendritic saturation; see Fig. 4B) could not accurately reproduce the data in Fig. 1 C and D (Fig. 6). Thus, synaptic noise and dendritic saturation both influence how synaptic input is transduced into firing rate, and both must be taken into account to fully explain the outcome of shunting inhibition on firing rate. | |||||
Discussion Data presented here indicate that shunting inhibition can indeed divisively modulate firing rate, but that this ability relies on two effects of synaptic input: the noise it causes in somatic Vm and the saturation it causes in dendrites. In the absence of those effects, shunting has a purely subtractive effect on fout (Fig. 5A). Threshold smoothing by noise allows a modest divisive reduction of fout by shunting (Fig. 5B). On its own, the nonlinearity caused by dendritic saturation does not allow a divisive effect of shunting on fout (Fig. 5C). However, when combined with threshold smoothing by noise, the effects of dendritic saturation serve to dramatically enhance divisive modulation of fout by shunting inhibition relative to the residual subtractive effect (Fig. 5D).
The results of this study explain the enigmatic link between shunting inhibition and gain control of firing rate. The discrepancy between our results and those of past modeling studies, which concluded that shunting inhibition could not mediate firing rate gain control (8–10), is accounted for by the presence or absence of synaptic noise in the model. This is supported by a recent modeling study that has also implicated noise in determining the effect of shunting inhibition (44) and is consistent with data in ref. 17. Neurons in vivo experience significant levels of synaptic noise (17, 33, 34, 40) and therefore exist under conditions where shunting is likely to act divisively on fout. How does noise allow introduction of a divisive effect? Studies on stochastic resonance have shown that noise can enhance the detection of weak signals (45, 46). This increased sensitivity reflects altered transduction of graded input into spikes: Noise smoothes the threshold nonlinearity, converting the probability of spiking in response to a given input from a step function into a sigmoidal one (34), with the result that firing rate more smoothly encodes the underlying depolarization (32, 35, 47). This smoothing allows for the contrast invariance of orientation tuning in the visual cortex (33) and, as shown in the present study and by Chance et al. (18), it also allows firing rate to reflect the divisive influence of shunting on Vm. One benefit of our approach was that we made no assumptions a priori as to what factors might be involved in allowing gain control by shunting. Instead, we dissected out those factors and tested whether and to what degree shunting inhibition could divisively modulate fout when factors were taken into account independently or in combination. This approach revealed that a second factor, the nonlinearity caused by dendritic saturation, combined with synaptic noise to determine the final outcome of shunting inhibition on fout. This nonlinearity stems from the fact that electrotonic attenuation causes only a fraction of the excitation produced in dendrites to make it to the soma to mediate somatic depolarization (48, 49); therefore, strong dendritic excitation is necessary to achieve even moderate somatic excitation. But the necessity for strong dendritic excitation results in dendritic saturation; in other words, dendrites becomes depolarized to the extent that the driving force associated with excitatory synaptic conductance becomes significantly reduced (50). Consequently, excitatory conductance becomes progressively less capable of generating excitatory current as overall excitation (and, by extension, dendritic saturation) increases. Compounding this, the requisite high level of synaptic input significantly alters the electrotonic structure of the neuron and exacerbates the attenuation (16). The end result is a highly nonlinear relation between somatic Vm and dendritic gexc, which, because of the shape of that nonlinearity relative to the region of probabilistic firing, enhances the divisive modulation of fout by shunting inhibition. During revisions of this paper, Chance et al. (18) independently demonstrated the role of noise in firing rate gain control using dynamic clamp. Our data are consistent with their findings but argue that in addition to synaptic noise, dendritic saturation, which they did not consider, enhances shunting inhibition's divisive effect on fout relative to its subtractive effect and thereby diminishes the necessity of some of the assumptions they made to demonstrate gain control (see Fig. 6). The present study has shown that the ability of shunting inhibition to divisively modulate firing rate depends critically on synaptic noise. Moreover, divisive modulation of firing rate is greatly enhanced relative to the residual subtractive effect by the nonlinearity caused by dendritic saturation. The consequences of noise and dendritic saturation, both of which result from the strong background synaptic input characteristic of in vivo conditions, demonstrate the impact of network activity on the computational capabilities of individual neurons. | |||||
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Acknowledgments We thank Drs. A. Destexhe, P. Drapeau, H. Kröger, S. Ratté, M. Rudolph, and M. W. Salter for helpful comments on the manuscript and N. T. Carnevale and M. L. Hines for assistance with neuron. S.A.P. is a recipient of an M.D./Ph.D. studentship from the Canadian Institutes of Health Research. Y.D.K. is a Scholar of the Fonds de la Recherche en Santé du Québec. This work was supported by the Natural Sciences and Engineering Research Council of Canada. | |||||
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