Figure 1 Effect of shunting inhibition on firing rate depends on stimulus conditions. (A) Hyperpolarization has a subtractive effect on subthreshold depolarization, whereas shunting has a divisive effect. (B) Image of the model cell with a sample of Poisson-distributed (more ...)

However, neurons communicate with each other through spike trains. One must, therefore, consider how V_m 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.

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 mScm⁻²): soma and primary dendrites, 48, 40; distal dendrites (≥ secondary), 6, 5; initial segment, axon hillock, and nodes, 240, 200; axon internodes, 6, 5. Kinetics of these currents were taken from ref. 21.

Synaptic Input. Synaptic input was modeled as a nonsaturating conductance with a dual exponential time course of the form [1-exp(-t/τ_rise)]exp(-t/τ_decay). For excitatory input, τ_rise = 0.2 ms, τ_decay = 2 ms, g_max = 1 nS, and E_rev = 0 mV; for inhibitory input, τ_rise = 0.8 ms, τ_decay = 8 ms, g_max = 0.5 nS, and E_rev = −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 V_m 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.

Figure 2 Synaptic input introduces significant noise into the postsynaptic neuron, with ramifications for spike generation. (A) Sample of somatic Vm fluctuations in model neuron excited by random synaptic input (fexc = 2 Hz/synapse) for two levels of (more ...)

Nonsynaptic Stimulation. Constant current (I_clamp) 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 (g_exc) and/or inhibitory conductance (g_inh) simulated as either a single-point process in the soma or as multiple-point processes at synaptic sites. Reported values of g_exc and g_inh refer to total excitatory or inhibitory conductance, based on the time-averaged synaptic conductances scaled by the number of synapses. Contrariwise, to recreate V_m 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 − I_avg) + 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 V_m 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 (f_out) values and 1 s for V_m values.

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 (f_exc) = 1.0 Hz/synapse and rate of inhibitory synaptic events (f_inh) = 5.6 Hz/synapse] (17), our model neuron was depolarized to −60.7 mV, standard deviation of V_m was 2.5 mV, and R_in 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 f_out-f_exc 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 f_out (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 I_clamp and shunting with constant somatic g_inh. Under these conditions, shunting inhibition had a purely subtractive effect on f_out (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 f_out 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 V_m produced by random synaptic input. Increased shunting had two effects: reduction of average somatic depolarization caused by excitatory input (Fig. 2B) and reduction of V_m fluctuation amplitude (Fig. 2C). Notably, standard deviation of V_m 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 V_m fluctuation amplitude was relatively unaffected by the strength of excitation, consistent with previously reported data (17, 18, 30).

Because of its fluctuations, somatic V_m occasionally crosses spike threshold despite average V_m remaining below threshold (Fig. 2A). Therefore, probability of crossing threshold depends jointly on average V_m (i.e., distance from threshold) and amplitude of V_m fluctuations. The consequences of this for spiking are seen in Fig. 2D after reinsertion of HH channels. Constant I_clamp gives rise to an abrupt increase in f_out at rheobase (i.e., minimum constant I_clamp eliciting spikes), where probability of spiking jumps from 0 to 1. By contrast, f_out increases more gradually in response to noisy input because the V_m fluctuations allow for probabilistic or noise-driven spiking before average depolarization reaches threshold. This amounts to smoothing the threshold nonlinearity, allowing f_out to encode the average underlying V_m even while average V_m remains below threshold (32–35). Given that smoothing allows f_out to more closely parallel V_m, divisive modulation of V_m may be predicted to manifest as divisive modulation of f_out under noisy conditions.

Spiking recorded in vivo is normally highly irregular (36, 37). Fig. 2E shows that spiking caused by V_m fluctuations (which we refer to as probabilistic spiking) is irregular compared with spiking generated by raising average V_m above threshold (27, 38–41). Under in vivo conditions, neurons typically operate in the lower range of f_out, 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 V_m 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 f_out. To isolate the influence of noise, V_m 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), V_m fluctuations generated by noisy I_clamp smoothed the threshold nonlinearity in proportion to the intensity of the noise (Fig. 3B).

Figure 3 Effect of Vm fluctuations on spike generation (smoothing the threshold nonlinearity) is critical for allowing divisive modulation of firing rate by shunting inhibition but cannot fully account for the divisive effect seen with synaptic input. (A) Sample (more ...)

Under noisy conditions, shunting had some divisive effect on f_out (Fig. 3C) in contrast to the purely subtractive effect observed with constant I_clamp (Fig. 1E). However, Fig. 3C shows that the reduction in slope of f_out-I_clamp curves was restricted to probabilistic firing (i.e., lower values of f_out), with curves becoming parallel at f_out > 40 Hz. Furthermore, based on responses with f_out < 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 V_m fluctuations allow f_out to more accurately encode average V_m 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 I_clamp (Fig. 4A Upper), and the relationship becomes only modestly sublinear for excitation by somatic g_exc (Fig. 4B Upper). In contrast, somatic depolarization is nonlinearly related to dendritic g_exc (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.

Figure 4 The nonlinearity caused by dendritic saturation enhances divisive modulation of fout by shunting inhibition. Gray shading in A–C represents the probability of spiking within an arbitrary time interval: darkest gray → probability = 1; white (more ...)

Analysis of the effects of shunting inhibition on somatic V_m-excitation curves (Fig. 4 A Upper–C Upper) allows one to roughly predict the subtractive effect of shunting inhibition on f_out for comparison to f_out-excitation curves (Fig. 4 A Lower–C Lower). Without noise, the subtractive effect of shunting on f_out is predicted by where V_m-excitation curves cross spike threshold. Introducing noise reduces the effective threshold for spiking such that the subtractive effect on f_out can be predicted from where V_m-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 V_m-excitation curves through the region of probabilistic spiking. Acting on the linear V_m-somatic I_clamp curves and near-linear V_m-somatic g_exc 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 V_m-dendritic g_exc curves (Fig. 4C). The relative changes in trajectory for V_m-somatic I_clamp and V_m-dendritic g_exc curves are summarized in Fig. 4D and correctly predict the enhanced divisive effect of shunting on f_out (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 f_out 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 f_out. We sought to quantitatively test the adequacy of this explanation by reconstituting a family of f_out-g_exc curves representing different levels of inhibition for comparison to the f_out-f_exc curves in Fig. 1C. In other words, these f_out-g_exc 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 f_out-g_exc curves directly comparable to f_out-f_exc curves is explained in Fig. 4E. The calculated f_out-g_exc 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 g_exc (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.

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 V_m and the saturation it causes in dendrites. In the absence of those effects, shunting has a purely subtractive effect on f_out (Fig. 5A). Threshold smoothing by noise allows a modest divisive reduction of f_out by shunting (Fig. 5B). On its own, the nonlinearity caused by dendritic saturation does not allow a divisive effect of shunting on f_out (Fig. 5C). However, when combined with threshold smoothing by noise, the effects of dendritic saturation serve to dramatically enhance divisive modulation of f_out by shunting inhibition relative to the residual subtractive effect (Fig. 5D).

Figure 5 Effect of shunting inhibition on fout under different conditions. (A) Without noise or dendritic saturation, the divisive effect of shunting on Vm-Iclamp curves manifests as a subtractive effect on fout-Iclamp curves. (B) Smoothing the threshold nonlinearity (more ...)

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 f_out.

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 V_m.

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 f_out 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 f_out. 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 V_m and dendritic g_exc, which, because of the shape of that nonlinearity relative to the region of probabilistic firing, enhances the divisive modulation of f_out 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 f_out 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.

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.