Different Roles for Simple-Cell and Complex-Cell Inhibition in V1

Model overview

The model contained the following three cortical cell types found in cat V1 layer 4: orientation-tuned excitatory and inhibitory simple cells with separate ON and OFF subfields, and inhibitory complex cells untuned for orientation with mixed ON-OFF receptive fields (Hirsch et al., 2000, 2003). All cells received thalamocortical input selected probabilistically, simple cell inputs chosen from ON-center inputs overlying ON subregions and OFF-center inputs overlying OFF subregions (Hubel and Wiesel, 1962; Tanaka, 1983; Reid and Alonso, 1995; Alonso et al., 2001), and complex cell inputs chosen from all LGN cells whose RFs overlap. Cortical connectivity was also determined probabilistically, with the most probable connections illustrated in the cartoon in Figure 1a. The simple cells formed synapses primarily to other cells with similar preferred orientation, according to the correlation of their RFs. Excitatory synapses were formed to cells with correlated RFs (similar preferred orientation and similar absolute spatial phase; meaning that ON subregions of the two cells tended to overlap in visual space and similarly OFF subregions tended to overlap), and inhibitory synapses were formed to cells with anticorrelated RFs (similar preferred orientation and opposite absolute spatial phase; meaning that ON subregions of one cell tended to overlap OFF subregions of the other cell). The complex cells project randomly with equal (10%) probability to all simple cells within 150 μm and, for simplicity, do not receive any cortical input. The model contained simple cells preferring all orientations and all spatial phases and a range of retinotopic positions. In particular, the relative spatial phase (which refers to the locations of ON and OFF subregions relative to the receptive field center) of each simple cell is assigned randomly. Only receptive fields with even symmetry are illustrated in Figure 1, but these are not favored over other relative spatial phases in the model.

We designed the architecture so that complex inhibitory cells would lack orientation tuning, as observed, and the feedforward inhibition from the complex cells would cause both inhibitory and excitatory simple cells to have sharp orientation tuning without responses to orientations orthogonal to the preferred. As described in the Introduction, some simple inhibitory cells, although sharply tuned, respond to all orientations (and thus could provide some of the untuned inhibition needed to explain sharp orientation tuning of other simple cells). Similarly, some excitatory simple cells in cats respond to all orientations (Azouz et al., 1997). For simplicity, we ignored these facts and assumed that the only orientation-untuned component of inhibition came from complex cells (see Discussion for additional information).

Background: the LGN input to a simple cell

Before examining the effects of the model circuit, it is useful to review the nature of the LGN input received by a simple cell in response to drifting gratings of various orientations (Troyer et al., 1998, 2002). In response to a preferred orientation grating, the cell receives LGN input that is strongly temporally modulated [see excitatory traces (black lines) in Figure 1, b-d, left; although, these include intracortical as well as LGN excitation]. This occurs because the preferred orientation grating causes all of the LGN inputs to the simple cell to modulate their firing rates approximately in phase with one another, peaking together and reaching low or zero firing rates together, so that the temporal modulations of the individual LGN inputs add constructively to produce a strongly modulated total LGN input. In response to a grating-oriented orthogonal to the preferred (which we will refer to as the null orientation of the cell), the cell instead receives an approximately constant, temporally unmodulated rate of LGN input (Fig. 1b-d, right, excitatory traces). This input is temporally unmodulated because the null-oriented grating drives different LGN inputs to the cell at different times (some are reaching their peak rates when others are at their lowest rates and others are in the middle) and so the temporal modulations of the individual LGN inputs add destructively. Thus, the total LGN input arrives at a constant, temporally unmodulated rate corresponding to the sum of the mean firing rates of all LGN inputs to the cell. This mean rate of total LGN input is the same for gratings of any orientation, because the mean rates of the individual LGN cells are independent of orientation. Thus, gratings of different orientations are distinguished, not by the mean rate of LGN input they elicit but by the temporal modulation of that input, with the strongest temporal modulation elicited by a grating of the preferred orientation of the cell and little or no modulation at the null orientation.

The mean rate of total LGN input to a simple cell grows with stimulus contrast, because the mean rates of the individual LGN cells grow with contrast. This is a nonlinear effect. Because the mean luminance does not change with changes in contrast, the mean response also would not change if LGN responses were linear in the image. It is induced by the fact that LGN firing rates cannot decrease below zero, whereas they can increase to 100 Hz or higher, which is many times their background level (10-15 Hz) (Kaplan et al., 1987).

Because the mean input grows with contrast for all orientations, even a null-oriented stimulus can elicit a high rate of LGN input at high contrast, even higher than the peak of the temporally modulated input elicited by a preferred orientation grating at sufficiently low contrast. Yet, simple cells show contrast-invariant orientation tuning (Sclar and Freeman, 1982; Skottun et al., 1987; Anderson et al., 2000b), so that they respond to their preferred orientation even at very low contrast and often do not respond to the null orientation even at high contrast. This problem is solved by the feedforward inhibition in our model network.

Orientation tuning

In simulations using drifting grating stimuli, we can block the inhibition coming from one of the two inhibitory cell types. If we block the complex-cell inhibition leaving only simple-cell inhibition, we also must turn off the I_s → I_s connections to avoid instabilities (Krukowski, 2000). (Unless these connections are very weak, they lead to a winner-take-all competition between inhibitory cells of similar preferred orientation and opposite spatial phase. Between inhibitory cells of two such opposite phases, only the one receiving more LGN input at any given time would be active. Orientation tuning in the simple-cell-only model depends on the graded strength of antiphase inhibition; thus, such winner-take-all behavior would greatly broaden the orientation tuning of the excitatory cells.)

In this case, excitatory cells stimulated at their preferred orientation receive excitatory and inhibitory synaptic input currents that modulate out of phase with one another, thus allowing the cells to reach threshold and fire at some point in their cycle (Fig. 1b, left). Cells preferring the orthogonal orientation will receive a temporally unmodulated synaptic input from the LGN, regardless of their preferred phase. Thus, an excitatory cell will receive both temporally unmodulated LGN excitation and temporally unmodulated inhibition from inhibitory neurons of similar preferred orientation and approximately opposite absolute spatial phase, which are equally driven by the LGN stimulus. Because the inhibition is dominant, the excitatory cells will not fire (Fig. 1b, right). This yields contrast-invariant orientation tuning for excitatory cells, whereas the inhibitory simple cells, which do not receive inhibition, have responses that mirror the tuning of the LGN input: a response component tuned for orientation atop a component untuned for orientation (Fig. 2a,b). As reported previously (Troyer et al., 1998; Krukowski, 2000), this result holds and responses are stable for a wide parameter regime.

These inhibitory simple cells are primarily simple according to the classical criteria of Hubel and Wiesel (1962), because they have spatially segregated ON and OFF subregions from which their responses can be primarily understood. Furthermore, their responses to a preferred orientation drifting grating have a ratio of first harmonic to mean (F1/DC ratio) >1, another standard criterion for simple-cell behavior (Skottun et al., 1991). However, because they do not receive feedforward inhibition to counter the nonlinear nature of their LGN input, they show several nonlinear response characteristics that are not typical of simple cells. Their orientation tuning is not invariant with stimulus contrast (Troyer et al., 1998), and they show a period-doubled response to a counter-phase grating 90° out of phase with the receptive field of the cell (Wielaard et al., 2001).

If we instead block the simple-cell inhibition (onto both other simple inhibitory cells and simple excitatory cells), leaving only complex-cell inhibition, the inhibitory synaptic current from the complex cells into the excitatory simple cells at the preferred orientation is temporally unmodulated. Because the excitatory synaptic current is temporally modulated, the membrane potential is still able to periodically reach threshold, so the cell will respond (Fig. 1c, left). For cells preferring the orthogonal orientation, both synaptic currents are unmodulated, so that the dominant inhibition prevents responses (Fig. 1c, right). The only difference from the previous case (Fig. 1b, right) is that the inhibitory synaptic current is now coming from the complex inhibitory cells rather than the simple inhibitory cells. The inhibitory simple cells receive the same type of input, both excitatory and inhibitory, as the excitatory simple cells, so both types of simple cells now display contrast-invariant orientation tuning (Fig. 2c,d).

If both types of inhibition are used, there is an extra contribution of temporally modulated inhibitory synaptic current at the preferred orientation (Fig. 1d, left). This extra current is antiphase with the excitatory synaptic current at the preferred orientation and goes away at nonpreferred orientations (because the inhibitory simple cells do not respond at these orientations) (Fig. 2d), and thus does not affect the orientation-tuning curves, which remain the same as in Figure 2, c and d (data not shown).

Achievement of sharp, contrast-invariant, orientation-tuned responses via complex-cell inhibition requires relatively tight tuning of inhibitory strength (Fig. 3). If the complex-cell inhibition is too weak, responses at nonpreferred orientations are not suppressed and orientation tuning becomes very wide (Fig. 3, black line). If the complex-cell inhibition is too strong, responses at preferred orientations are suppressed. Thus, overall responses become very weak (Fig. 3, gray line), and contrast invariance of tuning is lost at low contrasts. Only a relatively narrow range of complex-cell inhibitory strengths approximately the value of 4 nA msec used in our simulations achieves sharp contrast-invariant tuning with realistic response levels. This is quite different from the case in which inhibition comes only from simple cells. In that case, because simple-cell inhibition is antiphase to excitation, even very strong inhibition does not interfere with preferred-orientation responses; thus, a wide range of inhibitory strengths gives realistic responses (Troyer et al., 1998). Physiologically, homeostatic mechanisms (Turrigiano and Nelson, 2000) can adjust the overall level of inhibition (and excitation) of a cell to keep its firing rate within a reasonable range. Such mechanisms may provide the relatively fine tuning of overall inhibitory strength required by complex-cell inhibition.

Spatial frequency tuning

Given complex-cell inhibition, what role does simple-cell inhibition play? For simple-cell inhibition to play a role in shaping the responses of the cells, it must interfere with the excitatory drive underlying responses. At the preferred orientation and spatial frequency, the antiphase simple-cell inhibition is significant only during temporal phases in which complex-cell inhibition already dominates excitation, and thus it has little effect. However, if the spatial frequency of the stimulating drifting grating does not match the spatial frequency preference of the cells (approximately given by the spatial frequency component of the Gabor filter; 0.8 cycles per degree), simple-cell inhibition can impact responses. For example, for a drifting grating stimulus of 0.4 cycles per degree, simple-cell inhibition rises (Fig. 4b) at times when complex cell inhibition and excitation are comparable (Fig. 4a), so that responses away from the preferred spatial frequency are somewhat suppressed by the addition of simple-cell inhibition (Fig. 4c,d). Thus, simple-cell inhibition serves to sharpen the spatial frequency tuning of the simple cells.

To study how this inhibition serves to sharpen the spatial frequency tuning, we examined the dependence of spatial frequency tuning on the strength of simple-cell inhibition. The HWHH of the spatial frequency tuning curve decreases with increasing simple-cell inhibitory strength until this strength reaches ∼8 nA msec, the default level of simple-cell inhibitory strength used in our simulations with a mix of simple- and complex-cell inhibition, at which point the HWHH reaches a plateau level (Fig. 5). This sharpening is primarily attributable to an increase in the low spatial frequency cutoff of the cells, defined as the spatial frequency below the peak at which spike response is 50% of the peak response. There was no systematic change in the high spatial frequency cutoff with simple-cell inhibition (data not shown). Thus, there is a maximal level of simple-cell inhibition, beyond which increases do not further sharpen spatial frequency tuning, and this sharpening is primarily achieved by reducing the response to gratings of low spatial frequency.

Temporal frequency tuning

We have shown previously that the model with only simple-cell inhibition can explain the fact that cortical neurons have temporal frequency high-cutoffs at much lower frequencies than LGN cells (Krukowski and Miller, 2001). The explanation relied on the following two elements: NMDA receptor-mediated conductances (which we will refer to as NMDA) in synapses from LGN to excitatory cells act as a low-pass filter, reducing the temporal modulation of the excitatory input at higher frequencies and thus primarily reducing this input to its mean; and the dominant inhibition prevents responses to the mean excitation. Note that this explanation requires that some inhibitory neurons respond to the high temporal frequencies that drive LGN cells.

This explanation of temporal frequency tuning also works with complex-cell inhibition, and is even more powerful in this case. Because complex-cell inhibition is not modulated at the opposite phase of the excitation, less demodulation of the excitation is required for the inhibition to dominate the excitation, and temporal frequency high-cutoffs occur at lower frequencies than in the previous model. Assuming that NMDA constitutes 75% of the current in LGN synapses onto excitatory cells, which corresponds to observations in rat thalamocortical slices (Crair and Malenka, 1995), the resulting temporal frequency tuning of excitatory simple cells shows little or no response above 4 Hz (Fig. 6a). In contrast, the complex inhibitory cells have temporal frequency tuning that follows that of our model LGN input, cutting off above 16 Hz (Fig. 6b). Defining the high-frequency cutoff as the temporal frequency above the peak at which the mean spike response is reduced to 50% of the peak response, the cutoff of simple cells is ∼2 Hz. This is low but still within the range of cells reported experimentally (Movshon et al., 1978; Freeman et al., 2002). Decreasing the fraction of NMDA in the thalamocortical synapses to the excitatory cells decreases the demodulation of the excitatory input at higher temporal frequencies and hence causes the high-frequency cutoff to increase (Fig. 6d). If the proportion of NMDA is halved from the amount we assumed, the cutoff is ∼6 Hz, which is a representative experimental cutoff value for cortical cells (Ikeda and Wright, 1975; Movshon et al., 1978; Holub and Morton-Gibson, 1981; Saul and Humphrey, 1992b; DeAngelis et al., 1993; Freeman et al., 2002). In simulations without NMDA, we found that cortical temporal tuning closely follows LGN tuning (data not shown).

The addition of simple-cell inhibition up to ∼6 nA msec causes a decrease in the response to gratings at low temporal frequencies (Fig. 6c). Because the simple-cell inhibition does not affect the response at higher temporal frequencies, it does not have a significant effect on the temporal frequency cutoff (Fig. 6d).

Temporal frequency tuning of inhibitory simple cells in the model is primarily identical to that of excitatory simple cells (data not shown). This is easily understood because the inhibitory simple cells in the model receive the majority of their excitatory drive from excitatory simple cells, and their overall tuning follows that of this excitatory drive. We did not include NMDA in LGN synapses onto inhibitory cells; thus, the LGN-driven component of excitation does not demodulate with increasing temporal frequency and, hence, yields tuning like that of the LGN inputs, adding a small excitatory component at higher temporal frequencies to inhibitory simple-cell responses. Both factors, the relative strength of their feedback versus feedforward excitatory drive and the proportion of NMDA, if any, in their LGN synapses, shape the temporal frequency tuning of inhibitory simple cells. Because these are not well experimentally constrained, we cannot make strong predictions as to the exact temporal frequency tuning of the inhibitory simple cells other than to point out how it will be controlled by these two factors.

Network stability

The network including only simple-cell inhibition is stable for a fairly large parameter regime (Troyer et al., 1998). Here, we show that the network including only complex-cell inhibition is fragile and prone to at least one form of instability, and that simple-cell inhibition can help to expand the stable parameter regime.

If we increase all intracortical synaptic weights, both excitatory and inhibitory, by 20% in a network with only complex-cell inhibition, a drifting grating of one orientation evokes tonic firing from cells preferring all orientations (Fig. 7a). The recurrent excitatory synaptic current builds over time, resulting in uncontrolled firing (Fig. 7c,e). Adding simple-cell inhibition restores the network to a stable firing state (Fig. 7b,d,f). The increased inhibitory synaptic current from the simple cells acts to reset the network during nonresponsive phases of the stimulus, shutting down overactive cells and stabilizing the network.

To explore the conditions for this stabilization effect to occur, we constructed a toy model of a simple cell receiving the following four input currents: a thalamocortical input modulated with time, a recurrent excitatory input proportional to the activity of the cell, a complex-cell inhibitory input constant with time, and an antiphase simple-cell inhibitory input modulated with time. The response of the cell, r, is the difference between its excitatory and inhibitory inputs, rectified over a threshold, θ. For fixed threshold and simple-cell inhibitory strength, we can determine the stability of the model for a range of excitatory recurrent strengths and inhibitory complex strengths. By defining the stable regime as one in which the response of the cell is larger than a minimum response but does not grow indefinitely, we can compare the size of the stable regime both with and without simple-cell inhibition (Fig. 8c). For both scenarios, the minimum response is found for the same parameters, because simple-cell inhibition is out of phase with excitation. However, the size of the parameter regime in which the activity does not grow indefinitely is significantly increased when including antiphase simple-cell inhibition. For the fixed parameters used here, the ratio of the area of the two stable regimes is ∼2:1. The degree of this increase depends on the strength of the simple-cell inhibition, but even very weak simple-cell inhibition causes an increase in the area of the stable regime (data not shown). Thus, we can conclude qualitatively that simple-cell inhibition increases the size of the stable regime, although we cannot draw a parameter-free conclusion as to the size of this increase. Comparing the relative size of the inhibitory and excitatory traces in Figure 7f to those in Figure 8b indicates that the simple-cell inhibitory strength in the toy model, which achieves a 2:1 increase in stability, is comparable with the strength in the full model.

Model predictions

Predictions of the model include the following: (1) Simple cells receiving significant thalamic excitation should receive a comparable level of orientation-untuned inhibition, yielding detectable conductance changes in response to a stimulus oriented perpendicular to the preferred. Such conductance changes were seen in some but not all simple cells studied by Anderson et al. (2000a). Cells receiving little thalamic input could inherit their tuning from cells receiving such input and hence would not require such conductance changes. (2) Complex inhibitory cells should have temporal frequency tuning like that of the LGN inputs, responding to high temporal frequencies to which few other cortical cells respond. This gains some support from results in rabbit somatosensory (Swadlow, 1989, 1991) and motor (Swadlow, 1994) cortex that suspected interneurons (SINS) follow much higher temporal frequencies of peripheral stimulation than other cortical cells. (3) Thalamocortical synapses onto excitatory cells should include a significant NMDA receptor-mediated component. This agrees with slice studies of thalamocortical synapses (Crair and Malenka, 1995; Gil and Amitai, 1996) and with some (Miller et al., 1989) but not other (Fox et al., 1989) studies of V1 physiology.

In addition, assumptions as to circuit structure need testing. We assumed that simple inhibitory cells in layer 4 receive strong LGN input that is specifically arranged like that of other simple cells (Tanaka, 1983; Reid and Alonso, 1995; Alonso et al., 2001) and make connections that tend to be antiphase (Ferster, 1988; Hirsch et al., 1998; Anderson et al., 2000a), and that complex inhibitory cells in layer 4 receive strong LGN input that is nonspecific and project nonspecifically to other cortical cells.

Complex-cell feedforward inhibition as a solution to the problem posed by the LGN inputs

The problem posed by the LGN inputs to a simple cell is that the mean rate of LGN input grows with contrast at both nonpreferred orientations and high temporal frequencies, yet most excitatory simple cells do not respond to these stimuli. The solution we proposed is that broadly tuned feedforward inhibition cancels this mean input, allowing responses only to the peaks of temporally modulated excitatory input. This inhibition might arise from orientation-untuned complex cells (this study) or simple cells that respond to all orientations (previous studies). Because Hirsch et al. (2003) found that a minority of inhibitory simple cells responds strongly to all orientations (see Introduction), both types of inhibitory cells may actually contribute; we focused on a single type for simplicity.

We argue that a significant part of this inhibition must be feedforward for two reasons. First, in response to the positive pulse of LGN input evoked by transient nonpreferred stimuli such as a high-contrast flashed bar or grating of the null orientation of the cell, the inhibition must arrive in time to prevent spiking [see Gillespie et al. (2001) and Felsen et al. (2002) for evidence that many cat V1 cells do not respond to such stimuli]. The stimulus evokes a positive pulse of input because it excites some LGN inputs to the simple cell and inhibits others, and those that are excited can raise their firing rates much more strongly than those that are inhibited can lower theirs. Fast-spiking inhibitory cells have higher resistances and lower thresholds than excitatory cells (McCormick et al., 1985) and thus can respond very rapidly (e.g., within 0-1 msec after thalamic action potentials reach cortex in rabbit somatosensory cortex) (Swadlow, 1995). Hence, these cells can spike in time to prevent spiking in excitatory cells (Somers et al., 1995). Consistent with this, shocks to LGN elicit in LGN-recipient cortical cells a short depolarization followed by, before spike threshold is reached, a massive hyperpolarization (Ferster and Jagadeesh, 1992) induced by inhibitory neuronal firing. Second, the required inhibition seems likely to be driven by LGN rather than cortical excitation, because it must be driven by stimuli that excite LGN cells but do not excite most cortical cells. However, the minority of excitatory simple cells that does respond to the stimulus (a very small minority in the case of high temporal frequency stimuli) (Freeman et al., 2002) might provide this excitatory drive for steady state rather than transient stimuli. The inhibitory cells may of course also receive excitation from cortical cells. We only argue that the LGN-driven component of inhibition seems necessary and is sufficient to shape simple-cell orientation and temporal frequency tuning.

The solution via complex-cell inhibition requires relatively fine tuning of overall complex inhibitory strength. However, no additional specificity of these connections is needed; it suffices that complex cells receive topographic but otherwise nonspecific LGN input and project nonspecifically to nearby cortical cells. Thus, homeostatic mechanisms that can control overall inhibitory strength (Turrigiano and Nelson, 2000) could suffice to guide development of this inhibition. Hebbian mechanisms can guide development of inhibitory and excitatory simple-cell connections (Kayser and Miller, 2002).

The inclusion of complex-cell inhibition is consistent with experiments suggesting a push-pull arrangement of excitation and inhibition onto simple cells. Dark elicits inhibition in the RF where light elicits excitation and vice versa (Ferster, 1988; Hirsch et al., 1998). This is consistent with inhibition that is antiphase to excitation but also with partly or entirely phase-nonspecific inhibition along with phase-specific excitation; excitation, when evoked, overwhelms inhibition, and otherwise inhibition is seen. In response to optimal drifting gratings, the evoked inhibitory conductance in simple cells shows both a mean, or DC component, and a temporal modulation that is nearly antiphase to that of excitatory conductance (Anderson et al., 2000a). This is consistent with a circuit containing only antiphase simple-cell inhibition, because factors that cause inhibitory simple-cell firing rates to increase from background more strongly than they decrease, such as excitatory feedback and the minimum firing rate of zero, induce a DC. However, it is also consistent with a combination of antiphase simple-cell inhibition and complex-cell inhibition.

The orientation tuning of inhibition to simple cells is strongly peaked at the preferred orientation, dropping to low values at the orthogonal orientation (Ferster, 1986; Anderson et al., 2000a; Martinez et al., 2002; but see Monier et al., 2003). This suggests that orientation-tuned simple-cell inhibition is much stronger than orientation-untuned complex-cell inhibition.

Alternative solutions to the problem posed by the LGN inputs

With respect to orientation tuning, an alternative possible source of inhibition is cross-orientation inhibition (inhibition, in response to nonpreferred orientations, arising from more distant tuned neurons preferring those orientations). Such inhibition would have to be weak because inhibition driven by the cross orientation is typically weak in layer 4 (Ferster, 1986; Anderson et al., 2000a; Martinez et al., 2002). There is no need to invoke such inhibition, given the presence of untuned local inhibition (Hirsch et al., 2000, 2003). Such local inhibition can also explain temporal frequency tuning, which could not be explained by cross-orientation inhibition.

Related proposals were made in a model of monkey V1 layer 4 (McLaughlin et al., 2000; Wielaard et al., 2001). In this model, all cortical cells, both excitatory and inhibitory, received simple-cell patterns of LGN input and nonspecific, inhibition-dominated input from other cortical cells, both excitatory and inhibitory, within a given radius. The result was that cells received phase-nonspecific feedforward inhibition, similar to our complex cell model. However, the absence of a phase-specific inhibitory component is inconsistent with findings in cats that inhibitory conductance to a drifting bar or grating is temporally modulated, approximately antiphase to excitatory conductance (Ferster, 1986, 1988; Anderson et al., 2000a). Furthermore, this connectivity scheme predicts that cells in linear zones of the orientation map respond to all orientations, whereas cells near orientation pinwheels show sharp orientation tuning (but see Kang et al., 2003). This is incorrect in cats, in which pinwheel and linear zone cells show spiking orientation tuning of the same width (Maldonado et al., 1997; Schummers et al., 2002). Pinwheel cells show broader voltage orientation tuning than linear zone cells (Schummers et al., 2002); although it is unclear whether this holds in layer 4, this is suggestive of a distance-dependent component of connectivity that is excitation-dominated, rather than inhibition-dominated, which we neglected. Finally, this connectivity scheme predicts that simple cells showing sharp orientation tuning should receive approximately equal intracortical inhibition in response to stimuli of all orientations, whereas cat layer 4 simple cells have inhibitory tuning that is strongly peaked at the preferred orientation (Ferster, 1986; Anderson et al., 2000a; Martinez et al., 2002).

In some respects, synaptic depression acts as a high-pass filter (Abbott et al., 1997; Tsodyks and Markram, 1997) and thus might eliminate changes in the mean LGN input, leaving only temporal modulations of that input. This could reduce or eliminate the need for feedforward inhibition. Our explorations to date suggest that, although synaptic depression can reduce the stimulus-induced mean LGN input to a simple cell, it cannot eliminate it, except at the lowest temporal frequencies (1-2 Hz) with very strong depression [“train” parameters of Kayser et al. (2001) vs the weaker “pulse” parameters used here, which do not eliminate the untuned mean LGN input] (Fig. 2b). In the course of a previous study (Kayser et al., 2001), we found that even with strong (train) depression, inhibition was required to obtain contrast-invariant orientation tuning. Furthermore, including such strong depression tends to eliminate the differences between stimulus contrasts, so that after incorporating the contrast saturation of LGN responses (Sclar, 1987; Cheng et al., 1995), cortical cells tend to saturate at significantly lower contrasts than observed experimentally (Kayser et al., 2001).

Strong depression might render voltage responses to high temporal frequency stimuli small enough that an appropriate threshold could explain the high-frequency cutoff of cortical cells without requiring that such stimuli evoke inhibition. However, preliminary investigations suggest that, in this scenario, an appropriate threshold for high-contrast stimuli will unrealistically suppress responses to low-contrast stimuli.

Inhibitory neurons in other systems and the function of simple inhibitory neurons

Studies in a variety of cortical systems in rodents and rabbits have identified SINS in extracellular recordings and found that SINS are broadly tuned or untuned for the parameters for which other nearby cells are tuned (Swadlow and Weyand, 1987; Swadlow, 1988, 1989, 1990, 1991, 1994, 1995, 2003; Simons and Carvell, 1989; Miller et al., 2001; Bruno and Simons, 2002; Swadlow and Gusev, 2002). In particular, in rabbit V1, SINS are untuned for orientation and have mixed ON-OFF receptive fields similar to the complex cells studied here, whereas most other cells are tuned (Swadlow, 1988). The basis of this, at least in rat S1 whisker barrels (layer 4), is that SINS unselectively pool large numbers of functionally diverse thalamic inputs, receiving input on average from 65% of the thalamic neurons representing the same whisker (Bruno and Simons, 2002; Swadlow and Gusev, 2002). The high probability (≥0.6) of gap junction connections between nearby interneurons of a given physiological type, fast-spiking (FS) or low-threshold spiking (LTS) (Galarreta and Hestrin, 1999, 2001, 2002; Gibson et al., 1999), also suggests that rodent interneurons are functionally nonspecific. Note that the simple-complex interneuron distinction in cat V1 does not correspond to this FS-LTS distinction (Hirsch et al., 2000, 2003).

These unselective inhibitory neurons receiving unselective thalamic input seem analogous to the complex inhibitory neurons reported by Hirsch et al. (2000, 2003). In contrast, simple inhibitory neurons, which have equally selective orientation tuning and subregion structure as their excitatory neighbors, are likely to receive specifically arranged ON and OFF LGN inputs like other simple cells (Tanaka, 1983; Reid and Alonso, 1995). It seems unlikely that they could show extensive gap junction coupling without losing their phase specificity, because simple cells of all preferred spatial phases are likely to be present in a local region (DeAngelis et al., 1999). We suggest that the complex-cell interneurons correspond to the rodent and rabbit SINS, whereas simple-cell interneurons either correspond to an unreported type of interneuron in rodents and rabbits, or are an evolutionarily new type of interneuron not found in rodents and rabbits.

If simple-cell interneurons are evolutionarily new, the question arises what function they serve. The answers we found here, sharpening spatial frequency tuning, reducing low temporal frequency responses, and increasing network stability, are not large effects. The question remains open.

Model homogeneity versus physiological heterogeneity

In our model, all LGN cells have identical response properties, except for ON versus OFF structure and retinotopic position, and all cortical cells receive LGN inputs specified by an identical Gabor function, except for orientation, phase, and retinotopic position. In reality, there is much variability in LGN spatial (So and Shapley, 1981; Cheng et al., 1995) and temporal (Derrington and Fuchs, 1979; Lehmkuhle et al., 1980; Sclar, 1987; Saul and Humphrey, 1990; Hamamoto et al., 1994; Mukherjee and Kaplan, 1995; Freeman et al., 2002) response properties and in cortical Gabor filters (Jones and Palmer, 1987; DeAngelis et al., 1993, 1999) and temporal response properties (Ikeda and Wright, 1975; Movshon et al., 1978; Saul and Humphrey, 1992b; DeAngelis et al., 1993, 1999; Albrecht, 1995; Freeman et al., 2002). By ignoring this diversity, we are able to focus on basic principles: how feedforward inhibition can eliminate the orientation-untuned and high temporal frequency mean LGN input to yield sharp orientation tuning and, along with thalamocortical NMDA receptors, low-pass temporal frequency tuning.

These same principles would apply in the presence of response heterogeneity. However, heterogeneity might lessen the problem to be solved. For example, some simple cells respond to all orientations (Azouz et al., 1997), suggesting that they receive less feedforward inhibition relative to feedforward excitation (and perhaps less recurrent excitation to keep response strength in a reasonable range) or have more broadly tuned Gabor functions describing their LGN inputs. Simple cells with lower temporal frequency cutoffs might receive LGN inputs with lower cutoffs. Consistent with this, Alonso et al. (2001) found significant correlation between the temporal properties of a cortical cell and its LGN inputs. This could lessen the amount of NMDA or strength of inhibition needed to explain the discrepancy between LGN and cortical cell tuning. However, a significant discrepancy would almost certainly remain, given the large overall difference between LGN and cortical temporal tuning (Freeman et al., 2002). Variation in integration times of LGN inputs would cause dispersal in phase of LGN responses at higher temporal frequencies, yielding temporal demodulation of LGN input and thus reducing the required NMDA. However, one can estimate (see Materials and Methods) that this decrease in amplitude between 2 and 8 Hz should be 27% or less for the majority of V1 cells. In contrast, NMDA levels that mimic cortical tuning (one-half of our default level; see Results) yield 60% diminution under comparable conditions (LGN firing rate amplitudes constant across frequency; no synaptic depression), suggesting significant NMDA would still be required. Diversity in cortical temporal tuning could be explained by inter-cell diversity in the tuning or the range of integration times of LGN inputs, in the degree of NMDA on thalamocortical synapses, and in the strength of feedforward inhibition received. In summary, the principles found here would not be undermined by considering heterogeneity, and conversely they suggest means by which heterogeneity might be achieved.

Thalamic-recipient simple cells receiving relatively little feedforward inhibition should both respond well to all orientations and have higher temporal frequency cutoffs than typical. Although each response property can have other causes, it would be intriguing to search for a correlation between these response properties in layer 4 cells.