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The Journal of Neuroscience, August 1, 1998, 18(15):5908-5927

Contrast-Invariant Orientation Tuning in Cat Visual Cortex: Thalamocortical Input Tuning and Correlation-Based Intracortical Connectivity

Todd W. Troyer2, 6, Anton E. Krukowski5, 6, Nicholas J. Priebe4, 6, and Kenneth D. Miller1, 3, 4, 5, 6, 7

Departments of 1 Physiology, 2 Psychiatry, and 3 Otolaryngology, 4 Neuroscience and 5 Biophysics Graduate Programs, 6 W. M. Keck Center for Integrative Neuroscience, 7 Sloan Center for Theoretical Neurobiology at UCSF, University of California, San Francisco, California 94143-0444

    ABSTRACT
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References

The origin of orientation selectivity in visual cortical responses is a central problem for understanding cerebral cortical circuitry. In cats, many experiments suggest that orientation selectivity arises from the arrangement of lateral geniculate nucleus (LGN) afferents to layer 4 simple cells. However, this explanation is not sufficient to account for the contrast invariance of orientation tuning.

To understand contrast invariance, we first characterize the input to cat simple cells generated by the oriented arrangement of LGN afferents. We demonstrate that it has two components: a spatial-phase-specific component (i.e., one that depends on receptive field spatial phase), which is tuned for orientation, and a phase-nonspecific component, which is untuned. Both components grow with contrast.

Second, we show that a correlation-based intracortical circuit, in which connectivity between cell pairs is determined by the correlation of their LGN inputs, is sufficient to achieve well tuned, contrast-invariant orientation tuning. This circuit generates both spatially opponent, "antiphase" inhibition ("push-pull"), and spatially matched, "same-phase" excitation. The inhibition, if sufficiently strong, suppresses the untuned input component and sharpens responses to the tuned component at all contrasts. The excitation amplifies tuned responses. This circuit agrees with experimental evidence showing spatial opponency between, and similar orientation tuning of, the excitatory and inhibitory inputs received by a simple cell. Orientation tuning is primarily input driven, accounting for the observed invariance of tuning width after removal of intracortical synaptic input, as well as for the dependence of orientation tuning on stimulus spatial frequency.

The model differs from previous push-pull models in requiring dominant rather than balanced inhibition and in predicting that a population of layer 4 inhibitory neurons should respond in a contrast-dependent manner to stimuli of all orientations, although their tuning width may be similar to that of excitatory neurons. The model demonstrates that fundamental response properties of cortical layer 4 can be explained by circuitry expected to develop under correlation-based rules of synaptic plasticity, and shows how such circuitry allows the cortex to distinguish stimulus intensity from stimulus form.

Key words: visual cortex; LGN; contrast invariance; cerebral cortical circuitry; orientation selectivity; model; simple cell; layer 4; V1; push-pull; opponent inhibition; spatial phase

    INTRODUCTION
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References

Thirty-five years ago, Hubel and Wiesel (1962) discovered that cells in cat primary visual cortex (V1) are tuned for the orientation of light/dark borders. The inputs to V1 come from the lateral geniculate nucleus (LGN), whose cells are not significantly orientation selective (Hubel and Wiesel, 1961). The origin of orientation selectivity in visual cortex has been one of the most thoroughly investigated questions in neuroscience and serves as a model problem for understanding how the cortex processes and represents information.

In cats, orientation selective responses appear in cortical layer 4. Cat layer 4 is composed of simple cells (Hubel and Wiesel, 1962; Gilbert, 1977; Bullier and Henry, 1979): cells with receptive fields (RFs) composed of oriented subregions, each giving exclusively ON or OFF responses (response to light onset/dark offset or light offset/dark onset). Hubel and Wiesel (1962) proposed that the orientation selectivity of these cells derives from an oriented arrangement of inputs from the LGN: ON-center LGN inputs have RF centers aligned over the simple cell's ON subregions, and similarly for OFF-center inputs. Such an input arrangement has been confirmed experimentally (Tanaka, 1983; Reid and Alonso, 1995). Because the total LGN input grows with increasing contrast for stimuli of all orientations, this model by itself is insufficient to explain the invariance of orientation tuning under change in stimulus contrast (Sclar and Freeman, 1982; Skottun et al., 1987). A threshold for spiking responses might narrow the tuning at any one contrast, but higher contrast would require a higher threshold to prevent broadening of tuning.

Two major approaches to achieving contrast invariance have been proposed. Many authors have suggested that responses in simple cells are approximately linear, i.e., the response can be predicted by linear summation of stimulus luminance (relative to background), weighted by the cell's RF (Movshon et al., 1978; Glezer et al., 1982; Tolhurst and Dean, 1990; Albrecht and Geisler, 1991; Heeger, 1992; Carandini and Heeger, 1994; Carandini et al., 1997, 1998). Contrast change in such a model simply multiplies responses by a constant; contrast-invariant tuning follows automatically. It has been proposed that linear responses might be achieved by a balanced "push-pull" arrangement of inputs, in which an ON subregion shows equal excitation (push) to light stimuli as inhibition (pull) to dark stimuli, and conversely for OFF subregions (Glezer et al., 1982; Tolhurst and Dean, 1990; Carandini and Heeger, 1994; Carandini et al., 1997, 1998). However, there are two problems with achieving linear response in an actual neural circuit. First, spike thresholds are non-zero, and therefore oriented stimuli that at low contrast give positive but subthreshold input would yield spike responses at higher contrast. Second, at contrasts above ~5%, LGN responses increase more than they decrease, because spike rates cannot decrease below zero (i.e., responses "rectify"). This input nonlinearity alters the balance between push and pull.

Other authors have proposed that orientation tuning emerges from orientation-specific short-range excitation and longer-range inhibition in cortex (Ben-Yishai et al., 1995; Somers et al., 1995), despite evidence that in cat layer 4, excitation and inhibition show similar orientation tuning (Ferster, 1986). The width of orientation tuning in these models is an emergent property of intracortical circuitry, and so it does not depend on the parameters of the stimulus, including stimulus contrast. These proposals appear inconsistent with the fact that orientation tuning widths in cats do depend on at least one stimulus parameter: the spatial frequency of sinusoidal grating stimuli (Vidyasagar and Sigüenza, 1985; Webster and De Valois, 1985; Jones et al., 1987; Hammond and Pomfrett, 1990).

We propose a new model for cat layer 4 cortical circuitry that yields contrast-invariant orientation tuning. Our model examines two basic questions. First, what is the nature of the thalamocortical input to cortical simple cells? We assume that thalamocortical connectivity can be modeled by a Gabor function: a two-dimensional Gaussian multiplied by a sinusoid (Jones et al., 1987; Reid and Alonso, 1995). The spatial phase of the sinusoid determines the location of ON and OFF subregions within the thalamocortical RF. Using a simple model of LGN responses, we show that the total LGN input has two components: a spatial-phase-specific component (a component that varies with the spatial phase of a cell's RF) that is tuned for orientation, and a phase-nonspecific component that is entirely untuned. Both components grow with contrast. Separating these input components helps clarify the debate over whether the LGN input to simple cells is well or poorly tuned. In response to drifting gratings, the phase-specific component corresponds to the temporally modulated input component, which Ferster et al. (1996) recently demonstrated to be tuned. However, the total input includes the phase-nonspecific, temporally unmodulated component; this should be untuned and was not measured by Ferster et al. (1996). Separating the input components also clarifies the problem that cortical circuitry must solve to achieve contrast-invariant orientation tuning: eliminating the untuned component of the LGN input in a contrast-dependent manner while extracting and sharpening the tuned component.

Second, what patterns of intracortical connectivity are sufficient to yield contrast-invariant orientation tuning? We arrive at a surprisingly simple answer: "correlation-based" connectivity yields contrast invariance. By correlation-based connectivity we mean that intracortical connection strengths between two cells are fixed on the basis of the correlation in their thalamocortical RFs. Thus, inhibitory connections occur between cells with anticorrelated RFs, whereas excitatory connections occur between cells with correlated RFs. The "antiphase" inhibition eliminates the untuned input component and sharpens responses to the tuned component, whereas "same-phase" intracortical excitation amplifies the tuned response. As a result, our model achieves contrast-invariant tuning in the presence of positive thresholds and LGN rectification.

Our model uses a form of push-pull circuitry but differs from other such models in that inhibition dominates rather than balances excitation, and responses are not linear. Furthermore, we predict that a population of inhibitory neurons in cat layer 4 should respond in a contrast-dependent manner to stimuli of all orientations, although they may be tuned for orientation. The model has both developmental and functional implications for understanding the layer 4 cortical circuit, and suggests a general means of separating stimulus intensity (here represented by contrast) from stimulus form (represented by orientation).

This work has been published previously in abstract form (Krukowski et al., 1996).

    MATERIALS AND METHODS
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References

We study both a very simple ("conceptual") model and a more realistic ("computational") model. We first present the elements common to both, and then present each model.

Elements common to both conceptual and computational models

LGN model. Our model was based on cat V1 at ~5° eccentricity. LGN spatial RFs were center-surround difference of Gaussians, with cells responding either to light onset (ON cells) or light offset (OFF cells) in their RF centers. LGN spatial filter parameters [(17/sigma center2)e-x2/sigma 2center - (16/sigma surround2)e-x2/&sfgr;2surround; sigma center = 15', sigma surround = 1°] were taken from Peichl and Wassle (1979) and Linsenmeier et al. (1982). Firing rates in response to sinusoidal gratings were calculated on the assumption of linear rectified responses (unrectified firing rate was a sinusoid of the same temporal frequency as the stimulus; negative rates were then set to zero), using contrast-response curves from Cheng et al. (1995) (see Fig. 1). Assuming background firing rates of 10 Hz (ON cells) and 15 Hz (OFF cells) [modified from Kaplan et al. (1987), considering the lower mean luminance of 20 cd/m2 used in Cheng et al. (1995)], we calculated the sinusoidal amplitude that would lead to the reported values of the first harmonic (F1) after rectification. [Throughout, we will use F1 to denote the amplitude of the sinusoidal component at the frequency of the grating stimulus, although this value is twice as large as the value obtained using the Fourier transform normalized so that the F0 or DC component is the mean level (Skottun et al., 1991)]. The amplitudes were then fit to R = RmaxCn/(C50n + Cn), where R is response amplitude and C is contrast (ON cells: Rmax = 53.0 Hz, n = 1.20, C50 = 13.3%; OFF cells: Rmax = 48.6 Hz, n = 1.29, C50 = 7.18%). LGN responses for gratings of nonoptimal spatial frequencies were calculated by reducing modulation amplitudes by the factor predicted from the application of LGN spatial filters. ON and OFF cells had temporal phases offset by 180°. To calculate the firing rates in response to moving bars, LGN cell spatiotemporal RFs were used. Temporal filters were taken from the central RF pixel in reverse correlation data from 100% contrast M-sequences (supplied by R. C. Reid, Harvard Medical School); center and surround temporal filters were assumed equal for simplicity.

Cortical receptive fields. Cat cortical layer 4 simple cell RFs were modeled as Gabor functions (see Fig. 2A). A Gabor function is a two-dimensional Gaussian, here with peak value 1, multiplied by a sinusoid. Positive regions of the Gabor correspond to ON subregions and yield connections from ON-center LGN cells, and negative regions correspond to OFF subregions and yield OFF-center inputs; the strength of the connection depends on the magnitude of the Gabor. The number of subregions is defined as the ratio of the width of the Gaussian envelope (at 5% of peak) to the width of a half-cycle of the sinusoid. The aspect ratio of a single subfield is defined as the ratio of the Gaussian envelope length to the sinusoid half-cycle width. Two sets of Gabor parameters were used. "Default" parameters were the mean values for simple cell physiological RFs reported in Jones and Palmer (1987): 2.65 subregions and an aspect ratio of 4.54. (Care must be taken when comparing these numbers with other experimental estimates, e.g., using a 10% cutoff for the Gaussian reduces these numbers by nearly one-fourth.) All RFs have 0.625° half-cycle width, corresponding to a spatial frequency of 0.8 cycles/degree, the approximate mean preferred spatial frequency of cortical cells at 5° eccentricity (Movshon et al., 1978). Gaussian 5% envelope length and width are equal to 2.84 and 1.65°, respectively. The measurements of Ferster et al. (1996) suggest that the net LGN input to a simple cell has broader orientation tuning than results from the default parameters (see Results). To model this broader tuning, we used a second set of Gabor parameters, identical to those above except that the Gaussian envelope was compressed by a factor of 0.7 in both length and width. This yields 1.85 subfields, a subfield aspect ratio of 3.18, and a 5% envelope length and width of 1.99 and 1.15°, respectively.

Conceptual model

To explore the basic concepts underlying our results, we constructed a conceptual model designed to be as simple as possible. The model contains two "rate-coded" cortical neurons, one excitatory and one inhibitory; the inhibitory cell inhibits the excitatory cell. The activity of each cell is represented by a scalar value corresponding to average firing rate. The LGN was modeled as a uniform sheet of cells, approximated as a dense lattice (lattice spacing = 0.05°). The two cortical RFs were determined by Gabor RFs with identical Gaussian shape and location but having sinusoids of opposite spatial phase (thus, the inhibitory cell provides antiphase inhibition).

For computational convenience in obtaining orientation tuning curves, rather than showing many gratings to one pair of cells, we showed one grating to many independent cell pairs. Thus, we constructed multiple pairs of cortical RFs with identical retinotopic positions and with orientation and spatial phases spaced at 10 and 20° intervals, respectively.

For each time step, we first calculated the LGN input to each RF by summing LGN firing rates, weighted by the Gabor function, to give the excitatory input A(theta , phi ) to the cell of orientation theta  and phase phi . The net input to an excitatory cell with parameters (theta , phi ) was the weighted sum A(theta , phi - wA(theta , phi  + 180°); A(theta , phi + 180°) is the LGN input to the (inhibitory cell) RF having the same orientation but opposite (180° difference) spatial phase. The inhibitory gain factor w is unitless and represents the transformation from LGN excitatory current to inhibitory spike rate to inhibitory current in the excitatory cell. w is the only free cortical parameter in this model and controls the width of orientation tuning (see Fig. 5). A match to experimental tuning widths of ~20° is given by w = 1.5 for default Gabor parameters (see Figs. 4, 7), and w = 4.5 for broadly tuned Gabor parameters.

The output rate of an excitatory cell was obtained by thresholding the net input, i.e., spike rate is proportional to [A(theta , phi - wA(theta , phi  + 180°) - xi ]+. For each set of Gabor parameters, the threshold xi  was set automatically according to the following algorithm (thus, xi  is not a free parameter). For a given level of inhibition w, orientation tuning curves were constructed by determining the peak input over a stimulus cycle for cells of each orientation preference, averaged over cells of all spatial phases. Such curves were obtained for gratings of 5, 10, 25, and 50% contrast. Linear interpolation was used to sample these tuning curves at 0.1° intervals, and the orientation that gave the smallest variance in peak input across contrasts was determined (see Fig. 7). The threshold xi (w) was then set to the average across contrasts of the peak input for that orientation and level of inhibition. The excitatory cell's total response was determined by integrating its activity (calculated every 10 msec) over the course of one cycle. A single stimulus cycle was sufficient because the conceptual model is completely deterministic.

The inhibition level wbest that gave a best match to experimental tuning widths (w = 1.5 or w = 4.5 depending on Gabor parameters, as just described) was determined by constructing tuning curves for a range of w. Note that by the procedure just described, each value of w yields a different threshold xi (w). To test the robustness of the model to variations in w (see Fig. 5), for each set of Gabor parameters, we fixed xi  to the level appropriate for wbest and calculated all responses using this fixed threshold.

Computational model

Most simulations were carried out in a computational model incorporating details of cortical cells and maps.

Computational LGN model. For the computational model, a realistically dense lattice of LGN cells was used. We restricted our attention to LGN X-cells, which dominate central cat V1 physiology (Ferster, 1990). At 5° of eccentricity, 1 mm2 = 5 × 5° of visual field in retina (Bishop et al., 1962) and retinal ganglion X-cells (X-RGCs) have density 1000/mm2 (Peichl and Wassle, 1979), including both ON and OFF cells. We assume that each X-LGN cell receives input from a single X-RGC and each X-RGC projects to four X-LGN cells [as in Worgotter and Koch (1991); this value is intermediate between values from Sherman (1985) and Peters and Yilmaz (1993)]. We thus use 7200 LGN cells to cover 6.8 × 6.8° of the visual field, arranged in four overlying sheets of ON cells (30 × 30 cells each) and four sheets of OFF cells (30 × 30), with ON and OFF lattices offset by one-half lattice spacing. After LGN spike rates were calculated as above, spikes were produced in a random (Poisson) fashion: firing rates were converted into the probability of producing a spike in each simulated time step (0.25 msec). To match data showing correlations among LGN cells with overlapping RFs (Alonso et al., 1996), overlaying cells had 25% correlations in their spike trains (each of four overlaying cells picked spikes with probability one-fourth from a common set of four Poisson processes). These correlations made no detectable difference in model behavior.

The connection strength to a given cortical cell from each LGN cell was determined by a repeated probabilistic sampling of the Gabor function describing the cortical RF (see Fig. 2B). LGN synaptic strengths were equal to (<A><AC>g</AC><AC>&cjs1171;</AC></A>exff/npickff)Sigma nffpicki=1 pi where npickff = 3, <A><AC>g</AC><AC>&cjs1171;</AC></A>exff = 0.89 nS, and pi = 1 with probability determined by the absolute value of the Gabor function; pi = 0 otherwise. The number of picks, npickff, determines the degree of sampling of the Gabor function: for npickff right-arrow infinity , the RF becomes a perfect Gabor function. A typical sampled RF is shown in Figure 2B. With this sampling, cortical cells received input from 125 ± 8 (mean ± SD) LGN cells using the default Gabor. Using the more broadly tuned Gabor, cortical cells received input from 61 ± 5 LGN cells.

Cortical model. Cortical cells were modeled as simple integrate-and-fire neurons as described in Troyer and Miller (1997a,b), with parameters matched to experimental data from McCormick et al. (1985). Excitatory cells were fitted to responses from regular spiking cells, and inhibitory cells were fitted to responses from fast spiking neurons. Briefly, each cell is a single compartment with a capacitance C, leak conductance gleak, resting potential Vleak, and two synaptic conductances: fast (AMPA) excitation, gex (reversal potential Vex = 0 mV), and fast (GABA-A) inhibition, gin (Vin = -70 mV). Excitatory cells also have a spike-triggered adaptation conductance gadapt (Vadapt = -90 mV). Each time varying conductance, g, is modeled as a difference of exponentials: g(t) Sigma tj<t <A><AC>g</AC><AC>&cjs1171;</AC></A>(e-(t-tj)/tau fall - e-(t-tj)/tau rise), where the sum is over spike times tj (presynaptic spike times for gex, gin; postsynaptic for gadapt). When V crosses threshold, Vthresh = -52.5 mV, synaptic events are triggered after a delay (randomly chosen for each spike from a uniform distribution, 0.25 msec <=  tdelay <=  2.25 msec), adaptation is triggered (excitatory cells only), and V is set to Vreset and held there for trefract. Vreset was fit to the experimentally measured DC gain of cortical cells [the curve of firing rate vs level of DC injected current (Troyer and Miller, 1997a,b)]. All cells receive nonthalamocortical background excitatory input (Poisson with a mean rate of 5800 Hz and synaptic conductances equal to <A><AC>g</AC><AC>&cjs1171;</AC></A>exbg). The magnitude of this input was set to give low mean background firing rates for excitatory cells (0.16 Hz) at default values of the parameters; identical background input was given to inhibitory cells and resulted in mean background firing rates of 12.2 Hz. Parameters are as follows for excitatory cells: C = 500 pF, gleak = 25 nS, Vleak = -73.6 mV, Vreset = -56.5 mV, trefract = 1.5 msec; for inhibitory cells: C = 214 pF, gleak = 18.0 nS, Vleak = -81.6 mV, Vreset = -57.8 mV, trefract = 1.0 msec; for conductances: tau exrise = 0.25 msec, tau exfall = 1.75 msec, tau inrise = 0.75 msec, tau infall = 5.25 msec, tau adaptrise = 1 msec, tau adaptfall = 83.3 msec, <A><AC>g</AC><AC>&cjs1171;</AC></A>adapt = 3 nS, <A><AC>g</AC><AC>&cjs1171;</AC></A>exbg = 0.89 nS. <A><AC>g</AC><AC>&cjs1171;</AC></A>exctx, <A><AC>g</AC><AC>&cjs1171;</AC></A>exff, and <A><AC>g</AC><AC>&cjs1171;</AC></A>in were free parameters and set as described below.

The model contains 1600 excitatory and 400 inhibitory layer 4 simple cells, representing a <FR><NU>2</NU><DE>3</DE></FR> × <FR><NU>2</NU><DE>3</DE></FR> mm patch of cortex and 0.75 × 0.75° in visual angle [0.9 mm = 1° of visual field at 5° eccentricity (Tusa et al., 1978)]. A 20 × 20 grid of inhibitory cells was interspersed within a 40 × 40 grid of excitatory neurons, with each inhibitory RF center aligned with every other excitatory cell. Gabor-shaped RFs were defined by three parameters in addition to those described above: preferred orientation, determined by an optically measured cortical map from cat V1 [provided by Michael Crair and Michael Stryker (University of California, San Francisco); shown in Fig. 8A]; retinotopic position, progressing uniformly across the sheet; and spatial phase, assigned randomly to each cell (DeAngelis et al., 1992; Ghose et al., 1993).

The probability that any two cortical cells were connected depended on the correlation between their RFs. The following scheme was used for both excitatory and inhibitory connections. Raw correlation c'(a,b) between RFs of cortical cells a, b is c'(a, b) = Sigma i,jepsilon LGN g(i, a)g(j, b)c(i, j). Here, i, j are LGN cells, g(i, a) and g(j, b) are the thalamocortical weights from i to a and j to b, and c(i, j) is the cross-correlation of the spatial RFs of i and j, where OFF spatial RFs are negative of ON. Correlation is then c(a,b)=c'(a,b)/<RAD><RCD><IT>c′(a,a)c′(b,b)</IT></RCD></RAD>. A connectivity function C(a, b)---roughly, the probability of a connection from a to b---is defined as C(a, b) = [sgn(a)c(a, b)npow]+ where sgn(a) = 1 if a is excitatory, -1 if a is inhibitory; [x]+ = x, x > 0, [x]+ = 0 otherwise. npow is a parameter that determines connectivity strength as a function of correlation. Smaller values of npow lead to broader connectivity and more intracortical connections per cell; larger values have the opposite effect (see Fig. 8B). At the default value, npow = 6, a cortical cell receives connections from 132 ± 38 (mean ± SD) other cortical cells (80% from excitatory cells, 20% from inhibitory cells, on average). Just as the thalamocortical connections were sampled from the Gabor function, the intracortical connections were sampled from C(a, b): the strength of intracortical connection from a to b, g(a, b), is g(a, b) = (<A><AC>g</AC><AC>&cjs1171;</AC></A>/npickctx)Sigma nctxpicki=1 pi, where pi = 1 with probability C(a, b) (<A><AC>g</AC><AC>&cjs1171;</AC></A> = <A><AC>g</AC><AC>&cjs1171;</AC></A>exctx or <A><AC>g</AC><AC>&cjs1171;</AC></A> = <A><AC>g</AC><AC>&cjs1171;</AC></A>in, npickctx = 10). As npickctx right-arrow infinity , the connectivity becomes exactly <A><AC>g</AC><AC>&cjs1171;</AC></A>C(a, b).

The main parameters controlling model behavior were the total synaptic strength for each type of connection: thalamocortical (LGN), intracortical excitation (e right-arrow {e, i}), and intracortical inhibition onto excitatory cells (i right-arrow e). The total synaptic strength is obtained by (1) assuming the cell is voltage-clamped at threshold; (2) for each synapse, integrating over time the synaptic current induced by one presynaptic spike; and (3) summing over all synapses of the given type. Thus, total synaptic strength is expressed in units of nanoampere millisecond. The parameters were chosen to satisfy various experimental constraints such as orientation tuning width. We used two different parameter sets: the "feedforward" set with LGN and intracortical inhibitory connections only, and the "full circuit" set, which also included feedback intracortical excitation. For simplicity, inhibitory cells received only excitation; we have not yet explored the influence of inhibitory-to-inhibitory connections. For most simulations, the total intracortical excitatory synaptic strength onto each excitatory cell (e right-arrow e connections) and onto each inhibitory cell (e right-arrow i connections) was identical. Some simulations were run with intracortical excitatory connections onto excitatory cells only (e right-arrow e, but no e right-arrow i). After the pattern of synaptic strengths was determined by probabilistic sampling, synaptic conductances were multiplicatively scaled so that the total conductance from each synaptic type received by each cell was set to its respective mean across cells. This avoids large differences in the amount of input to different cells resulting from the unequal representation of orientations in our spatially limited sample of an orientation map. For the feedforward parameter set (see Figs. 3, 4, 7), total synaptic strengths received by a cell from each type of connection were 10 nA msec (LGN) and 3.75 nA msec (i right-arrow e), yielding mean values for unitary conductances of <A><AC>g</AC><AC>&cjs1171;</AC></A>exff = 2.1 nS, <A><AC>g</AC><AC>&cjs1171;</AC></A>in = 8.3 nS. For the full circuit parameter set (see Figs. 8-12), total synaptic strengths received by a cell from each type of connection were 5 nA msec (LGN), 4.25 nA msec (e right-arrow {e, i}), and 7.5 nA msec (i right-arrow e), yielding mean values for unitary conductances of <A><AC>g</AC><AC>&cjs1171;</AC></A>exctx = 2.0 nS, <A><AC>g</AC><AC>&cjs1171;</AC></A>exff = 1.0 nS, and <A><AC>g</AC><AC>&cjs1171;</AC></A>in = 16.6 nS. The effects of varying these values were also explored (see Fig. 13). Note that we have realistic numbers of LGN cells but unrealistically small numbers of cortical cells; therefore, intracortical connections are unrealistically strong relative to thalamocortical.

Simulations. A typical simulation consisted of three cycles of a 3 Hz sinusoidal grating. During each time step (0.25 msec), values for time-varying conductances were updated, and the membrane time constant and the equilibrium voltage for each cell were then calculated from the cell's conductances. Each cell's voltage was then adjusted according to an exponential decay. Finally, threshold crossings were detected, and subsequent synaptic, adaptation, and refractory events were registered. Simulations were written as C subroutines (mex files) in the MATLAB simulation environment. Initial conditions were determined by simulating 1 sec of model behavior at default parameter values and with LGN cells at background firing rates.

All orientation preferences are represented in the cortical network. Orientation tuning curves were constructed from the presentation of a single stimulus, by binning responses from all cells in the network according to their preferred orientation in 10° bins. Most results used as a stimulus a grating oriented at 128°. This orientation was chosen to avoid artifacts that might result from alignment of the stimulus with the axes of the LGN grid, but we saw no evidence of such behavior.

When displaying synaptic conductances and currents, we show "stimulus-induced" curves in which we have subtracted the mean values of these conductances and currents at background. These mean values were determined by running "blank stimulus" trials in which LGN firing rates were unmodulated.

To reproduce the results of Nelson et al. (1994), we ran simulations in which the inhibition and adaptation currents were blocked in a single cell (see Fig. 12). To accomplish this in a computationally convenient way, we ran a single simulation without any blockade, but monitored the behavior of an additional "blocked cell" for each cell in the network. The blocked cell made no connections. It received identical excitatory input as its unblocked partner cell, but had no inhibitory or adaptation current and was injected with sufficient hyperpolarizing current to bring the background firing rates back to normal. Thus each blocked cell received input from a network in which all other cells were normal (unblocked), but did not itself affect any other cells in the network. Under the assumption that altering a single cell does not affect network behavior, this method allows us to simulate numerous experiments in which one cell undergoes intracellular inhibitory blockade.

    RESULTS
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
References

Modeling approach

We pursued two parallel approaches to modeling contrast-invariant orientation tuning. To explore the basic ideas underlying such tuning, we constructed a conceptual model, designed to be as simple as possible. This model considered two cortical simple cells, one excitatory and one inhibitory, with a monosynaptic connection from inhibitory to excitatory. The RFs of the two cells had identical position and preferred orientation but opposite spatial phase (see Materials and Methods). The neurons were "rate-coded": the average firing rate of each cell was determined by a linear thresholding operation applied to the weighted sum of input cell firing rates. For simplicity, the inhibitory threshold was set to zero (i.e., the inhibitory cell's response was a linear function of its input). The excitatory cell's threshold was set automatically to the level that best produced contrast-invariant tuning for contrasts of 5% and above (see Materials and Methods). Therefore, after the structure of the cortical receptive fields was determined, the conceptual model had only a single free parameter: the strength of intracortical inhibition relative to the strength of thalamocortical excitation.

To study the robustness of our ideas to the complexity of real cortical circuits, we also constructed a computational model that incorporated known details of cortical cells and maps. The cortical component of this model consisted of 1600 excitatory and 400 inhibitory layer 4 simple cells, arranged in a <FR><NU>2</NU><DE>3</DE></FR> × <FR><NU>2</NU><DE>3</DE></FR> mm cortical sheet. Preferred orientations were determined by a measured V1 map, and intrinsic connectivity was determined probabilistically based on correlations in input RFs. Excitatory and inhibitory cells were modeled as conductance-based integrate-and-fire neurons, with parameters matched to those measured in cortical regular-spiking and fast-spiking cells, respectively, including a spike-rate adaptation current in the excitatory cells (McCormick et al., 1985; Troyer and Miller, 1997a,b) (details in Materials and Methods). We considered only the effects of fast synaptic conductances (AMPA and GABA-A); the role of slow conductances (NMDA and GABA-B) will be explored in future work.

LGN input

We focused our research on the response to full-field sinusoidal gratings, because these are the only stimuli for which contrast dependence of orientation tuning has been studied (Sclar and Freeman, 1982; Skottun et al., 1987). Our model was based on cat V1 at ~5° eccentricity. Circularly symmetric, center-surround LGN spatial receptive fields were used (Peichl and Wassle, 1979; Linsenmeier et al., 1982), and LGN firing rates were determined as rectified linear filterings of the input luminance using experimentally measured contrast gain curves (see Materials and Methods) (Fig. 1B) (Peichl and Wassle, 1979; Cheng et al., 1995). To determine whether our model would yield well tuned responses to transient stimuli, we also modeled responses to moving bars.


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Figure 1.   LGN cell responses to 3 Hz, 0.8 cycles/degree moving gratings. A, Instantaneous firing rate. Straight line is background. B, Contrast response functions. Top shows amplitude of first harmonic (F1); bottom shows mean (DC) firing rate. The mean rate increases at contrasts >5%, attributable to rectification as seen in A. Data modified from Cheng et al. (1995) (see Materials and Methods).

LGN cells responded to sinusoidal grating stimuli with a sinusoidal modulation in firing rate (Fig. 1A). The temporal responses of ON-center and OFF-center cells with spatially overlapping RFs were 180° out of phase. Increasing the stimulus contrast resulted in a larger modulation of firing rate. At contrasts above ~5%, the spike rate modulation exceeds the background firing rate. For these contrasts, responses are no longer purely sinusoidal, because spike rate cannot be negative (Fig. 1A, solid lines); that is, LGN responses rectify. Once responses rectify, mean (DC) firing rates increase with increasing contrast (Fig. 1B, DC curves), because peak firing rates continue to increase and minimal firing rates cannot decrease below zero. This contrast-dependent increase in mean LGN firing rates has important consequences for contrast-invariant orientation tuning that will be discussed in detail below.

The oriented arrangement of LGN inputs to simple cell RF subregions was modeled using a Gabor function, a two-dimensional Gaussian multiplied by a sinusoid (Fig. 2A). In the conceptual model, the Gabor function directly determined the weights of geniculocortical connections: positive values corresponded to the weights of ON-center inputs, negative values to the weights of OFF-center inputs. In the computational model, geniculocortical synaptic strengths were determined by probabilistic sampling of the Gabor function from a realistically dense lattice of LGN cells (Fig. 2B).


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Figure 2.   Gabor-shaped cortical RFs. Lighter grays to white indicate positive values of Gabor function, corresponding to weights of ON-center LGN cells with centers at corresponding spatial positions; darker grays to black indicate negative values of Gabor function, corresponding to weights of OFF-center cells. A, A full Gabor function, used to determine LGN inputs to a cortical cell in the conceptual model. B, Typical LGN inputs to a cortical cell in the computational model, after probabilistic sampling from the full Gabor (see Materials and Methods). These receptive fields are typical; different cortical cells may have different preferred orientations, spatial phase (relative locations of ON or OFF subregions), spatial location, and, in the computational model, different outcomes of the probabilistic sampling. Spatial frequency of sinusoid in Gabor function is 0.8 cycles/degree.

We considered two different sets of Gabor parameters to describe geniculocortical connections. The first set was matched to RF parameters taken from physiological measurements of cat simple cells (Jones and Palmer, 1987). The use of the Jones and Palmer parameters as a measure of LGN connectivity in simple cells is based on the experiments of Reid and Alonso (1995), which show that physiological RF parameters at least roughly correspond to the pattern of geniculocortical connections in cat layer 4. These will be used as our "default" parameters. We also considered a second set of parameters representing more broadly tuned LGN input, for several reasons. If cortical circuitry plays a significant role in sharpening simple cell orientation tuning, then the LGN input to a cell would have broader tuning than the cell's responses. Furthermore, the parameters of Jones and Palmer (1987) represent an average of simple cells from all layers, whereas layer 4 cells may be, on average, more broadly tuned for orientation than other layers (Tolhurst and Thompson, 1981). We base our more broadly tuned parameter set on the experiments of Ferster et al. (1996), who cooled the cortex to largely eliminate cortical inputs. Using intracellular electrodes, they then measured the direct LGN input for gratings presented at 30° intervals. The tuning of this input was quantified by measuring the first harmonic (F1) of the voltage response, as a function of stimulus orientation. Although orientation was sampled only coarsely, the figures presented in Ferster et al. (1996) show average orientation tuning half-width at half-height (HWHH) of ~35°. This is significantly broader than the input F1 tuning under our default Gabor parameters, which we find to be 24°. To mimic the broader tuning observed by Ferster et al. (1996), we artificially shrunk the default RFs by a factor of 0.7, leaving the width of each subregion unchanged. This resulted in an input F1 tuning width of 34.8°.

In the conceptual model, the excitatory and inhibitory cells had identical Gabor RFs, except that their sinusoids were 180° out of phase. In the computational model, a distribution of receptive fields was obtained from variations in three parameters: preferred orientation, determined by a measured cortical map (see Fig. 8A); retinotopic position, progressing uniformly across the sheet; and spatial phase, assigned randomly to each cell (DeAngelis et al., 1992).

Tuning of the LGN input to a simple cell

At the preferred orientation, the bright and dark portions of a sinusoidal grating stimulus align with the cortical cell's ON and OFF subregions simultaneously. Thus, all of the cortical cell's LGN inputs fire relatively synchronously and the temporal modulation of this input is large (Fig. 3A). At the null orientation, the inputs are stimulated asynchronously, so the temporal modulation of the total input is small. Note that the mean rate of LGN input does not depend on stimulus orientation. This follows from the assumption that LGN cells are untuned for orientation: because the mean LGN input received by a simple cell is the (weighted) sum of the mean rates of the LGN cells projecting to it, this mean input must also be untuned for orientation (Ferster, 1987). Therefore, only the temporally modulated component of the LGN input is orientation-tuned.


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Figure 3.   Tuning of total LGN input. A, Input to cortical cells in response to high (50%) and low (2.5%) contrast gratings at the preferred and null (orthogonal to preferred) orientations. High (50%) contrast and low (2.5%) contrast are shown. Curved traces show input in response to preferred orientation; black traces, average input (40 presentations) from computational model, using a sampled Gabor RF (as in B); gray curves, input for conceptual model, using connections from the full Gabor function (A). Gray straight lines show response in the conceptual model to a stimulus at the null orientation; in inset, these lines are repeated and compared to average input to null stimuli in computational model (black traces). Note that input to null stimulus at 50% contrast typically exceeds peak input to preferred stimulus at 2.5% contrast. Agreement of the two models for both preferred and null stimuli indicates that RF sampling and Poisson firing of LGN inputs have little effect. B, Tuning of mean (dashed lines) and mean plus first harmonic (solid lines) of thalamic input conductance. Lines show results from the conceptual model; solid circles show results from the computational model; error bars represent ±1 SD. Sum of mean plus first harmonic represents peak input during a cycle of the grating stimulus. Note that mean input is untuned for orientation, and mean input at high contrasts exceeds peak input to preferred orientation at low contrasts. Thus, although the first harmonic is well tuned, no single spike threshold can give tuned responses at both high and low contrasts. In this and subsequent figures showing orientation tunings, cells are grouped by preferred orientation in 10° bins, and orientation axis represents difference of stimulus orientation from preferred.

The untuned mean input presents the primary problem for a purely thalamocortical explanation of contrast-invariant orientation tuning. As a result of LGN rectification, mean LGN firing rates increase with increasing contrast (Fig. 1). This contrast-dependent increase in firing rate is sufficiently large that the mean LGN input at the null orientation at high contrasts exceeds the peak LGN input at the preferred orientation at low contrast (Fig. 3). No single-spiking threshold level can yield well tuned responses for stimuli of all contrasts.

Therefore, to achieve contrast-invariant orientation tuning in response to sinusoidal gratings, the cortex must cancel the untuned, mean input component in a contrast-dependent manner, while it extracts the tuned, modulated component. We will show that this decomposition of the input into a tuned and an untuned component generalizes to stimuli such as flashed and moving bars.

Antiphase inhibition can achieve contrast-invariant orientation tuning

The main purpose of this paper is to demonstrate that correlation-based intracortical inhibition can achieve contrast-invariant orientation tuning (the effects of correlation-based intracortical excitation will also be considered below). By correlation-based inhibition, we mean that the probability of a connection from an inhibitory cell to an excitatory cell is an increasing function of the degree of anticorrelation between their RFs, i.e., the strongest inhibitory connections are made between cells with the most anticorrelated RFs (see Materials and Methods). This implies that an excitatory cell receives the strongest inhibition from inhibitory cells with identical Hubel-Wiesel RFs but of opposite spatial phase. We will call such an inhibitory neuron the cell's "antiphase partner." (By "spatial phase" of an RF, we refer to absolute position in visual space of the ON or OFF subregions, rather than to their position relative to each cell's Gabor function; thus, two RFs have "opposite spatial phase" if the ON subregions of one tend to overlap the OFF subregions of the other in visual space.) The existence of such "spatially opponent" or antiphase inhibition in cat layer 4 is well supported experimentally: at ON locations, where a light stimulus evokes excitation (EPSPs), dark stimuli evoke inhibition (IPSPs), and vice versa for OFF locations (Palmer and Davis, 1981; Ferster, 1988; Hirsch et al., 1995). Note that because simple cells with orthogonal orientation preference have weakly correlated or uncorrelated RFs, correlation-based connectivity results in little or no inhibition from cells with orthogonal tuning. Instead, inhibition comes from cells with similar preferred orientations.

Our model is not a developmental model: we first determined the pattern of LGN input to cortical cells and then fixed the pattern of intracortical connections according to the above correlation-based rule. However, this pattern of inhibition would be expected to arise from a Hebb-type synaptic modification rule, generalized to apply to inhibitory synapses. Such a rule states that synaptic strengths grow more negative (more strongly inhibitory) when presynaptic and postsynaptic firings are anticorrelated, or equivalently, that synapses strengthen when they are effective, i.e., when the inhibitory presynaptic cell is active and the postsynaptic cell is inactive. Such generalization of Hebb-type learning rules to inhibitory synapses is only a hypothesis; plasticity of inhibitory synapses is not well understood [but see Komatsu (1996)]. This intracortical connectivity could also emerge without inhibitory synaptic plasticity. In models in which only thalamocortical synapses undergo correlation-based plasticity, the presence of a fixed inhibitory connection from one cortical cell to another tends to cause the two to develop anticorrelated thalamocortical RFs (Miller, 1994).

The sufficiency of correlation-based inhibition for contrast-invariant tuning is demonstrated in Fig. 4, which shows tuning curves for both the computational and conceptual models, for gratings of 2.5, 5, 10, 25, and 50% contrast. Both models display contrast-invariant orientation tuning above 5% contrast. By choosing the appropriate level of inhibition, both models were able to match experimental estimates of mean orientation tuning width for simple cells. For example, Heggelund and Albus (1978) report that simple cells have a mean tuning width (HWHH) of 19.5°. Model tuning widths (HWHH) above 5% contrast were between 18.7 and 20.8° for both the computational and conceptual models.


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Figure 4.   Contrast-invariant tuning. Response versus orientation for gratings of 2.5, 5, 10, 25, and 50% contrast. A, Computational model. B, Conceptual model. Both models yield contrast-invariant tuning at 5% contrast and above.

The width of the tuning is largely determined by the strength of the inhibition and the tuning of the LGN input. Fig. 5 shows cortical tuning half-widths, using either narrowly or broadly tuned LGN inputs, for various levels of inhibition. Tuning narrows with stronger inhibition but remains contrast-invariant above 5% contrast. Tuning to a long moving bar (width 0.62°, velocity 3.75°/sec) is slightly broader but shows identical sharpening with increasing levels of inhibition: filled circles show bar tuning at 50% contrast for the narrowly tuned LGN inputs.


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Figure 5.   Increasing inhibition leads to sharper tuning. Tuning half-width at half-height (HWHH) versus level of inhibition for gratings of 2.5, 5, 10, 25, and 50% contrast. Thick solid (bottom) curve shows mean tuning HWHH above 5% for RFs with large subfield aspect ratios and narrow LGN tuning (matched to data from Jones and Palmer, 1987). Thick dashed (top) curve shows mean tuning HWHH for RFs with small subfield aspect ratios and broad LGN tuning (matched to data from Ferster et al. 1996). Level of inhibition is normalized so that 1 is the level that produces physiological half-widths for narrow LGN input (Fig. 4). Overlapping symbols indicate contrast-invariance. Tuning gradually sharpens with increased levels of inhibition. A, Computational model. B, Conceptual model. In conceptual model, tuning narrows slightly at 5% contrast for large levels of inhibition. This is attributable to the fact that spike threshold is optimized for default parameters, i.e., inhibition level of 1 (see Materials and Methods). Responses to 2.5% contrast gratings at high inhibition levels for both narrow (solid) and broad (dashed) LGN tuning are shown using thin lines. At very low contrast, conceptual model predicts much narrower tuning.

For higher levels of inhibition and broadly tuned input, tuning at 5% contrast narrows slightly in the conceptual model. This is attributable to the fact that spike threshold was optimized for the default level of inhibition (see Materials and Methods) and could be corrected if spike thresholds were separately optimized for each set of parameters. At 2.5% contrast and high levels of inhibition (Fig. 5B, thin lines), the conceptual model predicts much narrower tuning, for reasons that are more general (see below).

The conceptual and computational models yield qualitatively similar results. Simple additions to the conceptual model led to progressively closer quantitative matches to computational model behavior. A significantly improved match was obtained by adding inhibitory thresholds and using correlation-based inhibitory connectivity from cells with a range of RF properties (rather than from only the single cell with precisely opposite spatial phase). Using simulated synaptic noise (and hence changing the threshold linear function to a smoother function near spike threshold) led to an even closer match between the models and nearly eliminated the difference in responses to 2.5% contrast gratings (see below). However, incorporation of these features required additional unconstrained parameters, and we began to lose the simplicity that was the strength of the conceptual model. Therefore, the results of these investigations are not further reported.

The behavior of our correlation-based model is presented below, in three steps. First, we analyze the reasons why antiphase inhibition achieves contrast-invariant tuning, using the simple conceptual version of the model. Second, we incorporate correlation-based intracortical excitation into the computational model and present results from this completed model. Finally, we explore the robustness of this computational model to variations in the key parameters controlling model behavior. A schematic representing the behavior of both models is shown in Fig. 6, and will be referred to throughout the text.


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Figure 6.   Behavior of model using correlation-based connectivity. Schematic representing behavior of the model in response to preferred (A) and null (B) stimuli. The excitatory cell described in Results is in the top left; its inhibitory antiphase partner is in the bottom right. E, Excitatory cells; I, inhibitory cells. Solid lines represent excitation and depolarization; open lines represent inhibition and hyperpolarization. Line thickness and size of RF icon represent magnitude of activity. Dashed lines represent correlation-based excitation, which is included in the complete computational model only (see Figs. 8-11). Some simulations were performed without cortical excitatory projections onto inhibitory neurons (gray dashed lines), but this did not substantially affect network behavior (see Fig. 13B).

Conceptual model: antiphase inhibition cancels the untuned component of the input

Recall that the main obstacle to achieving contrast-invariant tuning is the untuned component of the LGN input, which increases with contrast as a result of the rectification of LGN responses at higher contrasts. The ability of antiphase inhibition to overcome this problem is most easily demonstrated in the context of the two-cell conceptual model. Here we introduce the term feedforward, by which we mean input from LGN to cortical cells not mediated by cortical excitatory cells. Thus, the geniculocortical input represents feedforward excitation, whereas the pathway from LGN to cortical inhibitory cell to cortical excitatory cell represents feedforward inhibition.

Suppose an excitatory simple cell receives total input Ae from the LGN, and its inhibitory antiphase partner receives LGN input Ai. Assuming for simplicity that inhibitory cell response is linear, the total feedforward input to the excitatory cell is Ae - wAi, where w > 1 is the total strength of the inhibitory synaptic connection multiplied by the gain of the inhibitory cells. During the peak response to the preferred orientation, LGN excitation Ae is large, whereas the antiphase inhibition wAi is weak (Figs. 6A, 7A, top). Thus, the cell gives a strong response. At the null orientation, cells at all spatial phases are receiving an intermediate level of feedforward excitation Ae approx  Ai, and the inhibition wAi > Ae is sufficient to prevent excitatory cell spiking (Figs. 6B, 7A, bottom). Because Ai and Ae both rise with contrast at the same rate, the dominance of inhibition over excitation is maintained for null stimuli of all contrasts.


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Figure 7.   Inputs to a cortical cell given antiphase inhibition (inputs shown relative to background). A, Averaged computational model responses (40 presentations) to 50% contrast gratings. Excitatory LGN input is marked Ex.; intracortical inhibitory input is marked Inh. To compare excitatory and inhibitory inputs, synaptic conductances were converted to currents obtained if the cell was voltage-clamped at threshold. B, Peak synaptic current versus orientation for computational model. Responses are to single presentations of 50, 10, and 5% contrast gratings at 128°. Peak current is the first harmonic (F1) plus the mean (DC) of the stimulus-induced current (including excitation and inhibition). Error bars for 50% contrast are ±1 SD. Dotted line shows approximate threshold level that would lead to contrast-invariant tuning; actual threshold in computational model is determined independently from in vitro data (see Materials and Methods). C, Peak synaptic current versus orientation for conceptual model. Because there is no noise, true peak current is shown. Dotted line shows automatically selected threshold (see Materials and Methods). For both models, mean input decreases and modulation increases with contrast. Thresholds near the crossover point of net input tuning curves result in sharp, contrast-invariant tuning.

The ability of antiphase inhibition to achieve contrast-invariant tuning for a wide variety of stimuli can be best understood by dividing the LGN input into two components: the phase-nonspecific component Anon = (Ae + Ai)/2---the average of the input to the cell and to its antiphase partner---and the remaining phase-specific component, Aspec = (Ae - Ai)/2. The total input to the cell, Ae - wAi, can then be rewritten (1 - w)Anon + (1 + w)Aspec. Thus, antiphase inhibition acts to eliminate the phase-nonspecific component of the LGN input while it amplifies the phase-specific component. For all of the commonly presented oriented stimuli (moving or flashed bars, flashed or counterphased gratings), Hubel-Wiesel RFs yield a phase-specific component tuned for orientation and a phase-nonspecific component that is nearly or completely untuned. Thus, the effectiveness of the antiphase model in achieving contrast-invariant tuning generalizes across stimuli.

This can be summarized by noting that the schematic circuit (Fig. 6) acts as a "differential phase filter": with inhibition sufficiently large, any stimulus that gives similar excitation to each of two opposite phases will cause more inhibition than excitation in excitatory cells and hence will be "filtered out." Only stimuli that predominantly excite one phase and not its opposite can "pass" through this "filter" and cause the excitatory cells to fire. The only stimuli that can accomplish this are stimuli near the preferred orientation; stimuli far from the preferred will give similar input to both phases. This argument applies to any type of oriented stimulus.

Conceptual model: dominant antiphase inhibition provides a contrast-dependent effective threshold

Although the most important effect of antiphase inhibition is to eliminate the phase-nonspecific component of the LGN input, this is not sufficient to achieve contrast-invariant tuning. This can be seen by setting w = 1, thereby causing (1 - w)Anon = 0. In this case, contrast invariance can be achieved only if spike threshold is negligible, i.e., if any positive input leads to spiking. Otherwise, orientations that at low contrast give positive but subthreshold phase-specific input would yield spike responses at higher contrast, because Aspec grows with contrast; thus, orientation tuning would broaden with contrast.

This problem is remedied by including relatively strong inhibition, (w > 1). Then the phase-nonspecific component (1 - w)Anon has a net inhibitory influence that increases with contrast. Because the phase-nonspecific input is untuned for orientation, it serves as a "plateau"---an input identical for stimuli of all orientations---to which the orientation-tuned, phase-specific component is added. The distance from this plateau to the cell's spike threshold can be thought of as a contrast-dependent effective threshold for the tuned input component (Bonds, 1989; Ben-Yishai et al., 1995). With w > 1, this plateau is inhibitory and moves farther from spike threshold with increasing contrast (Fig. 7B,C). By "pulling down" the tuned component, so that only a portion of it is above the spike threshold, this inhibition serves to sharpen the spiking orientation tuning relative to the tuning of the phase-specific input. If spike threshold falls near the crossover point of the net input tuning curves for varying contrasts (Fig. 7B,C, dotted lines), this inhibition sharpens the feedforward input in a contrast-invariant manner.

In the conceptual model, spike threshold for excitatory cells was automatically set at this crossover point in the input current (see Materials and Methods). Somewhat surprisingly, we have found that in the computational model, simply using a physiologically based spiking neuron model (Troyer and Miller, 1997a,b) was adequate to robustly attain contrast-invariant tuning; no parameter adjustments were required. One possible explanation is that synaptic noise "smears out" spike threshold, making it relatively easy to match threshold with the crossover. Also, with inhibition dominant, the orientation tuning curves cross one another where input changes rapidly as a function of orientation, so moderate changes in threshold should make little difference in tuning. Simulations with the conceptual model show that moving threshold by as much as 10% of the peak-to-peak variation in the input driven by 5% gratings changes tuning by <1.5°.

At very low contrasts, the conceptual model predicts that orientation tuning will narrow. Below ~5% contrast, LGN responses do not rectify and therefore the plateau, (1 - w)Anon, does not change with contrast. Orientation tuning narrows with further decreases in contrast (Fig. 5B), because the tuned input component is reduced and the non-zero "effective threshold" is left unchanged. It is unclear whether one could expect to see narrower tuning in the experimental data. As mentioned above, synaptic noise eliminates sharp thresholds, and the effect may be lost in the noise. Computational model results bear this out: tuning for 2.5% contrast has HWHH similar to that at higher contrast (Fig. 5C). This conclusion is supported further by simulations in which synaptic noise was added to the conceptual model. As mentioned above, in this case the conceptual model behavior matched the broader tuning of the computational model, even at 2.5% contrast (data not shown).

Computational model: adding correlation-based excitation

Up to this point, we have not considered the effect of intracortical excitation. We have seen that correlation-based inhibition is sufficient to achieve sharp, contrast-invariant tuning. Here we show that the addition of correlation-based excitation "amplifies" these contrast-invariant responses, without altering their tuning. The conceptual model, which contains only two cortical neurons, is too simple to explore the effects of intracortical excitation in any meaningful way. Hence, the remainder of this paper will present results from the computational model only.

Intracortical excitation was incorporated using a correlation-based rule analogous to that used for intracortical inhibition: excitatory connections were determined probabilistically, such that the strongest connections are found between cells whose RFs are most strongly correlated, i.e., those with similar preferred orientation and similar spatial phase. This is illustrated schematically by the dashed lines in Fig. 6. That intracortical excitation comes primarily from cells of similar orientation preference and similar spatial phase is supported by the fact that EPSPs are evoked only by stimuli of appropriate position and phase, with opposite phase to the stimuli that evoke IPSPs (Ferster, 1988; Hirsch et al., 1995). More direct support is provided by Freeman et al. (1997), who recorded from pairs of cat V1 simple cells isolated on a single electrode. Cell pairs had similar preferred orientations but randomly varying spatial phases. However, cross-correlations indicative of a monosynaptic excitatory connection were found only when the cells had similar absolute spatial phase (G. Ghose, personal communication).

Because the dependence on correlation of intracortical inhibition and excitation differs only in sign, excitatory and inhibitory connections in our model have precisely the same average distribution in terms of orientation preference; they differ only in spatial phase. An example is shown in Fig. 8A, which illustrates the experimental V1 orientation map used to assign preferred orientations to cortical cells in the computational model. In this figure, white squares show the locations of cells making excitatory connections to the excitatory cell at the X, whereas black squares show the locations of cells making inhibitory connections. Excitatory and inhibitory connections to this cell have similar distributions as a function of orientation. Fig. 8B shows the theoretical average distribution of connections for retinotopically identical RFs, as a function of orientation difference (top) and spatial phase difference (bottom). The tightness of tuning as a function of correlation is determined by the parameter npow (see Materials and Methods). Large values of npow lead to tighter connectivity as a function of correlation, whereas smaller values of npow lead to broader connectivity. Increasing and decreasing npow had only minor effects on the behavior of the model.