 |
||||
|
||||
|
||||
|
||||
Previous Article | Next Article
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)
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
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
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
In cats, orientation selective responses appear in cortical layer 4. Cat layer 4 is composed of simple cells (Hubel and Wiesel, 1962
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
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
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
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
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/center2)e
x2/
2center
(16/
2surround;
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(,
) to the cell of orientation
]+. For each set of Gabor parameters, the 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, 1990exff/npickff)
nffpicki=1 pi where npickff = 3,
, 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
fall
tdelay
exrise = 0.25 msec,
×
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)/
. A connectivity function C(a, b)
roughly, the probability of a connection from a to b
RESULTS
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
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
View larger version (22K):[in this window]
[in a new window]
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)
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).
View larger version (67K):[in this window]
[in a new window]
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
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
View larger version (20K):[in this window]
[in a new window]
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
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)
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)
View larger version (21K):[in this window]
[in a new window]
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.
View larger version (24K):[in this window]
[in a new window]
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
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.
View larger version (38K):[in this window]
[in a new window]
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
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.
View larger version (23K):[in this window]
[in a new window]
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
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
This problem is remedied by including relatively strong inhibition, (w > 1). Then the phase-nonspecific component (1
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
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
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
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.
