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The Journal of Neurophysiology Vol. 87 No. 6 June 2002, pp. 2741-2752
Copyright ©2002 by the American Physiological Society
1Department of Psychology, Neuroscience and Cognitive Science Program, University of Maryland, College Park, Maryland 20742; 2National Aeronautics and Space Administration Ames Research Center, Moffett Field 94035-1000; and 3Departments of Physiology and Otolaryngology, W. M. Keck Center for Integrative Neuroscience, Sloan-Swartz Center for Theoretical Neurobiology, University of California, San Francisco, California 94143-0444
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ABSTRACT |
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Troyer, Todd W., Anton E. Krukowski, and Kenneth D. Miller. LGN Input to Simple Cells and Contrast-Invariant Orientation Tuning: An Analysis. J. Neurophysiol. 87: 2741-2752, 2002. We develop a new analysis of the lateral geniculate nucleus (LGN) input to a cortical simple cell, demonstrating that this input is the sum of two terms, a linear term and a nonlinear term. In response to a drifting grating, the linear term represents the temporal modulation of input, and the nonlinear term represents the mean input. The nonlinear term, which grows with stimulus contrast, has been neglected in many previous models of simple cell response. We then analyze two scenarios by which contrast-invariance of orientation tuning may arise. In the first scenario, at larger contrasts, the nonlinear part of the LGN input, in combination with strong push-pull inhibition, counteracts the nonlinear effects of cortical spike threshold, giving the result that orientation tuning scales with contrast. In the second scenario, at low contrasts, the nonlinear component of LGN input is negligible, and noise smooths the nonlinearity of spike threshold so that the input-output function approximates a power-law function. These scenarios can be combined to yield contrast-invariant tuning over the full range of stimulus contrast. The model clarifies the contribution of LGN nonlinearities to the orientation tuning of simple cells and demonstrates how these nonlinearities may impact different models of contrast-invariant tuning.
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INTRODUCTION |
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The tuning of visual neurons
for the orientation of contrast edges is the most thoroughly explored
response property of cortical neurons. Cortical layer 4 of cat primary
visual cortex (V1) is composed primarily of simple cells
(Bullier and Henry 1979
; Gilbert 1977
;
Hubel and Wiesel 1962
), i.e., cells with receptive
fields (RFs) containing oriented subregions each responding exclusively to either light onset/dark offset (ON subregions) or dark
onset/light offset (OFF subregions). Hubel and
Wiesel (1962)
proposed that these response properties arose
from a corresponding, oriented arrangement of inputs to simple cells
from ON-center and OFF-center cells in the
lateral geniculate nucleus (LGN) of the thalamus; such an arrangement
is referred to as Hubel-Wiesel thalamocortical connectivity. The
usual formulation of this so-called "feed-forward" model assumes
that simple cell response properties arise from a linear summation of
these LGN inputs, followed by a rectification nonlinearity due to spike
threshold (Carandini and Ferster 2000
; Ferster 1987
; Ferster and Miller 2000
).
The feed-forward model is challenged by the fact that spiking responses
in simple cells are invariant to changes in stimulus contrast
(Sclar and Freeman 1982
; Skottun et al.
1987
): under this model, inputs at nonoptimal orientations are
expected to be subthreshold at low contrast but become
suprathreshold at higher contrast. We have previously presented
simulation results showing that a simple form of strong "push-pull"
inhibition (inhibition induced by light in OFF subregions
or dark in ON subregions), combined with Hubel-Wiesel
thalamocortical connectivity, is sufficient to overcome
this difficulty and robustly yield contrast-invariant orientation
tuning (Troyer et al. 1998
). In this paper, we analyze the conditions required to achieve contrast-invariant orientation tuning in such a push-pull model.
In our previous work, we studied two versions of the push-pull model. In one version ("network model"), the cortex was modeled as a fairly realistic network of spiking neurons, each modeled as a single-compartment conductance-based integrate-and-fire neuron. The LGN responses were modeled as Poisson spike trains sampled from the stimulus-driven LGN firing rates. The second version ("conceptual model") was much simpler. In this model, both cortical and LGN neuronal activities were represented by firing rates, and the only nonlinearity was the rectification of firing rates at some threshold level of input (rates could not go below zero). While the network model also included intracortical connections from excitatory neurons, the conceptual model included only direct thalamic excitation and thalamic-driven feed-forward inhibition (meaning inhibition driven by LGN via inhibitory cortical interneurons).1 Despite the differences, the two models produced quantitatively similar orientation tuning curves.2 This suggests that the simpler conceptual model retained the key elements responsible for contrast-invariant orientation tuning in the more complex model, and in particular that the rectification or threshold nonlinearity is the primary nonlinearity that is essential for an understanding of this tuning.
In this article, we characterize the conditions required for contrast
invariance in the conceptual model, which is simple enough to allow
analysis. We begin by deriving a general equation for the total LGN
input to a cortical simple cell receiving Hubel-Wiesel thalamocortical
connections, making minimal assumptions other than that LGN responses
can be well-described by an instantaneous firing rate and that the
total LGN input to the simple cell is given by an appropriately
weighted sum of the LGN firing rates. We then show that, in the case of
a periodic grating stimulus, this equation is dominated by two terms: a
linear term, representing the sinusoidally modulated part of the input,
and a nonlinear term, representing the mean input. Except at very low
stimulus contrasts, this nonlinear term grows with stimulus contrast
due to the rectification of LGN firing rates. We then examine how the
combination of this LGN input, LGN-driven push-pull inhibition, and a
cortical cell threshold can yield contrast-invariant orientation tuning
in two regimes. In one regime, representing all but very low contrasts,
contrast invariance of orientation tuning can arise if the growth of
the linear and the nonlinear input terms have the same shape as a
function of contrast. Further analysis demonstrates that this condition
should be at least approximately true for a wide range of LGN models.
In the second regime, representing very low contrasts, the nonlinear
input term does not change with contrast, so that the stimulus-induced
input is simply given by the linear term, which scales with contrast.
In this regime, where the total input is small, input noise results in
a smoothed threshold. Over a wide range of thresholds, this smoothing
results in an input/output function that is approximated by a power-law
function (Miller and Troyer 2002
). The combination of
input that scales with contrast and a power-law input/output function
yields contrast-invariance of tuning. Finally, we demonstrate that
these two mechanisms can combine to yield contrast-invariant tuning
over all contrasts.
An abstract of this work has appeared (Troyer et al.
1999
).
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RESULTS |
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LGN input to simple cells
Previous investigations into the origins of orientation selectivity have made a variety of simplifying assumptions regarding the nature of the LGN input to cortical simple cells. Often, the visual stimulus is transformed directly into a pattern of cortical input, ignoring important nonlinearities contributed by LGN responses. In this paper we focus on periodic gratings, i.e., stimuli that are spatially periodic in one dimension and uniform in the other dimension. Our model is a purely spatial model and ignores cortical temporal integration. We consider an "instantaneous" pattern of LGN activity across a sheet of cells indexed by the two dimensional vector x representing the center of each LGN receptive field. Cortical output is derived as a static function of this pattern of activity.
In this section, we demonstrate that the total LGN input to a simple
cell in response to a grating stimulus with contrast C,
orientation
, and spatial location given as a phase variable
stim, can be well approximated by a function of the
following form
|
(1) |
2 For periodic grating stimuli, the second term represents the mean (DC) level of input averaged over all spatial phases, while the first term represents a sinusoidal (1st harmonic or F1) modulation of the input as a function of the grating's spatial phase.
3 The level of the mean input depends only on contrast. The amplitude of the modulation can be factored into the product of a function that depends only on contrast and a function that depends only on orientation.
Input from a Gabor RF: general expression
We view the LGN as a uniform sheet of ON-center and
OFF-center X cells, and let
LON(x)
[LOFF(x)] denote the response of
the LGN ON (OFF) cell at position vector
x at a particular time t [for simplicity, we
omit the time dependence in LON(x,
t) and LOFF(x,
t)]. Initially, we make no explicit assumptions regarding the relationship between ON and OFF cells
responses. However, we expect ON and OFF cells
at the same location to have roughly opposite responses to changes in
luminance. To extract the ON/OFF difference we let
Ldiff(x) = [LON(x)
LOFF(x)]/2. Letting
Lavg(x) = [LON(x) + LOFF(x)]/2 denote the
average LGN response at position x, we can rewrite
LON(x) = Lavg(x) + Ldiff(x) and
LOFF(x) = Lavg(x)
Ldiff(x).
We assume that the spatial pattern of LGN connections to a simple cell
can be described by a Gabor function (Jones and Palmer 1987b
; Reid and Alonso 1995
). For simplicity, we
will refer to this pattern of LGN connectivity as the receptive field
(RF) of the cell. We let the vector fRF
represent the preferred spatial frequency and orientation of the RF,
and choose our x coordinates so that
fRF is parallel to the
x1 axis and so that the RF center is at the origin. Thus we write the Gabor RF as
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1 and
2 determine the size, and
RF the spatial
phase of the Gabor RF. Positive values of the Gabor, G+(x) = max
[G(x), 0], give the connection strength from
ON cells, whereas the magnitude of the negative values,
G
(x) = |min
[G(x), 0]|, gives the connection strength from OFF cells. Note that
G(x) = G+(x)
G
(x), while
|G(x)| = G+(x) + G
(x). Using a linear summation
model, the total input to the cortical cell is given by
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(2) |
In the common linear model of LGN input, the firing rate modulation of OFF cells is assumed equal and opposite to ON cell modulations at a given point x. Therefore Lavg(x) is a constant equal to the average background firing rate of LGN cells, and the ON/OFF-averaged term contributes only a stimulus-independent background input to the cortex. As a result, the linear model only considers the ON/OFF-specific component of the LGN input. However, since LGN firing rates cannot be modulated below 0 Hz, at higher contrasts the balance between ON and OFF cell modulations cannot be maintained, and the ON/OFF-averaged term grows with increasing contrast. Hence it becomes important to retain both terms in modeling LGN input.
Further insight into this decomposition can be gained by considering, for any given cortical cell, the cell's antiphase partner: an imaginary cell that has identical Gabor receptive field except for an overall sign reversal, so that ON subregions (connections from ON cells) are replaced with OFF subregions (connections from OFF cells) and vice versa. Then the ON/OFF-specific term represents the component of LGN input that is equal and opposite to a cell and to its antiphase partner, while the ON/OFF-averaged term represents the input component that is identical to a cell and to its antiphase partner. It is in this sense that we call these the ON/OFF-specific and ON/OFF-averaged input components, respectively. The decomposition into these two components is key to the results presented below.
Input from grating stimuli
In this paper, we focus on responses to periodic gratings, i.e.,
stimuli that are spatially periodic in one dimension and uniform in the
other dimension. The periodicity of the gratings is described by a
two-dimensional spatial frequency vector
fstim, and we assume that LGN cells
have circularly symmetric receptive fields. Therefore the response of
an LGN ON-center cell is determined by its position
relative to the grating: LON(x) = lON(fstim
· x), where lON(
) is some
function of a scalar variable
determining a cell's location
relative to the periodic modulation of the overall LGN activity
pattern. Similarly, LOFF(x) = lOFF(fstim
· x). Throughout we will assume
|fstim| = |fRF|. LGN spatial frequency tuning
can be included by writing
LON(x) = FON(|fstim|)lON(fstim
· x), where
FON(|fstim|)
is the spatial frequency tuning of ON cells scaled so that FON(|fRF|) = 1. Similar definitions apply for OFF cells.
Given that LGN activity is periodic with spatial frequency
fstim, the
Lavg(x) and
Ldiff(x) components of LGN activity
can each be written as a cosine series
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and 0


represent the phase of each component.
Using these expansions, we can evaluate the integral in Eq. 2 and rewrite the total LGN input to a simple cell as the sum of a
pair of cosine series
|
(3) |
(nfstim) and
|
|(nfstim) are the
amplitudes of the Gabor and absolute Gabor filters, respectively,
evaluated at the spatial frequency vector
nfstim (i.e., the amplitudes of
the Fourier transforms of G and |G| at this
frequency). The phases are 















We reiterate that our model is a spatial model. The instantaneous spatial distribution of activity across a two-dimensional sheet of LGN cells (described by adiff and aavg) is multiplied by the appropriate Gabor filters to determine the total LGN input to a cortical cell at a given time, and the cortical response is assumed to depend instantaneously on this input. For temporal patterns of input, such as a drifting or counterphased grating, we simply assume that the cortical responses can be computed as instantaneous responses to a sequence of patterns of LGN activity that change in time. For flashed stimuli, the model will apply to the degree that cortical responses simply track the changing patterns of LGN activity triggered by the flash.
Reduction to two terms
In this subsection, we argue that each of the two sums in
Eq. 3 should be dominated by the contribution of a single
term. In particular, the ON/OFF-averaged sum is
dominated by the zero frequency (DC) term, and the
ON/OFF-specific sum is dominated by the first
harmonic (F1) term. Having shown that this is the case, we will have
demonstrated that
|
(4) |
stim


We consider a simple rectified-linear model of LGN responses.
ON and OFF cells are assumed to have background
firing rates of 10 and 15 Hz, respectively, and response modulations
are assumed to result from linear filter properties of the LGN cells.
Therefore a drifting sinusoidal grating stimulus leads individual LGN
cells to sinusoidally modulate their firing rates with time, rectified at 0 Hz (Fig. 1, A and
B). This means that at each instant the spatial pattern of
response across the sheet of LGN cells is a rectified sinusoidal
modulation. The amplitude of the modulation of activity across LGN
depends on the contrast of the grating (Fig. 1C) and was
calculated using measured contrast response functions from cat LGN
X cells in response to drifting sinusoidal gratings. The
phase of the modulation is assumed opposite (180° apart) for an
ON cell and an OFF cell at the same position.
This ignores the actual spread of response phases for both
ON and OFF cells (Saul and Humphrey
1990
; Wolfe and Palmer 1998
). At low contrasts,
the stimulus-induced modulations of firing rates do not exceed the
background firing rates, and so the mean input does not grow with
contrast. However, once the stimulus-induced modulation is as large as
the background firing rates
which occurs at about 5% contrast
then
the LGN firing rates rectify at 0 Hz for each trough of the modulation
(Fig. 1), and so increases in contrast above 5% lead to an increase in
the mean LGN input.
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Using the measured contrast response functions, we compute the
magnitudes of a





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We now turn our attention to the Gabor filters. Since a Gabor function
is obtained by multiplying a sinusoid and a Gaussian, the Fourier
transform of G consists of a Gaussian convolved with a pair
of delta functions located at the positive and negative frequencies of
the sinusoid (Fig. 2C). The specific parameter values used
have been extracted from two sets of experimental data. We started with
Gabor parameters taken as the mean values from measured simple cell RFs
[Jones and Palmer (1987b)
; full parameters are given in
METHODS section of Troyer et al. (1998)
]. The length of each subregion was then reduced so that the orientation tuning width of the F1 of the total LGN input (38.0° half-width at
half-height) matched the mean tuning width of experimentally measured
intracellular voltage (Carandini and Ferster 2000
). This was accomplished by multiplying the standard deviation of the Gaussian
envelope in the direction parallel to the subregion by a factor of
0.58. One important feature to note is that the width of the Gaussian
in Fourier space is similar to the preferred spatial frequency of the
Gabor RF. This condition will hold when the simple cell RF has roughly
two subregions; more subregions will narrow this width. The
absolute value of the Gabor, |G|, is obtained by
multiplying a Gaussian times the absolute value of a sinusoid. Thus the
Fourier transform |
| consists of a Gaussian convolved with the
Fourier series for the absolute value of a sinusoid, which has only
even harmonics (Fig. 2C). We focus on the response at the
optimal orientation, and fix the stimulus spatial frequency to match
the optimal spatial frequency of the Gabor filter, i.e., fstim = fRF. In this case, the multiples of the
stimulus spatial frequency vector,
nfstim, fall on the peaks of the
Gaussian-shaped humps in the Gabor and absolute Gabor filters (Fig. 2,
A and B, marked x). The result is that
the
(nfstim) coefficients are
dominated by the F1 (n = 1) component (Fig. 2A, gray bars), while the
|
|(nfstim) coefficients
are dominated by the DC (n = 0) component (Fig. 2B, gray bars).
Having calculated the values a

(nfstim) and
|
| (nfstim), we need
only multiply the corresponding components to calculate the
ON/OFF-specific and
ON/OFF-averaged components of the total LGN
input using Eq. 3. The results, shown by the black bars in Fig. 2, A and B, demonstrate that only two terms
contribute significantly to the total LGN input. Since both the Gabor
filter
and the Fourier series for
Ldiff are dominated by the F1 component,
99.9% of the total power in the diff terms is concentrated in the
first harmonic. Similarly, the avg terms are dominated by the DC, with
98.6% of the power concentrated in the zero frequency (mean) component
of the input. Thus the approximation given by Eq. 4 is valid
for the rectified-linear model of LGN response used here, with DC = |
|(0)a
) =
(fstim)a
Thus we have shown that the total LGN input to a cortical simple cell in response to a periodic grating (Eq. 3) is expected to be well-approximated by a sinusoidal modulation about some mean level (Eq. 4). This was derived assuming a linear-rectified LGN response model and a sinusoidal grating, but is likely to hold for more general LGN response models and for more general types of gratings. So long as the F1 of Ldiff is at least as large as the other Fourier coefficients, the dominance of the F1 in the Gabor filter will lead the ON/OFF-specific terms to be dominated by the F1. Similarly, the ON/OFF-averaged components will be dominated by the DC so long as the DC of Lavg is at least as large as the other Fourier coefficients.
Orientation tuning
We now examine the orientation and contrast dependence of the DC
and F1 terms of the total LGN input. We express the orientation
in
degrees from the preferred orientation, i.e., 0 degrees is the optimal
stimulus. Since the responses of individual LGN cells are untuned for
orientation, changing the orientation of the stimulus will rotate this
activity pattern, but will leave the shape of the activity modulation
unchanged. Therefore the coefficients a





|
(5) |
We now turn our attention to the F1 term. First, we write the spatial
frequency vector of the stimulus in polar coordinates, fstim = {|fstim|,
} and let
0 =
({|fstim|, 0}). Then we
factor the amplitude of the modulation as follows
|
(6) |
|
|
0a
) =
({|fstim|,
})/
0 captures the orientation tuning of the Gabor
RF, normalized so that at the preferred orientation,
h(0) = 1. Therefore the total LGN input can be written
in its final form
|
(7) |
) is determined by the evaluation of the filter G
along the circle of radius |fstim|
(Fig. 2C). If the Gabor RF has two or more subregions and
the spatial frequency |fstim| = |fRF|, h(
) is
dominated by the contribution from the Gaussian centered at
fRF. In this case
|
(8) |

x and 
y determine the dimensions of the Gaussian
envelope of the Gabor filter in Fourier space. For
|fstim|
|fRF|, h(
) takes on a
more complex form. Qualitatively, tuning narrows for
|fstim| > |fRF| and broadens for
|fstim| < |fRF|.
This concludes our analysis of the LGN input to a simple cell. We now turn to the analysis of the conditions yielding contrast-invariant orientation tuning.
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CONTRAST-INVARIANT ORIENTATION TUNING I: HIGHER-CONTRAST REGIME |
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Intracortical inhibition
A key to our model of contrast-invariant orientation tuning at
higher contrasts is the inclusion of strong, contrast-dependent feed-forward inhibition. We will take the inhibition a cell receives to
be spatially opponent to, or antiphase relative to, the excitation a
cell receives; this is also known as a push-pull arrangement of
excitation and inhibition. By spatial opponency of inhibition and
excitation, we mean that in an ON subregion of the simple cell RF, where increases in luminance evoke excitation [i.e., where
G+(x) > 0], decreases in
luminance evoke inhibition; the opposite occurs in OFF
subregions [where G
(x) > 0]. Since the LGN projection to cortex is purely excitatory
(Ferster and Lindström 1983
), this inhibition must come from inhibitory interneurons in the cortex. A simple hypothesis that is consistent with a broad range of experimental data is that a
given simple cell receives input from a set of inhibitory simple cells
that collectively 1) have RFs roughly overlapping that of
the simple cell, 2) are tuned to a similar orientation as
the simple cell, and 3) have OFF subregions
roughly overlapping the simple cell's ON subregions, and
vice versa (Troyer et al. 1998
); that is, collectively a
cell's inhibitory input has a receptive field like that of a cell's
"antiphase partner." This circuitry leads to inhibition that, like
the inhibition observed physiologically, is spatially opponent to the
excitation a cell receives (Anderson et al. 2000a
;
Ferster 1988
; Hirsch et al. 1998
), and
has the same preferred orientation and tuning width as a cell's
excitatory inputs from the LGN (Anderson et al. 2000a
;
Ferster 1986
).
The role of antiphase inhibition in cortical orientation tuning can be understood quite generally by returning to our decomposition of the LGN input into an ON/OFF-specific term and an ON/OFF-averaged term. The ON/OFF-specific component typically is tuned for stimulus parameters such as orientation while the ON/OFF-averaged component typically is untuned or poorly tuned. The ON/OFF-specific term represents input that is equal and opposite to a cell and to its antiphase partner. Thus, for this component, the antiphase inhibition goes down when LGN excitation goes up, and vice versa. The net result is that strong antiphase inhibition serves to amplify the well-tuned ON/OFF-specific component of the LGN input. The ON/OFF-averaged term represents input that is identical to a cell and to its antiphase partner. If the antiphase inhibition is stronger than the direct LGN excitation, the ON/OFF-averaged term elicits a net inhibitory input that is poorly tuned for orientation but depends on contrast. Thus antiphase inhibition serves to eliminate the untuned component of the LGN input and replace it with a net inhibition that sharpens the spiking responses driven by the ON/OFF-specific tuned component of the input. This argument generalizes to any type of stimulus, transient or sustained. We now work out the details of this for the case of a drifting sinusoidal grating stimulus.
In our reduced model, we represent the set of inhibitory cells
providing inhibition to a cell by a single inhibitory simple cell. This
cell receives input from the LGN specified by a Gabor function
identical to that which specifies the input to the excitatory cell
except that the sinusoidal modulation of the RF is 180° out of phase.
The total LGN input to the inhibitory cell is then
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(9) |
inh is spike threshold, and
binh is the amount of nonspecific input received
at background. For simplicity, we assume that input to the inhibitory
cell never dips below threshold [i.e., DC(100)
F1max(100) + binh
inh
0], allowing us to ignore inhibitory cell
rectification. This assumption is not unreasonable given the
experimental evidence suggesting that some inhibitory
interneurons have high background firing rates (Brumberg et al.
1996It is important to note that the model inhibitory cell responds
to all orientations, although it also shows orientation
tuning. For nonpreferred orientations, for which h(
)
0, the cell still receives positive input due to the term
DC(C) in Eq. 9, which drives inhibitory response.
This inhibition is critical for overcoming the strong,
contrast-dependent LGN excitation that would otherwise drive excitatory
simple cells at the null orientation. The DC term contributes an
untuned platform to the inhibitory cell orientation tuning curve,
identical for all orientations, on which the tuned response component
due to the term containing h(
) is superimposed. That is,
the inhibitory cell tuning follows the tuning of the total LGN input to
a simple cell, which includes both an untuned and a tuned component.
One of the key predictions of our model (Troyer et al.
1998
) was that there should be at least a subset of layer 4 interneurons that show contrast-dependent responses to all orientations
(see DISCUSSION), which is embodied here in the response of
our single model inhibitory neuron.
To facilitate analysis, we assume that excitation and inhibition
combine linearly, at least in their effect on spiking output. Because
this assumption ignores reversal potential nonlinearities, the total
input in our model should not be equated with the membrane voltage,
particularly for voltages near the inhibitory reversal potential (see
DISCUSSION). Letting 
|
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|
(10) |

Inhibition has opposite effects on the DC and F1 terms of the total
input: it acts to suppress the mean input [DC
(1
w)DC], while it enhances the modulation [F1
(1 + w)F1]. The increase in the modulation is due to the fact
that, at any given contrast, increases in LGN excitation are
accompanied by a withdrawal of cortical inhibition, while decreases of
excitation are accompanied by an increase of inhibition. A key
requirement of our model of contrast-invariant tuning is that the
inhibition be dominant (w > 1), i.e., the inhibitory
input is stronger than the direct input from the LGN. This implies that
the contrast-dependent increase in the mean input from the LGN is not
only suppressed, it is actually reversed (1
w < 0), so that the mean feed-forward input to the cortical simple cell
decreases with increasing contrast.
Contrast invariance
Given the assumption that output spike rate results from a
rectified linear function of the input, the excitatory spike rate is
|
(11) |
eff =
exc
bexc + w(binh
inh) is
an effective threshold that incorporates tonic inhibitory activity,
nonspecific input to the excitatory cell bexc
and the excitatory-cell spike threshold
exc. Note that,
after specifying the Gabor RF, the cortical component of our model has
only two free parameters, w and
eff;
the gain factor g is simply a scale factor that determines
the magnitude of response without affecting tuning.
For the orientation tuning of the response to be contrast invariant, we
must be able to write r(C,
,
stim) as the product of a contrast response function
p(C) and an orientation tuning function
q(
,
stim). Since the F1 term is the only
term that depends on the orientation and phase of the stimulus, we look for an expression of the form
|
|
(12) |
|
|
(13) |
w), guarantees that the response is contrast invariant
|
(14) |
stim) = 1],
suprathreshold responses require h(
) > k(w
1)/(w + 1). For
larger values of inhibitory strength w, h(
)
has to be closer to its maximum value before threshold is crossed,
i.e., tuning width is narrower.
Equation 14 implies a strong version of contrast-invariant tuning. Not only is the tuning of average spike rate contrast invariant, but the spike rate at any given phase of the response at a given orientation multiplicatively scales with changes in contrast.
Parameter dependence
Equation 14 is equivalent to the statement that the DC
and F1 terms have the same shape as functions of contrast,
up to a scale factor determined by k and an offset
determined by
eff/(w
1). Using
Eqs. 5 and 6, Eq. 14 can be rewritten
|
(15) |







|
Figure 3 shows, for our linear-rectified model of LGN response, that
a

eff, are chosen to satisfy Eq. 15 for some
constant k. To determine the robustness of our model, we
simply varied these two parameters and measured the resulting contrast
dependence of orientation tuning. We consider three levels of
inhibition (left to right columns of Fig.
4) and three levels of effective
threshold (thick, thin, and dashed lines in Fig. 4, where the thick
line represents the optimal threshold for achieving contrast-invariant
tuning for that level of inhibition). The orientation tuning width, as measured by half width at half height of the orientation tuning curve,
is contrast invariant down to about 5% contrast for the optimal
threshold, as expected (Fig. 4B). With nonoptimal
thresholds, modest deviations from contrast invariance arise in the
range of 5-10% contrast.
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Contrast invariance: an intuitive explanation
From Fig. 4A we can obtain an intuitive picture of how our model achieves contrast-invariant orientation tuning in the higher-contrast regime. Since the magnitude of the input modulation increases with increasing contrast, the tuned component of the total input also increases. However, with dominant inhibition, the mean input decreases with increasing contrast, so that the growing tuned component rides on a "sinking" untuned platform. Thus there will be some orientation where the input tuning curve for a higher contrast stimulus will cross the corresponding curve for a low contrast stimulus (Fig. 4A). Placing spike threshold at the level where the contrast-dependent tuning curves cross then yields contrast-invariant responses (Fig. 4, A and B, thick lines). Note that as the level of inhibition increases, the crossing point of the input tuning curves occurs closer to the peak, resulting in narrower tuning.
The fact that this crossing point occurs at the same orientation across
a range of contrasts is exactly equivalent to the condition that
a