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The Journal of Neuroscience, July 15, 2001, 21(14):5203-5211
How Simple Cells Are Made in a Nonlinear Network Model of the
Visual Cortex
D. J.
Wielaard,
Michael
Shelley,
David
McLaughlin, and
Robert
Shapley
Center for Neural Science and Courant Institute of Mathematical
Sciences, New York University, New York, New York 10012
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ABSTRACT |
Simple cells in the striate cortex respond to visual stimuli in an
approximately linear manner, although the LGN input to the striate
cortex, and the cortical network itself, are highly nonlinear. Although
simple cells are vital for visual perception, there has been no
satisfactory explanation of how they are produced in the cortex. To
examine this question, we have developed a large-scale neuronal network
model of layer 4C in V1 of the macaque cortex that is
based on, and constrained by, realistic cortical anatomy and
physiology. This paper has two aims: (1) to show that neurons in the
model respond like simple cells. (2) To identify how the model
generates this linearized response in a nonlinear network. Each neuron
in the model receives nonlinear excitation from the lateral geniculate
nucleus (LGN). The cells of the model receive strong (nonlinear)
lateral inhibition from other neurons in the model cortex. Mathematical
analysis of the dependence of membrane potential on synaptic
conductances, and computer simulations, reveal that the nonlinearity of
corticocortical inhibition cancels the nonlinear excitatory input from
the LGN. This interaction produces linearized responses that agree with
both extracellular and intracellular measurements. The model correctly
accounts for experimental results about the time course of simple cell
responses and also generates testable predictions about variation in
linearity with position in the cortex, and the effect on the linearity
of signal summation, caused by unbalancing the relative strengths of
excitation and inhibition pharmacologically or with extrinsic current.
Key words:
primary visual cortex; neuronal network model; simple
cells; linearity; synaptic inhibition; phase averaging
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INTRODUCTION |
Neurons in the primary visual cortex
are classified as simple or complex, depending on how they respond to
visual stimuli. If the response of the cell depends on the stimulus in
an approximately linear fashion, the cell is termed "simple";
otherwise, "complex." Specifically, in response to visual
stimulation by the temporal modulation of standing grating patterns,
the linear-like behavior of simple cells includes: (1) a sensitive
dependence on the spatial phase (position) of the grating, (2) very
little presence in a neuron's response of nonlinear distortion
components such as second (and higher) temporal harmonics. (This is
aside from distortions arising from threshold to firing.) The responses
of complex cells are very different: (1) they are spatial phase
(position)-insensitive, and (2) their responses are predominantly
second harmonic.
The linear dependence on visual stimuli of the simple cell might be
assumed to be a simple consequence of convergence of excitatory drive
from lateral geniculate nucleus (LGN) cells. However, this ignores the
nonlinearities of the LGN cells. For example, rectification caused by
the spike-firing threshold produces nonlinear distortion of LGN
responses for stimulus contrast >0.2, that is, even at relatively low
contrast (Tolhurst and Dean, 1990 ; Shapley,
1994). In the numerical simulations of the model (see below),
this nonlinearity is evident in the responses of cortical cells with
only LGN excitation. Such responses contain significant nonlinear
components. Therefore, it is an open and important question, how can
there be simple cells in the visual cortex?
Surprisingly, there has been as yet no explanation, based on known
cortical architecture, for the existence of simple cells in the
cerebral cortex. Here we offer an answer to this question by studying a
large-scale neuronal network model of layer 4C in macaque
primary visual cortex, V1. Our choice of lateral connectivity within
this model is motivated not by Hebbian-based ideas of activity-driven correlations (Troyer et al., 1998), but by our
interpretation of the anatomical and physiological evidence concerning
cortical architecture, which is known better for macaque V1 than for
almost any other cortical area. The crucial distinguishing features of the model, derived from biological data, are that the local lateral connectivity is nonspecific and isotropic, and that lateral
monosynaptic inhibition acts at shorter length scales than excitation
(Fitzpatrick et al., 1985; Lund, 1987;
Callaway and Wiser, 1996; Callaway, 1998). In the model, orientation preference is conferred on
cortical cells from the convergence of output from many LGN cells
(Reid and Alonso, 1995), with that preference laid out
in pinwheel patterns (Bonhoeffer and Grinvald, 1991;
Blasdel, 1992a,b; Maldonado et al., 1997). In
McLaughlin et al. (2000), we show that orientation selectivity of cells in such a model of 4C is greatly
enhanced by lateral corticocortical interactions. Here we show that:
(1) neurons in the network model can behave like simple cells; (2) cancellation of nonlinear LGN excitation by corticocortical inhibition causes the linear-like responses of simple cells in this nonlinear network.
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MATERIALS AND METHODS |
Simple cells: experimental classification
Precise characterization of the linear and nonlinear summation
of visual signals of cortical cells was achieved by experiments with
drifting and contrast reversal gratings, such as in Movshon et
al. (1978), De Valois et al. (1982), and
Spitzer and Hochstein (1985), in which experimental
techniques that proved useful for studying the linearity of spatial
signal summation in retinal ganglion cells (Enroth-Cugell and
Robson, 1966; Hochstein and Shapley, 1976) and
LGN cells (Kaplan and Shapley, 1982), were applied to
visual cortex. Figure 1, A and
B (De Valois et al., 1982), shows
experimental data, based on extracellular recordings of spikes, that
illustrate linearity of spatial summation in a simple cell located in
macaque V1 (Movshon et al., 1978; Reid et al.,
1991). (Note also that simple cell responses are not
necessarily linear in all quantities of interest, such as stimulus
intensity.) Figure 1A shows the response to contrast
reversal stimulation of the cell by a standing pattern at optimal
orientation, spatial, and temporal frequency. (Defined more precisely
below, "contrast reversal" is the temporal modulation of a standing
grating pattern by a sinusoidal modulation of the contrast.) Response
to contrast reversal has proven to be a critical test of linearity in
simple cells (Spitzer and Hochstein, 1985). The response
of the simple cell depends on the spatial phase or position of the
standing grating pattern relative to the midpoint of the receptive
field of the neuron with a large-amplitude response at the fundamental driving frequency at one spatial phase (the "in-phase" condition) and very little response to "orthogonal phase" stimulation 90° away. Both responses show little or no generation of the higher temporal harmonics that might be expected for a nonlinear system. However, nonlinear harmonic distortion products are apparent in the
responses of cortical complex cells (De Valois et al.,
1982), an example of which is shown in Figure 1B.
Note, in particular, the phase insensitivity and the frequency-doubled
(2nd harmonic) response of the complex cell. It is worth noting that
the simple cells in De Valois et al. (1982) were not
assigned to a cell layer and that subsequent experimental work in
recording the activity of neurons across all layers of macaque V1 has
found many neurons in layer 4C that behave just the same
as the simple cell illustrated in Figure 1A (Ringach et al.,
2001).
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Figure 1.
Simple cell responses to grating contrast
reversal. A and B are from De Valois et
al. (1982) (with author's permission), whereas data in
C are from Jagadeesh et al. (1997) (with
author's permission). A, Macaque monkey simple cell, spike
rate response to contrast reversal of a sine grating at 2 Hz
modulation. Position of the standing wave in the visual field is
specified in degrees of spatial phase in which one spatial cycle of the
grating pattern is 360°. At ~180°, the response goes to zero.
B, Macaque complex cell response to the same contrast
reversal stimulus. The response amplitude shows little variation with
spatial phase, and there are two response peaks per cycle of temporal
modulation this is the second harmonic or F2
component. C, Intracellular responses of a cat simple cell
to sine wave contrast reversal at 2 Hz, shown over two cycles. This
represents only half a cycle of spatial phase. The temporal modulation
waveform is shown below the neural responses. The membrane potential
response is predominantly at the fundamental frequency of temporal
modulation, with very little modulation at the 0° phase.
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There have been some measurements, in the cat visual cortex, of
intracellular responses of simple cells to such stimuli (Ferster et al., 1996; Jagadeesh et al., 1997). Such data
are shown in Figure 1C, which shows the membrane potential
(with spikes filtered) of a cat simple cell responding to contrast
reversal stimulation, for in-phase and orthogonal phase spatial
patterns. When the stimulus grating is in-phase, there is a large
component of the membrane potential that appears to be approximately
sinusoidal in time, at the same frequency as the temporal modulation of
the stimulus. When the stimulus grating is moved to the orthogonal
phase position, the modulation of the membrane potential is small in
amplitude, with a very small second harmonic. These results are
consistent with the linearity of an extracellular response of a simple cell.
Computational model
Description. Our model is a large-scale neuronal
network of layer 4C, comprised of excitatory and
inhibitory integrate-and-fire (I&F) point neurons. The simulated
neurons and the conductance-based interactions in the model are like
those used by many others before us. What distinguishes this model is
its reliance on cortical architecture to specify the corticocortical
connections, and in the choice of connection strengths that yield
responses that match physiological data. The architecture of the model
derives from cortical anatomy (Fitzpatrick et al., 1985;
Callaway and Wiser, 1996; Callaway, 1998)
and optical imaging experiments. Optical imaging (Bonhoeffer and
Grinvald, 1991; Blasdel,
1992a,b;
Maldonado et al., 1997) reveals orientation hypercolumns
with "pinwheel" patterns of orientation preference in the
superficial layers 2/3 of the cortex; neurons of like-orientation
preference reside along the same radial spoke of a pinwheel, with the
preferred angle sweeping through 180° as the center of the pinwheel
is encircled. We assume that there are pinwheel patterns in layer
4C, parallel to those in layers 2/3. This assumption is
based on the classical concept that there are orientation columns in V1
cortex (Hubel and Wiesel, 1962). The orientation
preference map is assumed to be hard-wired into the cortex during
development, through the orientation preference of each group of LGN
cells that converge onto each cortical cell (Reid and Alonso,
1995).
The model [described in more detail in McLaughlin et al.
(2000)] is of a small patch of cortex (1 mm2, containing four hypercolumns and four
orientation pinwheels) of input layer 4C. It is a
conductance-based model that consists of a two-dimensional lattice of
1282 coupled I&F neurons, of which 75% are
excitatory and 25% are inhibitory.
Basic equations of the model. Let
v (v ) be the
membrane potentials of excitatory (inhibitory) neurons. In the model,
they evolve by the coupled system of differential equations,
![<FR><NU>dv<SUP>j</SUP><SUB>P</SUB></NU><DE>dt</DE></FR>=<UP>−</UP>&lgr;v<SUP>j</SUP><SUB>P</SUB>−g<SUP>j</SUP><SUB>PE</SUB>(t)[v<SUP>j</SUP><SUB>P</SUB>−V<SUB>E</SUB>]−g<SUP>j</SUP><SUB>PI</SUB>(t)[v<SUP>j</SUP><SUB>P</SUB>−V<SUB>I</SUB>],](/content/vol21/issue14/fulltext/5203/img003.gif)
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(1)
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where P = E, I and the superscript j = (j1, j2) indexes the
spatial location of the neuron within the cortical layer. We specified
the cellular biophysical parameters, using commonly accepted values:
the capacitance C = 10 6 F
cm2, the leakage conductance
gR = 50 × 106
 1 cm2, the leakage reversal
potential VR =  70 mV, the excitatory
reversal potential VE = 0 mV, and the
inhibitory reversal potential VI = 80 mV.
We took the spiking threshold as 55 mV and the reset potential to be
equal to VR. The membrane potential and reversal potentials were normalized to set the spiking threshold to unity and
the reset potential (and thus VR) to
zero, so that VE = 14/3, VI = 2/3, and generally 2/3  v , v 1. The
capacitance does not appear in Equation 1 because all conductances were
redefined to have units of sec1 by dividing
through by C. This was done to emphasize the time scales
inherent in the conductances; For instance the leakage time-scale is
 1 = 20 msec. True conductances are found by
multiplication by C.
Conductances. The time-dependent conductances arise from the
input forcing (through the LGN) and from noise to the layer, as well as
from the cortical network activity of the excitatory and inhibitory
populations. They have the form:
with similar expressions for
g and
g , and where
FPE(t) = g (t) + f (t), P = E, I. Here t (T) denotes the time of the lth spike of the kth
excitatory (inhibitory) neuron.
The conductances f (t) are
stochastic. Unless stated otherwise, their means and SDs were taken as
f = f = 6 ± 6 sec1,
f = f = 85 ± 35 sec1. These conductances have
an exponentially decaying autocorrelation function with time constant 4 msec. The constant background g0 of the LGN
drive glgn is taken as 35 sec1. The kernels (a, b,
..)k represent the spatial coupling between neurons.
Only local cortical interactions (i.e. on scales <500 µm) are
included in the model, and these are assumed to be isotropic (Fitzpatrick et al., 1985; Lund, 1987;
Callaway and Wiser, 1996; Callaway,
1998), with Gaussian profiles for the kernels (a, b, ..)k. Based on the same anatomical studies, we
estimate that the spatial length scale of excitation exceeds that of
inhibition and that excitatory radii are of order 200 µm and
inhibitory radii of order 100 µm.
The cortical temporal kernels G (t) model the
time course of synaptic conductance changes in response to arriving
spikes from the other neurons. They are of the form:
where H(t) is the unit step function. The time
constants are based on experimental observations (Koch,
1999; Azouz et al., 1997) (A. Reyes, personal
communication). The time constant for excitation
( E = 0.6 msec; time to peak is 3 msec)
is shorter than that for inhibition (I = 1.0 msec; time to peak is 5 msec). In addition, based on recent
experimental findings (Gibson et al., 1999), we add a
second, longer time-course of inhibition (~30 msec in duration).
The behavior of the computational model depends on the choice of the
corticocortical synaptic coupling coefficients:
SEE, SEI,
SIE, SII. All cortical kernels have
been normalized to unit area. Hence, the coupling coefficients
represent the strength of interaction and are treated as adjustable
parameters in the model. In the numerical experiments reported here,
the strength matrix (SEE,
SEI, SIE,
SII) was set to be (0.8, 9.4, 1.5, 9.4). This
matrix means that excitatory neurons excite inhibitory neurons almost
twice as much as they excite other excitatory neurons but that
inhibitory neurons inhibit excitatory neurons and other inhibitory neurons with equal strength. Also, inhibitory neurons have much stronger coupling to all other cortical neurons than do excitatory neurons. We explored many strength matrices in many numerical experiments. If the corticocortical excitation was too strong, oscillations resulted. If the corticocortical inhibition was too weak,
the responses of the cells were nonlinear and not selective enough. If
inhibition was too strong, the response of the network became too
small. The matrix given here generated simple cells that had the
orientation selectivity, and the magnitude and dynamics of response,
seen in physiological experiments (McLaughlin et al.,
2000). This seems contrary to anatomical studies that show V1
cortex has a preponderance of excitatory synapses (Beaulieu et
al., 1992). However, the biological cortex is filled with
orientation-selective cells that are not only simple, but also complex,
as well as cells whose responses lie between these classifications. It
seems likely that once the sources of this diversity are understood and
properly accounted for, the constraints on coupling strengths to
produce simple cells will be different and the role of corticocortical excitation will be elucidated.
Contrast reversal stimuli. Let I( , t) be
the space- and time-dependent intensity of the visual stimulus. A
"contrast reversal" stimulus is given by:
![I(<A><AC>x</AC><AC>&cjs1164;</AC></A>, t)=I<SUB>0</SUB>[1+&egr; <UP>sin</UP>(&ohgr;t)<UP>cos</UP>(<A><AC>k</AC><AC>&cjs1164;</AC></A>·<A><AC>x</AC><AC>&cjs1164;</AC></A>−&phgr;)],](/content/vol21/issue14/fulltext/5203/img019.gif)
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(2)
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with parameters I0 (intensity),  (contrast),  (temporal frequency), and  (phase). The parameter
  k(cos  , sin ), where k
denotes the spatial frequency and the orientation of the grating
pattern. In the computational experiments, we used k = 3 cycles/°.
LGN response to contrast reversal stimuli. The total input
into the jth cortical neuron arrives from N (=17)
LGN cells:
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(3)
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Here {R}+ = R if R > 0; {R}+ = 0 if R 0;
g represents the maintained (background) activity of the LGN neurons feeding into the jth cortical
neuron, in the absence of visual stimulation. The summed LGN input,
g(t), into a cortical neuron depends
nonlinearly on the visual stimulus I(, t), because
of rectification. There may be some additional nonlinear input from the
magnocellular Y cells (Kaplan and Shapley, 1982). We
have not modeled this group of cells because the percentage of such
cells is small (<25%) and because the cortical mechanisms we propose
will tend to linearize the input of Y cells to cortex also, so no new
principle is involved.
The temporal kernel Glgn(t) and spatial kernel
A() of an LGN cell are chosen to agree with
experimental measurements (Benardete and Kaplan, 1999)
(R. Shapley and R. C. Reid, unpublished observations). Their
functional forms are:
where 0 = 3 msec, 1 = 5 msec,  a = 0.066°, b = 0.093°, a = 1, and b = 0.74
where + represents an "on-center," and an "off-center" LGN cell. The constant c1 is
determined so that the kernel G(t) integrates to zero, as is
approximately the case for LGN neurons in the magnocellular pathway.
The spatial arrangement of LGN cell receptive field centers,
 , is as segregated on-off
subregions (Reid and Alonso, 1995)here a center
subregion of like-polarity cells with twin flanks of opposite polarity. This segregation confers an orientation preference on the input to each
cortical cell, and this preference is laid out in pinwheel patterns.
Additionally, the center of the receptive field of each cortical cell
(created through the aggregate LGN input) is randomized. This was done
to account for diversity in the location of this receptive field center
and random variations in the spatial symmetry of the on-off
subregions. It confers a preferred spatial phase on the LGN input of
each cortical cell. From cortical cell to cell this spatial phase
preference is distributed randomly over a broad range, as has been
found in recent measurements (DeAngelis et al.,
1999).
When the stimulus is contrast reversal, Equation 3 for the input
conductance simplifies (for t  0)
to:
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(4)
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where tS, the temporal phase shift
caused by the retina and LGN, depends only on the choice of
Glgn(t) and on the temporal grating frequency
. An individual term in this sum is simply a rectified sinusoid (if
|p| > g), which
takes on absolute maxima either at 1/4 cycle
(p > 0) or at 3/4 cycle
(p < 0), with respect to
·tS. For convenience, we set
tS = 0.
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RESULTS |
Contrast reversal
Contrast reversal of a grating pattern is an effective stimulus
for classifying cells as simple or complex. In response to contrast
reversal, the summed LGN drive in the model has (for 100% contrast
modulation) the generic spatial phase and time dependence shown in
Figure 2. Notice that (1) for each phase,
the sinusoidal shape is significantly distorted, (2) the absolute
maxima occur at either 1/4 cycle or 3/4 cycle, and (3)
the orthogonal phase case (with the lowest peak heights of response)
possesses two absolute maxima per cycle, resulting in a pronounced
frequency doubling. This is produced by the rectification in Equation 4
and occurs in particular when a line of constant luminance
(I = I0 in Eq. 2) lies down the middle of a
segregated subregion of on-center (or off-center) LGN cells. In this
case, the stimulus modulation is elevated first on one side of this
line, then on the other, during one temporal period of the stimulus.
And so, during one temporal period, the modulation brings to fire first
the on-cells on one side of this line, then brings to fire the on-cells
on the other side, producing a frequency-doubled aggregate LGN
response.
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Figure 2.
In the model, the conductance received by the
jth cortical neuron from the LGN,
g(t): From contrast reversal
gratings (at preferred orientation, 100% contrast, optimal spatial and
temporal frequencies). Responses to nine different grating spatial
phases ( =  + i /8, i = 0, 1, 2, ... , 8) are shown. One thick curve is the
maximal "in-phase" case ( = ); and the second thick curve is
the minimal "orthogonal phase" case ( = + /2). For contrast reversal results, the
time axis has been translated so that t = 0 corresponds
to the initial arrival of excitations in V1 from the LGN. On the
right side of the figure, the different response waveforms
have been separated vertically to correspond to the data format of
Figure 1.
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Unlike their LGN input, cortical neurons in the model behave like
simple cells in the contrast reversal experiment. Figure 3a-c shows data from an
excitatory model neuron located near a pinwheel center. In Figure
3a both the "in-phase" and "orthogonal phase"
membrane potential responses are shown. Here, the spike and reset
mechanism of this neuron has been turned offblockedso that the
waveform of stimulus-modulated potential,
vb, can be seen more easily and compared
with the experimental data (where spikes have been filtered). Thus, the
averaged waveforms of the membrane potentials in Figure 3a
should be compared with those shown in Figure 1C. There is a
good degree of similarity. Extracellular spike counts for this same
neuron (spike and reset now on) are displayed in Figure 3, b
and c, as cycle averaged histograms, and these are
comparable to the simple cell data in Figure 1A.
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Figure 3.
Responses in the model to in-phase and orthogonal
phase, 4 Hz contrast reversal gratings (at 100% contrast, preferred
angle and optimal spatial frequency), for an excitatory neuron near a
pinwheel center. The left column (a-c) shows
responses for a representative fully coupled neuron, and the middle
column (d-f) when this neuron is uncoupled from the
network. The right column (g-i) shows responses for a
feedforward uncoupled neuron, for which the background mean and noise,
and the LGN drive, have been adjusted downward to give spike rates in a
normal range. The first row (a, d, g) shows cycle-averaged
membrane potentials, with spike and reset blocked. Cycle-averaged spike
histograms (when spikes are not blocked) are shown below the membrane
potentials [in-phase (b, e, h) and orthogonal phase
(c, f, i)]. The cycle averages were computed over 24 cycles
of the stimulus. Dashed horizontal lines are the background
responses.
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Figure 3a-c shows that the computational model captures the
linearity seen experimentally in simple cells. The in-phase response is
predominantly at the fundamental driving frequency. The spike rate is
not modulated at the second harmonic when the stimulus is at the
orthogonal phase, and the membrane potential shows very little second
harmonic (or F2) component in its
orthogonal phase response, consistent with experimental measurements.
Removing corticocortical interactions in the model
To study the effect of lateral corticocortical interactions in the
model, we shut them off and observed the consequences. Figure
3d-f shows the results of a simulation with all network interactions shut off but with the LGN and noise terms the same as in
the full network simulation shown in Figure 3a-c. For the orthogonal phase case, notice the large amplitude of the second harmonic in both the spike rate response and the membrane potential. This second harmonic response is inherited from the LGN input (as seen
in Fig. 2 in the orthogonal phase LGN stimuli).
The in-phase response of the uncoupled neuron in Figure 3d
has a very large peak spike rate relative to that of the fully coupled
network. This result indicates the presence of very strong corticocortical inhibition in the fully coupled network.
The responses of an uncoupled model neuron are much larger than seen in
the living cortex, because of the removal of strong inhibition in the
model. Another approach to cortical modeling is to choose different
input and internal noise parameters for the uncoupled model neurons to
fit the background and peak firing rates of the real cortex. We did
this and investigated the responses of what we called a
"feedforward" neuron with much weaker LGN drive
(glgn) and stochastic background
(fEE, fEI) than in the full model. In this case the LGN drive and the means and SDs of the
noise were adjusted downward, as follows: now g0 = 10 sec1, f = f = 6 ± 6 sec1, and
f = f = 45 ± 25
sec1. The results of the simulation for the
feedforward neuron are shown in Figure 3g-i. Compared with
both the responses of the feedforward and uncoupled neurons, the
membrane potential of the fully coupled neuron has a much smaller
F2 component, because of corticocortical interactions.
Mechanisms of linearization
To understand the mechanisms by which the model cortex produces
simple cells, we return to the governing equation for the jth cortical excitatory cell, and write it as:
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(5)
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where
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(6)
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(7)
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Here g is the total conductance and
is the inverse of an effective "integration time" for this neuron.
We call I the difference current, because
it is the difference of currents generated by the excitatory and
inhibitory conductances. As the membrane potential
vj was normalized by the difference
between resting and threshold potential, it is dimensionless.
Furthermore, having scaled by the fixed membrane capacitance, both the
total conductance gT and difference current
ID have units of sec1. We
now study these two quantities.
The total conductance
The total conductance gT, at in- and
orthogonal phases, is shown in Figure 4,
a and b, as a function of time within the cycle of contrast reversal. This figure is from data for the fully coupled model neuron of Figure 3a-c. There are two key points to
note: (1) the conductance gT is large, exceeding
400 sec1 under stimulation. The model is working
in this high conductance regime to achieve known properties of the
biological visual cortex: stability, high responsiveness, and
relatively high stimulus selectivity (McLaughlin et al.,
2000). Indeed, large conductances have been measured under
visual stimulation in the cat visual cortex (Borg-Graham et al.,
1998; Hirsch et al., 1998). One can calculate
from Equation 5 that the high conductance implies that, when spikes are
blocked, the membrane potential v is well
approximated by:
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(8)
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We find numerically that this equality holds in very good
approximation for the cycle averaged quantities, that is,
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(9)
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where  (t) = N1
  f(t + 2n/). (Henceforth we
drop the overbar, and assume it, unless stated otherwise.) A conclusion
from this analysis is that one can understand the behavior of the
membrane potential by studying the dependence of
ID and gT on the visual
stimuli. And (2), as can be seen in Figure 4, the conductance
gT has a waveform in response to contrast
reversal that is highly distorted and, in particular, contains a large
F2 component at the orthogonal phase.
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Figure 4.
The total conductance for the in-phase stimulus
(a), the orthogonal phase stimulus (b), and the
blank stimulus (c). The cycle averaged conductance
 T (averaged over 24 cycles) is shown as a
thick gray curve, superimposed on the (less smooth)
instantaneous conductance gT over one cycle (4 Hz contrast reversal stimulus). These are simulations for the near
neuron in the fully-coupled network in Figure 3.
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Of course the origin of these features lies in the constituent
conductances that make up gT. These conductances
also comprise ID, which we now consider.
The difference current
Plots of the difference current ID,
together with its constituent currents from Equation 7, are shown in
Figure 5. ID and its components are shown at both the in-phase (a) and
orthogonal phase (b) for contrast reversal stimulation. The
current contributed by the LGN drive is graphed in green, the
corticocortical excitatory current is graphed in red, and the
corticocortical inhibitory current is graphed in blue. From Equation 7
the difference current ID (graphed in black) is
simply the sum of these three currents.
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Figure 5.
Cycle-averaged currents (averaged over 24 cycles) comprising  D, for an
excitatory neuron in the coupled network, in response to in-phase
(a) and orthogonal phase (b) stimulus. Plotted
are the LGN (green), cortical excitatory
(red) and cortical inhibitory (blue), and grand
total (black) currents. Also plotted are the mean values of
the excitatory (red dotted) and inhibitory noises
(blue dotted), and total background current (black
dotted line).
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Again, there are key points to note: (1) Figure 5b shows
that corticocortical currents, whether at in- or orthogonal phase, have
primarily second harmonic distortions, with inhibitory corticocortical currents significantly larger than excitatory. (2) By comparing Figure
5, a and b, it is clear that the corticocortical
currents are mostly insensitive to the spatial phase of the grating.
And (3), it is consequently only the LGN drive that provides the large modulation at the first harmonic in both gT and
ID for the in-phase stimulus.
It is interesting to compare the components of the conductance for the
contrast reversal experiment at orthogonal spatial phase. This is
displayed in Figure 6. There it can be
seen that the corticocortical inhibitory conductance of the model is
the predominant component. This figure also shows that the inhibitory conductance is stronger for neurons far from the pinwheel
singularities.
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Figure 6.
Constituent conductances for the near and far
neurons illustrated in this paper, at the orthogonal spatial phase of
contrast reversal. The LGN input excitation is the lowest thin
curve. Near corticocortical excitation and inhibition are
remaining thin curves; far curves are thick. Note
the large inhibitory components of the conductances and that the far
neuron has significantly larger F2 modulation in its inhibitory
conductance than does the near neuron.
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What underlies the absence of modulation of the membrane potential
vb at orthogonal phase? Recall that at
orthogonal phase, the total conductance gT is
modulated at F2. This modulation is in
phase with the F2 modulation of the
difference current ID, as is evident in
Figure 5b. Then vb  ID/gT is approximately constant in
time because ID and gT
are approximately proportional. And the proportionality of
ID and gT is partly a
consequence of the fact that corticocortical inhibition is the
predominant term in gT.
Thus, simple-cell intracellular responses occur in the model because
its corticocortical conductances have significant
F2 modulations that cancel the
F2 coming from the input. But why are such
modulations present in the cortex? The reason is that the
jth neuron receives spikes from many other cortical neurons, each of which is responding individually in a manner sensitive to the
phase of its own LGN drive. This individual phase dependence arises
because each of these cortical neurons is driven by LGN excitation, and
each summed LGN drive will have its own temporal waveform that will be
one of those sketched in Figure 2. The excitation of each LGN cell is
maximal at 1/4 or 3/4 temporal cycle. Because the corticocortical input
to the jth neuron is an average over many such
phase-sensitive responses, some of which peak at 1/4, some at 3/4
cycle, this results in a total corticocortical conductance, which peaks
at both the 1/4 and 3/4 temporal cycle, and consequently has
significant F2 content. In summary, the
corticocortical modulations have large, phase-insensitive,
F2 modulations because the isotropic cortical
architecture of the model allows an averaging over the activity of many
cortical neurons, and thus, indirectly averages over the many preferred
spatial phases of the LGN input [as suggested by the results in
DeAngelis et al. (1999)], which peak at 1/4 and 3/4
cycle. This "phase averaging" by the network is similar to that
used in a model for complex cells (Chance et al., 1999). It should be emphasized that although we have invoked phase averaging as the mechanism for producing frequency-doubled cortical input, this
state of cortical activity arises from the dynamics of the system in a
way consistent with its architecture. It is not imposed a priori.
Drifting grating stimulus
So far, we have analyzed the response of the model to contrast
reversal. We also studied the responses of the model to other stimuli,
in particular drifting sine gratings that have also often been used to
characterize simple and complex cells (Movshon et al.,
1978; Spitzer and Hochstein, 1985; De
Valois et al., 1982). As shown in Figure
7, the spike rate and (blocked) membrane
potential of a model neuron are modulated approximately sinusoidally
when a sine grating at the optimal orientation is drifted over the receptive field of the neuron, as seen in real neurons. As also shown
in Figure 7, the total conductance gT is also
modulated sinusoidally at the drift rate of the grating but has a large DC offset. The modulation arises primarily from the modulation in the
excitatory conductance gE, while the
offset arises from the inhibitory conductance
gI, which is essentially unmodulated but
elevated over its background value of 180 sec1.
The corticocortical contribution to gE,
like gI, is also mostly unmodulated (data
not shown). This is consistent with the cortical excitatory and
inhibitory conductances of the model in response to contrast reversal,
shown in Figure 6. The same phase averaging that produces spatial phase
insensitivity of the inhibitory conductance, and current, to contrast
reversal also causes no modulation (but an elevated average level)
during stimulation with a drifting grating. This is an important
consideration in comparing our model with "push-pull" models, as
discussed below.
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Figure 7.
Responses to drifting gratings. This stimulus was
a drifting sinusoidal grating at optimal spatial frequency and
orientation, at a drift rate of 8 Hz and 100% contrast. From
left to right, the panels shown are
cycle-averaged spike rate, blocked membrane potential, total
conductance, excitatory conductance, and inhibitory conductance.
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Two predictions of the model
There are two predictions that are revealing about the mode of
action of the model. The first prediction concerns a greater nonlinear
temporal modulation expected in corticocortical conductances for
neurons farther from pinwheel centers. This is a consequence of the
lateral coupling having length scales well below the width of the
orientation hypercolumn. The second is a prediction that the
F2 component in the membrane potential of an
individual neuron should get larger when the membrane is hyperpolarized
with current.
The first prediction is illustrated in Figure
8. These are data from the same network
simulation as in Figure 3 but from a different model neuron, one
located far from a pinwheel center. These responses should be compared
with Figure 3a-c for a neuron near the pinwheel. There is
an overall resemblance in the spike rate waveforms between the two
neurons in the model, but there is a significant difference in
intracellular response: the neuron in Figure 8 has an evident
F2 component in its membrane potential response
at orthogonal phase. The F2 component in this
cortical cell is opposite in sign from the LGN drive. It is caused by
inhibition, which is so strong in this neuron to this stimulus that it
overrides the sources of excitation. The model predicts there should be strong inhibition in far neurons, as is evident in Figure 6. In the
contrast reversal experiment, only one spatial grating at one
orientation is exciting the cortex. This means only one spoke of the
pinwheel is being driven hard, most especially by the LGN input near
the maximal in-phase case. (Along the spoke at orthogonal preferred
angle, the LGN input is relatively phase-insensitive and of smaller
amplitude.) Each neuron in the model receives its inhibitory input
predominantly from cortical neurons within a 100 µm radius. Far from
the pinwheel center, near the spoke that is excited maximally by the
orientation of the grating stimulus, this disk of inhibitory input
covers many excited neurons of similar orientation preference. But near
the pinwheel center, where many preferred orientations are represented,
the disk then covers a smaller fraction of neurons well excited by this
one orientation, and so the summed inhibition is relatively weaker.
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Figure 8.
Responses to in-phase and orthogonal phase, 4 Hz
contrast reversal gratings (at 100% contrast, preferred angle and
optimal spatial frequency), for a neuron far from a pinwheel center.
The format for this figure is the same as for Figure 3. The
cycle-averaged membrane potential is shown, with spike and reset
blocked, as in Figure 3, and below it are the cycle-averaged spike
rates for in-phase and orthogonal phase stimuli, respectively. The
cycle averages were computed for 24 cycles.
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A second prediction concerns changes in the response of a simple cell
after injection of a constant transmembrane current. The
F2 component of the membrane potential of a
neuron depends on extrinsic current that polarizes the membrane. Figure
9 shows the membrane potential of the
model neuron that was used in Figure 3, at the orthogonal phase, for
five different choices of a constant holding current
Ihold. As this current becomes more
negative, there is an increasing temporal modulation at
F2 that is approximately linear with
Ihold. Indeed, the subthreshold potential is well described by the large conductance approximation v ID/gT + Ihold/gT. For this neuron,
ID/gT is approximately
constant (Fig. 3a), whereas
Ihold/gT contributes an
F2 term from gT that
grows as Ihold becomes more negative.
Similarly, other predictions of the model could be tested
experimentally: for values of the holding current for which the
potential is subthreshold, one could use linear regression to directly
estimate the time-dependent total conductance gT
(Borg-Graham et al., 1998; Hirsch et al., 1998), as well as its excitatory and inhibitory components
(assuming Eqs. 6 and 7). In this way the presence and nature of
F2 components could be checked experimentally.
In particular, by using visual stimuli with different spatial phases,
the phase (in)sensitivity of corticocortical inhibition could be
examined.
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Figure 9.
Membrane potential at different holding currents
for a neuron near a pinwheel center. The five modulated waveforms are
membrane potential responses to sinusoidal contrast reversal in the
orthogonal phase condition. These are averaged response waveforms to 24 cycles of stimulus contrast reversal. The top curve is the
potential when there is no extrinsic current. The curves below that
are, from top to bottom, measured with a constant
hyperpolarizing current of 100, 200, 400, 600
sec1. The dashed horizontal lines for
each curve are average values of the membrane potential when the
uniform background is shown under the same conditions.
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DISCUSSION |
The importance of simple cells
Our work establishes that the linear behavior of simple cells
arises as a consequence of network activity. However, this leads one to
ask why does the biological cortical network seem to have this
linearity as a goal? The generally accepted view is that, for visual
perception, cortical cells must resolve and represent key spatial
properties such as surface brightness and color and the perceptual
spatial organization of a scene that is the basis of form. The
existence of simple cells that respond selectively to spatial phase and
monotonically to signed contrast are required for the representation of
such surface properties and perceptual organization. The large body of
work on spatial pattern vision requires linear spatiotemporal neural
mechanisms to explain how patterns are detected separately and in
combination (for review, see Graham, 1989;
Wandell, 1995). Also, theories of color vision implicitly assume the existence of simple cells whenever they postulate
the necessity of numerical computations of (signed) edge contrast (for
review, see Wandell, 1995). Scene organization requires
computation of depth order that in turn depends on computation of
stereoscopic depth and also of pictorial occlusion. Both stereo (Anzai et al., 1999a,b) and occlusion (Anderson, 1997)
computations require cortical representation of signed edge contrast.
Furthermore, the perception of salient contours also has been shown to
be sensitive to spatial phase and thus contrast sign (Field et
al., 2000). Such neural computations would seem to require the
linearity that only simple cells provide. These considerations lead to
the conclusion that visual perception needs simple cells for basic functions.
Simple and complex cells were first discovered in cat visual cortex
(Hubel and Wiesel, 1962), and their existence confirmed subsequently in macaque V1(Hubel and Wiesel,
1968;De Valois et al., 1982). Simple cells have
been found in the primary visual cortex of many other species of
mammals: owl monkeys (O'Keefe et al., 1998), baboons
(Kennedy et al., 1985), tree shrews (Kaufmann and
Somjen, 1979), rats (Burne et al., 1984;
Girman et al., 1999), mice (Drager,
1975), rabbits (Glanzman, 1983), and sheep
(Kennedy et al., 1983). Their ubiquity in the animal
kingdom may be an indicator of their importance.
Neurons in other sensory cortices have linear signal processing
properties that resemble those seen in simple cells of the visual
cortex. Some neurons in the primary auditory cortex have been
characterized as linear transducers of auditory patterns (Kowalski et al., 1996). Similar characterizations in
the primary somatosensory cortex have also identified linearly summing
neurons with receptive fields similar in many ways to visual simple
cells (DiCarlo and Johnson, 2000). The same processes
that give rise to the creation of simple cells in visual cortex will be
important for understanding how they may be produced elsewhere by the
cortical circuitry.
Mechanisms for the production of simple cells
Our computational model of layer 4C in macaque V1 is
a nonlinear network of I&F neurons, driven by LGN input, which is
itself nonlinear. Yet, the neurons of the model respond in an
approximately linear manner, which is characteristic of simple cells,
including (1) a sensitive dependence of the responses of the neurons on spatial phase or position and (2) very little presence of nonlinear distortion (such as second temporal harmonics) in the responses of the
neurons. Stimulation by contrast reversal of a standing grating
pattern, with the phase of the grating orthogonal to the preferred
phase of the simple cell, constitutes a most stringent test of this
linearity. Although the temporal profile of the total LGN excitatory
drive to each neuron in the network contains significant second
harmonic content, the membrane potential of the output of each cell (as
measured intracellularly) contains little second harmonic distortion.
Given the nonlinearity of the full network and of the LGN drive, the
linear behavior of simple cells is not a simple consequence of
feedforward convergence. The active presence of the corticocortical interactions in the network significantly reduces the second harmonic content of the membrane potential of the simple cell, when compared with its temporal waveform in the absence of network interactions. The
work reported here has identified the two properties of the network
model that are responsible for this linearization of the response: (1)
phase averaging of the individual corticocortical inputs to the cell,
which collectively produce a frequency-doubled temporal component (as
used in Chance et al., 1999); (2) strong inhibition, so
that corticocortical inhibitory input tends to cancel the
frequency-doubled excitatory input from the LGN. These two properties
produce the linear responses of simple cells within the model cortex,
and they are likely to be the key network mechanisms that cause the
linear behavior of simple cells in the biological cortex. Both phase
averaging and strong inhibition are caused by the nature of
corticocortical connections in the model.
This model of simple cells as resulting from the cancellation of
nonlinear excitation by strong nonlinear inhibition can account for
many experiments. It explains why there is a large increase in
conductance in a simple cell both at the onset and offset of a flashing
bar in the receptive field of a simple cell (Borg-Graham et al.,
1996, 1998). As
the measurements of Borg-Graham et al. (1996, 1998)
indicate, the conductance increase is dominated by the inhibitory
conductance, as in our model, and corticocortical inhibition is on-off
as in the model. The model also provides a convincing explanation why
pharmacological weakening of inhibition would make simple cells appear
to be complex (Sillito, 1975; Frégnac and
Shulz, 1999; Murthy and Humphrey, 1999), because
weakening the inhibition should prevent it from cancelling the
nonlinear component of excitation. The data of Murthy and
Humphrey (1999) are particularly relevant. They stimulate their
simple cells with grating contrast reversal as in our modeling and
observe marked frequency doubling in spike rates of simple cells when
bicuculline is infused.
Our model is very different from previous attempts to explain cortical
linearization in terms of a "push-pull" model (Palmer and
Davis, 1981; Tolhurst and Dean, 1990) that
requires direct or indirect phase-sensitive inhibition from the LGN.
Direct inhibition from LGN to cortical neurons is ruled out by anatomy;
LGN-to-cortex inhibitory synapses do not exist. One might attempt to
preserve the push-pull concept by postulating that disynaptic
inhibition from inhibitory neurons in the cortex could provide
phase-sensitive inhibition (as instantiated in the model of
Troyer et al., 1998). But then one would have to explain
how phase-sensitive inhibition is consistent with the anatomy of
inhibitory interneurons in the cortex: such neurons receive many
synaptic connections from other cortical cells (Lund,
1987), and there is apparently indiscriminate arborization of
axonal branching within the cortex (Fitzpatrick et al.,
1985). Nevertheless, previous physiological studies have been
interpreted to mean that there is phase-sensitive or push-pull inhibition somehow generated intracortically (Hirsch et al.,
1998; Anderson et al., 2000). However, much of
this evidence has been indirect.
There is recent evidence on this point (Anderson et al.,
2000). From measurements of simple cell responses to drifting
gratings, the authors infer that the temporal modulation of synaptic
inhibition in opposition to the modulation of synaptic excitation is
indicative of push-pull interactions between inhibition and
excitation. However, scrutiny of the measurements in Anderson et
al. (2000) indicates that there usually is a large
phase-insensitive component of the inhibitory conductance, consistent
with the phase-insensitive inhibition that is observed in the response
of our model to drifting gratings (Fig. 7). Furthermore, the authors
saw modulation of the measured corticocortical inhibition primarily
when the cell was above threshold and firing. It is possible that their
measurements of synaptic conductances were made inaccurate by the
spiking. Other direct intracellular measurements by Borg-Graham
et al. (1998) indicate that inhibition in simple cells is more
often spatial phase-insensitive than phase-sensitive (or push-pull), as Borg-Graham et al. (1998) indeed noted. Our model
produces unmodulated cortical inhibition in response to drifting
gratings because neurons are excited by inhibitory neighbors of
different spatial phase preference. In this way our model differs from
that of Troyer et al. (1998), whose couplings are
explicitly phase-specific. Perhaps the real cortex has inhibitory
neurons that are neither wholly p |
TOP
ABSTRACT
INTRODUCTION
![G<SUB>lgn</SUB>(t)=c<SUB>0</SUB>t<SUP>5</SUP>[<UP>exp</UP>(<UP>−</UP>t/&tgr;<SUB>0</SUB>)−c<SUB>1</SUB><UP>exp</UP>(<UP>−</UP>t/&tgr;<SUB>1</SUB>)],](/content/vol21/issue14/fulltext/5203/img022.gif)
![A(<A><AC>y</AC><AC>&cjs1164;</AC></A>)=<UP>±</UP><FENCE><FR><NU>a</NU><DE>&pgr;&sfgr;<SUP>2</SUP><SUB>a</SUB></DE></FR> <UP>exp</UP>[<UP>−</UP>‖<A><AC>y</AC><AC>&cjs1164;</AC></A>/&sfgr;<SUB>a</SUB>‖<SUP>2</SUP>]−<FR><NU>b</NU><DE>&pgr;&sfgr;<SUP>2</SUP><SUB>b</SUB></DE></FR> <UP>exp</UP>[<UP>−</UP>‖<A><AC>y</AC><AC>&cjs1164;</AC></A>/&sfgr;<SUB>b</SUB>‖<SUP>2</SUP>]</FENCE>,](/content/vol21/issue14/fulltext/5203/img023.gif)
