Cerebral Cortex Advance Access originally published online on December 22, 2005
Cerebral Cortex 2006 16(11):1531-1545; doi:10.1093/cercor/bhj090
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Extraclassical Receptive Field Phenomena and Short-Range Connectivity in V1
Laboratory for Intelligent Imaging and Neural Computing, Department of Biomedical Engineering, Columbia University, New York, NY 10027, USA
Address correspondence to Jim Wielaard or Paul Sajda, Laboratory for Intelligent Imaging and Neural Computing, Department of Biomedical Engineering, Columbia University, New York, NY 10027, USA. Email addresses: djw21{at}columbia.edu (Jim Wielaard), ps629{at}columbia.edu (Paul Sajda).
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Abstract |
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Extraclassical receptive field phenomena in V1 are commonly attributed to long-range lateral connections and/or extrastriate feedback. We address 2 such phenomena: surround suppression and receptive field expansion at low contrast. We present rigorous computational support for the hypothesis that the phenomena largely result from local short-range (<0.5 mm) cortical connections and lateral geniculate nucleus input. The neural mechanisms of surround suppression in our simulations operate via (A) enhancement of inhibition, (B) reduction of excitation, or (C) action of both simultaneously. Mechanisms (B) and (C) are substantially more prevalent than (A). We observe, on average, a growth in the spatial summation extent of excitatory and inhibitory synaptic inputs for low-contrast stimuli. However, we find this is neither sufficient nor necessary to explain receptive field expansion at low contrast, which usually involves additional changes in the relative gain of these inputs.
Key Words: model • receptive field • simulation • spatial summation • surround suppression • visual cortex
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Introduction |
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In mammals, the very first stage of cortical visual processing takes place in the striate cortex (area V1). Already at this level, spatial summation of visual input is considerably complex. This is manifest from the fact that cells in V1 display a set of phenomena that are conventionally referred to as "extraclassical receptive field phenomena." In this paper, we focus on 2 such phenomena, surround suppression (suppression for increasing stimulus size, "size tuning") and the increase of the receptive field size at low contrast. These 2 phenomena are observed throughout the striate cortex, including all cell types in all layers and at all eccentricities (Schiller and others 1976
; Dow and others 1981; Silito and others 1995; Kapadia and others 1999; Sceniak and others 1999, 2001; Anderson and others 2001; Cavanaugh and others 2002; Ozeki and others 2004). The suppression seen is substantial, with cells in macaque V1 showing, on average, a 30–40% reduction in their firing rates at large stimulus sizes (Cavanaugh and others 2002). Similarly significant is the receptive field expansion at low contrast. Typical is a doubling in receptive field size when stimulus contrast is reduced by a factor of 2–3 (Sceniak and others 1999). Apparently, at low contrast, neurons in V1 sacrifice spatial sensitivity in return for a gain in contrast sensitivity (Sceniak and others 1999). Neural mechanisms responsible for these 2 phenomena are at present poorly understood. Understanding these mechanisms is potentially important for developing a theoretical model of early signal integration and neural encoding of visual features in V1.
Popular working hypotheses are that these 2 extraclassical receptive field phenomena are a product of long-range horizontal connections (DeAngelis and others 1994; Dragoi and Sur 2000; Hupé and others 2001; Stettler and others 2002) and/or feedback from extrastriate areas (Sceniak and others 2001; Angelucci and others 2002; Cavanaugh and others 2002; Bair and others 2003). Arguments in support of these hypotheses are based on the observed surround sizes and the cortical magnification factor and claim that short-range (<0.5 mm) and even long-range horizontal (<5 mm) connections in V1 would not have sufficient spatial extent to be responsible for surround suppression or receptive field expansion (Sceniak and others 2001; Cavanaugh and others 2002). Further, support along this line was presented using anterograde and retrograde tracer injections (Angelucci and others 2002) and timing experiments (Bair and others 2003). So far, however, these hypotheses are based on indirect experimental observations and also lack rigorous computational support.
The hypothesis that the phenomena result from local short-range (<0.5 mm) cortical connections and lateral geniculate nucleus (LGN) input is largely ignored or dismissed. However, support for it can be found in the experimental data. For instance, surround suppression and receptive field expansion at low contrast are significant throughout V1 (Sceniak and others 1999, 2001), including in layers that do not receive extrastriate feedback and do not have long-range horizontal connections. Both phenomena have been observed in the LGN and are likely to be partially inherited by V1 via the feed-forward input from the LGN (Solomon and others 2002; Ozeki and others 2004). Finally, there is experimental evidence for contextual modulations mediated by local short-range connections in cats (Das and Gilbert 1999).
In this paper, we suggest neural mechanisms for surround suppression and receptive field expansion in layers 4C
and 4Cß of macaque V1. We show that local short-range cortical connections and LGN input are, in principle, sufficient to explain a major part of the phenomena. We do this by means of a large-scale neural network model that is constructed, as much as possible, from established experimental data. We suggest neural mechanisms for the phenomena by analyzing the synaptic inputs that generate them in the model. An illustration of the model's architecture is given in Figure 1. A brief summary of the model is given in Methods.
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Methods |
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The Model
Our model consists of 8 ocular dominance columns and 64 orientation hypercolumns (i.e., pinwheels), representing a 16-mm2 area of a macaque V1 input layer 4C or 4Cß. The model contains approximately 65 000 cortical cells and the corresponding appropriate number of LGN cells. Our cortical cells are modeled as conductance-based integrate-and-fire point neurons, 75% are excitatory cells and 25% are inhibitory cells. Our LGN cells are half-wave rectified spatiotemporal linear filters. The model is constructed with isotropic short-range cortical connections (<500 µm), realistic LGN receptive field sizes and densities, realistic sizes of LGN axons in V1, and cortical magnification factors and receptive field scatter that are in agreement with experimental observations. We will only give a very brief description of the model here; it is explained in detail in Supplementary Materials. Some background information can also be found in previous works (McLaughlin and others 2000; Wielaard and others 2001) by one of the authors (J.W.).
Dynamic variables of a cortical model cell i are its membrane potential vi(t) and its spike train 
i(t) = 
k
(t–ti,k), where t is the time and ti,k is its kth spike time. Membrane potential and spike train of each cell obey a set of N equations of the form
 | (1) |
The quantities gL,i, gE,i(t,[],
E), and gI,i(t,[],I) are the leakage (rest), excitatory, and inhibitory conductances of neuron i. They are defined by interactions with the other cells in the network, external noise E(I), and, in the case of gE,i possibly by LGN input. The notation []E(I) stands for the spike trains of all excitatory (inhibitory) cells connected to cell i. Both the excitatory and inhibitory populations consist of 2 subpopulations 
k(E) and k(I), k = 0, 1, a population that receives LGN input (k = 1) and one that does not (k = 0). In the model presented here, 30% of both the excitatory and inhibitory cell populations receive LGN input. We assume noise, cortical interactions, and LGN input act additively in contributing to the total conductance of a cell,
 | (2) |
for 
The terms 
are the contributions from the cortical excitatory (µ = E) and inhibitory (µ = I) neurons and include only isotropic connections,
 | (3) |
k'(µ'). Here 
is the spatial position (in cortex) of neuron i, the functions Gµ,j(
) describe the synaptic dynamics of cortical synapses, and the functions 
describe the cortical spatial couplings (cortical connections). The length scale of excitatory and inhibitory connections is about 200 and 100 µm, respectively.
An important class of parameters is the geometric parameters, which define and relate the model's geometry in visual space and cortical space. Geometric properties are different for the 2 input layers 4C and 4Cß and for the 2 eccentricities. As said, the 2 extraclassical phenomena we seek to explain are observed to be largely insensitive to those differences (Kapadia and others 1999; Sceniak and others 1999, 2001; Cavanaugh and others 2002). In order to verify that our explanations are consistent with this observation, we have performed numerical simulations for 4 sets of parameters, corresponding to the 4C and 4Cß layers at parafoveal eccentricities <5° and at eccentricities around 10°. These different model configurations are referred to as M0, M10, P0, and P10 in the text. Reported results are qualitatively similar for all 4 configurations unless otherwise noted.
In agreement with experimental findings (see references in McLaughlin and others 2000), the LGN neurons are modeled as half-wave rectified center–surround linear spatiotemporal filters. A cortical cell, j 1(µ) is connected to a set 
of left eye LGN cells or to a set 
of right eye LGN cells,
 | (4) |
0 and [x]+ = 0 if x 
0, 
and 
are the spatial and temporal LGN kernels, respectively, and 
is the receptive field center of the 
left or right eye LGN cell, which is connected to the jth cortical cell, 
is the visual stimulus. The parameters 
represent the maintained activity of LGN cells, and the parameters 
measure their responsiveness to visual stimuli. Their numerical values are taken to be identical for all LGN cells in the model, 
s–1 and 
cd–1 m2 s–1. The LGN kernels are of the form (Benardete and Kaplan 1999)
 | (5) |
 | (6) |
and 
are the center and surround sizes, respectively, and 
is the integrated surround–center sensitivity. The temporal kernels are normalized in Fourier space, 
For the magno cases (M0, M10), the time constants 1 = 2.5 ms, 2 = 7.5 ms, and c = (1/2)6 so that 
in agreement with experiment (Benardete and Kaplan 1999). For the parvo cases (P0, P10), the time constants 1 = 8 ms, 2 = 9 ms, and c = 0.7(1/2)5. The delay times 
are taken from a uniform distribution between 20 and 30 ms, for all cases. Sizes for center and surround were taken from the experimental data (Hicks and others 1983; Derrington and Lennie 1984; Spear and others 1984; Shapley 1990; Croner and Kaplan 1995) and were 
0.2°, 0.04°, 0.0875° (centers) and 
1.4°, 0.32°, 0.7° (surrounds), for M0, M10, P0, and P10, respectively. The integrated surround–center sensitivity was in all cases 
(Croner and Kaplan 1995). By design, no diversity has been introduced in the center and surround sizes in order to demonstrate the level of diversity resulting purely from the cortical interactions and the connection specificity between LGN cells and cortical cells (i.e., the sets 
see specifications below). Further, no distinction was made between ON-center and OFF-center LGN cells other than the sign reversal of their receptive fields (± sign in eq. 6). The LGN receptive field centers 
were organized on a square lattice with lattice constants 
c/2, c, c/2, and c/2 for M0, M10, P0, and P10, respectively. These lattice spacings and consequent LGN receptive field densities imply LGN cellular magnification factors that are in the range of the experimental data available for macaques (Conolly and van Essen 1984; Malpeli and others 1996). The connection structure between LGN cells and cortical cells, given by the sets is made so as to establish ocular dominance bands and a slight orientation preference, which is organized in pinwheels (Blasdel 1992). It is further constructed under the constraint that the LGN axonal arbor sizes in V1 do not exceed the anatomically established values of 1.2 mm for magno and 0.6 mm for parvo cells (Blasdel and Lund 1983; Freund and others 1989). A sketch of the model is given in Figure 1. Further details are provided in Supplementary Materials.
Some of the geometric differences between the different model configurations can be expressed by the dimensionless parameter 
where v–1 is the cortical magnification factor, c is the LGN receptive field size (center size), and 
is a characteristic length scale for the excitatory cortical connectivity. Substituting numerical values taken from the experimental data, this parameter is 1, 0.57, 0.4, and 0.25 for M0, M10, P0, and P10, respectively. At 30° eccentricity, the experimental data suggest values for this parameter not very different from its values at 10° (
= 0.5 for M30 and = 0.25 for P30).
In the construction of the model, our objective has been to keep the parameters deterministic and uniform as much as possible. This enhances the transparency of the model while at the same time provides insight into what factors may be essential for the considerable diversity observed in the responses of V1 cells. Important parameters which are not subject to cell-specific variability are
- Parameters related to the integrate-and-fire mechanism, such as threshold, reset voltage, and leakage conductance. These are identical for all cells (eq. 1).
- The cortical interaction strengths and connectivity length scales. These are presented by the functions
which are not cell specific but only specific with respect to the 4 cell populations. Note: the functions are also not configuration specific (eq. 3).
- Maintained activity and responsiveness to visual stimulation of LGN cells (eq. 4).
- Receptive field sizes of LGN cells. These are neither cell nor population specific (i.e., where "population" in this case refers to the ON and OFF LGN cell populations) but are only specific with respect to the 4 model configurations, that is, receptive field sizes of all LGN cells are identical for a particular configuration (eq. 6).
- The cortical interaction strengths and connectivity length scales. These are presented by the functions
Important model parameters which are subject to a cell-specific variability are
- The external noisy conductances E,i(t) (excitatory) and I,i(t) (inhibitory) (eq. 2).
- The cortical synaptic dynamics as described by the kernels Gµ,j(
) (eq. 3).
- The LGN temporal kernels
(eq. 4).
- The LGN connectivity to our model cortex as described by
and 
(eq. 4).
- The cortical synaptic dynamics as described by the kernels Gµ,j(
Visual Stimuli and Data Collection
The stimulus used in this paper to analyze surround suppression and contrast-dependent receptive field size is a drifting grating confined to a circular aperture, surrounded by a blank (mean luminance) background. The luminance of the stimulus is given by 
for 
and 
for 
with average luminance I0, contrast 
, temporal frequency 
, spatial wave vector 
and aperture radius rA. The aperture is centered on the receptive field of the cell and varied in size, whereas the other parameters are kept fixed at close to preferred values for the cell. The stimuli are presented monocularly (other eye I = 0). As the aperture size increases, the response of a real V1 cell to such stimuli typically reaches a maximum, after which it settles down to a steady level.
Surround suppression is typically characterized by comparing the neuron's maximum firing rate to its firing rate at large aperture sizes. The aperture size for which the response reaches its maximum (fmax) is sometimes referred to as "the classical receptive field" size (DeAngelis and others 1994; Levitt and Lund 1997; Sceniak and others 1999). We will simply refer to the minimum aperture radius for which the response f(rA) is >95% of its maximum as "the receptive field" size (r). We define the surround size (R) as the minimum aperture radius > r for which the suppression fs(rA) = fmax – f(rA) is >95% of its maximum. We define the asymptotic response f
as the average response beyond R. We define the suppression index SI1 as the relative surround suppression,
 | (7) |
but different from the integrated suppression index used in Sceniak and others (1999) (see below).
The primary data in this paper, that is, responses and conductances as a function of aperture size for monocular stimulation, are obtained with the temporal and spatial frequencies of the grating set equal to the averaged preferred values for each model configuration (M0, M10, P0, and P10). Data used for analysis are from cells that have their preferred orientation equal to the grating angle, their preferred temporal frequency within 2 Hz of the grating frequency, a preferred spatial frequency kp that satisfies 
where k is the grating spatial frequency, a receptive field center that is less than 1/20th of the average receptive field size away from the aperture center, a maximum response at low contrast that is greater than fb + 5 where fb is the mean blank response (in spikes per second), and, finally, a central cortical location confined to the dashed white rectangle in Figure 1. Samples consist of approximately 200 cells, with about equal numbers of simple and complex cells. Each stimulus was presented for 3 s and preceded by a 1 s blank stimulus. The procedure was repeated five times with different initial conditions and noise realizations. Standard errors in cycle-trial average responses and conductances are negligible. The experiments were performed at "high" contrast, = 1, and "low" contrast, = 0.3. Additional details are provided in Supplementary Materials.
Difference-of-Gaussians and Ratio-of-Gaussians Models
In the Difference-of-Gaussians (DOG) model (DeAngelis and others 1994; Sceniak and others 1999, 2001), the response f(rA) is fit to
 | (8) |
E), and "inhibition" (spatial scale I). The integrated suppression index SI2 is defined as
 | (9) |
As is true for SI1, SI2 can be greater than one, indicating surround suppression beyond the background response. The limitations of the DOG model can be made more apparent by noting that, given the validity of the half-wave rectification model equation (13), one "derives" the DOG model by the substitutions 
The Ratio-of-Gaussians (ROG) model (Cavanaugh and others 2002) is defined by
 | (10) |
As for the DOG model, the limitations of the ROG model can be made more apparent by noting that, from equation (12), it may be "derived" from the standard half-wave rectification model. Equation (12) can be rewritten such that the numerator (N) and the denominator (D) represent a half-wave rectified weighted difference of the excitatory and inhibitory conductances, and the total conductance gT, respectively. The ROG model used in Cavanaugh and others (2002) is then obtained by the substitutions 
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Results |
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Classical Response Properties
One of our model's strong accomplishments is that it produces, with the same fixed parameters, a variety of response properties in good agreement with the experimental data. It also displays a great diversity in each of its response properties, quite similar to what is seen experimentally. Cells in the model exhibit in addition to the 2 extraclassical response properties central in this paper many realistic "classical response properties." By this we mean response properties, such as orientation tuning, spatial and temporal frequency tuning, and response modulations for drifting and contrast reversal grating stimuli, obtained for large, homogeneous (grating) stimuli, so that size effects of the stimulus are no longer observable. Our motivation for referring to these properties as "classical" stems from the fact that this is how they are traditionally obtained, that is, with use of large homogeneous stimuli. Classical response properties generally show modest changes when the stimuli are confined to the classical receptive field. These changes, in fact, form an interesting class of extraclassical response properties by themselves, of which surround suppression, as defined in this paper, is just 1 example. The classical receptive field is approximately equivalent to the minimum response field (Hubel and Wiesel 1962; Henry and others 1978; Gilbert 1997) and is precisely defined in Methods. Throughout this paper, we will refer to the classical receptive field simply as the receptive field.
Classical response properties are important when considering extraclassical phenomena. One of the reasons is that extraclassical phenomena are evoked from outside the receptive field but are not known to occur without sufficient stimulation of the receptive field. Extraclassical responses are thus defined relative to responses from within the receptive field, or equivalently, relative to classical responses. For example, consider orientation tuning. A cell's orientation-tuning curve obtained for a large, homogeneous grating (classical response property) will, in general, be modestly different from its tuning curve obtained when the same grating is confined to the receptive field. In general, there will be an overall increase in response and orientation selectivity will be somewhat less for the latter stimulus, whereas the preferred orientation will remain the same. Differences between the cell's orientation tuning for the 2 stimuli are precisely what constitute the extraclassical response properties (phenomena) in this case. It would therefore be less meaningful to present an analysis of extraclassical receptive field phenomena based on a cell population of a model if the basic classical responses for that cell population are nonexistent or show poor agreement with the experimental data.
Another reason for the importance of classical response properties is that responses of cortical cells depend strongly on how the cell's environment is responding. The responses of the cells that make up this environment will, in general, display an enormous diversity to any particular fixed stimulus. A cell's cortical environment generally consists of cells that have vastly different orientation and spatial and temporal frequency tuning widths and preferences. The fact that our model's classical responses, including their diversity, agree well with the experimental data thus provides some guarantee for a reasonable response of a cell's environment, not only for classical stimuli but, more importantly perhaps, also for extraclassical stimuli.
A selection of classical response properties of the model is illustrated in Figures 2–4. All plots are for the M0 configuration (see Methods), but the other configurations yield similar results. The spatial distribution of the circular variance (CV) for our model cortex is shown in Figure 2A. The CV is a measure for orientation selectivity and is defined as
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Here r(
) is the mean firing rate and the orientation. Smaller CV indicates a higher orientation selectivity. Cells with CV = 0 respond at just 1 orientation and hence are very selective (sharply tuned). Cells with CV = 1 respond identically at all orientations and hence are not selective for orientation. In Figure 2A, we color coded the CV for all cells within the white dashed rectangle in Figure 1. The stimulus (drifting grating) was presented monocularly; the other eye received no visual input. Pixels colored black indicate cells that do not show a significant response for this monocular stimulation and are mostly cells that respond to stimulation of the other eye. Notice that our model cortex is filled with very selective cells, moderately selective cells, and nonselective cells, as is the primary visual cortex of macaques. Notice also that there is no particular spatial organization of orientation selectivity in our model. Our model yields a diversity in spatial and temporal frequency tuning properties that is quite similar to what is observed experimentally. This is illustrated in Figure 2B–E. In considering this diversity, it is important to note that, in this respect, all LGN cells in a particular configuration are identical by construction. Their spatio-temporal tuning properties are indicated by the thick black curves. As in reality, the preferred temporal frequencies of our cortical cells are lower than the preferred temporal frequency of our LGN cells. The main reason for this is the inclusion of slow, N-methyl-
-aspartate, excitation in the model (Krukowski and Miller 2001) (see also Supplementary Materials). The diversity seen in the spatial frequency tuning of our cortical cells, when compared with the LGN cells, originates in part from the spatial diversity in the feed-forward LGN connections. However, by far the largest contributions, particularly for what concerns the increase in selectivity (sharpening), are cortical in origin. That is, the spatial frequency selectivity of the LGN inputs gLGN is substantially sharpened by the cortical interactions, similar to the sharpening of orientation selectivity as explained in McLaughlin and others (2000).
A partial list of the sources of variability (and of invariability) in the model's construction is given in Methods. In observing the diversity of responses in our model, however, one should also bear in mind that our model is a high-dimensional (many dynamic variables, i.e., vi(t) and i(t)) nonlinear system. The behavior of such systems is almost by definition nontrivial. Note in this respect, for instance, that the cortical interaction term, equation (3), contains the dynamic variables i(t). This means that, although some diversity may have been introduced in the construction of the model (LGN connectivity for instance), it is not correct to conclude that therefore the diversity observed in the model's response would simply be a direct reflection of that diversity. Because of the dynamic variables i(t), diversity (and in fact, by far the largest) is also introduced by the dynamics of the model. A good example of this, and at the same time of the nontrivial behavior of such large systems, is orientation tuning. As can be seen in Figure 2A, the system displays a large diversity for orientation tuning, attaining a CV anywhere between 0 and 1. If one considers the CV in our model without the cortical interactions, however, one finds that now there is practically no orientation tuning or diversity, CV's range anywhere between 0.8 and 1. Thus, the diversity introduced, by construction, in the LGN connectivity provides by itself little diversity in orientation tuning because the CV as a result of these inputs alone is always greater than 0.8 (and less than 1). The diversity of orientation tuning in the model is thus largely dynamical in origin and due to the cortical interactions, that is, to the dynamics of these interactions. We cannot discuss such issues in too much detail as it would lead us away from the main points. A nice demonstration of the above example, however, can be found in McLaughlin and others (2000). In that paper, the diversity in orientation tuning generated by the cortical interactions is shown explicitly, with practically no diversity in the LGN connectivity.
Another example of classical response properties of the model is provided in Figure 3. Shown in Figure 3A are averaged response waveforms of spike train and membrane potential in response to a drifting grating. These are responses of a simple and a complex cell, for several grating orientations, at the cells' preferred spatial and temporal frequencies. The modulation in the spike train at the preferred orientation is frequently used to classify simple and complex cells in V1. A cell is "complex" whenever F1s/F0s < 1 and "simple" otherwise, where F1s is the first harmonic of the spike response and F0s the mean. The distribution of F1s/F0s over a cell population is shown in Figure 3B (top). Our model contains about an equal number of simple and complex cells. The bimodal shape of the distribution of F1s/F0s agrees well with the experimental data (Ringach and others 2002). In fact, the availability of this distribution provided us with a useful criterion for setting the cortical interaction strength parameters in the model (see Supplementary Materials for details).
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It is easy to understand the origins of the diversity in response modulations in our model. The modulations enter our model cortex via the LGN input, which targets 30% of the cortical cells. The phases of these LGN inputs vary randomly (approximately uniform distribution) on [0, 2
]. This is due to the receptive field offsets of the clusters of LGN cells connected to different cortical cells, the difference in shape (symmetry) of the clusters themselves, and the diversity in temporal delays in the LGN kernels. A cell receives input from many other cells, thus a cell's excitatory and inhibitory inputs will show stronger or weaker modulations depending on its specific environment in the network and whether or not it receives LGN input. Interplay between the strengths and phases of the modulations in these inputs ultimately determine the modulation in the cell's firing rate and membrane potential. Most cells that receive LGN input are simple cells (80% in our model), and most cells that do not receive LGN input are complex (70% in our model).
The distribution of the subthreshold modulation ratio F1v/F0v, where the membrane potential is measured with respect to a blank stimulus, is shown in Figure 3B (bottom). Notice that the bimodality we observe in the distribution of F1s/F0s is not present in the distribution of F1v/F0v. However, we find that (not shown) the classification of simple and complex cells can be equally well made in terms of the distribution of F1v/F0v, the 2 modes in this case being its "core" (|F1v/F0v|<2, complex cells) and its "tails" (|F1v/F0v|>2, simple cells). Also notice that we observe a "gap" in the distribution at small negative values. Details regarding the F1v/F0v distribution for our model will be published in a separate note. The distribution of modulations in the membrane potential has not yet been observed experimentally for macaques. Some data for cats have recently been published (Priebe and others 2004), and they do not contradict the predictions based on our model. The spatial distribution of F1s/F0s (spike train) across all cells within the white dashed rectangle in Figure 1 is shown in Figure 3C. Once again, the stimulus was presented monocularly, and pixels colored black indicate cells that do not show a significant response for this stimulation. The figure shows that simple and complex cells are randomly distributed across our model cortex, that is, there is no particular spatial organization of F1s/F0s.
A final example of classical response properties of our model is provided in Figure 4. Averaged response waveforms of spike train and membrane potential in response to a standing (contrast reversal) grating 
at the preferred orientation are shown in Figure 4A. Shown are the responses of a simple and a complex cell in the model for several spatial phases 
of the grating. Simple cells perform an approximately linear spatial summation, that is, their responses contain a dominant first harmonic (F1s, F1v), and the spatial phase dependence of their response waveform is similar to the spatial phase dependence of the magnitude of the intensity modulations of the stimulus at any given fixed position. Complex cells respond nonlinearly, and their response waveform is relatively insensitive to spatial phase and contains a dominant second harmonic (F2s, F2v). The distribution of the ratio of second to first harmonic of the response, averaged over the spatial phase , is shown in Figure 4B. For what concerns the spike train waveforms (top), the distribution of F2s/F1s displays a weak bimodality and its behavior for our model cells agrees with the experimental data (Hawken and Parker 1987), complex cells having mostly F2s/F1s > 1 and simple cells F2s/F1s < 1. Note that this property of our model cells follows naturally, without any parameter adjustments, after the strength parameters have been set to achieve essentially only orientation tuning and a proper distribution of response modulations in response to a drifting grating (Fig. 3B, see also Wielaard and others 2001).
It is easy to understand the origins of the diversity in F2s/F1s (and F2v/F1v) in the model. As explained in Wielaard and others (2001), for a contrast reversal grating stimulus, each total LGN input into a cortical cell has, in general, a dominant first harmonic with a phase close to either 0 or , determined by the relative positions of the ON and OFF subfields of the corresponding cluster. The cortical excitatory and inhibitory inputs in a cell will thus have a relatively strong second harmonic component because they arise from many other cells. The actual strengths of first and second harmonic in a cell's excitatory and inhibitory inputs thus depend on the cell's specific environment in the network and on whether it receives LGN input or not. Interplay of these inputs determines the ratios of first to second harmonic in the cell's spike and membrane potential responses. Clearly, most cells that receive LGN input (simple) will have F2s/F1s, F2v/F1v < 1 and most cells that do not receive LGN input (complex) will have F2s/F1s, F2v/F1v > 1.
No experimental data are available for the distribution of F2v/F1v of the membrane potential waveforms. The distribution of this quantity (averaged over the spatial phase ) for the model is shown in Figure 4B (bottom). Our model predicts that, quite contrary to the situation for F1v/F0v, the (weak) bimodality of the distribution of F2s/F1s for spike waveforms is not eliminated but, rather, becomes more pronounced in the F2v/F1v distribution for membrane potential waveforms. This, in fact, can be understood quite simply from a standard half-wave rectification model, in which the membrane potential waveforms are subjected to a threshold to give the spike waveforms. Specifically, for complex cells, both the membrane potential and spike responses will contain a strong second harmonic component. In this case, practically all of the membrane potential waveform will be above threshold, so that evaluation of F2s/F1s for spike waveforms will yield about the same result as F2v/F1v for membrane potential waveforms. This is also apparent in Figure 4B: the F2s/F1s > 1 and the F2v/F1v > 1 sections of the 2 distributions (in top and bottom panels) are very similar. For simple cells, the membrane potential and spike responses will contain a dominant first harmonic and for both responses about an equally small second harmonic component. Because of the half-wave rectification, the F1v component in the membrane potential waveform is substantially reduced in the spike waveform. Hence, F2v/F1v will turn out substantially smaller than F2s/F1s. This is again apparent in Figure 4B: the F2v/F1v < 1 section (simple cells) of the distribution for membrane potential waveforms (bottom) is markedly shifted to the left with respect to the F2s/F1s < 1 section of the distribution for spike responses (top).
Extraclassical Response Properties
In this section, we summarize the extraclassical results for our model and compare them with the experimental data. Contrary to classical response properties, the 2 extraclassical response properties focused on in this paper are, as for experimental data, not substantially different for simple and complex cells and are therefore not made type specific in what follows. Figure 5A,B shows examples of the surround suppression and receptive field expansion observed in our model. Responses are shown for both firing rate and membrane potential, at high (solid) and low (dashed) contrast.
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Distributions of receptive field and surround sizes for the 4Cß, 10°-eccentricity model (P10) are shown in Figure 5C,D. The distributions for the other model configurations are given in Supplementary Materials. Receptive field sizes and surround sizes in our model show excellent agreement with the experimental data (Sceniak and others 2001
; Cavanaugh and others 2002). This is true for the mean values, for the diversity, as well as for their dependence on eccentricity. Specifically, Figure 3A of Sceniak and others (2001) shows that surround sizes are for about 95% of the cells <4° in radius, with an average surround radius of approximately 2°. Note that Sceniak and others (2001) actually use the inhibitory space constant of the DOG model as a measure of surround radius, though this is a minor detail. Further, Figure 2 of Cavanaugh and others (2002) shows that 100% of the cells have surround radii <6°. Average surround radius at 10° eccentricity is about 2.5°; at para-foveal eccentricities, the average surround radius is 1.25°. In Figure 5A of Levitt and Lund (2002) about 85% of the cells have surround radii <2.5°. Average surround radius is about 1.5°. In Figure 2D of Angelucci and others (2002) about 70% of the cells have surround radii <6°, whereas the average surround radius is 2.5°. We have pointed to the relevant figures from these references for extra clarity, because the precise sizes of the surrounds in V1 are the cause of some ongoing debate.
The distribution of surround suppression and receptive field growth for the M0 configuration of our model is given in Figure 5E,F. The suppression index SI1 is defined in Methods. Briefly, it gives the relative suppression: cells without surround suppression have SI1 = 0 and cells with fully suppressed response for large stimuli have SI1 = 1. In agreement with the experimental data, the shape of the distribution of the suppression index SI1 is skewed to low suppression (Cavanaugh and others 2002). Further, in agreement with the experimental data, we observe a small increase in the mean suppression at low contrast (Sceniak and others 1999; Cavanaugh and others 2002). The change in suppression, 
SI1 = SI1(–) – SI1(+), is broadly distributed (Sceniak and others 1999) around a mean 0.06 (see Figure 6A,B). The average suppression index (over all eccentricities) is SI1 
0.2, and this is about half of what is observed experimentally (Cavanaugh and others 2002). The receptive field and surround growths (Fig. 5F) are expressed as ratios, r–/r+ and R–/R+, respectively. The indices + and – refer to high and low contrast, respectively. We observe an average growth by about a factor of 2 in both receptive field size 
and surround size 
This receptive field growth is a little less than what is observed in experiments (Kapadia and others 1999; Cavanaugh and others 2002).
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We also fitted our data with the DOG and ROG (Sceniak and others 1999
; Cavanaugh and others 2002) models (Methods). We obtain for the integrated suppression index SI2 0.4. Average growth ratio for the excitatory space constant is 
(both DOG and ROG, averaged over all eccentricities). Again, in agreement with the experimental data (Sceniak and others 1999), we observe a small increase in the mean suppression at low contrast, and the change in suppression, SI2 = SI2(–) – SI2(+), is broadly distributed around a mean 0.04 (see Figure 6A,B). The suppression index and growth ratio are again less than those experimentally observed (0.6 and 2.3, respectively; Sceniak and others 1999).
All the above findings are based on spike responses. Membrane potential responses yield qualitatively similar results, but due to the spike threshold, suppression in the membrane potential is systematically smaller. This is illustrated in Figure 6C. The same observation has also been made experimentally in cats (Anderson and others 2001). A more extensive summary of the data generated by our model data, for different eccentricities and including receptive field sizes and surround sizes, is given in Supplementary Materials.
Mechanisms of Surround Suppression
The DOG and ROG models are phenomenological models and provide limited insight into the neural mechanisms of the phenomena. Both models miss an essential feature of the excitatory and inhibitory inputs, which is that these inputs generally show surround suppression themselves (Anderson and others 2001). In our model, we similarly observe a significant suppression in both conductances, as shown in Figure 6D. This cell shows that, unlike what is suggested by the DOG and ROG models, surround suppression in the spike response takes place entirely in the region of decreasing synaptic inputs (conductances). We can say that the surround suppression of this cell is caused by a decrease of excitation because the decrease of inhibition could not by itself suppress the cell's response. This cell is not uncommon in our model, and the above scenario is indeed how surround suppression works in about 50% of the cells.
Analysis of the surround suppression in our model is based on the fact that the average membrane potential 
vk(t, rA)
and instantaneous firing rate k(t, rA) (of the kth neuron) are well approximated by (Wielaard and others 2001)
 | (12) |
 | (13) |
0 and [x]+ = 0 if x 0, and for most cells, good approximations are obtained with a gain k and threshold k that depend neither on the aperture radius rA nor on time. The total conductance gT,k and difference current ID,k are given by
 | (14) |
 | (15) |
·, assuming it unless stated otherwise. Given equations (12) and (13), there are 3 ways in which surround suppression of spike train and membrane potential could arise, namely, (A) 
gE,k/rA 0 and gI,k/rA > 0, (B) gE,k/rA < 0 and gI,k/rA 0, or (C) gE,k/rA < 0 and gI,k/rA > 0. In other words, surround suppression is caused by (A) an increase in the inhibitory conductance, (B) a decrease in the excitatory conductance, or (C) both (A) and (B) simultaneously. Examples of this analysis for a (simple) cell receiving LGN input and a (complex) cell that does not receive LGN input are given in Figure 7. The cycle-trial–averaged conductances for consecutive apertures around the aperture of maximum response (rA = r, marked by an asterisk) are shown in Figure 7C–F. For example, by comparing the conductances for aperture "asterisk" and the aperture for which the suppression is maximum (rA = R, for instance, the third aperture to the right of aperture asterisk in Fig. 7C), we see that at high contrast the suppression mechanism for the simple cell (Fig. 7C) is (A) and for the complex cell (Fig. 7D) is (B). At low contrast the suppression mechanisms are (C) and (B) (Fig. 7E,F, respectively).
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We observe all 3 mechanisms (A, B, and C) in our model. By comparing the responses and conductances for apertures from the receptive field size rA = r to the surround size rA = R, we can identify the suppression sequence, that is, the mechanisms that act sequentially as the aperture size rA increases from receptive field size r to surround size R. We observe a rich variety of suppression sequences in the model. In some cases, we find that different mechanisms are active during different times in the stimulus cycle.
As may be clear from Figure 7, identifying the mechanisms for surround suppression based on equations (12) and (13) can be rather more subtle than just comparing the mean (F0) conductance, its first harmonic (F1), or the peak conductance (F0 + F1). However, we find that for most cells, an analysis using the sum of first and second harmonic (F0 + F1) of the conductances allows the identification of the suppression mechanisms. Comparing conductances at rA = r and at rA = R in this way, we find that at low contrast all 3 mechanisms are about equally prevalent, whereas at high-contrast mechanism (A) is somewhat more likely than (B) and (C).
Mechanisms of Receptive Field Expansion
The DOG model suggests that receptive field growth at low contrast is due to an increase of the spatial summation extent of excitation (Sceniak and others 1999) (for our data, correlation coefficient between r–/r+ and 
is r = 0.55, correlation coefficient between r–/r+ and 
is r = 0.17). This was partially confirmed experimentally in cat's primary visual cortex (Anderson and others 2001). Although it has been claimed (Cavanaugh and others 2002) that the ROG model would explain receptive field growth solely from a change in the relative gain parameter ks, we believe this is incorrect. Because there is a one-to-one relation between ks and the surround suppression, this would imply that receptive field growth (increase) at low contrast simply results from contrast-dependent (decrease) surround suppression, which contradicts the experimental data (Sceniak and others 1999; Cavanaugh and others 2002). As does the DOG model, the ROG model, based on analysis of our data, also predicts that receptive field growth at low contrast is due to a growth of the spatial summation extent of excitation at low contrast. As we will now show, our simulations do confirm an average growth of spatial summation extent of excitation (and inhibition) at low contrast, but this growth is neither sufficient nor necessary to explain receptive field growth.
From equations (12) and (13) it follows that a change in receptive field size, in general, results from a change in behavior of the relative gain parameter, defined as
 | (16) |
A change in the spatial summation extent of gE and/or gI is just one of the many ways to change the behavior of G and consequently the receptive field size. For example, some other possibilities are illustrated by the 2 cells in Figure 7. These cells show, both in spike and membrane potential responses, a receptive field growth of a factor of 2 (left) and 3 (right) at low contrast. However, for both cells, the spatial summation extent of excitation at low contrast is 1 aperture less than at high contrast.
The receptive field expansion at low contrast is apparent in Figure 8A. In agreement with the experiment (Sceniak and others 1999; Cavanaugh and others 2002), we observe little correlation between this growth and the surround suppression (correlation coefficient between r–/r+ and SI1 is r = 0.01). In a similar way as for spike train responses, we obtained receptive field sizes for the conductances. As shown in Figure 8B,C, both excitation and inhibition also show, on the average, an increase in their spatial summation extent as contrast is decreased. This increase is, however, rather arbitrary and bears not much relation with the receptive field growth based on spike responses (correlation c