Vision Research 50 (2010) 2261–2273

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Vision Research

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Correlation between spatial frequency and orientation selectivity in V1 cortex:Implications of a network model

Wei Zhu a,b,*, Dajun Xing c,*, Michael Shelley a, Robert Shapley c

a Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, United Statesb Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487, United Statesc Center for Neural Science, New York University, 4 Washington Place, NY 10003, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 October 2009Received in revised form 8 January 2010

Keywords:Spatial frequency selectivityOrientation selectivityLarge-scale neuronal networkCortical excitationCortical inhibitionSimple cells

0042-6989/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.visres.2010.01.007

* Corresponding authors.E-mail addresses: wzhu7@bama.ua.edu (W. Zhu),

We addressed how spatial frequency and orientation selectivity coexist and co-vary in Macaque primaryvisual cortex (V1) by simulating cortical layer 4Ca of V1 with a large-scale network model and then com-paring the model’s behavior with a population of cells we recorded in layer 4Ca. We compared the dis-tributions of orientation and spatial frequency selectivity, as well as the correlation between the two, inthe model with what we observed in the 4Ca population. We found that (1) in the model, both spatialfrequency and orientation selectivity of neuronal firing are greater and more diverse than the LGN inputsto model neurons; (2) orientation and spatial frequency selectivity co-vary in the model in a way verysimilar to what we observed in layer 4Ca neurons; (3) in the model, orientation and spatial frequencyselectivity co-vary because of intra-cortical inhibition. The results suggest that cortical inhibition pro-vides a common mechanism for selectivity in multiple dimensions.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Neuronal responses in primary visual cortex (V1) are selectivein different spatial dimensions, such as orientation, spatial fre-quency, and size. Most theoretical work has focused on orientationselectivity, because it is the most well-studied functional propertyin V1. In most models (McLaughlin, Shapley, Shelley, & Wielaard,2000; Tao, Cai, McLaughlin, Shelley, & Shapley, 2006; Tao, Shelley,McLaughlin, & Shapley, 2004; Troyer, Krukowski, Priebe, & Miller,1998), neurons have diverse orientation selectivity but either noor fixed selectivity in other domains. In this paper, we study V1neurons’ orientation selectivity and spatial frequency selectivitycoexisting in a large-scale realistic model.

V1 neurons are much more sharply-tuned to stimulus orienta-tion and spatial frequency than are neurons in the Lateral Genicu-late Nucleus (LGN). Orientation selectivity is generated in V1 (DeValois, Albrecht, & Thorell, 1982; Hubel & Wiesel, 1962). V1 neu-rons inherit spatial frequency selectivity from LGN cells (So &Shapley, 1981), which in turn inherit it from retinal ganglion cellsdue to the center-surround organization of retinal receptive fields(Enroth-Cugell & Robson, 1966; Kuffler, 1953; Rodieck, 1965).However, the spatial-frequency tuning of V1 neurons (Campbell,Cooper, & Enroth-Cugell, 1969; De Valois et al., 1982; Movshon,Thompson, & Tolhurst, 1978) is often much narrower than that

ll rights reserved.

xdj@cns.nyu.edu (D. Xing).

of LGN cells. Traditional explanations of the increase in corticalspatial selectivity involve additive convergence of LGN input (Day-an and Abbott (2001) for spatial frequency selectivity; De Valoisand De Valois (1988) and Hubel and Wiesel (1962) for orientationselectivity). However, intracellular data indicate that intra-corticalinhibition also could play a role in generating orientation selectiv-ity (Anderson, Carandini, & Ferster, 2000; Borg-Graham, Monier, &Fregnac, 1998; Marino et al., 2005; Monier, Chavane, Baudot, Gra-ham, & Fregnac, 2003; Pei, Vidyasagar, Volgushev, & Creutzfeldt,1994; Schummers, Marino, & Sur, 2002). Suppression of responsesto non-preferred orientations (Malone & Ringach, 2008; Ringach,Bredfeldt, Shapley, & Hawken, 2002; Ringach, Shapley, & Hawken,2002; Xing, Shapley, Hawken, & Ringach, 2005) and non-preferredspatial frequencies (cf. Bredfeldt & Ringach, 2002; Ringach, Bred-feldt, et al., 2002) in reverse correlation experiments is additionalevidence that inhibition may enhance selectivity.

The idea of a common mechanism, intra-cortical inhibition, forspatial selectivities seems plausible because there is a correlationbetween V1 neurons’ orientation and spatial frequency selectivity(De Valois et al., 1982; Xing, Ringach, Shapley, & Hawken, 2004).De Valois et al. (1982) found a moderate correlation (r = 0.5) be-tween the bandwidths of V1 neurons’ orientation tuning and spa-tial-frequency tuning. Xing et al. (2004) found a strongercorrelation (r = 0.78) between measures of spatial selectivity thatcompare responses at the peak of the tuning curve with responsesto non-preferred stimuli: at non-optimal orientations, or at lowspatial frequencies. The strong correlation between response

2262 W. Zhu et al. / Vision Research 50 (2010) 2261–2273

attenuation of non-optimal stimuli in the two domains of orienta-tion and spatial frequency led to the suggestion that there was acommon mechanism: cortical response suppression (Xing et al.,2004).

This paper investigates a realistic model of V1 cortex and com-pares the model’s performance with cortical data. We found that(1) in the model, both spatial frequency and orientation selectivityof neuronal firing are greater and more diverse than the LGN inputsto model neurons; (2) orientation and spatial frequency selectivityco-vary in the model in a way very similar to what is observed inlayer 4Ca; (3) in the model, orientation and spatial frequencyselectivity co-vary because they both depend on intra-cortical inhi-bition. The results suggest that cortical inhibition provides a com-mon mechanism for feature selectivity in multiple dimensions.

2. Methods

2.1. Large-scale network model

Complete details of the large-scale model we used can be foundin Zhu, Shelley, and Shapley (2009). Our model resembled themodels described in McLaughlin et al. (2000), Tao et al. (2004,2006), and Wielaard, Shelley, McLaughlin, and Shapley (2001): alarge-scale neuronal network of V1 layer 4Ca, consisting of16,384 integrate-and-fire neurons, 75% of which are excitatoryand 25% of which are inhibitory. It was a recurrent excitatory,inhibitory model with feedforward inputs from the LGN only. Eachmodel neuron’s spike-firing was governed by a standard integrate-and-fire equation (Eq. (1)) where the synaptic inputs to the neurondrove the membrane potential towards or away from the spike-fir-ing threshold.

dv jr

dt¼ �gR v j

r � VR� �

� gjrEðtÞ v j

r � VE� �

� gjrIðtÞ v j

r � VI� �

¼ �gjTðtÞ v j � Vj

SðtÞh i

: ð1Þ

In Eq. (1), the subscript r = E, I denotes excitatory, inhibitory neu-rons respectively; for example v j

E represents the membrane poten-tial of an excitatory neuron at the location within the cortical layerindexed by j. When v j

r > VThreshold, a spike was fired. Membranecapacitance was assumed to be a fixed constant, and was absorbedinto the conductances so that gR, gE, gI and gT in Eq. (1) have theunits s�1. In Eq. (1), gT = gR + gE + gI is the total membrane conduc-tance and VS = (VEgE + VIgI)/gT is the effective reversal potential. Volt-age was normalized so that the resting potential VR = 0 andVThreshold = 1. Therefore, the reversal potentials of the excitatoryand inhibitory synaptic currents, respectively VE and VI, were nor-malized to have the values 14/3 and 2/3. The leakage conductancegR was set to 50 s�1.

The synaptic currents that appear in Eq. (1) were set by the pat-tern of LGN and intra-cortical connectivity built into the modelthrough expressions for the synaptic conductances (Eq. (2)).

gjEEðtÞ ¼ gj

lgnðtÞ þ gjnoiseðtÞ þ Sj

EE

Xk

aj�kPj�k

Xm

GEðt � tkmÞ;

gjEIðtÞ ¼ Sj

EI

Xk

bj�k

Xm

GIðt � tkmÞ;

gjIEðtÞ ¼ gj

lgnðtÞ þ gjnoiseðtÞ þ Sj

IE

Xk

aj�kPj�k

Xm

GEðt � tkmÞ;

gjIIðtÞ ¼ Sj

II

Xk

bj�k

Xm

GIðt � tkmÞ: ð2Þ

The spatial coupling coefficients aj�k in Eq. (2) were approxi-mated by Gaussian functions of cortical distance between cells jand k The Gaussian length scales for excitation and inhibition were200 lm and 100 lm, respectively, derived from neuroanatomical

measurements as in McLaughlin et al. (2000). The term gnoise inEq. (2) was a constant noise source presumed to come from cor-tico-cortical inputs that are not visually-driven, and was neededto explain the higher spontaneous firing rates of complex corticalcells that receive only weak or no LGN excitatory drive (Zhuet al., 2009).

The spatial pattern of intra-cortical connectivity in the modelwas local, via intra-hypercolumn connections, based on neuroana-tomical measurements of V1 circuitry (Callaway, 1998; Fitzpatrick,Lund, & Blasdel, 1985). Each model neuron received hundreds ofsynaptic inputs, both excitatory and inhibitory, through randomconnections from its neighbors with no spatial phase preferenceand with an orientation preference given by the map of orientationonto the cortex (McLaughlin et al., 2000; Tao et al., 2004).

In the model, the spatial pattern of LGN input to a single modelneuron was side-by-side elongated sub-regions. We put spatial fre-quency information into the construction of those elongated re-gions by allowing different sub-region widths (Zhu et al., 2009).There are optical imaging data (Sirovich & Uglesich, 2004) and sin-gle unit data (DeAngelis, Ghose, Ohzawa, & Freeman, 1999) fromV1 that suggest that nearby cells have similar spatial frequencypreferences but that beyond a fairly short distance the cells’ pref-erences are unrelated. To be consistent with these data, we dividedthe network into many clusters of cells, inside each of which the V1neurons share the same width of their LGN input arrays. Clusters ofdifferent width were arranged randomly across the V1 layer. More-over, the spatial pattern of LGN inputs to each V1 neuron was setrandomly to have even or odd symmetry, i.e., for an even symmet-ric neuron, the LGN input to the elongated sub-regions could be ofthe form ON–OFF–ON or OFF–ON–OFF. Such a random assignmentof symmetry is also compatible with cortical data (DeAngelis et al.,1999; Ringach, 2002). The choice of the pattern of LGN input wasstrictly determined by experimental data, as explained more fullyin Zhu et al. (2009), following the approach of incorporating bio-logical data in the modeling that was used by McLaughlin et al.(2000). The LGN cell firing rates were modeled as inhomogeneousPoisson processes where the rate parameter of each Poisson pro-cess was modulated by a spatio-temporally filtered version of thevisual input (cf. Tao et al., 2004; Zhu et al., 2009).

The cortico-cortical conductance coupling matrix was a crucialpart of the model. The coupling coefficients were chosen to obtaina model that responded to visual stimuli as V1 cortex did, both interms of selectivity but also in terms of firing rates. The values inthe conductance coupling matrix in Eq. (2) were: recurrent excita-tion’s coefficient SEE = 0.1 s�1 at its minimum (for cells that receivemaximal LGN input), and SEE = 45 s�1 for its maximum value (forcells that receive no LGN input); the coupling strength for inhibi-tory synapses onto excitatory neurons SEI = 80 s�1; the strengthof cortical excitation of inhibitory neurons, SIE = 0.1–46 s�1 forthe range between its minimum and maximum values; inhibition’sstrength on inhibitory neurons, SII = 65 s�1. These coupling coeffi-cients represent the amount of synaptic conductance increase pernerve impulse from the source neurons. With this conductancematrix, the cortical model was in a high conductance state (Shelley,McLaughlin, Shapley, & Wielaard, 2002), considered in Section 4.

2.2. Animal experiments

2.2.1. PreparationAcute experiments of several days duration were performed on

adult old-world monkeys (Macaca fascicularis) in compliance withNational Institutes of Health (NIH) and New York University (NYU)guidelines, with the approval of the New York University AnimalWelfare Committee. Procedures were like those described in detailin Xing et al. (2004). Animals were initially tranquilized with ace-promazine (50 lg/kg). After the tranquilizer, the animal was anes-

W. Zhu et al. / Vision Research 50 (2010) 2261–2273 2263

thetized by ketamine (30 mg/kg, intramuscularly, im) for venouscannulation and tracheotomy. Additional ketamine was given dur-ing this surgery if needed. After cannulation and tracheotomy, theanimal was placed in a stereotaxic frame for craniotomy and sub-sequent visual experiments. Further surgery was performed undersufentanyl (6–18 lg/kg/h, intravenously, iv) anesthesia (infusedthrough a leg vein, usually the left-leg vein). A craniotomy(5 mm or smaller in radius) was made in one hemisphere 4 mmposterior to the lunate sulcus (15–20 mm anterior to the occipitalridge) and 15 mm lateral to the middle line. Then the dura was cut(less than 1 mm in radius) to provide access for the electrode. Tri-ple antibiotic ointment was applied surrounding the incision. Dur-ing the whole acute experiment, anesthesia was continued withsufentanyl (6–18 lg/kg/h, iv) and the animal was paralyzed withvecuronium bromide (0.1 mg/kg/h, iv) (infused through a venouscannula on another leg). After the animal was paralyzed, respira-tion was supported by a respirator (Harvard Apparatus) to main-tain expired CO2 close to 5%. Temperature was kept at a constant37 �C. A broad-spectrum antibiotic (Bicillin, 50,000 iu/kg, im) andanti-inflammatory steroid (dexamethasone, 0.5 mg/kg, im) weregiven on the first day of the experiment and every other day duringthe recording period. Experiments were terminated with a lethaldose of pentobarbital (60 mg/kg, iv). Expired CO2, blood pressure,EKG, EEG and core body temperature were monitored continu-ously, and were used to make sure that anesthesia was maintainedat a steady level. Ophthalmic atropine sulfate (1%) was adminis-tered to the eyes at the start of the experiment in order to dilatethe pupils. Throughout the experiment, the eyes were protectedby clear, gas-permeable contact lenses and a topical antibioticsolution (gentamicin sulfate, 3%). The foveae were mapped ontoa tangent screen using a reversing ophthalmoscope. Glass-coatedtungsten microelectrodes were advanced through a craniotomyover occipital cortex. Cells were recorded in V1, typically in the re-gion that represents 2–6� eccentricity. Extracellular spikes werediscriminated and time-stamped with 0.1 ms resolution via cus-tom software running on a Silicon Graphics O2. The visual recep-tive fields of isolated single neurons were mapped onto thetangent screen with reference to the foveae. Using histologicaltechniques we assigned cells to layer 4Ca. This paper is the firststudy that includes this population of 4Ca neurons.

2.3. Histology

We compared orientation and spatial frequency selectivity ofthe model population with cell data from layer 4Ca. Cells were as-signed to layer 4Ca by track reconstruction following proceduresdescribed by Hawken, Parker, and Lund (1988). Briefly, three tosix electrolytic lesions (2–3 lA for 2–3 s, tip negative) were madealong the length of each electrode track. The angle of the electrodetrack, relative to the surface normal, was approximately 60�. A typ-ical electrode track would extend for about 3–4 mm. Consecutivelesions were spaced by about 1 mm. Our electrode tracks resem-bled the one shown in Hawken and Parker (1984). The details offixation, sectioning, staining and reconstruction of electrode tracksare described by Hawken et al. (1988).

2.4. Visual stimulation

Each isolated single-cell was stimulated monocularly throughits dominant eye and characterized by measuring its steady-stateresponse to conventional drifting gratings (the non-dominant eyewas occluded). Using this method we recorded basic attributes ofthe cell, including spatial and temporal frequency tuning, orienta-tion tuning, contrast and color sensitivity, as well as area summa-tion curves. Receptive fields were located at eccentricities between

1� and 6�. The mean luminance of the screen was 50 cd/m2, theviewing distance 90–120 cm.

2.5. Spatial frequency and orientation tuning curves

In the present experiments, we measured tuning curves forsteady-state stimuli, drifting gratings, which vary in orientationand spatial frequency. We have described in detail the proceduresfor measuring the orientation tuning to drifting grating stimuli andthe analysis of the tuning curves (Ringach, Shapley, et al., 2002).Briefly, orientation was measured in steps of 20� or less, for stimuliof the optimal spatial and temporal frequency. In the standardexperiments contrast was usually 0.8 (though occasionally it wasset arbitrarily to 0.64, with no obvious difference in tuning curves).Spatial-frequency tuning was determined in half octave steps for arange of frequencies that covered the response range of each cell.For each cell, the size of the circular stimulus window was chosento optimize response. Each tuning function was fit with a differ-ence of Gaussians to obtain a smooth curve for calculating the spa-tial frequency bandwidth (Sceniak, Hawken, & Shapley, 2001).

2.6. Data analysis

2.6.1. Orientation tuning bandwidthGiven a cell’s orientation tuning curve, we smoothed the curve

with a Hanning window filter of width 18�. Then we found thepeak response in the smoothed curve, and looked for the pointson both sides of the peak at which the cell’s responses were halfof the peak response. Half of the distance between the two pointsis the orientation bandwidth.

Oribw ¼ ½Ori1=2;high � Ori1=2;low�=2 ð3Þ

If there was no response smaller than half of the peak, then thecell was called non-oriented and its orientation tuning bandwidthset to 180�.

2.6.2. Circular variance (CV)To determine a cell’s orientation tuning curve we measured 18

different responses of the cell to orientations over the range 0� and360� with 20� intervals. We denote the spike rate responses m(hi)(i = 1, 2, . . . , 18) for each orientation. Circular variance for a tuningfunction m(h) on a circle is defined by the following equation (Mar-dia, 1972):

CV ½m� ¼ 1�

R 2p0 mðhÞ expð2ihÞdh

��� ���R 2p0 mðhÞdh

ð4Þ

Circular variance is highly correlated with other measures of orien-tation selectivity like the ratio of orthogonal-to-preferred responsedivided by preferred response (Ringach, Shapley, et al., 2002). CVlies between 0 and 1. Highly selective cells have a CV � 0, whileunselective cells have CV = 1.

In the spatial-frequency tuning experiments, we used driftinggratings with optimal orientation, temporal frequency, radius,and high contrast to stimulate the cell’s receptive field. Cells’ re-sponses were measured to 10 gratings of different spatial frequen-cies evenly distributed between 0.1 and 10 cycles/deg on alogarithmic scale. Then we fit the data with a DOG (difference ofGaussians) model as in Eq. (5), by minimizing the square error be-tween the DOG curve and the data, with all parameters (R0, Ke, le,re, Ki, li, and ri) free.

RðSFÞ ¼ R0 þ Ke exp �ðSF � leÞ2

2r2e

!� Ki exp �ðSF � liÞ

2

2r2i

!ð5Þ

2264 W. Zhu et al. / Vision Research 50 (2010) 2261–2273

2.6.3. Spatial-frequency tuning bandwidthWe found the peak of the fitted spatial-frequency tuning curve,

and looked for the points where the curve dropped to half of thepeak, denoted Sfhigh and Sflow. The spatial frequency bandwidth(in octaves) is defined in the following equation:

Sfbw ¼ log2ðSfhighÞ � log2ðSflowÞ ð6Þ

If the cell’s response at the lowest spatial frequency measured(0.1 cycle/deg) was higher than half the best response, we definedthe cell as low-pass, without a bandwidth. In layer 4Ca, about90% of simple cells (28/30) had a measurable bandwidth in bothspatial frequency and orientation.

2.6.4. Low-spatial frequency variance (LSFV)Based on the fitted spatial-frequency tuning curve, the left

branch of the curve from 1/M (we usually chose M = 16 points) ofthe optimal spatial frequency to the optimal spatial frequencywas used to calculate low-spatial frequency variance (LSFV; intro-duced in Xing et al. (2004)) by means of Eq. (7).

LSFV ¼R Sfoptimal

Sfoptimal=16 RðSf Þ � ðlog16ðSf Þ � log16ðSfoptimalÞÞ2 � dlog16ðSf ÞR Sfoptimal

Sfoptimal=16 RðSf Þ � dlog16ðSf Þ

ð7Þ

LSFV is a global measure that assesses the degree of low-spatial fre-quency attenuation. It captures the global shape of a spatial-fre-quency tuning curve while the bandwidth characterizes the shapeof the tuning curve near the peak (preferred) spatial frequency.The LSFV measure is smaller when the cell is more selective for spa-tial frequency. LSFV has a value of 0 for the most spatial-frequency-selective cells; for a low-pass (unselective) neuron, LSFV = 1/3 (Xinget al., 2004).

3. Results

In this study, we simulated a 1 mm � 1 mm patch of corticallayer 4Ca of V1 with a large-scale network model consisting ofO(104) excitatory and inhibitory integrate-and-fire neurons withrealistic synaptic conductances (Zhu et al., 2009). Some featuresof our model were similar to other cortical network models(Chance, Nelson, & Abbott, 1999; McLaughlin et al., 2000; Taoet al., 2004, 2006; Troyer et al., 1998), but this new model allowedus to study model V1 neuronal responses in both orientation and

Modulati

# of

cel

ls

Experimental data

0 1 2

0

10

20

N=52 4Cα

A

Fig. 1. Segregation of simple and complex cells in the model vs. experimental data. (A)distribution. (B) In our large-scale network model, the distribution of F1/F0 is very simi

spatial frequency domains. The model successfully simulated sev-eral functional properties similar to those in the real 4Ca data thatwe present for comparison, including: (1) functionally distinct sim-ple and complex cells; (2) diversity of orientation and spatial fre-quency selectivity similar to experimental results; (3) correlationbetween the orientation and spatial frequency selectivity.

In this work, the visual stimuli considered were driftinggratings. We tested the large-scale model under 64 experimentalconditions, by applying stimuli with eight orientations (0�, 45�,90�, 135�, 180�, 225�, 270�, 315�) and eight spatial frequencies(1/16, 1/8, 1=4, ½, 1, 2, 4, 8 cycles/deg) and with the highest con-trast (100%). For each model neuron, the preferred orientationand preferred spatial frequency were taken to be those valueswhere the spike rate was highest among the 64 simulated experi-ments.

3.1. Simple and complex cells

Although it is not the main result of our current paper, we needto point out that our large-scale model generated simple and com-plex cells with a distribution like that seen in the biological cortex(Fig. 1). We show this result at the beginning, because the follow-ing results are mainly focused on simple cells’ behavior in the net-work. A traditional way to differentiate simple cells from complexcells with respect to drifting grating patterns is by means of themodulation ratio F1/F0 (Skottun et al., 1991): cells with F1/F0 > 1are called simple while cells F1/F0 < 1 are called complex, whereF1 and F0 are the first harmonic amplitude and one half of themean firing rate, respectively. Experimentally we found a bi-modaldistribution of F1/F0 in layer 4Ca (Fig. 1A) that is similar to the dis-tribution of modulation ratio observed throughout V1 (Ringach,Shapley, et al., 2002; Xing et al., 2004). The large-scale model gen-erated a bi-modal distribution that was similar to experimentaldata (Fig. 1B). The bi-modal distribution of F1/F0 in the V1 modelwas in part a consequence of the nonlinearity of spike thresholdas hypothesized by Mechler and Ringach (2002) and Priebe, Mech-ler, Carandini, and Ferster (2004). The distribution of F1/F0 in themodel neurons’ membrane potentials was unimodal (cf. Taoet al., 2004). It is useful to examine the feature selectivity of simpleand complex cells separately because there was a strong correla-tion of feature selectivity with F1/F0 ratio in the model as in thereal cortex (Ringach, Shapley, et al., 2002; Xing et al., 2004) aswe will show.

0 1 20

1000

2000

on Ratio F1/F0

Model dataB

Modulation ratio F1/F0 of individual neurons from layer 4Ca in V1 has a bi-modallar to that for experimental data in A.

0

5

10

15

20

0

50

100

150

0

5

10

15

20

0

50

100

150 CV=0.37

0

5

10

15

20

0

50

100

150

Peak

dev

iatio

n of

LG

N (

sec

−1

)

0

5

10

15

20

Rep

sons

e (s

pike

s/se

c)

0

50

100

150 CV=0.70

0 90 180 270 3600

6

12

18

24

00

60

120

180

Orientation (degree)

2 80

6

12

18

24

1/8 1/20

60

120

180

Spatial Frequency (cycles/degree)

CV=0.81

LSFV=0.16

LSFV=0.20

A B

C D

LSFV=0.27

E F

Fig. 2. Simple cells’ orientation and spatial-frequency tuning curves in the model neurons. Spatial-frequency tuning curves (solid blue curves in A, C and E) were plotted forthree example cells from the network. Corresponding orientation tuning curves were also plotted (solid curves in B, D and F). Dashed curves in A–F represent the orientationor spatial-frequency tunings of the three cells’ LGN inputs. The dashed horizontal lines indicate the background spike rate, the firing rate when there was no stimuluscontrast. The spatial frequency LSFV and orientation CV for the three examples were respectively: (0.16, 0.37), (0.2, 0.7), (0.27, 0.81). In A, C, E the LSFVs of the LGN inputswere 0.26, 0.27 and 0.26; in B, D, F the CVs were 0.73, 0.87 and 0.84.

W. Zhu et al. / Vision Research 50 (2010) 2261–2273 2265

3.2. Diversity of orientation selectivity and spatial frequencyselectivity: examples

The tuning curves for selected example simple cells in the mod-el illustrate how the model achieves selectivity for spatial fre-quency and orientation (Fig. 2). The spatial-frequency tuning(solid curves in Fig. 2A, C and E) and orientation tuning (solidcurves in Fig. 2B, D and F) of the spike rates of simple cells in themodel network resembled experimental results in V1 (Ringach,Shapley, et al., 2002; Xing et al., 2004). In order to understandthe mechanism of spatial frequency and orientation selectivity,we plotted in the same graphs the orientation and spatial-fre-quency tuning curves of excitatory synaptic current from thesummed LGN inputs to the model simple cells (dashed curves inFig. 2). The difference between model neurons’ tuning curves andthose of their LGN inputs can be seen mainly in the attenuationof the spike-firing rate to non-optimal stimuli. In general, theLGN inputs from cell to cell had similar, rather low, selectivity be-cause the LGN inputs in Fig. 2 had much higher responses to non-optimal orientations relative to their peak responses, and relativelyhigher responses to low spatial frequencies than the model V1neurons.

To summarize the tuning properties of neurons, one needs ameasure of selectivity for each tuning curve. It has been shown thatcircular variance (CV) for orientation tuning and low-spatial fre-quency variance (LSFV) for spatial-frequency tuning are measuresof feature selectivity that indicate the degree of attenuation of re-sponses to non-optimal stimuli, as well as the global selectivity inthese two domains (see Section 2; Ringach, Bredfeldt, et al., 2002;Xing et al., 2004). It has been shown previously CV is highly corre-lated with other measures of orientation selectivity like the orthog-onal/preferred ratio (Ringach, Shapley, et al., 2002), and that LSFV isa measure of the shape of the spatial-frequency tuning curve thatcorrelates with other measures like low-spatial-frequency-selectiv-ity index (Xing et al., 2004). The relatively weak selectivity of theLGN inputs can be quantified by the LSFV and CV of the LGN inputs.In A, C, E the LSFVs of the LGN inputs were 0.26, 0.27 and 0.26; in B,D, F the CVs were 0.73, 0.87 and 0.84. Fig. 2 indicates that cell spike-firing at non-optimal orientations and lower spatial frequencieswas suppressed in the cortex for the most selective cells. Thus,the LSFV’s for the spike-firing rates in examples A, C, E were 0.16,0.2, and 0.27 and the CVs in B, D, and F were 0.37, 0.7, and 0.81respectively. The firing rates of more selective cells as in Fig. 2ABhad lower LSFV and CV than their LGN inputs.

0

600

0

600

0 0.5 1

0

1000

0 0.2 0.4

0

1000

Circular Variance

# of

cel

ls

LSFV

V1 Simple cells

0.73 +/- 0.13

V1 Simple cells

0.25 +/- 0.05

LGN input

0.82 +/- 0.06

LGN input

0.29 +/- 0.03

A B

DC

0

10E FExperimental data

0.54 +/- 0.24

Experimental data

0.15 +/- 0.09

Fig. 3. Distribution of CV and LSFV for V1 simple cells vs the distribution of CV and LSFV for their LGN inputs (simple cells). The orientation selectivity and spatial frequencyselectivity of the population of model simple cells (A and B) were greater than those of their LGN inputs (C and D). The LGN inputs were the summed excitatory synapticcurrents coming from LGN inputs. The range of selectivity for V1 simple cells is also wider than that for simple cells’ LGN inputs; the distributions in A and B are broader thanin C and D. The distributions of orientation selectivity and spatial frequency selectivity for the dataset of 4Ca neurons from the real cortex are plotted in E and F respectively.

2266 W. Zhu et al. / Vision Research 50 (2010) 2261–2273

The examples in Fig. 2 were representative of the whole popu-lation, as we will show. But different cortical cells varied in selec-tivity as Fig. 2 also illustrates. The variation was mainly in howweak were the non-preferred responses compared to preferred re-sponses. As Fig. 2 also illustrates, when spatial frequency selectiv-ity was high, so also was orientation selectivity (Fig. 2AB), andwhen the selectivity for spatial frequency was less, also the orien-tation selectivity was less (Fig. 2EF). The correlation seen in theexamples of Fig. 2 also applied to the entire V1 model population,as shown below in Fig. 6.

3.3. Diversity of orientation selectivity and spatial frequencyselectivity: population analysis

For a population analysis of orientation and spatial frequencyselectivity we plotted the distributions of CV and LSFV for the pop-ulation of model simple cells and their LGN inputs (Fig. 3). Theshifts of the population distributions to the left for the firing ratescompared to LGN inputs, for both CV and LSFV, mean that the firingrates of model simple cells (Fig. 3A and B) were more selective on

average than their LGN inputs (Fig. 3C and D) in the dimensions oforientation and spatial frequency. Also, the model V1 cells had awider range of selectivity than their LGN inputs – the distributionsfor cell firing rates were broader than for the LGN inputs. Thediversity of selectivity in the model network resembled the diver-sity seen in the real cortex in layer 4Ca (Fig. 3E and F) but the 4Capopulation sampled in the experiments had a wider range of diver-sity of spatial frequency and orientation selectivity than the modelpopulation.

The model’s diversity is generated by interactions within thecortical network. It is evident in the simple cell examples fromthe model in Fig. 2 that, for selective neurons, LGN excitation atnon-preferred orientations and spatial frequencies had a reducedinfluence on the model cells’ spike rates. In the orientation domainthis suppression was caused by broadly-tuned inhibition in themodel (cf. McLaughlin et al., 2000). In the spatial frequency do-main, suppression of low-spatial frequency responses was causedby intra-cortical inhibition that was stronger than excitation atlow-spatial frequency, as analyzed in detail by (Zhu et al., 2009).The combination of broadly-tuned inhibition and the spike-firing

0 0.2 0.4 0.6 0.8 10

0.1

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tion

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euro

ns

0.73 +/- 0.130.83 +/- 0.07

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0.25 +/- 0.050.30 +/- 0.03

Simple excitatory cells All inhibitory cells

Fig. 4. Distribution of CV and LSFV for excitatory and inhibitory cells in the model. Dashed contours show the distribution of inhibitory neurons’ circular variance (A: mean0.83 ± 0.05) and LSFV (B: mean 0.30 ± 0.028). Solid contours show the distribution of excitatory simple cells’ circular variance or CV (A: mean 0.73 ± 0.13) and LSFV (B:0.25 ± 0.05).

W. Zhu et al. / Vision Research 50 (2010) 2261–2273 2267

threshold of V1 neurons caused the spike rates of many simplecells in the model to be much more selective for spatial frequencyand orientation than their LGN inputs were.

Individual inhibitory neurons in the model were considerablyless selective than model simple excitatory cells (Fig. 4) as indi-cated by the higher CV of inhibitory neurons for orientation selec-tivity (Fig. 4A) and as indicated by their higher LSFV for spatialfrequency selectivity (Fig. 4B). The different tuning characteristicsof inhibitory neurons in the model was in part a consequence oftheir higher spontaneous firing rates and also their higher firingrates in response to visual stimulation (McLaughlin et al., 2000;Zhu et al., 2009), as well as to their denser intra-cortical connectiv-ity. The existence of inhibitory neurons in visual cortex that werebroadly-tuned (or untuned) for orientation was reported recently(Cardin, Palmer, & Contreras, 2007; Nowak, Sanchez-Vives, &McCormick, 2008), consistent with model results. We also ob-served relatively higher spike rate responses of model inhibitoryneurons for lower spatial frequencies (cf. Fig. 12 of Zhu et al.(2009)), consistent with experimental data (Cardin et al., 2007;Nowak et al., 2008).

The convergence of many inhibitory neurons onto excitatoryneurons was what made the orientation and spatial-frequency tun-ing of intracellular inhibition in model neurons almost constant

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tion

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Fig. 5. Distribution of CV and LSFV for the inhibitory current in excitatory cells. The CVmodel. The uniformly high values of CV and LSFV indicate that inhibitory current in modeCV for the inhibitory current was 0.94 ± .028. The mean LSFV was 0.327 ± 0.007.

with orientation and strong at low-spatial frequency (Zhu et al.,2009). The tuning characteristics of the summed inhibitory currenton excitatory cells were very broad – almost flat with orientationand low-pass in spatial frequency as quantified by the distributionsof inhibition’s LSFV and CV across the V1 population in Fig. 5. Weconjecture that such broadly-tuned inhibition from the localhypercolumn circuit is probably what causes the untuned suppres-sion observed in reverse correlation experiments (Xing et al.,2005), and also probably is the source of low-spatial-frequencysuppression that is observed in reverse correlation experiments(Bredfeldt & Ringach, 2002; Ringach, Bredfeldt, et al., 2002).

3.4. Correlation between orientation selectivity and spatial frequencyselectivity

The presence of diversity in selectivity as illustrated in Fig. 3 en-abled us to study correlations between orientation and spatial fre-quency selectivity in the real cortex and in the model. This is thecrucial comparison in this paper. The distributions of orientationand spatial frequency selectivity in the model (Fig. 3) resembledexperimental data. The correlation between simple cells’ spatialfrequency and orientation selectivity in the model network (illus-

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700.0 -/+ 723.0

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CBA

Fig. 6. Correlation between orientation selectivity and spatial frequency selectivity (simple cells). The network model generates a correlation pattern (B) for simple cells’orientation and spatial frequency selectivity. This pattern resembles the pattern from experimental data (A) for simple cells in layer 4Ca in V1. But the correlation pattern (B)for simple cells is very different from the correlation pattern for simple cells’ summed LGN synaptic input currents in (C). The strengths of correlation are indicated by thecorrelation coefficients written within each figure panel.

2268 W. Zhu et al. / Vision Research 50 (2010) 2261–2273

trated with the scatter plots in Fig. 6) also was similar to experi-mental results (cf. De Valois et al., 1982; Xing et al., 2004).

As reported previously (Xing et al., 2004), LSFV was stronglycorrelated with CV in experimental data from all of V1 cortex.Experimental data from cells in layer 4Ca also had a strong corre-lation between CV and LSFV (Fig. 6A for cells in layer 4Ca; cf. Xinget al., 2004 for all cortical cells in V1); the correlation for layer 4Casimple cells was 0.81. Fig. 6B indicates that in the model also, ori-entation selectivity measured by circular variance, CV, was highlycorrelated with spatial frequency selectivity measured by low-spa-tial frequency variance, LSFV, with a correlation coefficient of 0.59.We found that the strong correlation between CV and LSFV in thelarge-scale network model was not due to simple cells’ LGN inputs(the summed excitatory current from LGN synaptic inputs ontomodel neurons), which had a much weaker correlation betweenorientation and spatial frequency selectivity, only 0.27 (Fig. 6C).Therefore, a cortical mechanism was required to explain the corre-lation between orientation and spatial frequency selectivity.

The correlation between orientation and spatial frequencybandwidth throughout V1 was noted by De Valois et al. (1982)and replicated by Xing et al. (2004). As in the whole of V1, thebandwidth correlation was weaker than between CV and LSFV inlayer 4Ca simple cells (Fig. 7A). The large-scale model emulatedlayer 4Ca in this respect too (Fig. 7B).

Our explanation for the resemblance of the correlations in mod-el and real cortex is the hypothesis that a single mechanism, whichwe suppose is the suppression caused by cortical inhibition, plays asimilar role in orientation selectivity and spatial frequency selec-

SF−Band

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idth

(de

gree

)

700

50

100

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A

Fig. 7. Correlation between orientation bandwidth and spatial frequency bandwidth (simspatial frequency bandwidth similar to the pattern in real 4Ca data (A).

tivity (cf. Ringach, Bredfeldt, et al., 2002). In the model, local corti-cal inhibition acts like shunting inhibition (cf. Shelley et al., 2002),a point we take up in Section 4. Next we explore this inhibitionhypothesis by examining the link between variations in inhibitoryconductance and selectivity in the V1 model.

3.5. Synaptic conductance and orientation and spatial selectivity

The model emulated the behavior of V1 in having a strong cor-relation between orientation and spatial frequency selectivities, asshown in Fig. 6. This suggests that we could understand the rela-tion between these selectivities in V1 by trying to understand theirrelation in the model. The way we did this was by plotting spatialfrequency LSFV and orientation CV versus the inhibitory and excit-atory conductance strength for the population of model neurons.The conductance strength for each model cell was quantified asthe average synaptic conductance during visual stimulation, aver-aged across all stimuli in one dimension, for instance averagedacross all eight spatial frequencies used as stimuli to obtain themodel cell’s spatial-frequency tuning curve. Fig. 8A and C showthe relation between strengths of inhibitory and excitatory synap-tic conductances and the orientation CV for simple cells (Zhu et al.,2009). Higher amounts of average synaptic inhibition were associ-ated with lower CV (more orientation selectivity). There appearedto be no consistent relation between variation in excitatory synap-tic conductance and CV for model simple cells. Fig. 8B and D showthe relation between strengths of inhibitory and excitatory synap-tic conductances and spatial frequency LSFV for simple cells. High-

70

width (octaves)

N=3035r=0.401

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B

ple cells). The network model generates a correlation pattern (B) for orientation and

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rage

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ance

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Fig. 8. Inhibition, excitation and selectivity for simple cells. Plots of mean inhibition and excitation vs circular variance and LSFV. Standard errors indicated by the error bars.There is a correlation between simple cells’ selectivity and the inhibition they received in both orientation (A) and spatial frequency (B) domain. However, there is no suchrelationship between simple cells’ selectivity and their excitation (C and D). The mean excitation and inhibition value for each model neuron is the average over time and overthe responses to the eight stimuli used to obtain a tuning curve in each domain.

W. Zhu et al. / Vision Research 50 (2010) 2261–2273 2269

er amounts of selectivity, lower LSFV, were associated with stron-ger mean inhibition. Synaptic conductance strengths in the modelvaried because of location within the layer of V1, because of vari-ations in the total amount of cortical input from neighboring neu-rons. What Fig. 8 indicates is that the variation of inhibitorysynaptic strength across the population was an underlying variablethat caused the observed correlation between orientation and spa-tial frequency selectivity.

Another way to analyze the influence of cortical mechanisms onthe correlation of selectivities is to seek differential patterns ofamount of synaptic input in the scatter plots of orientation vs spa-tial frequency selectivity. To do this, we re-plotted the scatter plotoriginally shown in Fig. 6B but used only a randomly selected sub-sample of 10% of the cells in the model so that individual pointscould be visualized. We marked each point (each model simplecell) with a color that signified the average amount of inhibition(Fig. 9) or excitation (Fig. 10). Fig. 9 shows that the model simplecells that had higher selectivities for both orientation and spatialfrequency tended to have higher average inhibitory input. Modelsimple cells that had lower average inhibitory input tended to belocated in the scatter plot in the region of low selectivity. Therewas no consistent pattern for excitation (Fig. 10). These cell-by-cellanalyses support the conclusions based on population averages inFig. 8.

Another way to illustrate this important point is to divide thecell population along the regression line of correlation betweenorientation and spatial frequency selectivity, as in Fig. 11.

A line perpendicular to the regression line divided the sub-sam-pled population of model simple cells approximately into twoequal groups. Those cells lying to the left of and below the dividingline were the more selective group. The more selective group was

colored purple and the less selective group colored green. Then wecomputed the mean inhibitory conductance (averaged over the 15runs that were used for the orientation and spatial-frequency tun-ing curves) and averaged the inhibitory conductance across thetwo groups, the more and less selective. The more selective grouphad a higher average inhibitory conductance. When we computedthe average excitations for the two groups, there was no difference.This computational result supports the idea that the common fac-tor that controlled correlated selectivity was cortico-corticalinhibition.

3.6. Complex cells: feature selectivity and correlation

Complex cells, the cells with modulation ratio <1, tended to beless selective for orientation (Ringach, Shapley, et al., 2002; Xinget al., 2004) and spatial frequency (Xing et al., 2004) throughoutall layers of V1. Complex cells in layer 4Ca obeyed this rule too(Fig. 12A and B). The complex cells in the large-scale model alsotended to be less selective than the simple cells (Fig. 12D and E).Furthermore, the correlation between orientation and spatial fre-quency selectivity was weaker among complex than simple cells,in the V1 4Ca data (Fig. 12C) and also in the large-scale model(Fig. 12F). As for the simple cells, the model’s selectivity and corre-lation of selectivity was less than for the data, but the tendencieswere in the same direction. Previously (Zhu et al., 2009), wepointed out that the spatial-frequency tuning of model complexcells depended on both inhibitory and excitatory network activityand also on the sparseness of intra-cortical excitatory–excitatoryconnections as implemented in this large-scale model followingthe lead of Tao et al. (2006). Similar reasoning applies to under-standing the orientation selectivity of complex cells in layer 4Ca.

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N = 337

r = 0.65

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Inhibitory conductance

Fig. 9. Correlation between orientation and spatial frequency selectivity andinhibition. The correlation pattern of orientation and spatial frequency selectivityfor a randomly selected subset (N = 337) of model simple cells. Each cell is markedby a color that indicates its average inhibitory conductance across the fifteenexperimental runs that were used to generate the orientation and spatial-frequencytuning curves. The cells with higher average conductances tend to plot to the lowerleft hand region of the cloud of points while the cells with lower average inhibitoryconductance tend to plot to the upper right hand region of the plot, the region oflow selectivity in both domains.

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r = 0.65

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Excitatory conductance

Fig. 10. Correlation between orientation and spatial frequency selectivity andexcitation. The correlation pattern of orientation and spatial frequency selectivityfor a randomly selected subset (N = 337) of model simple cells. Each cell is markedby a color that indicates its average excitatory conductance across the fifteenexperimental runs that were used to generate the two tuning curves for orientationand spatial frequency. There is no clear pattern of clustering of cells with low orhigh average excitatory conductance.

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r = 0.65

Average Inh = 403; Average Exc = 86Average Inh = 443; Average Exc = 86

Fig. 11. Correlation between orientation and spatial frequency selectivity andexcitation/inhibition. Along the linear regression line, we decomposed the set ofpoints into two sets of roughly equal numbers of points. Then we averaged theinhibition across each of the two sets. For the upper, less selective set the averageinhibitory conductance = 403 s�1 while for the lower set of higher selective pointsthe average inhibitory conductance = 443 s�1. Then we averaged the excitationacross each of the two sets. For the upper, less selective set the average excitatoryconductance = 86 s�1 and the lower set of had the same average excitatoryconductance = 86 s�1.

2270 W. Zhu et al. / Vision Research 50 (2010) 2261–2273

4. Discussion

4.1. A mechanism for correlated spatial selectivity

In this paper we have presented a realistic model of V1 that pro-duces orientation selectivity, spatial frequency selectivity and thecorrelation of selectivity similar to experimental results in V1 cor-tex. The major conclusion of this paper is the importance of corticalinhibition as the source of the correlation of orientation and spatialfrequency selectivity in V1. As we have shown in this study, thespatial summation of LGN cells onto V1 neurons provides both ori-entation and spatial frequency preferences but not high selectivity(see also Zhu et al., 2009). Furthermore, the selectivity of LGN in-puts is only weakly correlated in the orientation and spatial fre-quency domains (Fig. 6C). Therefore, it is very unlikely that astrong correlation between orientation selectivity and spatial fre-quency selectivity is directly due to the spatial summation ofLGN inputs. Our result also suggests that the correlation of orienta-tion and spatial frequency selectivity is not mainly due to the exci-tation that a model neuron receives, because neurons withdifferent selectivities, on average, receive a similar amount of exci-tation (Figs. 8, 10 and 11).

In our model the mechanism that is predominantly responsiblefor spatial selectivities and their correlation is intra-cortical inhibi-tion. In the model, inhibition suppresses V1 neurons’ responses tonon-optimal stimuli and generates high selectivity to orientationand spatial frequency. The variability of cortical inhibition for indi-vidual cells naturally explains the correlation of orientation andspatial frequency selectivity in the V1 population. Broadly-tunedinhibition also is the mechanism that sharpens orientation selec-tivity in other large-scale cortical models (McLaughlin et al.,2000; Troyer et al., 1998); we show here that local-circuit inhibi-tion is a mechanism that can heighten selectivity in both orienta-tion and spatial frequency domains, as was hypothesized earlier

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0.30 +/- 0.03

Fig. 12. Orientation and spatial frequency selectivity and their correlation (complex cells). CV and LSFV in 4Ca complex cells (A and B) and in model complex neurons (D andE). The network model generates a correlation pattern (F) for complex cells’ orientation and spatial frequency selectivity that is similar to that seen in the data (C).

W. Zhu et al. / Vision Research 50 (2010) 2261–2273 2271

from reverse correlation data (Ringach, Bredfeldt, et al., 2002). Themodel’s inhibition is more broadly-tuned in orientation and stron-ger at low spatial frequencies than are the responses of modelexcitatory cells (compare Figs. 4 and 5). The predicted broader tun-ing of visual cortical inhibitory neurons in the orientation domainis consistent with recent experimental data (Cardin et al., 2007;Nowak et al., 2008) as is the predicted broader tuning of inhibitoryneurons in the spatial frequency domain (Cardin et al., 2007).

It is important to note that the theoretical explanation offeredhere for spatial-frequency tuning, orientation tuning, and their cor-relation in the visual cortex is very different from the classical pic-ture that explains cortical spatial selectivity as a consequence ofquasi-linear filtering by the neurons’ spatio-temporal receptivefield (Dayan & Abbott, 2001; De Valois & De Valois, 1988; DeAnge-lis, Ohzawa, & Freeman, 1995; Lampl, Anderson, Gillespie, & Fer-ster, 2001; Robson, 1975). Sharpening of cortical tuning in thelarge-scale model we studied comes about because of nonlinearsuppression caused by shunting intra-cortical inhibition. The sim-ilarity of the correlation between orientation and spatial frequencyselectivity in the model and in the real cortex supports the hypoth-esis that, in the real cortex, spatial selectivity in different dimen-sions is greatly sharpened by cortical (suppressive) nonlinearinhibition. This theoretical result is similar to what was proposedfrom experimental results on the time-evolution of feature selec-tivities in the orientation (Ringach, Shapley, et al., 2002; Shapley,Hawken, & Ringach, 2003; Xing et al., 2005) and spatial frequency(Bredfeldt & Ringach, 2002; Ringach, Bredfeldt, et al., 2002) do-mains. Most other realistic cortical models have not addressedthe issue of spatial frequency and orientation selectivity and theirhigh correlation (McLaughlin et al., 2000; Tao et al., 2004; Troyeret al., 1998). One exception is the model developed by Wielaardand Sajda (2006) to study extra-classical receptive field propertiesof V1 neurons. The model of Wielaard and Sajda (2006) resemblesour present model in that it is based on a network of conductance-based neurons in which strong inhibitory coupling leads to corticalsharpening of selectivity for orientation and spatial frequency. It isnot known whether or not the Wielaard–Sajda model generatesthe correlation between orientation and spatial frequency selectiv-ity we found because they did not ask this question of their model.

Finn, Priebe, and Ferster (2007) revived the feedforward modelto try to explain orientation selectivity. Specifically, they reportedthat contrast-invariant orientation selectivity could be observed ina model that includes no visually-driven inhibition but that doeshave a high spike threshold and variability in the membrane po-tential. With respect to the importance of noise and threshold,our model agrees with Finn et al.’s (2007) results in that spikesin our model are evoked by fluctuations in the membrane potentialand the spike threshold plays an important role in making spikerates more selective than the LGN inputs. In this way the presentmodel also resembles the simple cell model of Wielaard et al.(2001) in which neurons fired spikes only when noise fluctuationscaused the membrane potential to exceed threshold (Shelley et al.,2002). However, above we showed that, in the present model, highselectivity for orientation and spatial frequency was correlatedwith high values of inhibitory conductance. Therefore one test ofour model is to test a strong prediction: that blocking local, corticalinhibition should markedly reduce spatial frequency and orienta-tion selectivity together. This would also test Finn et al. (2007)modified feedforward model that predicts no effect on selectivityof blocking inhibition.

It is difficult to understand how a feedforward model would ex-plain sharpening of the cortical spatial frequency responses byselectively reducing the amplitude of response at low-spatial fre-quency (as in Fig. 2C) because the tuning mechanism in the feed-forward model depends only on the amplitude of response andspike threshold, not on stimulus parameters. In our model, the spa-tial frequency response is selectively reduced at low-spatial fre-quency because of the strong spatial frequency responses ofinhibitory neurons at low-spatial frequency (as implied by the re-sults in Figs. 2 and 4; further documented in Zhu et al. (2009). InZhu et al. (2009), we discussed other experimental tests whereour model and the model in Finn et al. (2007) make differentpredictions.

4.2. Diversity of feature selectivity

Most theoretical work on the V1 cortical network has focusedon mechanisms that can generate sharply-tuned V1 cells. However,

2272 W. Zhu et al. / Vision Research 50 (2010) 2261–2273

very little work has addressed the fact that feature selectivity in V1is very diverse even within one cortical layer, such as 4Ca. To us,the diversity of feature selectivity is as important as the featureselectivity itself. Experimental studies have shown that both orien-tation selectivity (Ringach, Shapley, et al., 2002) and spatial fre-quency selectivity (Xing et al., 2004) are widely distributed inthe population of macaque V1 neurons in all layers. The large-scalenetwork model proposed in this paper generates diversity in theV1 network by variation from one cell to another in the patternof LGN inputs and also by variation in the strength of intra-corticalsynaptic interactions.

4.3. The role of local-circuit inhibition in the visual cortex

The role of inhibition in the large-scale model has implicationsfor the function of the cerebral cortex generally. Shelley et al.(2002) showed that models of V1 are in a high conductance state,and this conclusion is supported again in this study (the high aver-age conductance values in Fig. 8 for instance). For the high conduc-tance state, Shelley et al. (2002) obtained the result that the time-modulated response of a model V1 neuron’s membrane potentialwas to a good approximation a constant plus the ratio of excitatoryconductance divided by inhibitory conductance. Therefore, in themodel local-circuit inhibition acts like shunting inhibition. There-fore, one may conclude that local-circuit inhibition acts like a localdivider of excitation, like a gain control. In our model, variation inthe value of this gain signal across the population is what causesthe variations in selectivity and induces correlations in spatialselectivity (Figs. 8–10). The conclusion that local-circuit inhibitionis a local gain control emerges naturally from the model of V1 as arecurrent network in a high conductance state, justifying theassumption of a contrast gain control or normalization mechanismto fit data (Heeger, 1992). But the cortical architecture that gener-ates the local-circuit inhibition in V1 cortex is found throughoutcortex, and therefore inhibition’s role throughout the cortex maybe the same as it is in V1.

4.4. Modeling considerations

We had to make a choice in the model about how spatial fre-quency preference or bias, induced from the LGN input, is orga-nized spatially in the input to the model. The idea of randomclusters is our interpretation of the optical imaging paper aboutspatial frequency maps in V1 by Sirovich and Uglesich (2004).Completely different optical imaging maps have been reportedby others (Issa, Trepel, & Stryker, 2000). Our judgment was thatthe quasi-random clustering of spatial frequency bias deducedfrom the maps in Sirovich and Uglesich (2004) were most consis-tent with the electrophysiological recording literature (DeAngeliset al., 1999). Furthermore, Xing et al. (2004) comment that thereappears to be little correlation between preferred spatial frequencyand the circular variance of orientation selectivity. This result alsomade us believe the more random maps suggested by Sirovich andUglesich (2004). The resulting model does account for correlationsbetween spatial selectivities in a way that makes the choice wemade seem reasonable, but further experiments and modeling willbe needed to validate this choice.

What took most of the theoretical effort for this paper was theexploration of different values of the ratio of recurrent cortico-cor-tical excitation to inhibition, SEE/SEI. We found that if SEI was tooweak in the model, there were not enough simple cells and themodel’s spatial-frequency tuning and orientation tuning distribu-tions (as in Fig. 3) were very different from the real distribution.This theoretical finding about the importance of cortico-corticalinhibition echoes physiological results that the classification of acell as simple or complex can be changed by pharmacological

manipulation of cortical inhibition (Murthy & Humphrey, 1999).The results of this paper reinforce the model’s reliance on cor-tico-cortical inhibition because they show that the correlation be-tween what could be unrelated tuning parameters is quite robustand that a model with strong cortico-cortical inhibition can explainthis correlation.

Acknowledgments

This work was supported by the Swartz Foundation, by NIHTraining Grant T32-EY007158, and by NIH Grant R01 EY-01472.We would like to thank Chun-I Yeh, Patrick Williams, and DarioRingach for their helpful comments on manuscript drafts. Thanksto Michael Hawken, J. Andrew Henrie, Patrick Williams, SiddharthaJoshi, Christopher Henry, and Elizabeth Johnson for their help withthe physiological experiments.

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