Toward a Single-Cell Account for 
Binocular Disparity Tuning: An Energy 
Model May be Hiding in Your Dendrites 
Bartlett W. Mel 
Department of Biomedical Engineering 
University of Southern California, MC 1451 
Los Angeles, CA 90089 
mel@quake.usc.edu 
Daniel L. Ruderman 
The Salk Institute 
10010 N. Torrey Pines Road 
La Jolla, CA 92037 
ruderman@salk.edu 
Kevin A. Archie 
Neuroscience Program 
University of Southern California 
Los Angeles, CA 90089 
karchie@quake.usc.edu 
Abstract 
Hubel and Wiesel (1962) proposed that complex cells in visual cor- 
tex are driven by a pool of simple cells with the same preferred 
orientation but different spatial phases. However, a wide variety of 
experimental results over the past two decades have challenged the 
pure hierarchical model, primarily by demonstrating that many 
complex cells receive monosynaptic input from unoriented LGN 
cells, or do not depend on simple cell input. We recently showed us- 
ing a detailed biophysical model that nonlinear interactions among 
synaptic inputs to an excitable dendritic tree could provide the non- 
linear subunit computations that underlie complex cell responses 
(Mel, Ruderman, & Archie, 1997). This work extends the result 
to the case of complex cell binocular disparity tuning, by demon- 
strating in an isolated model pyramidal cell (1) disparity tuning 
at a resolution much finer than the the overall dimensions of the 
cell's receptive field, and (2) systematically shifted optimal dispar- 
ity values for rivalrous pairs of light and dark bars--both in good 
agreement with published reports (Ohzawa, DeAngelis,  Free- 
man, 1997). Our results reemphasize the potential importance of 
intradendritic computation for binocular visual processing in par- 
ticular, and for cortical neurophysiology in general. 
Single- Cell Account for Binocular Disparity Tuning 
Introduction 
209 
Binocular disparity is a powerful cue for depth in vision. The neurophysiological 
basis for binocular disparity processing has been of interest for decades, spawned 
by the early studies of Hubel and Wiesel (1962) showing neurons in primary visual 
cortex which could be driven by both eyes. Early qualitative models for disparity 
tuning held that a binocularly driven neuron could represent a particular disparity 
(zero, near, or far) via a relative shift of receptive field (RF) centers in the right 
and left eyes. According to this model, a binocular cell fires maximally when an 
optimal stimulus, e.g. an edge of a particular orientation, is simultaneously centered 
in the left and right eye receptive fields, corresponding to a stimulus at a specific 
depth relative to the fixation point. An account of this kind is most relevant to the 
case of a cortical "simple" cell, whose phase-sensitivity enforces a preference for a 
particular absolute location and contrast polarity of a stimulus within its monocular 
receptive fields. 
This global receptive field shift account leads to a conceptual puzzle, however, when 
binocular complex cell receptive fields are considered instead, since a complex cell 
can respond to an oriented feature nearly independent of position within its monoc- 
ulax receptive field. Since complex cell receptive field diameters in the cat lie in the 
range of 1-3 degrees, the excessive "play" in their monocular receptive fields would 
seem to render complex cells incapable of signaling disparity on the much finer scale 
needed for depth perception (measured in minutes). 
Intriguingly, various authors have reported that a substantial fraction of complex 
cells in cat visual cortex are in fact tuned to left-right disparities much finer than 
that suggested by the size of the monocular RF's. For such cells, a stimulus deliv- 
ered at the proper disparity, regardless of absolute position in either eye, produces 
a neural response in excess of that predicted by the sum of the monocular responses 
(Pettigrew, Nikara, & Bishop, 1968; Ohzawa, DeAngelis, & Freeman, 1990; Ohzawa 
et al., 1997). Binocular responses of this type suggest that for these cells, the left 
and right RF's are combined via a correlation operation rather than a simple sum 
(Nishihara & Poggio, 1984; Koch & Poggio, 1987). This computation has also been 
formalized in terms of an "energy" model (Ohzawa et al., 1990, 1997), building 
on the earlier use of energy models to account for complex cell orientation tuning 
(Pollen & Ronner, 1983) and direction selectivity (Adelson & Bergen, 1985). In 
an energy model for binocular disparity tuning, sums of linear Gabor filter out- 
puts representing left and right receptive fields are squared to produce the crucial 
multiplicative cross terms (Ohzawa et al., 1990, 1997). 
Our previous biophysical modeling work has shown that the dendritic tree of a cor- 
tical pyramidal cells is well suited to support an approximative high-dimensional 
quadratic input-output relation, where the second-order multiplicative cross terms 
arise from local interactions among synaptic inputs carried out in quasi-isolated 
dendritic "subunits" (Mel, 1992b, 1992a, 1993). We recently applied these ideas 
to show that the position-invariant orientation tuning of a monocular complex cell 
could be computed within the dendrites of a single cortical cell, based exclusively 
upon excitatory inputs from a uniform, overlapping population of unoriented ON 
and OFF cells (Mel et al., 1997). Given the similarity of the "energy" formulations 
previously proposed to account for orientation tuning and binocular disparity tun- 
ing, we hypothesized that a similar type of dendritic subunit computation could 
underlie disparity tuning in a binocularly driven complex cell. 
210 B. W. Mel, D. L. Ruderman and K. A. Archie 
Parameter Value 
Rm 10kFtcm 2 
Ra 200Ftcm 
Cm 1.0pF/cm 2 
Vrest -70 mV 
Compartments 615 
Somatic Na, DR 0.20, 0.12 S/cm 2 
Dendritic a, 0Drt 0.05, 0.03 S/cm 2 
Input frequency 0 - 100 Hz 
AMPA 0.027 nS - 0.295 nS 
TAMPA (on, off) 0.5 ms, 3 ms 
NMDA 0.27 nS - 2.95 nS 
TNMDA (on, off) 0.5 ms, 50 ms 
Esyn 0 mV 
Table 1: Biophysical simulation parameters. Details of HH channel implementa- 
tion are given elsewhere (Mel, 1993); original HH channel implementation cour- 
tesy Ojvind Bernander and Rodney Douglas. In order that local EPSP size be 
held approximately constant across the dendritic arbor, peak synaptic conduc- 
tance at dendritic location x was approximately scaled to the local input resis- 
tance (inversely), given by syn(X) -- C/lin(X), where c was a constant, and 
/in(X) = max(Rin(X), 200MFt). Input resistance Rin(X) was measured for a pas- 
sive cell. Thus syn Was identical for all dendritic sites with input resistance below 
200Mft, and was given by the larger conductance value shown; roughly 50% of the 
tree fell within a factor of 2 of this value. Peak conductances at the finest distal tips 
were smaller by roughly a factor of 10 (smaller number shown). Somatic input resis- 
tance was near 24Mft. The peak synaptic conductance values used were such that 
the ratio of steady state current injection through NMDA vs. AMPA channels was 
1.2 + 0.4. Both AMPA and NMDA-type synaptic conductances were modeled using 
the kinetic scheme of Destexhe et al. (1994); synaptic activation and inactivation 
time constants are shown for each. 
2 Methods 
Compartmental simulations of a pyramidal cell from cat visual cortex (morphol- 
ogy courtesy of Rodney Douglas and Kevan Martin) were carried out in NEURON 
(Hines, 1989); simulatior, parameters are summarized in Table 1. The soma and den- 
dritic membrane contained Hodgkin-Huxley-type (HH) voltage-dependent sodium 
and potassium channels. Following evidence for higher spike thresholds and decre- 
mental propagation in dendrites (Stuart &; Sakmann, 1994), HH channel density was 
set to a uniform, 4-fold lower value in the dendritic membrane relative to that of the 
cell body. Excitatory synapses from LGN cells included both NMDA and AMPA- 
type synaptic conductances. Since the cell was considered to be isolated from the 
cortical network, inhibitory input was not modeled. Cortical cell responses were 
reported as average spike rate recorded at the cell body over the 500 ms stimulus 
period, excluding the 50 ms initial transient. 
The binocular LGN consisted of two copies of the monocular LGN model used 
previously (Mel et al., 1997), each consisting of a superimposed pair of 64x64 ON 
and OFF subfields. LGN cells were modeled as linear, half-rectified center-surround 
filters with centers 7 pixels in width. We randomly subsampled the left and right 
LGN arrays by a factor of 16 to yield 1,024 total LGN inputs to the pyramidal cell. 
A Single-Cell Account for Binocular Disparity Tuning 211 
A developmental principle was used to determine the spatial arrangement of these 
1,024 synaptic contacts onto the dendritic branches of the cortical cell, as follows. 
A virtual stimulus ensemble was defined for the cell, consisting of the complete set 
of single vertical light or dark bars presented binocularly at zero-disparity within 
the cell's receptive field. Within this ensemble, strong pairwise correlations existed 
among cells falling into vertically aligned groups of the same (ON or OFF) type, 
and cells in the vertical column at zero horizontal disparity in the other eye. These 
binocular cohorts of highly correlated LGN cells were labeled mutual "friends". 
Progressing through the dendritic tree in depth first order, a randomly chosen LGN 
cell was assigned to the first dendritic site. A randomly chosen "friend" of hers 
was assigned to the second site, the third site was assigned to a friend of the site 2 
input, etc., until all friends in the available subsample were assigned (4 from each 
eye, on average). If the friends of the connection at site i were exhausted, a new 
LGN cell was chosen at random for site i + 1. In earlier work, this type of synaptic 
arrangement was shown to be the outcome of a Hebb-type correlational learning 
rule, in which random, activity independent formation of synaptic contacts acted 
to slowly randomize the axo-dendritic interface, shaped by Hebbian stabilization of 
synaptic contacts based on their short-range correlations with other synapses. 
3 Results 
Model pyramidal cells configured in this way exhibited prominent phase-invariant 
orientation tuning, the hallmark response property of the visual complex cell. Mul- 
tiple orientation tuning curves are shown, for example, for a monocular complex cell, 
giving rise to strong tuning for light and dark bars across the receptive field (fig. 1). 
The bold curve shows the average of all tuning curves for this cell; the half-width at 
half max is 25 , in the normal range for complex cells in cat visual cortex (Orban, 
1984). When the spatial arrangement of LGN synaptic contacts onto the pyra- 
midal cell dendrites was randomly scrambled, leaving all other model parameters 
unchanged, orientation tuning was abolished in this cell (right frame), confirming 
the crucial role of spatially-mediated nonlinear synaptic interactions (average curve 
from left frame is reproduced for comparison). 
Disparity-tuning in an orientation-tuned binocular model cell is shown in fig. 2, com- 
pared to data from a complex cell in cat visual cortex (adapted from Ohzawa et al. 
(1997)). Responses to contrast matched (light-light) and contrast non-matched 
(light-dark) bar pairs were subtracted to produce these plots. The strong diagonal 
structure indicates that both the model and real cells responded most vigorously 
when contrast-matched bars were presented at the same horizontal position in the 
left and right-eye RF's (i.e. at zero-disparity), whereas peak responses to contrast- 
non-matched bars occured at symmetric near and far, non-zero disparities. 
4 Discussion 
The response pattern illustrated in fig. 2A is highly similar to the response generated 
by an analytical binocular energy model for a complex cell (Ohzawa et al., 1997)' 
RC(XL,XR) 
= {exp(-kX)cos(2rfXz) +exp(-kX)cos(2rfXR)} 2 + 
{exp (-kX})sin (2rfX) + exp (-kX)sin (2,rfXn)}2, 
(1) 
where X and Xn are the horizontal bar positions to the two eyes, k is the factor 
212 B. W. Mel, D. L. Ruderman and K. A. Archie 
Orientation Tuning 
Ordered vs. Scrambled 
70 
60 
50 
40 
30 
20 
10 
0 
-90 
/  average -4-- 
Ight 0 -4- - 
/  dark4 -B - 
: _/ light8 -)4- 
/ ,'r , light 16 
i, , da 6 --- 
-0 -0 0 0 0 0 
Oritabon (dre) 
55 
50 
45 
40 
35 
30 
25 
20 
15 
10 
5 
-90 
 ordered o 
scrambled -4- - 
-60 -30 0 30 60 90 
Orientation (degrees) 
Figure 1: Orientation tuning curves are shown in the left frame for light and dark 
bars at 3 arbitrary positions. Essentially similar responses were seen at other re- 
ceptive field positions, and for other complex cells. Bold trace indicates average 
of tuning curves at positions 0, 1, 2, 4, 8, and 16 for light and dark bars. Similar 
form of 6 curves shown reflects the translation-invariance of the cell's response to 
oriented stimuli, and symmetry with respect to ON and OFF input. Orientation 
tuning is eliminated when the spatial arrangement of LGN synapses onto the model 
cell dendrites is randomly scrambled (right frame). 
Complex Cell Model 
Complex Cell in Cat V1 
Ohzawa, Deangelis, & Freeman, 1997 
Right eye position 
Right eye position 
Figure 2: Comparison of disparity tuning in model complex cell to that of a binoc- 
ular complex cell from cat visual cortex. Light or dark bars were presented simul- 
taneously to the left and right eyes. Bars could be of same polarity in both eyes 
(light, light) or different polarity (light, dark); cell responses for these two cases were 
subtracted to produce plot shown in left frame. Right frame shows data similarly 
displayed for a binocular complex cell in cat visual cortex (adapted from Ohzawa 
et al. (1997)). 
A Single-Cell Account for Binocular Disparity Tuning 213 
that determines the width of the subunit RF's, and f is the spatial frequency. 
In lieu of literal simple cell "subunits", the present results indicate that the subunit 
computations associated with the terms of an energy model could derive largely 
from synaptic interactions within the dendrites of the individual cortical cell, driven 
exclusively by excitatory inputs from unoriented, monocular ON and OFF cells 
drawn from a uniform overlapping spatial distribution. While lateral inhibition 
and excitation play numerous important roles in cortical computation, the present 
results suggest they are not essential for the basic features of the nonlinear disparity 
tuned responses of cortical complex cells. Further, these results address the paradox 
as to how inputs from both unoriented LGN cells and oriented simple cells can 
coexist without conflict within the dendrites of a single complex cell. 
A number of controls from previous work suggest that this type of subunit process- 
ing is very robustly computed in the dendrites of an individual neuron, with little 
sensitivity to biophysical parameters and modeling assumptions, including details of 
the algorithm used to spatially organize the geniculo-cortical projection, specifics of 
cell morphology, synaptic activation density across the dendritic tree, passive mem- 
brane and cytoplasmic parameters, and details of the kinetics, voltage-dependence, 
or spatial distribution of the voltage-dependent dendritic channels. 
One important difference between a standard energy model and the intradendritic 
responses generated in the present simulation experiments is that the energy model 
has oriented RF structure at the linear (simple-cell-like) stage, giving rise to ori- 
ented, antagonistic ON-OFF subregions (Movshon, Thompson, &; Tolhurst, 1978), 
whereas the linear stage in our model gives rise to center-surround antagonism only 
within individual LGN receptive fields. Put another way, the LGN-derived subunits 
in the present model cannot provide all the negative cross-terms that appear in the 
energy model equations, specifically for pairs of pixels that fall outside the range of 
a single LGN receptive field. 
While the present simulations involve numerous simplifications relative to the full 
complexity of the cortical microcircuit, the results nonetheless emphasize the po- 
tential importance of intradendritic computation in visual cortex. 
Acknowledgements 
Thanks to Ken Miller, Allan Dobbins, and Christof Koch for many helpful comments 
on this work. This work was funded by the National Science Foundation and the 
Office of Naval Research, and by a Sloan Foundation Fellowship (D.R.). 
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