Anatomical origin and computational role 
of diversity in the response properties of 
cortical neurons 
Kalanit Grill Spector t Shimon Edelman t Rafael Malach; 
Depts of tApplied Mathematics and Computer Science and ;Neurobiology 
The Weizmann Institute of Science 
Rehovot 76100, Israel 
{kalanit,edelman,malach}wisdom.weizmann.ac.il 
Abstract 
The maximization of diversity of neuronal response properties has been 
recently suggested as an organizing principle for the formation of such 
prominent features of the functional architecture of the brain as the corti- 
cal columns and the associated patchy projection patterns (Malach, 1994). 
We show that (1) maximal diversity is attained when the ratio of dendritic 
and axonal arbor sizes is equal to one, as found in many cortical areas 
and across species (Lund et al., 1993; Malach, 1994), and (2) that maxi- 
mization of diversity leads to better performance in systems of receptive 
fields implementing steerable/shiftable filters, and in matching spatially 
distributed signals, a problem that arises in many high-level visual tasks. 
1 Anatomical substrate for sampling diversity 
A fundamental feature of cortical architecture is its columnar organization, mani- 
fested in the tendency of neurons with similar properties to be organized in columns 
that run perpendicular to the cortical surface. This organization of the cortex was ini- 
tially discovered by physiological experiments (Mouncastle, 1957; Hubel and Wiesel, 
1962), and subsequently confirmed with the demonstration of histologically defined 
columns. Tracing experiments have shown that axonal projections throughout the 
cerebral cortex tend to be organized in vertically aligned clusters or patches. In par- 
ticular, intrinsic horizontal connections linking neighboring cortical sites, which may 
extend up to 2 - 3 ram, have a striking tendency to arborize selectively in preferred 
sites, forming distinct axonal patches 200- 300 yra in diameter. 
Recently, it has been observed that the size of these patches matches closely 
the average diameter of individual dendritic arbors of upper-layer pyramidal cells 
118 Kalanit Grill Spector, Shimon Edelman, Rafael Malach 
2500 
1000 
0 
0 
02 04 06 O I 12 14 16 16 
Figure 1: Left: histograms of the percentage of patch-originated input to the neurons, 
plotted for three values of the ratio r between the dendritic arbor and the patch 
diameter (0.5, 1.0, 2.0). The fiattest histogram is obtained for r = 1.0 Iight: the 
diversity of neuronal properties (as defined in section 1) vs. r. The maximum is 
attained for r = 1.0, a value compatible with the anatomical data. 
(see Malach, 1994, for a review). Determining the functional significance of this 
correlation, which is a fundamental property that holds throughout various cortical 
areas and across species (Lund et al., 1993), may shed light on the general principles 
of operation of the cortical architecture. One such driving principle may be the 
maximization of diversity of response properties in the neuronal population (Malach, 
1994). According to this hypothesis, matching the sizes of the axonal patches and the 
dendritic arbors causes neighboring neurons to develop slightly different functional 
selectivity profiles, resulting in an even spread of response preferences across the 
cortical population, and in an improvement of the brain's ability to process the variety 
of stimuli likely to be encountered in the environment. 1 
To test the effect of the ratio between axonal patch and dendritic arbor size on 
the diversity of the neuronal population, we conducted computer simulations based 
on anatomical data concerning patchy projections (Rockland and Lund, 1982; Lund 
et al., 1993; Malach, 1992; Malach et al., 1993). The patches were modeled by disks, 
placed at regular intervals of twice the patch diameter, as revealed by anatomical 
labeling. Dendritic arbors were also modeled by disks, whose radii were manipulated 
in different simulations. The arbors were placed randomly over the axonal patches, 
at a density of 10,000 neurons per patch. We then calculated the amount of patch- 
related information sampled by each neuron, defined to be proportional to the area of 
overlap of the dendritic tree and the patch. The results of the calculations for three 
! Necessary conditions for obtaining dendritic sampling diversity are that dendritic arbors cross 
freely through column borders, and that dendrites which cross column borders sample with equal 
probability from patch and inter-patch compartments. These assumptions were shown to be valid 
in (Malach, 1992; Malach, 1994). 
Diversity in the Response Properties of Cortical Neurons I 19 
values of the ratio of patch and arbor diameters appear in Figure 1. 
The presence of two peaks in the histogram obtained with the arbor/patch ratio 
r - 0.5 indicates that two dominant groups are formed in the population, the first 
receiving most of its input from the patch, and the second - from the inter-patch 
sources. A value of r - 2.0, for which the dendritic arbors are larger than the axonal 
patch size, yields near uniformity of sampling properties, with most of the neurons 
receiving mostly patch-originated input, as apparent from the single large peak in 
the histogram. To quantify the notion of diversity, we defined it as diversity ~< 
I[ >-l, where n(p) is the number of neurons that receive p percent of their inputs 
from the patch, and < > denotes average over p. Figure 1, right, shows that 
diversity is maximized when the size of the dendritic arbors matches that of the 
axonal patches, in accordance with the anatomical data. This result confirms the 
diversity maximization hypothesis stated in (Malach, 1994). 
2 
Orientation tuning: a functional manifestation 
of sampling diversity 
The orientation columns in V1 are perhaps the best-known example of functional 
architecture found in the cortex (Hubel and Wiesel, 1962). Cortical maps obtained 
by optical imaging (Grinvald et al., 1986) reveal that orientation columns are patchy 
rather then slab-like: domains corresponding to a single orientation appear as a 
mosaic of round patches, which tend to form pinwheel-like 
changes in the orientation of the stimulus lead to smooth 
these domains. We hypothesized that this smooth variation 
found in V1 originates in patchy projections, combined with 
properties of neurons sampling from these projections. The 
the rest of this section substantiate this hypothesis. 
structures. Incremental 
shifts in the position of 
in orientation selectivity 
diversity in the response 
simulations described in 
Computer simulations. The goal of the simulations was to demonstrate that a 
limited number of discretely tuned elements can give rise to a continuum of responses. 
We did not try to explain how the original set of discrete orientations can be formed 
by projections from the LGN to the striate cortex; several models for this step can 
be found in the literature (Hubel and Wiesel, 1962; Vidyasagar, 1985). 2 In setting 
the size of the original discrete orientation columns we followed the notion of a point 
image (Macllwain, 1986), defined as the minimal cortical separation of cells with 
non-overlapping RFs. Each column was tuned to a specific angle, and located at 
an approximately constant distance from another column with the same orientation 
tuning (we allowed some scatter in the location of the RFs). The RFs of adjacent units 
with the same orientation preference were overlapping, and the amount of overlap 
eIn particular, it has been argued (Vidyasagar, 1985) that the receptive fields at the output of the 
LGN are already broadly tuned for a small number of discrete orientations (possibly just horizontal 
and vertical), and that at the cortical level the entire spectrum of orientations is generated from the 
discrete set present in the geniculate projection. 
120 Kalanit Grill Spector, Shimon Edelman, Rafael Malach 
007 
O06 
O05 
0 04 
OO3 
O02 
00 
0 
number oi hfftlng biters 
nase=3% noe n coef-2 5% eateenng lJtera=11 
20 
70 o lO 2o 30 4o 5o 6o 7o 
ntrnber ol shlltng ille ra 
Figure 2: The effects of (independent) noise in the basis RFs and in the steer- 
ing/shifting coefficients. Left: the approximation error vs. the number of basis RFs 
used in the linear combination. Right: the signal to noise ratio vs. the number of ba- 
sis RFs. The SNR values were calculated as 101og10 (signal energy/noise energy). 
Adding RFs to the basis increases the accuracy of the resultant interpolated RF. 
was determined by the number of RFs incorporated into the network. The preferred 
orientations were equally spaced at angles between 0 and r. The RFs used in the 
simulations were modeled by a product of a 2D Gaussian G, centered at Fy, with 
orientation selectivity G2, and optimal angle Oi' G( F, Fy, O, Oi ) = G ( F, Fy )G2(O, Oi ). 
According to the recent results on shiftable/steerable filters (Simoncelli et al., 
1992), a RF located at 0 and tuned to the orientation b0 can be obtained by a linear 
combination of basis RFs, as follows: 
M-1N-1 
= 
j=0 i=0 
M-1 N-1 
-- Z bj(r')C(r) Z ki(Oo)G2(O, Oi) (1) 
j=O i=0 
From equation 1 it is clear that the linear combination is equivalent to an outer 
_ $h.[ "*l M-1 
product of the shifted and the steered RFs, with {ki((o))__  and tv3r0//=0 de- 
noting the steering and shifting coefficients, respectively. Because orientation and 
localization are independent parameters, the steering coefficients can be calculated 
separately from the shifting coefficients. The number of steering coefficients depends 
on the polar Fourier bandwidth of the basis RF, while the number of steering filters 
is inversely proportional to the basis RF size (Grill-Spector et al., 1995). In the pres- 
ence of noise this minimal basis has to be extended (see Figure 2). The results of the 
simulation for several RF sizes are shown in Figure 3, left. As expected, the number 
of basis RFs required to approximate a desired RF is inversely proportional to the 
Diversity in the Response Properties of Cortical Neurons 121 
i so 
0 45 f 
40 
5 
-2 i 
30 tO 20 30 40 SO 60 70 O0 
number of RFs 
The dependency o the number c RFs on the vane, nce 
I data 
L.-.- - 
OS 1 1'5 2 25 3 315 
Figure 3' Left: error of the steering/shifting approximation for several basis RF sizes. 
Right: the number of basis RFs required to achieve a given error for different sizes of 
the basis RFs. The dashed line is the hyperbola nurn lFs x size = const. 
size of the basis RFs (Figure 3, right). 
Steerability and biological considerations. The anatomical finding that the 
columnar "borders" are freely crossed by dendritic and axonal arbors (Malach, 1992), 
and the mathematical properties of shiftable/steerable filters outlined above suggest 
that the columnar architecture in V1 provides a basis for creating a continuum of RF 
properties, rather that being a form of organizing RFs in discrete bins. Computation- 
ally, this may be possible if the input to neurons is a linear combination of outputs 
of several RFs, as in equation 1. The anatomical basis for this computation may 
be provided by intrinsic cortical connections. It is known that long-range (~ lmm) 
connections tend to link cells with like orientation preference, while the short-range 
(~ 400 ttm) connections are made to cells of diverse orientation preferences (Malach 
et al., 1993). We suggest that the former provide the inputs necessary to shift the po- 
sition of the desired RF, while the latter participate in steering the RF to an arbitrary 
angle (see Grill-Spector et al., 1995, for details). 
3 Matching with patchy connections 
Many visual tasks require matching between images taken at different points in space 
(as in binocular stereopsis) or time (as in motion processing). The first and foremost 
problem faced by a biological system in solving these tasks is that the images to be 
compared are not represented as such anywhere in the system: instead of images, 
there are patterns of activities of neurons, with RFs that are overlapping, are not 
located on a precise grid, and are subject to mixing by patchy projections in each 
successive stage of processing. In this section, we show that a system composed of 
scattered RFs with smooth and overlapping tuning functions can, as a matter of 
122 Kalanit Grill Spector, Shimon Edelman, Rafael Malach 
fact, perform matching precisely by allowing patchy connections between domains. 
Moreover, the weights that must be given to the various inputs that feed a RF carrying 
out the match are identical to the coefficients that would be generated by a learning 
algorithm required to capture a certain well-defined input-output relationship from 
pairs of examples. 
DOMAIN A 
 DOMAlII 
NEU ROIq C 
Figure 4: Unit C receives patchy input from areas A and B which contain receptors 
with overlapping RFs. 
Consider a unit C, sampling two domains A and B through a Gaussian-profile 
dendritic patch equal in size to that of the axonal arbor of cells feeding A and B 
(Figure 4). The task faced by unit C is to determine the degree to which the activity 
patterns in domains A and B match. Let (jp be the response of the j'th unit in A 
to an input xp 
bjp -- exp{ --( -- )) (2) 
2er 2 
where  be the optimal pattern to which the j'th unit is tuned (the response Oip of 
a unit in B is of similar form). If, for example, domains A and B contain orientation 
selective cells, then  would be the optimal combination of orientation and location 
of a bar stimulus. For simplicity we assume that all the RFs are of the same size a, 
that unit C samples the same number of neurons N from both domains, and that 
the input from each domain to unit C is a linear combination of the responses of the 
units in each area. The input to C from domain A, with  presented to the system 
is then: 
N 
Aim = E ajejp 
j=l 
Diversity in the Response Properties of Cortical Neurons 123 
The problem is to find coefficients {aj} and {bj } such that on a given set of inputs 
{} the outputs of domains A and B will match. We define the matching error as 
follows: 
Em= E aic)ip - E biOip (4) 
p--1 i--1 i--1 
Proposition I The desired coefficients, minimizing Era, can be generated by an al- 
gorithm trained to learn an input/output mapping from a set of examples. 
This proposition can be proved by taking the derivative of Era with respect to the 
coefficients (Grill-Spector et al., 1995). Learning here can be carried out by radial 
basis function (RBF) approximation (Poggio and Girosi, 1990), which is particularly 
suitable for our purpose, because its basis functions can be regarded as multidimen- 
sional Gaussian RFs. 
4 Summary 
Our results show that maximal diversity of neuronal response properties is attained 
when the ratio of dendritic and axonal arbor sizes is equal to 1, a value found in 
many cortical areas and across species (Lund et al., 1993; Malach, 1994). Maxi- 
mization of diversity also leads to better performance in systems of receptive fields 
implementing steerable/shiftable filters, which may be necessary for generating the 
seemingly continuous range of orientation selectivity found in V1, and in matching 
spattally distributed signals. This cortical organization principle may, therefore, have 
the double advantage of accounting for the formation of the cortical columns and 
the associated patchy projection patterns, and of explaining how systems of receptive 
fields can support functions such as the generation of precise response tuning from 
imprecise distributed inputs, and the matching of distributed signals, a problem that 
arises in visual tasks such as stereopsis, motion processing, and recognition. 
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