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The Journal of Neuroscience, April 15, 2001, 21(8):2842-2850
Neuronal Population Codes and the Perception of Object Distance
in Weakly Electric Fish
John E.
Lewis and
Leonard
Maler
Department of Cellular and Molecular Medicine, University of
Ottawa, Ottawa, Ontario, Canada K1H 8M5
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ABSTRACT |
Weakly electric fish use an electric sense to navigate and capture
prey in the dark. Objects in the surroundings of the fish produce
distortions in their self-generated electric field; these distortions
form a two-dimensional Gaussian-like electric image on the skin
surface. To determine the distance of an object, the peak amplitude and
width of its electric image must be estimated. These sensory features
are encoded by a neuronal population in the early stages of the
electrosensory pathway, but are not represented with classic
bell-shaped neuronal tuning curves. In contrast, bell-shaped tuning
curves do characterize the neuronal responses to the location of the
electric image on the body surface, such that parallel two-dimensional
maps of this feature are formed. In the case of such two-dimensional
maps, theoretical results suggest that the width of neural tuning
should have no effect on the accuracy of a population code. Here we
show that although the spatial scale of the electrosensory maps does
not affect the accuracy of encoding the body surface location of the
electric image, maps with narrower tuning are better for estimating
image width and those with wider tuning are better for estimating image amplitude. We quantitatively evaluate a two-step algorithm for distance
perception involving the sequential estimation of peak amplitude and
width of the electric image. This algorithm is best implemented by two
neural maps with different tuning widths. These results suggest that
multiple maps of sensory features may be specialized with different
tuning widths, for encoding additional sensory features that are not
explicitly mapped.
Key words:
depth perception; electrolocation; electrosensory system; neuronal tuning; population coding; sensory coding
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INTRODUCTION |
In many sensory systems, neurons in
the early processing stages are tuned to a specific two-dimensional
(2D) location of a stimulus. In the visual system, this corresponds to
the 2D projection of the visual world onto the retina; in the
somatosensory system, this is the location of a touch on the skin. The
neurons in these systems respond maximally for one location, with their
activity decreasing for locations away from this preferred location;
hence the neural responses are described by 2D bell-shaped tuning
curves. Typically, these neurons have preferred locations distributed over a wide space such that a neural map of stimulus location is formed
(Konishi, 1986 ; Knudsen et al., 1987). This is often referred to as a
coarse code for stimulus location (Churchland and Sejnowski, 1992).
Populations of neurons can also carry information about sensory
features to which its component neurons are not explicitly tuned in
this manner. In somatosensory processing, the 2D location of a skin
probe is coarse-coded by peripheral mechanosensory neurons; yet humans
cannot only determine the location of the probe, but can also
accurately determine its shape, a feature that is not encoded with
bell-shaped tuning curves (Wheat et al., 1995; Khalsa et al., 1998). We
refer to such population codes, which involve multiple coding
strategies, as combined codes.
Weakly electric fish must use a combined population code during
electrosensory processing. These fish can accurately determine the
locations of objects in their surroundings using an active electric
sense, a behavior called electrolocation (Heiligenberg, 1991; von der
Emde et al., 1998). Objects with electrical properties that differ from
those of the ambient water produce distortions in the fish's
self-generated electric field. On the body surface, these distortions
form a 2D electric image (Fig. 1) and
provide the sensory input required to accurately encode object location in 3D (Rasnow, 1996; von der Emde et al., 1998). The electric image is
initially encoded in the activity of skin electroreceptors. These
receptors contact primary afferents that project somatotopically to the
hindbrain and terminate in parallel on four maps in the electrosensory
lateral line lobe (ELL). Each map is different in size and comprises
pyramidal neurons with distinct physiological properties, including
tuning curve width (Shumway, 1989a,b; Metzner, 1999; Turner and Maler,
1999). The necessary features of the electric image must be encoded in
these 2D arrays of ELL pyramidal neurons. Object location in the 2D
body plane is coarse-coded. Object distance (i.e., the third dimension)
must be estimated indirectly from population activity related to the
width and peak amplitude of the electric image (Rasnow, 1996; Assad et
al., 1999).
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Figure 1.
Computation of object distance in the
electrosensory system. a, A schematic of the
two-dimensional electric image on the surface of the fish for
objects of two different sizes and lateral distances. Although the
widths of the images are different, the peak amplitudes are the same
(measured in grayscale, with white being the largest).
Thus, detecting object distance based only on amplitude leads to
ambiguities. b, One-dimensional slices of the electric
images caused by conducting spheres of different sizes
(ro = 0.5 cm and
ro = 1.0 cm) and different lateral
distances (z* = 1.0 cm and z* = 1.26 cm).
The schematic (top right, fish not to scale) illustrates
the combinations of ro and
z* (by line type, size, and location) that relate to the
graph below.
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Theoretical results suggest that the accuracy of a 2D coarse code
should be unaffected by the width, and overlap, of the tuning curves
(Snippe and Koenderink, 1992; Abbott and Dayan, 1999; Zhang and
Sejnowski, 1999). Nonetheless, multiple parallel maps, exhibiting neuronal tuning with different widths and extents of overlap, are
universal in sensory systems, even when they do not exist at the
sensory periphery (Konishi, 1986). Different maps may provide multiple
samples of information that can be averaged by downstream networks for
higher accuracy. Alternatively, the different maps may be optimized to
encode additional stimulus features using other strategies. The
different ELL maps appear to be specialized; in some situations,
information from each map is used to produce distinct behaviors
(Metzner and Juranek, 1997). Here, we use theoretical analyses and
modeling to investigate the influence of ELL pyramidal neuron tuning
width in 2D on the accuracy of encoding object location in 3D. In doing
so, we suggest that the different ELL maps may be specialized for
encoding the different stimulus features used for computing object distance.
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MATERIALS AND METHODS |
A model description of the electric image. The
electric image caused by a spherical conductor is well approximated by
a Gaussian-shaped surface with width and peak amplitude given by 2
and Ao. Using data and simulations from a
previous study (Rasnow, 1996), we have developed a parametric model of
the electric image that enables our present analysis. We describe the
electric image produced by a sphere of radius,
ro, at a location (x*,
y*, z*) by the function S (Eqs.
1-3):
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(1)
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The half-width of the image () increases linearly with
lateral distance (Eq. 2) in the range of available data
(ro = [0.125, 0.7]; z* = [1.0, 2.0]; , ro, x*,
y*, z* have units in centimeters). The peak
amplitude of the image, Ao (units in
millivolts), decreases as the third power of lateral distance,
and although it is actually proportional to the volume of the sphere
(Rasnow, 1996), in the range considered,
Ao is approximately linear with
ro (Eq. 3):
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(2)
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(3)
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With c1 = ( 0.055) and
c2 = (0.79), Equations 2 and 3 provide a
good description of the data ( 2 < 10 4). This
model is not meant to be a detailed reproduction of the electric image,
but rather a simple description that allows us to gain insight into the
nature of the electrosensory information available for electrolocation.
The exact parameter choices do not affect our general conclusions.
A model of the ELL network. To describe the response of the
population of ELL pyramidal neurons, we convolve the stimulus, S, with 2D Gaussian-shaped tuning curves of width,  , and
where (xi,yj) is the tuning
curve (or receptive field) center of the neuron labeled ij.
Because we have assumed that the electric image is also Gaussian, the
convolution and hence the response of the pyramidal neuron population
is given by Equation 4 (after rescaling to obtain a physiologically
appropriate spike count for a 1 sec time window, and accounting for a
baseline activity level, go = 100, Ebaseline = 20) (Bastian, 1986b). We
include additive noise, Enoise, which has
a normal distribution with zero mean and SD,  . The response of the
neuron ij, Eij, is described in
Equation 5:
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(4)
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(5)
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The network we consider consists of an N × N square grid of pyramidal neurons
(i,j = 1,... ,N) with
their locations on the grid defining the centers of their evenly spaced
tuning curve centers (xi,
yj). Although we allow the grid size N and
the grid dimensions (x, y) to vary, we specify the grid
spacing (xi + 1 xi = yj + 1 yj=  = 0.15 cm) so that the
density of tuning curve centers ( = 46.7) is in the
physiological range of 40-50 neurons/cm2
[expressed in relation to body surface area (Shumway, 1989a,b) (J. Lewis and L. Maler, unpublished observations)]. The center of the grid
is the origin, (x, y) = (0, 0). Although
receptive field sizes of ELL pyramidal neurons have been reported
previously (Bastian, 1981; Shumway, 1989a), the methods used (different
combinations of object size and direct electrical stimulation) make it
difficult to directly obtain values of . However, estimates for the
physiological range of are between ~0.3 and 0.7 cm depending on
the particular ELL map (the centromedial map has the narrowest, and the
lateral map has the widest tuning curves).
For simulations of this network, we calculate a neuron response profile
using Equation 4 for a given set of object features. Gaussian random
numbers (Enoise) with zero mean and SD of
are generated (Press et al., 1993) for each neuron and added to the response profile Eij. These responses are
then rounded to the nearest integer value to give the single trial
response of the population in terms of spike count. A typical single
trial response is shown in Figure
2c. For the open symbols
plotted in Figure 3, we estimate the
image features from this noisy profile using a least-squares fit to
Equation 4 with the free parameters being either
ro, x*, y*,
z* (Fig. 3a) or ,
Ao, x*, y* (Fig.
3b). This is equivalent to a maximum-likelihood (ML)
estimate of the free parameters (Kay, 1993; Deneve et al., 1999). The
estimation error over a number of trials is given by the mean-squared
difference between the estimated and true values of each parameter
(equivalent, in this case, to the variance of the estimated values).
For all of the results shown, we use additive noise (Eq. 5) with
= 7 in agreement with preliminary data (J. Bastian, J. Lewis,
and L. Maler, unpublished data); however, the exact value of does not affect our conclusions.
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Figure 2.
A model of the ELL network response.
a, The electric image for an object of radius
(ro = 0.5 cm) at a location
(x*, y*, z*) = (0, 0, 1) is shown on a spatial grid. Image amplitude is in grayscale
(white = 0.3 mV; black = 0 mV).
b, The 41 × 41 neuronal grid with the tuning curve
size of one neuron denoted by the gray shaded circle
( = 0.6). The position of each neuron on the grid is given by
its tuning curve center (xi,
yj) in register with the image in
a. The neuronal density is = 46.7 neurons/cm2. c, A typical realization
of the neural response produced by the image in a is
shown in grayscale (white = 65 Hz;
black = 0 Hz). Other parameter values are:
Ebaseline = 20;
go = 100; =7. d, The
broadening of the average neuronal response (plotted vs the tuning
curve centers xi, open
circles) compared with the electric image (solid
line) illustrated in one-dimension for the above parameters.
The half-widths of the image and response profile are and
 (2 + 2), respectively.
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Figure 3.
Accuracy of estimating electric image
features. a, The error in estimating the size
(ro) and (x*,
y*, z*) location of a conducting sphere
as a function of pyramidal neuron tuning width, . b,
The error in estimating the corresponding image features
Ao and as a function of tuning
width, . In both panels, the continuous lines indicate
the analytically computed error given by the minimum variance of the
estimate (Cramer-Rao lower bound, see Results). The open
squares and open circles [for
ro (a) and
z* (b)
Ao and , respectively] show the
errors from network simulations (5000 trials each point; 41 × 41 neuronal grid; = 46.7 neurons/cm2;
Ebaseline = 20;
go = 100; = 7; see
Results). For the theoretical calculations, a larger grid
(101 × 101) was used (with the neuronal density preserved)
to avoid edge effects for the larger tuning widths. For the features
ro,
Ao, and z*, the error
is normalized to the true values of the feature. The true values are
ro = 0.5, Ao = 0.289, = 1.00, and
(x*, y*, z*) = (0, 0, 1.2).
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We consider two different network implementations of a two-step
algorithm for determining and Ao; one
in which the same network is used to estimate both stimulus features
(model 1) and the other consisting of two networks (model 2), with each
used to estimate a single feature (see Results) (see Fig. 5). Our
initial comparison involves specific, previously proposed (Assad et
al., 1999), mechanisms to implement the algorithm, but to also compare these models in a general decoding framework we used a variation of the
ML method described earlier (see Results). In this case, two networks
of the same size (41 × 41 grid) were used. The first network was
used to estimate Ao by using a
least-squares fit to the noisy neuronal profile with
Ao and as free parameters. A similar
procedure was then performed on the second network but with only as
a free parameter, with Ao fixed to the
value estimated by the first network. This can be viewed as an optimal
implementation of the two-step algorithm.
The Cramer-Rao lower bound and Fisher information.
Estimation of object size and location in the present context is
formulated as the estimation of a vector parameter,  = (1,2,3,4).
In Figure 3a,
(1,2,3,4)
corresponds to (ro, x*,
y*, z*), whereas in Figure 3b
(1,2,3,4)
equals (, Ao, x*,
y*). The accuracy of an estimator can be assessed by its
bias and variance. An estimator is considered unbiased if its average
value is equal to the true value of the estimated parameter. The
variance of an unbiased estimator is equivalent to the mean-squared
estimation error; the lower the variance the more accurate the
estimator. The theoretical lower limit on the variance of any unbiased
estimator is given by the Cramer-Rao lower bound (Kay, 1993). The
Cramer-Rao bound is the reciprocal of the Fisher information,
IF (Eqs. 6, 7). The more accurate an
estimator is, the more information it provides about the parameter that
is estimated; this information is quantified by
IF:
![<UP>var</UP>(ϕ<SUP>est</SUP><SUB>k</SUB>)≥[I<SUP>−1</SUP><SUB>F</SUB>(ϕ)]<SUB>kk</SUB>](/content/vol21/issue8/fulltext/2842/img006.gif)
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(6)
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![[I<SUB>F</SUB>(ϕ)]<SUB>km</SUB>=<FR><NU>1</NU><DE>&eegr;<SUP>2</SUP></DE></FR> <LIM><OP>∑</OP><LL>i=1</LL><UL>N</UL></LIM> <LIM><OP>∑</OP><LL>j=1</LL><UL>N</UL></LIM> <FR><NU>∂F<SUB>ij</SUB></NU><DE>∂ϕ<SUB>k</SUB></DE></FR> <FR><NU>∂F<SUB>ij</SUB></NU><DE>∂ϕ<SUB>m</SUB></DE></FR>](/content/vol21/issue8/fulltext/2842/img007.gif)
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(7)
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In Equations 6 and 7, est is the
estimate and is the true value of the vector parameter, is the
SD of Enoise,
N2 is the number of neurons in
the population, k = (1,... ,4), and m = (1,... ,4) for each of the four parameters.
Thus, when four parameters are estimated simultaneously,
IF is a 4 × 4 matrix. The Fisher
information has previously been used to measure the accuracy of
neuronal population codes (Abbott and Dayan, 1999; Deneve et al., 1999;
Zhang and Sejnowski, 1999). Assuming (x* = 0, y* = 0) the Fisher information for the parameter alone can be
rewritten in terms of the grid spacing, (Eq. 8):
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(8)
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In situations in which multiple but similar neuronal populations
are involved in estimation (e.g., multiple maps), the Cramer-Rao bound
can be calculated from the Cramer-Rao bound for the individual networks. If 1 and
2 are estimates from the two
different networks, and the com- bined estimate is
1-2, then the variance of 1-2 can be described by Equation 9 (Rosner, 1995):
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(9)
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In the case of two identical networks (i.e., same size and same
tuning widths, etc), taking the average of the two independent estimates is optimal; in this case k1 = k2 = 0.5, and because var(1) = var(2), the net Cramer-Rao bound
is exactly half that for the individual networks. To similarly evaluate
the combinations of networks with different properties (as in Fig. 7),
a weighted average is best, so we choose the constants
k1 and k2 to
be the reciprocals of the single network variances
[k1 = 1/var(1) and k2 = 1/var(2)]. A similar procedure
was used for Ao estimates as well.
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RESULTS |
Estimating object distance
From a 2D electric image on their body surface, electric fish are
able to determine the 3D location (x*, y*,
z*) of the object producing the image (von der Emde et al.,
1998). The object location in the body plane (x-y plane)
can be estimated from the location at which the image has its peak
amplitude. However, the peak amplitude of the electric image provides
ambiguous information about the third dimension, lateral distance away
from the fish (z*). In Figure 1a, two spherical
objects of different sizes (and otherwise identical) are located at the
same (x*, y*) location, but the larger object is
farther away. For this, and many other combinations of object size and
lateral distance, the peak amplitude of the image is the same and thus
cannot be used to unambiguously determine the lateral distance of each
object (see Materials and Methods; Eq. 3) (Rasnow, 1996). The image
produced by the larger object is wider than the other (Fig.
1a). This is shown more clearly by a one-dimensional slice
through the image (Fig. 1b). When the image is normalized to
its peak amplitude, its width can then be used to estimate lateral
distance, z* (Rasnow, 1996; Assad et al., 1999).
To enable our analyses, we used a simplified description of the
electric image. We assume the electric image has a 2D Gaussian shape,
with its peak amplitude and half-width given by the parameters Ao and (see Materials and Methods)
(Eqs. 1-3). Because provides a measure of normalized width and
varies linearly with lateral distance, z* (Rasnow, 1996), it
then can be used to estimate lateral distance (we use and image
width interchangeably, although is actually the half-width).
Another image feature proposed as an indicator of object distance is
the maximum slope of the image normalized to its peak amplitude (von
der Emde et al., 1998; von der Emde, 1999). For a Gaussian image, this
quantity varies as 1/ and also fits the published maximum slope data
(von der Emde et al., 1998) very well (Lewis and Maler, unpublished
observations). Because of the direct relationship between , maximum
slope, and previously reported data, we have discussed our results in
terms of alone.
Estimation accuracy and tuning curve width
In the present context, downstream electrosensory networks must
extract information about object location given a noisy profile of
activity in the ELL pyramidal neuron population. We have formulated a
simple model of the ELL population response to a stereotyped electric
image (see Materials and Methods). Figure 2a shows the electric image produced by a small sphere (Eq. 1), which provides the
input to the 2D grid of model neurons that constitute the ELL network
(Fig. 2b). Each neuron on the grid integrates input from the
electric image over a restricted range or receptive field (shown
schematically by the shaded region in Fig. 2b; Eq. 4), such
that for a point stimulus each neuron has a 2D Gaussian-shaped tuning
curve (in the x-y plane). The tuning width is given by 2
(measured at a height corresponding to
e1/2 of the tuning curve peak). We ignore
any contributions that dynamics may provide, with the response of each
neuron given by a spike count over an integration time of 1 sec. After
the addition of noise the ELL population response profile resembles a
noisy replication of the electric image (Fig. 2c). Because
the electric image is not a point stimulus, the actual response profile
of the ELL population is wider than the image, to an extent that
depends on the relative values of and (Fig. 2d) (see
next section).
Given the noisy response profile of the ELL population, the typical
population decoding problem is to determine the features of the object
(x*, y*, z*,
ro) that produced the response (Abbott, 1994; Salinas and Abbott, 1994; Deneve et al., 1999; Zhang and Sejnowski, 1999). As discussed before in a functional context, to
unambiguously determine z* and
ro, two features of the electric image
produced by the object must be estimated: the image width and the
amplitude of the image peak ( and Ao,
respectively). The accuracy of estimation is limited by the accuracy
with which the ELL neurons jointly encode these different features.
Using a common approach from statistical estimation theory (Kay, 1993), we can determine an upper limit on this accuracy by computing the
Cramer-Rao lower bound for estimating each object feature x*, y*, z*, and
ro, as well as the image features and
Ao (Eq. 6) (see Materials and Methods).
Accuracy in this context is given by the mean-squared estimation error,
or equivalently, the variance of the estimate. We investigated the
influence of two parameters on estimation accuracy: the lateral
distance of the object z* and the tuning curve width .
The error bound for estimating all features increases with
z* (data not shown). This is not surprising because the
image amplitude (and thus the effective signal-to-noise ratio)
decreases fairly quickly with distance (Eq. 3). More interestingly, the
effects of changing differ between the features (Fig. 3). There is
no effect on estimating the x*-y* location (Fig.
3a); the same result has been found previously for point
stimuli (Snippe and Koenderink, 1992; Abbott and Dayan, 1999; Zhang and
Sejnowski, 1999). On the other hand, increasing tuning width results in worse estimation of ro and z*
(Fig. 3a). For estimating electric image features (Fig.
3b), increasing results in worse estimation of (larger error), but better estimation of
Ao (smaller error). Intuitively, this
makes sense, wider tuning curves allow more neurons to accurately
contribute to the estimation of Ao, and by
averaging across neurons, a better estimate results. Estimating image
width is different although, because the ELL neurons distort the image
through a convolution with their tuning curves (Eq. 4, Fig.
2d). This distortion increases with tuning width, resulting in more neurons that do not accurately represent image width, nonetheless influencing the estimate.
Shown also in Figure 3 are the results of network simulations. Using a
network grid consisting of a physiological number and density of
neurons (41 × 41 neuronal grid, density = 46.7 neurons/cm2), we estimated the image
features from the noisy neural responses using an ML approach (see
Materials and Methods). The estimation error for this method is very
close to the corresponding lower bound (Fig. 3, compare open
symbols with solid lines). Note that for larger however, there is a slight deviation from the theoretical bound
attributable mainly to edge effects (i.e., the neuronal image has above
baseline values beyond the limits of the grid edges).
The relationship between the accuracy of image width estimation and
tuning width can be made explicit by expressing the Fisher information
for , IF(), in terms of the grid
spacing, , the distance between tuning curve centers (Eq. 8).
Differentiating IF() with respect to
reveals that IF() decreases with
(i.e., the derivative is negative and thus the estimation error
increases) as long as 2 + 2 > (/2)2. This condition will hold as long
as the tuning curve width is greater than the grid spacing (i.e., if
2 > ). A similar calculation shows that the Fisher
information for Ao,
IF(Ao),
increases with for all > 0.
A simple neural algorithm for determining object distance
Estimating , in the present context, is equivalent to
estimating the half-width of the image at a level of
Aoe1/2. To provide an
unambiguous estimate of z*, a measure of image width must be
calculated from an image normalized by Ao.
So in such a practical situation, peak amplitude
Ao must be estimated first, before image width.
One simple algorithm to calculate image width is to first normalize the
neural responses to the maximal response and then count the number of
neurons that are active above a certain threshold (Assad et al., 1999).
One way to formalize this two-step algorithm is to first compute the
average activity Eave of all the neurons firing above a threshold,  a (Fig.
4a). This step (step 1)
provides an estimate of the peak response in the population, which can be used to normalize all neural activity. Then in step 2, the fraction
of neurons (Nw) firing above a
different threshold (w) can be determined
(Fig. 4b). These two thresholds are distinct in that
a is fixed and not relative to any neural
response, whereas w comes after the
normalization step and is relative to the maximum response in the
network. Figure 4, c and d shows how these
measures vary with the features they are supposed to estimate. The
actual peak neural activity
(goAo) differs from
Ao by a constant factor and thus varies with
z* in parallel with Ao (Fig.
4c). However, Eave underestimates
goAo but still varies linearly with
Ao (Fig. 4c, inset). Similarly,
Nw varies in a near linear manner with (Fig. 4d).
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Figure 4.
Two-step algorithm for estimating electric image
width. a and b show representative
response profiles Ei of a
one-dimensional slice through the neuronal grid. In a
the peak amplitude of the image is estimated by the average activity
Eave of all neurons firing above a threshold
level of Ebaseline+a
(Ebaseline = 20). In b
the response profile is normalized by
Eave, and the image width is estimated
by the number of neurons Nw firing above a
threshold level of
(Ebaseline/Eave)+w.
c, The average spike count of the neurons above
threshold (Eave), as well as the decay in
actual peak activity
(goAo;
go = 100) with increasing object distance
z* (a = 2 = 14; = 0.6). Over this range of z*,
Eave varies linearly with
Ao (inset).
d, The fraction of total neurons activated above a
threshold level of w = e1/2 (i.e., the number of
neurons with a preferred location within a radius of the object
location) plotted versus . For different values of , this measure
increases in an almost linear manner with . The solid
lines are the theoretical curves derived for a continuous
distribution of neurons (see Results), and the open
symbols show the measure for an actual model network (41 × 41 neuronal grid; = 46.7).
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We consider two specific neural implementations of this algorithm (Fig.
5). Model 1 uses the same map (i.e.,
network), with tuning width 1, for estimating
both and Ao. Model 2 uses two maps,
one with a relatively large tuning width (2 = 1) for estimating Ao and another with
narrower tuning widths (1  1) for estimating . Model 1 is analogous to a single sensory map for all computations, and Model 2 is analogous to having two specialized sensory maps, one
for estimating peak amplitude Ao, with
larger tuning widths, and the other for estimating width , with
smaller tuning widths. It is critical to note that both models use the
same number of neurons in each processing step (each map is a 41 × 41 neuronal grid). The critical difference is that model 2 has two
different tuning widths for each processing step.
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Figure 5.
Schematic representation of two models for
implementing the two-step algorithm for estimating electric image
width. In model 1 (left), both
Ao and are estimated using the
same map, map 1 with tuning widths 1. In model 2 (right), separate maps are used for each estimation
step: map 2, with wide tuning curves (2 = 1), is
used to obtain an estimate of Ao,
which is then used to normalize the activity in map 1, with narrow
tuning curves (1 < 2), from
which an estimate of is obtained. Because
Ao and are estimated separately in
this two-step algorithm, both models use the same number of neurons to
estimate each feature, although model 2 has two maps, and model 1 has
only one.
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To compare the performance of these models, we compute
Nw for many simulated presentations of
an object over a range of values of 1 and
z* (a = 2;
w = e1/2). In this
situation, the true value of Nw is
given by the number of neurons with preferred locations within a circle
of radius ( 2). The estimate
of Nw from the present neural
algorithm, however, is biased. This is in part because of its
dependence on Eave and also because it is
determined from a neural profile that has an effective width of
(2+2)
caused by the tuning curve convolution (Eq. 4, Fig. 2d).
Because in the context of these models, downstream networks would have to use Nw to estimate object distance,
and Nw is directly related to and
object distance (Fig. 4d), we evaluate model performance from the bias and variance in Nw. In
all cases tested (1 = 0.15-1.0; z* = 1.0-1.4), model 2 outperforms model 1. For z* = 1.2, the estimation
variance for both models is shown in Figure
6a. The biases in
Nw estimation are nearly identical for
both models (data not shown), but the variance for model 2 is
substantially less than that for model 1. Because model 2 is better at
estimating peak amplitude (by virtue of its wide tuning curves for this
step, 2 = 1), the variance is dominated by the
width estimator, and thus increases with 1 in
the same manner as the Cramer-Rao bound for (Fig. 3b).
Model 1 must use a network with the same 1 for all steps, so there is a trade-off between accuracy of peak amplitude estimation and accuracy of width estimation. Peak amplitude estimation is better for larger 1 when more neurons are
activated close to peak levels, but width estimation is better for
smaller 1. In the case shown (Fig.
6a), the amplitude estimate dominates even for small
1 and thus the overall variance decreases with 1, similar to the Cramer-Rao bound for
Ao (Fig. 3b). When
1 = 1, both models have similar overall
accuracy. Although it would seem that model 2 effectively has twice as
many neurons as model 1, as stated earlier, it really uses the same
number of neurons as model 1 for each processing step. The slightly
better performance of model 2 for 1 = 1 is
caused by the independence of the responses between the different maps.
In other words, if the noise in the neuronal responses was exactly
correlated between the two maps of model 2, the accuracy would be
identical to that of model 1.
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Figure 6.
Performance of the two models in
implementing the two-step algorithm for estimating electric image
width. a, This panel shows the variance in the estimate
of Nw for each model. Model 1 (open
symbols) results in higher variance than model 2 (2 = 1; closed symbols) for all
tuning widths 1. Nw is the
fraction of neurons above threshold (i.e., the actual number normalized
by the total number of neurons N2;
parameter values are N = 41, w = e1/2, a = 2, ro = 0.5, (x*, y*,
z*) = (0, 0, 1.2), = 46.7, Ebaseline = 20, go = 100, = 7. The true value
of Nw = 2/N2 ~0.09.
b, The variance of the estimate for a generalized
decoding scheme in both models (see Results). All parameter
values are the same in a and b,
except that in b two independent maps are used for
both models (see Results). Note that when 1 = 1 in this case, both models are identical so the variances are
necessarily the same. Each point in a and
b represents the variance calculated from 3000 simulated
trials.
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|
The general trends shown in Figure 6a are similar for
neuronal densities within ~50-150% of that used in the simulations
shown. We also tested several combinations of values for
a (range, 2-4) and
w (range, 0.25-0.75) for z* = 1.2 and 1 = 0.3. Similar trends resulted, so the
increased accuracy of model 2 over model 1 does not depend critically
on these threshold values. In addition, we also considered conditions
in which the noise term Enoise was such that the SD of the ELL neuron responses was equal to their mean
response Fij (Eq. 4), rather than constant
( = 7) and independent of Fij.
This type of noise resulted in similar results (data not shown) and
does not change our conclusions.
Although the previous analysis demonstrates a clear difference between
the two models, it is important to prove that this difference is
fundamental and is not simply attributable to the details of the
algorithm implementation or the fact that model 2 uses two independent
networks. We now consider two independent maps composed of 41 × 41 neuronal grids with tuning widths of 1 and
2, respectively. Map 2 provides an estimate of
Ao using ML estimation; this estimate is
used to normalize the activity in map 1. Then map 1 is used to find the
ML estimate of . For model 1, both maps have the same tuning width
1 = 2. Model 2 is
identical to model 1 except that map 2 has a fixed tuning
width 2 = 1. This constitutes a test of the
two models in a general decoding framework in which the only difference
is in the tuning width of map 2. The results are similar to those
previous, with model 2 providing a better estimate of (Fig.
6b). In this case however, the error increases with
1 for both model 1 and 2, in the same manner
as the Cramer-Rao bound for (Fig. 3b), suggesting that
estimation dominates the overall estimation error. This provides a
theoretical validation of our conclusions, but it is certainly not an
option for the fish. The ELL does not have multiple maps with the same
tuning widths, and thus the fish does not have access to identical
information from two identical maps. Our initial analysis (Fig.
6a) shows how the specialized use of an additional map can
improve the computation performed by a single map.
The two-step algorithm we have considered is based on previous
ideas (Assad et al., 1999) and practical constraints (i.e., peak
amplitude must be estimated before normalization can occur). But it is
also interesting to ask how maps can be combined in the context of
optimal estimation as defined by the Cramer-Rao bound. We again
consider the two-map configuration analyzed in Figure 6b. We
calculated the Cramer-Rao bounds for and
Ao for two maps (see Materials and
Methods) and compared them to that for a single map (Fig.
7a,b). Two maps with identical
tuning widths are twice as good as one map with that tuning width
(i.e., the error decreases by half for two maps). The neuronal density
is a critical factor in determining population coding accuracy (Zhang and Sejnowski, 1999). Having two identical maps is the same, in terms
of accuracy, as having a single map with twice the density, not
necessarily twice the number of neurons. Also shown in Figure 7
is that having one of the maps with a fixed tuning
width (2 = 1) is better than two identical
maps for estimating Ao, but worse for
estimating . Thus, the relative importance of these parameters will
influence the optimal configuration of the two maps; if a premium is
placed on estimating independently of Ao, then narrow tuning in both maps is
better. In the two-step algorithm considered in this paper, accurately
estimating Ao is critical for the overall
accuracy of estimating , so a combination of tuning widths is best.
View larger version (13K):
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|
Figure 7.
Performance of two maps in the context of the
Cramer-Rao bound. For two different combinations of two maps (similar
to those in Fig. 6b), the analytically calculated
Cramer-Rao bounds for estimating Ao
(a), and (b) (see
Materials and Methods), are shown by the thick solid and
dotted lines, respectively. In one configuration
(2 = 1), both maps have the
same tuning width, and in the other configuration
(2 = 1), one map has a tuning width of
1, and the other is fixed at
2 = 1. Also shown (thin solid lines)
are the analytically calculated Cramer-Rao bounds for a single map
(same as those in Fig. 3b).
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DISCUSSION |
Multiple maps and population coding
Weakly electric fish can accurately electrolocate objects
in their surroundings using sensory information contained in a 2D electric image (Bastian, 1987; Heiligenberg, 1991; Nelson and MacIver,
1999; von der Emde, 1999). To unambiguously extract 3D object location,
the fish must compute the width and location of the peak of the
electric image that is normalized to its peak amplitude (Rasnow, 1996).
The electric image is initially encoded in four populations of
pyramidal neurons that comprise the four parallel maps in ELL. One map
(ampullary system) is specialized for low-frequency signals. The
neurons within the three remaining maps (tuberous system) can be
distinguished, among other characteristics, by their distinct spatial
response properties: the lateral map with large receptive fields,
centromedial map with small receptive fields, and the centrolateral map
with intermediate-sized receptive fields (Shumway, 1989a,b; Metzner,
1999; Turner and Maler, 1999). Our results suggest a novel function for
the parallel sensory maps in ELL, as well as the occurrence of parallel
maps in other sensory systems. Namely, in addition to coarse coding
stimulus features on different scales, parallel sensory maps may also
be optimized to encode features of a sensory stimulus to which the component neurons are not tuned in the same manner. Specifically, in
addition to encoding the 2D electric image at different spatial scales,
different ELL maps can also be specialized to accurately represent the
sensory features required to compute the third dimension, i.e., object distance.
Previous theoretical studies have found that neuronal tuning width (or
spatial resolution) should not affect encoding accuracy in 2D (Snippe
and Koenderink, 1992; Abbott and Dayan, 1999; Zhang and Sejnowski,
1999). This is also the case for the coarse-coded features of a
spatially extended stimulus (i.e., the electric image) (Fig.
3a). This result does not apply when multiple 2D stimuli are
given simultaneously, as in two-point discrimination, where narrower
tuning curves are better (Snippe and Koenderink, 1992). We show that
depending on the encoding strategy for a particular stimulus feature,
either wider or narrower tuning curves improve encoding accuracy. We
illustrate the impact of tuning width on the accuracy of determining
object distance from the electric image using two simple models. To
encode the peak amplitude of the electric image, wider tuning curves in
the two coarse-coded dimensions (x and y) result
in higher accuracy; whereas, to encode image width, narrower tuning
curves are more accurate. This suggests that the lateral map in ELL (in
addition to its other functions, such as processing high-frequency
signals like chirps) (Shumway, 1989a; Metzner and Juranek, 1997), may
provide information about image amplitude that can be used to normalize
the activity in the centromedial map, which is then used to compute
image width and object distance. Normalization could be mediated by the
extensive cerebellar-like feedback that projects to ELL, through
shunting inhibition or synaptic depression (Maler and Mugnaini, 1994;
Bastian, 1996; Berman and Maler, 1999). This simple hypothesis can be
readily tested with established experimental techniques (Bastian, 1987; Metzner and Juranek, 1997; Nelson and MacIver, 1999). For example, ablating the lateral map of ELL (wide tuning) should disrupt the accurate estimation of image amplitude and the subsequent normalization step, resulting in an ambiguous estimation of object distance. So,
predictable behavioral errors should occur when animals attempt to
distinguish objects with certain combinations of size and distance.
The locus of computation of object distance is not known, and it need
not be the centromedial map itself, because information from all ELL
maps could be combined in higher brain regions (i.e., torus
semicircularis or optic tectum) (Heiligenberg, 1991). Indeed, it is not
necessary that there be a locus of computation, or explicit neural map,
of object distance in electric fish. Such information could remain in a
combined population code throughout its processing stream. However,
there is some evidence of neurons in both the tectum and cerebellum
that are tuned to object distance (Bastian, 1986a). Similar
"distance-tuned" neurons exist in the optic tectum of frogs and
toads (House, 1989). There is also evidence that information from the
different maps is treated very differently in the torus (Metzner and
Juranek, 1997). So apparently the same information from different maps
is not simply being averaged. Other constraints could lead to the
formation of differently sized maps, such as specialized roles in
temporal processing and communication (Metzner, 1999), as well as those
proposed in the present paper.
The present analyses have primarily considered the location of an
object. However, with an estimate of object distance (z*), the
approximate size of the object (ro) can
then be decoded from the amplitude estimate (Eq. 3). The accuracy of
this estimate will be constrained by the Cramer-Rao bound shown in
Figure 3a. Thus, extensive cross-talk between ELL maps
(either within ELL or in their projections to higher centers) may be
required to identify the complete array of necessary object properties
(Assad et al., 1999).
Combined strategies in population coding
The population coding literature has primarily dealt with how
neuronal populations encode features to which its component neurons
exhibit bell-shaped tuning curves. These studies often focus on how a
single value of the feature in question can be extracted from the
neuronal population response. There can be more information in the
population response than that one value; for example, the entire
probability distribution of a stimulus feature can be decoded from the
population response (Zemel et al., 1998). There is recent evidence, in
the case of visual motion perception, that such information is actually
used to form a specific percept (Treue et al. 2000). This information
is still related to the coarse-coded stimulus features. To our
knowledge, extracting information from a combined population code in a
functional context, has not been previously considered.
Cues for electrosensory depth perception
Our study of electrosensory depth perception has considered only
static cues of object distance. In the context of visual processing,
the problem is analogous to judging the depth of a stationary object
using only monocular information. Electric images resulting from near
objects are narrower and of greater peak amplitude than those of far
objects, and thus can be considered as having less blur and higher
contrast. Blur and contrast can have a significant influence on visual
depth perception and are commonly used by artists in the pictorial
depiction of depth (O'Shea et al., 1994; Mather, 1997). In normal
visual processing, such cues are usually effective only in the absence
of others such as those resulting from stereopsis and motion. Although
there is no binocular analog in electrosensory processing, electric
fish certainly have many motion cues available. Indeed, some species of
electric fish exhibit a back-and-forth hovering motion that could
be used to generate specific cues. Also, looming cues, such as those
resulting from a changing electric image as an object approaches, could
also be used for computing a parameter such as the time-to-collision, often discussed in the context of visual looming (Sun and Frost, 1998;
Gabbiani et al., 1999; Rind and Simmons, 1999). As yet, there is little
known about electrosensory motion processing and how electric fish
might use such information for electrolocation.
| |
FOOTNOTES |
Received Sept. 22, 2000; revised Jan. 18, 2001; accepted Jan. 23, 2001.
This study was supported by the Canadian Institutes of Health Research
through an operating grant to L.M. and a postdoctoral fellowship to
J.E.L. Thanks to T. Lewis and S. Kealey for helpful comments on this manuscript.
Correspondence should be addressed to John E. Lewis, Department of
Cellular and Molecular Medicine, University of Ottawa, 451 Smyth Road,
Ottawa, Ontario, Canada K1H 8M5. E-mail: jlewis{at}uottawa.ca.
| |
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TOP
ABSTRACT
INTRODUCTION
