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Biological Cybernetics 77, 41–47 (1997)
Biological
Cybernetics
c Springer-Verlag 1997
Coarse coding: applications to the visual system of salamanders
Christian W. Eurich1,2 , Helmut Schwegler2 , Richard Woesler2
1
2
The University of Chicago, Department of Neurology, MC 2030, 5841 South Maryland Avenue, Chicago, IL 60637, USA
Institut für Theoretische Neurophysik, Universität Bremen, D-28334 Bremen, Germany
Received: 7 January 1996 / Accepted: 16 April 1997
Abstract. In a previous study, we calculated the resolution
obtained by a population of overlapping receptive fields, assuming a coarse coding mechanism. The results, which favor large receptive fields, are applied to the visual system
of tongue-projecting salamanders. An analytical calculation
gives the number of neurons necessary to determine the direction of their prey. Direction localization and distance determination are studied in neural network simulations of the
orienting movement and the tongue projection, respectively.
In all cases, large receptive fields are found to be essential
to yield a high sensory resolution. The results are in good
agreement with anatomical, electrophysiological and behavioral data.
1 Introduction
Many species are capable of highly precise object localization (Roth 1987; Konishi 1993; Dinse et al. 1995) even
though the mesencephalic or telencephalic neurons involved
in the sensory information processing have remarkably large,
overlapping receptive fields (Wiggers et al. 1995a, Gaillard
1985; Knudsen and Konishi 1978; Dinse et al. 1995).
In a recent paper (Eurich and Schwegler 1997) we considered a coarse coding mechanism for the representation
of the position of a single object in the nervous system in
order to account for these observations. In the coarse coding scheme (Hinton et al. 1986), the position of an object
is encoded by a population of binary neurons with overlapping receptive fields. In calculating the resolution obtained
by such a neuron population, we came to the encouraging
result that large receptive fields are advantageous for the encoding of stimulus position, indicating that they can indeed
play a role in sensory object localization.
In the following, the results derived in Eurich and Schwegler (1997) are applied to the visual system of tongueprojecting salamanders (Bolitoglossini) using analytical calculations and neural network implementations.
In Sect. 2, the resolution in direction determination is
calculated analytically using empirical data about the poCorrespondence to: Christian Eurich, Universität Bremen, Institut für Theoretische Physik, NW 1, Postfach 330 440, D-28334 Bremen, Germany
(Fax: +49 421 218 4869, e-mail: [email protected])
sitions and sizes of receptive fields of neurons in the optic tectum of the tongue-projecting salamander Hydromantes
italicus. In Sect. 3, the resolving capability of large receptive fields for direction determination is tested with a neural
network model called Simulander I which tracks prey with
head movements. Since the model has been described in detail elsewhere (Eurich et al. 1995), it is evaluated only with
respect to its coarse coding properties. In Sect. 4, a coarse
coding scheme for depth perception is developed comprised
of large, overlapping three-dimensional receptive fields in
the binocular visual field. The scheme is implemented in
Simulander II, which is a neural network model for the control of the projectile tongue of salamanders. The results are
discussed in Sect. 5, where we also address the question
of the coding and decoding of information in the nervous
system and the problem of relating special tasks to certain
neurons in distributed information-processing systems.
2 Coarse coding of object direction in the optic tectum
of salamanders
The optic tectum is considered to be the main brain center
for visual information processing in amphibians. In tongueprojecting salamanders, each tectum hemisphere receives direct input from both retinae (Rettig and Roth 1986). Three
pathways extend from the optic tectum to the brainstem and
spinal cord, where efferent tectal neurons innervate the motor
nuclei of the muscles involved in the control of the projectile
tongue (Roth and Wake 1985; Dicke and Roth 1994). The
fast reaction of Bolitoglossini and their high success rates
in catching small and fast prey may be due partly to a fast
connection from the retinae to the brainstem (retinal ganglion cell – tectum neuron – motor neuron) with only two
synapses in between (Matsushima and Roth 1990).
Here we are interested in the properties of the tectal
neurons. Cell counts reveal that in some salamander species
there are as few as about 1000 tectal neurons descending
to the brainstem and spinal cord in each brain hemisphere
(U. Dicke, personal communication). These neurons must
convey the information about the position of a prey in the
visual field.
42
100
a
80
1
1000
(deg)
60
900
40
20
800
0
700
-20
-40
-150
-100
-50
0
50
100
500
400
b
0.06
0.05
w( )
2
600
150
(deg)
-150 -100
-50
0
50
100
150
0.04
0.03
Fig. 2. Angular resolution per neuron, N αe , as a function of the horizontal
angle, ϕ̃0 , on the horizontal median (ϑ̃0 ≡ 0). Curve 1, result for a population of receptive fields all having the same angular size %. Curve 2, result
for the same population if the size distribution is taken into account
0.02
0.01
0.0
0
20
40
60
80
100
120
140
160
180
2 (deg)
Fig. 1a,b. Properties of tectal neurons in Hydromates italicus. a Distribution
of receptive field centers in the visual field. Coordinates (ϕ̃, ϑ̃) = (0, 0)
correspond to the center of the visual field. b Distribution of receptive field
sizes. Data from Wiggers (1991)
tion (2) is then replaced by
Le (x0 ; w) =
Figure 1a shows the distribution of receptive field centers in the visual field for the tongue-projecting salamander
Hydromantes italicus. Angular coordinates ϑ̃, ϕ̃ are chosen
so that the point (ϑ̃, ϕ̃) = (0, 0) corresponds to the center
of the visual field. All data have been doubled by mirroring
at the ϑ̃ axis in order to make the problem symmetric for
calculations. Typically, the receptive field density is high in
the center of the visual field and decreases as a function of
the mediolateral and dorsoventral angles. Figure 1b shows
the distribution of receptive field diameters. The mean diameter is 40.6◦ ± 39.9◦ , and there are large receptive fields
measuring 180◦ in diameter, i.e., they occupy half of the surrounding. The mean diameter shows a tendency to increase
with ϕ̃ (data not shown). Large receptive fields exist only
in the lateral visual field, which is a natural consequence of
binocularity: a monocular visual field does not extend over
180◦ around the center of the visual field.
In order to calculate the accuracy of direction localization
we follow the considerations derived in Eurich and Schwegler (1997). The sensory space for direction determination is
2
. For a population of receptive
idealized to the 2-sphere, SR
fields with mean angular diameter 2%, the angular resolution
2
for the direction e is given by
αe (x0 ; %) at x0 ∈ SR
αe (x0 ; %) =
1
RLe (x0 ; %)
(1)
where Le (x0 ; %) is the corresponding density of receptive
field boundaries. Le (x0 ; %) is obtained by integrating the den2
), over all points consity of receptive fields, σ(x) (x ∈ SR
tributing with a receptive field boundary at x0 :
Z2π
Le (x0 ; %) = R sin % σ(ϑ(β), ϕ(β))| cos β| dβ
(2)
0
2
; β: angle parameterizing
(ϑ, ϕ: polar coordinates on SR
the integration path; R sin %| cos β|: weighting factor which
takes into account the local geometry of the integration path).
In a more general case, instead of the mean receptive
field diameter 2% a receptive field size distribution w(%|x)
2
. Equacan be considered which is a function of x ∈ SR
Zπ Z2π
R sin % σ(ϑ(β), ϕ(β))
0
0
×w(%|(ϑ(β), ϕ(β)))| cos β| dβd%
(3)
The resolution has been shown to be inversely proportional
to the number N of neurons.
The angular variables ϑ̃, ϕ̃ introduced above can be calculated from polar coordinates according to the formulae
π
ϑ̃ = − ϑ , ϑ ∈ [0; π]
2
π
− ϕ,
ϕ ∈ [0; 32 π]
(4)
ϕ̃ = 2
2π + π2 − ϕ , ϕ ∈ [ 32 π; 2π]
For the following calculations, a subset of the receptive
fields shown in Fig. 1 is chosen whose properties have been
empirically determined with appropriate accuracy. The mean
diameter of the receptive fields in this subset is 2% = 35.9◦ .
In the first step, the empirical distribution of receptive
fields in the visual field (see Fig. 1a) is considered while
all receptive fields are assumed to have the mean diameter, 2%. A method for obtaining a continuous density of
receptive fields, σ(ϑ, ϕ), from a finite sample of receptive
field positions has been described in Eurich and Schwegler
(1997, appendix A). Basically, each receptive field center is
2
; the functions are
replaced by a Gaussian function on SR
summed, and the result is normalized and multiplied by the
number of receptive fields, N . With σ(ϑ, ϕ) given, (2) can
2
and
be evaluated numerically for an arbitrary point x0 ∈ SR
direction e, and (1) yields the corresponding angular resolution. Curve 1 in Fig. 2 shows the angular resolution per
neuron, N αe (x0 ; %), as a function of the horizontal angle ϕ̃0
for points x0 on the horizontal median (ϑ̃0 ≡ 0); e is the
horizontal direction. As expected, the resolution reaches its
optimum in the middle of the visual field. The corresponding
value is ≈ 385◦ per neuron, i.e., with N neurons an angular
resolution of about 385◦ /N is achieved.
In the second step, the receptive field size distribution
(Fig. 1b) is also taken into account. In addition, we allow
this distribution to be dependent on the position in the visual
field: the mean receptive field diameter increases with |ϕ̃| because more large receptive field neurons are found laterally
43
than centrally. Since the available empirical data are not sufficient to infer the function w(%|(ϑ, ϕ)) directly, additional
assumptions for the distribution are necessary. The construction of an appropriate function w(%|(ϑ, ϕ)) is described in the
Appendix. w(%|(ϑ, ϕ)) is inserted into (3); again the density
of receptive field boundaries can be evaluated numerically
2
and e, and the angular resolution is
for arbitrary x0 ∈ SR
given by (1). Curve 2 of Fig. 2 shows the angular resolution
per neuron on the horizontal median as a function of ϕ̃0 (as
before, e is the horizontal direction).
The calculations reveal the following result: the existence
of large receptive fields in the periphery of the visual field
leads to a lateral increase in resolution compared with the
situation of curve 1, where all receptive fields are assumed
to have the same size. These findings suggest that the large
receptive field neurons that are found in urodeles as well
as in anurans (Grüsser-Cornehls and Himstedt 1976; Roth
and Himstedt 1978; Grüsser-Cornehls 1984) contribute to
the localization of small objects such as prey and are no
mere ‘predator detectors’ signaling the emergence of large
objects somewhere in the visual field (e.g., Ewert 1984).
Curve 2 also shows a slight decrease in the center which
may be due to the animal’s snout hiding a part of the visual
field. However, the finer features of the resolution curve
are not considered to be significant, since predictions of the
angular resolution to an accuracy of several degrees would
require a much larger set of empirical data.
Finally, the number of neurons necessary to yield the
angular resolution observed in behavioral experiments can
be estimated. We assume that the salamander can localize a
0.05-cm-sized prey at the distance of the maximal protraction length of the tongue, which is 5 cm in Hydromantes
italicus (Roth 1976; Wiggers 1991). This corresponds to an
angular resolution of 0.57◦ . Curve 2 of Fig. 2 yields an angular resolution per neuron of about 450◦ at ϕ̃0 = 0. Thus,
the necessary number of neurons is N ≈ 450/0.57 ≈ 790.
This is in good agreement with the empirically estimated
number of 2 000 descending neurons in both brain hemispheres. The result can be compared with the performance
of the local coding scheme in which the sensory space is
divided into small receptive fields whose sizes correspond
to the sensory resolution; the firing of a single neuron then
indicates the position of an object. Assuming the visual resolution of 0.57◦ , about 162 000 neurons would be necessary
to cover the visual field! The estimated small number of receptive fields in the coarse coding scheme and the presence
of descending neurons with large receptive fields in the optic tectum suggest that a coarse coding mechanism might
indeed be reponsible for conveying the information on prey
position from the midbrain to the brainstem and spinal cord.
3 Comparison with the performance of Simulander I
In catching prey, tongue-projecting salamanders exhibit a
‘sit-and-wait’ (or ambush) strategy. Their first visible reaction to a small object is a turning movement of the head, by
which the object is centered in the visual field. The turning
movement, also called the orienting movement, requires an
appropriate direction localization.
Simulander I is a neural network model for the orienting movement which incorporates electrophysiological, neuroanatomical and behavioral empirical data (Eurich et al.
1995; Wiggers et al. 1995b). The network has a feedforward
topology and consists of three layers of 100 neurons each.
The input layer of the network represents a population of tectal neurons whose receptive fields have the actual properties
shown in Fig. 1. Two hidden layers represent populations of
excitatory and inhibitory neurons in the brainstem and spinal
cord. The output consists of contractions of two pairs of antagonistic neck muscles by which the salamander moves its
head in order to track the prey. In this way, the performance
of the network in direction determination can be judged by
the accuracy of the motor behavior.
The network is prewired according to the topology of
tectofugal connections (Dicke and Roth 1994); the strengths
of the couplings, however, remain to be determined. They
are adapted in a training procedure which is not considered
to be biologically motivated but which is a mere optimization strategy searching for optima of performance in weight
space. We have found that evolution strategies (Rechenberg
1973; Bäck et al. 1991) yield good results.
After the training, Simulander I is capable of centering a model prey, which moves irregularly inside a threedimensional terrarium, in about 92% of the simulation time.
The results show that the necessary angular resolution is
achieved with only 100 tectum neurons, which confirms
our contention in the previous section that high resolution
can be attained through neurons with large receptive fields.
Moreover, from these simulations it becomes clear that the
decoding of information about prey position in the form of
motor commands requires a number of neurons which does
not significantly exceed the number of the encoding neurons.
4 Coarse coding of depth by binocular neurons
and the network Simulander II
Depth perception in urodeles and anurans is mainly due to
the evaluation of binocular disparity, whereas mechanisms
such as accommodation, vergence, and motion parallax are
either absent or play only a minor role (e.g., Ingle 1976;
Collett 1977; Roth 1987). In tongue-projecting salamanders,
binocular neurons in the rostral optic tectum receive direct
input from both retinae; some of them project to the brainstem and spinal cord and activate motor networks responsible for the control of the projectile tongue.
A binocular receptive field is defined as the overlap of
the monocular receptive fields of a binocular neuron. A suitable object in this region of the visual field elicits a strong
reaction in the neuron. From the size distribution of monocular fields (Fig. 1b) it becomes clear that binocular receptive
fields also tend to be large. The question arises under these
circumstances as to whether an exact distance determination
can be achieved at all.
Figure 3 shows the two-dimensional projections of three
typical binocular receptive fields; they occupy a high percentage of the binocular visual field. A discrimination of
small regions of space is nevertheless possible with an ensemble coding. This becomes clear from the following qualitative example. Consider the small rhombus V in Fig. 3. An
44
Tectum opticum
144 tectum neurons
V
Medulla oblongata /
Medulla spinalis
50%
1
12 interneurons
100%
50%
12 motor neurons
2
3
100%
Muscles
SAR1
RCP
Fig. 3. An example of the discrimination of a small region of space, V ,
with large binocular receptive fields (dark grey). V is characterized by the
fact that it lies within receptive field 1 but not within receptive fields 2 and
3. The light grey shading indicates the binocular visual field
1
O
1
I0
Is
I
excitatory parts
object in the visual field is positioned within V if and only
if the neuron which corresponds to receptive field 1 fires,
but the neurons which correspond to receptive fields 2 and
3 do not fire. It follows that a non-firing neuron conveys as
much information as a firing one.
From these considerations depth perception with large
receptive fields seems to be possible. In the Simulander II
model, information about the distance of a prey as evaluated
by binocular neurons is translated into commands for the
control of the projectile tongue.
The Simulander II network is shown in Fig. 4. The input
layer consists of 144 binocular tectum neurons. The positions and sizes of their receptive fields are adopted from
electrophysiological measurements in Hydromantes italicus
(Wiggers 1991). There are two more layers in the network
which are totally connected: an inhibitory interneuron layer
and an excitatory motor neuron layer. These model neuron
populations in the brainstem and spinal cord which form the
motor nuclei of the muscles of the projectile tongue. The
layers consist of 12 neurons each which are modeled with
a nonlinear (sigmoid) activation function. Each tectum neuron projects to 50% of both the interneurons and the motor
neurons.
There are no connections within the neural layers; the
network has a feedforward structure. The exact number of
neurons does not play a role and has merely been chosen
so as to yield a convenient connection structure which can
be easily evaluated numerically. The neuron layers have a
symmetry resembling the left-right symmetry of the brain in
order to reduce the number of weights, i.e., the number of
free parameters.
The motor neurons innervate the two main muscles controlling the tongue: The subarcualis rectus 1 (SAR1) whose
contraction leads to the protraction of the tongue, and the
rectus cervicis profundus (RCP) responsible for tongue retraction. Actually, there are several more muscles involved,
including some by which lateral protractions are achieved
(Roth 1987). Here we confine ourselves to central shots assuming an appropriate head position which can be thought of
as the result of the action of the Simulander I network. Empirical data show that both muscles are activated at roughly
Rectus cervicis profundus
Subarcualis rectus 1
O
I0
I
inhibitory parts
Fig. 4. The Simulander II network. Insets under the muscles boxes show
the muscular activation functions, i.e., muscle contraction, O, as a function
of the input, I, which is a weighted sum of the motor neurons’ firing rates.
In the simulations, I0 = 1.5 and Is = 12. RCP, rectus cervicis profundus;
SAR1, subarcualis rectus 1
the same time, and the protraction takes place on a short
time scale of 8–10 ms. The SAR1 has an all-or-nothing
characteristic whereas the RCP shows a smooth contraction
strength as a function of neural input (Roth 1987). The activity of the RCP thus determines the protraction length:
minimal RCP contraction corresponds to maximal protrusion (approximately 5 cm) and vice versa. In the model, the
activation of both muscles, I, is given by the weighted sum
of the motor neuron outputs:
I=
12
X
w i xi
(5)
i=1
(wi : neuromuscular synapse of motor neuron i; xi : firing
rate of motor neuron i). According to the RCP and SAR1
characteristics, the muscle contractions O(I) are given by the
functions shown in the insets in Fig. 4. The relation between
the contraction of the RCP and the protraction length of the
tongue is assumed to be linear.
As in the case of the Simulander I model, the network
weights are trained with an evolution strategy. A stimulus with a radius of 0.3 cm is presented near the center
of the visual field at different distances within reach of the
tongue. The fitness function is defined as the square deviation between the desired and actual contraction of the RCP,
summed over all stimulus presentations.
Results are shown in Fig. 5. Three different network performances are plotted as a function of stimulus distance, y.
Each point is a mean value calculated from the network reactions to 100 slightly different object positions at the same
distance. The vertical line designates the reach of the projectile tongue, ymax . Figure 5a gives the relative frequency
of correct reactions, where a correct reaction is defined to
be a shot of the tongue if the prey is within its reach, and
relative frequency
of correct reactions
45
1.0
0.8
0.6
0.4
a
0.2
0.0
success rate
1.0
0.8
0.6
0.4
b
0.2
0.0
0.5
mean deviation (cm)
iological studies yield the resolution necessary to catch a
central prey; a simple motor scheme results in success rates
between 80% and 100%. Since the resolution is inversely
proportional to the number of encoding neurons, even higher
resolutions can be achieved with a modest additional number
of neurons.
The network can easily be extended to include lateral
shots of the tongue. The large binocular receptive fields are
expected to yield a good resolution in the lateral binocular
visual field also.
0.4
0.3
0.2
c
0.1
0.0
0
2
4
ymax6
8
10
12
14
y (cm)
Fig. 5a–c. Performance of the Simulander II network after training as a
function of prey distance, y. The vertical line represents the reach of the
projectile tongue, ymax . a The relative frequency of correct reactions. b The
relative frequency of successful shots at prey within reach of the tongue.
c The mean deviation between the tip of the tongue and the center of the
prey for prey within reach of the tongue
no reaction for prey which is too far away. For y < ymax ,
the Simulander exhibits a tongue protraction in almost 100%
of the cases. In the region immediately beyond ymax , however, the performance is inaccurate: although the object is
out of reach, a reaction frequently takes place. The behavior
of living salamanders at the sight of prey just outside their
reach has not been studied. They may make errors in this
case, just as the model predicts. For y > 7 cm, the network
shows no reaction – as should be the case. This feature is
an immediate consequence of the shape of the binocular receptive fields and the topology of the network: an object
which is far away elicits a reaction in many tectum neurons.
This leads to high activity in the interneuron layer which
causes inhibition of the motor neurons and thus suppression
of muscle activity. The missing reaction to distant prey is a
generalization property of the network, because object positions outside the reach of the tongue are not contained in
the training set. Figure 5b shows the relative frequency of
events in which a prey at distance y < ymax is caught. Apart
from a small region at ymax , the results range between 80%
and 100%. Figure 5c represents the mean deviation between
the tip of the tongue and the center of the prey. It lies between 0.1 cm and 0.2 cm in most cases. For small values of
y, however, the deviation is greater. The high success rates
in this region indicate that the tongue overshoots if the prey
is close. Again, this is a result which could be tested with a
high-speed camera in a behavioral experiment.
The network simulation shows that fewer than 150 binocular tectum neurons with the properties found in electrophys-
5 Discussion
Both in analytical calculations and in neural network simulations the coarse coding mechanism shows good performance in resolving the position of a single object in twoand three-dimensional sensory space. We suggest that coarse
coding plays an important role, especially in situations where
a clearly defined, single stimulus has to be localized. This
seems to be true for prey capture by salamanders, where
an attention mechanism is at work, selecting and defining a
single target: salamanders track a single prey even if several
stimuli are present. A second example is sound localization
by barn owls (Tyto alba), which have an auditory map in
their midbrain auditory area, the nucleus mesencephalicus
lateralis dorsalis. The map consists of large, overlapping receptive fields (Knudsen and Konishi 1978; Konishi 1993).
Barn owls hunting in darkness are able to recognize and isolate noise patterns originating from prey, and their success
depends on their ability to localize prey from such auditory
signals alone. Sound localization in barn owls has also been
described with a neural network model (Uhlenwinkel et al.
1994). Finally, in the somatosensory systems of rats and humans, where large receptive fields can also be found (Dinse
et al. 1995), an important task is the localization of single
tactile stimuli on the skin surface. The somatosensory system will be discussed in a forthcoming article (Eurich et al.,
in preparation).
Problems such as the performance of the coarse coding
mechanism for discriminating two stimuli or for analyzing
complex sensory environments have not been addressed so
far. It may be that in these cases small receptive fields (as,
for example, in the human retina) or a combination of large
and small receptive fields yield better results.
In the coarse coding scheme, the firing of a neuron alone
suffices to convey information about the position of an object. The mechanism is fast compared with firing rate coding since the mean firing rate is properly defined only for
a sufficiently high number of action potentials, which takes
a certain time to emerge. However, in the coarse coding
framework, the neural firing rate is likely to convey further
information about the object such as size, color, movement
and pattern.
The coarse coding mechanism employs massively parallel information processing with neurons which are nonspecific with respect to stimulus position. Clearly, in a piece
of neural tissue one finds differences in the neurons’ preferences and reactions. In the past, this has led to the definition
of neuron classes, for example in the amphibian optic tectum
(Grüsser-Cornehls and Himstedt 1976; Roth and Himstedt
46
∼∼
w(ρ|(ϑ,ϕ))
∼
a(ϕ)
5
∼
∼ ρ (ϕ)
ρ1(ϕ)
2
We assume that it takes the values 15◦ and 35◦ at ϕ̃ = 0◦ and ϕ̃ = 135◦ ,
respectively. The ramp-shaped part of the distribution extends from % = 60◦
to % = 90◦ .
Taking into account these constraints, a rather tedious calculation yields
the position-dependent size distribution
∼
1-a(ϕ)
w(%|(ϑ̃, ϕ̃)) =
ρ (deg)
60

58320π−15552|ϕ̃|
1620π 2 −432π|ϕ̃|
%−
,


 320πϕ̃2 +240π2 |ϕ̃|+45π3 320πϕ̃22+240π2 |ϕ̃|+45π3 2
90
−
Fig. A1. The position-dependent receptive field size distribution w(%|(ϑ̃, ϕ̃))
as a function of receptive field size, %. For further explanations see text
58320π−15552|ϕ̃|
320π ϕ̃2 +240π 2 |ϕ̃|+45π 3

96|ϕ̃|
32|ϕ̃|

 5π3 %− 5π2 ,
%+
4860π +7344π|ϕ̃|−2304ϕ̃
320π ϕ̃2 +240π 2 |ϕ̃|+45π 3
(A3)
π ≤%≤% (ϕ̃)
1
36
,
%1 (ϕ̃)<%≤%2 (ϕ̃)
π ≤%≤ π
3
2
0,
otherwise
where
1978; Grüsser-Cornehls 1984). This does not mean that specific tasks can be assigned to single neurons: usually, single
neuron behavior is too nonspecific to represent distinctive
features or reactions. Instead, indiviual neurons seem to be
involved in multiple tasks. Conversely, specific tasks are
performed with many neurons.
This holds, for example, for the decoding of information
about the position of an object stored in a population of neurons with large receptive fields. In the local coding scheme,
the extraction of object position in the form of Cartesian
or polar coordinates is straightforward, whereas this seems
to be difficult in the coarse coding scheme. However, such
extraction of information does not appear to take place in
the nervous system. Instead, there are always populations of
neurons processing and conveying information, beginning
from the cells of a receptor surface to pools of motor neurons from which subpopulations are recruited in order to
yield specific muscle contractions. The network simulations
in Sects. 4 and 5 show that under these circumstances, the
evaluation of stimulus position is successfully achieved with
a fairly small number of broadly tuned neurons.
Appendix. The construction of a function w(%|(ϑ, ϕ))
The number of empirical data is not sufficiently high to yield a size dis2 , which would be necessary for a function
tribution for all (ϑ, ϕ) ∈ SR
w(%|(ϑ, ϕ)). Instead, we consider the overall size distribution given by a
subset of the receptive fields shown in Fig. 1, and make additional assumptions concerning the increase in large receptive field neurons in the
lateral visual field. Instead of w(%|(ϑ, ϕ)), the distribution w(%|(ϑ̃, ϕ̃)) is
constructed, to which the transformation (4) can be applied.
First, we assume that the size distribution does not depend on ϑ̃:
∂
∂ ϑ̃
w(%|(ϑ̃, ϕ̃)) = 0
(A1)
which is a reasonable simplification since we are mainly interested in the
resolution along the horizontal meridian.
Fig. A1 shows the structure of w(%|(ϑ̃, ϕ̃)). It consists of two parts:
a tent-shaped part representing the majority of the receptive fields, and
a ramp-like part representing large receptive fields. The areas under the
curves are a(ϕ̃) and 1 − a(ϕ̃), respectively, to satisfy the normalization
Zπ/2
w(%|(ϑ̃, ϕ̃)) d% = 1
2
∀(ϑ̃, ϕ̃) ∈ SR
(A2)
0
a(ϕ̃) is a linearly decreasing function of |ϕ̃| where a(ϕ̃ = 0◦ ) = 1 (no large
receptive fields in the central visual field) and a(ϕ̃ = 135◦ ) = 0.8 (20%
large receptive fields in the lateral visual field). The tent-shaped part of the
distribution is a piecewise linear function extending from 5◦ to %2 (%̃); it
is symmetric with respect to %1 (ϕ̃). %2 (%̃) increases linearly with |ϕ̃|, thus
taking into account a broadening of the distribution from central to lateral.
%1 (ϕ̃) =
2|ϕ̃|
π
+
,
27
18
%2 (ϕ̃) =
4|ϕ̃|
π
+
27
12
(A4)
Acknowledgements. For numerous stimulating discussions and the supply
of salamander data we are indebted to Gerhard Roth, Ursula Dicke and
Wolfgang Wiggers. We also thank Hermann Wagner for useful comments.
This work was supported by the Deutsche Forschungsgemeinschaft with a
grant from the Schwerpunktprogramm ‘Physiologie und Theorie neuronaler
Netzwerke’. One of us (C.W.E.) has benefited from support by the Studienstiftung des deutschen Volkes and also acknowledges financial support
from the BASF corporation while writing this paper.
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