Download Biological Cybernetics

Biol. Cybern. 76, 357–363 (1997)
Biological
Cybernetics
c Springer-Verlag 1997
Coarse coding: calculation of the resolution achieved by a population
of large receptive field neurons
Christian W. Eurich1 , Helmut Schwegler2
1
2
The University of Chicago, Department of Neurology, MC 2030, 5841 South Maryland Avenue, Chicago, IL 60637, USA
Institut für Theoretische Physik, Universität Bremen, D-28334 Bremen, Germany
Received: 7 January 1996 / Accepted in revised form: 7 January 1997
Abstract. Electrophysiological studies in various sensory
systems of different species show that many neurons involved in object localization have large receptive fields. This
seems to contradict the high sensory resolution and the behavioral precision observed in localization experiments. Assuming a coarse coding mechanism, the resolution obtained
by an ensemble of neurons is analytically calculated as a
function of receptive field size. It is shown that particularly
large receptive fields yield a high resolution.
1 Introduction
A substantial amount of work in the neurosciences has been
dedicated to the question of how information is represented
in neural systems. In this paper, we consider tasks of object localization such as visual or auditory localization of
an object in the surroundings, or tactile localization of an
object on the skin surface. The space in which a localization
takes place will be referred to as the sensory space. Theoretical calculations of the sensory resolution for different forms
of internal representation of object position lead to results
which can be tested quantitatively in behavioral experiments.
At the same time, electrophysiological and anatomical data
about the neurons involved in the processing of sensory information are available, relating behavioral patterns to the
underlying anatomy and physiology.
Many species show a high accuracy in object localization. Tongue-projecting salamanders (Bolitoglossini ), for example, are able to localize mites of 0.5 mm size at distances
of 15–20 cm and snap at them with high success rates (Roth
1987). Barn owls (Tyto alba) determine the direction of the
sound of a prey with an accuracy of a few degrees (Konishi
1983, 1993). In human subjects, the two-point discrimination threshold for tactile stimulation is 1.4 mm, decreasing
to 1.16 mm after paired peripheral tactile stimulation (PPTS;
Stauffenberg et al. 1994; Dinse et al. 1995).
Interestingly, in all of these cases central neurons involved in object localization have receptive fields which are
Correspondence to: C. Eurich
(Fax: +(312) 702-9076; e-mail: [email protected])
large compared with the sensory resolution observed in behavioral experiments. By a receptive field of a central neuron we mean the subset of the sensory space in which an
appropriate stimulus elicits a reaction in the corresponding
neuron. Tectal neurons in the tongue-projecting salamander
Hydromantes italicus have a mean receptive field diameter of 41◦ , with a minimum of 10◦ and a maximum near
360◦ (Wiggers et al. 1995). The receptive fields of auditory
neurons in the nucleus mesencephalicus lateralis dorsalis of
Tyto alba range from 23◦ to “unrestricted” in elevation and
have a mean diameter of 25◦ in azimuth (Knudsen and Konishi 1978; Konishi 1993). In the case of tactile stimulation,
electrophysiological recordings have been obtained in the
primary somatosensory cortex of the rat. The neurons react to a skin surface of 45.4(±4.6) mm2 with an increase
to 79.9(±8.2) mm2 after PPTS, resulting in a higher receptive field overlap (Dinse et al. 1995). These findings suggest
that large receptive field neurons may well be involved in
object localization, a task which has often been ascribed exclusively to small receptive field neurons (Grüsser-Cornehls
1984; Gaillard 1985).
Several mechanisms have been proposed for the neural
coding of the position of a stimulus in a sensory space,
X (Milner 1974; Feldman and Ballard 1982; Hinton et al.
1986; Snippe and Koenderink 1992). The main concepts are
the local coding scheme, the intensity coding scheme, and
the ensemble coding scheme.
In the local coding scheme, X is divided into small cells
the size of the space which can be resolved by the receptor
cells. Each cell corresponds to one neuron which fires whenever a stimulus appears in its cell. Although a discrimination
of several stimuli is fairly easy in this coding scheme, the
number of neurons necessary to cover X grows exponentially with the dimension of X. This combinatorial explosion makes it unlikely that local coding is exclusively used
as a localization mechanism in neural systems (Feldman and
Ballard 1982).
In the intensity coding scheme, each neuron encodes the
position of an object along one dimension of X by firing
at different rates. Thus, the number of neurons involved is
relatively small, and the resolution is determined by the reliability of the neuron reaction (e.g., by the amount of noise).
358
The main drawback of the intensity coding is the time needed
for defining a precise firing rate with a neural spike train;
this time is much too long to explain fast animal reactions
in many cases, particularly in prey capture activity. Another
difficulty lies in the fact that the same firing rate may be
elicited in a neuron either by a single object or by a combined
stimulation originating from two or more objects; the neural
system cannot discriminate between one or more stimuli in
sensory space. This property of the convergence of information channels, known as metamery, takes an extreme form
in the case of intensity coding and prevents an appropriate
representation of the world.
Finally, in the ensemble coding scheme, each neuron has
a receptive field covering a region which is large compared
with the sensory resolution, resulting in a receptive field
overlap everywhere on X. The position of a stimulus is
encoded by the ensemble of neurons contributing to the respective overlap. The abovementioned large receptive fields
found empirically suggest an ensemble coding mechanism
in various sensory systems. This is supported by the fact
that parallel information processing plays an important role
in the brain, from the simultaneous activations of many receptors to the population code for muscle control. Although
metamery is also present in the ensemble coding scheme,
different stimuli can still be discriminated if they are not too
close to each other or if the neuron density is sufficiently
high.
Several attempts have been made to understand the properties of ensemble coding. Heiligenberg (1987) and Baldi
and Heiligenberg (1988) considered the hyperacuity properties of an infinite, one-dimensional array of large, overlapping receptive fields. They assumed a Gaussian sensitivity
profile for the neurons, i.e., the firing rate of a neuron depended on the position of the stimulus within the receptive
field (channel coding). Baldi and Heiligenberg concluded
that the resolution increases with receptive field size and
that the system is stable with respect to noise.
For the case of channel coding, Snippe and Koenderink
(1992) calculated the resolution obtained by receptive fields
arranged on an n-dimensional lattice in the presence of
noise. Their conclusion was that receptive field size does
not play a role for n = 2 but that large receptive fields and,
correspondingly, substantial receptive field overlap are only
advantageous for n > 2. Obviously, these findings do not
account for the large receptive fields described above: Direction localization (either visual or auditory) and somatosensory localization are two-dimensional problems.
Instead of channel coding, which like intensity coding
suffers from the problem of the definition of an appropriate
firing rate within a short amount of time, Hinton (1981) and
Hinton et al. (1986) considered the coarse coding mechanism in which the neurons do not have a Gaussian sensitivity
profile, but are binary in nature: They fire whenever a stimulus is within their receptive field, and are otherwise “silent”.
In the coarse coding scheme, the resolution is determined
by the number of different encodings (i.e., by the number of
different firing patterns) in the neural population as a stimulus is moved about in the sensory space. This directly leads
to the result that large receptive fields yield a high resolution, because they overlap extensively and therefore show
a high number of different encodings. In a rough estima-
tion, Hinton et al. (1986) showed that for X = IRk with
equally distributed receptive fields all having the shape of
k-dimensional spheres of radius r, the accuracy (which is the
reciprocal of the resolution) is proportional to N rk−1 , i.e.,
for k ≥ 2, large receptive fields are advantageous compared
with small receptive fields.
In a biologically relevant situation, however, the space
X = IRk has to be replaced by a space of finite size. In this
case, r cannot be arbitrarily large. Instead, we find that the
resolution is bad if r approaches the size of X, since receptive fields covering the whole sensory space (and hence,
neurons firing all the time) do not convey information. On
the other hand, Hinton’s formula shows that for r → 0, the
resolution is also bad. Therefore, an optimal value for the
receptive field size exists, the calculation of which requires a
more general formalism for the evaluation of the resolution
obtained by a population of neurons.
Concerning the accuracy in the coarse coding scheme,
Hinton et al. (1986, p. 91) wrote: “The accuracy is proportional to the number of different encodings that are generated
as we move a feature point along a straight line from one
side of the space to the other. Every time the line crosses
the boundary of a zone, the encoding of the feature point
changes because the activity of the unit corresponding to
that zone changes.” The mechanism developed here suggests
that it might be possible to calculate the resolution obtained
by a population of neurons by mapping combinatorics of receptive fields onto the density of receptive field boundaries.
Following this concept, we assume that (i) receptive field
boundaries are simple, one-dimensional lines and (ii) receptive fields are “compact” in shape, e.g., circular, elliptical,
or rectangular. The first assumption rules out pathological
cases such as fractal receptive boundaries which – despite
their infinite length – do not lead to an increase in resolution, because they do not increase the number of different
encodings in the neural population. The second assumption
rules out receptive fields which are widely ramified, or consist of several disconnected parts. This feature corresponds
to the existence of a topological map in the sense that nearby
positions in the sensory space have similar representations
in the neural population, whereas stimuli in positions which
are far apart activate very different ensembles of neurons.
Although in the presence of such a topological map the number of different encodings in a population of N neurons is
smaller than the combinatorial maximum of 2N , the evaluation of the information stored in the population – in the
form of motor commands or the activation of further neural
structures – is likely to be more efficient, especially in the
presence of noise. Receptive fields with simple shapes are
usually encountered in nature, e.g., in the visual (Gaillard
1985; Wiggers et al. 1995) or in the auditory (Knudsen and
Konishi 1978) system.
In the remainder of this article, the resolution obtained
by a population of receptive fields is calculated analytically by considering the density of receptive field boundaries. Thereby, we restrict ourselves to circular fields in twodimensional sensory spaces. In Sect. 2, the basic concepts
such as receptive field density and resolution are introduced.
Section 3 gives a simple example revealing the basic mechanism responsible for a high resolution. In Sect. 4, a local
formalism is developed which allows the derivation of the
359
i.e., to a different representation of the position of the object.
In this way, the resolution obtained by the neuron population
is related to the density of receptive field boundaries.
In order to obtain an analytical expression for the resolution, we idealize the situation to a continuous field approximation with a receptive field density σ(x) with the properties
g
3
f
e
1
d
b
γ
Z
σ(x) ≥ 0
σ(x) dx = N
c
2
a
Fig. 1. A sensory space, X = IR2 , is covered with three circular receptive
fields. A stimulus moving along a path, γ, elicits different reactions in
the neuron population depending on the actual set of overlapping receptive
fields
resolution for arbitrary receptive field distributions in space
and arbitrary receptive field size distributions. We close the
considerations with a short discussion in Sect. 5.
A forthcoming article (Eurich et al. 1997b) is concerned
with the applications to special biological systems, namely
the visually controlled orienting movement of salamanders
towards prey positions and the tongue-projection of salamanders which requires an accurate determination of prey
distance.
2 Basic notions
Let X be the sensory space, i.e., the space in which the
localization of an object takes place. Examples include a
three-dimensional subset of Euclidean space, IR3 , for the
localization of a visual or auditory object including distance,
or the two-sphere, S 2 , for the determination of direction
only. In the latter case, it is assumed that the two-sphere is
embedded in IR3 ; the observer is positioned in the center of
the sphere, so that each point x ∈ S 2 corresponds to a certain
direction. Consider N receptive fields Ri ⊆ X at positions
xi (i = 1, . . . , N ) where xi refers to a characteristic point
of Ri (e.g., the center of a circular receptive field). The
output function of a neuron is defined as
1, if x ∈ Ri
(i = 1, . . . , N )
(1)
fi (x) =
0, if x ∈
/ Ri
i.e., we consider binary neurons with the additional property
of a receptive field. Note that this restricted view of neuron
behavior only refers to the position of an object; a firing
rate coding can be supplemented corresponding to additional
features of the environment.
As an illustration, Fig. 1 shows a sensory space X = IR2
with three circular receptive fields. A single stimulus moves
along a path, γ, which can be divided into 7 parts, (a)–
(g), according to the different firing patterns elicited in the
neuron population. For example, when the stimulus is in
part (b) neuron 1 fires, and when the stimulus is in part
(c) neurons 1 and 2 fire. Whenever the stimulus crosses a
receptive field boundary, one of the neurons is switched on
or off, thus leading to a different active neuron ensemble,
(2)
(3)
X
and, for arbitrary A ⊆ X,
Z
σ(x) dx ≈ # (receptive fields in A)
(4)
A
where the approximation results from the fact that the number of real receptive fields is integer while the integral on
the left side takes continuous values. σ(x) can either be calculated from a probability distribution of receptive fields or
from a set of experimental data; see Appendix A.
In optics, resolution is defined as the minimal distance
of two points which can still be separately mapped by an
optical device. In our case, two points in X are separately
mapped (i.e., have different representations) if there is a
receptive field boundary running between them. This leads
to the definition of the resolution A(x)
A(x) =
1
L(x)
(5)
where L(x) is the density of receptive field boundaries. The
main objective in the remainder of this article is the inference
of a relation between the density σ(x) – which is usually
determined by experimental investigations – and the density
L(x). Here we confine ourselves to the geometric situation
of a two-sphere with radius R,
2
= (x, y, z) ∈ IR3 : x2 + y 2 + z 2 = R2
(6)
X = SR
covered with circular receptive fields, i.e., with receptive
fields having the shapes of spherical caps. Extensions to
X = IR2 and X = S 1 (Eurich 1995) or to the cylinder (Eurich et al. 1997a) are straightforward. The consideration of
arbitrary manifolds with a more general class of receptive
field morphologies requires a sophisticated mathematical approach and is not required for most biological applications
anyway.
An immediate consequence of (5) is
1
(7)
N
since σ(x) ∝ N and therefore L(x) ∝ N . Equation (7) is
advantageous compared with the situation of local coding
with nonoverlapping receptive
√ fields where the resolution is
only proportional to 1/ dim X N . The result is in agreement
with the abovementioned estimation in Hinton et al. (1986).
A(x) ∝
3 An introductory example
As an introductory example revealing the basic mechanism
responsible for the high resolution of large receptive fields,
360
γ
2ρ
R
1
a
b
2 . a Cross-section of S 2 showing a
Fig. 2. The geometry for X = SR
R
receptive field with angular diameter 2%. b The sphere is divided into two
sets, S1 (dark gray), a stripe of angular width 2% surrounding a great circle,
γ, and the remaining spherical caps, S2 (light gray)
2
consider the case X = SR
with uniformly distributed receptive fields of equal size. The normalization (3) yields
N
≡ σ ∀(x∈X)
(8)
4πR2
2
Figure 2a shows the cross-section of SR
with a recep2
tive field which has an angular diameter 2%. In Fig. 2b, SR
is depicted. The path, γ, is a great circle. The resolution
is determined by the number of receptive field boundaries
intersecting γ. According to the angular diameter of the re2
can be divided into two regions, S1 and
ceptive fields, SR
S2 , where S1 is a stripe of angular width 2% around γ, and
S2 is composed of the two remaining spherical caps. Thus,
receptive fields whose centers are situated within S1 intersect γ twice, whereas receptive fields in S2 do not intersect
γ at all. (Receptive fields sitting exactly on the boundary
between S1 and S2 are tangent to γ in one point, but they
form a set of measure zero.)
The number of receptive field boundaries intersecting γ,
E(%), is expected to be
Fig. 3. The angular resolution per neuron, N α(%), as a function of receptive
field size
e
σ(x) =
E(%) = σ (2 · F1 + 0 · F2 )
= 2N sin %
(9)
F1 (resp. F2 ) being the surface area of S1 (resp. S2 ). Division by the length of γ yields the density of receptive field
boundaries along γ:
E(%) N sin %
=
2πR
πR
This results in a resolution
1
πR
A(%) =
=
L(%) N sin %
L(%) =
(10)
(11)
x0
γ
κ(x0,ρ)
Fig. 4. The geometrical situation for the calculation of the density of re2 for the direction e which is tangential
ceptive field boundaries at x0 ∈ SR
to a curve γ. κ(x0 ; %) indicates the set of all positions of receptive fields
which contribute with a field boundary at x0 . One of these receptive fields
is shown (gray)
yield the best resolution is that they have the longest possible field boundaries: Given a fixed number of neurons, they
yield the highest possibility of being crossed by a stimulus
moving in X.
4 Extension to more general cases
In this section, a twofold extension of the previous example
is developed. First, a local formalism is introduced which
allows the calculation of the resolution for arbitrary receptive field densities σ(x) on the sphere. Second, instead of a
fixed receptive field size for all neurons, a size distribution
is considered. The distribution may vary from place to place
as observed, for example, in the visual system of salamanders, where large receptive fields tend to be situated mainly
in the lateral visual field.
The angular resolution is
α(%) =
A(%)
π
=
R
N sin %
(12)
In Fig. 3, the function N α(%) – which can be interpreted
as the angular resolution per neuron – is plotted against %. It
has a minimum at % = 90◦ , which means that the best resolution is achieved by a population of neurons with a receptive
field diameter of 180◦ . These are large field neurons whose
receptive fields cover half of the sphere. The corresponding
resolution is α(90◦ ) = 180◦ /N , i.e., with, say, N = 100
neurons a resolution of 1.8◦ is achieved. An illustrative explanation of the fact that the largest possible receptive fields
4.1 Arbitrary field densities
Assume that all receptive fields possess the same angular
2
as in the
diameter, 2%. Instead of a path γ across X = SR
previous example, consider a local direction, e, which can be
2
. For a
thought of being tangential to γ at some point x0 ∈ SR
given field density σ(x), we calculate the density of receptive
field boundaries at x0 , Le (x0 ; %). In general, Le (x0 ; %) varies
with e, which is indicated by the subscript.
The geometric situation is reproduced in Fig. 4. All receptive fields contributing with a field boundary at x0 are
361
situated on a circle, κ(x0 ; %), of angular diameter 2% around
x0 . For each point x ∈ κ(x0 ; %) a weighting factor σ(x)
has to be supplied. Furthermore, σ(x) has to be multiplied
by a geometrical factor R sin %| cos β|, where β is an angle parametrizing κ(x0 ; %) with β = 0 corresponding to the
direction e (see Fig. 5a). The geometrical factor takes into
account that receptive fields lying in the directions parallel
to e have a greater influence than receptive fields lying in
the directions perpendicular to e. The density of receptive
field boundaries finally results from an integration over β:
Z2π
Le (x0 ; %) = R sin % σ(ϑ(β), ϕ(β))| cos β| dβ
(13)
a different value of %. Ultimately, in the expression for the
density of receptive field boundaries (13), σ(x) has to be
replaced by σ̃(x, %), and an additional integration over the
range of values of % has to be performed. Considering (18),
the result is
(19)
Le (x0 ; w)
2π
π
ZZ
R sin % σ(ϑ(β), ϕ(β))w(%|(ϑ(β), ϕ(β)))| cos β| dβd%
=
0 0
The parameter w in Le (x0 ; w) indicates that L does not refer
to a single receptive field size but to a size distribution.
0
2
where ϑ, ϕ are spherical coordinates on SR
. For a more
detailed derivation of (13), see Appendix B.
According to (5), the resolution is the reciprocal value
of the density of receptive field boundaries, Ae (x0 ; %) =
1/Le (x0 ; %), while the angular resolution results from a division of Ae (x0 ; %) by R as in (12), αe (x0 ; %) = Ae (x0 ; %)/R.
As an example, let σ(x) again take the constant value
2
and e, it follows from (13)
(8). Then, for arbitrary x0 ∈ SR
sin %
L(%) = N
4πR
Z2π
| cos β| dβ
0
sin %
(14)
=N
πR
which is in agreement with the result (10) obtained above.
4.2 Receptive field size distribution
Receptive fields of sensory neurons usually have different
sizes. For a mathematical treatment, consider the joint den2
, % ∈ R ≡ [0; π]) where w(x, %)dxd%
sity w(x, %) (x ∈ SR
is the probability of finding a receptive field in the range
[%; % + d%] in the volume dx at x. The joint distribution is
normalized as follows:
Z Z
w(x, %) dxd% = 1
(15)
X R
In practice, instead of w(x, %), a size distribution is given
2
. This is mathematically described as the
for various x ∈ SR
conditional probability density
w(%|x) =
w(x, %)
w(x)
(16)
where w(x) is the marginal receptive field distribution (23)
introduced in Appendix A. Analogous to the case (28) of
constant field sizes, a receptive field density σ̃(x, %) can be
introduced which comprises the dependencies of both position and size:
σ̃(x, %) = N w(x, %)
(17)
(16), (17), and (28) yield
σ̃(x, %) = w(%|x) σ(x)
(18)
In the local formalism, instead of a single curve κ(x0 , %) a
set of such curves has to be taken into account, each for
5 Discussion
A coarse coding mechanism was assumed for the encoding
of a single position in a sensory space, X, by a population
2
of neurons. As a standard example for X, the two-sphere SR
was chosen, describing for example the visual or auditory
direction localization. For the case of “compact” receptive
fields, the notion of accuracy developed in Hinton et al.
(1986) was replaced by a definition of resolution based on
the density of receptive field boundaries. The resolution was
calculated analytically for circular receptive fields with arbitrary distributions in X and arbitrary size distributions. Even
the simplest case of equally distributed receptive fields all
having the same size shows that the best resolution is obtained with receptive fields which are 180◦ in diameter. In
this case, receptive field overlap is maximal, leading to the
highest possible number of overlap patterns, and hence to
the highest number of different neural representations of positions in the sensory space.
The results suggest that the large receptive field neurons
found in the visual, auditory, and somatosensory system of
various species contribute to the high accuracy observed in
behavioral experiments. According to a previously adopted
view, only small receptive field neurons are involved in object localization, whereas large receptive field neurons are
responsible for large-scale movements, predator detection,
etc. (Grüsser-Cornehls 1984; Gaillard 1985). However, only
the detailed analysis of a given sensory system can answer
the question of the relevance of neurons with different receptive field sizes. Examples for such analyses will be supplied
in forthcoming articles (Eurich et al. 1997a,b).
In the general case of arbitrarily distributed receptive
fields, the density of receptive field boundaries (13,19) depends on the direction e in the sensory space, i.e., the resolution is angle-dependent. This anisotropy is consistent with
psychophysical studies (Jastrow 1893; for a review, see Appelle 1972) showing that the performance of the visual system is poorer for diagonal stimuli than for horizontal and
vertical stimuli (oblique effect). A similar effect has been
demonstrated in the somatosensory system (Lechelt 1988).
Further analysis comparing theoretical prediction and empirical data may show that at least part of the phenomena can
be explained in terms of distributions of receptive fields.
Some further topics in connection with coarse coding remain to be worked out. First, the mechanism of decoding the
information carried by a population of neurons has not been
362
Appendix A Calculation of the receptive field density
eϑ
e
W (x1 , . . . , xN ) dx1 . . . dxN = 1
(20)
XN
where X N is the N -fold Cartesian product of X with itself. W is symmetric with respect to permutations of its arguments since the sample of
receptive fields is not ordered:
x0
{
R sin ρ
β
κ (x 0 ;ρ)
∼
β0
eϕ
R cos ρ
0
κ (x 0 ;ρ)
a
{
R sin ρ
ρ
x
b
dh
κ (x 0+dl e; ρ)
dF
dβ
e
The receptive field density, σ(x), can either be obtained from a probability
distribution of receptive fields on X, or from a limited set of (usually
experimental) data.
In the former case, consider the function W (x1 , . . . , xN ) giving the probability density of receptive field Ri being at xi (i = 1, . . . , N ) with respect
to an ensemble of individuals of the same species. The normalization yields
Z
0
x0+dl e
x0 dl
β
β
{
tackled analytically. However, numerical calculations in the
form of a neural network have been performed, indicating
that the neuron population which evaluates the distributed
information (e.g., in the form of muscle contractions) does
not considerably outnumber the encoding neuron population
(Eurich et al. 1995). Second, only a single stimulus has been
considered so far. It is an open question as to what preprocessing is necessary to analyse a complex sensory scene,
and eventually discriminate between several objects, with a
coarse coding mechanism.
R sin ρ
e
κ(x0; ρ)
κ(x0; ρ)
c
d
Fig. 5. a Topview and b sideview of the curve κ(x0 ; %). c The area dF
(gray) traversed by the curve κ when the stimulus moves from x0 to x0 +dle.
d The local geometry concerning a single stripe (gray) of dF . For further
explanations, see text
W (x1 , . . . , xi , . . . , xj , . . . , xN )
= W (x1 , . . . , xj , . . . , xi , . . . , xN )
∀i,j∈{1,...,N }
(21)
The marginal distribution with respect to the ith receptive field is
Z
W (x1 , . . . , xN ) dx1 . . . dxi−1 dxi+1 . . . dxN
wi (xi ) =
(22)
X N −1
(i = 1, . . . , N )
and (21) immediately yields
w1 (x1 ) = · · · = wN (xN ) =: w(x)
(23)
w(x)dx is the probability of finding an arbitrary receptive field in the volume
dx at x. Now define a function nA (x1 , . . . , xN ) giving the number of
receptive fields in A ⊆ X:
nA (x1 , . . . , xN ) = # (receptive fields in A ⊆ X)
(24)
Just like W (x1 , . . . , xN ), nA (x1 , . . . , xN ) is symmetric with respect to
permutations of its arguments. Let IA (x) be the indicator function for the
subset A ⊆ X,
n
IA (x) =
1, x ∈ A
0, x ∈
/A
(25)
Then,
A comparison of (27) and (4) leads to the definition of the density of
receptive fields, σ(x),
σ(x) := N w(x)
(28)
σ(x) fulfils the requirements (2) and (3).
In a slightly different interpretation, σ(x) in (28) can be regarded as
the receptive field density of a “mean system”. Under the assumption that
the members of the statistical ensemble do not differ significantly, σ(x) can
be obtained by looking at an actual, single system which is given in an
experimental situation.
The question arising in this situation is: Given a finite number of receptive field positions, x0i (i = 1, . . . , N ), what does the field density σ(x)
look like? Since a probability arises from a relative frequency in the limit
of an infinite number of trials only, σ(x) cannot be inferred uniquely. For
2 , the following procedure yields
X being the two-sphere with radius R, SR
good results: For all i, replace the receptive field at x0i by a Gaussian
function with variance π 2 /4 centered at x0i ,
w̃i (x; x0i ) = exp (−
2ε2 (x, x0i )
2
) (x ∈ SR
)
π2
where ε(x, x0i ) is the angle between x and x0i :
ε(x, x0i ) = arccos(x · x0i )
nA (x1 , . . . , xN ) =
N
X
IA (xi )
(26)
i=1
Applying (22), (23), and (26), the expected value of the number of receptive
fields in A is given by
Z
hnAi =
N Z
X
=
i=1
Z
=N
N
P
σ(x) = N
A
i=1
N
RP
X i=1
exp (− 2ε
2
(x,x0i )
)
π2
(31)
2
0i )
exp (− 2ε (x,x
)
π2
dx
A change of summation and integration in the denominator of (31) yields
N integrals all having the same value I which is obtained numerically as
I ≈ 7.579. The resulting formula for the density σ(x) is
IA (x) w(x) dx
N
P
X
w(x) dx
(30)
The receptive field density is obtained by summation of the Gaussian functions, normalization, and multiplication with N :
nA (x1 , . . . , xN )W (x1 , . . . , xN ) dx1 . . . dxN
XN
(29)
(27)
σ(x) =
exp (− 2ε
i=1
I
2
(x,x0i )
)
π2
(32)
363
Appendix B Derivation of Equation (13)
According to Fig. 4, all points x ∈ κ(x0 ; %) contribute with a receptive field
boundary at x0 . κ(x0 ; %) is a circle which forms the basis of a spherical
shell centered around x0 . A topview and a sideview of κ(x0 ; %) are shown
in Fig. 5a and 5b, respectively.
The circle is parametrized as follows. Each point x ∈ κ(x0 ; %) can be
written as
x = x0 cos % + R sin % cos(β + β0 ) eϕ0 + R sin % sin(β + β0 ) eϑ0
(33)
where eϕ0 and eϑ0 are the basis vectors corresponding to spherical coordinates at x0 ,
Ã
eϑ0 =
cos ϑ0 cos ϕ0
cos ϑ0 sin ϕ0
− sin ϑ0
!
Ã
,
eϕ0 =
− sin ϕ0
cos ϕ0
0
!
(34)
and β0 is the angle between e and eϕ0 ,
β0 = arccos(e · eϕ0 )
(35)
The circle is thus parametrized by the angle β ∈ [0; 2π] with β = 0 corresponding to the direction e (Fig. 5a). Given a fixed set (x0 , e, %), (33) yields
x = (x, y, z)T as a function of β which is necessary to evaluate σ(x(β)).
If the receptive field density is given as a function of spherical coordinates
instead, (33) has to be combined with the well-known transformation of
Cartesian and spherical coordinates to obtain σ(ϑ(β), ϕ(β)).
A stimulus moving from x0 to x0 + dle leads to the curve κ traversing
the area dF as indicated in Fig. 5c. For the purpose of integration, dF is
divided into small stripes parallel to e (Fig. 5d). The stripes have a length
dl and a width dh which depends on the angle β:
dh = R sin %| cos β|dβ
(36)
An integration of the receptive field density, σ(x(β)) or σ(ϑ(β), ϕ(β)), over
dF = dh dl yields the number of receptive field boundaries the stimulus
intersects on its way, while an integration over dh alone as given by (36)
results in an expression for the density of receptive field boundaries at x0
corresponding to the direction e. The latter immediately leads to (13).
Acknowledgements. We thank Gerhard Roth and Wolfgang Wiggers for
helpful discussions and the supply of salamander data. Manfred Fahle’s hint
on the oblique effect and the comments of an anonymous referee are gratefully acknowledged. 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.) was supported by
the Studienstiftung des deutschen Volkes.
References
1. Appelle S (1972) Perception and discrimination as a function of stimulus orientation: the “oblique effect” in man and animals. Psychol Bull
78: 266–278
2. Baldi P, Heiligenberg W (1988) How sensory maps could enhance
resolution through ordered arrangements of broadly tuned receivers.
Biol Cybern 59: 313–318
3. Dinse HR, Godde B, Spengler F (1995) Short-term plasticity of topographic organizaton of somatosensory cortex and improvement of
spatial discrimination performance induced by an associative pairing
of tactile stimulation. (Internal Report 95-01) Institut für Neuroinformatik, Bochum
4. Eurich CW (1995) Objektlokalisation mit neuronalen Netzen. Harri
Deutsch, Frankfurt
5. Eurich CW, Roth G, Schwegler H, Wiggers W (1995) Simulander: a
neural network model for the orientation movement of salamanders. J
Comp Physiol A 176: 379–389
6. Eurich CW, Dinse HR, Dicke U, Godde B, Schwegler H (1997a)
Coarse coding accounts for improvement of spatial discrimination after
plastic reorganization in rats and humans. (Submitted)
7. Eurich CW, Schwegler H, Woesler R (1997b) Coarse coding: applications to the visual system of salamanders. Biol Cybern (in press)
8. Feldman JA, Ballard DH (1982) Connectionist models and their properties. Cogn Sci 6: 205–254
9. Gaillard F (1985) Binocularly driven neurons in the rostral part of the
frog optic tectum. J Comp Physiol A 157: 47–55
10. Grüsser-Cornehls U (1984) The neurophysiology of the amphibian optic tectum. In: Vanegas H (ed) Comparative neurology of the optic
tectum. Plenum Press, New York, pp211–245
11. Heiligenberg W (1987) Central processing of sensory information in
electric fish. J Comp Physiol A 161: 621–631
12. Hinton GE (1981) Shape representation in parallel systems. In: Proceedings of the 7th International Joint Conference on Artificial Intelligence, Vancouver, BC, Canada, pp1088–1096
13. Hinton GE, McClelland JL, Rumelhart DE (1986) Distributed representations. In: Rumelhart DE, McClelland JL (eds) Parallel distributed
processing, Vol 1. MIT Press, Cambridge, Mass., pp77–109
14. Jastrow J (1893) On the judgment of angles and positions of lines. Am
J Psychol 5: 214–248
15. Knudsen EI, Konishi M (1978) A neural map of auditory space in the
owl. Science 200: 795–797
16. Konishi M (1983) Localization of acoustic signals in the owl. In: Ewert
J-P, Capranica RR, Ingle DJ (eds) Advances in vertebrate neuroethology. Plenum Press, New York, pp227–245
17. Konishi M (1993) Listening with two ears. Sci Am 268: 34–41
18. Lechelt EC (1988) Spatial asymmetries in tactile discrimination of line
orientation: a comparison of the sighted, visually impaired, and blind.
Perception 17: 579–588
19. Milner PM (1974) A model for visual shape recognition. Psychol Rev
81: 521–535
20. Roth G (1987) Visual behavior in salamanders. Springer, Berlin Heidelberg New York
21. Snippe HP, Koenderink JJ (1992) Discrimination thresholds for channel-coded systems. Biol Cybern 66: 543–551
22. Stauffenberg B, Godde B, Spengler F, Dinse HR (1994) Time course
and persistence of changes of human spatial discrimination performance induced by a paired tactile stimulation protocol. In: Elsner N,
Breer H (eds) Sensory transduction. Thieme, Stuttgart, p262
23. Wiggers W, Roth G, Eurich CW, Straub A (1995) Binocular depth perception mechanisms in tongue-projecting salamanders. J Comp Physiol
A 176: 365–377