Perceiving without Learning: from Spirals to 
Inside/Outside Relations 
Ke Chen* and DeLiang L. Wang 
Department of Computer and Information Science and Center for Cognitive Science 
The Ohio State University, Columbus, OH 43210-1277, USA 
{kchen,dwang} @cis.ohio-state.edu 
Abstract 
As a benchmark task, the spiral problem is well known in neural net- 
works. Unlike previous work that emphasizes learning, we approach 
the problem from a generic perspective that does not involve learning. 
We point out that the spiral problem is intrinsically connected to the in- 
side/outside problem. A generic solution to both problems is proposed 
based on oscillatory correlation using a time delay network. Our simu- 
lation results are qualitatively consistent with human performance, and 
we interpret human limitations in terms of synchrony and time delays, 
both biologically plausible. As a special case, our network without time 
delays can always distinguish these figures regardless of shape, position, 
size, and orientation. 
1 INTRODUCTION 
The spiral problem refers to distinguishing between a connected single spiral and discon- 
nected double spirals, as illustrated in Fig. 1. Since Minsky and Papert (1969) first intro- 
duced the problem in their influential book on perceptrons, it'has received much attention 
and has become a benchmark task in neural networks. Many solutions have been attempted 
using different learning models since Lang and Witbrock (1988) reported that the problem 
could not be solved with a standard multilayer perceptron. However, resulting learning 
systems are only able to produce decision regions highly constrained by the spirals de- 
fined in a training set, thus specific in shape, position, size, and orientation. Moreover, 
no explanation is provided as to why the problem is difficult for human subjects to solve. 
Grossberg and Wyse (1991) proposed a biologically plausible neural network architecture 
for figure-ground separation and reported their network can distinguish between connected 
and disconnected spirals. In their paper, however, no demonstration was given to the spiral 
problem, and their model does not exhibit the limitations that humans do. 
*Also with National Laboratory of Machine Perception and Center for Information Science, 
Peking University, Beijing 100871, China. E-mail: chen@cis.pku.edu.cn 
Perceiving without Learning 11 
There is a related problem in the study of visual perception, i.e., the perception of in- 
side/outside relations. Considering the visual input of a single closed curve, the task of 
perceiving the inside/outside relation is to determine whether a specific pixel lies inside or 
outside the closed curve. For the human visual system, the perception of inside/outside 
relations often appears to be immediate and effortless (see an example in Fig. 2(a)). As il- 
lustrated in Fig. 2(b), however, the immediate perception is not available for humans when 
the bounding contour becomes highly convoluted (Ullman 1984). Ullman (1984) suggested 
the computation of spatial relation through the use of visual routines. Visual routines result 
in the conjecture that the inside/outside is inherently sequential. As pointed out recently by 
Ullman (1996), the processes underlying the perception of inside/outside relations are as 
yet unknown and applying visual routines is simply one alternative. 
(a) (b) (b) 
(a) 
Fig. 1: The spiral problem. (a) a connected single 
spiral. (b) disconnected double spirals (adapted 
from Minsky and Papert 1969, 1988). 
Fig. 2: Inside/Outside relations. (a) an ex- 
ample (adapted from Julesz 1995). (b) an- 
other example (adapted from Ullman 1984). 
Theoretical investigations of brain functions indicate that timing of neuronal activity is a 
key to the construction of neuronal assemblies (Milner 1974, Malsburg 1981). In partic- 
ular, the discovery of synchronous oscillations in the visual cortex (Singer & Gray 1995) 
has triggered much interest to develop computational models for oscillatory correlation. 
Recently, Terman and Wang (1995) proposed locally excitatory globally inhibitory oscilla- 
tor networks (LEGION). They theoretically showed that LEGION can rapidly achieve both 
synchronization in a locally coupled oscillator group representing each object and desyn- 
chronization among a number of oscillator groups representing different objects. More 
recently, Campbell and Wang (1998) have studied time delays in networks of relaxation 
oscillators and analyzed the behavior of LEGION with time delays. Their studies show 
that loosely synchronous solutions can be achieved under a broad range of initial condi- 
tions and time delays. Therefore, LEGION provides a computational framework to study 
the process of visual perception from a standpoint of oscillatory correlation. 
We explore both the spiral problem and the inside/outside relations by oscillatory correla- 
tion in this paper. We show that computation through LEGION with time delays yields a 
generic solution to these problems since time delays inevitably occur in information trans- 
mission of a biological system. This investigation indicates that perceptual performance 
would be limited if local activation cannot be rapidly propagated due to time delays. As a 
special case, LEGION without time delays reliably distinguishes between connected and 
disconnected spirals and discriminates the inside and the outside regardless of shape, po- 
sition, size, and orientation. Thus, we suggest that this kind of problems may be better 
solved by a neural oscillator network rather than by sophisticated learning. 
2 METHODOLOGY 
The architecture of LEGION used in this paper is a two-dimensional network. Each os- 
cillator is connected to its four nearest neighbors, and the global inhibitor (GI) receives 
excitation from each oscillator on the network and in turn inhibits each oscillator (Terman 
12 K. Chen and D. L. Wang 
& Wang 1995). In LEGION, a single oscillator, i, is defined as 
dxi _ 3xi 3 (1 a) 
dt - xi - Yi q- Ii q- Si q- p 
dyi = e(,k + fftanh(/xi) - Yi). (lb) 
dt 
Here Ii represents external stimulation to the oscillator, and $i represents overall coupling 
from other oscillators and the GI in the network. The symbol p denotes the amplitude of a 
Gaussian noise. Other parameters e, /, A, and ' are chosen to control a periodic solution 
of the dynamic system. The periodic solution alternates between the silent and the active 
phases of near steady-state behavior (Terman & Wang 1995). The coupling term $i at time 
t is 
Si = Y WikS(xk(t-- ),Ox) - WzS(z, Oz), (2) 
kJV(i) 
where $oo(x, 0) = 1/(1 + exp[-tc(x - 0)]) and the parameter tc controls the steepness of 
the sigmoid function. Wi& is a synaptic weight from oscillator k to oscillator i, and N(i) 
is the set of its immediate neighbors. r is a time delay in interactions (Campbell & Wang 
1998), and 0x is a threshold over which an oscillator can affect its neighbors. Wz is the 
positive weight used for the inhibition from the global inhibitor z, whose activity is defined 
as 
dz 
= - z). (3) 
where rr = 0 if xi < 0 for every oscillator i, and rr = 1 if xi(t) _> O for at least 
one oscillator i. Here 0 represents a threshold to determine whether the GI z sends inhibi- 
tion to oscillators, and the parameter b determines the rate at which the inhibitor reacts to 
stimulation from oscillators. 
We use pattern formation to refer to the behavior that all the oscillators representing the 
same object are synchronous, while the oscillators representing different objects are desyn- 
chronous. Terman and Wang (1995) have analytically shown that such a solution can be 
achieved in LEGION without time delays. However, a solution may not be achieved when 
time delays are introduced. Although the loose synchrony concept has been introduced to 
describe time delay behavior (Campbell & Wang 1998), it does not indicate pattern forma- 
tion in an entire network even when loose synchrony is achieved because loose synchrony 
is a local concept defined in terms of pairs of neighboring oscillators. Here we intro- 
duce a measure called min-max difference in order to examine whether pattern formation is 
achieved. Suppose that oscillators Oi and Oj represent two pixels in the same object, and 
the oscillator Ot represents a pixel in a different object. Moreover, let t 8 denote the time 
at which oscillator O8 enters the active phase. The min-max difference measure is defined 
as I - tJl < rRB and [t i - tkl _> rRB, where rRB is the time period of an active phase. 
Intuitively, this measure suggests that pattern formation is achieved if any two oscillators 
representing two pixels in the same object have some overlap in the active phase, while any 
two oscillators representing two pixels belonging to different objects never stay in the active 
phase simultaneously. This definition of pattern formation applies to both exact synchrony 
in LEGION without time delays and loose synchrony with time delays. 
3 SIMULATIONS 
For a given image consisting of N x N pixels, a two-dimensional LEGION network with 
N x N oscillators is used so that each oscillator in the network corresponds to one pixel 
in the image. In the following simulations, the equations 1-3 were numerically solved 
using the fourth-order Runge-Kutta method. We illustrate stimulated oscillators with black 
squares. All oscillators were initialized randomly. A large number of simulations have 
Perceiving without Learning 13 
been conducted with a broad range of parameter values and network sizes (Chen & Wang 
1997). Here we report typical results using a specific set of parameter values. 
3.1 THE SPIRAL PROBLEM 
For simulations, the two images in Fig. 1 were sampled as two binary images with 29 x 29 
pixels. For these images, two problems can be addressed: (1) When an image is presented, 
can one determine whether it contains a single spiral or double spirals? (2) Given a point 
on a two-dimensional plane, can one determine whether it is inside or outside a specific 
spiral? 
S 
GI 
(a) (b) (c) 
Fig. 3: Results of LEGION with a time delay r = 0.002T (T is the period of oscillation) for 
the spiral problem. The parameter values used in this simulation are e = 0.003, /3 = 500, V = 
24.0, A = 21.5, o:r = 6.0, p = 0.03, t = 500, Ox = -0.5, Oz = 0.1, 4> = 3.0, Wz = 1.5, 
Is = 1.0, and I, = -1.0 where Is and I, are external input to stimulated and unstimulated 
oscillators, respectively. 
We first applied LEGION with time delays to the single spiral image in Fig. 1 (a). Fig. 3(a) 
illustrates the visual stimulus, where black pixels correspond to the stimulated oscillators 
and white ones correspond to the unstimulated oscillators. Fig. 3(b) shows a sequence of 
snapshots after the network was stabilized except for the first snapshot which shows the 
random initial state of the network. These snapshots are arranged in temporal order first 
from left to right and then from top to bottom. We observe from these snapshots that an 
activated oscillator in the spiral propagates its activation to its two immediate neighbors 
with some time delay, and the process of propagation forms a traveling wave along the 
spiral. We emphasize that, at any time, only the oscillators corresponding to a portion 
of the spiral stay in the active phase together, and the entire spiral can never be in the 
active phase simultaneously. Thus, based on the oscillatory correlation theory, our system 
cannot group the whole spiral together, which indicates that our system fails to realize that 
the pixels in the spiral belong to the same pattern. Note that the convoluted part of the 
background behaves similarly. Fig. 3(c) shows the temporal trajectories of the combined z 
activities of the oscillators representing the spiral (S) and the background (B) as well as the 
temporal activity of the GI. According to the min-max difference measure, Fig. 3(c) shows 
that pattern formation cannot be achieved. In order to illustrate the effects of time delays, 
we applied LEGION without time delays to the same image. Simulation results show that 
pattern formation is achieved, and the single spiral can be segregated from the background 
by the second period (Chen & Wang 1997). Thus, LEGION without time delays can readily 
solve the spiral problem in this case. The failure to group the spiral in Fig. 3 is caused by 
time delays in the coupling of neighboring oscillators. 
We also applied LEGION with time delays to the double spirals image in Fig. 1 (b). Fig. 
4(a) shows the visual stimulus. Fig. 4(b) shows a sequence of snapshots arranged in the 
same order as in Fig. 3(b). We observe from these snapshots that starting from an end of one 
spiral a traveling wave is formed along the spiral and the activated oscillators representing 
the spiral propagate their activation. Due to time delays, however, only the oscillators 
corresponding to a portion of the spiral stay in the active phase together, and the entire 
14 K. Chen and D. L. Wang 
spiral is never in the active phase simultaneously. The oscillators representing the other 
spiral have the same behavior. The results show that the pixels in any one of double spirals 
cannot be grouped as the same pattern. We mention that the behavior of our system for the 
convoluted part of the background is similar to that for the double spirals. It is also evident 
from Fig. 4(c) that the pattern formation is not achieved after the network was stabilized. 
We also applied LEGION without time delays to the double spirals image for the same 
purpose as described before. Simulation results also show that anyone of spirals can be 
segregated from both the other spiral and the background by the second period (Chen & 
Wang 1997). Once again, it indicates that the failure to group the double spirals in Fig. 4 
results from time delays. 
(a) (b) (c) 
Fig. 4: Results of LEGION without time delays for the spiral problem. The parameter values used 
are the same as listed in the caption of Fig. 3. In (c), S1 and S2 represent two disconnected spirals. 
B and GI denote background and the global inhibitor, respectively. 
For the spiral problem, pattern formation means that solutions to the two problems in ques- 
tion can be provided to the questions of counting the number of objects or identifying 
whether two pixels belong to the same spiral or not. No such solutions are available when 
pattern formation is not achieved. Hence, our system cannot solve the spiral problem in 
general. Only under the special condition of no time delay can our system solve the prob- 
lem. 
3.2 INSIDE/OUTSIDE RELATIONS 
For simulations, the two pictures in Fig. 2 were sampled as binary images with 43 x 43 
pixels. We first applied LEGION with time delays to the two images in Fig. 2. Figures 
5(a) and 6(a) show the visual stimuli, where black pixels represent areas A and B that cor- 
respond to stimulated oscillators and white pixels represent the boundary that corresponds 
to unstimulated oscillators. Figures 5(b) and 6(b) illustrate a sequence of snapshots after 
networks were stabilized except for the first snapshot which shows the random initial states 
of networks. Figures 5(c) and 6(c) show temporal trajectories of the combined z activities 
of the oscillators representing areas A and B as well as the GI, respectively. 
(a) (b) (c) 
Fig. 5: Results of LEGION with a time delay r = 0.002T for Fig. 2(a). The parameter values used 
in this simulation are e = 0.004, 3' = 14.0, ,X = 11.5 and the other parameter values are the same 
Perceiving without Learning 15 
as listed in the caption of Fig. 3. In (c), A, B, and GI denote areas A, B, and the global inhibitor, 
respectively. 
(a) (b) (c) 
Fig. 6: Results of LEGION with a time delay r = 0.002T for Fig. 2(b). The parameter values used 
and other statements are the same as listed in the caption of Fig. 5. 
We observe from Fig. 5(b) that the activation of an oscillator can rapidly propagate through 
its neighbors to other oscillators representing the same area, and eventually all the oscilla- 
tors representing the same area (A or B) stay together in the active phase simultaneously, 
though they generally enter the active phase at different times due to time delays. Thus, on 
the basis of oscillatory correlation, our system can group an entire area (A or B) together 
and recognize all the pixels in area A or B as elements of the same area. According to 
the min-max difference measure, Fig. 5(c) shows that pattern formation is achieved by the 
second period. In contrast, we observe from Fig. 6(b) that although an activated oscillator 
rapidly propagates its activation in open regions as shown in the last three snapshots, prop- 
agation is limited once the traveling wave spreads in spiral-like regions as shown in earlier 
snapshots. As a result, at any time, only the oscillators corresponding to a portion of either 
area stay in the active phase together, and the oscillators representing the whole area are 
never in the active phase simultaneously. Thus, on the basis of oscillatory correlation, our 
system cannot group the whole area, and fails to identify the pixels of one area as belonging 
to the same pattern. Furthermore, according to the min-max difference measure, Fig. 6(c) 
shows that pattern formation is not achieved after the network was stabilized. In order to 
illustrate the effects of time delays and show how to use an oscillator network to perceive 
inside/outside relations, we applied LEGION without time delays to the two images in Fig. 
2. Our simulations show that LEGION without time delays readily segregates two areas 
in both cases by the second period (Chen & Wang 1997). Thus, the failure to group each 
area in Fig. 6 is also attributed to time delays in the coupling of neighboring oscillators. 
In general, the above simulations suggest that oscillatory correlation provides a way to ad- 
dress inside/outside relations by a neural network; when pattern formation is achieved, a 
single area segregates from other areas that appear in the same image. For a specific point 
on the two-dimensional plane, the inside/outside relations can be identified by examining 
whether the oscillator representing the point synchronizes with the oscillators representing 
a specific area or not. 
4 DISCUSSION AND CONCLUSION 
It has been reported that many neural network models can solve the spiral problem through 
learning. However, their solutions are subject to limitations because generalization abil- 
ities of resulting learning systems highly depend on the training set. As pointed out by 
Minsky and Papert (1969), solving the spiral problem is equivalent to detecting connect- 
edness. They showed that connectedness cannot be computed by any diameter-limited or 
order-limited perceptrons (Minsky & Papert 1969). This limitation holds for multilayer 
perceptrons regardless of learning scheme (Minsky & Papert 1988, p.252). Unfortunately, 
16 K. Chen and D. L. Wang 
few people have discussed generality of their solutions. In contrast, our simulations have 
shown that LEGION without time delays can always distinguish these figures regardless 
of shape, position, size, and orientation. We emphasize that no learning is involved in LE- 
GION. In terms of performance, we suggest that the spiral problem may be better solved 
by a network of oscillators without learning. 
Our system provides an alternative way to perceive inside/outside relations from a neural 
computation perspective. Our method is significantly distinguished from visual routines 
(Ullman 1984, 1996). First, the visual routine method is described as serial algorithms, 
while our system is an inherently parallel and distributed process although its emergent be- 
havior reflects a degree of serial nature of the problems. Second, the visual routine method 
does not make a qualitative distinction between rapid effortless perception that corresponds 
to simple boundaries and slow effortful perception that corresponds to convoluted bound- 
aries - the time a visual routine, e.g. the coloring method, takes varies continuously. In 
contrast, our system makes such a distinction: effortless perception with simple boundaries 
corresponds to when pattern formation is achieved, and effortful perception with convo- 
luted boundaries corresponds to when pattern formation is not achieved. Third, perhaps 
more importantly conceptually, our system does not invoke high-level serial process to 
solve such problems like inside/outside relations; its solution involves the same mecha- 
nism as it does for parallel image segmentation (see Wang & Terman 1997). 
Acknowledgments: Authors are grateful to S. Campbell for many discussions. This work 
was supported in part by an NSF grant (IRI-9423312), an ONR grant (N00014-93-10335), 
and an ONR Young Investigator Award (N00014-96-1-00676)to DLW. 
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