Synchrony and Desynchrony 
in Neural Oscillator Networks 
DeLiang Wang 
Department of Computer and Information Science 
and Center for Cognitive Science 
The Ohio State University 
Columbus, Ohio 43210, USA 
dwang @cis.ohio-state.edu 
David Terman 
Department of Mathematics 
The Ohio State University 
Columbus, Ohio 43210, USA 
terman @ math. ohio - state. e du 
Abstract 
An novel class of locally excitatory, globally inhibitory oscillator 
networks is proposed. The model of each oscillator corresponds to a 
standard relaxation oscillator with two time scales. The network 
exhibits a mechanism of selective gating, whereby an oscillator 
jumping up to its active phase rapidly recruits the oscillators stimulated 
by the same pattern, while preventing others from jumping up. We 
show analytically that with the selective gating mechanism the network 
rapidly achieves both synchronization within blocks of oscillators that 
are stimulated by connected regions and desynchronization between 
different blocks. Computer simulations demonstrate the network's 
promising ability for segmenting multiple input patterns in real time. 
This model lays a physical foundation for the oscillatory correlation 
theory of feature binding, and may provide an effective computational 
framework for scene segmentation and figure/ground segregation. 
1 INTRODUCTION 
A basic attribute of perception is its ability to group elements of a perceived scene into 
coherent clusters (objects). This ability underlies perceptual processes such as 
figure/ground segregation, identification of objects, and separation of different objects, and 
it is generally known as scene segmentation or perceptual organization. Despite the fact 
200 DeLiang Wang, David Terman 
that humans perform it with apparent ease, the general problem of scene segmentation 
remains unsolved in the engineering of sensory processing, such as computer vision and 
auditory processing. 
Fundamental to scene segmentation is the grouping of similar sensory features and the 
segregation of dissimilar ones. Theoretical investigations of brain functions and feature 
binding point to the mechanism of temporal correlation as a representational framework 
(von der Malsburg, 1981; von der Malsburg and Schneider, 1986). In particular, the 
correlation theory of von der Malsburg (1981) asserts that an object is represented by the 
temporal correlation of the firing activities of the scattered cells coding different features 
of the object. A natural way of encoding temporal correlation is to use neural 
oscillations, whereby each oscillator encodes some feature (maybe just a pixel) of an 
object. In this scheme, each segment (object) is represented by a group of oscillators that 
shows synchrony (phase-locking) of the oscillations, and different objects are represented 
by different groups whose oscillations are desynchronized from each other. Let us refer to 
this form of temporal correlation as oscillatory correlation. The theory of oscillatory 
correlation has received direct experimental support from the cell recordings in the cat 
visual cortex (Eckhorn et al., 1988; Gray et al., 1989) and other brain regions. The 
discovery of synchronous oscillations in the visual cortex has triggered much interest 
from the theoretical community in simulating the experimental results and in exploring 
oscillatory correlation to solve the problems of scene segmentation. While several 
demonstrate synchronization in a group of oscillators using local (lateral) connections 
(K6nig and Schillen, 1991; Somers and Kopell, 1993; Wang, 1993, 1995), most of these 
models rely on long range connections to achieve phase synchrony. It has been pointed 
out that local connections in reaching synchrony may play a fundamental role in scene 
segmentation since long-range connections would lead to indiscriminate segmentation 
(Sporns et al., 1991; Wang, 1993). 
There are two aspects in the theory of oscillatory correlation: (1) synchronization within 
the same object; and (2) desynchronization between different objects. Despite intensive 
studies on the subject, the question of desynchronization has been hardly addressed. The 
lack of an efficient mechanism for desynchronization greatly limits the utility of 
oscillatory correlation to perceptual organization. In this paper, we propose a new class 
of oscillatory networks and show that it can rapidly achieve both synchronization within 
each object and alesynchronization between a number of simultaneously presented objects. 
The network is composed of the following elements: (1) A new model of a basic 
oscillator; (2) Local excitatory connections to produce phase synchrony within each 
object; (3) A global inhibitor that receives inputs from the entire network and feeds back 
with inhibition to produce desynchronization of the oscillator groups representing 
different objects. In other words, the mechanism of the network consists of local 
cooperation and global competition. This surprisingly simple neural architecture may 
provide an elementary approach to scene segmentation and a computational framework for 
perceptual organization. 
2 MODEL DESCRIPTION 
The building block of this network, a single oscillator i, is defined in the simplest form 
as a feedback loop between an excitatory unit x i and an inhibitory unit Yi: 
Synchrony and Desynchrony in Neural Oscillator Networks 201 
8 
 dx/dt = 0 
-2 0 
 ' dy/dt = 0 
2 
x 
Figure 1: Nullclines and periodic orbit of 
a single oscillator as shown in the phase 
plane. When the oscillator starts at a 
randomly generated point in the phase 
plane, it quickly converged to a stable 
trajectory of a limit cycle. 
Figure 2: Architecture of a two dimensional 
network with nearest neighbor coupling. 
The global inhibitor is indicated by the 
black circle. 
dx i 
dt - 3xi - xi3 + 2 - Yi + P + Ii + Si (la) 
dyi 
-  (y(1 + tanh(xi/fl)) - Yi) (lb) 
where p denotes the amplitude of a Gaussian noise term. I i represents external 
stimulation to the oscillator, and S i denotes coupling from other oscillators in the 
network. The noise term is introduced both to test the robustness of the system and to 
actively desynchronize different input patterns. The parameter e is chosen to be small. 
In this case (1), without any coupling or noise, corresponds to a standard relaxation 
oscillator. The x-nullcline of (1) is a cubic curve, while the y-nullcline is a sigmoid 
function, as shown in Fig. 1. If I > 0, these curves intersect along the middle branch of 
the cubic, and (1) is oscillatory. The periodic solution alternates between the silent and 
active phases of near steady state behavior. The parameter ?'is introduced to control the 
relative times that the solution spends in these two phases. If I < 0, then the nullclines 
of (1) intersect at a stable fixed point along the left branch of the cubic. In this case the 
system produces no oscillation. The oscillator model (1) may be interpreted as a model of 
spiking behavior of a single neuron, or a mean field approximation to a network of 
excitatory and inhibitory neurons. 
The network we study here in particular is two dimensional. However, the results can 
easily be extended to other dimensions. Each oscillator in the network is connected to 
only its four nearest neighbors, thus forming a 2-D grid. This is the simplest form of 
local connections. The global inhibitor receives excitation from each oscillator of the 
grid, and in turn inhibits each oscillator. This architecture is shown in Fig. 2. The 
intuitive reason why the network gives rise to scene segmentation is the following. 
When multiple connected objects are mapped onto the grid, local connectivity on the grid 
will group together the oscillators covered by each object. This grouping will be reflected 
202 DeLiang Wang, David Terman 
by phase synchrony within each object. The global inhibitor is introduced for 
desynchronizing the oscillatory responses to different objects. We assume that the 
coupling term $i in (1) is given by 
S i=  WikSoo(x k, 0 x)-W zSoo(z, Oxz) 
kN(i) 
(2) 
1 
soo(x, o) = 1+ exp[-K(x-O)] (3) 
where Wik is a connection (synaptic) weight from oscillator k to oscillator i, and N(i) is 
the set of the neighoring oscillators that connect to i. In this model, N(i) is the four 
immediate neighbors on the 2-D grid, except on the boundaries where N(O may be either 
2 or 3 immediate neighbors. 0 x is a threshold (see the sigmoid function of Eq. 3) above 
which an oscillator can affect its neighbors. W z (positive) is the weight of inhibition 
from the global inhibitor z, whose activity is defined as 
 z) (4) 
where croo = 0 if x i < Ozx for every oscillator, and croo = 1 if x i > Ozx for at least one 
oscillator i. Hence 0zx represents a threshold. If the activity of every oscillator is below 
this threshold, then the global inhibitor will not receive any input. In this case z --> 0 
and the oscillators will not receive any inhibition. If, on the other hand, the activity of at 
least one oscillator is above the threshold 0zx then, the global inhibitor will receive 
input. In this case z--> 1, and each oscillator feels inhibition when z is above the 
threshold zx' The parameter q determines the rate at which the inhibitor reacts to such 
stimulation. 
In summary, once an oscillator is active, it triggers the global inhibitor. This then 
inhibits the entire network as described in Eq. 1. On the other hand, an active oscillator 
spreads its activation to its nearest neighbors, again through (1), and from them to its 
further neighbors. In the next section, we give a number of properties of this system. 
Besides boundaries, the oscillators on the grid are basically symmetrical. Boundary 
conditions may cause certain distortions to the stability of synchrous oscillations. 
Recently, Wang (1993) proposed a mechanism called dynamic normalization to ensure 
that each oscillator, whether it is in the interior or on a boundary, has equal overall 
connection weights from its neighbors. The dynamic normalization mechanism is 
adopted in the present model to form effective connections. For binary images (each pixel 
being either 0 or 1), the outcome of dynamic normalization is that an effective connection 
is established between two oscillators if and only if they are neighbors and both of them 
are activated by external stimulation. The network defined above can readily be applied 
for segmentation of binary images. For gray-level images (each pixel being in a certain 
value range), the following slight modification suffices to make the network applicable. 
An effective connection is established between two oscillators if and only if they are 
neighbors and the difference of their corresponding pixel values is below a certain 
threshold. 
Synchrony and Desynchrony in Neural Oscillator Networks 203 
3 ANALYTICAL RESULTS 
We have formally analyzed the network. Due to space limitations, we can only list the 
major conclusions without proofs. The interested reader can find the details in Terman 
and Wang (1994). Let us refer to a pattern as a connected region, and a block be a subset 
of oscillators stimulated by a given pattern. The following results are about singular 
solutions in the sense that we formally set e = 0. However, as shown in (Terman and 
Wang, 1994), the results extend to the case e > 0 sufficiently small. 
Theorem 1. (Synchronization). The parameters of the system can be chosen so that all 
of the oscillators in a block always jump up simultaneously (synchronize). Moreover, 
the rate of synchronization is exponential. 
Theorem 2. (Multiple Patterns) The parameters of the system and a constant T can be 
chosen to satisfy the following. If at the beginning all the oscillators of the same block 
synchronize with each other and the temporal distance between any two oscillators 
belonging to two different blocks is greater than T, then (1) Synchronization within each 
block is maintained; (2) The blocks activate with a fixed ordering; (3) At most one block 
is in its active phase at any time. 
Theorem 3. (Desynchronization) If at the beginning all the oscillators of the system lie 
not too far away from each other, then the condition of Theorem 2 will be satisfied after 
some time. Moreover, the time it takes to satisfy the condition is no greater than N 
cycles, where N is the number of patterns. 
The above results are true with arbitrary number of oscillators. In summary, the network 
exhibits a mechanism, referred to as selective gating, which can be intuitively interpreted 
as follows. An oscillator jumping to its active phase opens a gate to quickly recruit the 
oscillators of the same block due to local connections. At the same time, it closes the 
gate to the oscillators of different blocks. Moreover, segmentation of different patterns is 
achieved very rapidly in terms of oscillation cycles. 
4 COMPUTER SIMULATION 
To illustrate how this network is used for scene segmentation, we have simulated a 20x20 
oscillator network as defined by (1)-(4). We arbitrarily selected four objects (patterns): 
two O's, one H, and one I; and they form the word OHIO. These patterns were 
simultaneously presented to the system as shown in Figure 3A. Each pattern is a 
connected region, but no two patterns are connected to each other. 
All the oscillators stimulated (covered) by the objects received an external input I = 0.2, 
while the others have I = -0.02. The amplitude p of the Gaussian noise is set to 0.02. 
Thus, compared to the external input, a 10% noise is included in every oscillator. 
Dynamic normalization results in that only two neighboring oscillators stimulated by a 
single pattern have an effective connection. The differential equations were solved 
numerically with the following parameter values: e = 0.02, p = 3.0; 7= 6.0,  = 0.1, K 
= 50, 0 x = -0.5, and 0zx = Oxz = 0.1. The total effective connections were normalized to 
6.0. The results described below were robust to considerable changes in the parameters. 
The phases of all the oscillators on the grid were randomly initialized. 
204 DeLiang Wang, David Terman 
Fig. 3B-3F shows the instantaneous activity (snapshot) of the network at various stages 
of dynamic evolution. The diameter of each black circle represents the normalized x 
activity of the corresponding oscillator. Fig. 3B shows a snapshot of the network a few 
steps after the beginning of the simulation. In Fig. 3B, the activities of the oscillators 
were largely random. Fig. 3C shows a snapshot after the system had evolved for a short 
time period. One can clearly see-the effect of grouping and segmentation: all the 
oscillators belonging to the left O were entrained and had large activities. At the same 
time, the oscillators stimulated by the other three patterns had very small activities. Thus 
the left O was segmented from the rest of the input. A short time later, as shown in Fig. 
3D, the oscillators stimulated by the right O reached high values and were separated from 
the rest of the input. Fig. 3E shows another snapshot after Fig. 3D. At this time, 
pattern I had its turn to be activated and separated from the rest of the input. Finally in 
Fig. 3F, the oscillators representing H were active and the rest of the input remained 
silent. This successive "pop-out" of the objects continued in a stable periodic fashion. 
To provide a complete picture of dynamic evolution, Fig. 3G shows the temporal 
evolution of each oscillator. Since the oscillators receiving no external input were 
inactive during the entire simulation process, they were excluded from the display in Fig. 
3G. The activities of the oscillators stimulated by each object are combined together in 
the figure. Thus, if they are synchronized, they appear like a single oscillator. In Fig. 
3G, the four upper traces represent the activities of the four oscillator blocks, and the 
bottom trace represents the activity of the global inhibitor. The synchronized oscillations 
within each object are clearly shown within just three cycles of dynamic evolution. 
5 DISCUSSION 
Besides neural plausibility, oscillatory correlation has a unique feature as an 
computational approach to the engineering of scene segmentation and figure/ground 
segregation. Due to the nature of oscillations, no single-object can dominate and 
suppress the perception of the rest of the image permanently. The current dominant 
object has to give way to other objects being suppressed, and let them have a chance to be 
spotted. Although at most one object can dominant at any time instant, due to rapid 
oscillations, a number of objects can be activated over a short time period. This intrinsic 
dynamic process provides a natural and reliable representation of multiple segmented 
patterns. 
The basic principles of selective gating are established for the network with lateral 
connections beyond nearest neighbors. Indeed, in terms of synchronization, more distant 
connections even help expedite phase entrainment. In this sense, synchronization with 
all-to-all connections is an extreme case of our system. With nearest-neighbor 
connectivity (Fig. 2), any isolated part of an image is considered as a segment. In an 
noisy image with many tiny regions, segmentation would result in too many small 
fragments. More distant connections would also provide a solution to this problem. 
Lateral connections typically take on the form of Gaussian distribution, with the 
connection strength between two oscillators falling off exponentially. Since global 
inhibition is superimposed to local excitation, two oscillators positively coupled may be 
desynchronized if global inhibition is strong enough. Thus, it is unlikely that all objects 
in an image form a single segment as the result of extended connections. 
Synchrony and Desynchrony in Neural Oscillator Networks 205 
Due to its critical importance for computer vision, scene segmentation has been studied 
quite extensively. Many techniques have been proposed in the past (Haralick and Shapiro, 
1985; Sarkar and Boyer, 1993). Despite these techniques, as pointed out by Haralick and 
Shapiro (1985), there is no underlying theory of image segmentation, and the techniques 
tend to be adhoc and emphasize some aspects while ignoring others. Compared to the 
traditional techniques for scene segmentation, the oscillatory correlation approach offers 
many unique advantages. The dynamical process is inherently parallel. While 
conventional computer vision algorithms are based on descriptive criteria and many adhoc 
heuristics, the network as exemplified in this paper performs computations based on only 
connections and oscillatory dynamics. The organizational simplicity renders the network 
particularly feasible for VLSI implementation. Also, continuous-time dynamics allows 
real time processing, desired by many engineering applications. 
Acknowledgments 
DLW is supported in part by the NSF grant IRI-9211419 and the ONR grant N00014-93- 
1-0335. DT is supported in part by the NSF gram DMS-9203299LE. 
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206 DeLiang Wang, David Terman 
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Figure 3. A An image composed of four patterns which were presented (mapped) to a 
20x20 grid of oscillators. B A snapshot of the activities of the oscillator grid at the 
beginning of dynamic evolution. C A snapshot taken shortly after the beginning. D 
Shortly after C. E Shortly after D. F Shortly after E. G The upper four traces show the 
combined temporal activities of the oscillator blocks representing the four patterns, 
respectively, and the bottom trace shows the temporal activity of the global inhibitor. 
The simulation took 8,000 integration steps. 
