Recent experimental evidence and earlier theoretical considerations point to neural oscillations in the visual cortex as a possible mechanism by which the brain detects and binds features in a visual scene. It is known that neurons in the visual system respond to features such as color, orientation, and motion, and are arranged in regular structures called columns and hypercolumns. In addition, these neurons respond only to stimuli from a particular part of the visual field. Recent experimental work showed persistent stimulus-dependent oscillations around 40 Hz exhibited by neuronal groups in the primary visual cortex [10][8]. Furthermore, synchronized behavior was observed between spatially separate neuronal groups. This supports earlier theoretical considerations [14] which suggest that cells acting as visual feature detectors bind together through correlation of their firing activities. Previously proposed oscillator networks [19][4][12] represent an oscillatory event by a single phase variable. These networks are limited when applied to image segmentation. Oscillations in these networks are built into the system rather than stimulus-dependent. More substantially, these systems rely on fully connected network architectures to achieve synchronization, which results in indiscriminate grouping and loss of topological information. LEGION is able to overcome these deficiencies with stimulus-dependent oscillations and fast and long range synchrony by using local connections. In addition, LEGION achieves fast desynchrony with a global inhibitory mechanism.
LEGION was proposed by Terman and Wang [20][23] as a biologically plausible computational framework for image segmentation and has been used successfully to segment binary and grey-level image data [24]. It is a network of relaxation oscillators, each constructed from an excitatory unit x and an inhibitory unit y as shown in Fig. 1. Unit x sends excitation to unit y, which responds by sending inhibition back. When external input stimulus I is continuously applied to x, this feedback loop produces oscillations. Neighboring oscillators are connected via mutual excitatory coupling, as well as the global inhibitor (see (1a) below).
LEGION is formally defined and analyzed in [24][20]. The behavior of each oscillator, indexed by i in a network, is defined by the following equations:
The behavior of 
is defined in (1a) which contains a cubic function.
The subtractive term 
represents inhibition from unit y, 
is
external stimulus, and 
represents excitatory coupling with neighboring oscillators and coupling
with the global inhibitor. We call i stimulated if 
,
and unstimulated if 
.
The Heaviside step function H is defined as 
if 
, and 
if 
. The function
H determines oscillatory behavior by it multiplying 
.
The variable 
is called the lateral potential of the oscillator and is used to
suppress noisy regions. Parameter 
is set in the range 
and is used as a threshold for
and an exponential term which decays at rate 
.
The oscillator whose lateral potential exceeds
is referred to as the leading oscillator. Parameter 
is
the amplitude of Gaussian noise and plays a role in assisting the separation
of synchronized groups of oscillators. The behavior of 
is defined in (1b) which contains a sigmoid function
with 
chosen small.
Parameter 
is chosen
to be small, i.e. 
,
and determines that the oscillator defined in (1) is a relaxation oscillator
with two time scales.
The potential term
plays the role of removing the oscillations of noisy regions. Its value
is determined by the activities of its coupled neighbors. As shown in equation
(2), if the activity of each neighbor 
,
is larger than threshold 
,
,
which is a permanent connection weight defining the topology of the network,
is accumulated. If the sum is greater than threshold 
,
then the outer Heaviside function will be one and
will grow its value to 1 by the term 
,
where 
is a constant and chosen to be 
.
If H = 0, then the potential will decay to 0 with rate 
,
chosen to be 
. Following
Wang and Terman [24], 
is called the potential neighborhood of i and 
is called the recruiting neighborhood of i.
Equation (3) defines the coupling to oscillator
i, which includes excitation from neighboring oscillators and inhibition
from a global inhibitor z. A threshold
is applied to the activity of each coupled oscillator 
,
, using
the function H. The resultant value is weighted by 
,
which is a dynamic weight used to achieve weight normalization to improve
synchronization [22]. Note that
the neighborhood
may be different from
that is used to compute the potential. If the activity of the global inhibitor
z is above the threshold 
,
then the weight 
is subtracted (see (3)). The global inhibitor
is activated when at least one oscillator in the network is excited, i.e.
in the active phase as will be described. In (4),
is
1 if 
for at least one oscillator i, and 0 otherwise, and 
is a parameter. Fig. 2 shows
a 2D network architecture with 4-neighborhood coupling. The global inhibitor,
shown with a black circle, is coupled with the entire network.
The behavior of an oscillator is qualitatively shown in the phase
plane diagram (see Fig. 3),
which is an XY plot of the activities of x and y. Fig.
3a illustrates the oscillatory behavior by a limit cycle trajectory.
When equations (1a) and (1b)
are each set to zero, i.e. 
and 
, two nullclines
are defined called the x-nullcline and the y-nullcline, respectively.
The x-nullcline is a cubic with three branches called the left,
middle, and right branches (Fig.
3a). The left and middle branches connect at a point called the left
knee (LK) and the middle and right branches connect at the right
knee (RK). The y-nullcline is a sigmoid function, where 
is chosen to be small so that the sigmoid is close to a step function.
The two nullclines intersect at an unstable fixed point along the middle
branch of the cubic. A stimulated oscillator starting at some random position
will be attracted to a counter-clockwise limit cycle trajectory illustrated
in Fig. 3a. The section of the
orbit that lies on the left branch is called the silent phase, because
x has low activity. Similarly, the section on the right branch is
called the active phase. The temporal behavior of an oscillator
exhibits two time scales due to 
.
The slow time scale occurs during the two phases, denoted by the single
arrows in Fig. 3a. Parameter
in equation (1b)
controls the relative times an oscillator spends in the two phases, whereby
a larger leads to
a shorter time in the active phase. The fast time scale, denoted by double
arrows in Fig. 3a, occurs when
an oscillator alternates, or jumps, between the two phases at either
LK or RK.
In equation (1a), I and S have the effect to vertically shift the cubic. The position of the cubic relative to the sigmoid defines two states that an oscillator may be in at any given time. The first is an oscillatory state where the cubic intersects the sigmoid at exactly one point and a limit cycle exists (Fig. 3a). The non-oscillatory state occurs when the cubic shifts downward so that LK is below the left part of the sigmoid (Fig. 3b). Two fixed points are created on two sides of LK, and the stable fixed point to the left of LK will attract an oscillator travelling in the silent phase and prevent it from jumping up to the active phase and oscillating.
