LEGION: locally excitatory globally inhibitory oscillator networks

Post-publication activity

Curator: DeLiang Wang

Figure 1: LEGION architecture. An oscillator is indicated by an open circle and the global inhibitor by the filled circle.

The acronym LEGION comes from the initials of "Locally Excitatory Globally Inhibitory Oscillator Networks" introduced by Wang and Terman (1995). The term has also been used in the literature as a noun modifier, as in "a LEGION model" or "a LEGION network".

The purpose of LEGION is to serve as a framework in which the features of an object are grouped (e.g. pixels grouped into a visual object) and segregated from other objects through oscillatory correlation. Oscillatory correlation means that oscillators representing an object have a single phase of oscillation, and different objects have different phases.

LEGION differs from other types of neural networks in the following two aspects:

1. Its basic unit is a neural oscillator and
2. its connectivity consists of excitation between locally coupled oscillators and inhibition via a global inhibitor.

As a result, a LEGION network (see Figure 1) can rapidly achieve synchronization among locally coupled oscillator assemblies (groups) and desynchronization between these assemblies. LEGION has been successfully applied to a number of scene analysis tasks, including image segmentation, object selection, and speech segregation. Direct hardware implementation of LEGION also exists.

Basic Oscillator Model

Figure 2: Enabled state of a relaxation oscillator. The arrows indicate the direction of motion, and double arrows indicate jumping.
Figure 3: Excitable state of a relaxation oscillator.

In the standard formulation, Terman and Wang (1995) define the basic oscillator of a LEGION network as a reciprocally connected pair of excitatory variable $$x$$ and inhibitory variable $$y\ :$$ $\begin{matrix}\dot{x}&=&3x-x^3+2-y+I\\\dot{y}&=&\varepsilon(\alpha(1+\tanh(x/\beta))-y) \end{matrix}$ Here, $$I$$ denotes the external stimulation to the oscillator. The parameter $$\varepsilon$$ is positive and it is chosen so that $$0<\varepsilon<<1\ .$$ In this case, the above equations define a typical relaxation oscillator with two time scales induced by $$\varepsilon\ .$$ The $$x$$-nullcline (i.e., the set defined by the condition $$\dot{x}=0$$) is a cubic function. The $$y$$-nullcline is a sigmoidal function with parameters $$\alpha\,$$ and $$\beta\,\ .$$

If $$I > 0\ ,$$ the two nullclines intersect only at a point along the middle branch of the cubic, and in this case the oscillator produces a stable limit cycle and is called enabled. The limit cycle alternates between an active phase of relatively high $$x$$ values and a silent phase of relatively low $$x$$ values. This is illustrated in Figure 2. Within each of the two phases the oscillator exhibits near steady-state behavior, and in the silent phase it tracks the left branch (LB) of the cubic and in the active phase the right branch (RB). In contrast to the behavior within each phase, the transition between the two phases takes place rapidly, and it is referred to as jumping. Such alternations between fast change and slow change are characteristic of relaxation oscillations (van der Pol 1926). The parameter $$\alpha\,$$ determines the relative durations of the two phases - a larger $$\alpha\,$$ produces a relatively shorter active phase.

If $$I<0\ ,$$ the two nullclines of the above equations intersect at a stable equilibrium point on the LB of the cubic (see Figure 3). In this case no oscillation occurs, and the oscillator is called excitable. Whether the oscillator is enabled or excitable depends on external stimulation, and hence oscillations defined by the equations are stimulus-dependent.

The relaxation oscillator model defined above may be interpreted either as a model of action potential generation, where $$x$$ represents the membrane potential of a neuron and $$y$$ represents a recovery variable (e.g., the level of activation of ion channels), or an oscillating burst of neuronal spikes where $$x$$ represents the envelope of the burst. Figure 4 shows a typical trace of $$x$$ activity, akin to a spike train. In fact, the equations are dynamically very similar to standard neuronal models, including the FitzHugh-Nagumo model and the Morris-Lecar model. All of these models can be viewed as simplifications of the classic Hodgkin-Huxley model.

Figure 4: Activity of a relaxation oscillator with respect to time.

Connectivity

LEGION connectivity is characterized by local excitation and global inhibition. Each oscillator $$i$$ in a LEGION network is defined as: $\begin{matrix}\dot{x}_i&=&3x_i-x_i^3+2-y_i+I_i+S_i+\rho\\\dot{y}_i&=&\varepsilon(\alpha(1+\tanh(x_i/\beta))-y_i) \end{matrix}$ The parameter $$\rho$$ denotes the amplitude of Gaussian (white) noise, which is introduced to facilitate desynchronization. The term $$S_i$$ denotes the overall coupling from the rest of the network, given by $S_i=\sum_{k\in N(i)}W_{ik}H(x_k-\theta_x)-W_zH(z-\theta_z)$ where $$H$$ stands for the Heaviside step function, defined as $$H(v) = 1$$ for $$v \geq 0$$ and $$H(v) = 0$$ for $$v < 0\ .$$ Each coefficient $$W_{ik}$$ is the positive connection weight from oscillator $$k$$ to $$i\ .$$ Both $$\theta_x$$ and $$\theta_z$$ are thresholds, and $$\theta_x$$ is chosen between LB and RB along the $$x$$ direction. $$N(i)$$ denotes a set of $$i$$'s neighbors. In the simplest form of a two-dimensional (2-D) network, $$N(i)$$ contains just $$i$$'s 4 nearest neighbors except on boundaries where the number of the nearest neighbors is either 3 or 2. Figure 1 (top of the article) illustrates this network architecture. Following Wang (1995), dynamic weight normalization is typically used to ensure that each oscillator has equal overall weight of dynamic connections from its neighborhood.

Finally, $$W_z$$ (> 0) in the above equation denotes the weight of inhibition from global inhibitor $$z\ ,$$ defined as $\dot{z}=\phi(\sigma_\infty-z)$ where $$\phi$$ is a parameter. $$\sigma_\infty$$ = 1 if $$x_i\geq\theta_z$$ for at least one oscillator $$i$$ and $$\sigma_\infty$$ = 0 otherwise. This equation makes $$z$$ approach exponentially to $$\sigma_\infty$$ at rate $$\phi\ .$$ Note that in implementation the exponential approach can be replaced by a Heaviside function.

Figure 5: LEGION activity (from Wang & Terman, 1995). The upper left shows an input scene with four connected patterns on a 20x20 grid. The upper middle shows a snapshot of the LEGION network activity at the beginning of dynamic evolution. The diameter of each black circle indicates the $$x$$ activity of the corresponding oscillator. Subsequent snapshots are shown from the upper right to the middle right. The lower panel shows the combined temporal activities of the four oscillator assemblies and the global inhibitor. Unstimulated oscillators are excitable during the entire simulation, hence excluded from the display.

Synchrony and Desynchrony in LEGION

The LEGION system was extensively analyzed by Terman and Wang (1995). Prior to their analysis, Somers and Kopell (1993) found the fast threshold modulation mechanism for synchronizing locally coupled relaxation oscillators. Specifically, Somers and Kopell proved a theorem that a pair of relaxation oscillators approaches synchrony at a geometric (or exponential) rate. Furthermore, their numerical simulations suggest that fast synchronization also occur in 1-D arrays (Somers & Kopell 1993). Terman and Wang proved that the Somers-Kopell theorem for a pair of oscillators extends to an arbitrary network of locally coupled relaxation oscillators.

In addition, Terman and Wang (1995) found the selective gating mechanism for desynchronizing oscillator assemblies in a LEGION network, which works in the following way. Let an oscillator assembly refer to those stimulated by a connected region. When an enabled oscillator, say $$i\ ,$$ jumps to the active phase, it quickly activates the global inhibitor which then prevents the oscillators of different assemblies from jumping up but does not affect $$i$$'s ability to recruit the oscillators of the same assembly thanks to stronger, local excitation. Combining the selective gating and fast threshold modulation mechanisms, Terman and Wang (1995) proved the following theorem: A LEGION network achieves both synchronization within each oscillator assembly and desynchronization between different assemblies in no greater than $$m$$ periods of oscillations, where $$m$$ is the number of the assemblies. Desynchronization between two assemblies means that they are never in the active phase simultaneously. LEGION is the first neural network model to achieve rapid synchronization and desynchronization.

Figure 5 illustrates the process of synchronization and desynchronization in LEGION. An input pattern comprising four connected patterns, forming the word OHIO, is presented to a 20x20 LEGION network. The stimulated oscillators become enabled, while those without stimulation are excitable. To start, the phases of all the oscillators are randomized. The figure shows the snapshots of the network activity at different stages of dynamic evolution. Here, the effects of synchronization and desynchronization are clearly seen: Different patterns reach the active phase in synchrony and become desynchronized from the other assemblies. This "popout" of the assemblies continues in an approximately periodic fashion, as long as the external stimulation stays on. The lower panel of the figure provides a complete picture of dynamic evolution by showing the evolution of every oscillator. Synchronization within each assembly and desynchronization between the four assemblies emerge in about three periods.

Applications to Scene Analysis

The development of LEGION was mainly motivated by the binding problem in brain theory and perception, which refers to the problem of how the brain integrates responses of local feature detectors in different cortical areas to form perceived objects or organizations (von der Malsburg 1981; 1999; see also Dynamic Link Architecture). Von der Malsburg (1981) put forward a theoretical proposal, called temporal correlation theory, to address the binding problem. The temporal correlation theory asserts that the time structure of neural signals provides the key basis for binding. Oscillatory correlation as a special form of temporal correlation has since been supported by the discovery of neural oscillations and synchrony in the visual cortex in the late 1980s (Eckhorn et al. 1988; Gray et al. 1989). LEGION was intended to provide a neurocomputational foundation for oscillatory correlation theory.

LEGION's ability of rapid synchronization based on local excitation and desynchronization based on global inhibition has led to a number of applications to scene analysis tasks, including the following (see Wang 2005 for a comprehensive review).

Figure 6: Image segmentation using LEGION (courtesy of Xiuwen Liu). The left panel shows an intensity image with 256x256 pixels and the right panel the segmentation result, where each segment corresponds to a distinct color and the background is indicated by black.

Image segmentation

For image segmentation, a pixel is typically mapped to an oscillator and connection weights between neighboring oscillators are set to be proportional to feature similarity between the corresponding pixel locations (Wang & Terman 1997). In this case, the level of global inhibition controls the granularity of segmentation: Larger values of $$W_z$$ produce more but smaller segments. Figure 6 shows an example of LEGION segmentation. An input image (an MRI image of the human brain) on the left panel is segmented into the segments shown on the right panel, where different oscillator assemblies are indicated by different colors. In addition, a background, indicated by black, is also formed that consists of regions not corresponding to any segment.

Object selection

On the basis of LEGION segmentation, Wang (1999) proposed an oscillator network for object selection. The basic idea is that after oscillator assemblies are formed, competition between assemblies takes place in the following way. When an assembly jumps to the active phase, it leaves an inhibitory trace via a slow inhibitor, which can be overcome only by larger (or more salient) assemblies. After a number of oscillation cycles, the largest assembly will be the only one that oscillates, while all the others are suppressed and become excitable. Unlike traditional winner-take-all networks (Arbib 2003) where competition occurs at the cell (or location) level, in the object selection network competition occurs at the cell assembly (or object) level.

Speech segregation

Similar to visual analysis, a listener in an auditory environment is exposed to acoustic energy from different sound sources and must segregate the acoustic wave reaching the ears in order to understand the auditory environment. According to Bregman (1990), auditory scene analysis takes place in two stages. In the first stage, the acoustic mixture reaching the ears is decomposed into a collection of sensory elements (or segments). In the second stage, auditory segments that likely arise from the same source are grouped to form a stream, which is a perceptual representation of an auditory event. Wang and Brown (1999) studied the speech segregation problem: Separating target speech from acoustic interference. Echoing Bregman's two-stage notion, the Wang and Brown model first performs auditory segmentation by a LEGION network and then groups segments on the basis of common periodicity by a laterally connected network of relaxation oscillators.

Related Developments

Other oscillator models

While LEGION originally builds on relaxation oscillators, other types of oscillators may also be used with LEGION connectivity to produce rapid synchrony and desynchrony. In particular, Campbell et al. (1999) showed that a LEGION network with integrate-and-fire (or spike) oscillators can exhibit properties of synchrony and desynchrony similar to a corresponding LEGION network with relaxation oscillators. Wilson-Cowan oscillators have also been studied in this context and were also found to be capable of synchronization and desynchronization in a LEGION type network (Wang 1995; Campbell & Wang 1996), but the process of oscillator assembly formation is probably not as fast (Campbell 1997; Wang 2005). However, harmonic oscillators cannot synchronize with local connections (Mermin & Wagner 1966; see also Terman & Wang 1995).

Hardware implementation

In an effort to implement LEGION-like networks on electric circuits for intensity image segmentation, Ando et al. (2000) modified the original equations by making the coupling term $$S_i$$ bipolar. They introduced pulse-width modulation and pulse-phase modulation techniques to implement relaxation oscillator networks. Cosp et al. (1999) employed a hysteresis current comparator and a damped integrator for implementing a single relaxation oscillator. Later, Cosp and Madrenas (2003) designed an analog very large-scale integration (VLSI) chip that implements a 16x16 LEGION network using CMOS technology. Their evaluation of the chip showed that it performs segmentation with much higher speed as well as area and power consumption advantages compared to simulation on a general-purpose computer. Using a digital, reconfigurable device called Field Programmable Gate Array (FPGA), Torres-Huitzil and Girau (2006) implemented a 16x16 LEGION network with spike oscillators.

References

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Internal references

• Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
• Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.