CONCLUDING REMARKS

Relaxation oscillations are characterized by two time scales, and exhibit qualitatively different behaviors than sinusoidal or harmonic oscillations. This distinction is par­ticularly prominent in synchronization and desynchroniza­tion in networks of relaxation oscillators. The unique prop­erties in relaxation oscillators have led to new and promis­ing applications to neural computation, including scene analysis. It should be noted that networks of relaxation os­cillations often lead to very complex behaviors other than synchronous and antiphase solutions. Even with identical oscillators and nearest neighbor coupling, traveling waves and other complex spatiotemporal patterns can occur [31].

Relaxation oscillations with a singular parameter lend themselves to analysis by singular perturbation theory [32]. Singular perturbation theory in turn yields a geomet­ric approach to analyzing relaxation oscillation systems, as illustrated in Figs. 4 and 6. Also based on singular so­lutions, Linsay and Wang [33] proposed a fast method to numerically integrate relaxation oscillator networks. Their technique, called the singular limit method, is derived in the singular limit є ^ 0. A numerical algorithm is given for the LEGION network, and it produces large speedup com­pared to commonly used integration methods such as the Runge-Kutta method. The singular limit method makes it possible to simulate large-scale networks of relaxation os­cillators.

Computation using relaxation oscillator networks is in­herently parallel, where each single oscillator operates in parallel with all the other oscillators. This feature, plus continuous-time dynamics makes oscillator networks at­tractive for direct hardware implementation. Using CMOS technology, for example, Cosp and Madrenas [34] fabri­cated a VLSI chip for a 16 x 16 LEGION network and used the chip for a number of segmentation tasks. With its dy­namical and biological foundations, oscillatory correlation promise to offer a general computational framework.

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