Capacity of Oscillatory Associative-Memory Networks with Error-Free Retrieval

Networks of coupled periodic oscillators (similar to the Kuramoto model) have been proposed as models of associative memory. However, error-free retrieval states of such oscillatory networks are typically unstable, resulting in a near zero capacity. This puts the networks at disadvantage as compared with the classical Hopfield network. Here we propose a simple remedy for this undesirable property and show rigorously that the error-free capacity of our oscillatory, associative-memory networks can be made as high as that of the Hopfield network. They can thus not only provide insights into the origin of biological memory, but can also be potentially useful for applications in information science and engineering.

Figure 1

A pattern encoded in the phase deviation among the oscillators. Each cell representing an oscillator is painted in gray scale according to its phase.

Figure 2

The distribution of the maximum eigenvalue ${\lambda }_{\max }(J)$ for three types of solutions: a memory pattern, a memory pattern with single-bit error, and a random pattern. We chose each bit of each memory pattern to be $\pm 1$ at random with equal probabilities. The parameters of the system were $n=1000$ and $p=10$. Note that ${\lambda }_{\max }(J)=2\epsilon$ is the borderline of stability.

Figure 3

The final overlap after 1000 time units as a function of the initial overlap for $n=1000$, $p=40$, and different values of $\epsilon$ ($=0,0.2,0.4,0.8$). The final overlap is averaged over the results from 10 random initial conditions with the same initial overlap.