Oscillator-Based Associative Memory with Exponential Capacity: Theory, Algorithms, and Hardware Implementation

Abstract

Associative memory systems enable content-addressable storage and retrieval of patterns, a capability central to biological neural computation and artificial intelligence. Classical implementations such as Hopfield networks face fundamental limitations in memory capacity, scaling at most linearly with network size. We present an associative memory architecture based on Kuramoto oscillator networks with honeycomb topology in which memories are encoded as stable phase-locked configurations. The honeycomb network consists of multiple cycles that share nodes in a chain-like arrangement, creating a one-dimensional lattice of chained+loops. We prove that this architecture achieves exponential memory capacity: a network of $N$ oscillators can store $(2\lceil n_c/4 \rceil - 1)^m$ distinct patterns, where $m$ honeycomb cycles each contain $n_c$ oscillators. Moreover, we fully characterize all stable configurations and prove that each memory's basin of attraction maintains a guaranteed minimum size independent of network scale. Simulations using charge-density-wave (CDW) oscillators validate predicted phase-locking behavior, demonstrating practical realizability in neuromorphic hardware.

Figures (6)

• Figure 1: This figure shows the 1D honeycomb network described by the graph $\mathcal{H}_m^{n_\text{c}}$. In (a), the network has $m=5$ cycles, and each cycle is a pentagon with $n_c =5$ nodes. In (b), the network has $m=5$ cycles, and each cycle is a pentagon $n_c = 6$ nodes.
• Figure 2: Stable phase-locked configurations for a single cycle with $n_c=5$ ($m=1$). (a) Synchronization: all phases equal. (b) Phase differences of $-2\pi/5$ between consecutive oscillators. (c) Phase differences of $+2\pi/5$ between consecutive oscillators.
• Figure 3: Stable phase-locked configurations encoding two different memory patterns for a Kuramoto honeycomb network with $m=2$ cycles and $n_c=5$. (a) Configuration with $(k_1, k_2) = (0, -1)$: the first cycle has equal phases while the second has phase differences of $-2\pi/5$. (b) Configuration with $(k_1, k_2) = (1, 1)$: both cycles have phase differences of $+2\pi/5$.
• Figure 4: Numerical validation of basin of attraction bounds for honeycomb networks with $m=400$ cycles. The solid lines show the empirical success rate of memory retrieval under random perturbations of varying magnitude. The vertical dashed lines mark the theoretical lower bounds from Theorem \ref{['thm:basin']}. For each cycle size, we randomly selected 500 stable phase-locked configurations and applied independent random perturbations at each magnitude level, then simulated the Kuramoto dynamics to determine if the system converged to the original phase-locked configuration.
• Figure 5: Numerical simulation of CDW oscillator network implementing a honeycomb topology with $m=2$ cycles and $n_c=5$. (a) Circuit topology showing two cycles sharing node 5. (b,d) Steady-state voltage waveforms for cycles 1 and 2, demonstrating synchronized oscillations with characteristic phase delays. (c,e) Phase circle diagrams showing uniform $2\pi/5$ phase spacing: counterclockwise in cycle 1 and clockwise in cycle 2.

Theorems & Definitions (11)

• Definition 1
• Theorem 3.1
• Definition 2
• Theorem 3.2
• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 3
• proof