Chaotic Oscillatory Associative Memory

TL;DR

This work proposes Chaotic Oscillator Associative Memory (COAM), a brain-inspired memory framework that encodes patterns as phase-synchronized states in a network of chaotic Roessler oscillators. Memories are stored via Hebbian learning, with coupling incorporating higher-order Fourier modes to boost storage capacity, and a curvature-based phase definition enables phase-locked retrieval despite chaotic fluctuations. The results show robust phase synchronization and markedly enhanced memory capacity with higher Fourier-mode coupling, including near-perfect retrieval of MNIST-like patterns and capacity growth with chaos, while also revealing limits where excessive chaos can impair exact recall. Biologically, COAM offers a pathway to reconcile chaotic neural dynamics with stable memory retrieval, while also suggesting boundaries where phase-encoded memory remains viable in highly chaotic regimes.

Abstract

Associative memory models retrieve stored information through content-based addressing, mimicking the neural processes of animal brains. The classical Hopfield network-based models store memories as vectors of discrete values and have good storage capacity but do not consider the role of neuronal synchronization in memory storage and retrieval as observed in brains. This is addressed in phase-oscillator-based models which store memories as time-dependent phase-synchronized states, but suffer from instability and low capacity. The present study addresses these challenges through a novel chaotic oscillator-based associative memory model, by defining a phase relationship in chaotic systems and encoding memory as synchronized states of these phases. The underlying chaos in the network is shown to significantly improve both storage and retrieval and offer insights into the dynamics of memory retrieval.

Figures (11)

• Figure 1: Time evolution of phase difference for two coupled Rössler systems at (a) sparsely chaotic regime ($\mathfrak{c}=9$) and (b) strongly chaotic regime($\mathfrak{c}=14$). Black full line: numerical simulations; Green full line: Temporal moving average; red dash-dot line: Target phase difference $\Theta = 2\pi/3$. Identical initial values were used in both cases. Numerical values of parameters for Eq.(\ref{['eq:CAM_with_second_order_coupling']}): $N=2$, $K=1$, $\mathfrak{a}=0.1$, $\mathfrak{b}=0.1$, $r = 0.001$.
• Figure 2: Lyapunov spectrum of a network of 3 coupled Rössler systems, as a function of coupling strength $\epsilon$. Three zero LEs are observed for $\epsilon=0$; only one zero LE is seen for $\epsilon >0$. The 6 largest LEs are shown; the remaining 3 LEs are significantly negative to be shown in this scale. The numerical parameters usef in Eq.(\ref{['eq:CAM_with_second_order_coupling']}) are $N=3$, $K=1$, $\mathfrak{a} = 0.1$, $\mathfrak{f} = 0.1$, $\mathfrak{c} = 9$.
• Figure 3: Time evolution of the energy function for the COAM; the underlying Rössler systems correspond to periodic ($\mathfrak{c}=4$), sparsely chaotic ($\mathfrak{c}=9$) and strongly chaotic ($\mathfrak{c}=14$) regimes. The numerical values of the parameters for Eq.(\ref{['eq:CAM_with_second_order_coupling']}) are $N=50$, $\mathfrak{a} = \mathfrak{f} = 0.1$, $\epsilon = 0.001$.
• Figure 4: The quality of memory retrieval in terms of average overlap measure $m^\eta$ as a function of coupling strength $\epsilon$. The numerical values of the parameters for Eq.(\ref{['eq:CAM_with_second_order_coupling']}) are $N=100$, $K=1$,$\mathfrak{a} = 0.1$, $\mathfrak{f} = 0.1$, $\mathfrak{c} = 9$.
• Figure 5: Retrieval quality of memories, quantified through overlap measure $m$ as a function of time for COAM with the Rössler systems in the periodic ($\mathfrak{c}=4$) regime,for cases corresponding to number of stored memories varying from $p=1$ to $p=5$. Initial overlap with one of the stored patterns is 0.7. Numerical values for the parameters for Eq.(\ref{['eq:CAM_with_second_order_coupling']}): $N = 100$, $K = 1$, ($\mathfrak{a, b} = 0.1$), $\epsilon = 0.001$.