Autonomous Learning of Attractors for Neuromorphic Computing with Wien Bridge Oscillator Networks

TL;DR

This work tackles autonomous, energy-based neuromorphic learning by embedding associative memory in networks of Wien-bridge oscillators. By mapping the oscillator phases to an effective Kuramoto energy and implementing a local Hebbian rule, the system learns and recalls phase patterns without separate training phases. A 2-4-2 architecture with a hidden layer demonstrates richer attractor dynamics and shows the non-uniqueness of internal representations, while hardware experiments confirm robustness to component tolerances and dynamic switching via energy-surprise spikes. The results argue for coupled oscillator circuits as a practical platform for continuous, hardware-native learning and energy-based computation.

Abstract

We present an oscillatory neuromorphic primitive implemented with networks of coupled Wien bridge oscillators and tunable resistive couplings. Phase relationships between oscillators encode patterns, and a local Hebbian learning rule continuously adapts the couplings, allowing learning and recall to emerge from the same ongoing analog dynamics rather than from separate training and inference phases. Using a Kuramoto-style phase model with an effective energy function, we show that learned phase patterns form attractor states and validate this behavior in simulation and hardware. We further realize a 2-4-2 architecture with a hidden layer of oscillators, whose bipartite visible-hidden coupling allows multiple internal configurations to produce the same visible phase states. When inputs are switched, transient spikes in energy followed by relaxation indicate how the network can reduce surprise by reshaping its energy landscape. These results support coupled oscillator circuits as a hardware platform for energy-based neuromorphic computing with autonomous, continuous learning.

Figures (9)

• Figure 1: Physical circuit implementation of a Hebbian weight matrix. Each neuron is implemented as a Wien bridge oscillator with intrinsic frequency set by its series–parallel RC feedback network; for matched components $R$ and $C$ in the bridge $f\approx (2\pi RC)^{-1}$. Resistors between oscillators implement synaptic weights, determined by the Hopfield outer-product for the two encoded patterns, and potentiometers are used in the gain network to stabilize amplitude and minimize distortion. Analog sources inject the input pattern into the first two nodes, and all oscillator output voltages are measured using an oscilloscope.
• Figure 2: Plots of the cosine phase, $\cos(\phi)$, of an unbiased neuron in the $2 \times 2$ oscillator circuit (Figure \ref{['fig:circuit']}) are shown in blue, dynamics simulated by LTspice. The corresponding cosine phase determined by the fit Kuramoto model dynamics are shown in orange. Pattern 1 and pattern 2 correspond to distinct phases on the biased neurons, phase is set by the driving bias; with two oscillators driven at phases of $(0,\pi)$ and $(0,0)$ in pattern 1 and pattern 2, respectively.
• Figure 3: Hebbian learning in a $2\times 2$ network. A) The ideal weight matrix for the two desired patterns is obtained using the summed outer products of their $\pm 1$ representations; this matrix is used in numerical, SPICE and physical circuit implementations. B) Average learning curve over 100 numerically simulated runs, with individual runs using a different pair of randomized patterns. A continuous Hebbian learning rule with a decay term is applied to a randomly initialized weight matrix. During training, input oscillators (0,1) are clamped and alternate between two desired patterns, while output oscillators (2,3) are nudged towards the correct phase relative to the inputs. At each time step, the MSE is computed by taking the element-wise difference between the current weight matrix and the ideal weight matrix, squaring the differences, and averaging them into a single scalar. Shaded region displays the standard deviation across runs.
• Figure 4: Oscilloscope voltage measurements showing transitions between stored patterns: the circuit is driven to the first stored pattern and subsequently switched to the second, and the resulting waveforms capture the transient switching behavior and convergence to the corresponding attractor defined by Hopfield weight matrix. After switching, both patterns settle into stable steady states.
• Figure 5: Trajectories of oscillators in model simulations, SPICE, and hardware. Pattern 1 (blue) encoded as [1,-1,1,-1], and pattern 2 (black) as [1,1,-1,-1]. A) A $2\times 2$ oscillator architecture is simulated numerically using ideal Kuramoto dynamics. Voltage trajectories are plotted from startup with color representing the encoded pattern. B) The same architecture is implemented in SPICE using Wien bridge oscillators, with 2% capacitor and 1% resistor tolerances sampled uniformly for the RC frequency-setting network of each oscillator. C) The same Wien bridge oscillator circuit is implemented in hardware, including 10% capacitors and 1% resistors; smoothed voltage traces are plotted from oscilloscope measurements.