Synapses That Remember Two Moments: Equilibrium Propagation Moves Its Contrastive Update Inside the Device, Matching Standard MNIST Error

A synapse no longer needs external circuitry to hold the local correlation reached at the end of the free phase; the temporal comparison is carried by the synapse's internal dynamics. Earlier physical implementations either retained the free-phase value until the nudged phase or used two copies of the network to make both states available at once, both requiring extra memory or duplicated hardware. On MNIST with a single-hidden-layer network, the energy-coupled and decoupled forms reach test errors of 2.01% and 1.98%, against 2.00% for standard phase-based EqProp, a difference within one standard deviation.

Each synapse carries two internal variables, one relaxing with neural activity and one storing the learned parameter over the long term. The volatile state follows a Hebbian-like rule toward the product of pre- and postsynaptic activities and is reset between examples, while the non-volatile state is a pure integrator with no decay term, written only during learning. Both forms are trained on a 784–512–10 multilayer perceptron, and in the decoupled form the volatile state does not enter the effective weight, so inference uses ordinary multiply–accumulate operations.

The continual form writes the successive increments of the volatile state directly into the non-volatile state, removing the need to read back a previous value. The phase-based form still retains the free-phase value until the nudged phase ends, whereas the continual form shortens the retention interval to a single time step. Continual updates give test errors 0.2% to 0.3% higher than phase-based updates, consistent with the trend reported by Ernoult et al., attributed to the learning rate having to stay small during the nudged phase.

The result validates on MNIST with a single-hidden-layer network that both update forms preserve the learning capability of equilibrium propagation, with the energy-coupled and decoupled forms performing almost identically and leaving flexibility for hardware implementation. The authors frame the intended target as a synaptic device with volatile and non-volatile internal states, and list four dynamical requirements: a volatile state driven by the activities at the two terminals with a finite relaxation time, a non-volatile state that dominates the effective weight, transfer of the volatile state's change rather than its instantaneous value, and the ability to reset the volatile state between examples. The stated next step is to realize or simulate at the device level a synaptic element with the required coupling and to determine how learning is affected by realistic non-idealities.

Continual updates give test errors 0.2% to 0.3% higher than phase-based updates, which the authors attribute to the learning rate having to stay small during the nudged phase, slowing convergence and potentially limiting final performance. The equivalence between continual and phase-based updates requires the non-volatile weights to change only weakly during the nudged relaxation, and the simulations apply per-layer norm clipping to the update gradients, so the steps no longer sum exactly to the phase-based update. Existing higher-order memristors already combine volatile and non-volatile states, but their coupling laws typically transfer the instantaneous value of the volatile state, which does not match the required transfer of its change.