On Biologically Plausible Learning in Continuous Time

TL;DR

This work introduces a continuous-time neural framework in which inference and learning unfold together via coupled ODEs, removing phase separation and aligning learning dynamics with biological timescales. By tuning propagation, plasticity, and decay timescales ($ au_{ ext{prop}}$, $ au_{ ext{plas}}$, $ au_{ ext{dec}}$) and using local two-signal plasticity rules, the authors show that SGD, FA, DFA, and KP emerge as limiting cases, with learning governed by the temporal overlap between inputs and error signals. A key contribution is the explicit overlap (eligibility trace) mechanism, predicting that robust learning requires the plasticity window to outlast the stimulus by about an order of magnitude, placing eligibility traces in the few-second range for cortical-like inputs. The results bridge neuroscience and hardware, offering a physically grounded path to continuous-time learning in biological and neuromorphic systems, and yielding testable predictions about the duration of eligibility traces and the impact of propagation delays on learning depth.

Abstract

Biological learning unfolds continuously in time, yet most algorithmic models rely on discrete updates and separate inference and learning phases. We study a continuous-time neural model that unifies several biologically plausible learning algorithms and removes the need for phase separation. Rules including stochastic gradient descent (SGD), feedback alignment (FA), direct feedback alignment (DFA), and Kolen-Pollack (KP) emerge naturally as limiting cases of the dynamics. Simulations show that these continuous-time networks stably learn at biological timescales, even under temporal mismatches and integration noise. Through analysis and simulation, we show that learning depends on temporal overlap: a synapse updates correctly only when its input and the corresponding error signal coincide in time. When inputs are held constant, learning strength declines linearly as the delay between input and error approaches the stimulus duration, explaining observed robustness and failure across network depths. Critically, robust learning requires the synaptic plasticity timescale to exceed the stimulus duration by one to two orders of magnitude. For typical cortical stimuli (tens of milliseconds), this places the functional plasticity window in the few-second range, a testable prediction that identifies seconds-scale eligibility traces as necessary for error-driven learning in biological circuits.

Figures (8)

• Figure 1: Neuron design with two archetypal topologies. At left (zoom): continuous-time heterosynaptic neuron model whose weights evolve according to plasticity rules. Neurons receive forward input $\textbf{x}$ and produce activated output $z$ with weights $\textbf{w}$. Error signals $\textbf{e}$ enter through modulatory weights $\textbf{v}$ and drive plasticity of $\textbf{w}$. In center: layerwise error propagation topology, where errors are propagated to previous layers via backward weights. At right: direct error propagation topology, where error signals are broadcast to previous layers directly.
• Figure 2: Zoomed-in view of output neuron dynamics during early training. The network input and corresponding error term changes during the buffer time periods. Over a single sample, the neuron outputs move only minutely.
• Figure 3: Sample train/test accuracy during training. Training set accuracy is measured as the moving average of instantaneous correct/incorrect predictions on the network input signals. Test set accuracy is measured by evaluating the network with frozen weight dynamics.
• Figure 4: Single-neuron dynamics with different relative timings of the error and input signals. Left: error signal arrives early. Right: error signal is delayed. The bottom panel of both plots depicts the cumulative weight change and shows that in the presence of delay, the weight accumulates a biased gradient update compared to the case where there was no error delay.
• Figure 5: Evaluation of the direct error routing topology on the $7\times7$ downsampled MNIST dataset. The $x$-axis denotes the temporal delay between input signal and label: negative values indicate the label arrives before the input, while positive values indicate the label arrives after the input. At left is a network with 1 hidden layer of 49 neurons, and at right is a network with 2 hidden layers of 49 and 32 neurons. The learning is fairly robust until the delay exceeds the sample time.