Continuous-Time Biologically Plausible Learning

• The paper demonstrates that continuous-time learning unifies gradient descent, feedback alignment, DFA, and weight-mirror dynamics using ODE frameworks.
• The model quantifies how temporal overlap between presynaptic input and error signals, with plasticity windows exceeding stimulus durations, is crucial for effective synaptic updates.
• The approach predicts testable mechanisms in neural circuits, including the necessity of seconds-long eligibility traces and robustness against timing delays.

Biologically plausible learning in continuous time encompasses a class of theories and models in which synaptic plasticity—the modification of connection strengths between neurons—unfolds in real time, driven by local, temporally correlated signals and without the artificial staging, global error transport, or dual-phase delineations of classic gradient-based deep learning. This paradigm is motivated by the fact that synaptic changes in the brain occur as a direct and ongoing consequence of streams of sensory input and feedback, modulated by biophysical mechanisms that obey strict locality and timing constraints.

1. Continuous-Time Neural Networks: Principles and Formalism

Biologically plausible continuous-time neural models treat both the neural activity and the synaptic weights as dynamical variables governed by ordinary (and occasionally stochastic) differential equations. Each neuron's state $z_{\ell}$ and each synaptic weight matrix $W_{\ell}$ is updated continuously in time, with timescales for neural dynamics ($\tau$) generally much faster than for plasticity ($\tau_{W}$). The full system couples rapid inference (the computation of neural responses) with slow, overlapping synaptic changes:

$$\frac{d z_{\ell}}{dt} = \frac{-z_{\ell} + \sigma\left(W_{\ell}^T z_{\ell-1}\right)}{\tau}$$

$$\frac{d W_{\ell}}{dt} = -\frac{W_{\ell}}{\tau_W} + \frac{1}{\tau_W} \left[z_{\ell-1} (V_{\ell}^T \varepsilon_{\ell})^T\right]$$

$$\frac{d V_{\ell}}{dt} = -\frac{V_{\ell}}{\tau_V} + \frac{1}{\tau_V} \left[(W_{\ell}^T z_{\ell-1}) \varepsilon_\ell^T \right]$$

where $V_{\ell}$ are feedback or modulatory weight matrices and $\varepsilon_{\ell}$ are local error signals (Bacvanski et al., 21 Oct 2025). These coupled equations allow inference and learning to evolve on their respective timescales without discrete phase alternations or explicit backward passes.

2. Emergence and Unification of Learning Algorithms

Several well-known biologically plausible learning rules are unified within the above continuous-time framework by appropriate choices of the feedback matrix $V_{\ell}$ and the error drive:

• Stochastic Gradient Descent (SGD): When $V_{\ell} = W_{\ell}^T$ and the true backpropagated error is available, the model reduces to classic gradient descent.
• Feedback Alignment (FA): When $V_{\ell}$ is a fixed, random, sign-constrained, or otherwise asymmetric matrix, learning proceeds via local, feedforward synaptic plasticity steered by nonmatching feedback weights. This mechanism has been shown numerically to drive alignment with the true error gradient over time (Xiao et al., 2018, Bacvanski et al., 21 Oct 2025).
• Direct Feedback Alignment (DFA): When the feedback is a direct projection from the output to each layer (often with fixed random weights), error signals reach all layers simultaneously without sequential backpropagation.
• Kolen–Pollack/Weight Mirror Dynamics: When $V_{\ell}$ themselves evolve to mirror the transposed forward weights (with appropriate plasticity timescales and decay), weight-symmetric error propagation is recovered as a limiting case (Bacvanski et al., 21 Oct 2025).

All these mechanisms, when implemented as continuous dynamics with co-evolving weights, obviate the need for temporally separated forward/backward phases and align with key constraints observed in biological circuits.

3. Temporal Overlap and the Plasticity Window

A critical result from the continuous-time perspective is the realization that effective synaptic updates demand substantial temporal overlap between presynaptic activity and the local error signal. When a synapse's input spike or activity and the associated error are temporally offset, the efficacy of plasticity diminishes according to the overlap interval:

$\mathbb{E}[\Delta W] \propto (T - |\Delta|)_+$

where $T$ is the stimulus duration and $\Delta$ is the delay between input and error (Bacvanski et al., 21 Oct 2025). This triangular relationship implies that learning becomes fragile as the error signal arrives later relative to the input, an effect exacerbated in deep networks with inter-layer delays. Notably, robust learning requires the plasticity timescale $\tau_W$ (the functional "eligibility trace" duration) to be significantly greater than $T$—by at least one to two orders of magnitude.

In quantitative terms, for cortical stimulus durations on the order of 20–50 ms, the model predicts that eligibility traces should persist for 2–5 s to ensure proper credit assignment and synaptic updates. This value is compatible with experimental evidence for seconds-long eligibility traces in dopaminergic and other modulatory systems.

4. Implications for Synaptic Plasticity Mechanisms

The continuous-time perspective directly constrains the plausible biophysical properties required for learning in real neural circuits:

• Necessity of Seconds-Scale Eligibility Traces: To bridge the temporal gap between presynaptic activity and potentially delayed reward or feedback signals, intracellular eligibility traces—sustained, for example, by calcium transients, molecular second messengers, or receptor modulations—must operate on the time scale of seconds (Bacvanski et al., 21 Oct 2025).
• Locality of Plasticity Signals: All updates in these models are functions only of signals locally available at the synapse (presynaptic activity, postsynaptic membrane potential, and the local error drive), consistent with constraints that prohibit network-wide error transport or parameter mirroring.
• Robustness to Noise and Delay: Simulations demonstrate stability and learning even under temporal mismatches, integration noise, and biologically relevant variability, provided the overlap principle and sufficient plasticity timescales are respected.

A plausible implication is that anatomical features such as short-cut feedback connections, dendro-somatic integration, and fast subcortical pathways may have evolved to bolster the temporal overlap between signal and error in deep brain structures.

5. Model Predictions and Testable Consequences

The analysis yields several experimentally testable predictions:

• Plasticity Window Duration: Error-driven synaptic updates in cortical and striatal regions should persist over seconds-long windows. This aligns with observations from dopamine-dependent learning and points to specific molecular candidates underlying behavioral learning rules.
• Depth-Dependent Temporal Constraints: Deeper biological networks, or those with more inter-laminar delays, should require longer stimulus presentations or enhanced anatomical feedback to maintain robust plasticity.
• Failure under Temporal Mismatch: If the error signal is delayed beyond the plasticity window, learning efficacy will sharply decline or fail—a phenomenon that can be quantitatively probed in both computational and neurophysiological settings.

Moreover, the instantiation of these dynamics as analog, neuromorphic hardware models must also respect the hierarchy of timescales, providing guidance for hardware-software co-design.

6. Simulation Results and Computational Validation

Numerical experiments confirm the theoretical principles: networks trained with various plausible error-propagation schemes exhibit stable learning curves and high accuracy for appropriately chosen plasticity timescales. Heatmaps demonstrate rapid drop-off in learning as the delay between input and error approaches the duration of stimulus presentation, especially in deeper networks where layerwise error routing causes cumulative propagation delays. The existence of a broad regime of robust learning is conditioned specifically on plasticity timescales being an order of magnitude longer than typical input durations (Bacvanski et al., 21 Oct 2025).

7. Outlook, Limitations, and Future Directions

The continuous-time approach to biologically plausible learning closes the conceptual gap between algorithmic models of credit assignment and the real-time, local, and noisy environment of biological brains. By showing that SGD, FA, DFA, and weight-mirror methods are unified in a dynamical ODE framework, and by quantifying the temporal constraints necessary for synaptic plasticity, it connects theoretical modeling, neurobiology, and hardware realization.

Several open directions remain:

• The current models employ rate-based neurons; extensions to spike-based or event-driven architectures will require adaptation of the overlap and filtering principles.
• The explicit modeling of gating mechanisms or advanced eligibility traces may further enhance robustness and selectivity for error-driven learning.
• Large-scale, multi-area, or closed-loop experiments may test the model’s predictions regarding the dependence of learning on timescale hierarchies and anatomical feedback shortcuts.

In summary, continuous-time, biologically plausible learning models ground error-driven plasticity in local, temporally extended, and dynamically co-evolving processes, offering a general mechanism that integrates rich temporal dynamics with the constraints and capabilities of real neural circuits (Bacvanski et al., 21 Oct 2025).