How does our motor system solve the problem of anticipatory control in spite of a wide spectrum of response dynamics from different musculo-skeletal systems, transport delays as well as response latencies throughout the central nervous system? To a great extent, our highly-skilled motor responses are a result of a reactive feedback system, originating in the brain-stem and spinal cord, combined with a feed-forward anticipatory system, that is adaptively fine-tuned by sensory experience and originates in the cerebellum. Based on that interaction we design the counterfactual predictive control (CFPC) architecture, an anticipatory adaptive motor control scheme in which a feed-forward module, based on the cerebellum, steers an error feedback controller with counterfactual error signals. Those are signals that trigger reactions as actual errors would, but that do not code for any current of forthcoming errors. In order to determine the optimal learning strategy, we derive a novel learning rule for the feed-forward module that involves an eligibility trace and operates at the synaptic level. In particular, our eligibility trace provides a mechanism beyond co-incidence detection in that it convolves a history of prior synaptic inputs with error signals. In the context of cerebellar physiology, this solution implies that Purkinje cell synapses should generate eligibility traces using a forward model of the system being controlled. From an engineering perspective, CFPC provides a general-purpose anticipatory control architecture equipped with a learning rule that exploits the full dynamics of the closed-loop system.

In order to illustrate the main idea from our paper in a simplified context, we show in this section how observed hidden-layer activity in a linear feedforward network can be used to infer the learning rule that is used to train the network. Consider the simple feedforward network shown in Fig. S1. N (0, Σ) is noise injected into the network. This is similar to learning with Feedback Alignment [4], except that here we do not assume that the readout weights are being learned. Equations (11) and (13) provide predictions for how the hidden-layer activity is expected to evolve under either SL or RL.

Here, we propose that extra-synaptic diffusion of local neuromodulators such as neuropeptides may afford an effective mode of backpropagation lying within the bounds of biological plausibility.

How does our motor system solve the problem of anticipatory control in spite of a wide spectrum of response dynamics from different musculo-skeletal systems, transport delays as well as response latencies throughout the central nervous system? To a great extent, our highly-skilled motor responses are a result of a reactive feedback system, originating in the brain-stem and spinal cord, combined with a feed-forward anticipatory system, that is adaptively fine-tuned by sensory experience and originates in the cerebellum. Based on that interaction we design the counterfactual predictive control (CFPC) architecture, an anticipatory adaptive motor control scheme in which a feed-forward module, based on the cerebellum, steers an error feedback controller with counterfactual error signals. Those are signals that trigger reactions as actual errors would, but that do not code for any current of forthcoming errors. In order to determine the optimal learning strategy, we derive a novel learning rule for the feed-forward module that involves an eligibility trace and operates at the synaptic level. In particular, our eligibility trace provides a mechanism beyond co-incidence detection in that it convolves a history of prior synaptic inputs with error signals. In the context of cerebellar physiology, this solution implies that Purkinje cell synapses should generate eligibility traces using a forward model of the system being controlled. From an engineering perspective, CFPC provides a general-purpose anticipatory control architecture equipped with a learning rule that exploits the full dynamics of the closed-loop system.

However, even if a plant has negligible transmission delays, it may still have sizable inertial latencies. For example, if we apply a force to a visco-elastic plant, its peak velocity will be achieved after a certain delay; i.e. the velocity itself will lag the force.

The Free Energy Principle (FEP) states that self-organizing systems must minimize variational free energy to persist (Friston, 2010, 2019), but the path from principle to implementable algorithm has remained unclear. We present a constructive proof that the FEP can be realized through exact local credit assignment. The system decomposes gradient computation hierarchically: spatial credit via feedback alignment, temporal credit via eligibility traces, and structural credit via a Trophic Field Map (TFM) that estimates expected gradient magnitude for each connection block. We prove these mechanisms are exact at their respective levels and validate the central claim empirically: the TFM achieves 0.9693 Pearson correlation with oracle gradients. This exactness produces emergent capabilities including 98.6% retention after task interference, autonomous recovery from 75% structural damage, self-organized criticality (spectral radius ρ 1.0), and sample-efficient reinforcement learning on continuous control tasks without replay buffers. The architecture unifies Pri-gogine's dissipative structures (Prigogine, 1977), Fris-ton's free energy minimization (Friston, 2010), and Hopfield's attractor dynamics (Hopfield, 1982; Amit et al., 1985a,b), demonstrating that exact hierarchical inference over network topology can be implemented with local, biologically plausible rules.

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.

Spiking Neural Networks (SNNs) provide an efficient framework for processing dynamic spatio-temporal signals and for investigating the learning principles underlying biological neural systems. A key challenge in training SNNs is to solve both spatial and temporal credit assignment. The dominant approach for training SNNs is Backpropagation Through Time (BPTT) with surrogate gradients. However, BPTT is in stark contrast with the spatial and temporal locality observed in biological neural systems and leads to high computational and memory demands, limiting efficient training strategies and on-device learning. Although existing local learning rules achieve local temporal credit assignment by leveraging eligibility traces, they fail to address the spatial credit assignment without resorting to auxiliary layer-wise matrices, which increase memory overhead and hinder scalability, especially on embedded devices. In this work, we propose Traces Propagation (TP), a forward-only, memory-efficient, scalable, and fully local learning rule that combines eligibility traces with a layer-wise contrastive loss without requiring auxiliary layer-wise matrices. TP outperforms other fully local learning rules on NMNIST and SHD datasets. On more complex datasets such as DVS-GESTURE and DVS-CIFAR10, TP showcases competitive performance and scales effectively to deeper SNN architectures such as VGG-9, while providing favorable memory scaling compared to prior fully local scalable rules, for datasets with a significant number of classes. Finally, we show that TP is well suited for practical fine-tuning tasks, such as keyword spotting on the Google Speech Commands dataset, thus paving the way for efficient learning at the edge.

It is straightforward to compute. The following derivation leads to our learning rule. We expand on our explanation for step (a) approximation. Here, we simply use the i.i.d. To put this derivation in terms of biological components, we make the following further approximations.