Event-driven eligibility propagation in large sparse networks: efficiency shaped by biological realism

TL;DR

This work addresses scalable, energy-efficient learning in brain-inspired recurrent SNNs by converting time-driven e-prop updates to an event-driven formulation. The authors introduce e-prop+, a biologically richer variant that preserves online, local credit assignment while incorporating transmission delays, dynamic firing-rate regulation, per-spike updates, and a smoother surrogate gradient. They validate the approach on pattern generation, evidence accumulation, and neuromorphic MNIST, demonstrating learning performance that closely matches the time-driven baseline and strong scaling up to millions of neurons. The resulting framework, implemented in NEST and benchmarked on neuromorphic data, offers a practical path toward sustainable, brain-like AI with scalable, biologically plausible learning rules and hardware-compatible architectures.

Abstract

Despite remarkable technological advances, AI systems may still benefit from biological principles, such as recurrent connectivity and energy-efficient mechanisms. Drawing inspiration from the brain, we present a biologically plausible extension of the eligibility propagation (e-prop) learning rule for recurrent spiking networks. By translating the time-driven update scheme into an event-driven one, we integrate the learning rule into a simulation platform for large-scale spiking neural networks and demonstrate its applicability to tasks such as neuromorphic MNIST. We extend the model with prominent biological features such as continuous dynamics and weight updates, strict locality, and sparse connectivity. Our results show that biologically grounded constraints can inform the design of computationally efficient AI algorithms, offering scalability to millions of neurons without compromising learning performance. This work bridges machine learning and computational neuroscience, paving the way for sustainable, biologically inspired AI systems while advancing our understanding of brain-like learning.

Figures (13)

• Figure 1: Mathematical basis and technical implementation of e-prop with event-driven weight updates.(a) Network layout and dynamic variables for weight updates in input, recurrent, and output synapses. The recurrent network may include LIF, ALIF, or both neuron types. (b) The first spike (highlighted with yellow circle) of the neural response to input data sample $n+1$ triggers the retrieval of the archived history for the previous sample $n$ (yellow arrow) and the computation of the corresponding gradients. Arrows indicate the surrogate gradient entry $\psi_j^t$ and presynaptic spike $z_i^{t-1}$ associated with a learning signal entry $L_j^t$. (c) Propagation of spikes and signals within a single step (dotted box). Over e-prop synapses (gray arrows): (I) Transmit spikes from input neurons (green circles) to recurrent neurons (blue circles), (II) transmit spikes to output neurons (red circles). (III) Transmit signals (gray arrows) between output neurons to compute the softmax. (IV) Send the learning signal (red arrow) from the output layer to the recurrent layer. (d) Representation of transmission I-IV as a pipeline across four steps. The number of incomplete operations (gray circles) at the boundary (dashed black line) increases with pipeline depth. In this case, three learning signals have not arrived yet at update time, corresponding to the pipeline depth minus one.
• Figure 2: Comparison of learning performance between time-driven and event-driven models. Regression task (pattern generation): (a) Absolute difference in loss between the two models during the first 4 training iterations with a mini-batch size of 1 in a single trial and (b) loss of both models over 900 training iterations with a mini-batch size of 1 averaged over 10 trials. Classification task (evidence accumulation): (c) Absolute difference in loss between the two models during the first 4 training iterations with a mini-batch size of 1 in a single trial and (d) prediction error of both models over 900 training iterations with a mini-batch size of 32 averaged over 10 trials. Curves represent the mean and shaded areas the standard deviation across 10 trials with different random seeds. Bars in the inset show the mean across all trials and 10 test iterations per trial, and error bars represent the combined standard deviation, calculated as the square root of the sum of within-trial and between-trial variances.
• Figure 3: Learning performance of event-driven e-prop models. Learning performance measured as prediction errors on the N-MNIST task using event-driven e-prop models. Curves represent the mean and shaded areas the standard deviation across 10 trials with different random seeds. Bars in the inset show the mean across all trials and 10 test iterations per trial, and error bars represent the combined standard deviation, calculated as the square root of the sum of within-trial and between-trial variances. The convergence speed and test error of e-prop+ match those of e-prop, confirming that learning remains effective. Learning converges slightly faster for e-prop+, possibly because the identical parameters across both models produce dynamics more favorable to learning. A precise conclusion would require fair parameter tuning for both models.
• Figure 4: Dynamics and weight distribution before and after training N-MNIST using e-prop+.(a) Time traces of dynamic variables before and after training. Spike states $z_i$ for the input neurons; spike state $z_j$, membrane voltage $v_j$, surrogate gradient $\psi_j$, and learning signal $L_j$ for an example recurrent LIF neuron; and membrane voltages $v_k$, target signals $y_k^{*}$, readout signals $y_k$, and their differences $y_k - y_k^*$ for the ten output neurons. After training, the output neuron with the highest membrane voltage (and thus the highest readout signal) matches the target output neuron (red curves), correctly predicting the class. (b) Distributions of input, recurrent, and output weights. Points on the diagonal indicate no change. The largest changes occur in the input and recurrent weights.
• Figure 5: Scaling of e-prop models.(a) Weak scaling and (b) strong scaling results shown as wall-clock runtime of the simulation excluding the network-building phase, and the real-time factor defined as the ratio of simulated biological time to runtime, both as a function of the number of cores. The results reveal sub-linear strong scaling, with runtime differences diminishing as the number of cores increases, and near-linear weak scaling. The gray line indicates ideal linear speedup where runtime decreases inversely with number of cores. As expected, plasticity calculations increase runtime compared to static synapses, and the third factor in e-prop adds overhead relative to the two-factor STDP rule. Points and error bars show mean and standard deviation across runs, though the error bars are barely visible due to low variance.