Event-Driven E-Prop in Spiking Neural Networks

• Event-Driven E-Prop is a biologically plausible, event-driven learning rule that leverages sparse spike events and strict synaptic locality for efficient online credit assignment.
• It replaces time-driven gradient updates with event-triggered eligibility traces, reducing computational complexity and enhancing scalability in deep, recurrent architectures.
• The method demonstrates significant energy savings and real-time performance on benchmarks like N-MNIST, supporting practical deployment on neuromorphic hardware.

Event-Driven E-Prop is a biologically plausible, event-synchronous learning rule for recurrent spiking neural networks that leverages sparse spiking activity and strict synaptic locality to enable scalable, efficient, and online credit assignment. It generalizes the eligibility propagation (e-prop) algorithm by replacing traditional time-driven gradient and eligibility trace updates with a fully event-driven mechanism, matching both the constraints of large-scale simulation and the known properties of synaptic plasticity in biological circuits. The methodology extends to arbitrarily deep recurrent networks, providing a framework that is both computationally and memory-efficient and adaptable to neuromorphic hardware deployments (Korcsak-Gorzo et al., 26 Nov 2025, Millidge, 30 Dec 2025).

1. Mathematical Foundation and Gradient Derivation

Event-Driven E-Prop builds on the factorization of the loss gradient in recurrent spiking neural networks. Denoting by $z_j^t$ the postsynaptic spike output of neuron $j$ at time $t$, the synaptic gradient for weight $W_{ji}$ is expressed as:

$\frac{dL}{dW_{ji}} = \sum_{t=1}^T L_j^t\,e_{ji}^t,$

where $L_j^t$ is a "learning signal" broadcast to neuron $j$ and $e_{ji}^t$ is the eligibility trace quantifying the instantaneous influence of $W_{ji}$ on $z_j^t$ (Korcsak-Gorzo et al., 26 Nov 2025).

The eligibility trace is computed recursively:

$$\boldsymbol\epsilon_{ji}^t = \frac{\partial \mathbf{h}_j^t}{\partial \mathbf{h}_j^{t-1}}\,\boldsymbol\epsilon_{ji}^{t-1} + \frac{\partial \mathbf{h}_j^t}{\partial W_{ji}},$$

with $\mathbf{h}_j^t = (v_j^t,a_j^t)$ denoting the hidden state variables (membrane voltage, adaptation). The spike output’s sensitivity to hidden-state changes is handled through a surrogate gradient $\psi_j^t$:

$$e_{ji}^t \approx \psi_j^t \left(\epsilon_{ji}^{v,t} - \beta^a\,\epsilon_{ji}^{a,t}\right).$$

For temporal credit assignment, a filtered (low-pass) eligibility trace $$\bar{e}_{ji}^t = \kappa\,\bar{e}_{ji}^{t-1} + e_{ji}^t$$ is maintained, with decay parameter $\kappa$ aligned with the output neuron’s time constant (Korcsak-Gorzo et al., 26 Nov 2025).

This derivation extends to deep recurrent networks by formulating eligibility traces $e_{ji}^{(\ell),t}$ for each layer $\ell$ and incorporating recursions both across time and depth. For the parameter $W_{ji}^{(\ell)}$ in layer $\ell$:

$e_{ji}^{(\ell),t} = \frac{\partial h_j^{(\ell)}_t}{\partial W_{ji}^{(\ell)}} + \sum_k \frac{\partial h_j^{(\ell)}_t}{\partial h_k^{(\ell)}_{t-1}} e_{ki}^{(\ell),t-1} + \sum_m \frac{\partial h_j^{(\ell)}_t}{\partial h_m^{(\ell-1)}_t} e_{mi}^{(\ell-1),t}$

with the update

$$\frac{\partial E}{\partial W_{ji}^{(\ell)}} \approx \sum_{t=1}^T L_j^{(\ell),t} \cdot e_{ji}^{(\ell),t}$$

(Millidge, 30 Dec 2025).

2. Transition from Time-Driven to Event-Driven Implementation

In conventional e-prop, all neurons and synapses update at every simulation step, resulting in $O(N_{\text{syn}})$ computational complexity per time step, which is inefficient given the sparse activity in biological and neuromorphic networks (Korcsak-Gorzo et al., 26 Nov 2025).

Event-driven E-Prop instead synchronizes synaptic updates to spike events:

• Eligibility traces and gradient accumulators are updated and stored only at the occurrence of spikes (presynaptic or modulatory).
• History buffers archive relevant neuron-local variables (e.g., $\psi_j^t$, $L_j^t$) in time-stamped arrays.
• When a presynaptic spike arrives at synapse $(j, i)$, the synapse retrieves the required history over the interval since its last spike, computes the cumulative gradient update

$$g_{ji}^{[t_{\text{last}}, t]} = \sum_{t'=t_{\text{last}}+1}^t L_j^{t'}\,\bar{e}_{ji}^{t'-d}$$

and applies a weight adjustment immediately (Korcsak-Gorzo et al., 26 Nov 2025).

• Continuous-time dynamics are preserved: neuron states integrate with fine-grained steps, but synaptic updates are event-triggered, decoupling spike transmission delays from the simulation clock.

3. Event-Driven Algorithmic Realization

Efficient event-driven e-prop requires synchronization between neuron activity, synaptic plasticity, and distributed simulation for large-scale networks:

For each neuron j: Maintain History_j[t] = {ψ_j^t, L_j^t} Maintain UpdateCount_j[t] = number of synapses yet to process at t For each synapse (j←i): On presynaptic spike at time t: For τ = last_spike_time+1 to t: Access L_j^τ, ψ_j^{τ-d} Compute e_{ji}^{τ-d} g += L_j^τ ⋅ Filter_{κ}[e_{ji}^{τ-d}] ΔW_{ji} ← optimizer_step(g) last_spike_time ← t # History cleanup when UpdateCount_j[...] reaches zero

(Korcsak-Gorzo et al., 26 Nov 2025)

In the multilayer context, eligibility traces are updated at each spike or non-trivial activation in a forward sweep across time and depth, with immediate per-synapse adjustment utilizing only presynaptic timing, locally cached eligibility, and error broadcast (Millidge, 30 Dec 2025).

4. Biological Constraints and Locality

The scheme enforces several constraints inspired and validated by biological networks:

• Strict locality: Each synapse requires only its eligibility trace, the local error broadcast $L_j^t$, the local surrogate gradient $\psi_j^t$, and presynaptic spike timing—no global weight symmetry or backward traversal through network activations.
• Continuous dynamics: Neural state integration occurs without artificial resets, enabling uninterrupted, physically realistic evolution of network activity.
• Sparse connectivity: Updates are only carried out when a presynaptic spike and a nonzero eligibility trace coincide, naturally exploiting low biological firing rates (typically $\sim3$ Hz) for computational and energy efficiency.
• The approach eliminates dependencies on global replay or weight transport, rendering the implementation compatible with biologically plausible substrate constraints (Korcsak-Gorzo et al., 26 Nov 2025).

5. Computational Complexity and Scalability

Event-driven e-prop achieves substantial computational and memory savings for sparse networks:

• Time-driven: $O(T\times N_{\text{syn}})$ operations, updating all synapses at every time step.
• Event-driven: $O(\lambda T \times N_n)$, where $\lambda$ is the mean firing rate, $N_n$ the number of neurons, and each synaptic update is performed only at spike times. For typical biological sparsity, $\lambda \ll 1/\Delta t$, giving one to two orders of magnitude computational acceleration.
• Memory: Only active synapses maintain history buffers, and memory usage does not scale with the total number of time steps, unlike BPTT.
• Parallel scaling: Demonstrated near-ideal weak scaling to millions of neurons with constant in-degree. On JURECA, strong scaling yielded $3$–$5\times$ speedup for a 2 million neuron network across 32 to 256 cores. The real-time factor improved by $10$–$20\times$ over time-driven implementations (Korcsak-Gorzo et al., 26 Nov 2025).

6. Benchmark Results and Applications

Event-driven e-prop has been validated on several key tasks:

• Pattern Generation and Evidence Accumulation: Demonstrates exact reproduction of time-driven output up to floating-point precision; minor divergences in highly sensitive regimes do not affect mean performance.
• N-MNIST Benchmark: Matches time-driven e-prop performance (test error $\sim3$–$5\%$) with only 1ms time resolution and single-sample updates. Addition of biological enhancements preserved or improved accuracy with faster convergence.
• Energy and Efficiency: Achieved up to an order of magnitude reduction in CPU utilization for sparse spiking regimes, enabling deployment on neuromorphic hardware with local, online learning (Korcsak-Gorzo et al., 26 Nov 2025).

A plausible implication is that by adhering to strictly local information and event-driven update rules while retaining gradient-based optimization, event-driven e-prop offers a unifying approach that scales to deep, large-scale networks and is compatible with both biological and artificial substrates (Millidge, 30 Dec 2025).

7. Theoretical Extensions to Deep Architectures

The event-driven e-prop framework generalizes to deep recurrent networks by introducing a two-dimensional recursion along temporal and depth axes. Eligibility traces propagate both through time (recurrent influence) and depth (feedforward and feedback) layers:

• Eligibility trace, per synapse in each layer: $e_{ji}^{(\ell),t}$, updated via local Hebbian terms, recurrent propagation, and upward transfer across depth.
• Learning signals, $L_j^{(\ell),t}$, constructed from local loss gradients and feedback from higher layers.
• Computational cost: per-step, per-layer operations remain linear in depth and quadratic in layer width, with memory requirements constant in sequence length, unlike BPTT (Millidge, 30 Dec 2025).

This extension enables fully online, local credit assignment in arbitrarily deep recurrent hierarchies without backward replay, making event-driven e-prop suitable for both biological modeling and deployment in deep neuromorphic network implementations.

---

References:

• "Event-driven eligibility propagation in large sparse networks: efficiency shaped by biological realism" (Korcsak-Gorzo et al., 26 Nov 2025)
• "Generalising E-prop to Deep Networks" (Millidge, 30 Dec 2025)