SpikingGamma: Surrogate-Gradient Free and Temporally Precise Online Training of Spiking Neural Networks with Smoothed Delays

TL;DR

SpikingGamma addresses the challenge of training Spiking Neural Networks at fine temporal granularity and mapping to neuromorphic hardware by introducing an online, surrogate-gradient-free training approach. It combines adaptive recursive memory with sigma-delta spike-coding to enable direct error backpropagation through a feedforward SNN, without requiring surrogate gradients. The method demonstrates competitive accuracy on temporal benchmarks (DVS Gesture, SHD, SSC) and learns precise timing with sparse spiking, while maintaining robustness to temporal discretization and scalable memory usage. This work suggests a practical route for energy-efficient neuromorphic AI and paves the way for further sparsity and scalable sequence-learning extensions.

Abstract

Neuromorphic hardware implementations of Spiking Neural Networks (SNNs) promise energy-efficient, low-latency AI through sparse, event-driven computation. Yet, training SNNs under fine temporal discretization remains a major challenge, hindering both low-latency responsiveness and the mapping of software-trained SNNs to efficient hardware. In current approaches, spiking neurons are modeled as self-recurrent units, embedded into recurrent networks to maintain state over time, and trained with BPTT or RTRL variants based on surrogate gradients. These methods scale poorly with temporal resolution, while online approximations often exhibit instability for long sequences and tend to fail at capturing temporal patterns precisely. To address these limitations, we develop spiking neurons with internal recursive memory structures that we combine with sigma-delta spike-coding. We show that this SpikingGamma model supports direct error backpropagation without surrogate gradients, can learn fine temporal patterns with minimal spiking in an online manner, and scale feedforward SNNs to complex tasks and benchmarks with competitive accuracy, all while being insensitive to the temporal resolution of the model. Our approach offers both an alternative to current recurrent SNNs trained with surrogate gradients, and a direct route for mapping SNNs to neuromorphic hardware.

Figures (12)

• Figure 1: Overview of the neural processing model. At the synapses, incoming spikes generate weighted currents that evolve over multiple timescales. Within the neuron, the resulting synaptic responses are weighted according to their timescales and summed to produce a continuous neuronal signal. This signal is then converted back into spikes through $\Sigma\Delta$-coding, allowing downstream synapses to reconstruct an estimate of the original signal.
• Figure 2: Visualization of temporal kernel computation using a cascade of leaky "buckets" that drain into one another at different rates ($\alpha_k$). Each bucket represents a temporal kernel.
• Figure 3: Signal evolution and error propagation in the neuron model. (a) At each input synapse, the neuron receives a signal. During training this is the signal estimate $\hat{y}_i^k$ of the presynaptic neuron, while during spiking inference (e.g., on a neuromorphic chip) this estimate is reconstructed from the incoming spike train $s_i$ following Eq. \ref{['eq:bucket_dynamics']}. (b) Each input is scaled by its synaptic weight $w_{ij}$, producing a current response per synapse and bucket. (c) These responses are weighted per bucket by the bucket weights $v_j^k$ (or $v_{ij}^k$ if weighted on synapse level), then accumulated across the buckets and finally rectified, forming the neuron signal $y_j$. (d) For discretizing this analog signal back into spikes, the neuron maintains a running estimate $\hat{y}_j$ that is encoded by the output spikes. (e) This estimate is expressed in the same kernel basis as the input, ensuring consistency across layers. Because $\hat{y}_j^k$ is mathematically identical to the input representation $\hat{y}_i^k$ (Eq. \ref{['eq:bucket_dynamics']}), it can be passed directly to downstream synapses during training, without spike decoding. (f) As in sigma-delta coding, whenever the mismatch $z_j = y_j - \hat{y}_j$ exceeds a threshold, a correction is triggered. (g) This results in a spike output $s_j$, which is added back into the estimate. The red arrows indicate the error pathway during training: the error flows from the signal estimate back to the neuron signal and further to the inputs and parameters. Notably, the error bypasses the spikes, eliminating the need for surrogate gradients.
• Figure 4: Transition of neuron states over time in recurrent architectures (a) versus SpikingGamma (b). Recurrent models rely on internal states (e.g., membrane potential $u$) and therefore require BPTT for training, while SpikingGamma has access to the entire history via buckets for each timestep and so does not need BPTT.
• Figure 5: Comparison between SpikingGamma (top) and BPTT with SGs (bottom). Left shows the neuron dynamics after converging, and right the bucket-weight evolution during training.