Filtered-Trace Online Variational Training for Probabilistic Spiking Neural Networks

Probabilistic spiking neural networks (SNNs) model spike trains as temporal point processes and provide a principled framework for learning with latent spikes. Recent differentiable point-process methods enable path-wise variational learning, but their training still relies on full-sequence backpropagation through time (BPTT), leading to memory costs that grow with the temporal horizon. In this paper, we develop an online variational training framework for probabilistic SNNs based on discrete-time spike response model dynamics. By representing synaptic history with finite-dimensional Markovian traces, our method updates both generative and variational parameters without storing the full temporal computation graph. To handle future-dependent credit assignment from recurrent spike histories, we introduce a horizon-$R$ family of online estimators. Theoretically, we show that the truncation-induced bias decays exponentially with $R$ under the SRM kernel-decay condition. Synthetic experiments validate the predicted horizon-dependent behavior and bias decay, while N-MNIST experiments show competitive classification accuracy and avoid the sequence-length-dependent memory growth of BPTT.