Facilitating online learning in spiking neural networks (SNNs) is a key step in developing event-based models that can adapt to changing environments and learn from continuous data streams in real-time. Although forward-mode differentiation enables online learning, its computational requirements restrict scalability. This is typically addressed through approximations that limit learning in deep models. In this study, we propose Online Training with Postsynaptic Estimates (OTPE) for training feed-forward SNNs, which approximates Real-Time Recurrent Learning (RTRL) by incorporating temporal dynamics not captured by current approximations, such as Online Training Through Time (OTTT) and Online Spatio-Temporal Learning (OSTL). We show improved scaling for multi-layer networks using a novel approximation of temporal effects on the subsequent layer's activity. This approximation incurs minimal overhead in the time and space complexity compared to similar algorithms, and the calculation of temporal effects remains local to each layer. We characterize the learning performance of our proposed algorithms on multiple SNN model configurations for rate-based and time-based encoding. OTPE exhibits the highest directional alignment to exact gradients, calculated with backpropagation through time (BPTT), in deep networks and, on time-based encoding, outperforms other approximate methods. We also observe sizeable gains in average performance over similar algorithms in offline training of Spiking Heidelberg Digits with equivalent hyper-parameters (OTTT/OSTL - 70.5%; OTPE - 75.2%; BPTT - 78.1%).

Biological neural networks are equipped with an inherent capability to continuously adapt through online learning. This aspect remains in stark contrast to learning with error backpropagation through time (BPTT) applied to recurrent neural networks (RNNs), or recently to biologically-inspired spiking neural networks (SNNs). BPTT involves offline computation of the gradients due to the requirement to unroll the network through time. Online learning has recently regained the attention of the research community, focusing either on approaches that approximate BPTT or on biologically-plausible schemes applied to SNNs. Here we present an alternative perspective that is based on a clear separation of spatial and temporal gradient components. Combined with insights from biology, we derive from first principles a novel online learning algorithm for deep SNNs, called online spatio-temporal learning (OSTL). For shallow networks, OSTL is gradient-equivalent to BPTT enabling for the first time online training of SNNs with BPTT-equivalent gradients. In addition, the proposed formulation unveils a class of SNN architectures trainable online at low time complexity. Moreover, we extend OSTL to a generic form, applicable to a wide range of network architectures, including networks comprising long short-term memory (LSTM) and gated recurrent units (GRU). We demonstrate the operation of our algorithm on various tasks from language modelling to speech recognition and obtain results on par with the BPTT baselines. The proposed algorithm provides a framework for developing succinct and efficient online training approaches for SNNs and in general deep RNNs.