Optimal Supervised Learning in Spiking Neural Networks for Precise Temporal Encoding

Precise spike timing as a means to encode information in neural networks is biologically supported, and is advantageous over frequency-based codes by processing input features on a much shorter time-scale. For these reasons, much recent attention has been focused on the development of supervised learning rules for spiking neural networks that utilise a temporal coding scheme. However, despite significant progress in this area, there still lack rules that are theoretically justified, and yet can be considered biologically relevant. Here we examine the general conditions under which optimal synaptic plasticity takes place to support the supervised learning of a precise temporal code. As part of our analysis we introduce two analytically derived learning rules, one of which relies on an instantaneous error signal to optimise synaptic weights in a network (INST rule), and the other one relying on a filtered error signal to minimise the variance of synaptic weight modifications (FILT rule). We test the optimality of the solutions provided by each rule with respect to their temporal encoding precision, and then measure the maximum number of input patterns they can learn to memorise using the precise timings of individual spikes. Our results demonstrate the optimality of the FILT rule in most cases, underpinned by the rule's error-filtering mechanism which provides smooth convergence during learning. We also find the FILT rule to be most efficient at performing input pattern memorisations, and most noticeably when patterns are identified using spikes with sub-millisecond temporal precision. In comparison with existing work, we determine the performance of the FILT rule to be consistent with that of the highly efficient E-learning Chronotron rule, but with the distinct advantage that our FILT rule is also implementable as an online method for increased biological realism