  Most normative models in computational neuroscience describe the task of
learning as the optimisation of a cost function with respect to a set of
parameters. However, learning as optimisation fails to account for a time
varying environment during the learning process; and the resulting point
estimate in parameter space does not account for uncertainty. Here, we frame
learning as filtering, i.e., a principled method for including time and
parameter uncertainty. We derive the filtering-based learning rule for a
spiking neuronal network - the Synaptic Filter - and show its computational and
biological relevance. For the computational relevance, we show that filtering
in combination with Bayesian regression improves performance compared to a
gradient learning rule with optimal learning rate in terms of weight
estimation. Furthermore, the filtering-based rule outperforms gradient-based
rules in the presence of model mismatch, indicating a better generalisation
performance. The dynamics of the mean of the Synaptic Filter is consistent with
the spike-timing dependent plasticity (STDP) while the dynamics of the variance
makes novel predictions regarding spike-timing dependent changes of EPSP
variability. Moreover, the Synaptic Filter explains experimentally observed
negative correlations between homo- and heterosynaptic plasticity.
