  Recently, it has been shown that spiking neural networks (SNNs) can be
trained efficiently, in a supervised manner, using backpropagation through
time. Indeed, the most commonly used spiking neuron model, the leaky
integrate-and-fire neuron, obeys a differential equation which can be
approximated using discrete time steps, leading to a recurrent relation for the
potential. The firing threshold causes optimization issues, but they can be
overcome using a surrogate gradient. Here, we extend previous approaches in two
ways. Firstly, we show that the approach can be used to train convolutional
layers. Convolutions can be done in space, time (which simulates conduction
delays), or both. Secondly, we include fast horizontal connections \`a la
Den\`eve: when a neuron N fires, we subtract to the potentials of all the
neurons with the same receptive the dot product between their weight vectors
and the one of neuron N. As Den\`eve et al. showed, this is useful to represent
a dynamic multidimensional analog signal in a population of spiking neurons.
Here we demonstrate that, in addition, such connections also allow implementing
a multidimensional send-on-delta coding scheme. We validate our approach on one
speech classification benchmarks: the Google speech command dataset. We managed
to reach nearly state-of-the-art accuracy (94%) while maintaining low firing
rates (about 5Hz). Our code is based on PyTorch and is available in open source
at http://github.com/romainzimmer/s2net
