We propose a spiking neural network model that encodes information in the relative timing of individual neuron spikes and performs classification using the first output neuron to spike. This temporal coding scheme allows the supervised training of the network with backpropagation, using locally exact derivatives of the postsynaptic with respect to presynaptic spike times. The network uses a biologically-inspired alpha synaptic transfer function and trainable synchronisation pulses as temporal references. We successfully train the network on the MNIST dataset encoded in time. Our spiking neural network outperforms comparable spiking models and achieves similar accuracy to a fully connected conventional network. During training, our network displays a speed-accuracy trade-off, with either slow and highly-accurate or very fast but less accurate classification. The results demonstrate the computational power of spiking networks with biological characteristics that encode information in the timing of individual neurons. Our code is publicly available.