  Recent experimental work has suggested that the neural firing rate can be
interpreted as a fractional derivative, at least when signal variation induces
neural adaptation. Here, we show that the actual neural spike-train itself can
be considered as the fractional derivative, provided that the neural signal is
approximated by a sum of power-law kernels. A simple standard thresholding
spiking neuron suffices to carry out such an approximation, given a suitable
refractory response. Empirically, we find that the online approximation of
signals with a sum of power-law kernels is beneficial for encoding signals with
slowly varying components, like long-memory self-similar signals. For such
signals, the online power-law kernel approximation typically required less than
half the number of spikes for similar SNR as compared to sums of similar but
exponentially decaying kernels. As power-law kernels can be accurately
approximated using sums or cascades of weighted exponentials, we demonstrate
that the corresponding decoding of spike-trains by a receiving neuron allows
for natural and transparent temporal signal filtering by tuning the weights of
the decoding kernel.
