4. A Novel Reconstruction Framework in Shift-Invariant Spaces for Signals Encoded with Integrate-and-Fire Neurons

A shift-invariant space (SIS) is more general than a bandlimited space and can be used to model signals with more realistic characteristics. The state-of-the-art reconstruction algorithm for IF neurons with inputs belonging to a SIS is based on an iterative computation of the stimulus. This chapter introduces new theoretical results that form the basis for a new non-iterative reconstruction algorithm, more accurate than the state-of-the-art method. Both methods above perform reconstruction directly from the nonuniform samples generated by the neuron, where the sampling times are different for every stimulus. Based on the results above and on the new IF encoding framework introduced in Chap. 3, two new indirect methods are introduced for reconstructing the input u of a IF neuron belonging to a SIS via the auxiliary function \( \bar{\psi }'\). The methods reconstruct function \( \bar{\psi }'\) iteratively and non-iteratively, respectively. The direct and indirect algorithms in this chapter have been evaluated through numerical simulations. The results show that the indirect methods have a similar level of accuracy but are significantly faster than the direct ones.