The Computational Power of Spiking Neurons Depends on the Shape of the Postsynaptic Potentials

Recently one has started to investigate the computational power of spiking neurons (also called \integrate and re neurons"). These are neuron models that are substantially more realistic from the biological point of view than the ones which are traditionally employed in arti cial neural nets. It has turned out that the computational power of networks of spiking neurons is quite large. In particular they have the ability to communicate and manipulate analog variables in spatio-temporal coding, i.e. encoded in the time points when speci c neurons \ re" (and thus send a \spike" to other neurons). These preceding results have motivated the question which details of the ring mechanism of spiking neurons are essential for their computational power, and which details are \accidental" aspects of their realization in biological \wetware". Obviously this question becomes important if one wants to capture some of the advantages of computing and learning with spatio-temporal coding in a new generation of arti cial neural nets, such as for example pulse stream VLSI. The ring mechanism of spiking neurons is de ned in terms of their postsynaptic potentials or \response functions", which describe the change in their electric membrane potential as a result of the ring of another neuron. We consider in this article the case where the response functions of spiking neurons are assumed to be of the mathematically most elementary type: they are assumed to be step-functions (i.e. piecewise constant functions). This happens to be the functional form which has so far been adapted most frequently in pulse stream VLSI as the form of potential changes (\pulses") that mimic the role of postsynaptic potentials in biological neural systems. We prove the rather surprising result that in models without noise the computational power of networks of spiking neurons with arbitrary piecewise constant response functions is strictly weaker than that of networks where the response functions of neurons also contain short segments where they increase respectively decrease in a linear fashion (which is in fact biologically more realistic). More precisely we show for example that an addition of analog numbers is impossible for a network of spiking neurons with piecewise constant response functions (with any bounded number of computation steps, i.e. spikes), whereas addition of analog numbers is easy if the response functions have linearly increasing segments.