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TR96-025 | 22nd March 1996 00:00
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#### The Computational Power of Spiking Neurons Depends on the Shape of the Postsynaptic Potentials

**Abstract:**
Recently one has started to investigate the

computational power of spiking neurons (also called ``integrate and

fire neurons''). These are neuron models that are substantially

more realistic from the biological point of view than the

ones which are traditionally employed in artificial 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 specific neurons ``fire'' (and thus send a ``spike''

to other neurons).

These preceding results have motivated the question which details of the

firing 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 artificial neural nets, such

as for example pulse stream VLSI.

The firing mechanism of spiking neurons is defined in terms of their

postsynaptic potentials or ``response functions'', which describe the

change in their electric membrane potential as a result of the firing 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.