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NeuroCOLT
Technical Report NC-TR-96-040
The Computational
Power of Spiking Neurons Depends on the Shape of the Postsynaptic
Potentials
Wolfgang
Maass
Technische Universitaet Graz
Austria
Berthold
Ruf
Technische Universitaet Graz
Austria
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.
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