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NeuroCOLT
Technical Report NC-TR-97-020
The
Computational Power of Spiking Neurons Depends on the Shape of the
Postsynaptic Potentials
Wolfgang
Maass, 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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