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NeuroCOLT
Technical Report NC-TR-96-014
Analog Computations
on Networks of Spiking Neurons
Wolfgang
Maass
Institute for Theoretical Computer Science
Technische Universitaet Graz
Austria
Abstract
We characterize the class of functions with real-valued input and
output which can be computed by networks of spiking neurons with piecewise
linear response- and threshold-functions and unlimited timing precision.
We show that this class coincides with the class of functions computable
by recurrent analog neural nets with piecewise linear activation functions,
and with the class of functions computable on a certain type of random
access machine (N-RAM) which we introduce in this article. This result
is proven via constructive real-time simulations. Hence it provides
in particular a convenient method for constructing networks of spiking
neurons that compute a given real-valued function $f$: it now suffices
to write a program for computing $f$ on an N-RAM; that program can
be ``automatically'' transformed into an equivalent network of spiking
neurons (by our simulation result). Finally, one learns from
the results of this paper that certain very simple piecewise linear
response- and threshold-functions for spiking neurons are {\it universal},
in the sense that neurons with these particular response- and threshold-functions
can simulate networks of spiking neurons with arbitrary piecewise
linear response- and threshold-functions. The results of this paper
also show that certain very simple piecewise linear activation functions
are in a corresponding sense universal for recurrent analog neural
nets.
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