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algebraic formula, but the model requires that synapses be slow (i.e., that they have
long time constants) (Ermentrout, 1994). Ermentrout does not suggest that the
reduced model will necessarily be a replacement for the full model, but rather that
it can be used to find parameter ranges of interest before simulating the full system
(Ermentrout, 1994). However, the slow synapse assumption limits the usefulness of
this technique.
In 2003 a new technique was proposed by Shriki, Hansel, and Sompolinsky to
deal with fast synapses as well (Shriki et al., 2003). The firing rate of a network
of conductance-based cells can be characterized using far fewer equations if the f-I
curve (the firing rate versus stimulus strength curve) is approximately linear, and if
the network is in an asynchronous state (Shriki et al., 2003). There is no assumption
on the synaptic time constants, and thus this technique has opened up new brain
regions to model reduction, specifically cortical networks, which they studied. Yet,
in order to achieve this reduction, the extra assumptions of the f-I curve and the
network state prohibit their technique from general use. One can debate that the f-I
curve assumption is not too strict, for it is possible that the presence of certain ionic
currents can have a linearizing effect (Morel and Levy, 2009). However, restricting the
output to be only asynchronous is strong, because it effectively limits the behaviors
that can possibly be recovered.
Most recently, Stefanescu and Jirsa have developed a reduction technique based
on mode decomposition which gives good qualitative reproductions of network dy-