  Several efforts are currently underway to decipher the connectome or parts
thereof in a variety of organisms. Ascertaining the detailed physiological
properties of all the neurons in these connectomes, however, is out of the
scope of such projects. It is therefore unclear to what extent knowledge of the
connectome alone will advance a mechanistic understanding of computation
occurring in these neural circuits, especially when the high-level function of
the said circuit is unknown. We consider, here, the question of how the wiring
diagram of neurons imposes constraints on what neural circuits can compute,
when we cannot assume detailed information on the physiological response
properties of the neurons. We call such constraints -- that arise by virtue of
the connectome -- connectomic constraints on computation. For feedforward
networks equipped with neurons that obey a deterministic spiking neuron model
which satisfies a small number of properties, we ask if just by knowing the
architecture of a network, we can rule out computations that it could be doing,
no matter what response properties each of its neurons may have. We show
results of this form, for certain classes of network architectures. On the
other hand, we also prove that with the limited set of properties assumed for
our model neurons, there are fundamental limits to the constraints imposed by
network structure. Thus, our theory suggests that while connectomic constraints
might restrict the computational ability of certain classes of network
architectures, we may require more elaborate information on the properties of
neurons in the network, before we can discern such results for other classes of
networks.
