  In this paper we study a simple model of a purely excitatory neural network
that, by construction, operates at a critical point. This model allows us to
consider various markers of criticality and illustrate how they should perform
in a finite-size system. By calculating the exact distribution of avalanche
sizes we are able to show that, over a limited range of avalanche sizes which
we precisely identify, the distribution has scale free properties but is not a
power law. This suggests that it would be inappropriate to dismiss a system as
not being critical purely based on an inability to rigorously fit a power law
distribution as has been recently advocated. In assessing whether a system,
especially a finite-size one, is critical it is thus important to consider
other possible markers. We illustrate one of these by showing the divergence of
susceptibility as the critical point of the system is approached. Finally, we
provide evidence that power laws may underlie other observables of the system,
that may be more amenable to robust experimental assessment.
