lower probability (e.g., those in which more cells spike) are more numerous (because there are more ways to arrange $K$ spikes among $N$ cells as $K$ increases from very low values). But the usual result is that this tradeoff—which is precisely the tradeoff between energy and entropy in thermodynamics—selects “typical” states that all have roughly the same probability. The statement that $S(E) =E$, as suggested in Fig \[fig3\], is the statement that states which are ten times less probable are [*exactly*]{} ten times more numerous, and so there is no typical value of the probability. This balancing of probability and numerosity extends over a range of at least $0.2$ along the $E/N$ axis, corresponding to a factor of $\sim e^{(0.2)N}\sim 10^{10}$ in probability for networks of $N=120$ neurons.

Heat capacities
===============

The divergence of the specific heat is one of the classical signs of criticality in equilibrium thermodynamic systems. Although the neurons obviously are not an equilibrium system, the model probability distribution in Eqs (\[PH\], \[HKpair\]) is mathematically identical to the Boltzmann distribution for a system in equilibrium at temperature $k_BT =1$. Thus we can take this model seriously as a statistical mechanics problem, and compute the specific heat in 