Higher-order interactions renormalize the thermodynamic signature of neural population codes

Abstract

Maximum-entropy models of neural recordings can be treated as Boltzmann distributions at a fictitious temperature $T$, with the fitted model at $T=1$. Their specific-heat peak grows with population size and approaches that temperature, which has been read as a sign of criticality. The case rests on pairwise models, yet retinal codes require higher-order interactions. The thermodynamics of a model with such interactions fitted cell by cell has not been computed. Here I compute it exactly for pairwise and third-order models of the same populations of up to $20$ retinal ganglion cells, in salamander and marmoset. The triplet terms, required by the held-out likelihood, shift the peak by more than the finite-sample bias, with opposite sign in the two retinas. The peak is lower and grows more slowly in salamander, higher and grows faster in marmoset, and lies closer to $T=1$ in both. An exact decomposition traces most of the shift to the energy variance across population spike counts. A constraint on the spike-count distribution alone reproduces all of it in salamander and over half in marmoset. A second recording of each species, re-binning, and a common-input surrogate show that the sign is set by the spike-count structure of a recording at its temporal resolution, not by species. The thermodynamic signature of a neural code is therefore set not by the data alone but by the interaction order of the model used to read it.