Pairwise Maximum Entropy Model for Neural Populations

A neural population can be modeled as an Ising spin system: P(s) ∝ exp(Σᵢ hᵢsᵢ + Σᵢⱼ Jᵢⱼsᵢsⱼ), where hᵢ captures firing rates and Jᵢⱼ captures pairwise correlations. This maximum entropy model (Jaynes principle) is exactly as constrained as the data warrants. Real neural populations often sit near a critical point where the heat capacity (variance of energy) peaks — suggesting criticality is neurally relevant.

Neurons N 8 Coupling J 0.80 Bias h -0.30 Temperature T 1.0

Population statistics:
Mean firing rate: —
Mean pairwise corr: —
Multi-information: —
Heat capacity C: —

Ising model:
E(s) = −Σᵢhᵢsᵢ − Σᵢ<ⱼ Jᵢⱼsᵢsⱼ
P(s) = e^(−E/T) / Z

Critical point: heat capacity
peaks when T ≈ J (near T=1)
Real retina data: critical point
matches T ≈ 0.7–1.2 × J