Ising models for networks of real neurons

Ising models with pairwise interactions are the least structured, or maximum–entropy, probability distributions that exactly reproduce measured pairwise correlations between spins. Here we use this equivalence to construct Ising models that describe the correlated spiking activity of populations of 40 neurons in the retina, and show that pairwise interactions account for observed higher–order correlations. By first finding a representative ensemble for observed networks we can create synthetic networks of 120 neurons, and find that with increasing size the networks operate closer to a critical point and start exhibiting collective behaviors reminiscent of spin glasses. Physicists have long explored analogies between the statistical mechanics of Ising models and the functional dynamics of neural networks [1, 2]. Recently it has been suggested that this analogy can be turned into a precise mapping [3]: In small windows of time, a single neuron i either does (σ i = +1) or does not (σ i = −1) generate an action potential or " spike " [4]; if we measure the mean probability of spiking for each cell (σ i) and the correlations between pairs of cells (C ij = σ i σ j − σ i σ j), then the maximum entropy model consistent with these data is exactly the Ising model P ({σ i }) = 1 Z exp   N i=1 h i σ i + 1 2 N i =j J ij σ i σ j   , (1) where the magnetic fields {h i } and the exchange couplings {J ij } have to be set to reproduce the measured values of {σ i } and {C ij }. We recall that maximum entropy models are the least structured models consistent with known expectation values [5, 6]; thus the Ising model is the minimal model forced upon us by measurements of mean spike probabilities and pairwise correlations. The surprising result of Ref [3] is that the Ising model provides a very accurate description of the combinatorial patterns of spiking and silence in retinal ganglion cells as they respond to natural movies, despite the fact that the model explicitly discards all higher order interactions among multiple cells. This detailed comparison of theory and experiment was done for groups of N ∼ 10 neu-rons, which are small enough that the full distribution P ({σ i }) can be sampled experimentally. Here we extend these results to N = 40, and …