Ising model for neural data: Model quality and approximate methods for extracting functional connectivity

We study pairwise Ising models for describing the statistics of multineuron spike trains, using data from a simulated cortical network. We explore efficient ways of finding the optimal couplings in these models and examine their statistical properties. To do this, we extract the optimal couplings for subsets of size up to 200 neurons, essentially exactly, using Boltzmann learning. We then study the quality of several approximate methods for finding the couplings by comparing their results with those found from Boltzmann learning. Two of these methods—inversion of the Thouless-Anderson-Palmer equations and an approximation proposed by Sessak and Monasson—are remarkably accurate. Using these approximations for larger subsets of neurons, we find that extracting couplings using data from a subset smaller than the full network tends systematically to overestimate their magnitude. This effect is described qualitatively by infinite-range spin-glass theory for the normal phase. We also show that a globally correlated input to the neurons in the network leads to a small increase in the average coupling. However, the pair-to-pair variation in the couplings is much larger than this and reflects intrinsic properties of the network. Finally, we study the quality of these models by comparing their entropies with that of the data. We find that they perform well for small subsets of the neurons in the network, but the fit quality starts to deteriorate as the subset size grows, signaling the need to include higher-order correlations to describe the statistics of large networks.

Figure 1

(Color online) Checking the reliability of the Boltzmann results. (a) The couplings of a set of $N=200$ found using Boltzmann learning with 20000 learning steps versus those found after 40000 learning steps. (b) Couplings learned from half of the data (200000 time bins) versus those learned from all the data (400000 time bins)

Figure 2

(Color online) Scatter plots comparing the solutions found from different approximations with the Boltzmann learning results for $N=20$. (a) Naive mean-field approximation, (b) independent-pair approximation, (c) low-rate limit, (d) TAP, (e) SM, and (f) a hybrid approximation obtained by averaging TAP and SM.

Figure 3

(Color online) The same as Fig. 2, but for $N=200$ neurons.

Figure 4

(Color online) Quantifying the performance of different approximation by computing $R^2$ [defined in Eq. (14)] and the rms error [defined in Eq. (15)] between the approximate solutions and the result of Boltzmann learning, as a function of $N$. Black (long-dashed line), SM; red (short-dashed line), TAP; blue (full curve), hybrid SM-TAP; green (dashed dotted line), nMF; magenta (dashed double dotted), low-rate approximation; and gray (small dotted), independent-pair approximation.

Figure 5

(Color online) The $N$ dependence of the mean and standard deviation of the solutions found from different approximations and Boltzmann learning. Black (long-dashed line), SM; red (short-dashed line), TAP; green (dashed dotted line), nMF; blue (full curve), hybrid SM-TAP; magenta (dashed double dotted), low-rate approximation; gray (small dotted), independent-pair approximation; and brown (large dots), Boltzmann.

Figure 6

(Color online) The couplings among 20 neurons inferred (a) when no additional neurons are considered (a), and when inferred (b) as a part of a network of size 200. This figure shows that the $N$ dependence of the mean and standard deviation of the couplings in Fig. 5 arises from the scaling of individual couplings.

Figure 7

(Color online) The behavior of the ten largest and ten smallest couplings in the set of 20 neurons, when more and more neurons are added. The blue (full) curves show how the ten largest weights in the population of 20 neurons change their values as we add more and more cells, and the red (dashed) curves show the same thing for the ten smallest couplings. The weights are inferred using Boltzmann learning.

Figure 8

(Color online) Comparing the $N$ dependence of the mean and standard deviation of the solutions found from the first three good approximations [SM (black, long dashed], TAP (red, short dashed), and SM-TAP hybrid (solid blue), with the SK prediction (magenta, dotted). The Boltzmann results for individual samples up to $N=200$ are replotted from Fig. 5 for comparison (brown stars).

Figure 9

(Color online) The $N$ dependence of the mean couplings in the stimulus-driven case as found from Boltzmann learning on single samples (brown squares), SM (black long-dashed lines), TAP (red short-dashed lines), SM-TAP hybrid (blue full curve), and the prediction of Eq. (19) (magenta dotted line). The thick curves are the results for the stimulus-driven case, and the results of the tonic case are plotted using thin curves.

Figure 10

(Color online) The behavior of the KL distances and $G$ versus the set size $N$ [(a) $d_{\text{ind}}$], the KL distance between the independent and the true distributions versus $N$ [(b) $d_{\text{Ising}}$], and the KL distance between the Ising and the true distribution versus $N$ [(c) the goodness of fit $G$] versus $N$. Magenta circles, red crosses, and blue squares represent estimations of these quantities from $T={10}^6$, $T=1.5\times {10}^6$, and $T=1.8\times {10}^6$ samples, respectively, while black stars show the bias-corrected $\left(T\to \infty \right)$ estimates.