Fig 1.
This is a summary of our method for approximating the minimum information bipartition (MIB) of large systems, which is necessary for calculating integrated information, without a brute-force search.
We assume that the MIB of a brain network is not random, but instead is delineated by the network’s functional architecture. To identify the functional architecture of brain networks from time-series data, we draw on work from functional brain connectomics, in which “functional brain networks” are often constructed by taking correlation matrices of neural time-series data, thresholding those correlation matrices to produce weighted adjacency matrices, and applying community detection algorithms like spectral clustering to those adjacency matrices. This procedure partitions the brain into functionally distinct sub-networks [35]. Our hypothesis is that the MIB of a brain network should be delineated by the functional boundaries identified through graph clustering. Out of the range of approaches to clustering brain networks, we chose spectral clustering because it is particularly well-suited for normalized partitioning problems, in which (just as with the search for the MIB), the goal is to find sub-networks of roughly equal size (i.e., to avoid partitioning a network into one node isolated from the rest of the network). see Methods for details on how spectral clustering was used to approximate the minimum information bipartition of brain networks.
Fig 2.
We first tested our spectral clustering-based approach in small simulations.
A This is an example of a small brain-like network we generated using a novel algorithm based on Hebbian plasticity. This algorithm produces networks that are loosely brain-like, in that they are modular, show rich cross-module connectivity, and display a log-normal degree distribution with long right tails. We used this algorithm to generate 50 14- and 16-node networks. see Methods for more details on network generation. B This is a sample of oscillatory data generated from the network in A. We generated these data using a stochastic coupled Rössler oscillator model. In the Rössler oscillator model, each node stochastically oscillates according to its own intrinsic frequency, and dynamically synchronizes with other nodes it is connected to. The resulting data are multivariate normal (S2 Fig), allowing for the fast computation of integrated information. C As a first test of our spectral clustering-based approach to identifying the MIB from time-series data, we subtracted ΦG (normalized) across the ground-truth MIB, identified through a brute-force search through all possible bipartitions, from ΦG (normalized) across the partitions identified through spectral clustering. In this test, a perfect match between values would yield a difference of 0 bits. Red squares indicate the mean across 50 networks, and the blue bars indicate standard error of the mean. D As a second test of our spectral clustering-based approach, we computed the Rand Index [45], which is a common measure of partition similarity, between the spectral partitions and the ground-truth MIBs of these networks. A Rand Index of 1 indicates a perfect match between partitions, and a Rand Index of 0 indicates maximum dissimilarity between partitions. Red squares indicate the mean across 50 networks, and the blue bars indicate standard error of the mean. These results show that spectral clustering finds the MIB of small networks of coupled oscillators. We found similar results using the same networks but with autoregressive network dynamics (S7 Fig).
Fig 3.
A Having shown that spectral clustering can find the MIB in time-series data from small networks, we next asked whether it could find the MIB of large simulated networks. While large networks cannot be exhaustively searched for their MIB, the MIB can be forced onto them by cutting them in half. We generated 40 such cut networks for each network size. Network sizes ranged from 50 nodes to 300 nodes. B Here, we show ΦG (normalized) across the ground-truth cut subtracted from ΦG (normalized) across the partition identified through spectral clustering. Red squares indicate the mean across 40 networks, the absolute value of which never exceeded 0.001 bits (normalized), and the blue bars indicate standard error of the mean. C Here, we show the mean and standard error of the Rand Index between the ground-truth cut and the spectral clustering-based partition of the correlation matrix estimated from each network. The Rand Index between the spectral partition and the ground-truth cut was greater than 0.8 (indicating high similarity) for the majority of networks of all network sizes, except for the 200- to 300-node networks. Despite this dip in Rand Index, spectral clustering still found partitions across which ΦG (normalized) was extremely close to ΦG (normalized) across the ground-truth cut in the 300-node networks (A), which suggests that in these networks, there was sometimes several possible partitions that minimized normalized integrated information. We found that spectral clustering performed even better (nearly perfectly) using the same networks but with autoregressive network dynamics (S8 Fig).
Fig 4.
A We split the available ECoG electrodes in two macaque monkeys into overlapping sets of 14 electrodes. The ground-truth MIB of 14 electrodes can be identified through a brute-force search, and compared to the spectral partition estimated from the correlation matrix of data from those electrodes. Here, we subtracted ΦG (normalized) across the ground-truth MIB from ΦG (normalized) across the spectral partition. There was a difference of 0 bits for 67/112 (mean difference = 0.0002 bits) datasets from George’s brain, and a difference of 0 bits in 46/112 (mean difference = 0.0001 bits) datasets from Chibi’s brain. Red squares indicate the mean difference in ΦG (normalized) across all datasets from one brain, and blue bars indicate standard error of the mean. B Spectral clustering found the exact MIB for the same 67/112 datasets in George’s brain (mean Rand Index = 0.87) and 46/112 datasets in Chibi’s brain (mean Rand Index = 0.79). C We used our spectral clustering approach to estimate the MIB of Chibi’s entire left cortex, and found that it split posterior sensory areas from anterior association areas. Electrodes are colored according to the community in which they are clustered; the electrodes that were excluded from the analysis because they displayed consistent artifacts are colored grey. D ΦG (normalized) across the spectral partition of Chibi’s left cortex (solid green line) was lower than it was across 4/6 partitions identified by the Replica Exchange Markov Chain Monte Carlo (REMCMC) method (yellow dashed lines) [21]. The other 2/6 partitions yielded values of normalized integrated information that were very slightly lower (0.0002 bits) than the value across the spectral clustering-based partition, and were dissimilar both to each other (Rand Index = 0.5) and to the spectral partition (Rand Indices = 0.5, 0.55), suggesting that there were several local minima of normalized integrated information in Chibi’s brain. We ran the REMCMC algorithm for 10 days. E Our estimate of the MIB of George’s left cortex using spectral clustering also (largely) split posterior sensory areas from anterior association areas. F ΦG (normalized) across the spectral partition of George’s left cortex was lower than it was across all bipartitions identified by the REMCMC method. Note the difference in scale on the x-axes of D and F; it is unclear why this scale should differ between the two brains.
Fig 5.
The method presented in this paper for quickly identifying a network’s MIB using spectral clustering makes it possible to quickly measure integrated information in large brain networks.
A straightforward first-pass at an application for our method is to evaluate the long-held and untested assumptions that the “global efficiency” of a network reflects its capacity for information integration and that the modularity of a network underpins the segregation of information. A Following the procedure introduced by Watts and Strogatz [53], we systematically increased the global efficiency of our networks by increasing their rewiring probability p. Following Watts and Strogatz [53], we varied p on a log-scale between 0.001 and 0.1; to explore the full parameter space, we also linearly varied p between 0.1 and 1. For each value of p, we generated 50 100-node networks, and generated time-series data for each of those networks using the stochastic Rössler oscillator model. We then used our spectral clustering-based technique to measure geometric integrated information in these networks. B As expected [49], increasing p increased the global efficiency of the networks. Here, each dot corresponds to the global efficiency of one network of coupled Rössler oscillators with that particular value of p. The green line passes through the mean across networks. C Increasing p also systematically decreased the modularity Q of the networks. D A higher probability p of forming long-distance network connections, which increases global efficiency, led to higher integrated information (non-normalized). E There was a strong negative correlation between the networks’ structural modularity and how much information they integrate, in bits (Spearman’s ρ = -0.90, p < 10−324). Note that the gap around Q = 0.65 occurs at the transition from the log variance of p to the linear variance of p (C). F There was a strong positive correlation between the networks’ global efficiency and how much information they integrate, in bits (Spearman’s ρ = 0.91, p < 10−324). Note that the gap around E = 0.32 occurs at the transition from the log variance of p to the linear variance of p (B). These results support the hypothesis that network modularity supports the segregation of information, while global efficiency supports the integration of information.
Fig 6.
Average run time across for the three algorithms as a function of network size.
Error bars indicate standard error of the mean across five networks of coupled oscillators of a given size. For very small brain-like networks (10-14 nodes), our spectral clustering-based approach is slower than either the Queyranne algorithm or a brute-force search for the MIB. This is because our algorithm searches through a fixed number of candidate graph cuts (see Methods). But, this feature is also the algorithm’s strength: because our algorithm searches through the same number of candidate partitions for large systems as it does for small systems, its computation time scales much less steeply than that of the other two algorithms. If our algorithm were to search through more partitions (for e.g. by iterating through more threshold values of the correlation matrices—see Methods), then it would be slower, but its run time would still scale far less steeply than the other two algorithms, because the number of candidate partitions would remain fixed.