Fig 1.
Cross-embedding analysis of complex network dynamics based on state-space reconstruction.
(A) Example of randomized delay-coordinate reconstruction assuming we observe dynamics determined with three-dimensional differential equation. (1) Global state is embedded using (2) temporal sequence of single observable of system. Attractor topology in global state space is fully recovered in (3) delay-coordinate of single observable. This relationship is maintained after (4) random linear projection of delay-coordinate space. Thus, there exists unique one-to-one map from (4) to (1), given sufficient number of coordinates. (B) The reconstruction needs sufficient dimensionality. Whether we have embedding or not depends on the relation between the reconstruction dimension (d) and the underlying attractor dimension (dA). For example, a two-dimensional delay-coordinate space generically embeds one-dimensional attractor (e.g., a limit cycle, top row), except for a finite number of points. Higher dimensionality is required to embed more complex dynamics (bottom row). Generally, a dimensionality larger than that of an attractor (d > dA) is necessary and sufficient for the embedding—to distinguish almost all states of the system [16] although perfect embedding without no self-intersection requires d > 2dA [15]; the current cross-embedding protocol utilizes the former property. The Figure explores 1, 2 or 3 embedding dimensions after randomly projecting original delay-coordinates. Red regions indicate self-intersection with unignorable volume, reflecting incomplete embedding. (C) Unidirectional interaction between two modeled systems (directionally coupled Rössler systems, see Materials and Methods for equations). Downstream has information for whole system (light green) whereas upstream only has information on itself (light magenta). (D) Example of observed upstream and downstream signals. (E) The cross-embedding based on the random-coordinate state-space reconstructions. Mx and My respectively represent the downstream (x) and upstream (y) attractors based on single observed variable from each system. Colored arrows depict how nearby states in one attractor (disk: 20 nearest-neighbor states) are mapped onto states (dots) in other attractor (green: from x to y and magenta: from y to x). (F) Relationship between coordinate dimensions (number of randomized coordinates used for attractor reconstruction) and embeddedness (how accurately one node’s history can predict another node’s state). The circled dots represent complexity estimates, which were determined as minimum embedding dimension that provided ≥ 95% of optimal embeddedness measure (correlation between actual and forecast signals. See Materials and Methods).
Fig 2.
Cross-embedding with random coordinates dissociates interaction-relevant complexity from timescale or noise variations.
(A) Complexity estimates based on embedding y with x in the system used in Fig 1C–1F, where timescale of signal was varied. The estimates with each method were shown as relative values to those at timescale = 0.5. The results were similar in embedding x with y. (B) The results where the observation noise level was varied. The noise level corresponds to the standard deviation of the (zero-mean) normal distribution, from which noise were generated. The results are shown as relative values to those at noise level = 0. Although both the standard- and random-coordinate embedding methods (circle symbols) are invariant to noise level in terms of the relative values, the random-coordinate method estimates the true complexity more accurately in absolute values (see panel C). (C) Coordinate randomization avoids overestimation of complexity. The markers and error bars indicate the averages ±s.e.m. across 10 trials with different initial states of dynamics. Here, the pre-normalized complexity estimates using regular delay coordinates, (xt, xt-τ …) (open symbols), are compared with results in Fig 1C (y embedded by x) which used random delay coordinates (closed symbols); other conventions follows those in Fig 1F. The theoretically predicted embedding dimension of x (shaded bar) were accurately estimated using random coordinates, while the previous method using standard delay-coordinates tends to overestimate it, even exceeding the total number of nodes (dotted line) in the simulated system, which upper bounds the theoretically possible attractor dimensions by construction.
Fig 3.
Cross-embedding analysis reveals large-scale cortical interaction, which can be missed by correlation-based statistics.
(A) ECoG electrode loci in a representative subject. (B) Illustration of cortical areas covered by the present recording system. (C) Example of ECoG signals. Gray traces represent the signal from the all electrodes superimposed; green and magenta traces show the signals in an example electrode pair (electrodes 41 and 118). (D) Cross-embedding analysis based on the state-space reconstructions for the example electrode pair, corresponding to Fig 1E and 1F. (E) Relationship between coordinate dimensions and embeddedness in the example electrode pair, corresponding to Fig 1F. (F) Optimal embeddedness values as functions of data length (number of data points) used in the analysis. The line colors corresponds to those in panel E. Note that the embeddedness values improve as data length increases, which is a hallmark of causally coupled deterministic dynamics [21]. (G-K) Cross-embedding results. (G) The complexity of area-to-area interaction are shown in the matrix formats, which show the results of interaction from areas specifying columns to areas specifying rows. (H) Scatter plot of complexity in awake vs. anesthetized conditions; each dot represents a complexity value for each “effect” electrode, averaged across “cause” electrodes. Dashed red line shows equality. (I) (Top) The directionality of area-to-area interaction are shown in the matrix formats. (Bottom) The directionality among individual electrodes in a single subject. (J) Scatter plot of directionality in awake vs. anesthetized conditions; each dot represents the values for each electrode pair. (K) The same as panels I, except for that the directionality values were computed based on the cross-embedding analysis with limiting the embedding dimension to 1. (L-N) Correlation analysis of snapshot electrode signals. (L) (Top) Average correlation among areas. (Bottom) Correlation among individual electrodes in a single subject. (M) Scatter plots of complexity, directionality, and correlation coefficient in awake vs. anesthetized conditions; each dot represents the values for each electrode pair. (N) Comparison between cross-embedding-based directionality and correlation. Dashed red line shows zero value of correlation. Panels G-L, N and O show the pooled results across the four subjects. LVC: lower visual cortex; HVC: higher visual cortices; TC: temporal cortex; PC: parietal cortex; MSC motor and somatosensory cortex; PMC: premotor cortex; lPFC: lateral prefrontal cortex; mPFC: medial prefrontal cortex
Fig 4.
Dynamical complexity and directed interaction within awake and anesthetized brain.
Unlike anesthetized brain, awake brain exhibited clear differentiation between frontoparietal and occipital areas. This figure provides results in a single representative experiment, where subject state was changed from awake (A and B), eyes-closed condition to anesthesia by injection of propofol (C and D). Loci of bubbles correspond to electrode positions; the colors of bubbles indicate (A and C) the complexity or (B and D) the directionality averaged across forecast electrodes. (The absolute values of complexity/directionality are also indicated by sizes of bubbles, for visibility.) (E) Area-wise complexity and directionality. Thin lines correspond to the individual electrodes, and thick lines indicate averages and standard errors across electrodes within each area.
Fig 5.
Universality of complex dynamics across different awake conditions.
(A) Conscious and unconscious states are differentiated in the space of complexity averaged across all areas inside and outside the visual cortices. Markers represent individual experiments. Results from four monkeys have been superimposed. Note that the data from Reaching, Awake-Eyes-Open, and Awake-Eyes-Closed conditions are overlapped. (B) In conscious states, the complexity and directionality simultaneously increase in the outside visual cortex relative to the inside visual cortex. The convention follows that of panel A. (C) Complexity and directionality are correlated even in anesthetized condition. Each marker shows averaged complexity and directionality values (as in Fig 4) in each single electrode. (D) Anesthetization decreases the bottom-up interaction. The relative embeddedness reflects the dynamical coupling after the baseline change correlation is subtracted (Materials and Methods). The black and gray bars indicate the strength of bottom-up (from visual areas to other areas) and top-down (form other areas to visual areas), respectively. In panels A–C, the ellipses indicate the data covariance within individual conditions; the darker gray ellipse indicates the covariance for the data pooling the ketamine-medetomidine- and propofol-induced anesthesia. The dotted line indicates equality of average complexity. VC: visual cortex.
Fig 6.
Increased complexity at the cortical downstream differentiates conscious form unconscious brain states.
(A) Summary of task conditions. (B) Cortical hierarchy defined by complexity. Areas aligned by complexity and relative directed interaction between them have been visualized. Arrows indicate relative information flows from one area to another, which were quantified by average directionality for interactions between two areas (only strong information flows exceeding 0.01 are shown). Complexity in visual areas was smaller than that in other areas during awake conditions (reaching: P<0.004; awake-eyes-open and awake-eyes-closed: P<0.00007; sign test, matched samples), and this inter-area difference in complexity was reduced by anesthesia (P<3×10−8; Wilcoxon rank sum test, independent samples). (C) Schematic summary of the main findings.
Fig 7.
Robustness to variations in timescale.
The main result is not affected by lengthening the unit delay although forecasting accuracy (embeddedness) was generally degraded by using suboptimal unit delay size. (A) The replications of the results shown in Fig 5A, respectively, where a longer unit delay (100 ms) was used for the state space reconstruction. (B) Optimal embeddedness in the all electrode pairs decreased by 0.21 times with the longer unit delay (p<10−5, sign test, paired sample across all electrode pairs). The figure shows the embeddedness averaged inside (dashed lines) or outside (solid lines) visual cortex. (C) Complexity (averaged inside or outside visual cortex) showed only small or not significant dependency to the unit delay variation (inside visual cortex: changed by 0.90 times, p <10−4; outside visual cortex: by 0.80 times, p = 0.5; sign test, paired sample, across all experimental sessions). This robustness validates that complexity reflects dimensions of dynamical attractor rather than variations in timescale of signals. In panels D-F, the color code follows that of panel B.
Fig 8.
Possible mechanisms of changing frontoparietal dynamics in the conscious brain state.
(A) Simplified architecture of nervous system under unconscious (anesthetized) state. In the unconscious state, three clusters (frontoparietal, visual, and subcortical systems) have no strong nonlinear interaction. (B) Model 1: the frontoparietal complexity is increased by the enhanced bottom-up interaction. (C) Model 2: the frontoparietal complexity is increased by subcortical input. (D) Model 3: the frontoparietal complexity is increased by change in the connectivity inside of the frontoparietal system itself. Only Model 1 accounts for the increase in the directionality from the visual to the frontoparietal cortex, which was observed in the experimental data. (From left to right columns) Network architecture, mechanistic connectivity matrices, directionality matrices derived based on cross-embedding analysis with simulated dynamics, average complexity in the frontoparietal cortex, the averaged directionality from the visual to the frontoparietal cortex. V: visual cortex. FP: frontoparietal cortex.