Existence of information transfer between recording sites

Before analyzing differences in information transfer induced by isoflurane, we tested for the existence of significant information transfer between the recording sites (Table 1). For both animals, TE_SPO was significant in the top-down direction, while the bottom-up direction was significant for animal 1 only. A non-significant information transfer in bottom-up direction for animal 2 may be explained by the dark experimental environment, i.e., the lack of visual input.

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Table 1. Results significance test of TE_SPO estimates in both animals and for both directions of interaction.

https://doi.org/10.1371/journal.pcbi.1005511.t001

Changes in information theoretic measures under anesthesia

Overall, for higher isoflurane levels both locally available information and information transfer were decreased, while information storage in local activity increased.

As the estimation of information theoretic measures from finite length neural recordings poses a considerable challenge we present detailed, converging results from two complementary strategies to deal with this challenge—nearest-neighbor based estimators, and a Bayesian approach to entropy estimation suggested by Nemenman, Shafe, and Bialek (NSB-estimator) [17, 18]. This latter approach required a discretization of the continuous-valued LFP data, but yields principled control of bias, while the first approach allows the estimation of information-theoretic measures directly from continuous data, and thus conserves the information originally present in those data. Statistical testing was performed using a nonparametric permutation ANOVA (pANOVA), and a linear mixed model (LMM). The LMM approach was used in addition to the main pANOVA for the purpose of comparison to older studies using parametric statistics.

Results based on next neighbor-based estimation from continuous data.

For higher isoflurane levels, we found an overall reduction in the locally available information (H), and in the information transfer (TE_SPO). We found an increase in the locally predictable (stored) information (AIS) (Table 2 and Fig 1).

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Fig 1. pANOVA results for nearest-neighbor based estimates of transfer entropy (TE_SPO), active information storage (AIS), and entropy (H).

Left columns show interactions isoflurane level x direction and isoflurane level x recording site for both animals; right columns show main effects isoflurane level. Grey lines in interaction plots indicate TE_SPO from prefrontal cortex (PFC) to primary visual areas (V1), or H and AIS in PFC; black lines indicate TE_SPO from V1 to PFC, or H and AIS in V1. Error bars indicate the standard error of the mean (SEM); stars indicate significant interactions or main effects (*p < 0.05; **p < 0.01; ***p < 0.001). Axis units for all information theoretic measures based on continuous variables are z-normalized values across conditions.

https://doi.org/10.1371/journal.pcbi.1005511.g001

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Table 2. Results of permutation analysis of variance for information theoretic measures (p-values).

https://doi.org/10.1371/journal.pcbi.1005511.t002

In general, H decreased in both animals under higher isoflurane levels (main effect isoflurane level, p < 0.01** for ferret 1 and p < 0.001*** for ferret 2), indicating a reduction of locally available information for higher isoflurane concentrations. Yet, an isoflurane concentration of 0.5% (abbreviated as iso 0.5%, with other concentrations abbreviated accordingly) led to a slight increase in H for ferret 1, followed by a decrease for concentration iso 1.0%, which was below initial entropy values. This rise in H in condition iso 0.5% was present only in V1 of ferret 1, while H decreased monotonically in PFC. In ferret 2, H increased monotonically in both recording sites, with a stronger decrease in PFC. The interaction effect (isoflurane level x brain region) was significant for both animals (p < 0.001*** for ferret 1 and p < 0.01** for ferret 2).

The information transfer as measured by the self prediction optimal transfer entropy (TE_SPO, [15]) decreased significantly with higher levels of isoflurane in both animals (main effect isoflurane level, p < 0.001***), indicating an overall reduction in information transfer. This reduction was stronger in the top-down direction PFC → V1 (significant interaction effect isoflurane level x direction in ferret 1, p < 0.01**). In ferret 2 this interaction was not significant in the permutation ANOVA (pANOVA) on aggregated data, but was highly significant using the LMM approach (see next section).

The stored information as measured by the active information storage (AIS, [19]) increased in both animals under higher isoflurane levels (main effect isoflurane level, p < 0.01** for ferret 1 and p < 0.001*** for ferret 2), indicating more predictable information in LFP signals under higher levels of isoflurane. In ferret 1 the concentration iso 0.5% led to a slight decrease in AIS, followed by an increase compared to initial levels for concentration iso 1.0%. This initial decrease in AIS in condition iso 0.5% was present only in V1 of this animal, while in its PFC AIS increased monotonically. In ferret 2, AIS increased monotonically in both recording sites, with a stronger increase in PFC. The interaction effect was significant for both animals (p < 0.001*** for ferret 1 and p < 0.01** for ferret 2). Overall, AIS behaved complementary to H for all animals and isoflurane levels, despite the fact that AIS is one component of H [19].

Alternative statistical testing using linear mixed models.

We additionally performed a parametric test, using linear mixed models (LMM) on non-aggregated data from individual epochs of recording sessions (adding recording as additional random factor to encode the recording session) to enable a comparison to results from earlier studies using parametric testing.

For both animals, model comparison showed a significant main effect of factor isoflurane level on TE_SPO, AIS, and H as well as a significant interaction of factors isoflurane level and direction (see Table 3). The only exception to this was the factor direction in the evaluation of TE_SPO in Ferret 2. Thus, the results of this alternative statistical analysis were in agreement with those from the pANOVA. The detailed Table 3 reports the Bayesian Information Criterion with , where x are the observed realizations of the data, are the parameters that optimize the likelihood for a given model M, k(a) is the number of parameters and N the number of data points. The BIC becomes smaller with better model fit. In addition, Table 3 gives the deviance, , which is higher for better model fit, and the χ² with the corresponding p-value, describing the likelihood ratio, which follows a χ²-distribution.

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Table 3. Results of parametric statistical testing using model comparison between linear mixed models.

Simple effects of factors isoflurane level and direction or recording site were tested against the null model; interaction effects isoflurane level times direction and isoflurane level times recording site were tested against the additive models isoflurane level + direction and isoflurane level + recording site, respectively. *** as defined in Table 2.

https://doi.org/10.1371/journal.pcbi.1005511.t003

Bayesian estimation on discretized data.

In addition to the neighbor-distance based estimators for TE_SPO, AIS, and H used above, we also applied Bayesian estimators recently proposed by Nemenman, Shafe, and Bialek (NSB) [17, 18] to our data. The Bayesian approach promises to yield unbiased estimators when priors are chosen appropriately. However, one has to keep in mind that these estimators currently require the discretization of continuous data, and therefore may loose important information.

When applying the NSB estimator to the discretized LFP time series with N_bins discretization steps, we observed that for N_bins ≥ 8 the results were qualitatively consistent for different choices of numbers of bins. We present results for N_bins = 12 (Fig 2), which provides a reasonable resolution of the signal while still allowing for a reliable estimation of entropies within the scope of available data.

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Fig 2. pANOVA results for Bayesian estimates of transfer entropy (TE_SPO), active information storage (AIS), and entropy (H).

Left columns show interactions isoflurane level x direction and isoflurane level x recording site for both animals; right columns show main effects isoflurane level. Grey lines in interaction plots indicate TE_SPO from prefrontal cortex (PFC) to primary visual areas (V1), or H and AIS in PFC; black lines indicate TE_SPO from V1 to PFC, or H and AIS in V1. Error bars indicate the standard error of the mean (SEM); stars indicate significant interactions or main effects (*p < 0.05; **p < 0.01; ***p < 0.001).

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We confirmed the reliability of the estimator by systematically reducing the sample size and found no substantial impact on our estimates (Fig 3).

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Fig 3. Examples for estimates of entropy (H) terms for transfer entropy calculation (Eq 12) by number of data points N, using the Nemenman-Shafee-Bialek-estimator (NSB).

(A) entropies for ferret 1, V1, iso 0.5%: estimates are stable for N ≥ 100,000; (B) entropies for ferret 2, PFC, iso 0.0%: an insufficient number of data points does not allow for verification of the estimate’s robustness (recording was excluded from further analysis). Variables labeled X reflect data from the source variable, Y from the target variable. t is an integer time index, and bold typeface indicates the state of a system (see Methods).

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The estimation of H, AIS, and TE_SPO by Bayesian techniques for the binned signal representations provided results that were qualitatively consistent with results from the neighbor-distance based estimation techniques (compare Figs 1 and 2, and Tables 2 and 4). While the Bayesian estimates showed larger variances across different recordings and sample sizes, on average, we saw a decrease of TE_SPO and H for higher concentrations of isoflurane (main effect isoflurane level), while AIS increased for higher concentrations. For ferret 1 we also found an interaction effect for TE_SPO, with a stronger reduction in information transfer in top-down direction.

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Table 4. Results of permutation analysis of variance for information theoretic measures obtained through Bayesian estimation (p-values).

https://doi.org/10.1371/journal.pcbi.1005511.t004

Note that we also applied an alternative Bayesian estimation scheme based on Pitman-Yor-process priors [20]. However, for this estimation procedure, we observed that the data were insufficient to allow for a robust estimation of the tailing behavior of the distribution as indicated by large variances and unreasonably high estimates across different sample sizes.

Simulated effects of changed source entropy on transfer entropy.

To test the effect of reduced source entropy on TE_SPO between a source and target process, we simulated two test cases with high and low source entropy respectively. In both test cases signals were based on the original recordings, and the coupling between source and target was held constant (see Methods).

We found significantly higher TE_SPO in the case with high source entropy than in the case with low source entropy (Fig 4A). The simulation thus demonstrated that a lower source entropy does indeed lead to a reduction in TE_SPO despite an unchanged coupling. Information transfer in the high-entropy case was similar to the average information transfer found in recordings under an isoflurane concentration of 0.0%, indicating that the simulation scheme mirrored information transfer found in the experimental recordings.

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Fig 4. Simulated effects of changed source entropy on transfer entropy (TE_SPO).

(A) TE_SPO for two simulated cases of high (H-high) and low (H-low) source entropy (**p < 0.01, error bars indicate the standard error of the mean, SEM, over data epochs). Dashed lines indicate the average TE_SPO estimated from the original data under 0.0% isoflurane ± SEM; (B) source entropy (dashed bars) and target entropy H(Y_t) (empty bars) for the two simulated test cases of high (gray bars) and low entropy (white bars), error bars indicate the SEM over data epochs. Source entropy was higher in the high-entropy simulation, while target entropy was approximately the same for both cases. Results are given as z-values across estimates for all epochs from both simulations.

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Optimized embedding parameters for TE_SPO and AIS estimation

As noted in the introduction, the estimation of information theoretic measures from finite data is challenging. For the measures that describe distributed computation in complex systems, such as transfer entropy, estimation is further complicated because the available data are typically only scalar observations of a process with multidimensional dynamics. This necessitates the approximate reconstructions of the process’ states via a form of embedding [21], where a certain number of past values of the scalar observations spaced by an embedding delay are taken into account (e.g. for a pendulum swinging through its zero position one additional past position values will clarify whether it’s going left or right). An important part of proper transfer entropy estimation is thus optimization of this number of past values (embedding dimension), and of the embedding delay. These two embedding parameters then approximately define past states, whose correct identification is crucial for the estimation of transfer entropy [22], but also for the estimation of active information storage. Without it information storage may be underestimated and, erroneous values of the information transfer will be obtained; even a detection of information transfer in the wrong direction is likely (see [15, 22, 23] and Methods section). Existing studies using transfer entropy often omitted the optimization of embedding parameters, and instead used ad-hoc choices, which may have had a detrimental effect on the estimation of transfer entropy—hence the need for a confirmation of previous results in this study.

In the present study, we therefore used a formal criterion proposed by Ragwitz [21], to find an optimal embedding defined by the embedding dimension d (the number of values collected) and delay τ (the temporal spacing between them), to find embeddings for the TE_SPO and AIS past states. We used the implementation of this criterion provided by the TRENTOOL toolbox. Since the bias of the estimators used depends on d, we used the maximum d over all conditions and directions of interaction as the common embedding dimension for estimation from each epoch to make the estimated values statistically comparable. The resulting dimension used was 15 samples, which is considerably higher than the value commonly used in the literature [6, 7], when choosing the embedding dimension ad-hoc or using other criteria as for example the sample size [9]. The embedding delay τ was optimized individually for single epochs in each condition and animal as it has no influence on the estimator bias.

Relevance of individual information transfer delay estimation

Several previous studies on information transfer under anesthesia reported the sign of the so called net transfer entropy (TE_net) as a measure of the predominant direction of information transfer between two sources. TE_net is essentially just the normalized difference between the transfer entropies measured in the two direction connecting a pair of recording sites (see Methods). When calculating TE_net, it is particularly important to individually account for physical delays δ in the two potential directions of information transfer, because otherwise the sign of TE_net may become meaningless (see examples in [15]). As this delay is unknown a priori it has to be found prior to the actual estimation of information transfer. We have recently shown that this can be done by using a variable delay parameter u in a delay sensitive transfer entropy estimator TE_SPO (see Methods); here, the u that maximizes the transfer entropy over a range of assumed values for u reflects the physical delay [15].

The necessity to individually optimize u for each interaction is not a mere theoretical concern but was clearly visible in the present study: Fig 5C shows representative results from ferret 1 under 0.0% isoflurane, where apparent TE_SPO values strongly varied as a function of the delay u. As a consequence, TE_net values also varied if a common delay u was chosen for both directions. In other words, the sign of the TE_net varied as a function of individual choices for u_{PFC → V1} and u_{V1 → PFC} for each direction of information transfer and hence became uninterpretable (Fig 5D).

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Fig 5. Estimation of information transfer delays.

(A, modified from [15]) estimation of transfer entropy (TE_SPO) depends on the choice of the delay parameter u, if u is much smaller or bigger than the true delay δ, information arrives too late or too early in the target time series and information transfer is not correctly measured; (B, modified from [15]) TE_SPO values estimated from two simulated, bidirectionally coupled Lorenz systems (see [15] for details) as a function of u for both directions of analysis, TE_SPO(X → Y, t, u) (black line) and TE_SPO(Y → X, t, u) (gray line): TE_SPO values vary with the choice of u and so does the absolute difference between values; using individually optimized transfer delays for both directions of analysis yields the meaningful difference Δ (dashed lines), where TE_SPO(X → Y, t, u_opt) > TE_SPO(Y → X, t, u_opt); (C) example transfer entropy analysis for one recorded epoch: TE_SPO values vary greatly as a function of u, optimal choices of u are marked by dashed lines; (D) sign (gray: negative, white: positive) of TE_net for different values of u_{PFC → V1} (x-axis) and u_{V1 → PFC} (y-axis), calculated from TE_SPO values shown in panel C: the sign varies with individual choices of u; the black frame marks the combination of individually optimal choices for both parameters that yields the correct result.

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As a consequence of the above, we here individually optimized u for each direction of information transfer in each condition and animal to estimate the true delay of information transfer following the mathematical proof in [15]. We individually optimized u to obtain estimates of transfer entropy that were not biased by a non-optimal choice for u. We used the implementation in TRENTOOL [23] and scanned values for u ranging from 0 to 20 ms. Averages for optimized delays ranged from 4–7 ms across animals and isoflurane levels (Fig 6).

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Fig 6. Optimized information transfer delays u for both directions of interaction and three levels of isoflurane, by animal.

Bars denote averages over recordings per condition; error bars indicate the standard error of the mean (SEM). There was a significant main effect of isoflurane level for ferret 1 (p < 0.05).

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Note that in Fig 5C, TE_SPO as a function of u shows multiple peaks, especially for the direction V1 → PFC. Since these peaks resembled an oscillatory pattern close to the low-pass filter frequency used for preprocessing the data, we investigated the influence of filtering on delay reconstruction by simulating two coupled time series for which we reconstructed the delay with and without prior band-pass filtering (Fig 7). For filtered, simulated data TE_SPO as a function of u indeed showed an oscillatory pattern for certain filter settings. However, broadband filtering—as used here for the original data—did not lead to the reconstruction of an incorrect value for the delay δ (Fig 7C). Yet, when filtering using narrower bandwidths, the reconstruction of the correct information transfer delay failed (Fig 7D–7F). This finding supports the general notion that filtering, especially narrow-band filtering, may only be applied with caution before estimating connectivity measures [24, 25].

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Fig 7. Simulated effects of filtering on the reconstruction of information transfer delays (u).

(A) Cross-correlation R_X,Y between two simulated, coupled time series (N = 100000, drawn from a uniform random-distribution over the open interval (0, 1), true coupling delay 10 samples, indicated by the red dashed line), the simulation was run 50 times, black lines indicate the mean over simulation runs, gray lines indicate individual runs; (B) TE_SPO as a function of information transfer delay u before filtering the data; (C–D) TE_SPO as a function of u after filtering the data with a bandpass filter (fourth order, causal Butterworth filter, implemented in the MATLAB toolbox FieldTrip [26]); red dashed lines indicate the simulated delay δ, blue lines indicate the average reconstructed delay u over simulation runs, histograms (inserts) show the distribution of reconstructed values for u over simulation runs in percent; (C) broadband filtering (0.1–300 Hz) introduced additional peaks in TE_SPO values, however, the maximum peak indicating the optimal u was still at the simulated delay for all simulated runs; (D) filtering within a narrower band (0.1–200 Hz) led to an imprecise reconstruction of the correct δ in each run with an error of 1 sample, i.e. δ tended to be underestimated; (E–F) narrow-band filtering in the beta range (12–30 Hz) and theta range (4–8 Hz) led to a wide distribution of u with a large absolute error of up to 10 samples.

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Yet, since broad-band filtering did not lead to the observed oscillatory patterns in the simulated data, alternative explanations for multiple peaks in TE_SPO are more likely: one alternative cause of multiple peaks in the TE_SPO are multiple information transfer channels with various delays between source and target (see [15], especially Fig 6). Each information transfer channel with its individual delay will be detected when estimating TE_SPO as a function of u (see simulations in [15]). A further possible cause for multiple peaks in the TE_SPO is the existence of a feedback loop between the source of TE_SPO and a third source of neural activity [15]. In such a feedback loop, information in the source is fed back to the source such that it occurs again at a later point in time, leading to the repeated transfer of identical bits of information. Both causes are potential explanations for the occurrence of multiple peaks in the TE_SPO in the present study—yet, deciding between these potential causes requires additional research, specifically interventional approaches to the causal structure underlying the observed information transfer.

In sum, our results clearly indicate the necessity to individually optimize information transfer delays and that often employed ad-hoc choices for u may result in spurious results in information transfer. The additional simulations of filtering effects showed that narrow-band filtering may have detrimental effects on this optimization procedure and should thus be avoided.

Relation of information-theoretic measures with other time-series properties

The information-theoretic measures used in this study are relatively novel and have been applied in neuroscience research rarely. Thus, it is conceivable that these measures quantify signal properties that are more easily captured by traditional time-series measures, namely, the autocorrelation decay time (ACT), the signal variance, and the power in individual frequency bands. To investigate the overlap between information-theoretic and traditional measures, we calculated correlations between AIS and ACT, between H and signal variance, between TE_SPO and ACT, between TE_SPO and signal variance, and between AIS, H, and TE_SPO and the power in various frequency bands, respectively. Fig 8 shows average ACT, signal variance, and power over recordings, for individual isoflurane levels and recording sites, and for both animals.

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Fig 8. Autocorrelation decay time (ACT), signal variance, and power by isoflurane level and recording site for both animals.

Averages over recordings per isoflurane level and recording site; error bars indicate one standard deviation.

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Detailed tables showing correlation coefficients and explained variances are provided as supporting information S1 to S6 Tables. In summary, for AIS and ACT we found significant correlations in both animals, however the median of the variance explained, , over individual recordings was below 0.1 for each significant correlation in an animal, recording site and isoflurane level. Hence, even though there was some shared variance between ACT and AIS, there remained a substantial amount of unexplained variance in AIS, indicating that AIS quantified properties other than the signal’s ACT—in line with results from our earlier studies [27, 28]. This is expected because the autocorrelation (decay) time ACT measures how long information persists in a time series in a linear encoding, while AIS, in contrast, measures how much information is stored (at any moment), and also reflects nonlinear transformations of this information. The fact that there is still some shared variance between the two measures here may be explained by the construction of the AIS’s past state embedding, where the time steps between the samples in the embedding vector are defined as a fractions of the ACT.

Between H and signal variance, we found no significant correlation; moreover, correlations for individual recordings were predominantly negative.

The TE_SPO was significantly correlated with source ACT for an isoflurane level of 1.0%. Also for the correlation of TE_SPO and the source’s variance, we found significant correlations for the bottom-up direction under 0.5% and 1.0% isoflurane in animal 1, and for both directions under 1.0% isoflurane in animal 2. For all significant correlations we found , again indicating a substantial amount of variance in TE_SPO that was not explained by ACT or signal variance.

When correlating band power with information-theoretic measures, we found significant correlations between AIS and all bands in both animals animals (see supporting information S4 Table, ); for H and band power, we found correlations predominantly in higher frequency bands (beta and gamma, see supporting information S5 Table, ); for TE_SPO and band power, we found significant correlations predominantly in the gamma band and one significant correlation in the delta band for animal 2 and in the beta band for animal 1, respectively (see supporting information S6 Table, ). In sum, we found relationships between the power in individual frequency bands and all three information-theoretic measures. For all measures, the variance explained was below 0.2, indicating again that band power did not fully capture the properties measured by TE_SPO, AIS, and H.

Reduction in transfer entropy may be caused by changes in local information processing

To test our alternative hypothesis we evaluated two of its predictions about changes in locally available information, as measured by signal entropy, under administration of different isoflurane concentrations. First, entropy should be reduced; second, the strongest decrease in information transfer should originate from the source node with the strongest decrease in entropy, rather than end in this node. Indeed, we found that signal entropy decreased; the most pronounced decrease in signal entropy was found in PFC. In accordance with our prediction, we found that PFC—the node with the larger decrease in entropy—was also at the source, not the target, of the most pronounced decrease in transfer entropy. This is strong evidence against the theoretical possibility that in a recording site, entropy decreased due to a reduced influx of information—because in this latter scenario the strongest reduction in entropy should have been found at the target (end point) of the most pronounced decrease in information transfer. Hence, our hypothesis that long-range cortico-cortical information transfer is reduced because of changes in local processing must be taken as a serious alternative to the currently prevailing theories of anesthetic action based on disruptions of long range interactions. We suggest that a renewed focus on local information processing in anesthesia research will be pivotal to advance our understanding of how consciousness is lost.

Our predictions that reductions in entropy should potentially be reflected in reduced information transfer derives from the simple principle that information that is not available at the source cannot be transferred. We may thus in principle reduce TE_SPO to arbitrarily low values by reducing the entropy of one of the involved processes, without changing the physical coupling between the two systems, just by changing their internal information processing. This trivial but important fact has been neglected in previous studies when interpreting changes in information transfer as changes in coupling strength. Even when this bound is not attained, e.g. because only a certain fraction of the local information is transferred even under 0.0% isoflurane, it seems highly plausible that reductions in the locally available information affect the amount of information transfered. This line of reasoning generalizes to applications beyond anesthesia research, i.e., every application that observes changes in information transfer between two or more experimental conditions should consider changes in local information processing as a potential alternative cause.

A possible indication of how exactly local information processing is changed is given by the observation of increased active information storage in PFC and V1 (also see the next paragraph). This means that more ‘old’ information is kept stored in a source’s activity under anesthesia, rather than being dynamically generated. Such stored source information will not contribute to a measurable transfer entropy under most circumstances because it is already known at the target (see [29], section 5.2.3, for an illustrative example).

Relation between changed locally available information and information storage

In general, AIS increases if a signal becomes more predictable when knowing it’s past, but is unpredictable otherwise. This also means that the absolute AIS is upper-bounded by the system’s entropy H (see Methods, Eq 5). Thus, a decrease in H can in principle lead to a decrease in AIS, i.e., fewer possible system states may lead to a decrease in absolute AIS. However, in the present study, we observed an increase in AIS while H decreased—this indicates an increase in predictability that more than compensates for the decrease in locally available information. In other words, the system visited fewer states in total but the next state visited became more predictable from the system’s past. Thus, a reduction in entropy and increase in predictability points at highly regular neural activity for higher isoflurane concentrations. Such a behavior in activity is in line with existing electrophysiological findings: under anesthesia signals have been reported to become more uniform, exhibiting repetitive patterns, interrupted by bursting activity (see [5] for a review). For example, Purdon and colleagues observed a reduction of the median frequency and an overall shift towards high-power, low-frequency activity during LOC [30]. In particular, slow-wave oscillatory power was more pronounced during anesthesia-induced LOC. LOC was also accompanied by a significant increase in power and a narrowing of oscillatory bands in the alpha frequency range in their study.

Limitations of the applied estimators and measures

Unfortunately a quantitative comparison between different information theoretic estimates that would directly relate H, AIS and TE_SPO is not possible using the continuous estimators applied in this study. Their estimates are not comparable because the bias properties of each estimator differ. For each estimator, the bias depends on the number of points used for estimation as well as on the dimensionality of involved variables—however, the exact functional relationship between these two quantities and the bias is unknown and may differ between estimators. (In our application, the dimensionality is determined mainly by the dimension of the past state vectors, and by how many different state variables enter the computation of a measure).

This lack of comparability makes it impossible to normalize estimates; for example, transfer entropy is often normalized by the conditional entropy of the present target state to compare the fraction of transferred information to the fraction of stored information. We forgo this possibility of comparison for a greater sensitivity and specificity in the detection of changes in the individual information theoretic measures here. New estimators, e.g. Bayesian estimators like the ones tested here, promise more comparable estimates by tightly controlling the biases. Yet, these estimators were not as reliable as expected in our study, displaying a relatively high variance.

A further important point to consider when estimating transfer entropy between recordings from neural sites, is the possibility of third unobserved sources influencing the information transfer. In the present study, third sources (e.g., in the thalamus) may influence the information transfer between PFC and V1—for example, if a third source drove the dynamics in both areas, the areas would become correlated, leading to non-zero estimates of TE_SPO. This TE_SPO is then attributable to the correlation between source and target, but does not measure an actual information transfer between sources. Hence, information transfer estimated by transfer entropy should in general not be directly equated with a causal connection or causal mechanism existing between the source and target process (see also [12] for a detailed discussion of the difference between transfer entropy and measures of causal interactions).

Potential physiological causes for altered information transfer under anesthesia

We here tested the possibility that changes in local information processing lead to the frequently observed reduction in information transfer between cortical areas under isoflurane administration, instead of altered long-range coupling. Results on entropies and active information storage suggest that this is a definite possibility from a mathematical point of view.

This is supported by the neurophysiology related to the mode of action of isoflurane, because a dominant influence of altered long-range coupling on TE_SPO would mandate that synaptic terminals of the axons mediating long range connectivity should be targets of isoflurane. Such long range connectivity is thought to be dominated by glutamatergic AMPA receptors for inter-areal bottom-up connections, and glutamatergic NMDA receptors for inter-areal top-down connections, building upon findings in [31] and [32] (but see [33] for some evidence of GABAergic long range connectivity). Yet, evidence for isoflurane effects on AMPA and NMDA receptors is sparse to date (table 2 in [14]). In contrast, the receptors most strongly influenced by isoflurane seem to be GABA_A and nicotinic acetylcholine (nAChR) receptors. More specifically, isoflurane potentiates agonist interactions at the former, while inhibiting the latter.

Thus, if one adopts the current state of knowledge on the synapses involved in long-range inter-areal connectivity, evidence speaks against a dominant effect of modulation of effective long-range connections by isoflurane. This, in turn, points at local information processing as a more likely reason for changed transfer entropy under isoflurane anesthesia. This interpretation is perfectly in line with our finding that decreases in source entropy seem to determine the transfer entropy decreases, instead of decreases in transfer entropy determining the target entropies.

Nevertheless, targeted local interventions by electrical or optogenetic activation of projection neurons, combined with the set of information theoretic analyses used here, will most likely be necessary to reach final conclusions on the causal role of local entropy changes in reductions of transfered information.

How may altered long-range information transfer lead to loss of consciousness?

Investigating long-range information transfer under anesthesia is motivated by the question how changed information transfer may cause LOC. To that effect, our findings—a dominant decrease in top-down information transfer under anesthesia, and a decrease in locally available information possibly driving it—may be interpreted in the framework of predictive coding theory [34–36]. Predictive coding proposes that the brain learns about the world by constructing and maintaining an internal model of the world, such that it is able to predict future sensory input at lower levels of the cortical hierarchy. Whether predictions match actual future input is then used to further refine the internal model. It is thus assumed that top-down information transfer serves the propagation of predictions [37]. Theories of conscious perception within this predictive coding framework propose that conscious perception is “determined” by the internal prediction (or ‘hypothesis’) that matches the actual input best [35, p. 201]. It may be conversely assumed that the absence of predictions leads to an absence of conscious perception.

In the framework of predictive coding theory two possible mechanisms for LOC can be inferred from our data: (1) the disruption of information transfer, predominantly in top-down direction, may indicate a failure to propagate predictions to hierarchically lower areas; (2) the decrease in locally available information and in entropy rates in PFC may indicate a failure to integrate information in an area central to the generation of a coherent model of the world and the generation of the corresponding predictions. These hypotheses are in line with findings reviewed in [4] and [38], which discuss activity in frontal areas and top-down modulatory activity as important to conscious perception.

Future research should investigate top-down information transfer more closely; for example, recent work suggests that neural activity in separate frequency bands may be responsible for the propagation of predictions and prediction errors respectively [37, 39]. Future experiments may target information transfer within a specific band to test if the disruption of top-down information transfer happens in the frequency band responsible for the propagation of predictions.

Last, it should be kept in mind that different anesthetics may lead to loss of consciousness by vastly different mechanisms. Ketamine, for example, seems to increase, rather than decrease, overall information transfer—at least in sub-anesthetic doses [40].

Comparison of the alternative approaches to the estimation of information theoretic measures and to statistical testing

The main analysis of this study was based on information theoretic estimators relying on distances between neighboring data points and on a permutation ANOVA. As each of these has its weaknesses we used additional alternative approaches, first, a Bayesian estimator for information theoretic measures, and second, statistical testing of non-aggregated data using LMMs. Both approaches returned results qualitatively very similar to those of the main analysis. Specifically, replacing neighbor-distance based estimators with Bayesian variants, we replicated the main finding of our study—a reduction in information transfer and locally available information, and an increase in information storage under anesthesia. However, using the Bayesian estimators we did not find a predominant reduction of top-down compared to bottom up information transfer. In contrast, using LMMs for statistical testing instead of a pANOVA additionally revealed a more pronounced reduction in top-down information transfer also in ferret 2 (in this animal, this effect was not significant when performing permutation tests based on the aggregated data).

The Bayesian estimators performed slightly worse than the next neighbor-based estimators in terms of their higher variance in estimates across recordings. Thus, even though Bayesian estimators are currently the best available estimators for discrete data, they may be a non-optimal choice for continuous data. A potential reason for this is the destruction of information on neighborhood relationships through data binning. This is, however, necessary to make the current Bayesian estimators applicable to continuous data.

In sum, we obtained similar results through three different approaches. This makes us confident that locally available information and information transfer indeed decrease under anesthesia, while the amount of predictable information increases.

On information theoretic measures obtained from from continuous time processes via time-discrete sampling

When interpreting the information theoretic measures presented in this work, it must be kept in mind that they were obtained via time-discrete sampling of processes that unfold in continuous time (such as the LFP, or spike trains). This time-discrete sampling was taken into account in recent work by Spinney and colleagues [41], who could show that the classic transfer entropy as defined by Schreiber is indeed ambiguous when applied to sampled data from time-continuous processes—it can either be seen as the integral of a transfer entropy rate in continuous time over one sampling interval, and is then given in bits, or it can be seen as an approximation to the continuous time transfer entropy rate itself, and be given in bits/sec. In our study we stick with the first notion of transfer entropy, and note that our results will numerically change when using different sampling intervals. In contrast to the case of transfer entropy covered in [41], corresponding analytical results for AIS are not known at present, as this is still a field of active research. To nevertheless elucidate the practical dependency of AIS on the sampling rate, when using our estimators, we have taken the original data and have up- and down-sampled them. Indeed the empirical AIS depended on the sampling rate (supporting information S1 Fig). Yet, all relations of AIS values between different isoflurane concentrations, i.e., the qualitative results, are independent of sampling. Thus, sampling effects do not affect the conclusions of the current study.

A cautionary note on the interpretation of information theoretic measures evaluated on local field potential data

When interpreting the results obtained in this study it should be kept in mind that LFP signals are not in themselves the immediate carriers of information relevant to individual neurons. This is because from a neuron’s perspective information arrives predominantly in the form postsynaptic potentials generated by incoming spikes and chemical transmission in the synaptic cleft (but see [42] for a potential influence of LFPs on neural dynamics and computation). Thus, LFP signals merely reflect a coarse grained view of the underlying neural information processing. As a consequence, our results only hold in as far as at least some relevant information about the underlying information processing survives this coarse graining in the recording process, and little formal mathematical work has been carried out to estimate bounds on the amount of information available after coarse graining (but see [43]).

Yet, the enormous success that brain reading approaches had when based on local field potentials or on even more coarse grained magnetoencephalography (MEG) recordings (e.g. [44]) indicates that relevant information on neural information processing is indeed available at the level of these signals. However, successful attempts at decoding neural representations of stimuli or other features of the experimental setting should not lead us to misinterpret the information captured by information theoretic measures of neural processing as necessarily being about something we can understand and link to the outside world. Quite to the contrary, the larger part of information captured by these measures may be related to intrinsic properties of the unfolding neural computation.

Conclusion

Using two different methods for transfer entropy estimation, and two different statistical approaches, we found that locally available information and information transfer are reduced under isoflurane administration. The larger decrease in the locally available information was found at the source of the larger decrease of information transfer, not at its end point, or target. Therefore, previously reported reductions in information transfer under anesthesia may be caused by changes in local information processing rather than a disruption of long range connectivity. We suggest to put this hypothesis more into the focus of future research effort to understand the loss of consciousness under anesthesia. This suggestion receives further support from the fact that the synaptic targets of the anesthetic isoflurane, as used in this study, are most likely located in local circuits.

Electrophysiological recordings

We conducted simultaneous electrophysiological recordings of the local field potential (LFP) in primary visual cortex (V1) and prefrontal cortex (PFC) of two female ferrets (17 to 20 weeks of age at study onset) under different levels of isoflurane (Fig 9). Raw data are available from [45]. The choice of the animal model is discussed further in [46]. Recordings were made in a dark environment during multiple, individual sessions of max. 2 h length, during which the animals’ heads were fixed. For recordings, we used single metal electrodes acutely inserted in putative layer IV, measured 0.3–0.6 mm from the surface of cortex (tungsten micro-electrode, 250-μm shank diameter, 500-kΩ impedance, FHC, Bowdoin, ME). The hardware high pass filter was 0.1 Hz and the low pass filter was 5,000 Hz. A silver chloride wire placed between the skull and soft tissue was used as the reference electrode. The reference electrode was located approximately equidistant between the recording electrodes. This location was selected in order to have little shared activity with either recording electrode. The same reference was used for both recording locations; also the same electrode position was used for both animals and all isoflurane concentrations. To verify that electrode placement was indeed in V1, we mapped receptive fields by eliciting visually evoked potentials in a separate series of experiments. We confirmed electrode placement in PFC by lesioning through the recording electrode after completion of data collection and post-mortem histology (as described in [46]). Details on surgical procedures can be found in [47]. Unfiltered signals were amplified with gain 1,000 (model 1800, A-M Systems, Carlsborg, WA), digitized at 20 kHz (Power 1401, Cambridge Electronic Design, Cambridge, UK), and digitally stored using Spike2 software (Cambridge Electronic Design). For analysis, data were low pass filtered (300 Hz cutoff) and down-sampled to 1000 Hz.

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Fig 9. Recording sites in the ferret brain.

Prefrontal cortex (PFC, gray dot): anterior sigmoid gyrus; primary visual area (V1, black dot): lateral gyrus. Arrows indicate analyzed directions of information transfer (gray: top-down; black: bottom-up).

https://doi.org/10.1371/journal.pcbi.1005511.g009

All procedures were approved by the University of North Carolina-Chapel Hill Institutional Animal Care and Use Committee (UNC-CH IACUC) and exceed guidelines set forth by the National Institutes of Health and U.S. Department of Agriculture.

LFPs were recorded during wakefulness (condition iso 0.0%, number of recording sessions: 8 and 5 for ferret 1 and 2, respectively) and with different concentrations of anesthetic: 0.5% isoflurane with xylazine (condition iso 0.5%, number of sessions: 5 and 6), as well as 1.0% isoflurane with xylazine (condition iso 1.0%, number of sessions: 10 and 11). In the course of pilot experiments, both concentrations iso 0.5% and iso 1.0% lead to a loss of the righting reflex; however, a systematic assessment of this metric during recordings was technically not feasible. Additionally, animals were administered 4.25 ml/h 5% dextrose lactated Ringer and 0.015 ml/h xylazine via IV.

LFP recordings from each session were cut into epochs of 4.81 s length to be able to remove segments of data if they were contaminated by artifacts (e.g., due to movement). We chose a relatively short epoch length to avoid removing large chunks of data when there was only a short transient artifact. This resulted in 196 to 513 epochs per recording (mean: 428.6) for ferret 1, and 211 to 526 epochs (mean: 472.8) for ferret 2. epochs with movement artifacts were manually rejected (determined by extreme values in the LFP raw traces). In the iso 0.0% condition, infrared videography was used to verify that animals were awake during the whole recording; additionally, iso 0.0% epochs with a relative delta power (0.5 to 4.0 Hz) of more than 30% of the total power from 0.5 to 50 Hz were rejected to ensure that only epochs during which the animal was truly awake entered further analysis.

Information theoretic measures

To measure information transfer between recording sites V1 and PFC, we estimated the transfer entropy [10] in both directions of possible interactions, PFC → V1 and V1 → PFC. To investigate local information processing within each recording site, we estimated active information storage (AIS) [19] as a measure of predictable information, and we estimated differential entropy (H) [48] as a measure of information available locally. We will now explain the applied measures and estimators in more detail, before we describe how these estimators were applied to data from electrophysiological recordings in the next section. To mathematically formalize the estimation procedure from these data, we assume that neural time series recorded from two systems and (e.g. cortical sites) can be treated as collections of realizations x_t and y_t of random variables X_t, Y_t of two random processes X = {X₁, …, X_t, …, X_N} and Y = {Y₁, …, Y_t, …, Y_N}. The index t here indicates samples in time, measured in units of the dwell time (inverse sampling rate) of the recording.

Entropy as an upper bound on information transfer

The transfer entropy from Eq 1 can be rewritten as: (7) By dropping the negative term on the right hand side we obtain an upper bound (as already noticed by [54, P. 65]), and by realizing that a conditional entropy is always smaller than the corresponding unconditional one, we arrive at (8) This indicates that the overall entropy of the source states is an upper bound. Several interesting other bounds on information transfer exist as detailed in [54], yet these are considerably harder to interpret and were not the focus of the current presentation.

Estimation of information theoretic measures

In this section we will describe how the information theoretic measures presented in the last section may be estimated from neural data. In doing so, we will also describe the methodological pitfalls mentioned in the introduction in more detail and we will describe how these were handled here. If not stated otherwise, we used implementations of all presented methods in the open source toolboxes TRENTOOL [23] and JIDT [53], called through custom MATLAB^® scripts (MATLAB 8.0, The MathWorks^® Inc., Natick, MA, 2012). Time series were normalized to zero mean and unit variance before estimation.

Finding optimal embedding parameters.

The second methodological problem raised in the introduction was the choice of embedding parameters for transfer entropy estimation. One important parameter here is the choice of the total signal history when constructing past states for source and target signal (see Eqs 1 and 2). Failure to properly account for signal histories may lead to a variety of errors, such as underestimating transfer entropy, failure to detect transfer entropy altogether, or the detection of spurious transfer entropy. Transfer entropy is underestimated or missed if the past state of the source time series does not cover all the relevant history, i.e., the source is under-embedded. In contrast, spurious transfer entropy may be detected if the past state of the target time series is under-embedded, such that spurious detection of transfer entropy is a false positive and therefore the most serious error. One scenario where spurious transfer entropy results from under-embedding is shown in Fig 10.

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Fig 10. Spurious information transfer resulting from under-embedding (modified from [22]).

(A) Actual coupling between processes X and Y. (B) Mutual information between the present value in X, x_t, and a value in the far past of X, x_t−(τ⋅d), conditional on all intermediate values (shaded box), the mutual information is non-zero, i.e., x_t−(τ⋅d) holds some information about x_t. (C) Both directions of interaction are analyzed; (D) Information in x_t−(τ⋅d) (white sample point) is transferred to Y (solid arrow), because of the actual coupling X → Y. The information in x_t−(τ⋅d) about x_t is thus transferred to the past of Y, , which thus becomes predictive of x_t as well. Assume now, we analyzed information transfer from Y to X, , without a proper embedding of X, : Because of the actually transferred information from x_t−(τ⋅d) to , the mutual information is non-zero. If we now under-embed X, such that the information in x_t−(τ⋅d) is not contained in and is not conditioned out, will be non-zero as well. In this case, under-embedding of the target X will lead to the detection of spurious information transfer in the non-coupled direction Y → X. (E) Information transfer is falsely detected for both directions of interaction, the link from Y to X is spurious (dashed arrow).

https://doi.org/10.1371/journal.pcbi.1005511.g010

The choice of an optimal embedding is also relevant for the estimation of AIS, where under-embedding leads to underestimation of the true AIS. Note that on the other hand, we can not increase the embedding length to arbitrarily high values because this leads to computationally intractable problem sizes and requires exponentially more data for estimation.

Optimal embedding parameters d and τ may be found through the optimization of a local predictor proposed by Ragwitz [21]. Ragwitz’ criterion tests different combinations of a range of values for d and τ. The current combination is used to embed each point in a time series, then, the future state for each point is predicted from the future states of its neighbors. The parameter combination that leads to the best prediction on average is used as the optimal embedding. To determine the neighbors of a point, we used a k-nearest-neighbor search with k = 4, i.e., the same value for k as was used for k-nearest-neighbor searches when estimating information-theoretic measures. We minimized the mean squared error when optimizing Ragwitz’ local predictor. Further details on Ragwitz’ criterion can be found in the documentations of the TRENTOOL [23] and JIDT [53] toolbox.

Other approaches for embedding parameter optimization have been proposed, see for example non-uniform embedding using mutual information to determine all relevant past samples as proposed by [64].

Statistical testing using permutation testing

To test for statistically relevant effects of isoflurane levels and direction of interaction or recording site on estimated measures, we performed two-factorial permutation analyses of variance (pANOVA) for each animal and estimated measure [39, 65–67]. We used a MATLAB^® implementation of the test described in [67], which is compatible with the FieldTrip toolbox data format [26] and is available from [68].

The permutation ANOVA can be used if the normality assumptions of parametric ANOVA are violated or—as in the present study—if assumptions are not testable due to too few data points per factor level. The permutation ANOVA evaluates the significance of the main effect of individual factors or their interaction effect under the null-hypothesis of no experimental effect at all. The significance is evaluated by calculating a F-ratio for the effect from the original data; this original F-ratio is then compared against a distribution of F-ratios obtained from permuted data. The F-ratio’s p-value is calculated as the fraction of ratios obtained from permuted data that is bigger than the original F-ratio. When permuting data, it is crucial to only permute data in such a way that the currently investigated effect is destroyed while all other effects are kept intact [65]: for example, consider a two-factorial design with factors A and B—if the main effect of factor A is to be tested, the assignment of levels of A to data points has to be permuted; yet, the permutation of levels of A can only happen within levels of factor B such as to not simultaneously destroy the effect of factor B. Thus, when testing for the effect of one factor, the effect of all other factors is preserved, making sure that variability due to one factor is tested for while the variability due to the other factor is held constant. However, this permutation scheme is not applicable to the interaction effect because it leaves no possible permutations—destroying the interaction effect through permutation always also destroys the main effects of individual factors. Here, both factors have to be permuted, which yields an approximative test of the interaction effect (see [67] for a discussion of permutation strategies and simulations).

The factors in the permutation ANOVA were isoflurane level and direction for TE_SPO estimates, and isoflurane level and recording site for AIS and H estimates. The number of permutations was set to 10,000. In the present study, data was recorded in different sessions and segmented into epochs. As a result, estimates for individual epochs should not be pooled over recordings for statistical analysis (see for example [69], and also the next paragraph for a discussion). We therefore aggregated estimates over epochs to obtain one estimate per recording session. We used the median to aggregate the estimated values for each recording session over individual epochs, because the distribution of measures over epochs was skewed and we considered the median a more exact representation of the distributions’ central tendencies. The aggregation of data resulted in relatively few observations per ANOVA cell and also in unequal number of observations between cells, thus violating two basic assumptions of parametric ANOVA (too few observations make it impossible to test for parametric assumptions like homogeneity of variances). Therefore, we used the non-parametric permutation approach over a parametric one, because the permutation ANOVA does not make any assumptions on data structure.

Statistical testing using linear mixed models

As described above, LFP recordings were conducted in epochs over multiple recording sessions. This introduces a so-called “nested design” [69], i.e., a hierarchical structure in the data, where data epochs are nested within recordings, which are nested in animals. Such structures lead to systematic errors or dependence within the data. This violates the assumption of uncorrelated errors made by the most common tests derived from the general linear model and leads to an inflation of the type I error [69]. A measure of the degree of dependence is the intraclass correlation coefficient (ICC), which was 0.35 for ferret 1 and 0.19 for ferret 2, indicating a significant dependency within the data (see [69] for a discussion of this measure). Thus, we performed an additional statistical tests where we again tested for a significant effect of the two factors isoflurane level and direction as well as their interaction, but we additionally modeled the nested structure in our data by a random factor recordings. Such a model is called a linear mixed effects model [70], and may yield higher statistical power than aggregating data within one level of the nested design (as was done for the permutation ANOVA). We used the following model for both animals separately: where y_ij is the TE_SPO value from the i-th epoch in recording j, modeled as a function of isoflurane level and direction of interaction, where β₀ describes the model intercept, β_k are the regression coefficients and describe fixed effects, γ_j is the random deviation of recording j from the intercept β₀, and ϵ_ij describes random noise. D_i, , and are predictor variables, encoding factors direction as (13) and factor isoflurane level as (14) (15) Note that we used dummy coding for factor direction so that the the estimated effect β₁ can be interpreted like the simple or main effect in a standard ANOVA framework. We used contrast coding for factor isoflurane level which allows to interpret estimated effects β₂ and β₃ as deviations from a reference group (in this case the condition iso 0.0%). We further assume that noise was i.i.d. and and .

We used the R language [71] and the function lmer from the lme4-package [72] for model fitting. We assessed statistical significance of individual factors by means of model comparison using the maximum likelihood ratio between models [73]. To allow for this model comparison, we used maximum likelihood estimation, instead of restricted maximum likelihood estimation, of random and fixed effects. To test for main effects, we compared models including individual factors isoflurane level (fm_a) and direction (fm_d) to a Null model (fm_0) including only the random effect; to furthermore test for an interaction effect, we compared the model including an interaction term (fm_axb) to a model where both factors only entered additively (fm_ab).

The models were fitted to 15,973 TE_SPO values from ferret 1 and 18,202 TE_SPO values from ferret 2, respectively.

Simulating the effect of reduced source entropy on transfer entropy

To test whether changes in source entropy influenced the transfer entropy despite unchanged coupling, we simulated two test cases, with high and low source entropy, respectively, while the coupling between source and target process were held constant. To simulate the two test cases, we randomly selected two recordings—one from the iso 0.0% condition, which on average showed higher source entropy, and one from the iso 1.0% condition, which on average showed lower source entropy (Figs 1 and 2). In the recording from the iso 0.0% condition, we permuted epochs in the target time series to destroy all information transfer present in the original data. From the permuted data, we simulated two cases of artificial coupling, first, using the high-entropy source time course from the iso 0.0% condition; and second, using the low-entropy source time course from the iso 1.0% condition. The coupling was simulated by adding a filtered, scaled and delayed version of the respective source time course to the target time course for each epoch. For filtering, we used a Gaussian filter with a smoothing of 10 samples; for the scaling factor, we used a value of 0.2, which resulted in a TE_SPO value for the high-entropy test case that was close to the TE_SPO in the iso 0.0% condition. By replacing the original coupling in both recordings with a simulated coupling, we made sure that the coupling was constant for both test cases.

For both test cases, we estimated TE_SPO and following the estimation procedures described above. We tested differences in TE_SPO using a permutation independent samples t-test with 10,000 permutations. To make sure that our simulation reflected information transfer found in the original data, we further tested for a significant difference between TE_SPO in the original data from the iso 0.0% recording and TE_SPO in the high-entropy test case (using the source time iso 0.0% condition). Here, we found no significant difference, indicating that the transfer entropy in our test case did not differ significantly from the transfer entropy found in the original data.

Correlating information-theoretic measures with other time-series properties

We correlated TE_SPO, AIS, and H with more conventional measures from time-series analysis, namely, the autocorrelation decay time (ACT), signal variance, and power in individual frequency bands. This correlation was performed to investigate whether information-theoretic measures captured signal properties that could be described equally well by more simple measures.

The ACT was calculated by finding the lag at which the autocorrelation coefficient decayed below e⁻¹. The signal variance was calculated as the variance over time for each recording after subtracting the mean of the time series. The band power was calculated for individual frequency bands (delta = 0.5–4 Hz, theta = 4–8 Hz, alpha = 8–12 Hz, beta = 12–30 Hz, gamma = 30–40 Hz, following [47]), using the multitaper method with discrete prolate spheroidal sequences (Slepian sequences) as tapers (smoothing = 1 Hz) implemented in the MATLAB toolbox FieldTrip [26].

We calculated correlations between information-theoretic measures and conventional measures for individual isoflurane levels and recording sites or direction of interaction, respectively. For each correlation, we pooled data over recording sessions for the respective isoflurane level and calculated Spearman’s rank correlation. We tested the correlation for significance using a restricted permutation test, where permutations were allowed only within one recording, accounting for the nested experimental design (see [69] and section Statistical testing using linear mixed models, below). Additionally, we calculated the correlation as well as the variance explained, R², for individual recordings, because calculating R² for the coefficient calculated from pooled data does not yield interpretable results.