# Breakdown of local information processing may underlie isoflurane anesthesia
effects

Patricia Wollstadt111MEG Unit, Brain Imaging Center, Goethe University,
Frankfurt/Main, Germany Kristin K. Sellers22footnotemark: 233footnotemark: 3
Lucas Rudelt444Max Planck Institute for Dynamics and Self-Organization,
Göttingen, Germany Viola Priesemann555Max Planck Institute for Dynamics and
Self-Organization, Göttingen, Germany666Bernstein Center for Computational
Neuroscience, BCCN, Göttingen, Germany Axel Hutt777Deutscher Wetterdienst,
Section FE 12 - Data Assimilation, Offenbach/Main, Germany888Department of
Mathematics and Statistics, University of Reading, Reading, United Kingdom
Flavio Fröhlich99footnotemark: 91010footnotemark: 101111footnotemark:
111212footnotemark: 121313footnotemark: 131414footnotemark: 14 Michael
Wibral151515MEG Unit, Brain Imaging Center, Goethe University, Frankfurt/Main,
Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The disruption of coupling between brain areas has been suggested as the
mechanism underlying loss of consciousness in anesthesia. This hypothesis has
been tested previously by measuring the information transfer between brain
areas, and by taking reduced information transfer as a proxy for decoupling.
Yet, information transfer is a function of the amount of information available
in the information source—such that transfer decreases even for unchanged
coupling when less source information is available. Therefore, we reconsidered
past interpretations of reduced information transfer as a sign of decoupling,
and asked whether impaired local information processing leads to a loss of
information transfer. An important prediction of this alternative hypothesis
is that changes in locally available information (signal entropy) should be at
least as pronounced as changes in information transfer. We tested this
prediction by recording local field potentials in two ferrets after
administration of isoflurane in concentrations of 0.0 %, 0.5 %, and 1.0 %.

We found strong decreases in the source entropy under isoflurane in area V1
and the prefrontal cortex (PFC)—as predicted by our alternative hypothesis.
The decrease in source entropy was stronger in PFC compared to V1. Information
transfer between V1 and PFC was reduced bidirectionally, but with a stronger
decrease from PFC to V1. This links the stronger decrease in information
transfer to the stronger decrease in source entropy—suggesting reduced source
entropy reduces information transfer. This conclusion fits the observation
that the synaptic targets of isoflurane are located in local cortical circuits
rather than on the synapses formed by interareal axonal projections. Thus,
changes in information transfer under isoflurane seem to be a consequence of
changes in local processing more than of decoupling between brain areas. We
suggest that source entropy changes must be considered whenever interpreting
changes in information transfer as decoupling.

††journal: PLoS Computational Biologyunc_psychunc_psychfootnotetext:
Department of Psychiatry, University of North Carolina at Chapel Hill, Chapel
Hill NC 27599, USAunc_neurobiolunc_neurobiolfootnotetext: Neurobiology
Curriculum, University of North Carolina at Chapel Hill Chapel Hill NC 27599,
USAunc_cellbiolunc_cellbiolfootnotetext: Department of Cell Biology and
Physiology, University of North Carolina at Chapel Hill, Chapel Hill NC 27599,
USAunc_biomedengunc_biomedengfootnotetext: Department of Biomedical
Engineering, University of North Carolina at Chapel Hill, Chapel Hill NC
27599, USAunc_neurosciunc_neuroscifootnotetext: Neuroscience Center,
University of North Carolina at Chapel Hill, Chapel Hill NC 27599,
USAunc_neurolunc_neurolfootnotetext: Department of Neurology, University of
North Carolina at Chapel Hill, Chapel Hill NC 27599, USA

## Author Summary

Currently we do not understand how anesthesia leads to loss of consciousness
(LOC). One popular idea is that we loose consciousness when brain areas lose
their ability to communicate with each other – as anesthetics might interrupt
transmission on nerve fibers coupling them. This idea has been tested by
measuring the amount of information transferred between brain areas, and
taking this transfer to reflect the coupling itself. Yet, information that
isn’t available in the source area can’t be transferred to a target. Hence,
the decreases in information transfer could be related to less information
being available in the source, rather than to a decoupling. We tested this
possibility measuring the information available in source brain areas and
found that it decreased under isoflurane anesthesia. In addition, a stronger
decrease in source information lead to a stronger decrease of the information
transfered. Thus, the input to the connection between brain areas determined
the communicated information, not the strength of the coupling (which would
result in a stronger decrease in the target). We suggest that interrupted
information processing within brain areas has an important contribution to
LOC, and should be focused on more in attempts to understand loss of
consciousness under anesthesia.

## Introduction

To this day it is an open question in anesthesia research how general
anesthesia leads to loss of consciousness (LOC). Several recent theories agree
in proposing that anesthesia-induced LOC may be caused by the disruption of
long range inter-areal information transfer in cortex [1, 2, 3, 4, 5]—a
hypothesis supported by a series of recent studies [3, 6, 7, 8, 9]. In all of
these studies, information transfer is quantified using transfer entropy [10],
an information theoretic measure, which has become a quasi-standard for the
estimation of information transfer in anesthesia research, or by transfer
entropy’s linear implementation as a Granger causality. In many of these
studies reduced information transfer has been interpreted as a sign of inter-
areal long range connectivity being disrupted by anesthesia.

Yet, information transfer between a source of information and a target depends
on information (entropy) being available at the source in the first place.
Considering constraints of this kind we can easily conceive of cases where a
decrease in information transfer under anesthesia is observed despite
unchanged long range coupling, e.g., when the available information at the
source decreases due to an anesthesia-related change in local information
processing. Ultimately, this dissociation between information transfer and
causal coupling just reflects that information transfer is one possible
consequence of physical coupling, but not identical to it [11, 12, 13].

Therefore, we consider it necessary to evaluate the hypothesis that the
reduced inter-areal information transfer observed under anesthesia possibly
originates from disrupted information processing in local circuits rather than
from disrupted long range connectivity. This alternative hypothesis receives
additional support for the case of isoflurane, which potentiates agonist
actions at $GABA_{A}$-receptors and inhibits nicotinic acetylcholine (nAChR)
receptors. Conversely, evidence on direct inhibitory effects of isoflurane on
AMPA and NMDA synapses, which are the dominant mediators of long-range
cortico-cortical interactions, is sparse at best (see table 2 in [14]). Under
the alternative hypothesis of changed local information processing, a decrease
in transfer entropy under anesthesia must be accompanied by:

  1. 1.

a reduction in locally available information per brain area, i.e. in the
sources of information transfer,

  2. 2.

and the fact that the strongest decrease in locally available information is
found at the source of the link with the strongest decrease in information
transfer, rather than at its target (i.e. the end point),

Here, we perform tests of these predictions by estimating local information
processing in and information transfer between local field potentials (LFPs)
simultaneously recorded from primary visual cortex (V1) and prefrontal cortex
(PFC) of two ferrets under different levels of isoflurane. We quantify local
information processing by estimating the signal entropy (measuring available
information) and quantified information transfer between recording sites by
estimating transfer entropy. Additionally, to demonstrate the effect of
reduced source entropy on transfer entropy, we estimated transfer entropy on
simulated data with a constant coupling between processes, but a varying
source entropy.

To better understand potential changes in local information processing we also
quantified the active information storage, a measure of the information
available at a recording site that can be predicted from past signals at that
site, i.e. the information stored from past to present.

Because the estimation of such information theoretic quantities from finite
data is difficult in general, we employ two complementary strategies: (i)
probability density estimation based on nearest-neighbor searches in
continuous data, and (ii) Bayesian estimation based on discretized data.

We also test whether the previously reported decrease of transfer entropy
under anesthesia can indeed be replicated when avoiding some recently
identified pitfalls in estimation of information transfer related to the use
of symbolic time series, suboptimal embedding of the time series, and the use
of net transfer entropy without identification of the individual information
transfer delays (for problems related to these approaches see [15]).

Our results provide first evidence for the alternative hypothesis of altered
local information processing causing reduced information transfer, as the
above predictions were indeed met. We suggest to consider the alternative
hypothesis as a serious candidate mechanism for LOC, and to use causal
interventions to gather further experimental evidence.

Preliminary results for this study were published in abstract form in [16].

## Results

### 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.

Table 1: Results significance test of $TE_{SPO}$ estimates in both animals and for both directions of interaction. animal | direction of interaction |  $p$-value
---|---|---
Ferret 1 | PFC $\rightarrow$ V1 | <0.01∗∗
| V1 $\rightarrow$ PFC | <0.05∗
Ferret 2 | PFC $\rightarrow$ V1 | <0.05∗
| V1 $\rightarrow$ PFC | 0.2262 n.s.
${}^{*}p<0.05$; ${}^{**}p<0.01$; ${}^{***}p<0.001$; Bonferroni-corrected |

### 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).

Figure 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. Table 2: Results of permutation analysis of variance for information theoretic measures ($p$-values). measure | effect | Ferret 1 | Ferret 2
---|---|---|---
$TE_{SPO}$ | isoflurane level | <0.0001∗∗∗ | <0.0001∗∗∗
| direction | 0.0003∗∗∗ | 0.9345
| interaction | 0.0052∗∗ | 0.2486a
$AIS$ | isoflurane level | 0.0017∗∗ | <0.0001∗∗∗
| recording site | 0.6774 | <0.0001∗∗∗
| interaction | <0.0001∗∗∗ | 0.0029∗∗
$H$ | isoflurane level | 0.0010∗∗ | <0.0001∗∗∗
| recording site | 0.9326 | <0.0001∗∗∗
| interaction | <0.0001∗∗∗ | 0.0243∗
${}^{*}p<0.05$; ${}^{**}p<0.01$; ${}^{***}p<0.001$;
aThis effect was significant when using LMM for statistical testing

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 $PFC\rightarrow V1$ (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 $BIC=-2\log
p(\mathbf{x}|M,\hat{a})+k(a)log(N)$, where $\mathbf{x}$ are the observed
realizations of the data, $\hat{a}$ 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, $-2\log p(\mathbf{x}|M,\hat{a})$, which
is higher for better model fit, and the $\chi^{2}$ with the corresponding
$p$-value, describing the likelihood ratio, which follows a
$\chi^{2}$-distribution.

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

animal measure effect BIC deviance $\mathcal{X}^{2}$ $\mathcal{X}$ df $p$
Ferret 1 $TE_{SPO}$ null model 345430.8 345401.7 NA NA NA isoflurane level
345421.5 345373.1 28.58 2 $<<$0.000∗∗∗ direction 342828.4 342789.7 2612.04 1
$<<$0.000∗∗∗ isoflurane l. \+ direction 342819.3 342761.3 NA NA NA isoflurane
l. $\times$ direction 342139.9 342062.4 698.84 2 $<<$0.000∗∗∗ $AIS$ null model
37581.2 37551.8 NA NA NA isoflurane level 37555.8 37506.8 46.00 2 $<<$0.000∗∗∗
recording site 33296.2 33256.9 4294.82 1 $<<$0.000∗∗∗ isoflurane l. \+
recording site 33270.9 33212.1 NA NA NA isoflurane l. $\times$ recording site
30754.3 30675.8 2536.29 2 $<<$0.000∗∗∗ $H$ null model 353650.8 353621.8 NA NA
NA isoflurane level 353653.1 353604.7 17.03 2 <0.000∗∗∗ recording site
353446.8 353408.1 213.64 1 $<<$0.000∗∗∗ isoflurane l. \+ recording site
353449.1 353391.1 NA NA NA isoflurane l. $\times$ recording site 347204.7
347127.2 6263.85 2 $<<$0.000∗∗∗ Ferret 2 $TE_{SPO}$ null model 131164.7
131135.3 NA NA NA isoflurane level 131152.3 131103.2 32.06 2 $<<$0.000∗∗∗
direction 131168.9 131129.6 5.66 1 0.020 isoflurane l. \+ direction 131109.5
131097.5 NA NA NA isoflurane l. $\times$ direction 130871.5 130793.0 304.56 2
$<<$0.000∗∗∗ $AIS$ null model 109203.1 109173.7 NA NA NA isoflurane level
109183.1 109134.1 39.63 2 $<<$0.000∗∗∗ recording site 105827.2 105788.0
3385.68 1 $<<$0.000∗∗∗ isoflurane l. \+ recording site 105807.3 105748.4 NA NA
NA isoflurane l. $\times$ recording site 104173.5 104095.0 1653.46 2
$<<$0.000∗∗∗ $H$ null model -14074.8 -14104.2 NA NA NA isoflurane level
-14088.6 -14137.7 33.43 2 <0.000∗∗∗ recording site -18423.0 -18462.3 4358.04 1
$<<$0.000∗∗∗ isoflurane l. \+ recording site -18436.7 -18495.6 NA NA NA
isoflurane l. $\times$ recording site -19867.2 -19945.7 1450.09 2 $<<$0.000∗∗∗

#### 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 $N_{bins}\geq 8$ 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.

Figure 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 (${}^{*}p<0.05$; ${}^{**}p<0.01$;
${}^{***}p<0.001$).

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

Figure 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 $N\geq 100,000$; (B) entropies for ferret 2, PFC, awake: 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).

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.

Table 4: Results of permutation analysis of variance for information theoretic measures obtained through Bayesian estimation ($p$-values). measure | effect | Ferret 1 | Ferret 2
---|---|---|---
$TE_{SPO}$ | isoflurane level | 0.0549 | 0.0013∗
| direction | 0.0272∗ | 0.0002∗∗∗
| interaction | 0.6627 | 0.0616
$AIS$ | isoflurane level | 0.2148 | <0.0001∗∗∗
| direction | 0.0788 | 0.0032∗∗
| interaction | 0.0026∗∗ | 0.0153∗
$H$ | isoflurane level | 0.0184∗ | <0.0001∗∗∗
| recording site | 0.2738 | <0.0001∗∗∗
| interaction | 0.0017∗∗ | 0.0053
${}^{*}p<0.05$; ${}^{**}p<0.01$; ${}^{***}p<0.001$

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.

Figure 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
$\pm$ SEM; (B) source entropy $H(\mathbf{X}_{t-u}^{d_{X}})$ (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.

### 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 [22, 23, 15] 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 $\tau$ (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 [7, 6], when choosing
the embedding dimension ad-hoc or using other criteria as for example the
sample size [9]. The embedding delay $\tau$ 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 $\delta$ 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 $u_{PFC\rightarrow V1}$ and
$u_{V1\rightarrow PFC}$ for each direction of information transfer and hence
became uninterpretable (Fig. 5D).

Figure 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 $\delta$,
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, $TE_{SPO}(X\rightarrow Y,t,u)$ (black line) and
$TE_{SPO}(Y\rightarrow X,t,u)$ (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 $\Delta$ (dashed lines), where
$TE_{SPO}(X\rightarrow Y,t,u_{opt})>TE_{SPO}(Y\rightarrow X,t,u_{opt})$; (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 $u_{PFC\rightarrow V1}$ (x-axis) and $u_{V1\rightarrow PFC}$
(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.

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).

Figure 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 ($p<0.05$).

Note that in Fig. 5C, $TE_{SPO}$ as a function of $u$ shows multiple peaks,
especially for the direction $V1\rightarrow PFC$. 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 $\delta$ (Fig. 7C). Yet,
when filtering using narrower bandwidths, the reconstruction of the correct
information transfer delay failed (Fig. 7D–F). This finding supports the
general notion that filtering, especially narrow-band filtering, may only be
applied with caution before estimating connectivity measures [24, 25].

Figure 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
$\delta$, 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 $\delta$ in each run with an error
of 1 sample, i.e. $\delta$ 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.

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.

Figure 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.

Detailed tables showing correlation coefficients and explained variances are
provided as supporting information S2. In summary, for $AIS$ and ACT we found
significant correlations in both animals, however the median of the variance
explained, $\widetilde{R}^{2}$, 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
$\widetilde{R}^{2}\leq 0.015$, 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 table 8 as part of supporting information S2, $\widetilde{R}^{2}<0.16$);
for $H$ and band power, we found correlations predominantly in higher
frequency bands (beta and gamma, see table 9 as part of supporting information
S2, $\widetilde{R}^{2}<0.1$); 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 table 10 as part of supporting information S2,
$\widetilde{R}^{2}<0.024$). 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 $H$.

## Discussion

We analyzed long-range information transfer between areas V1 and PFC in two
ferrets under different levels of isoflurane. We found that transfer entropy
was indeed reduced under isoflurane and that this reduction was more
pronounced in top-down directions. These results validate earlier findings
made using different estimation procedures [3, 8, 6, 7, 9]. As far as
information transfer alone was concerned our results are compatible with an
interpretation of reduced long-range information transfer due to reduced
inter-areal coupling. Yet, this interpretation provides no direct explanation
for the findings of reduced locally available information as explained below.
In contrast, the alternative hypothesis that the reduced long range
information transfer is a secondary effect of changes in local information
processing provides a concise explanation for our findings both with respect
to locally available information, and information transfer.

### 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, 35,
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 [39, 37]. 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 S3). 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.

## Methods

### 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-$\mu$m shank diameter,
500-k$\Omega$ 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.

Figure 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).

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, $PFC\rightarrow V1$ and $V1\rightarrow PFC$. 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 $\mathcal{X}$ and $\mathcal{Y}$ (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
$X=\\{X_{1},\ldots,X_{t},\ldots,X_{N}\\}$ and
$Y=\\{Y_{1},\ldots,Y_{t},\ldots,Y_{N}\\}$. The index $t$ here indicates
samples in time, measured in units of the dwell time (inverse sampling rate)
of the recording.

#### Transfer entropy

Transfer entropy [10, 22, 49] is defined as the mutual information between the
future of a process $Y$ and the past of a second process $X$, conditional on
the past of $Y$. Transfer entropy thus quantifies the information we obtain
about the future of $Y$ from the past of $X$, taking into account information
from the past of $Y$. Taking this past of $Y$ into account here removes
information redundantly available in the past of both $X$ and $Y$, and reveals
information provided synergistically by them [50]. In this study, we used an
improved estimator of transfer entropy presented in [15], which accounts for
arbitrary information transfer delays:

$\displaystyle TE_{SPO}(X\rightarrow
Y,t,u)=I\left(Y_{t};\mathbf{X}_{t-u}^{d_{X}}|\mathbf{Y}_{t-1}^{d_{Y}}\right),$
(1)

where $I$ is the conditional mutual information (or the differential
conditional mutual information for continuous valued variables) between
$Y_{t}$ and $\mathbf{X}_{t-u}^{d_{X}}$, conditional on
$\mathbf{Y}_{t-1}^{d_{Y}}$; $Y_{t}$ is the future value of random process $Y$,
and $\mathbf{X}_{t-u}^{d_{X}}$, $\mathbf{Y}_{t-1}^{d_{Y}}$ are the past states
of $X$ and $Y$, respectively. Past states are collections of past random
variables

$\displaystyle\mathbf{Y}_{t-1}^{d_{Y}}=\left(Y_{t-1},Y_{t-1-\tau},\ldots,Y_{t-1-(d_{Y}-1)\tau}\right),$
(2)

that form a delay embedding of length $d_{Y}$ [51], and that render the future
of the random process conditionally independent of all variables of the random
process that are further back in time than the variables forming the state.
Parameters $\tau$ and $d$ denote the embedding delay and embedding dimension
and can be found through optimization of a local predictor as proposed in [21]
(see next section on the estimation of information theoretic measures). Past
states constructed in this manner are then maximally informative about the
present variable of the target process, $Y_{t}$, which is an important
prerequisite for the correct estimation of transfer entropy (see also [15]).

In our estimator (Eq. 1), the variable $u$ describes the assumed information
transfer delay between the processes $X$ and $Y$, which accounts for a
physical delay $\delta_{X,Y}\geq 1$ [15]. The estimator thus accommodates
arbitrary physical delays between processes. The true delay $\delta_{X,Y}$
must be recovered by ’scanning’ various assumed delays and keeping the delay
that maximizes $TE_{SPO}$ [15]:

$\displaystyle\hat{\delta}_{X,Y}=\arg\max_{u}\left(TE_{SPO}\left(X\rightarrow
Y,t,u\right)\right).$ (3)

#### Active Information Storage

$AIS$ [19] is defined as the (differential) mutual information between the
future of a signal and its immediate past state

$\displaystyle AIS(Y_{t})=I\left(Y_{t};\mathbf{Y}_{t-1}^{d_{Y}}\right),$ (4)

where $Y$ again is a random process with present value $Y_{t}$ and past state
$\mathbf{Y}_{t-1}^{d_{Y}}$ (see Eq. 2). $AIS$ thus quantifies the amount of
predictable information in a process or the information that is currently in
use for the next state update [19]. $AIS$ is low in processes that produce
little information or are highly unpredictable, e.g., fully stochastic
processes, whereas $AIS$ is highest for processes that visit many equi-
probable states in a predictable sequence, i.e., without branching. In other
words, $AIS$ is high for processes with “rich dynamics” that are predictable
from the processes’ past [52]. A reference implementation of $AIS$ can be
found in the Java Information Dynamics Toolkit (JIDT) [53]. As for $TE_{SPO}$
estimation, an optimal delay embedding $\mathbf{Y}_{t-1}^{d_{Y}}$ may be found
through optimization of the local predictor proposed in [21].

Note, that $AIS$ is upper bounded by the entropy as:

$\displaystyle\begin{split}AIS(Y)&=I\left(Y_{t};\mathbf{Y}_{t-1}^{d_{Y}}\right)\\\
&=H\left(Y_{t}\right)-H\left(Y_{t}|\mathbf{Y}_{t-1}^{d_{Y}}\right).\end{split}$
(5)

#### Differential entropy

The differential entropy $H$ (see for example [48]) expands the classical
concept of Shannon’s entropy for discrete variables to continuous variables:

$\displaystyle
H=-\int_{\mathbb{X}_{t-u}^{d_{X}}}f(\mathbf{X}_{t-u}^{d_{X}})\log
f(\mathbf{X}_{t-u}^{d_{X}})\,d\mathbf{X}_{t-u}^{d_{X}},$ (6)

where $f(Y_{t})$ is the probability density function of $Y_{t}$ over the
support $\mathbb{Y}$. Entropy quantifies the average information contained in
a signal. Based on the differential entropy the corresponding measures for
mutual and conditional mutual information and, thereby, active information
storage and transfer entropy can be defined.

### Entropy as an upper bound on information transfer

The transfer entropy from Eq. 1 can be rewritten as:

$\displaystyle\begin{split}TE_{SPO}(X\rightarrow
Y,t,u)=&I\left(\mathbf{X}_{t-u}^{d_{X}}:Y_{t}|\mathbf{Y}_{t-1}^{d_{Y}}\right)\\\
=&H(\mathbf{X}_{t-u}^{d_{X}}|\mathbf{Y}_{t-1}^{d_{Y}})-H(\mathbf{X}_{t-u}^{d_{X}}|\mathbf{Y}_{t-1}^{d_{Y}},Y_{t})~{},\end{split}$
(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

$\begin{split}TE_{SPO}(X\rightarrow Y,t,u)\leq
H(\mathbf{X}_{t-u}^{d_{X}})\end{split}$ (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.

#### Estimating information theoretic measures from continuous data

Estimation of information theoretic measures from continuous data is often
handled by simply discretizing the data. This is done either by binning or the
use of symbolic time series—mapping the continuous data onto a finite
alphabet. Specifically, the use of symbolic time series for transfer entropy
estimation was first introduced by [55] and maps the continuous values in past
state vectors with length $d$ (Eq. 2) onto a set of rank vectors. Hence, the
continuous-valued time series is mapped onto an alphabet of finite size $d!$.
After binning or transformation to rank vectors transfer entropy and the other
information theoretic measures can then be estimated using plug-in estimators
for discrete data, which simply evaluate the relative frequency of occurrences
of symbols in the alphabet. Discretizing the data therefore greatly simplifies
the estimation of transfer entropy from neural data, and may even be necessary
for very small data sets. Yet, binning ignores the neighborhood relations in
the continuous data and the use of symbolic times series destroys important
information on the absolute values in the data. An example where transfer
entropy estimation fails due to the use of symbolic time series is reported in
[56] and discussed in [15]: In this example, information transfer between two
coupled logistic maps was not detected by symbolic transfer entropy [56]; only
when estimating $TE_{SPO}$ directly using an estimator for continuous data,
the information transfer was identified correctly [15]. To circumvent the
problems with binned or symbolic time series, we here used a nearest-neighbor
based $TE_{SPO}$-estimator for continuous data, the Kraskov-Stögbauer-
Grassberger (KSG) estimator for mutual information described in [57]. At
present, this estimator has the most favorable bias properties compared to
similar estimators for continuous data. The KSG-estimator leads to the
following expression for the estimation of $TE_{SPO}$ as introduced in Eq. 1
[58, 22]:

$\displaystyle\begin{split}TE_{SPO}(X\rightarrow&Y,t,u)=I(Y_{t}:\mathbf{X}^{d_{X}}_{t-u}|\mathbf{Y}^{d_{Y}}_{t-1})\\\
&=\psi(k)+\langle\psi(n_{\mathbf{y}_{t-1}^{d_{Y}}}+1)-\psi(n_{y_{t}\mathbf{y}_{t-1}^{d_{Y}}}+1)-\psi(n_{\mathbf{y}_{t-1}^{d_{Y}}\mathbf{x}_{t-u}^{d_{X}}}+1)\rangle_{r},\end{split}$
(9)

where $\psi$ denotes the digamma function, $k$ is the number of neighbors in
the highest-dimensional space spanned by variables $Y_{t}$,
$\mathbf{Y}^{d_{Y}}_{t-1}$, $\mathbf{X}^{d_{X}}_{t-u}$, and is used to
determine search radii for the lower dimensional subspaces; $n_{\cdot}$ are
the number of neighbors within these search radii for each point in the lower
dimensional search spaces spanned by the variable indicated in the subscript.
Angle brackets indicate the average over realizations $r$ (e.g. observations
made over an ensemble of copies of the systems or observations made over time
in case of stationarity, which we assumed here). We used a $k$ of 4 as
recommended by Kraskov [59, p. 23], such as to balance the estimator’s
bias—which decreases for larger $k$—and variance—which increases for larger
$k$ (see also [60] for similar recommendations based on simulation studies).
For a detailed derivation of $TE_{SPO}$-estimation using the KSG-estimator see
[57, 22, 58].

The KSG-estimator comes with a bias that is not analytically tractable [59],
hence, estimates can not be interpreted at face value, but have to be tested
for their statistical significance against the null-hypothesis of no
information transfer [23, 22]. A suitable test distribution for this null-
hypothesis can be generated by repeatedly estimating $TE_{SPO}$ from surrogate
data. In these surrogate data, the quantity of interest—potential transfer
entropy between source and target time series—should be destroyed while all
other statistical properties of the data are preserved such that the
estimation bias is constant between original and surrogate data. In the
present study we thus generated 500 surrogate data sets by permuting whole
epochs of the target time series while leaving the order of source time course
epochs unchanged (see [61] for a detailed account of surrogate testing).

We used the ensemble-method for transfer entropy estimation [61], implemented
in TRENTOOL [23]. The ensemble method allows to pool data over epochs, which
maximizes the amount of data entering the estimation, while providing an
efficient implementation of this estimation procedure using graphics
processing units (GPU). We tested the statistical significance of $TE_{SPO}$
in both directions of interaction in the iso 0.0 % condition. We only tested
this condition, because $TE_{SPO}$ was expected to be reduced for higher
isoflurane levels based on the results of existing studies. Because the
estimation of $TE_{SPO}$ is computationally heavy, we used a random subset of
50 epochs to reduce the running time of this statistical test (see supporting
information S4 for theoretical and practical running times of the used
estimators). We first tested $TE_{SPO}$ estimates for their significance
within individual recording sessions against 500 surrogate data sets, and then
used a binomial test to establish the statistical significance over
recordings. We used a one-sided Binomial test under the null hypothesis of no
significant $TE_{SPO}$ estimates, where individual estimates $l$ were assumed
to be $B(l,p_{0},n)$-distributed, with $p_{0}=0.05$ and $n=5$ for animal 1 and
$p_{0}=0.05$ and $n=8$ for animal 2.

As the KSG-estimator used for estimating $TE_{SPO}$ (Eq. 9) is an estimator of
mutual information it can also be used for the estimation of $AIS$:

$\displaystyle\begin{split}AIS(Y)&=I(Y_{t}:\mathbf{Y}^{d_{Y}}_{t-1})\\\
&=\psi(k)-1/k+\psi(N)-\langle\psi(n_{y_{t}})+\psi(n_{\mathbf{y}^{d_{Y}}_{t-1}})\rangle_{r},\end{split}$
(10)

where again $\psi$ denotes the digamma function, $k$ is the number of
neighbors in the highest-dimensional space, $N$ is the number of realizations,
and $n_{\cdot}$ denotes the number of neighbors for each point in the
respective search space. Again, we chose $k=4$ for the estimation of AIS (see
above). Note that the sampling rate has an effect on the estimated absolute
values of AIS (a simulation of the effect of sampling on AIS estimates are
shown as supporting information S3)—however, qualitative results are not
influenced by the choice of sampling rate; in other words, relative
differences between estimates are the same for different choices of sampling
rates. As a consequence, for AIS estimation the sampling rate should be
constant over data sets if the aim is to compare these estimates.

A conceptual predecessor of the KSG-estimator for mutual information is the
Kozachenko-Leonenko (KL) estimator for differential entropies [62]. The KL-
estimator also allows for the estimation of $H$ from continuous data and reads

$\displaystyle H(X)=-\psi(1)+\psi(N)+\sum_{i=1}^{N}{\log(\epsilon(i))},$ (11)

where $\epsilon(i)$ is twice the distance from data point $i$ to its $k$-th
nearest neighbor in the search space spanned by all points.

#### Bayesian Estimators for discretized data

For Bayesian estimation we converted the continuous LFP time series to
discrete data by applying voltage bins as follows: The voltages $\pm 3$
standard deviations around the mean of the LFP were subdivided into $N$
equally spaced bins. We added two additional bins containing all the values
that were either smaller or larger than the 3 SD region, amounting to a total
number of bins $N_{bins}=N+2$. We then calculated $H$, $AIS$ and $TE_{SPO}$
for the discrete data. For $AIS$ and $TE_{SPO}$ estimation states were defined
using the same dimension $d$ and $\tau$ as for the KSG-estimator, optimized
using the Ragwitz criterion. We decomposed $TE_{SPO}$ into four entropies (Eq.
9), and $AIS$ into three entropies, which we then estimated individually [63].
To reduce the bias introduced by the limited number of observed states, we
used the NSB-estimator by Nemenman, Shafee, and Bialek [17], which is based on
the construction of an almost uniform prior over the expected entropy using a
mixture of symmetric Dirichlet priors $\mathcal{P}_{\beta}$. The estimator has
been shown to be unbiased for a broad class of distributions that are typical
in $\mathcal{P}_{\beta}$ [18].

We further applied the recently proposed estimator by Archer et al. [20] that
uses a prior over distributions with infinite support based on Pitman-Yor-
processes. In contrast to the NSB prior, this prior also accounts for heavy-
tailed distributions that one encounters frequently in neuronal systems and
does not require knowledge of the support of the distribution.

When estimating entropies for the embedding dimensions used here, the number
of possible states or “words” is between $K=12^{14}$ and $K=12^{31}$. This is
much larger than the typical number of observed states per recording of around
$N=5\cdot 10^{5}$. As a consequence, a precise estimation of entropies is only
possible if the distribution is sparse, i.e. most words have vanishing
probability. In this case, however, the estimates should be independent of the
choice of support $K$ as long as $K$ is sufficiently large and does not omit
states of finite probability. We chose $K^{\prime}=10^{12}$ for the results
shown in this paper, which allowed a robust computation of the NSB estimator
instead of the maximum support $K$ that results from simple combinatorics.

#### 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 Eq. 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.

Figure 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 $X$, $x_{t-(\tau\cdot d)}$, conditional on all intermediate
values $\mathbf{x}_{t-1}^{d_{X}}$ (shaded box), the mutual information is non-
zero, i.e., $x_{t-(\tau\cdot d)}$ holds some information about $x_{t}$. (C)
Both directions of interaction are analyzed; (D) Information in
$x_{t-(\tau\cdot d)}$ (white sample point) is transferred to $Y$ (solid
arrow), because of the actual coupling $X\rightarrow Y$. The information in
$x_{t-(\tau\cdot d)}$ about $x_{t}$ is thus transferred to the past of $Y$,
$\mathbf{y}_{t-u}^{d_{Y}}$, which thus becomes predictive of $x_{t}$ as well.
Assume now, we analyzed information transfer from $Y$ to $X$,
$I(x_{t}:\mathbf{y}_{t-u}^{d_{Y}}|\mathbf{x}_{t-1}^{d_{X}})$, without a proper
embedding of $X$, $\mathbf{x}_{t-1}^{d_{X}}$: Because of the actually
transferred information from $x_{t-(\tau\cdot d)}$ to
$\mathbf{y}_{t-u}^{d_{Y}}$, the mutual information
$I(x_{t}:\mathbf{x}_{t-1}^{d_{X}})$ is non-zero. If we now under-embed $X$,
such that the information in $x_{t-(\tau\cdot d)}$ is not contained in
$\mathbf{x}_{t-u}^{d_{X}}$ and is not conditioned out,
$I(x_{t}:\mathbf{y}_{t-1}^{d_{Y}}|\mathbf{x}_{t-u}^{d_{X}})$ 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
$Y\rightarrow X$. (E) Information transfer is falsely detected for both
directions of interaction, the link from $Y$ to $X$ is spurious (dashed
arrow).

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 $\tau$ 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 $\tau$. 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].

#### Reconstruction of information transfer delays

The third methodological problem raised in the introduction was failure to
account for a physical delay $\delta$ between neural sites when estimating
transfer entropy. In our estimator $TE_{SPO}$ (Eq. 9) we account for $\delta$
by introducing the parameter $u$. The delay $u$ needs to be optimized to
correctly estimate $TE_{SPO}$. If $u$ is not optimal, i.e., $u$ is not
sufficiently close to $\delta$ (Fig. 5 and [15]), information transfer may be
underestimated or not measured at all. This is because choosing the parameter
$u$ too large ($u\gg\delta$) means that the information present in the
evaluated samples of the source is also present in the history of the target
already, and conditioned away. In contrast, choosing the parameter $u$ too
small means that the information of the evaluated samples of the source will
only arrive in the future of the current target sample, and is useless for
providing information about it (Fig. 5A).

It can be proven for bivariate systems that $TE_{SPO}$ becomes maximal when
the true delay $\delta$ is chosen for $u$ [15]. Therefore, the true delay
$\delta$—and thus an optimal choice for $u$—can be found by using the value
for $u$ that maximizes $TE_{SPO}$ [15]. This optimal $u$ can be found by
scanning a range of assumed values. In the present study, we scanned values
ranging from 0 to 20 ms. (Note that assumed values should be physiologically
plausible to keep the computations practically feasible.)

Accounting for the information transfer $\delta$ by finding optimal parameters
$u$ for $TE_{SPO}$ estimation has important consequences when calculating
indices from estimated $TE_{SPO}$, such as $TE_{net}$:

$\displaystyle TE_{net}=\frac{TE_{SPO}(X\rightarrow Y,t,u_{X\rightarrow
Y})-TE_{SPO}(Y\rightarrow X,t,u_{Y\rightarrow X})}{TE_{SPO}(X\rightarrow
Y,t,u_{X\rightarrow Y})+TE_{SPO}(Y\rightarrow X,t,u_{Y\rightarrow X})},$ (12)

or variations of this measure. The $TE_{net}$ is popular in anesthesia
research [7, 9] and indicates the predominant direction of information
transfer between two bidirectionally coupled processes $X$ and $Y$
($TE_{net}\geq 0$ if $TE_{SPO}(X\rightarrow Y,t,u_{X\rightarrow Y})\geq
TE_{SPO}(Y\rightarrow X,t,u_{Y\rightarrow X})$ and $TE_{net}<0$ if
$TE_{SPO}(Y\rightarrow X,t,u_{Y\rightarrow X})>TE_{SPO}(X\rightarrow
Y,t,u_{X\rightarrow Y})$). However, if values for $u_{X\rightarrow Y}$ and
$u_{Y\rightarrow X}$ are not optimized individually, $TE_{net}$ may take on
arbitrary signs: In Fig. 5B, we show a toy example of two coupled Lorenz
systems [15], where the absolute difference between raw $TE_{SPO}$ values
changes as a function of a common $u$ for both directions and where the
difference even changes signs for values $u>65$. To obtain a meaningful value
from $TE_{net}$ we thus need to find the individually optimal choices of $u$
for both directions of transfer—in the example, these optima are found at
$u_{X\rightarrow Y}=46$ and $u_{Y\rightarrow X}=76$, leading to the “true”
difference.

#### Simulating the effect of filtering on information transfer delay
reconstruction

To simulate the effects of filtering as a preprocessing technique on the
ability of the $TE_{SPO}$ estimator to reconstruct the correct information
transfer delay, we simulated two coupled time series, for which we estimated
transfer entropy before filtering and after band-pass filtering with different
bandwidths. The simulation was repeated 50 times.

We simulated two time series with 100,000 samples each, which were drawn from
a uniform random distribution over the open interval (0,1). We introduced a
coupling between the time series by adding a scaled version (factor 0.2) of
the first time series to the second with a delay of 10 samples. We estimated
$TE_{SPO}$ from the first to the second time series using the KSG-estimator
implemented in the JIDT toolbox, with $k=4$ and a history of one sample for
both source and target.

We estimated $TE_{SPO}$ with and without band-pass filtering the data for
different values of $u$, ranging from 1 to 20 samples. We filtered the data
using a fourth order, causal Butterworth filter, implemented in the MATLAB
toolbox FieldTrip [26]. We filtered the data using four different bandwidths:
from 0.1 to 300 Hz (corresponding to the filtering done in this study), from
0.1 to 200 Hz, from 12 to 30 Hz (corresponding to the beta frequency range),
and from 4 to 8 Hz (corresponding to the theta frequency range).

### 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 [65, 66, 67, 39]. 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:

$y_{ij}=\beta_{0}+\gamma_{j}+\beta_{1}D_{i}+\beta_{2}A^{(0.5)}_{i}+\beta_{3}A^{(1.0)}_{i}+\epsilon_{ij},$

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
$\beta_{0}$ describes the model intercept, $\beta_{k}$ are the regression
coefficients and describe fixed effects, $\gamma_{j}$ is the random deviation
of recording $j$ from the intercept $\beta_{0}$, and $\epsilon_{ij}$ describes
random noise. $D_{i}$, $A^{(0.5)}_{i}$, and $A^{(1.0)}_{i}$ are predictor
variables, encoding factors direction as

$D_{i}:=\begin{cases}1&\textrm{if }\textrm{ {direction} is }PFC\rightarrow
V1,\\\ -1&\textrm{if }\textrm{ {direction} is }V1\rightarrow PFC,\end{cases}$
(13)

and factor isoflurane level as

$A^{(0.5)}_{i}:=\begin{cases}1&\textrm{if {isoflurane level} is {iso 0.5
\%},}\\\ 0&\textrm{else,}\end{cases}$ (14)

$A^{(1.0)}_{i}:=\begin{cases}1&\textrm{if {isoflurane level} is {iso 1.0
\%},}\\\ 0&\textrm{else.}\end{cases}$ (15)

Note that we used dummy coding for factor direction so that the the estimated
effect $\beta_{1}$ 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 $\beta_{2}$ and $\beta_{3}$ as
deviations from a reference group (in this case the condition iso 0.0 %). We
further assume that noise was i.i.d. and
$\epsilon_{ij}\sim\mathcal{N}\left(0,\sigma^{2}_{\epsilon}\right)$ and
$\gamma_{j}\sim\mathcal{N}\left(0,\sigma^{2}_{\gamma}\right)$.

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 $H(\mathbf{X}_{t-u}^{d_{X}})$
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, $R^{2}$, for
individual recordings, because calculating $R^{2}$ for the coefficient
calculated from pooled data does not yield interpretable results.

## Acknowledgements

This study was partly inspired by the work of Vasily A. Vakorin on the
interplay of complexity and information transfer in [74]. The authors thank
Saskia Helbling for fruitful discussions on data analysis and for making the
permutation ANOVA code available. The authors acknowledge support by the Human
Brain Project (EU Grant No. 604102).

## Supporting Information

#### S1 Dataset

Group statistical data and estimated information-theoretic values entering
statistical tests.

#### S2 Correlation tables

Correlation between information-theoretic measures and traditional measures of
time-series properties.

Table 5: Correlation of $AIS$ with autocorrelation decay time (ACT). $\widetilde{R}^{2}$ and $max(R^{2})$ indicate the median and maximum of $R^{2}$ over recordings per condition, respectively. animal |  isoflurane level |  recording site | $p$ |  $\widetilde{R}^{2}$ ($max(R^{2})$
---|---|---|---|---
1 | iso 0.0 % | PFC | 1.0000 | 0.024 (0.289)
|  | V1 | 0.1028 | 0.003 (0.153)
| iso 0.5 % | PFC | 0.0013∗∗ | 0.001 (0.122)
|  | V1 | 0.0000∗∗∗ | 0.014 (0.114)
| iso 1.0 % | PFC | 0.0000∗∗∗ | 0.005 (0.271)
|  | V1 | 0.9998 | 0.004 (0.051)
2 | iso 0.0 % | PFC | 0.0000∗∗∗ | 0.022 (0.116)
|  | V1 | 0.0000∗∗∗ | 0.029 (0.175)
| iso 0.5 % | PFC | 0.7672 | 0.007 (0.018)
|  | V1 | 0.0000∗∗∗ | 0.017 (0.076)
| iso 1.0 % | PFC | 0.0000∗∗∗ | 0.008 (0.121)
|  | V1 | 0.8247 | 0.009 (0.214)
${}^{*}p<0.05$; ${}^{**}p<0.01$; ${}^{***}p<0.001$
Table 6: Correlation of $H$ with signal variance. $\widetilde{R}^{2}$ and $max(R^{2})$ indicate the median and maximum of $R^{2}$ over recordings per condition, respectively. animal |  isoflurane level |  recording site | $p$ |  $\widetilde{R}^{2}$ ($max(R^{2})$
---|---|---|---|---
1 | iso 0.0 % | PFC | 1.0000 | 0.087 (0.705)
|  | V1 | 1.0000 | 0.079 (0.452)
| iso 0.5 % | PFC | 1.0000 | 0.036 (0.093)
|  | V1 | 1.0000 | 0.062 (0.217)
| iso 1.0 % | PFC | 1.0000 | 0.010 (0.474)
|  | V1 | 1.0000 | 0.018 (0.083)
2 | iso 0.0 % | PFC | 1.0000 | 0.042 (0.699)
|  | V1 | 1.0000 | 0.162 (0.363)
| iso 0.5 % | PFC | 1.0000 | 0.041 (0.130)
|  | V1 | 1.0000 | 0.031 (0.289)
| iso 1.0 % | PFC | 1.0000 | 0.118 (0.602)
|  | V1 | 1.0000 | 0.018 (0.578)
${}^{*}p<0.05$; ${}^{**}p<0.01$; ${}^{***}p<0.001$
Table 7: Correlation of $TE_{SPO}$ with source autocorrelation decay time
(ACT) and source signal variance. $\widetilde{R}^{2}$ and $max(R^{2})$
indicate the median and maximum of $R^{2}$ over recordings per condition,
respectively.

animal isoflurane level direction ACT signal variance $p$ $\widetilde{R}^{2}$
($max(R^{2})$ $p$ $\widetilde{R}^{2}$ ($max(R^{2})$ 1 iso 0.0 % PFC
$\rightarrow$ V1 1.0000 0.035 (0.110) 0.9861 0.009 (0.060) V1 $\rightarrow$
PFC 0.9996 0.029 (0.215) 0.6644 0.024 (0.089) iso 0.5 % PFC $\rightarrow$ V1
1.0000 0.016 (0.110) 1.0000 0.024 (0.033) V1 $\rightarrow$ PFC 0.7310 0.003
(0.014) 0.0063∗ 0.005 (0.011) iso 1.0 % PFC $\rightarrow$ V1 1.0000 0.006
(0.047) 0.5836 0.002 (0.037) V1 $\rightarrow$ PFC 0.0069∗ 0.001 (0.032)
0.0000∗∗∗ 0.005 (0.092) 2 iso 0.0 % PFC $\rightarrow$ V1 1.0000 0.063 (0.105)
0.8315 0.008 (0.025) V1 $\rightarrow$ PFC 0.9996 0.006 (0.041) 0.0434 0.005
(0.031) iso 0.5 % PFC $\rightarrow$ V1 1.0000 0.115 (0.155) 0.0440 0.017
(0.103) V1 $\rightarrow$ PFC 1.0000 0.022 (0.125) 0.9985 0.005 (0.086) iso 1.0
% PFC $\rightarrow$ V1 0.0000∗∗ 0.011 (0.111) 0.0000∗∗∗ 0.012 (0.058) V1
$\rightarrow$ PFC 0.0076∗ 0.009 (0.031) 0.0000∗∗∗ 0.015 (0.032)
${}^{*}p<0.05$; ${}^{**}p<0.01$; ${}^{***}p<0.001$

Table 8: Correlation of $AIS$ with band power. $\widetilde{R}^{2}$ and
$max(R^{2})$ indicate the median and maximum of $R^{2}$ over recordings per
condition, respectively.

animal level site $\delta$ $\theta$ $\alpha$ $p$ $\widetilde{R}^{2}$
($max(R^{2})$ $p$ $\widetilde{R}^{2}$ ($max(R^{2})$ $p$ $\widetilde{R}^{2}$
($max(R^{2})$ 1 iso 0.0 % PFC 1.0000 0.068 (0.418) 0.0000∗∗∗ 0.021 (0.201)
0.0000∗∗∗ 0.022 (0.258) V1 0.0000∗∗∗ 0.041 (0.275) 0.0003∗∗ 0.020 (0.160)
0.0000∗∗∗ 0.029 (0.166) iso 0.5 % PFC 0.0000∗∗∗ 0.019 (0.174) 0.0000∗∗∗ 0.005
(0.040) 0.0000∗∗∗ 0.003 (0.075) V1 0.0000∗∗∗ 0.008 (0.129) 0.0093 0.024
(0.060) 0.0986 0.009 (0.090) iso 1.0 % PFC 0.0000∗∗∗ 0.006 (0.036) 0.8710
0.005 (0.093) 0.0445 0.003 (0.028) V1 0.2751 0.010 (0.303) 0.0000∗∗∗ 0.002
(0.054) 0.0000∗∗∗ 0.002 (0.030) 2 iso 0.0 % PFC 0.0000∗∗∗ 0.158 (0.359)
0.0000∗∗∗ 0.044 (0.089) 0.0000∗∗∗ 0.025 (0.075) V1 0.0000∗∗∗ 0.005 (0.303)
0.0000∗∗∗ 0.023 (0.228) 0.0000∗∗∗ 0.042 (0.238) iso 0.5 % PFC 0.0000∗∗∗ 0.019
(0.255) 0.0000∗∗∗ 0.007 (0.018) 0.0000∗∗∗ 0.006 (0.074) V1 0.0000∗∗∗ 0.025
(0.098) 0.6476 0.024 (0.068) 1.0000 0.017 (0.045) iso 1.0 % PFC 0.0000∗∗∗
0.005 (0.298) 0.9729 0.005 (0.021) 0.9999 0.004 (0.151) V1 0.0063∗ 0.043
(0.302) 0.7462 0.001 (0.039) 1.0000 0.003 (0.051) animal level site $\beta$
$\gamma$ $p$ $\widetilde{R}^{2}$ ($max(R^{2}$) $p$ $\widetilde{R}^{2}$
($max(R^{2})$ 1 iso 0.0 % PFC 0.1179 0.016 (0.154) 1.0000 0.036 (0.089) V1
0.0001∗∗∗ 0.039 (0.163) 1.0000 0.018 (0.331) iso 0.5 % PFC 0.0000∗∗∗ 0.003
(0.023) 0.5147 0.007 (0.068) V1 0.9981 0.025 (0.081) 1.0000 0.005 (0.009) iso
1.0 % PFC 0.7781 0.001 (0.044) 0.9392 0.004 (0.063) V1 0.1122 0.003 (0.045)
0.8076 0.003 (0.061) 2 iso 0.0 % PFC 0.0000∗∗∗ 0.005 (0.047) 1.0000 0.006
(0.077) V1 0.2450 0.015 (0.185) 1.0000 0.027 (0.060) iso 0.5 % PFC 0.0000∗∗∗
0.007 (0.094) 0.0000∗∗∗ 0.014 (0.088) V1 1.0000 0.036 (0.087) 1.0000 0.010
(0.080) iso 1.0 % PFC 1.0000 0.007 (0.073) 1.0000 0.008 (0.227) V1 1.0000
0.001 (0.062) 1.0000 0.017 (0.147) ${}^{*}p<0.05$; ${}^{**}p<0.01$;
${}^{***}p<0.001$

Table 9: Correlation of $H$ with band power. $\widetilde{R}^{2}$ and
$max(R^{2})$ indicate the median and maximum of $R^{2}$ over recordings per
condition, respectively.

animal level site $\delta$ $\theta$ $\alpha$ $p$ $\widetilde{R}^{2}$
($max(R^{2})$ $p$ $\widetilde{R}^{2}$ ($max(R^{2})$ $p$ $\widetilde{R}^{2}$
($max(R^{2})$ 1 iso 0.0 % PFC 1.0000 0.095 (0.458) 1.0000 0.015 (0.201) 1.0000
0.020 (0.271) V1 1.0000 0.091 (0.706) 0.9998 0.037 (0.109) 1.0000 0.041
(0.097) iso 0.5 % PFC 1.0000 0.046 (0.202) 1.0000 0.004 (0.036) 1.0000 0.003
(0.059) V1 1.0000 0.035 (0.133) 0.9100 0.040 (0.051) 0.8029 0.002 (0.086) iso
1.0 % PFC 1.0000 0.020 (0.079) 0.0359 0.005 (0.082) 0.9198 0.004 (0.023) V1
1.0000 0.010 (0.461) 0.9941 0.003 (0.061) 0.9991 0.004 (0.032) 2 iso 0.0 % PFC
1.0000 0.171 (0.395) 1.0000 0.058 (0.110) 1.0000 0.024 (0.060) V1 1.0000 0.031
(0.723) 1.0000 0.036 (0.135) 1.0000 0.048 (0.170) iso 0.5 % PFC 1.0000 0.035
(0.298) 1.0000 0.011 (0.016) 1.0000 0.015 (0.088) V1 1.0000 0.038 (0.135)
0.0428 0.021 (0.073) 0.0000∗∗∗ 0.016 (0.045) iso 1.0 % PFC 1.0000 0.015
(0.580) 0.0000∗∗∗ 0.006 (0.076) 0.0000∗∗∗ 0.008 (0.219) V1 1.0000 0.112
(0.601) 0.1329 0.003 (0.062) 0.0000∗∗∗ 0.009 (0.094) animal level site $\beta$
$\gamma$ $p$ $\widetilde{R}^{2}$ ($max(R^{2}$) $p$ $\widetilde{R}^{2}$
($max(R^{2})$ 1 iso 0.0 % PFC 0.9665 0.018 (0.165) 0.0000∗∗∗ 0.052 (0.097) V1
0.9999 0.046 (0.121) 0.0000∗∗∗ 0.014 (0.358) iso 0.5 % PFC 1.0000 0.007
(0.020) 0.5510 0.013 (0.064) V1 0.0000∗∗∗ 0.028 (0.089) 0.0000∗∗∗ 0.006
(0.011) iso 1.0 % PFC 0.2144 0.005 (0.056) 0.0269 0.006 (0.082) V1 0.4394
0.002 (0.040) 0.2250 0.003 (0.053) 2 iso 0.0 % PFC 1.0000 0.011 (0.043)
0.0000∗∗∗ 0.001 (0.068) V1 0.5631 0.022 (0.094) 0.0000∗∗∗ 0.037 (0.061) iso
0.5 % PFC 1.0000 0.013 (0.107) 1.0000 0.012 (0.095) V1 0.0000∗∗∗ 0.034 (0.096)
0.0000∗∗∗ 0.010 (0.102) iso 1.0 % PFC 0.0000∗∗∗ 0.007 (0.051) 0.0000∗∗∗ 0.012
(0.290) V1 0.0017∗ 0.011 (0.072) 0.0000∗∗∗ 0.013 (0.103) ${}^{*}p<0.05$;
${}^{**}p<0.01$; ${}^{***}p<0.001$

Table 10: Correlation of $TE_{SPO}$ with band power. $\widetilde{R}^{2}$ and
$max(R^{2})$ indicate the median and maximum of $R^{2}$ over recordings per
condition, respectively.

animal level direction $\delta$ $\theta$ $\alpha$ $p$ $\widetilde{R}^{2}$
($max(R^{2})$ $p$ $\widetilde{R}^{2}$ ($max(R^{2})$ $p$ $\widetilde{R}^{2}$
($max(R^{2})$ 1 iso 0.0 % PFC $\rightarrow$ V1 1.0000 0.024 (0.101) 1.0000
0.034 (0.122) 1.0000 0.061 (0.146) V1 $\rightarrow$ PFC 0.9662 0.030 (0.230)
0.8389 0.009 (0.045) 1.0000 0.008 (0.047) iso 0.5 % PFC $\rightarrow$ V1
1.0000 0.033 (0.090) 1.0000 0.073 (0.132) 1.0000 0.013 (0.117) V1
$\rightarrow$ PFC 0.1319 0.001 (0.010) 1.0000 0.018 (0.058) 0.9097 0.001
(0.015) iso 1.0 % PFC $\rightarrow$ V1 1.0000 0.009 (0.058) 1.0000 0.009
(0.029) 1.0000 0.012 (0.038) V1 $\rightarrow$ PFC 0.0947 0.002 (0.024) 1.0000
0.003 (0.116) 0.9996 0.004 (0.058) 2 iso 0.0 % PFC $\rightarrow$ V1 1.0000
0.060 (0.102) 1.0000 0.055 (0.121) 1.0000 0.036 (0.052) V1 $\rightarrow$ PFC
0.9998 0.006 (0.038) 1.0000 0.012 (0.056) 0.9998 0.010 (0.040) iso 0.5 % PFC
$\rightarrow$ V1 1.0000 0.108 (0.157) 1.0000 0.115 (0.166) 1.0000 0.079
(0.160) V1 $\rightarrow$ PFC 1.0000 0.026 (0.149) 1.0000 0.015 (0.052) 0.6675
0.003 (0.009) iso 1.0 % PFC $\rightarrow$ V1 0.0000∗∗∗ 0.009 (0.095) 1.0000
0.010 (0.037) 1.0000 0.007 (0.030) V1 $\rightarrow$ PFC 0.0153 0.008 (0.023)
1.0000 0.006 (0.035) 0.7369 0.001 (0.017) animal level site $\beta$ $\gamma$
$p$ $\widetilde{R}^{2}$ ($max(R^{2}$) $p$ $\widetilde{R}^{2}$ ($max(R^{2})$ 1
iso 0.0 % PFC $\rightarrow$ V1 1.0000 0.069 (0.157) 0.0352 0.005 (0.044) V1
$\rightarrow$ PFC 0.9612 0.010 (0.040) 0.0009∗∗ 0.010 (0.028) iso 0.5 % PFC
$\rightarrow$ V1 1.0000 0.033 (0.121) 0.0611 0.031 (0.038) V1 $\rightarrow$
PFC 0.9991 0.010 (0.039) 0.0014∗∗ 0.021 (0.028) iso 1.0 % PFC $\rightarrow$ V1
0.0000∗∗∗ 0.004 (0.019) 0.0000∗∗∗ 0.007 (0.054) V1 $\rightarrow$ PFC 0.0912
0.002 (0.063) 0.0000∗∗∗ 0.006 (0.109) 2 iso 0.0 % PFC $\rightarrow$ V1 1.0000
0.021 (0.040) 0.3062 0.015 (0.036) V1 $\rightarrow$ PFC 0.9998 0.002 (0.039)
0.2046 0.002 (0.041) iso 0.5 % PFC $\rightarrow$ V1 1.0000 0.107 (0.132)
0.6054 0.006 (0.013) V1 $\rightarrow$ PFC 0.0186 0.004 (0.019) 0.0000∗∗∗ 0.022
(0.041) iso 1.0 % PFC $\rightarrow$ V1 0.9940 0.004 (0.039) 0.0025∗ 0.003
(0.027) V1 $\rightarrow$ PFC 0.9122 0.002 (0.013) 0.0000∗∗∗ 0.002 (0.058)
${}^{*}p<0.05$; ${}^{**}p<0.01$; ${}^{***}p<0.001$

#### S3 The effect of sampling on AIS-estimation

AIS estimates for data sampled at different rates.

Figure 11: Estimates of active information storage (AIS) from data sampled at
different rates. AIS estimates from three random recordings in animal 1 under
three levels of Isoflurane; estimated from data with the sampling rate used
for analysis in the present work (1000 Hz) and re-sampled at 2000 and 600 Hz
respectively; note that qualitative results did not change due to re-sampling,
but absolute estimates increased for higher sampling rates; the number of data
points was held approximately constant by selecting a subset of trials for
estimation such that the number of points entering the analysis was equal to
the smallest number of points over all isoflurane levels.

#### S4 Theoretical and practical running time of nearest-neighbor based
estimators

The KSG-estimator used in this work is computationally demanding, because as a
nearest-neighbor based estimator it requires the execution of $k$-nearest-
neighbor as well as range searches for all data points $N$. Hence, the
algorithms used to perform these searches determine the asymptotic time
complexity of the estimator. In this work, we used two different
implementations published as part of the TRENTOOL toolbox: a CPU-based
estimator for the estimation of $TE_{SPO}$ from epochs or trials of data [23],
and a GPU-based estimator for the estimation of $TE_{SPO}$ from data pooled
over epochs or trials [61]. We used the CPU-based work-flow for the epoch-wise
estimation of $TE_{SPO}$ and the GPU-based work-flow when testing for
statistical significance of $TE_{SPO}$ to maximize the number of points
entering the estimation. The CPU-based estimator uses a neighbor-search
algorithm that—depending on data-properties—requires for neighbor-searches on
all $N$ points at maximum $O(kN\log(N))$ time. The GPU-based estimator does
not make use of fast data structures used for the CPU-based estimator, but
uses a linear search which results in a worse time complexity of $O(dN^{2})$.
However, in the implementation used in this study [61] the linear neighbor
search is performed in parallel over points and problem instances, which
significantly improves the overall running time when using the estimator on
multiple problem instances—a comparison of the running times of the serial
CPU-algorithm and the parallel GPU-algorithm can be found in [61].

Additionally, we used the KL-estimator for entropy estimation. This estimator
requires the execution of a $k$-nearest-neighbor search for all data points
$N$—we used the estimator’s implementation published as part of the JIDT
toolbox [53], which has a time complexity of $O(kN\log(N))$.

We also measured practical running times of the estimators to provide a point
of reference when planning similar analyses. Approximate, average running
times for the estimation of $AIS$, $TE_{SPO}$, and $H$ are presented in Table
11. The presented running times include the estimation of each measure for
both directions of interactions or recording site, for one recording;
presented running times are averages over recordings. We measured the total
time needed for estimation, including data preparation (e.g., the optimization
of embedding parameters for the estimation of $TE_{SPO}$). All estimation
procedures that did not require a GPU, were executed on a Intel(R) Xeon(R) CPU
clocked at 2.90 GHz. The GPU-implementation of the $TE_{SPO}$ estimator was
run on a Intel(R) Xeon(R) CPU clocked at 2.00 GHz and a NVIDIA GeForce GTX
TITAN. Both machines were running 64-bit Ubuntu Linux. Note that the GPU
estimation was performed on 50 trials only, because here the computational
demand was higher due to more data points entering one estimation of
$TE_{SPO}$ (pooled over epochs).

Table 11: Practical average running times for estimation of information-theoretic measures from one recording session and two recording sites or directions of interaction in animal 1. measure | toolbox/implementation | mean running time [min]
---|---|---
$TE_{SPO}$ | TRENTOOL/GPU-implementation | 2235.93 (314.27 SD)
$TE_{SPO}$ | TRENTOOL/CPU-implementation | 1871.51 (926.56 SD)
$AIS$ | TRENTOOL/CPU-implementation | 2.83 (1.51 SD)
$H$ | JIDT/CPU-implementation | 1.55 (0.56 SD)
SD = standard deviation

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