Neuroscience in Anesthesiology and Perioperative Medicine Section Editor: Gregory J. Crosby
Brain Electrical Activity Obeys Benford’s Law Matthias Kreuzer, MSc,* Denis Jordan, PhD,* Bernd Antkowiak, PhD,† Berthold Drexler, MD,† Eberhard F. Kochs, MD,* and Gerhard Schneider, MD‡ BACKGROUND: Monitoring and automated online analysis of brain electrical activity are frequently used for verifying brain diseases and for estimating anesthetic depth in subjects undergoing surgery. However, false diagnosis with potentially catastrophic consequences for patients such as intraoperative awareness may result from unnoticed irregularities in the process of signal analysis. Here we ask whether Benford’s Law can be applied to detect accidental or intended modulation of neurophysiologic signals. This law states that the first digits of many datasets such as atomic weights or river lengths are distributed logarithmically and not equally. In particular, we tested whether data obtained from electrophysiological recordings of human patients representing global activity and organotypic slice cultures representing pure cortical activity follow the predictions of Benford’s Law in the absence and in the presence of an anesthetic drug. METHODS: Electroencephalographic (EEG) recordings from human subjects and local field potential recordings from cultured cortical brain slices were obtained before and after administration of sevoflurane. The first digit distribution of the datasets was compared with the Benford distribution. RESULTS: All datasets showed a Benford-like distribution. Nevertheless, distributions belonging to different anesthetic levels could be distinguished in vitro and in human EEGs. With sevoflurane, the first digit distribution of the in vitro data becomes steeper, while it flattens for EEG data. In the presence of high frequency noise, the Benford distribution falls apart. CONCLUSIONS: In vitro and EEG data show a Benford-like distribution which is altered by sevoflurane or destroyed by noise used to simulate artefacts. These findings suggest that algorithms based on Benford’s Law can be successfully used to detect sevoflurane-induced signal modulations in electrophysiological recordings. (Anesth Analg 2014;118:183–91)
M
onitoring and automated online analysis of brain electrical activity are increasingly used as diagnostic tools to assess brain activity or detect brain diseases. For example, electroencephalographic (EEG) monitoring is increasingly used by anesthesiologists to assess depth of anesthesia, that is, the hypnotic component of anesthesia. This online monitoring may help to prevent episodes of intraoperative awareness in patients undergoing surgery.1–4 Still, these devices may not be superior to standard monitoring as demonstrated over the last several years.5,6 Nevertheless, the use of these monitoring systems may be beneficial in some situations. It is evident that irregularities in data sampling or poor quality of the recorded signals may lead to errors in data analysis and may therefore have serious consequences for patients. This may be caused by technical failure or handling From the *Department of Anesthesiology, Klinikum rechts der Isar, Technische Universität München, München; †Department of Anesthesiology, Experimental Anesthesiology Section, University of Tübingen, Tübingen; and ‡Department of Anesthesiology, Witten/Herdecke University, Helios Clinic Wuppertal, Germany. Accepted for publication September 19, 2013. Funding: None. The authors declare no conflicts of interest. Reprints will not be available from the authors. Address correspondence to Gerhard Schneider, MD, Department of Anesthesiology, Witten/Herdecke University, Helios Clinic Wuppertal, Heusnerstr, 40 D-42283, Wuppertal, Germany. Address e-mail to gerhard. [email protected]. Copyright © 2013 International Anesthesia Research Society DOI: 10.1213/ANE.0000000000000015
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errors and may occur in clinical practice every day. In serious cases, such erroneous data analysis would mimic an aberrant clinical patient state. To prevent such events, it is necessary to gain knowledge about expected changes of electrophysiological activity, for example, caused by anesthetic drugs, and to develop a robust method for detection of distortions or errors in the underlying signal. For this purpose, algorithms based on Benford’s Law may be a promising approach, given that neurophysiologic data may obey this law. This method may be used to detect irregularites, that is, distortion of physiological signals. It has been shown to be useful in the analysis of naturally occuring signals. For example, earthquakes can be detected by changes in the first digit distribution of a seismogram’s data.7 Benford’s Law states that the first digits of datasets are not equally but logarithmically distributed as presented in Figure 1. This finding goes back to 1881, when Newcomb8 detected this property in many scientific and statistical tables. In 1938, Frank Benford confirmed Newcomb’s observation by creating an extensive database containing different kinds of data such as length of rivers, atomic weights, specific heat constants, the numbers occurring in a newspaper, and many more.9 In all these collected datasets, the leading digits followed the logarithmic distribution reasonably well. A short example describing the nature of Benford’s Law can be found in the Appendix. Furthermore, Engel and Leuenberger10 showed that exponentially distributed numbers more or less obey Benford’s Law. Today this distribution is used in macroeconomic statistics11 to detect fraud in accounting data,12 income tax evasion,13 or for the screening of large datasets.14 Most recently, Benford’s Law was used to www.anesthesia-analgesia.org
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Figure 1. First digit distribution according to Benford’s Law. The height of each bar represents the percentage of each digit occurring as first digit in a Benford distributed dataset.
detect abnormalities in the 2009 Iranian election.15 The idea of using Benford’s Law to screen data is based on the observation that regular, “naturally generated” data usually follow a logarithmic distribution, while faked data show abnormalities in the distribution.16 In the presented experiments, electrophysiological activity, namely EEG from humans and local field potentials (LFP) from mouse neocortex, was analyzed following the idea that this “natural” activity should also follow the described distribution, whereas distorting influences such as noise or modulation of cerebral activity caused by sevoflurane may alter the distribution of the first digits of the signals’ amplitude values. If the hypothesis of Benford-like distributed datasets proves valid, the evaluation of the first digit distribution may also be used as a sensor for signal distortions which change the Benford-like distribution.
METHODS
EEG data recorded during a volunteer study and LFP recordings obtained from organotypic slice cultures (OTC) were used for analysis. All participating volunteers gave informed written consent for the protocol, which was approved by the ethics committee of the Technische Universität München, Munich, Germany (Application No. 942/03). All animal procedures were approved by the animal care committee (Eberhard Karls University, Tübingen, Germany) and were in accordance with the German Animal Welfare Act (TierSchG).
LFP Recordings from Organotypic Slice Cultures
To receive LFP recordings, spontaneous synaptic activity in OTC were analyzed. Neocortical slices were excised from 5-day old rat pups and grown in culture for at least 2 weeks according to the method described by Antkowiak et al.17 During this time, cultured cells establish new synaptic connections and form a spontaneously active network. These cultures were treated with sevoflurane in concentrations comparable with the levels used in humans. Sevoflurane was dissolved in an artificial cerebrospinal fluid and was applied via bath transfusion. The test solution was delivered
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via syringe pumps (ZAK, Marktheidenfeld, Germany) with a flow rate of 1 mL/min over a Teflon tubing (Lee, Frankfurt, Germany) connection. LFP were recorded 10 minutes after washing the culture with the new perfusate to ensure equilibrated concentrations. A 10-minute waiting time was established to ensure near steady-state conditions.17 LFP recordings were performed using artificial cerebrospinal fluid-filled glass electrodes with a resistance of about 5 MΩ at 34°C. The extracellular recordings were low-pass filtered at 5 kHz, digitized at a sampling frequency of fs = 10 kHz, and visualized and stored with the Axoscope 9 software tool (Axon Instruments, Molecular Devices Corp., Chicago, IL). LFP were separated from action potentials by digital band-pass filtering. The duration of each recording was 180 seconds. The influence of sevoflurane on the recorded LFP is displayed in Figure 2. OTC were first tested for general “healthiness” by visual inspection. After placement of extracellular electrodes, OTC were only used for an experiment in case the observed action potential firing pattern was typical and stable for >10 minutes. If the firing pattern changed during the recording time, these cultures were not used for further analysis. The sample size was n = 15 for control conditions, n = 5 for 0.75 minimum alveolar concentration (MAC) equivalent to a deep sedation in humans, n = 5 (1 MAC) equivalent to light anesthesia, and n = 5 for the 1.5 MAC experiments corresponding to deep general anesthesia. In cell cultures, extracellular LFP activity was recorded at these concentrations. A 10-second sequence showing spiking activity and a resting signal (up and down states) was cut at the end of each recording, followed by analysis of the first digit distribution of these sequences with respect to Benford’s Law. In addition, three 2 s segments of every recording without spiking activity were cut and also followed by an analysis of their first digit distribution. During these “silent” periods, the network is in a resting state, a so-called down state (Fig. 3). The information content of signals from the down state is affected by anesthetics.18 Experimental results indicate that the active epochs, the up states, can initiate structured information flow in local cortical columns19 and may be associated with entering specific memory states or with solving computational problems.20 Up and down states in vivo observed under ketamine– xylazine anesthesia could be directly associated with slow waves that are prominent in surface EEG.21 In sleep experiments with cats, the waking electrical activity was connected to sequences of silenced firing in the cortex at the onset of slow wave sleep.22 These findings pronounce that (cortical) LFP activity is directly connected to surface EEG.
EEG Data Recording
EEG was recorded during a volunteer study. Men participants 18 to 35 years old were anesthetized with sevoflurane according to a study protocol that has been described in detail by Horn et al.1 In short, EEG was recorded using ZipPrep electrodes (Aspect Medical Systems, Natick, MA) from positions AT1, M2, Fz(reference), and Fpz(ground) according to the international 10 to 20 system at equilibrated levels of anesthesia. The signal from AT1-Fz was used for the present analysis. The equilibrated levels of anesthesia were defined as follows: the “lighter” end point of anesthetic levels was defined by loss of consciousness
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Figure 2. Anesthetic-induced changes of the electroencephalographic (EEG) signal (left) and local field potentials (LFP) signal amplitudes recorded in the organotypic slice cultures (OTC) (right). Under the influence of sevoflurane, the EEG frequencies decrease and amplitudes increase. In LFP recordings, activity decreases with the administration of sevoflurane.
Figure 3. Sequences of 2-second length, where network activity was low, were cut out of the local field potentials (LFP) recording. These selected sequences represent cortical down states (DS).
(LOC), and the deepest level was indicated by the occurrence of EEG burst suppression. On the basis of individual anesthetic concentrations, 2 intermediate levels were then defined. The difference between end-expired sevoflurane concentration at burst suppression and LOC was divided into 3 intervals, resulting in the 2 intermediate levels. In
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the presented experiments, the intermediate level closer to LOC was termed “light general anesthesia” and the deeper intermediate level “deep anesthesia.” Five-minute sequences of EEG were recorded at the equilibrated anesthetic levels (1) awake (before administration of sevoflurane, n = 8), (2) light general anesthesia (n = 6), and (3)
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deep anesthesia (n = 6).1 The signals were recorded with an analog to digital converter resolution of 12 bit and sampling frequency of fs = 5 kHz, followed by digital low-pass filtering and decimation to 1 kHz. The final bandwidth of the signal was 0.5 to 400 Hz. Representative traces of EEG episodes are displayed in Figure 2.
Data Processing
Since the analyses were targeted toward detection of properties represented by activity of neuronal networks in the brain, electromyography (EMG) can be assumed to be a biological artefact that superposes the primary signal, that is, the EEG. Hence, minimizing the influence of EMG as spurious signal is necessary. In scalp recordings, frequencies up to 30 Hz may mainly represent EEG activity, while higher frequencies are prone to be disturbed by EMG.23 As a consequence, the recorded EEG was also analyzed after a 30 Hz low-pass filtering using MATLAB R2012a (the MathWorks, Natick, MA). The last 10 seconds were extracted from each recording, and their first digit distribution was tested for concordance with Benford’s Law. For the LFP recordings, the selected episodes of general and down state only activity were analyzed after adequate low-pass filtering and down sampling to 2 kHz.
datasets were compared with the Benford distribution using LabView 6i (National Instruments, Austin, TX), and the first digit distributions of recordings without sevoflurane administration were compared with the distributions with sevoflurane. Therefore, data of 1 anesthetic level were pooled to 1 dataset X for patient or animal data, respectively. If the first digit distribution of the analyzed dataset shows logarithmic behavior, the mantissas of the data’s logarithm is uniformly distributed.8 A set union can be defined as described in equation (1): 9
Ed = ∪ [d ⋅ 10 k ,(d + 1) ⋅ 10 k [for d ∈ {1,2,...,9}, i.e., ℜ+ = ∪ Ed k ∈Z
d =1
Hence, the probability of 1 data point having the first digit d is described in equation (2): P(X ∈ Ed ) = P (log(X ) ∈[log d , log( d + 1)]) 1 = log(d + 1) − log d = log(1 + ) d
Benford Analysis
The first digit distribution was determined from EEG and LFP recordings. The first digits (without leading zeroes) of the amplitude values at discrete time points were analyzed. If the amplitude at a certain discrete time was 58 μV, the first digit was 5. If the amplitude is 0.95 μV, the first digit is 9 because leading zeroes are ignored. All included
(2)
where X is the mantissa of x, that is, x mod 1.
Influence of Noise on EEG Recordings
To evaluate the influence of general signal distortions on EEG, a representative EEG sequence recorded during wakefulness was superposed with high-pass white filtered noise at 100 Hz using a third-order Butterworth low pass. Two different noise amplitudes were used, and the raw and resulting signals used for analysis are displayed in Figure 4. The procedure of superposing the EEG sequence was performed using LabView 6i (National Instruments, Austin, TX).
(1)
1 P(X ∈ Ed ) = log(1 + )(3) d is the definition of Benford’s Law. We compared the logarithmic, modulo 1 first digit distribution of the single datasets with a uniform distribution using the Kolmogorov– Smirnov test statistics. This test was performed using R 2.4.0 (R Foundation for Statistical Computing, Vienna, Austria).24 In addition, similarity of the distributions was indicated by QQ-plots plotted with MATLAB R2012a (the MathWorks, Natick, MA).
RESULTS
Both in vitro and EEG data led to test statistics of ≤0.11 (Kolmogorov–Smirnov test), indicating a good similarity to the Benford distribution, related to a maximum deviation of
Figure 4. A 10-second electroencephalographic (EEG) trace recorded from a patient being awake is superposed with white noise, low-pass filtered at 100 Hz, of different amplitudes. The top trace represents the raw EEG. The middle trace shows raw EEG superposed with noise (awake + noise level 1), and the lower trace shows EEG superposed with white noise of higher amplitude (awake + noise level 2)
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Figure 5. QQ-plots of the unfiltered electroencephalographic (EEG) data compared with the logarithmic, modulo 1 first digit distribution of the EEG sequences to a uniform distribution (angle bisector, thin line). All 3 EEG distributions are close to the bisecting line representing uniform distribution. The curvature of the EEG “awake” (black) distribution is inverse to the EEG distributions recorded under sevoflurane. (blue: light general anesthesia, red: deep general anesthesia).
Figure 7. First digit distribution of unfiltered (A) and 30 Hz lowpass filtered (B) electroencephalographic (EEG) data derived from sequences recorded at the levels “awake” (black, n = 8), “light general anesthesia” (blue, n = 6), and “deep general anesthesia” (red, dashed, n = 6). The 30 Hz low-pass filtering does not influence the first digit distribution. After administration of sevoflurane, the first digit distribution becomes flatter. There are no significant differences in the distribution of the states “light general anesthesia” and “deep general anesthesia.” Figure 6. QQ-plots of the in vitro up and down-state sequences data, compared with the logarithmic, modulo 1 first digit distribution of the local field potentials (LFP) sequences (black, bold line) to a uniform distribution (gray). The distribution of the LFP “awake” (black) sequences almost perfectly matches the Benford distribution, while the LFP distributions corresponding to sedation (blue), light (red), and deep anesthesia (green) are still in concordance with the bisecting line (thin black line).
1. This is confirmed by corresponding QQ-plots which only indicate a small deviation from the bisecting line (Figs. 5 and 6). If the plotted EEG/in vitro data comply with the bisecting line, it is uniformly distributed, that is, it shows Benford distribution. Small deviations of the data from the bisecting line still are in accordance with the Benford distribution. Notice that because of the large dataset and of possible dependence of measurements, the P-value may be biased and is therefore not included in the present analysis.
EEG Analysis
The QQ-plots derived from analysis of EEG amplitudes are close to the bisecting line for all anesthetic levels (Fig. 5), that is, the amplitudes of EEG recorded at different anesthetic levels is and remains Benford-like distributed. Still, the level “awake” can be distinguished from anesthetic levels by a different curvature in the QQ-plot. The awake plot starts below the bisecting line, while at anesthetic levels, the plot is initially above the bisecting line. The curvature flips with delivery of sevoflurane. The state awake can be
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even better separated from “light or moderate general anesthesia” and “deep general anesthesia” by looking at the distribution of probabilities of the first digit being d with d ∈{1, 2,..., 9}. The curve of the first digit distribution from awake is more pronounced than the distributions from the recordings performed under anesthesia (Fig. 7). The probability P(d = 1) and P(d = 2) is higher at the level awake than it is during anesthesia. The second plot of this figure shows the first digit distribution of the recorded data after 30 Hz low-pass filtering that was performed to exclude EMG activity from the analysis. The course of the plots is similar, that is, the low-pass filtering does not have an apparent effect on the outcome of the Benford analysis. The averaged distribution derived from each anesthetic level for the unfiltered and the 30 Hz low-pass filtered data was compared with a Kolmogorov–Smirnov test. For all comparisons, the P-value was 1, clearly rejecting the null hypothesis that the distributions are unequal and suggesting similar results.
EEG Superposed With Noise
In the presence of additional noise, the Benford-like first digit distribution is distorted. The unfiltered raw EEG’s first digit distribution is almost congruent with the Benford distribution as displayed in Figure 8. With additional noise, P(d = 1) decreases and falls apart from the Benford distribution.
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Figure 8. First digit distribution of an unfiltered raw electroencephalographic (EEG) trace from a patient who was awake (black). This EEG trace was artificially distorted with white noise, low-pass filtered at 100 Hz (blue). As a consequence, the first digit distribution becomes less steep for low first digits. If the noise amplitude becomes higher, the distribution becomes even flatter (red). For better comparison, the Benford distribution is additionally presented in gray. Further, the first digit distributions of the additional noise content are presented in cyan (100 Hz low-pass filtered noise level 1) and magenta (100 Hz low-pass filtered noise level 2)
In Vitro Analysis
The in vitro QQ-plot derived in the absence of sevoflurane almost completely fits the bisecting line. The plots from data with sevoflurane do not show this strong concordance but are still very close to the bisecting line, that is, agree with the Benford distribution (Fig. 6). The flip in the curvature caused by sevoflurane is hardly visible, and it is inverse to the EEG data. The probability distribution of the leading digits was more pronounced after the cells were washed with sevoflurane, that is, P(d = 1) and P(d = 2) were higher when sevoflurane was delivered. This analysis result is contradictory to the EEG observation. This distribution did not change significantly during analysis of the entire LFP recording, containing up and down states or only down states. Spiking activity did not have any influence on the resulting probability distribution (Fig. 9).
DISCUSSION
Our results indicate that neuronal network activity in the cortex, recorded as LFP and frontal EEG, follow Benford’s law. The Benford-like behavior does not disappear with sevoflurane delivered in concentrations corresponding to anesthetic levels such as “sedation,” “light general anesthesia,” and “deep general anesthesia.” Our assumptions on MAC equivalents of sevoflurane are based on data published by Franks and Lieb.25 These MAC values refer to the plasma or blood concentrations of volatile anesthetics in mammals, determined at 37°C. Based on these data, a MAC-equivalent of 0.35 mM sevoflurane was used in this study. The first digit distribution of the recorded data changes with sevoflurane, but it remains Benford-like. The QQ-plots of LFP and EEG recorded in the presence of sevoflurane reveal a different effect on a local, cortical network (LFP) compared
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with the EEG recordings that also contain information from subcortical areas. The results hold for unfiltered EEG and for 30 Hz low-pass filtered EEG as well as for LFP episodes containing both up and down states and down states only. If the EEG recordings are distorted by superposition with noise, the Benford-like distribution is destroyed.
Scale Invariance
The results indicate that the analyzed activity is scale invariant because a Benford distribution is scale-free, that is, the revealed properties are not used due to a distinct scaling but are independent from it. Other experiments also indicate that distinct properties of cerebral activity show scale-invariant behavior. There, scale invariance has been suggested to represent the network’s property to optimize its structure and enable advantageous synchronization and information transfer,26–28 hence reflect important cerebral tasks. In general, scale invariance seems to be associated with structured or organized behavior. LFP episodes of combined up and down states seem to follow structured activity as well as isolated down-state sequences. LFP are local activity of a small neuronal network. The fact that a Benford distribution can be observed for episodes containing both up and down states and for down states only leads to the assumption that cortical activity is structured in periods of activity, that is, during up states when working memory tasks may be performed20 as well as during inactive down-state periods. Surface EEG electrodes measure the averaged activity of up to 1010 neurons.29 The recorded EEG, representing “global” neuronal activity, also shows structured activity. The structured activity observed at all scales changes in the presence of sevoflurane but is not destroyed.
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Figure 9. First digit distribution of the in vitro data (A, up and downstate sequences; B, down state sequences) recorded at levels corresponding to “awake” (black, n = 15), “sedation” (green, n = 5), “light general anesthesia” (blue, n = 5), and “deep general anesthesia” (red, dashed, n = 5). With administration of sevoflurane, the first digit distribution of the recorded local field potentials (LFP) sequences becomes steeper. There is no significant difference in the distributions of the analyzed sequences containing up and down states and the down-state only sequences. The sevoflurane concentration also seems to have a small impact only on the distribution, because distributions recorded during light and deep general anesthesia only differ to a small degree.
In case of contamination of the EEG with noise, the present structure in the signal is distorted, and the first digits Benford-like behavior vanishes as opposed to the physiologic changes caused by sevoflurane. Other research also addressing the question of structured neuronal activity represented by scale invariance seems to support our results of structure in the signals. A global scalefree behavior, according to power laws of neuronal network activity in terms of coupling of different cortical areas, was observed in the absence and presence of propofol.30 Other results did also find power law distributions for neuronal network tasks such as LFP firing propagation.26 All these findings suggest that neuronal processes and interactions are scale-free and follow a distinct organization. Our results indicate that persistent scale invariance can be observed at all scales of neuronal networks from the isolated cortex up to EEG recordings.
Effects of Sevoflurane: Differences Between Isolated Neocortical Cultures and EEG
Visual inspection of the first digit distribution plot shows steepening of the LFP first digit distribution and flattening of the EEG first digit distribution with increasing concentrations of sevoflurane. Generally speaking, an effect of sevoflurane can be observed in the first digit distribution. Nevertheless,
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the Benford-like behavior of the distributions persists. This leads to the assumption that the organized structure of activity is altered but not destroyed by sevoflurane, meaning that during anesthesia, the communication structure of the neuronal network is altered but not destroyed. The LFP recordings were lacking subcortical inputs whereas EEG activity received input also from subcortical areas. These existing or missing inputs from deeper brain areas seem a plausible explanation for the change of distribution in opposite directions. Thalamic structures, for example, have strong influence on EEG recordings under anesthesia. During wakefulness, asynchronous sensory input is transferred via thalamic relay neurons into the cortex. In this mode, relay neurons do not display synchronized firing.31 As a consequence, cortical neurons receive irregular synaptic inputs. Therefore, cortical cells display irregular action potential firing, and the EEG is desynchronized. During deep sleep and anesthesia, thalamic relay neurons oscillate in synchrony in the delta frequency range (1–4 Hz). In this state, thalamic neurons produce strong and rhythmic synaptic excitation of cortical pyramidal neurons, forcing these cells to display a very similar activity pattern. These findings indicate that cortical and subcortical neuronal network activity is strongly influenced by anesthetics. The effects of sevoflurane on subcortical areas may influence network activity in the cortex, that is, sevoflurane may affect network activity differently in an isolated cortical network than in a cortical network which is connected to subcortical structures. Taken together, the different effects of sevoflurane on the distributions calculated from in vitro data and eeg recordings most likely reflect actions on different neuronal substrates. To conclude, the above analysis shows that cortical neuronal network activity follows the Benford distribution, reflecting organized activity. The effect of volatile anesthetic, a changed state of consciousness, alters the distribution compared with control conditions. Despite of this change, the Benford-like characteristics of the distributions are maintained even under sevoflurane. Sevoflurane influences the analyzed neuronal network but does not destroy structured activity, that is, the Benford distribution. Sevoflurane initiated the first digit distribution of isolated cortical LFP and surface EEG to react differently. This difference is probably caused by the absence or presence of subcortical structures that influence cortical network activity. In contrast to the physiological changes, adding white noise to the EEG recordings destroys the Benford-like distribution of the EEG. This finding may open the possibility of using this method to detect distortions in the signal that are not of electrophysiological origin.
Clinical and Scientific Impact
With the analyses performed, we have shown that electrophysiolgical activity, represented as surface EEG and LFP derived from isolated cortex, follows a certain pattern that is altered but not eliminated by sevoflurane. Even during anesthesia, the neuronal network activity is reflected in organized activity. This holds true for local cortical networks with a small number of contributing neurons as well as for global EEG recordings. Hence, our approach may be
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valuable in terms of evaluating datasets according to their validity. If neuronal activity is recorded, the dataset should show a logarithmic first digit distribution. This method may be applied in the clinical setting of EEG monitoring. High-quality EEG recordings should show a Benford-like first digit distribution, irrespective of the anesthetic level. If the signal is influenced by contaminating sources, the distribution can be affected. A recorded EEG signal that does not show this distribution may be distorted and should be handled with care for monitoring purposes. E
APPENDIX Example for the Nature of Benford’s Law:
The following example explains the nature of this law: Imagine the stock price is increasing from 10 to 20 points. The difference between both prices is 10 points, and the price is doubled. If the price increases from 80 to 90 points, the difference is still 10 points, but the price only increases ̭ by (9/8 − 1) ‗ 12.5%. Even if the difference for both examples is 10 points, it takes more effort to come from 10 to 20 than from 80 to 90. That means that if you have a stock course of 10, it will take a longer time to reach a course of 20 since you have to double your value than if your course is 80 and you only need to add another 12.5 percent of your value to reach 90. The wider the relative gap to bridge from one first digit to the next, the longer you will remain in that corridor and the higher the probability will be of detecting this first digit. This implies that the 1 will show up more often than any other digit. DISCLOSURES
Name: Matthias Kreuzer, MSc. Contribution: This author helped analyze the data, contribute reagents/materials/analysis tools, and wrote the paper. Attestation: Matthias Kreuzer approved the final manuscript. Name: Denis Jordan, PhD. Contribution: This author helped analyze the data and contribute reagents/materials/analysis tools. Attestation: Denis Jordan approved the final manuscript. Name: Bernd Antkowiak, PhD. Contribution: This author helped conceive and design the experiments, contribute reagents/materials/analysis tools, and wrote the paper. Attestation: Bernd Antkowiak approved the final manuscript. Name: Berthold Drexler, MD. Contribution: This author helped perform the experiments and contribute reagents/materials/analysis tools. Attestation: Berthold Drexler approved the final manuscript. Name: Eberhard F. Kochs, MD. Contribution: This author helped contribute reagents/materials/analysis tools. Attestation: Eberhard F. Kochs approved the final manuscript. Name: Gerhard Schneider, MD. Contribution: This author helped conceive and design the experiments and contribute reagents/materials/analysis tools. Attestation: Gerhard Schneider approved the final manuscript. This manuscript was handled by: Gregory J. Crosby, MD. ACKNOWLEDGMENTS
We thank Sandra Schmidt for linguistic review.
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REFERENCES 1. Horn B, Pilge S, Kochs EF, Stockmanns G, Hock A, Schneider G. A combination of electroencephalogram and auditory evoked potentials separates different levels of anesthesia in volunteers. Anesth Analg 2009;108:1512–21 2. Jordan D, Stockmanns G, Kochs EF, Pilge S, Schneider G. Electroencephalographic order pattern analysis for the separation of consciousness and unconsciousness: an analysis of approximate entropy, permutation entropy, recurrence rate, and phase coupling of order recurrence plots. Anesthesiology 2008;109:1014–22 3. Rampil IJ. A primer for EEG signal processing in anesthesia. Anesthesiology 1998;89:980–1002 4. Viertiö-Oja H, Maja V, Särkelä M, Talja P, Tenkanen N, Tolvanen-Laakso H, Paloheimo M, Vakkuri A, Yli-Hankala A, Meriläinen P. Description of the Entropy algorithm as applied in the Datex-Ohmeda S/5 Entropy Module. Acta Anaesthesiol Scand 2004;48:154–61 5. Avidan MS, Zhang L, Burnside BA, Finkel KJ, Searleman AC, Selvidge JA, Saager L, Turner MS, Rao S, Bottros M, Hantler C, Jacobsohn E, Evers AS. Anesthesia awareness and the bispectral index. N Engl J Med 2008;358:1097–108 6. Avidan MS, Jacobsohn E, Glick D, Burnside BA, Zhang L, Villafranca A, Karl L, Kamal S, Torres B, O’Connor M, Evers AS, Gradwohl S, Lin N, Palanca BJ, Mashour GA; BAGRECALL Research Group. Prevention of intraoperative awareness in a high-risk surgical population. N Engl J Med 2011;365: 591–600 7. Sambridge M, Tkalčić H, Jackson A. Benford’s law in the natural sciences. Geophys Res Lett 2010;37:L22301 8. Newcomb S. Note on the frequency of use of the different digits in natural numbers. Amer J Math 1881;4:39–40 9. Benford F. The law of anomalous numbers. Proc Amer Math Soc 1938;78:551–72 10. Engel H-A, Leuenberger C. Benford’s law for exponential random variables. Stat Probab Lett 2003;63:361–5 11. Nye J, Moul C. The political economy of numbers: on the application of benford’s law to international macroeconomic statistics. B E J Macroecon. 2007;7:1–14 12. Durtschi C, Hillison W, Pacini C. The effective use of benford’s law to assist in detecting fraud in accounting data. J Forensic Account 2004;5:17–34 13. Nigrini M. A taxpayer compliance application of Benford’s law. JATA 1996;18:72–91 14. Brown RJC. Benford’s Law and the screening of analytical data: the case of pollutant concentrations in ambient air. Analyst 2005;9:1280–5 15. Roukema B. Benford’s Law anomalies in the 2009 Iranian presidential election. arXiv 2009;0906.2789 16. Hill TP. “The difficulty of faking data”. Chance 1999;12:27–31 17. Antkowiak B. Different actions of general anesthetics on the firing patterns of neocortical neurons mediated by the GABA(A) receptor. Anesthesiology 1999;91:500–11 18. Drexler B, Kreuzer M, Jordan D, Antkowiak B, Schneider G. Sevoflurane-induced loss of consciousness is paralleled by a prominent modification of neural activity during cortical down-states. Neurosci Lett 2013;548:149–54 19. Hoffman KL, Battaglia FP, Harris K, MacLean JN, Marshall L, Mehta MR. The upshot of up states in the neocortex: from slow oscillations to memory formation. J Neurosci 2007;27: 11838–41 20. Cossart R, Aronov D, Yuste R. Attractor dynamics of network UP states in the neocortex. Nature 2003;423:283–8 21. Mahon S, Deniau JM, Charpier S. Relationship between EEG potentials and intracellular activity of striatal and cortico-striatal neurons: an in vivo study under different anesthetics. Cereb Cortex 2001;11:360–73 22. Timofeev I, Contreras D, Steriade M. Synaptic responsiveness of cortical and thalamic neurones during various phases of slow sleep oscillation in cat. J Physiol 1996;494:265–78 23. Jordan D, Stockmanns G, Kochs EF, Schneider G. Median frequency revisited: an approach to improve a classic spectral electroencephalographic parameter for the separation of
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consciousness from unconsciousness. Anesthesiology 2007;107: 397–405 24. R Development Core Team: R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing, 2006 25. Franks NP, Lieb WR. Temperature dependence of the potency of volatile general anesthetics: implications for in vitro experiments. Anesthesiology 1996;84:716–20 26. Beggs JM, Plenz D. Neuronal avalanches in neocortical circuits. J Neurosci 2003;23:11167–77 27. Linkenkaer-Hansen K, Nikouline VV, Palva JM, Ilmoniemi RJ. Long-range temporal correlations and scaling behavior in human brain oscillations. J Neurosci 2001;21:1370–7
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28. Worrell GA, Cranstoun SD, Echauz J, Litt B. Evidence for self-organized criticality in human epileptic hippocampus. Neuroreport 2002;13:2017–21 29. Nunez PL. Generation of human EEG by a combination of long and short range neocortical interactions. Brain Topogr 1989;1:199–215 30. Lee U, Oh G, Kim S, Noh G, Choi B, Mashour GA. Brain networks maintain a scale-free organization across consciousness, anesthesia, and recovery: evidence for adaptive reconfiguration. Anesthesiology 2010;113:1081–91 31. Steriade M, McCormick DA, Sejnowski TJ. Thalamocortical oscillations in the sleeping and aroused brain. Science 1993;262: 679–85
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Brain Electrical Activity Obeys Benford’s Law Matthias Kreuzer, MSc,* Denis Jordan, PhD,* Bernd Antkowiak, PhD,† Berthold Drexler, MD,† Eberhard F. Kochs, MD,* and Gerhard Schneider, MD‡ BACKGROUND: Monitoring and automated online analysis of brain electrical activity are frequently used for verifying brain diseases and for estimating anesthetic depth in subjects undergoing surgery. However, false diagnosis with potentially catastrophic consequences for patients such as intraoperative awareness may result from unnoticed irregularities in the process of signal analysis. Here we ask whether Benford’s Law can be applied to detect accidental or intended modulation of neurophysiologic signals. This law states that the first digits of many datasets such as atomic weights or river lengths are distributed logarithmically and not equally. In particular, we tested whether data obtained from electrophysiological recordings of human patients representing global activity and organotypic slice cultures representing pure cortical activity follow the predictions of Benford’s Law in the absence and in the presence of an anesthetic drug. METHODS: Electroencephalographic (EEG) recordings from human subjects and local field potential recordings from cultured cortical brain slices were obtained before and after administration of sevoflurane. The first digit distribution of the datasets was compared with the Benford distribution. RESULTS: All datasets showed a Benford-like distribution. Nevertheless, distributions belonging to different anesthetic levels could be distinguished in vitro and in human EEGs. With sevoflurane, the first digit distribution of the in vitro data becomes steeper, while it flattens for EEG data. In the presence of high frequency noise, the Benford distribution falls apart. CONCLUSIONS: In vitro and EEG data show a Benford-like distribution which is altered by sevoflurane or destroyed by noise used to simulate artefacts. These findings suggest that algorithms based on Benford’s Law can be successfully used to detect sevoflurane-induced signal modulations in electrophysiological recordings. (Anesth Analg 2014;118:183–91)
M
onitoring and automated online analysis of brain electrical activity are increasingly used as diagnostic tools to assess brain activity or detect brain diseases. For example, electroencephalographic (EEG) monitoring is increasingly used by anesthesiologists to assess depth of anesthesia, that is, the hypnotic component of anesthesia. This online monitoring may help to prevent episodes of intraoperative awareness in patients undergoing surgery.1–4 Still, these devices may not be superior to standard monitoring as demonstrated over the last several years.5,6 Nevertheless, the use of these monitoring systems may be beneficial in some situations. It is evident that irregularities in data sampling or poor quality of the recorded signals may lead to errors in data analysis and may therefore have serious consequences for patients. This may be caused by technical failure or handling From the *Department of Anesthesiology, Klinikum rechts der Isar, Technische Universität München, München; †Department of Anesthesiology, Experimental Anesthesiology Section, University of Tübingen, Tübingen; and ‡Department of Anesthesiology, Witten/Herdecke University, Helios Clinic Wuppertal, Germany. Accepted for publication September 19, 2013. Funding: None. The authors declare no conflicts of interest. Reprints will not be available from the authors. Address correspondence to Gerhard Schneider, MD, Department of Anesthesiology, Witten/Herdecke University, Helios Clinic Wuppertal, Heusnerstr, 40 D-42283, Wuppertal, Germany. Address e-mail to gerhard. [email protected]. Copyright © 2013 International Anesthesia Research Society DOI: 10.1213/ANE.0000000000000015
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errors and may occur in clinical practice every day. In serious cases, such erroneous data analysis would mimic an aberrant clinical patient state. To prevent such events, it is necessary to gain knowledge about expected changes of electrophysiological activity, for example, caused by anesthetic drugs, and to develop a robust method for detection of distortions or errors in the underlying signal. For this purpose, algorithms based on Benford’s Law may be a promising approach, given that neurophysiologic data may obey this law. This method may be used to detect irregularites, that is, distortion of physiological signals. It has been shown to be useful in the analysis of naturally occuring signals. For example, earthquakes can be detected by changes in the first digit distribution of a seismogram’s data.7 Benford’s Law states that the first digits of datasets are not equally but logarithmically distributed as presented in Figure 1. This finding goes back to 1881, when Newcomb8 detected this property in many scientific and statistical tables. In 1938, Frank Benford confirmed Newcomb’s observation by creating an extensive database containing different kinds of data such as length of rivers, atomic weights, specific heat constants, the numbers occurring in a newspaper, and many more.9 In all these collected datasets, the leading digits followed the logarithmic distribution reasonably well. A short example describing the nature of Benford’s Law can be found in the Appendix. Furthermore, Engel and Leuenberger10 showed that exponentially distributed numbers more or less obey Benford’s Law. Today this distribution is used in macroeconomic statistics11 to detect fraud in accounting data,12 income tax evasion,13 or for the screening of large datasets.14 Most recently, Benford’s Law was used to www.anesthesia-analgesia.org
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Figure 1. First digit distribution according to Benford’s Law. The height of each bar represents the percentage of each digit occurring as first digit in a Benford distributed dataset.
detect abnormalities in the 2009 Iranian election.15 The idea of using Benford’s Law to screen data is based on the observation that regular, “naturally generated” data usually follow a logarithmic distribution, while faked data show abnormalities in the distribution.16 In the presented experiments, electrophysiological activity, namely EEG from humans and local field potentials (LFP) from mouse neocortex, was analyzed following the idea that this “natural” activity should also follow the described distribution, whereas distorting influences such as noise or modulation of cerebral activity caused by sevoflurane may alter the distribution of the first digits of the signals’ amplitude values. If the hypothesis of Benford-like distributed datasets proves valid, the evaluation of the first digit distribution may also be used as a sensor for signal distortions which change the Benford-like distribution.
METHODS
EEG data recorded during a volunteer study and LFP recordings obtained from organotypic slice cultures (OTC) were used for analysis. All participating volunteers gave informed written consent for the protocol, which was approved by the ethics committee of the Technische Universität München, Munich, Germany (Application No. 942/03). All animal procedures were approved by the animal care committee (Eberhard Karls University, Tübingen, Germany) and were in accordance with the German Animal Welfare Act (TierSchG).
LFP Recordings from Organotypic Slice Cultures
To receive LFP recordings, spontaneous synaptic activity in OTC were analyzed. Neocortical slices were excised from 5-day old rat pups and grown in culture for at least 2 weeks according to the method described by Antkowiak et al.17 During this time, cultured cells establish new synaptic connections and form a spontaneously active network. These cultures were treated with sevoflurane in concentrations comparable with the levels used in humans. Sevoflurane was dissolved in an artificial cerebrospinal fluid and was applied via bath transfusion. The test solution was delivered
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via syringe pumps (ZAK, Marktheidenfeld, Germany) with a flow rate of 1 mL/min over a Teflon tubing (Lee, Frankfurt, Germany) connection. LFP were recorded 10 minutes after washing the culture with the new perfusate to ensure equilibrated concentrations. A 10-minute waiting time was established to ensure near steady-state conditions.17 LFP recordings were performed using artificial cerebrospinal fluid-filled glass electrodes with a resistance of about 5 MΩ at 34°C. The extracellular recordings were low-pass filtered at 5 kHz, digitized at a sampling frequency of fs = 10 kHz, and visualized and stored with the Axoscope 9 software tool (Axon Instruments, Molecular Devices Corp., Chicago, IL). LFP were separated from action potentials by digital band-pass filtering. The duration of each recording was 180 seconds. The influence of sevoflurane on the recorded LFP is displayed in Figure 2. OTC were first tested for general “healthiness” by visual inspection. After placement of extracellular electrodes, OTC were only used for an experiment in case the observed action potential firing pattern was typical and stable for >10 minutes. If the firing pattern changed during the recording time, these cultures were not used for further analysis. The sample size was n = 15 for control conditions, n = 5 for 0.75 minimum alveolar concentration (MAC) equivalent to a deep sedation in humans, n = 5 (1 MAC) equivalent to light anesthesia, and n = 5 for the 1.5 MAC experiments corresponding to deep general anesthesia. In cell cultures, extracellular LFP activity was recorded at these concentrations. A 10-second sequence showing spiking activity and a resting signal (up and down states) was cut at the end of each recording, followed by analysis of the first digit distribution of these sequences with respect to Benford’s Law. In addition, three 2 s segments of every recording without spiking activity were cut and also followed by an analysis of their first digit distribution. During these “silent” periods, the network is in a resting state, a so-called down state (Fig. 3). The information content of signals from the down state is affected by anesthetics.18 Experimental results indicate that the active epochs, the up states, can initiate structured information flow in local cortical columns19 and may be associated with entering specific memory states or with solving computational problems.20 Up and down states in vivo observed under ketamine– xylazine anesthesia could be directly associated with slow waves that are prominent in surface EEG.21 In sleep experiments with cats, the waking electrical activity was connected to sequences of silenced firing in the cortex at the onset of slow wave sleep.22 These findings pronounce that (cortical) LFP activity is directly connected to surface EEG.
EEG Data Recording
EEG was recorded during a volunteer study. Men participants 18 to 35 years old were anesthetized with sevoflurane according to a study protocol that has been described in detail by Horn et al.1 In short, EEG was recorded using ZipPrep electrodes (Aspect Medical Systems, Natick, MA) from positions AT1, M2, Fz(reference), and Fpz(ground) according to the international 10 to 20 system at equilibrated levels of anesthesia. The signal from AT1-Fz was used for the present analysis. The equilibrated levels of anesthesia were defined as follows: the “lighter” end point of anesthetic levels was defined by loss of consciousness
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Figure 2. Anesthetic-induced changes of the electroencephalographic (EEG) signal (left) and local field potentials (LFP) signal amplitudes recorded in the organotypic slice cultures (OTC) (right). Under the influence of sevoflurane, the EEG frequencies decrease and amplitudes increase. In LFP recordings, activity decreases with the administration of sevoflurane.
Figure 3. Sequences of 2-second length, where network activity was low, were cut out of the local field potentials (LFP) recording. These selected sequences represent cortical down states (DS).
(LOC), and the deepest level was indicated by the occurrence of EEG burst suppression. On the basis of individual anesthetic concentrations, 2 intermediate levels were then defined. The difference between end-expired sevoflurane concentration at burst suppression and LOC was divided into 3 intervals, resulting in the 2 intermediate levels. In
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the presented experiments, the intermediate level closer to LOC was termed “light general anesthesia” and the deeper intermediate level “deep anesthesia.” Five-minute sequences of EEG were recorded at the equilibrated anesthetic levels (1) awake (before administration of sevoflurane, n = 8), (2) light general anesthesia (n = 6), and (3)
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deep anesthesia (n = 6).1 The signals were recorded with an analog to digital converter resolution of 12 bit and sampling frequency of fs = 5 kHz, followed by digital low-pass filtering and decimation to 1 kHz. The final bandwidth of the signal was 0.5 to 400 Hz. Representative traces of EEG episodes are displayed in Figure 2.
Data Processing
Since the analyses were targeted toward detection of properties represented by activity of neuronal networks in the brain, electromyography (EMG) can be assumed to be a biological artefact that superposes the primary signal, that is, the EEG. Hence, minimizing the influence of EMG as spurious signal is necessary. In scalp recordings, frequencies up to 30 Hz may mainly represent EEG activity, while higher frequencies are prone to be disturbed by EMG.23 As a consequence, the recorded EEG was also analyzed after a 30 Hz low-pass filtering using MATLAB R2012a (the MathWorks, Natick, MA). The last 10 seconds were extracted from each recording, and their first digit distribution was tested for concordance with Benford’s Law. For the LFP recordings, the selected episodes of general and down state only activity were analyzed after adequate low-pass filtering and down sampling to 2 kHz.
datasets were compared with the Benford distribution using LabView 6i (National Instruments, Austin, TX), and the first digit distributions of recordings without sevoflurane administration were compared with the distributions with sevoflurane. Therefore, data of 1 anesthetic level were pooled to 1 dataset X for patient or animal data, respectively. If the first digit distribution of the analyzed dataset shows logarithmic behavior, the mantissas of the data’s logarithm is uniformly distributed.8 A set union can be defined as described in equation (1): 9
Ed = ∪ [d ⋅ 10 k ,(d + 1) ⋅ 10 k [for d ∈ {1,2,...,9}, i.e., ℜ+ = ∪ Ed k ∈Z
d =1
Hence, the probability of 1 data point having the first digit d is described in equation (2): P(X ∈ Ed ) = P (log(X ) ∈[log d , log( d + 1)]) 1 = log(d + 1) − log d = log(1 + ) d
Benford Analysis
The first digit distribution was determined from EEG and LFP recordings. The first digits (without leading zeroes) of the amplitude values at discrete time points were analyzed. If the amplitude at a certain discrete time was 58 μV, the first digit was 5. If the amplitude is 0.95 μV, the first digit is 9 because leading zeroes are ignored. All included
(2)
where X is the mantissa of x, that is, x mod 1.
Influence of Noise on EEG Recordings
To evaluate the influence of general signal distortions on EEG, a representative EEG sequence recorded during wakefulness was superposed with high-pass white filtered noise at 100 Hz using a third-order Butterworth low pass. Two different noise amplitudes were used, and the raw and resulting signals used for analysis are displayed in Figure 4. The procedure of superposing the EEG sequence was performed using LabView 6i (National Instruments, Austin, TX).
(1)
1 P(X ∈ Ed ) = log(1 + )(3) d is the definition of Benford’s Law. We compared the logarithmic, modulo 1 first digit distribution of the single datasets with a uniform distribution using the Kolmogorov– Smirnov test statistics. This test was performed using R 2.4.0 (R Foundation for Statistical Computing, Vienna, Austria).24 In addition, similarity of the distributions was indicated by QQ-plots plotted with MATLAB R2012a (the MathWorks, Natick, MA).
RESULTS
Both in vitro and EEG data led to test statistics of ≤0.11 (Kolmogorov–Smirnov test), indicating a good similarity to the Benford distribution, related to a maximum deviation of
Figure 4. A 10-second electroencephalographic (EEG) trace recorded from a patient being awake is superposed with white noise, low-pass filtered at 100 Hz, of different amplitudes. The top trace represents the raw EEG. The middle trace shows raw EEG superposed with noise (awake + noise level 1), and the lower trace shows EEG superposed with white noise of higher amplitude (awake + noise level 2)
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Figure 5. QQ-plots of the unfiltered electroencephalographic (EEG) data compared with the logarithmic, modulo 1 first digit distribution of the EEG sequences to a uniform distribution (angle bisector, thin line). All 3 EEG distributions are close to the bisecting line representing uniform distribution. The curvature of the EEG “awake” (black) distribution is inverse to the EEG distributions recorded under sevoflurane. (blue: light general anesthesia, red: deep general anesthesia).
Figure 7. First digit distribution of unfiltered (A) and 30 Hz lowpass filtered (B) electroencephalographic (EEG) data derived from sequences recorded at the levels “awake” (black, n = 8), “light general anesthesia” (blue, n = 6), and “deep general anesthesia” (red, dashed, n = 6). The 30 Hz low-pass filtering does not influence the first digit distribution. After administration of sevoflurane, the first digit distribution becomes flatter. There are no significant differences in the distribution of the states “light general anesthesia” and “deep general anesthesia.” Figure 6. QQ-plots of the in vitro up and down-state sequences data, compared with the logarithmic, modulo 1 first digit distribution of the local field potentials (LFP) sequences (black, bold line) to a uniform distribution (gray). The distribution of the LFP “awake” (black) sequences almost perfectly matches the Benford distribution, while the LFP distributions corresponding to sedation (blue), light (red), and deep anesthesia (green) are still in concordance with the bisecting line (thin black line).
1. This is confirmed by corresponding QQ-plots which only indicate a small deviation from the bisecting line (Figs. 5 and 6). If the plotted EEG/in vitro data comply with the bisecting line, it is uniformly distributed, that is, it shows Benford distribution. Small deviations of the data from the bisecting line still are in accordance with the Benford distribution. Notice that because of the large dataset and of possible dependence of measurements, the P-value may be biased and is therefore not included in the present analysis.
EEG Analysis
The QQ-plots derived from analysis of EEG amplitudes are close to the bisecting line for all anesthetic levels (Fig. 5), that is, the amplitudes of EEG recorded at different anesthetic levels is and remains Benford-like distributed. Still, the level “awake” can be distinguished from anesthetic levels by a different curvature in the QQ-plot. The awake plot starts below the bisecting line, while at anesthetic levels, the plot is initially above the bisecting line. The curvature flips with delivery of sevoflurane. The state awake can be
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even better separated from “light or moderate general anesthesia” and “deep general anesthesia” by looking at the distribution of probabilities of the first digit being d with d ∈{1, 2,..., 9}. The curve of the first digit distribution from awake is more pronounced than the distributions from the recordings performed under anesthesia (Fig. 7). The probability P(d = 1) and P(d = 2) is higher at the level awake than it is during anesthesia. The second plot of this figure shows the first digit distribution of the recorded data after 30 Hz low-pass filtering that was performed to exclude EMG activity from the analysis. The course of the plots is similar, that is, the low-pass filtering does not have an apparent effect on the outcome of the Benford analysis. The averaged distribution derived from each anesthetic level for the unfiltered and the 30 Hz low-pass filtered data was compared with a Kolmogorov–Smirnov test. For all comparisons, the P-value was 1, clearly rejecting the null hypothesis that the distributions are unequal and suggesting similar results.
EEG Superposed With Noise
In the presence of additional noise, the Benford-like first digit distribution is distorted. The unfiltered raw EEG’s first digit distribution is almost congruent with the Benford distribution as displayed in Figure 8. With additional noise, P(d = 1) decreases and falls apart from the Benford distribution.
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Figure 8. First digit distribution of an unfiltered raw electroencephalographic (EEG) trace from a patient who was awake (black). This EEG trace was artificially distorted with white noise, low-pass filtered at 100 Hz (blue). As a consequence, the first digit distribution becomes less steep for low first digits. If the noise amplitude becomes higher, the distribution becomes even flatter (red). For better comparison, the Benford distribution is additionally presented in gray. Further, the first digit distributions of the additional noise content are presented in cyan (100 Hz low-pass filtered noise level 1) and magenta (100 Hz low-pass filtered noise level 2)
In Vitro Analysis
The in vitro QQ-plot derived in the absence of sevoflurane almost completely fits the bisecting line. The plots from data with sevoflurane do not show this strong concordance but are still very close to the bisecting line, that is, agree with the Benford distribution (Fig. 6). The flip in the curvature caused by sevoflurane is hardly visible, and it is inverse to the EEG data. The probability distribution of the leading digits was more pronounced after the cells were washed with sevoflurane, that is, P(d = 1) and P(d = 2) were higher when sevoflurane was delivered. This analysis result is contradictory to the EEG observation. This distribution did not change significantly during analysis of the entire LFP recording, containing up and down states or only down states. Spiking activity did not have any influence on the resulting probability distribution (Fig. 9).
DISCUSSION
Our results indicate that neuronal network activity in the cortex, recorded as LFP and frontal EEG, follow Benford’s law. The Benford-like behavior does not disappear with sevoflurane delivered in concentrations corresponding to anesthetic levels such as “sedation,” “light general anesthesia,” and “deep general anesthesia.” Our assumptions on MAC equivalents of sevoflurane are based on data published by Franks and Lieb.25 These MAC values refer to the plasma or blood concentrations of volatile anesthetics in mammals, determined at 37°C. Based on these data, a MAC-equivalent of 0.35 mM sevoflurane was used in this study. The first digit distribution of the recorded data changes with sevoflurane, but it remains Benford-like. The QQ-plots of LFP and EEG recorded in the presence of sevoflurane reveal a different effect on a local, cortical network (LFP) compared
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with the EEG recordings that also contain information from subcortical areas. The results hold for unfiltered EEG and for 30 Hz low-pass filtered EEG as well as for LFP episodes containing both up and down states and down states only. If the EEG recordings are distorted by superposition with noise, the Benford-like distribution is destroyed.
Scale Invariance
The results indicate that the analyzed activity is scale invariant because a Benford distribution is scale-free, that is, the revealed properties are not used due to a distinct scaling but are independent from it. Other experiments also indicate that distinct properties of cerebral activity show scale-invariant behavior. There, scale invariance has been suggested to represent the network’s property to optimize its structure and enable advantageous synchronization and information transfer,26–28 hence reflect important cerebral tasks. In general, scale invariance seems to be associated with structured or organized behavior. LFP episodes of combined up and down states seem to follow structured activity as well as isolated down-state sequences. LFP are local activity of a small neuronal network. The fact that a Benford distribution can be observed for episodes containing both up and down states and for down states only leads to the assumption that cortical activity is structured in periods of activity, that is, during up states when working memory tasks may be performed20 as well as during inactive down-state periods. Surface EEG electrodes measure the averaged activity of up to 1010 neurons.29 The recorded EEG, representing “global” neuronal activity, also shows structured activity. The structured activity observed at all scales changes in the presence of sevoflurane but is not destroyed.
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Figure 9. First digit distribution of the in vitro data (A, up and downstate sequences; B, down state sequences) recorded at levels corresponding to “awake” (black, n = 15), “sedation” (green, n = 5), “light general anesthesia” (blue, n = 5), and “deep general anesthesia” (red, dashed, n = 5). With administration of sevoflurane, the first digit distribution of the recorded local field potentials (LFP) sequences becomes steeper. There is no significant difference in the distributions of the analyzed sequences containing up and down states and the down-state only sequences. The sevoflurane concentration also seems to have a small impact only on the distribution, because distributions recorded during light and deep general anesthesia only differ to a small degree.
In case of contamination of the EEG with noise, the present structure in the signal is distorted, and the first digits Benford-like behavior vanishes as opposed to the physiologic changes caused by sevoflurane. Other research also addressing the question of structured neuronal activity represented by scale invariance seems to support our results of structure in the signals. A global scalefree behavior, according to power laws of neuronal network activity in terms of coupling of different cortical areas, was observed in the absence and presence of propofol.30 Other results did also find power law distributions for neuronal network tasks such as LFP firing propagation.26 All these findings suggest that neuronal processes and interactions are scale-free and follow a distinct organization. Our results indicate that persistent scale invariance can be observed at all scales of neuronal networks from the isolated cortex up to EEG recordings.
Effects of Sevoflurane: Differences Between Isolated Neocortical Cultures and EEG
Visual inspection of the first digit distribution plot shows steepening of the LFP first digit distribution and flattening of the EEG first digit distribution with increasing concentrations of sevoflurane. Generally speaking, an effect of sevoflurane can be observed in the first digit distribution. Nevertheless,
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the Benford-like behavior of the distributions persists. This leads to the assumption that the organized structure of activity is altered but not destroyed by sevoflurane, meaning that during anesthesia, the communication structure of the neuronal network is altered but not destroyed. The LFP recordings were lacking subcortical inputs whereas EEG activity received input also from subcortical areas. These existing or missing inputs from deeper brain areas seem a plausible explanation for the change of distribution in opposite directions. Thalamic structures, for example, have strong influence on EEG recordings under anesthesia. During wakefulness, asynchronous sensory input is transferred via thalamic relay neurons into the cortex. In this mode, relay neurons do not display synchronized firing.31 As a consequence, cortical neurons receive irregular synaptic inputs. Therefore, cortical cells display irregular action potential firing, and the EEG is desynchronized. During deep sleep and anesthesia, thalamic relay neurons oscillate in synchrony in the delta frequency range (1–4 Hz). In this state, thalamic neurons produce strong and rhythmic synaptic excitation of cortical pyramidal neurons, forcing these cells to display a very similar activity pattern. These findings indicate that cortical and subcortical neuronal network activity is strongly influenced by anesthetics. The effects of sevoflurane on subcortical areas may influence network activity in the cortex, that is, sevoflurane may affect network activity differently in an isolated cortical network than in a cortical network which is connected to subcortical structures. Taken together, the different effects of sevoflurane on the distributions calculated from in vitro data and eeg recordings most likely reflect actions on different neuronal substrates. To conclude, the above analysis shows that cortical neuronal network activity follows the Benford distribution, reflecting organized activity. The effect of volatile anesthetic, a changed state of consciousness, alters the distribution compared with control conditions. Despite of this change, the Benford-like characteristics of the distributions are maintained even under sevoflurane. Sevoflurane influences the analyzed neuronal network but does not destroy structured activity, that is, the Benford distribution. Sevoflurane initiated the first digit distribution of isolated cortical LFP and surface EEG to react differently. This difference is probably caused by the absence or presence of subcortical structures that influence cortical network activity. In contrast to the physiological changes, adding white noise to the EEG recordings destroys the Benford-like distribution of the EEG. This finding may open the possibility of using this method to detect distortions in the signal that are not of electrophysiological origin.
Clinical and Scientific Impact
With the analyses performed, we have shown that electrophysiolgical activity, represented as surface EEG and LFP derived from isolated cortex, follows a certain pattern that is altered but not eliminated by sevoflurane. Even during anesthesia, the neuronal network activity is reflected in organized activity. This holds true for local cortical networks with a small number of contributing neurons as well as for global EEG recordings. Hence, our approach may be
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valuable in terms of evaluating datasets according to their validity. If neuronal activity is recorded, the dataset should show a logarithmic first digit distribution. This method may be applied in the clinical setting of EEG monitoring. High-quality EEG recordings should show a Benford-like first digit distribution, irrespective of the anesthetic level. If the signal is influenced by contaminating sources, the distribution can be affected. A recorded EEG signal that does not show this distribution may be distorted and should be handled with care for monitoring purposes. E
APPENDIX Example for the Nature of Benford’s Law:
The following example explains the nature of this law: Imagine the stock price is increasing from 10 to 20 points. The difference between both prices is 10 points, and the price is doubled. If the price increases from 80 to 90 points, the difference is still 10 points, but the price only increases ̭ by (9/8 − 1) ‗ 12.5%. Even if the difference for both examples is 10 points, it takes more effort to come from 10 to 20 than from 80 to 90. That means that if you have a stock course of 10, it will take a longer time to reach a course of 20 since you have to double your value than if your course is 80 and you only need to add another 12.5 percent of your value to reach 90. The wider the relative gap to bridge from one first digit to the next, the longer you will remain in that corridor and the higher the probability will be of detecting this first digit. This implies that the 1 will show up more often than any other digit. DISCLOSURES
Name: Matthias Kreuzer, MSc. Contribution: This author helped analyze the data, contribute reagents/materials/analysis tools, and wrote the paper. Attestation: Matthias Kreuzer approved the final manuscript. Name: Denis Jordan, PhD. Contribution: This author helped analyze the data and contribute reagents/materials/analysis tools. Attestation: Denis Jordan approved the final manuscript. Name: Bernd Antkowiak, PhD. Contribution: This author helped conceive and design the experiments, contribute reagents/materials/analysis tools, and wrote the paper. Attestation: Bernd Antkowiak approved the final manuscript. Name: Berthold Drexler, MD. Contribution: This author helped perform the experiments and contribute reagents/materials/analysis tools. Attestation: Berthold Drexler approved the final manuscript. Name: Eberhard F. Kochs, MD. Contribution: This author helped contribute reagents/materials/analysis tools. Attestation: Eberhard F. Kochs approved the final manuscript. Name: Gerhard Schneider, MD. Contribution: This author helped conceive and design the experiments and contribute reagents/materials/analysis tools. Attestation: Gerhard Schneider approved the final manuscript. This manuscript was handled by: Gregory J. Crosby, MD. ACKNOWLEDGMENTS
We thank Sandra Schmidt for linguistic review.
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