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Sparse asynchronous cortical generators can produce measurable scalp EEG signals
Sparse asynchronous cortical generators can produce measurable scalp EEG signals Nicol´as von Ellenrieder, Jonathan Dan, Birgit Frauscher, Jean Gotman PII: DOI: Reference:
S1053-8119(16)30188-4 doi: 10.1016/j.neuroimage.2016.05.067 YNIMG 13225
To appear in:
NeuroImage
Received date: Revised date: Accepted date:
16 February 2016 19 April 2016 26 May 2016
Please cite this article as: von Ellenrieder, Nicol´ as, Dan, Jonathan, Frauscher, Birgit, Gotman, Jean, Sparse asynchronous cortical generators can produce measurable scalp EEG signals, NeuroImage (2016), doi: 10.1016/j.neuroimage.2016.05.067
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ACCEPTED MANUSCRIPT Sparse asynchronous cortical generators can produce measurable scalp EEG signals
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Montreal Neurological Institute and Hospital, McGill University
École Polytechnique, Université Libre de Bruxelles
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3801 University Street, Montreal, Quebec, H3A 2B4, Canada
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Nicolás von Ellenriedera*, Jonathan Danab, Birgit Frauscherc, Jean Gotmana
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50 Avenue F. D. Roosevelt, 1050 Bruxelles, Belgium
Department of Medicine, Center for Neuroscience Studies, Queen’s University
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18 Stuart Street, Kingston, Ontario, K7L 3N6, Canada
* Corresponding author:
Montreal Neurological Institute
3801 University Street, Room 009e Montreal, QC H3A 2B4, Canada [email protected] phone: +1 514 398 6644, ext 00445 fax: +1 514 398 8106
ACCEPTED MANUSCRIPT Abstract
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We investigate to what degree the synchronous activation of a smooth patch of cortex is necessary for observing EEG scalp activity. We perform extensive simulations to compare the activity generated on the scalp by different models of cortical activation, based on intracranial EEG findings reported in the literature. The spatial activation is modeled as a cortical patch of constant activation or as random sets of small generators (0.1 to 3 cm 2 each) concentrated in a cortical region. Temporal activation models for the generation of oscillatory activity are either equal phase or random phase across the cortical patches. The results show that smooth or random spatial activation profiles produce scalp electric potential distributions with the same shape. Also, in the generation of oscillatory activity, multiple cortical generators with random phase produce scalp activity attenuated on average only 2 to 4 times compared to generators with equal phase. Sparse asynchronous cortical generators can produce measurable scalp EEG. This is a possible explanation for seemingly paradoxical observations of simultaneous disorganized intracranial activity and scalp EEG signals. Thus, the standard interpretation of scalp EEG might constitute an oversimplification of the underlying brain activity.
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Keywords
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Scalp EEG, cortical generators, asynchronous activity, modeling, boundary elements method.
ACCEPTED MANUSCRIPT Introduction
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In the interpretation of scalp EEG, the seminal paper by Gloor (1985) is to this day one of the most influential in clinical electrophysiology, given its largely qualitative but rigorous explanation of the volume conduction effects. Gloor concludes that when pyramidal neurons are activated synchronously within a cortical area of macroscopic extent, the fields of the individual neurons of the population combine to create measurable fields on the scalp. This conclusion is based on physical principles, and is a natural conclusion of any analysis of the EEG forward problem (e.g. Geselowitz, 1967; Nunez, 1981). This type of explanation for the generation of the EEG is given in several major reference books (e.g. Lopes da Silva chapter 5 in Niedermeyer). However, what was originally stated as a sufficient condition for the measurement of EEG on the scalp, has sometimes been interpreted as a necessary condition. It is therefore not uncommon to find the explicit or implicit notion that scalp EEG arises only when pyramidal neuron populations act in synchrony, both spatially and temporally.
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The presence of extended and synchronous cortical generators leading to scalp potentials has been observed at the time of interictal epileptic discharges when recording simultaneously subdural and scalp EEG (Tao et al., 2005). There is however evidence from experimental intracranial EEG studies that such an extended and synchronous patch of cortex acting as generator of scalp EEG may be an oversimplification in some situations. In the canine cortex, for example, phase coherence was observed only in relative small cortical regions during alpha rhythm bursts even though alpha activity could be recorded on the scalp (Lopes da Silva et al, 1973). At the time of sleep spindles in the human scalp EEG, it was found that there are frequently spindles in various intracerebral regions but they are not synchronous among themselves nor with the scalp spindle, and the time relationships between various regions is highly variable from spindle to spindle (Frauscher et al, 2015). In epileptic patients, simultaneous but non-synchronous high frequency oscillations (HFOs) were occasionally observed in non-neighboring channels of cortical grids at the time of scalp HFOs (Zelmann et al, 2014). All these studies point to asynchronous focal cortical generators combining in some way to produce measurable scalp EEG activity. We present simulations to determine if it is theoretically possible to observe scalp EEG produced by asynchronous cortical generators, and under what conditions. The model we propose is based on observations from the intracranial studies mentioned in the previous paragraph, and consists in cortical generators with irregular spatial morphology and non-coherent phase. The rationale for a model with irregular spatial profile of cortical activation is that for scalp EEG generation the total area of activation is likely important, but not its spatial contiguity. The inclusion of generators of oscillatory activity with random phase seems less intuitive, given that oscillatory signals with random phase cancel each other when averaged. However, this reasoning overlooks that complete cancellation is achieved only for an infinite number of generators, and that the EEG recorded on the scalp is not a mere averaging of cortical generators.
Methods A realistically shaped three-layer model was built based on the high-quality Colin MR data (AubertBroche et al., 2006), conductivity values of 0.33 S/m were adopted for the brain and scalp, and 0.015 S/m for the skull. The EEG forward problem was solved using the Boundary Element Method (BEM), with
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linear elements (de Munck, 1992) and the isolated skull approach (Meijs et al., 1989). Scalp electrodes were simulated positioned according to the 10-5 extension of the 10-10 system (Oostenveld and Praamstra, 2001). When simulating amplifier noise, independent and normally distributed noise was added to each scalp channel. The Signal to Noise Ratio (SNR) was defined based on the channel with highest amplitude. Additionally, four intracerebral electrodes with 5 mm distance between contacts were simulated in the left frontal lobe. A point approximation was adopted for the intracerebral contacts (von Ellenrieder et al., 2012), and bipolar channels between neighboring contacts were simulated with the same layered model and BEM formulation.
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Special attention was given to the modeling of the cortex, represented by a surface in the middle of the layer of cortical gray matter. Each hemisphere was tessellated in 60,000 triangular elements. Distributed generators were modeled as sets of dipolar sheets associated to the triangles of the tessellation, with a current density value for each vertex and linear variation of the current density on the triangular faces of the tessellation, with orientation normal to the surface (von Ellenrieder et al., 2009). This model was selected rather than elementary dipoles at the vertices of the cortical surface distribution, because with the adopted point approximation of the intracerebral electrodes, the dipoles would introduce singularities in the solution leading to inaccurate results for the intracerebral simulations. The source model based on dipolar sheets does not introduce singularities and leads to more accurate simulation results (von Ellenrieder et al., 2015).
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Most examples shown in this study correspond to generators modeled within a region of 100 cm2 as measured by geodesic distance on the cortical surface, i.e. over the highly folded cortical surface. The visible area when projecting the region onto the scalp would be smaller. The region was located in the left frontal lobe, below scalp electrode F3 as seen in Figure 1A. However, results are also presented for different generator extents and for 100 locations distributed over the cortical surface of the left hemisphere. Generators with random spatial profile The scalp and intracranial electric potential distribution produced by a generator of constant current density covering the complete 100 cm2 region was compared to the one produced by generators in the same region but with random spatial activation profile. These generators were built by assigning randomly a normally distributed value at each vertex of the cortical surface tessellation, applying a spatial smoothing based on geodesic distance, and comparing the resulting value to a threshold. Cortical regions where the threshold was exceeded were deemed active, and otherwise inactive. All active regions were then assigned the same current density value. This created a partition of the cortical region in active and inactive subsets, examples are shown in Figure 1. The spatial scale, or correlation length, of the random maps was controlled by varying the degree of spatial smoothing, and is an indication of the size of the active regions. The activation of any pair of points on the cortical surface is uncorrelated if they are separated by a distance larger than this spatial scale. The coverage density, or active to inactive surface area ratio, was controlled by changing the threshold. Spatial scales from 2 to 10 mm, and coverage density of 25, 50, and 75 % of the total 100cm2 surface were investigated. The described procedure results in an activation pattern equivalent to spreading small generators of radius equal to the spatial scale (although not necessarily with circular shape) until the desired proportion of the cortical region is covered. Examples are shown in Figure 1. For low coverage densities this results in scattered generator patches of 0.12 to 3.1 cm2 surrounded by mostly inactive regions, and
ACCEPTED MANUSCRIPT for high coverage densities it results in inactive cortical patches of the same extent immersed in a mostly active cortical region. Asynchronous generators
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Generators of oscillatory activity and random phase were also simulated. The random phase was obtained by defining two independent random variables in each vertex of the cortical surface, as inphase and quadrature terms (i.e. two terms with a 90 degree phase shift between them). After spatial smoothing the ratio between these two variables was used to define the phase of the generator at each point of the cortical surface through a 4-quadrant arc-tangent operation. With this approach the spatial scale, or correlation length, of the phase could be controlled by changing the degree of spatial smoothing. Values between 2 and 20 mm were explored.
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Quantification of differences
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Results
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To quantify the difference between two scalp electric potential distributions, two measures were used. The ratio of the maximum amplitude of each distribution, and the Normalized Relative Difference (NRDM). The maximum amplitude of the scalp distribution is of interest because it is directly related to the visibility of the distribution on the scalp assuming the noise level remains unchanged. The NRDM compares the shape of the distribution irrespectively of the absolute amplitude (Meijs et al., 1989), and takes values between zero (distributions of identical shape), 1.4 (distributions of uncorrelated shapes), and two (distributions of identical shape but opposite signs).
Generators with random spatial profile Generators with random spatial profiles covering the same cortical region produce similar potential distributions on the scalp, as shown by the examples in Figures 2 and 3. Figure 2 shows that the scalp distribution of random generators can be almost identical if the strength of the current density at source level is allowed to change. Variations in the current density could be due e.g. to a different proportion of involvement of the neuronal populations. In all the remaining simulations, the strength of the generators is kept fixed for simplicity. However, the results shown in Figure 2 are an important reminder that changes in the current density of the active regions directly affect the amplitude of the scalp potential. Then, any difference we report on the amplitude of the scalp distribution due to the random activation profile could be compensated or exacerbated by the uncertainty in the current density value at source level. In any case, the shape of the distributions remains almost unchanged in the examples, and the amplitude of the scalp EEG is related to source strength and to the proportion of the cortical surface covered by active regions. In Figure 3 the current density or strength of the cortical generators is assumed equal for the three shown examples, and the scalp distribution has similar shape but different amplitude. Figures 2 and 3 (G-I) also shows that since intracerebral channels have a very local sensitivity, when the cortical activation profile is random different intracerebral patterns can be observed associated to the same scalp EEG distribution. This can be appreciated in examples of real intracerebral
ACCEPTED MANUSCRIPT measurements in Figure 4, where sleep spindles observed on the scalp appear associated to different intracerebral activation patterns.
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The difference between scalp potential distributions is shown quantitatively in Figure 5A. The NRDM was computed between the scalp distribution produced by the smooth 100 cm2 cortical patch of constant current density and 1000 random generators for each investigated coverage density and spatial scale. The figure shows that coverage density has a larger influence than spatial scale. Lower coverage means a greater difference between the spatial distribution of the activation profile, and leads to larger variability of the scalp distribution. Lower spatial scale leads to slightly lower NRDM values because the low skull conductivity introduces some blurring, or attenuation of high spatial frequencies, thus reducing the ability to resolve finer spatial details of the activation profile. For 50 % coverage, median NRDM values close to 0.2 are observed. To interpret the NRDM values, consider as an example that the scalp distributions shown in Figure 3E and 3F have NRDM of 0.23 and 0.16 when compared to the visually similar distribution shown in Figure 3D. Additional examples of the electric potential distribution on the scalp from generators with random activation profile and their respective NRDM values are presented in Inline Supplementary Figure 1. Also, we include in Figure 5B the NRDM that can be expected due to instrumental noise present in scalp EEG recordings. An NRDM of 0.2 is expected for signal to noise ratios of 26 dB, corresponding to noise amplitude 20 times lower than the signal. This noise level can be found in some clinical situations, e.g. for spindles of 20 µV amplitude in systems with noise level in the order of 1 µV due to the electrode/skin electrochemical interface, and to a lesser degree to the electronic noise of the amplifiers (Huigen et al., 2012). Thus, the difference in NRDM of 0.2 between 100% and 50% coverage is around the level of noise and the two are essentially indistinguishable.
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Insert Inline Supplementary Figure 1 here.
We assessed whether the above results are independent of the specific cortical region that has been illustrated (below F3). Figure 5C shows the results for generators in 100 different cortical regions distributed on the cortical surface of the left hemisphere, and for different extents of the regions containing the random generators. For each location and extent 1000 generators were simulated. The figure shows that the example region analyzed above is typical, since the median NRDM value for 50 % coverage and 100 cm2 is close to 0.2. Note that the extent of the cortical regions corresponds to the geodesic surface area; as the cortical surface is highly folded, the visible area when projected on the scalp is much smaller. In the example in Figure 2A, the 100 cm2 geodesic extent corresponds to 21 cm2 when projected into the scalp, and the 5 to 200 cm2 range explored in Figure 5C corresponds to an average of 2 to 40 cm2 when projected onto the scalp. Asynchronous generators The possibility of observing scalp EEG as a result of cortical generators of oscillatory activity with noncoherent phase may seem unlikely. The argument behind this notion is that averaging sinusoidal signals of random phase results in cancellation leading to a signal with null amplitude. However, for the amplitude to be completely canceled out it is necessary to average an infinite number of sinusoids. In the case of the scalp EEG the number of generators is finite. Considering first the averaging of sinusoids only (independently of the generation of the scalp EEG from cortical generators), Figure 6A shows the attenuation resulting from averaging sinusoids with uniformly distributed random phase compared to averaging the same number of sinusoids with equal phase. Almost 100 sinusoids need to be averaged to
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get a median attenuation of ten times, and still in 5 % of the cases the attenuation will be lower than half that value. The figure also shows that if the amplitudes of the sinusoids are multiplied by a random factor uniformly distributed between -1 and 1 before averaging, no attenuation is observed (see below the rational for positive and negative weights). This is because in the coherent case half the sinusoids will be multiplied by negative numbers that lead to cancellation, and the level of attenuation is similar to the one observed due to uniformly distributed random phase.
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Considering now the transformation from multiple cortical generators to a scalp potential, the forward problem cannot be considered a simple average of multiple cortical generators, even though some degree of blurring is introduced by the skull. The EEG forward problem can be seen as a weighted average of the cortical generators, and while the weight is not uniformly distributed, it does take positive and negative values as shown in Figure 6B. The figure shows the sensitivity of scalp electrode F3 to the different points of the cortical region used in the examples. The convoluted shape of the cortex means that generators in different regions sometimes contribute with opposite signs (as would be the case for two walls of a sulcus). If sinusoids are averaged weighted with the distribution of observed sensitivities, the resulting attenuation of the scalp potential is between the results corresponding to constant weights and uniformly distributed random weights (Figure 6A). This is still a simplification of the problem, but helps to understand why the asynchronous generators can produce measurable scalp EEG. Figure 7 shows an example of the situation, with three generators located in different cortical regions contributing to the signal recorded in electrode C4. Contributions from generators A and B partially cancel out in the synchronous case but not necessarily so in the asynchronous cases.
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Results of extensive simulations of generators of oscillatory activity with random phase are shown in Figure 8. Panel A shows the attenuation for different coverage density of the sources and different spatial scale for the phase of the signals, i.e. how far two cortical points need to be to have an uncorrelated phase. Results correspond to 4000 different realizations for each coverage density and spatial scale, for the left frontal brain region shown in Figure 2A. Attenuation values decrease slightly with increasing spatial scale, and increase slightly for increasing coverage density. Median attenuation values are between 3 and 6 times. Figure 8B shows the results for 100 different locations of the cortical region distributed over the cortical surface of the left hemisphere, and for different extents of the surface area containing the generators. The attenuation increases with the extent of the cortical region, but stabilizes for extents larger than 100 cm2, with median attenuations lower than 4 times. Inline Supplementary Figure 2 shows a 2D representation of the random phase and activation distribution used in the simulations. The effect of random phase alone are shown by the 100 % coverage curves of Figure 8, and by the examples of generators with random phase shown in Figure 9 together with the resulting scalp potential distribution and intracerebral signals. The combination of independent random activation profile and phase leads to an increase in the attenuation of the scalp amplitude compared to the coherent case, as shown by the curves with less than 100 % coverage in Figure 8, and the examples shown in Figure 10, for random phase and 50 % coverage density. Insert Inline Supplementary Figure 2 here. Figures 9 and 10 show that the scalp distribution changes shape compared to the constant phase case, and is attenuated by factors between 3 and 5, but would still be visible in the context of an EEG dominated by a constant phase generator. Examples of real intracerebral EEG measurements that lend support to the proposed random phase model can be found in Figure 4 for sleep spindles (the phase
ACCEPTED MANUSCRIPT difference between the channels can be appreciated clearly in the zoomed version of Figure 4 presented in the Inline Supplementary Figure 3). An example involving high frequency oscillations is shown in Inline Supplementary Figures 4 and 5.
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Discussion
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Generators with random spatial profile
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Our results show that it is not necessary to have a large patch of smooth cortical activation to generate scalp EEG, and that asynchronous cortical generators of oscillatory activity produce scalp EEG with an amplitude usually of the same order as synchronous generators. The scalp EEG originates from the cortex through a mechanism of weighted average of a finite number of generators, with weights that can be positive or negative due to the folding of the cortex. In this situation only very mild cancellation of generators with random phase should be expected.
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Regarding the spatial profile of cortical activation, the difference between the scalp electric potential distribution resulting from random patches of activation and a smooth area of cortex with constant activation (constant current density) is restricted almost entirely to their amplitude. On average, the difference in the shape of the distribution is in many cases of the same order as the shape change caused by additive instrumental noise contaminating the signal in clinical situations. Beyond the general behavior, a relatively large variability is observed in the results associated to different random activation patterns. This high variability can be explained by the different proportion of activity on the gyral crowns and sulcal walls. The electric activity on the crown is in close proximity to the surface and will easily produce visible scalp potential, while the activity on the sulcal walls is not only further away but also can suffer from partial cancelation between neighboring walls with opposite orientation of pyramidal neuron populations. As for the amplitude, it cannot be used in practice to distinguish between random and constant activation profiles because it is directly related to the uncertainty in the typical current density value of the generators, which is at least of one order of magnitude (von Ellenrieder et al., 2014). We therefore do not know the amplitude of an EEG generated by a synchronous generator. Asynchronous generators In the case of generators of oscillatory activity with random phase, we found that in half of the cases the attenuation compared to equal phase (synchronous generators) would be lower than 2 to 6 times. If the spatial variation of the phase is more rapid, or the percentage of cortical area covered by active regions increases, the attenuation is slightly higher because this is equivalent to an increase in the number of averaged generators. For the same reason the attenuation grows with the area of the cortex where the generators are located, but this effects stabilizes for areas larger than 50 to 100 cm2 because the sensitivity of any point on the scalp only covers a limited cortical region. Note that the amplitude of the scalp distribution is proportional to the strength of the generators, and almost proportional to the skull conductivity. Both of these parameters are highly uncertain (see discussion in von Ellenrieder et al.,
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Experimental situations concordant with the model
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Lopes da Silva et al. (1973) studied the generation of alpha rhythm in the canine brain. They observed bursts of alpha activity that could be topographically very specific, with regions of high cortico-cortical coherence often of 5 or 6 mm, concordant with the modeled spatial scales. The degree of coherence between different areas did not seem to be deterministic. They proposed that the apparently simultaneous occurrence of bursts of alpha rhythm at different locations depended on a common modulating influence which would gate the multiple “alpha generators” at the same time without necessarily causing all alpha bursts to be synchronous.
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Sleep spindles observed on the scalp occur most often at the same time as spindles in several intracerebral locations but exhibit a remarkable lack of repetitive intracerebral patterns (Nir et al., 2011; Frauscher et al., 2015), see also Figure 4. The phase coherence of the intracranial channels is also low as shown in the Inline Supplementary Figure 3, and the intracranial measurements resemble the examples provided in Figures 9 and 10. This supports the hypothesis of non-repetitive generators with random profile, simultaneous but not synchronized, acting as generators of spindles with similar scalp topographies. MEG measurements also lend support to this hypothesis of asynchronous spindle generators (Dehghani et al., 2011).
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HFOs, a biomarker in epilepsy (Zijlmans et al, 2012; Jacobs et al., 2012), are commonly recorded in only one intracranial channel, suggesting that the underlying cortical generators are small (possibly around 1cm2, given the inter-electrode spacing). However, HFOs can also be recorded on scalp EEG (AndradeValença et al., 2012), which should require a larger cortical involvement. A solution of this apparent inconsistency could be the simultaneous but not necessarily synchronized activation of a set of small cortical generators producing a scalp HFO by the mechanisms described above. Zelman et al. (2014) found occasional multiple and non-synchronized HFOs in non-neighboring contacts of cortical grids and strips at the time of scalp HFOs, see also Inline Supplementary Figures 4 and 5. There is also evidence that this simultaneous but not perfectly synchronized cortical activity occurs at finer spatial scales. Schevon et al. (2009) found that at the time of HFOs recorded by macro-electrodes, there were simultaneous activations in non-neighboring contacts of a micro-electrode array. At an even finer spatial scale, no synchrony in neuronal firing was found during epileptic seizures characterized by gamma band activity in macroelectrodes (Truccolo et al., 2014). The activation of several small cortical generators at the same time could be explained by the combination of an external facilitator and a typical local cortical response modulated by an underlying status of the input of the local cortical networks. This was the hypothesis proposed by Lopes da Silva et al. (1973) for the alpha rhythm, with a proven pacemaker role by the thalamus. They introduced the concept of a stochastic pacemaker instead of a deterministic one, to account for the highly specific topographic patterns. In this model thalamic and cortical alpha rhythms would result from the filter properties of neural networks when submitted to several random inputs. Neural networks with similar design and frequency selectivity should exist in different inputs, and would give rise to alpha rhythms with the same frequency spectra. The degree of coherence between different areas would depend upon
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the correlation of the inputs with the different networks. The same hypothesis fits the other discussed situations. In the case of sleep spindles there is also proof that the thalamus could play the role of stochastic pacemaker (Fuentealba and Steriade, 2005; Steriade, 2006), and there is also some evidence of thalamic involvement in the generation of HFOs. In an EEG-fMRI study a differential activation was found in the thalamus between epileptic spikes likely and unlikely to be accompanied by HFOs (Fahoum et al., 2014).
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Conclusions
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Electrical activity on the scalp is generated by extended cortical generators, consisting in neuronal populations firing simultaneously. However, the cortical region does not need to be a large uniform patch, but can also be a set of small generators covering only partially some large cortical region. Also, while the neuronal activity should be simultaneous in a broad sense, phase coherence of the generators is not necessary for the generation of scalp oscillatory activity. Our study demonstrates how sparse generators with random phase can nevertheless add up to result in oscillatory scalp activity. This activity is surprisingly only slightly attenuated in amplitude compared to that generated by a uniform and synchronized generator. Because we do not know how to interpret the amplitude of the EEG, we have no way of knowing the type of underlying generator. Experimental evidence tends to favor the proposed “sparse-random phase” over the prevalent “uniform-synchronized” model at least in some situations.
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These observations are also a reflection of the well-known fact that the EEG inverse problem is ill-posed, with an infinite number of cortical activation patterns leading to the same scalp electric potential distribution. Assumptions regarding the low complexity of the generator morphology are routinely made to interpret the scalp potential distribution, either visually or by means of source localization algorithms. While the resulting interpretation can be useful, the presented results remind us that the fine spatial scale of distributed generators, and the phase coherence of oscillating generators cannot be determined uniquely from scalp measurements.
Acknowledgments
This work was supported by Canadian Institutes of Health Research grants MOP-102710 and FDN143208.
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ACCEPTED MANUSCRIPT Figure 1: Examples of random spatial activation patterns with different spatial scales and coverage densities. Each square represents an area of 5x5 cm of flattened cortex, with light gray indicating active regions and black indicating inactive regions. The spatial scale is related to the typical size of the active/inactive regions.
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Figure 2: Examples of cortical generators and their associated scalp electric potential distributions and intracerebral EEG signals. The maximum scalp potential is lower for generators with random spatial activation profile than for constant profile, but the shape of the distribution is similar. A: Cortical region of 100 cm2 geodesic surface area, located below scalp electrode F3. The white dots indicate the position of the contacts of four depth electrodes. B and C: Examples of random spatial activation profiles with 50% coverage of the cortical region, and a spatial scale of 4 mm, i.e. the activation between points on the cortex more than 4 mm apart is uncorrelated. The color shows the relative current density, constant over all active regions and equal in the three panels. D, E, and F: Scalp electric potential distribution produced by the generators shown in panels A, B, and C. Note that the amplitude is slightly lower in panels E and F, but the shape of the distribution is quite similar in the three cases. G, H, and I: EEG signals that would be observed in scalp channels F3-C3 and F7-T7, and in the intracerebral bipolar channels recording in the neighborhood of the neocortex (higher numbers indicate outer contacts). The scale of the scalp channels is magnified 5 times with respect to the intracerebral channels. The scale is the same in the three panels. The time scale is intentionally undefined to indicate that the signals could represent activity of different frequency. Note that very different intracerebral patterns can be associated to essentially the same scalp distribution. Observe that even in the case of a constant cortical activation profile, the signal in intracerebral channels can have different amplitude and polarity depending on the relative position of the contacts with respect to the cortex.
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Figure 3: Examples of cortical generators and their associated scalp electric potential distributions and intracerebral EEG signals. If the strength of the generators is allowed to vary by a factor lower than two, the distribution on the scalp is almost identical for constant and random activation profiles. A: Cortical region of 100 cm2 geodesic surface area, located below scalp electrode F3. The white dots indicate the position of the contacts of four depth electrodes. B and C: Examples of random spatial activation profiles with 50 % coverage of the cortical region, and a spatial scale of 4 mm, i.e. the activation between points on the cortex more than 4 mm apart is uncorrelated. The color indicates the relative current density, which was scaled to get the same amplitude for the scalp distributions. D, E, and F: Scalp electric potential distribution produced by the generators shown in panels A, B, and C. Note that the distribution is almost identical in the three cases. G, H, and I: EEG signals that would be observed in scalp channels F3-C3 and F7-T7, and in the intracerebral bipolar channels recording in the neighborhood of the neocortex (higher numbers indicate outer contacts). The scale of the scalp channels is magnified 5 times with respect to the intracerebral channels. The scale is the same in the three panels. The time scale is intentionally undefined to indicate that the signals could represent activity of different frequency. Note that very different intracerebral patterns can be associated to essentially the same scalp distribution. Figure 4: Example of sleep spindles measured with simultaneous scalp EEG and intracerebral EEG. The different panels show three very similar sleep spindles clearly visible in the scalp, and their intracranial correlates in the neocortical contacts of four depth electrodes. Underlined segments correspond to spindles marked by an expert scorer. The scale of the scalp channels is magnified 5 times with respect to the intracerebral channels. The scale is the same in the three panels. Different intracerebral patterns are visible in each case, as in panels G-I of Figures 2 and 3, suggesting that the cortical spatial activation
ACCEPTED MANUSCRIPT pattern of the generator could be random. A zoomed version of this figure is shown on Inline Supplementary Figure 3, illustrating the lack of phase synchrony between the scalp and intracranial signals.
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Figure 5: Quantification of differences between scalp electric potential distributions using the Normalized Relative Difference Measure (NRDM). In all the panels each point corresponds to the median of 1000 realizations and the bars extend from the 5th to the 95th percentiles. A NRDM value of 0 represents “no difference”; the maximum value of NRDM is 2. A: Difference between scalp distribution produced by a patch of cortex with constant current density and by generators with random spatial profiles covering different percentages of the total area, and with different spatial scale (i.e. spatial correlation length of the active regions). Random generators located in a 100 cm2 cortical region below electrode F3. Note that lower spatial coverage implies larger differences in the scalp distribution. However the variability is very high indicating that some random spatial activation profiles will generate a scalp distribution similar to that of a constant activation profile. B: NRDM values that can be expected for normally distributed additive noise at different Signal to Noise Ratios. An SNR as low as 26 dB can be expected in clinical situations. This indicates that NRDM values below 0.2 are indistinguishable in practice. C: Difference between scalp distributions of constant current density generators and generators with random spatial profile with 50 % coverage and 4 mm spatial scale in 100 different locations over the cortical surface, and in cortical regions of different extents. Overall, NRDM values indicate that the scalp distributions of generators with random spatial profiles and constant profile are similar.
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Figure 6: A: Attenuation observed when averaging a finite number of sinusoidal signals of uniformly distributed random phase compared to equal phase. Each point corresponds to the median of 1000 realizations, and the bars extend from the 5th to the 95th percentiles. Results are shown for averaging with constant weights, uniformly distributed random weights between -1 and 1, and weights with distribution given by the sensitivity to generators in different parts of the cortex, observed for scalp electrode F3. An attenuation is observed for constant weights, increasing with the number of sinusoids; note that with around 20 sinusoids, the attenuation is only by a factor of about 5. No attenuation is seen for uniformly distributed random weights, but the variability is large as some weight combinations can result in an increase in amplitude (attenuation lower than 1) and others in the more expected decrease. If the weights are taken at random from a distribution equal to that of the sensitivity profile of the scalp electrode F3 shown in panel B, the situation is intermediate, with 20 generators resulting in a median attenuation by a factor of only 2. B: Sensitivity of scalp electrode F3 to generators at different points of the cortical surfaces, as determined by the solution of the EEG forward problem. Observe that due to the folding of the cortex negative and positive sensitivities can be seen. Figure 7: Example of scalp signals arising from synchronous and asynchronous generators. On the left the color scale shows the sensitivity profile of electrode C4 to different regions of the cortex. Generators in three different locations marked as A, B and C are shown. The right panels show the summation of the contributions of the 3 generators, each with an amplitude corresponding to its sensitivity for C4. The top right panel shows that in the synchronous case there is some cancellation between the contributions of generators A and B, and the bottom right panel shows three examples of the same generators with different phases (randomly selected), leading to different levels of cancellation, of the same order as in the synchronous case.
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Figure 8: Attenuation observed for generators with random phase compared to generators with equal phase. In both panels each point corresponds to the median of 4000 realizations and the bars extend from the 5th to the 95th percentiles. A: Results for different coverage of the cortical region and different degrees of spatial scale of the phase, i.e. correlation length of the spatial distribution of the phase. Results correspond to a 100 cm2 cortical region on the left frontal lobe below electrode F3, and 4 mm spatial scale of the activations. B: Results for 100 cortical regions on different locations on the left hemisphere, and different extent of the cortical area containing the generators, for spatial scale of 4 mm for the activation profile and for the phase of the generators. Median attenuation values are between 2 and 4 times, indicating that in most cases if generators with coherent phase produce a measurable scalp distribution, generators with random phase also will.
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Figure 9: Examples of cortical generators with 100 % coverage density, and their associated scalp potential distributions and intracerebral EEG signals. The maximum scalp potential is attenuated only 3 to 5 times when the generators have random phase compared to constant phase. A, B, and C: Spatial distribution of the phase of cortical generators of 100 cm2. The white dots indicate the position of the contacts of four depth electrodes. The intensity of the current density is the same in the three panels. A: Constant phase. B and C: Two examples of random phase with 4 mm spatial scale, i.e. the phase between points on the cortex more than 4 mm apart is uncorrelated. D, E, and F: Absolute value of the amplitude of the scalp electric potential distribution produced by the generators shown in panels A, B, and C. Note that the shape of the scalp distribution varies, but their amplitudes are only three to five times lower in the case of random phase compared to constant phase. G, H, and I: EEG signals that would be observed in scalp channels F3-C3 and F7-T7, and in the intracerebral bipolar channels recording in the neighborhood of the neocortex (higher numbers indicate outer contacts). The scale of the scalp channels is magnified 10 times with respect to the intracerebral channels. The scale is the same in the three panels. The time scale is intentionally undefined to indicate that the signals could represent activity of different frequencies. Figure 10: Examples of cortical generators with 50 % coverage density and their associated scalp electric potential distributions and intracerebral EEG signals. The maximum scalp potential is attenuated only 3 to 5 times when the generators have random phase compared to constant phase. A, B, and C: Spatial distribution of the phase of cortical generators of 100 cm2, with random spatial activation profiles of 4 mm spatial scale, i.e. the activation between points on the cortex more than 4 mm apart is uncorrelated. The white dots indicate the position of the contacts of four depth electrodes. The intensity of the current density is the same in the three panels. A: Constant phase. B and C: Two examples of random phase with 4 mm spatial scale. D, E, and F: Absolute value of the amplitude of the scalp electric potential distribution produced by the generators shown in panels A, B, and C. Note that the shape of the scalp distribution varies, but the amplitude are only three to five times lower in the cases of random phase compared to constant phase. G, H, and I: EEG signals that would be observed in scalp channels F3-C3 and F7-T7, and in the intracerebral bipolar channels recording in the neighborhood of the neocortex (higher numbers indicate outer contacts). The scale of the scalp channels is magnified 10 times with respect to the intracerebral channels. The scale is the same in the three panels. The time scale is intentionally undefined to indicate that the signals could represent activity of different frequency. Inline Supplementary Figure 1: Examples of electric potential distribution on the scalp for different spatial scales and coverage density. The 100 % coverage case is taken as reference in the upper right corner. The examples serve as a guide to interpret NRDM values.
ACCEPTED MANUSCRIPT Inline Supplementary Figure 2: Examples of random phase distributions with different spatial scales, compounded with different spatial activation profiles. The examples shown correspond to the parameter values shown in Figure 8. The phase and spatial activation distributions are independent. The strength of the activation on the cortex is assumed constant.
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Inline Supplementary Figure 3: Example of spindles measured with simultaneous scalp EEG and intracerebral EEG. The different panels show three sleep spindles clearly visible in the scalp, and their intracranial correlates in the neocortical contacts of four depth electrodes. The scale of the scalp channels is magnified 5 times with respect to the intracerebral channels. The scale is the same in the three panels. This figure is a zoomed version of Figure 4 to show that the intracerebral spindles are not necessarily in phase with the scalp signal.
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Inline Supplementary Figure 4: Example of HFOs measured in the neocortical contacts of four intracerebral electrodes. The scale is the same in the three panels. In most cases HFOs are visible in only one or two neighboring channels (example in the right panel), but occasionally HFOs can be observed more or less simultaneously in many channels, with varying morphology. Underlined segments correspond to HFOs marked by an expert scorer. This supports the proposed model of cortical generators with random phase and activation profile. A zoomed version of this figure is shown on Inline Supplementary Figure 5, illustrating the lack of phase synchrony between the intracranial signals.
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Inline Supplementary Figure 5: Example of HFOs measured in the neocortical contacts of four intracerebral electrodes. This figure is a zoomed version center and right panel of of Inline Supplementary Figure 4 to show that the intracerebral HFOs are not necessarily in phase in different intracerebral channels.
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S1053-8119(16)30188-4 doi: 10.1016/j.neuroimage.2016.05.067 YNIMG 13225
To appear in:
NeuroImage
Received date: Revised date: Accepted date:
16 February 2016 19 April 2016 26 May 2016
Please cite this article as: von Ellenrieder, Nicol´ as, Dan, Jonathan, Frauscher, Birgit, Gotman, Jean, Sparse asynchronous cortical generators can produce measurable scalp EEG signals, NeuroImage (2016), doi: 10.1016/j.neuroimage.2016.05.067
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Sparse asynchronous cortical generators can produce measurable scalp EEG signals
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Montreal Neurological Institute and Hospital, McGill University
École Polytechnique, Université Libre de Bruxelles
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3801 University Street, Montreal, Quebec, H3A 2B4, Canada
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Nicolás von Ellenriedera*, Jonathan Danab, Birgit Frauscherc, Jean Gotmana
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50 Avenue F. D. Roosevelt, 1050 Bruxelles, Belgium
Department of Medicine, Center for Neuroscience Studies, Queen’s University
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18 Stuart Street, Kingston, Ontario, K7L 3N6, Canada
* Corresponding author:
Montreal Neurological Institute
3801 University Street, Room 009e Montreal, QC H3A 2B4, Canada [email protected] phone: +1 514 398 6644, ext 00445 fax: +1 514 398 8106
ACCEPTED MANUSCRIPT Abstract
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We investigate to what degree the synchronous activation of a smooth patch of cortex is necessary for observing EEG scalp activity. We perform extensive simulations to compare the activity generated on the scalp by different models of cortical activation, based on intracranial EEG findings reported in the literature. The spatial activation is modeled as a cortical patch of constant activation or as random sets of small generators (0.1 to 3 cm 2 each) concentrated in a cortical region. Temporal activation models for the generation of oscillatory activity are either equal phase or random phase across the cortical patches. The results show that smooth or random spatial activation profiles produce scalp electric potential distributions with the same shape. Also, in the generation of oscillatory activity, multiple cortical generators with random phase produce scalp activity attenuated on average only 2 to 4 times compared to generators with equal phase. Sparse asynchronous cortical generators can produce measurable scalp EEG. This is a possible explanation for seemingly paradoxical observations of simultaneous disorganized intracranial activity and scalp EEG signals. Thus, the standard interpretation of scalp EEG might constitute an oversimplification of the underlying brain activity.
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Keywords
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Scalp EEG, cortical generators, asynchronous activity, modeling, boundary elements method.
ACCEPTED MANUSCRIPT Introduction
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In the interpretation of scalp EEG, the seminal paper by Gloor (1985) is to this day one of the most influential in clinical electrophysiology, given its largely qualitative but rigorous explanation of the volume conduction effects. Gloor concludes that when pyramidal neurons are activated synchronously within a cortical area of macroscopic extent, the fields of the individual neurons of the population combine to create measurable fields on the scalp. This conclusion is based on physical principles, and is a natural conclusion of any analysis of the EEG forward problem (e.g. Geselowitz, 1967; Nunez, 1981). This type of explanation for the generation of the EEG is given in several major reference books (e.g. Lopes da Silva chapter 5 in Niedermeyer). However, what was originally stated as a sufficient condition for the measurement of EEG on the scalp, has sometimes been interpreted as a necessary condition. It is therefore not uncommon to find the explicit or implicit notion that scalp EEG arises only when pyramidal neuron populations act in synchrony, both spatially and temporally.
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The presence of extended and synchronous cortical generators leading to scalp potentials has been observed at the time of interictal epileptic discharges when recording simultaneously subdural and scalp EEG (Tao et al., 2005). There is however evidence from experimental intracranial EEG studies that such an extended and synchronous patch of cortex acting as generator of scalp EEG may be an oversimplification in some situations. In the canine cortex, for example, phase coherence was observed only in relative small cortical regions during alpha rhythm bursts even though alpha activity could be recorded on the scalp (Lopes da Silva et al, 1973). At the time of sleep spindles in the human scalp EEG, it was found that there are frequently spindles in various intracerebral regions but they are not synchronous among themselves nor with the scalp spindle, and the time relationships between various regions is highly variable from spindle to spindle (Frauscher et al, 2015). In epileptic patients, simultaneous but non-synchronous high frequency oscillations (HFOs) were occasionally observed in non-neighboring channels of cortical grids at the time of scalp HFOs (Zelmann et al, 2014). All these studies point to asynchronous focal cortical generators combining in some way to produce measurable scalp EEG activity. We present simulations to determine if it is theoretically possible to observe scalp EEG produced by asynchronous cortical generators, and under what conditions. The model we propose is based on observations from the intracranial studies mentioned in the previous paragraph, and consists in cortical generators with irregular spatial morphology and non-coherent phase. The rationale for a model with irregular spatial profile of cortical activation is that for scalp EEG generation the total area of activation is likely important, but not its spatial contiguity. The inclusion of generators of oscillatory activity with random phase seems less intuitive, given that oscillatory signals with random phase cancel each other when averaged. However, this reasoning overlooks that complete cancellation is achieved only for an infinite number of generators, and that the EEG recorded on the scalp is not a mere averaging of cortical generators.
Methods A realistically shaped three-layer model was built based on the high-quality Colin MR data (AubertBroche et al., 2006), conductivity values of 0.33 S/m were adopted for the brain and scalp, and 0.015 S/m for the skull. The EEG forward problem was solved using the Boundary Element Method (BEM), with
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linear elements (de Munck, 1992) and the isolated skull approach (Meijs et al., 1989). Scalp electrodes were simulated positioned according to the 10-5 extension of the 10-10 system (Oostenveld and Praamstra, 2001). When simulating amplifier noise, independent and normally distributed noise was added to each scalp channel. The Signal to Noise Ratio (SNR) was defined based on the channel with highest amplitude. Additionally, four intracerebral electrodes with 5 mm distance between contacts were simulated in the left frontal lobe. A point approximation was adopted for the intracerebral contacts (von Ellenrieder et al., 2012), and bipolar channels between neighboring contacts were simulated with the same layered model and BEM formulation.
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Special attention was given to the modeling of the cortex, represented by a surface in the middle of the layer of cortical gray matter. Each hemisphere was tessellated in 60,000 triangular elements. Distributed generators were modeled as sets of dipolar sheets associated to the triangles of the tessellation, with a current density value for each vertex and linear variation of the current density on the triangular faces of the tessellation, with orientation normal to the surface (von Ellenrieder et al., 2009). This model was selected rather than elementary dipoles at the vertices of the cortical surface distribution, because with the adopted point approximation of the intracerebral electrodes, the dipoles would introduce singularities in the solution leading to inaccurate results for the intracerebral simulations. The source model based on dipolar sheets does not introduce singularities and leads to more accurate simulation results (von Ellenrieder et al., 2015).
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Most examples shown in this study correspond to generators modeled within a region of 100 cm2 as measured by geodesic distance on the cortical surface, i.e. over the highly folded cortical surface. The visible area when projecting the region onto the scalp would be smaller. The region was located in the left frontal lobe, below scalp electrode F3 as seen in Figure 1A. However, results are also presented for different generator extents and for 100 locations distributed over the cortical surface of the left hemisphere. Generators with random spatial profile The scalp and intracranial electric potential distribution produced by a generator of constant current density covering the complete 100 cm2 region was compared to the one produced by generators in the same region but with random spatial activation profile. These generators were built by assigning randomly a normally distributed value at each vertex of the cortical surface tessellation, applying a spatial smoothing based on geodesic distance, and comparing the resulting value to a threshold. Cortical regions where the threshold was exceeded were deemed active, and otherwise inactive. All active regions were then assigned the same current density value. This created a partition of the cortical region in active and inactive subsets, examples are shown in Figure 1. The spatial scale, or correlation length, of the random maps was controlled by varying the degree of spatial smoothing, and is an indication of the size of the active regions. The activation of any pair of points on the cortical surface is uncorrelated if they are separated by a distance larger than this spatial scale. The coverage density, or active to inactive surface area ratio, was controlled by changing the threshold. Spatial scales from 2 to 10 mm, and coverage density of 25, 50, and 75 % of the total 100cm2 surface were investigated. The described procedure results in an activation pattern equivalent to spreading small generators of radius equal to the spatial scale (although not necessarily with circular shape) until the desired proportion of the cortical region is covered. Examples are shown in Figure 1. For low coverage densities this results in scattered generator patches of 0.12 to 3.1 cm2 surrounded by mostly inactive regions, and
ACCEPTED MANUSCRIPT for high coverage densities it results in inactive cortical patches of the same extent immersed in a mostly active cortical region. Asynchronous generators
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Generators of oscillatory activity and random phase were also simulated. The random phase was obtained by defining two independent random variables in each vertex of the cortical surface, as inphase and quadrature terms (i.e. two terms with a 90 degree phase shift between them). After spatial smoothing the ratio between these two variables was used to define the phase of the generator at each point of the cortical surface through a 4-quadrant arc-tangent operation. With this approach the spatial scale, or correlation length, of the phase could be controlled by changing the degree of spatial smoothing. Values between 2 and 20 mm were explored.
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Quantification of differences
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Results
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To quantify the difference between two scalp electric potential distributions, two measures were used. The ratio of the maximum amplitude of each distribution, and the Normalized Relative Difference (NRDM). The maximum amplitude of the scalp distribution is of interest because it is directly related to the visibility of the distribution on the scalp assuming the noise level remains unchanged. The NRDM compares the shape of the distribution irrespectively of the absolute amplitude (Meijs et al., 1989), and takes values between zero (distributions of identical shape), 1.4 (distributions of uncorrelated shapes), and two (distributions of identical shape but opposite signs).
Generators with random spatial profile Generators with random spatial profiles covering the same cortical region produce similar potential distributions on the scalp, as shown by the examples in Figures 2 and 3. Figure 2 shows that the scalp distribution of random generators can be almost identical if the strength of the current density at source level is allowed to change. Variations in the current density could be due e.g. to a different proportion of involvement of the neuronal populations. In all the remaining simulations, the strength of the generators is kept fixed for simplicity. However, the results shown in Figure 2 are an important reminder that changes in the current density of the active regions directly affect the amplitude of the scalp potential. Then, any difference we report on the amplitude of the scalp distribution due to the random activation profile could be compensated or exacerbated by the uncertainty in the current density value at source level. In any case, the shape of the distributions remains almost unchanged in the examples, and the amplitude of the scalp EEG is related to source strength and to the proportion of the cortical surface covered by active regions. In Figure 3 the current density or strength of the cortical generators is assumed equal for the three shown examples, and the scalp distribution has similar shape but different amplitude. Figures 2 and 3 (G-I) also shows that since intracerebral channels have a very local sensitivity, when the cortical activation profile is random different intracerebral patterns can be observed associated to the same scalp EEG distribution. This can be appreciated in examples of real intracerebral
ACCEPTED MANUSCRIPT measurements in Figure 4, where sleep spindles observed on the scalp appear associated to different intracerebral activation patterns.
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The difference between scalp potential distributions is shown quantitatively in Figure 5A. The NRDM was computed between the scalp distribution produced by the smooth 100 cm2 cortical patch of constant current density and 1000 random generators for each investigated coverage density and spatial scale. The figure shows that coverage density has a larger influence than spatial scale. Lower coverage means a greater difference between the spatial distribution of the activation profile, and leads to larger variability of the scalp distribution. Lower spatial scale leads to slightly lower NRDM values because the low skull conductivity introduces some blurring, or attenuation of high spatial frequencies, thus reducing the ability to resolve finer spatial details of the activation profile. For 50 % coverage, median NRDM values close to 0.2 are observed. To interpret the NRDM values, consider as an example that the scalp distributions shown in Figure 3E and 3F have NRDM of 0.23 and 0.16 when compared to the visually similar distribution shown in Figure 3D. Additional examples of the electric potential distribution on the scalp from generators with random activation profile and their respective NRDM values are presented in Inline Supplementary Figure 1. Also, we include in Figure 5B the NRDM that can be expected due to instrumental noise present in scalp EEG recordings. An NRDM of 0.2 is expected for signal to noise ratios of 26 dB, corresponding to noise amplitude 20 times lower than the signal. This noise level can be found in some clinical situations, e.g. for spindles of 20 µV amplitude in systems with noise level in the order of 1 µV due to the electrode/skin electrochemical interface, and to a lesser degree to the electronic noise of the amplifiers (Huigen et al., 2012). Thus, the difference in NRDM of 0.2 between 100% and 50% coverage is around the level of noise and the two are essentially indistinguishable.
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We assessed whether the above results are independent of the specific cortical region that has been illustrated (below F3). Figure 5C shows the results for generators in 100 different cortical regions distributed on the cortical surface of the left hemisphere, and for different extents of the regions containing the random generators. For each location and extent 1000 generators were simulated. The figure shows that the example region analyzed above is typical, since the median NRDM value for 50 % coverage and 100 cm2 is close to 0.2. Note that the extent of the cortical regions corresponds to the geodesic surface area; as the cortical surface is highly folded, the visible area when projected on the scalp is much smaller. In the example in Figure 2A, the 100 cm2 geodesic extent corresponds to 21 cm2 when projected into the scalp, and the 5 to 200 cm2 range explored in Figure 5C corresponds to an average of 2 to 40 cm2 when projected onto the scalp. Asynchronous generators The possibility of observing scalp EEG as a result of cortical generators of oscillatory activity with noncoherent phase may seem unlikely. The argument behind this notion is that averaging sinusoidal signals of random phase results in cancellation leading to a signal with null amplitude. However, for the amplitude to be completely canceled out it is necessary to average an infinite number of sinusoids. In the case of the scalp EEG the number of generators is finite. Considering first the averaging of sinusoids only (independently of the generation of the scalp EEG from cortical generators), Figure 6A shows the attenuation resulting from averaging sinusoids with uniformly distributed random phase compared to averaging the same number of sinusoids with equal phase. Almost 100 sinusoids need to be averaged to
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get a median attenuation of ten times, and still in 5 % of the cases the attenuation will be lower than half that value. The figure also shows that if the amplitudes of the sinusoids are multiplied by a random factor uniformly distributed between -1 and 1 before averaging, no attenuation is observed (see below the rational for positive and negative weights). This is because in the coherent case half the sinusoids will be multiplied by negative numbers that lead to cancellation, and the level of attenuation is similar to the one observed due to uniformly distributed random phase.
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Considering now the transformation from multiple cortical generators to a scalp potential, the forward problem cannot be considered a simple average of multiple cortical generators, even though some degree of blurring is introduced by the skull. The EEG forward problem can be seen as a weighted average of the cortical generators, and while the weight is not uniformly distributed, it does take positive and negative values as shown in Figure 6B. The figure shows the sensitivity of scalp electrode F3 to the different points of the cortical region used in the examples. The convoluted shape of the cortex means that generators in different regions sometimes contribute with opposite signs (as would be the case for two walls of a sulcus). If sinusoids are averaged weighted with the distribution of observed sensitivities, the resulting attenuation of the scalp potential is between the results corresponding to constant weights and uniformly distributed random weights (Figure 6A). This is still a simplification of the problem, but helps to understand why the asynchronous generators can produce measurable scalp EEG. Figure 7 shows an example of the situation, with three generators located in different cortical regions contributing to the signal recorded in electrode C4. Contributions from generators A and B partially cancel out in the synchronous case but not necessarily so in the asynchronous cases.
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Results of extensive simulations of generators of oscillatory activity with random phase are shown in Figure 8. Panel A shows the attenuation for different coverage density of the sources and different spatial scale for the phase of the signals, i.e. how far two cortical points need to be to have an uncorrelated phase. Results correspond to 4000 different realizations for each coverage density and spatial scale, for the left frontal brain region shown in Figure 2A. Attenuation values decrease slightly with increasing spatial scale, and increase slightly for increasing coverage density. Median attenuation values are between 3 and 6 times. Figure 8B shows the results for 100 different locations of the cortical region distributed over the cortical surface of the left hemisphere, and for different extents of the surface area containing the generators. The attenuation increases with the extent of the cortical region, but stabilizes for extents larger than 100 cm2, with median attenuations lower than 4 times. Inline Supplementary Figure 2 shows a 2D representation of the random phase and activation distribution used in the simulations. The effect of random phase alone are shown by the 100 % coverage curves of Figure 8, and by the examples of generators with random phase shown in Figure 9 together with the resulting scalp potential distribution and intracerebral signals. The combination of independent random activation profile and phase leads to an increase in the attenuation of the scalp amplitude compared to the coherent case, as shown by the curves with less than 100 % coverage in Figure 8, and the examples shown in Figure 10, for random phase and 50 % coverage density. Insert Inline Supplementary Figure 2 here. Figures 9 and 10 show that the scalp distribution changes shape compared to the constant phase case, and is attenuated by factors between 3 and 5, but would still be visible in the context of an EEG dominated by a constant phase generator. Examples of real intracerebral EEG measurements that lend support to the proposed random phase model can be found in Figure 4 for sleep spindles (the phase
ACCEPTED MANUSCRIPT difference between the channels can be appreciated clearly in the zoomed version of Figure 4 presented in the Inline Supplementary Figure 3). An example involving high frequency oscillations is shown in Inline Supplementary Figures 4 and 5.
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Discussion
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Generators with random spatial profile
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Our results show that it is not necessary to have a large patch of smooth cortical activation to generate scalp EEG, and that asynchronous cortical generators of oscillatory activity produce scalp EEG with an amplitude usually of the same order as synchronous generators. The scalp EEG originates from the cortex through a mechanism of weighted average of a finite number of generators, with weights that can be positive or negative due to the folding of the cortex. In this situation only very mild cancellation of generators with random phase should be expected.
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Regarding the spatial profile of cortical activation, the difference between the scalp electric potential distribution resulting from random patches of activation and a smooth area of cortex with constant activation (constant current density) is restricted almost entirely to their amplitude. On average, the difference in the shape of the distribution is in many cases of the same order as the shape change caused by additive instrumental noise contaminating the signal in clinical situations. Beyond the general behavior, a relatively large variability is observed in the results associated to different random activation patterns. This high variability can be explained by the different proportion of activity on the gyral crowns and sulcal walls. The electric activity on the crown is in close proximity to the surface and will easily produce visible scalp potential, while the activity on the sulcal walls is not only further away but also can suffer from partial cancelation between neighboring walls with opposite orientation of pyramidal neuron populations. As for the amplitude, it cannot be used in practice to distinguish between random and constant activation profiles because it is directly related to the uncertainty in the typical current density value of the generators, which is at least of one order of magnitude (von Ellenrieder et al., 2014). We therefore do not know the amplitude of an EEG generated by a synchronous generator. Asynchronous generators In the case of generators of oscillatory activity with random phase, we found that in half of the cases the attenuation compared to equal phase (synchronous generators) would be lower than 2 to 6 times. If the spatial variation of the phase is more rapid, or the percentage of cortical area covered by active regions increases, the attenuation is slightly higher because this is equivalent to an increase in the number of averaged generators. For the same reason the attenuation grows with the area of the cortex where the generators are located, but this effects stabilizes for areas larger than 50 to 100 cm2 because the sensitivity of any point on the scalp only covers a limited cortical region. Note that the amplitude of the scalp distribution is proportional to the strength of the generators, and almost proportional to the skull conductivity. Both of these parameters are highly uncertain (see discussion in von Ellenrieder et al.,
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Lopes da Silva et al. (1973) studied the generation of alpha rhythm in the canine brain. They observed bursts of alpha activity that could be topographically very specific, with regions of high cortico-cortical coherence often of 5 or 6 mm, concordant with the modeled spatial scales. The degree of coherence between different areas did not seem to be deterministic. They proposed that the apparently simultaneous occurrence of bursts of alpha rhythm at different locations depended on a common modulating influence which would gate the multiple “alpha generators” at the same time without necessarily causing all alpha bursts to be synchronous.
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Sleep spindles observed on the scalp occur most often at the same time as spindles in several intracerebral locations but exhibit a remarkable lack of repetitive intracerebral patterns (Nir et al., 2011; Frauscher et al., 2015), see also Figure 4. The phase coherence of the intracranial channels is also low as shown in the Inline Supplementary Figure 3, and the intracranial measurements resemble the examples provided in Figures 9 and 10. This supports the hypothesis of non-repetitive generators with random profile, simultaneous but not synchronized, acting as generators of spindles with similar scalp topographies. MEG measurements also lend support to this hypothesis of asynchronous spindle generators (Dehghani et al., 2011).
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HFOs, a biomarker in epilepsy (Zijlmans et al, 2012; Jacobs et al., 2012), are commonly recorded in only one intracranial channel, suggesting that the underlying cortical generators are small (possibly around 1cm2, given the inter-electrode spacing). However, HFOs can also be recorded on scalp EEG (AndradeValença et al., 2012), which should require a larger cortical involvement. A solution of this apparent inconsistency could be the simultaneous but not necessarily synchronized activation of a set of small cortical generators producing a scalp HFO by the mechanisms described above. Zelman et al. (2014) found occasional multiple and non-synchronized HFOs in non-neighboring contacts of cortical grids and strips at the time of scalp HFOs, see also Inline Supplementary Figures 4 and 5. There is also evidence that this simultaneous but not perfectly synchronized cortical activity occurs at finer spatial scales. Schevon et al. (2009) found that at the time of HFOs recorded by macro-electrodes, there were simultaneous activations in non-neighboring contacts of a micro-electrode array. At an even finer spatial scale, no synchrony in neuronal firing was found during epileptic seizures characterized by gamma band activity in macroelectrodes (Truccolo et al., 2014). The activation of several small cortical generators at the same time could be explained by the combination of an external facilitator and a typical local cortical response modulated by an underlying status of the input of the local cortical networks. This was the hypothesis proposed by Lopes da Silva et al. (1973) for the alpha rhythm, with a proven pacemaker role by the thalamus. They introduced the concept of a stochastic pacemaker instead of a deterministic one, to account for the highly specific topographic patterns. In this model thalamic and cortical alpha rhythms would result from the filter properties of neural networks when submitted to several random inputs. Neural networks with similar design and frequency selectivity should exist in different inputs, and would give rise to alpha rhythms with the same frequency spectra. The degree of coherence between different areas would depend upon
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the correlation of the inputs with the different networks. The same hypothesis fits the other discussed situations. In the case of sleep spindles there is also proof that the thalamus could play the role of stochastic pacemaker (Fuentealba and Steriade, 2005; Steriade, 2006), and there is also some evidence of thalamic involvement in the generation of HFOs. In an EEG-fMRI study a differential activation was found in the thalamus between epileptic spikes likely and unlikely to be accompanied by HFOs (Fahoum et al., 2014).
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Electrical activity on the scalp is generated by extended cortical generators, consisting in neuronal populations firing simultaneously. However, the cortical region does not need to be a large uniform patch, but can also be a set of small generators covering only partially some large cortical region. Also, while the neuronal activity should be simultaneous in a broad sense, phase coherence of the generators is not necessary for the generation of scalp oscillatory activity. Our study demonstrates how sparse generators with random phase can nevertheless add up to result in oscillatory scalp activity. This activity is surprisingly only slightly attenuated in amplitude compared to that generated by a uniform and synchronized generator. Because we do not know how to interpret the amplitude of the EEG, we have no way of knowing the type of underlying generator. Experimental evidence tends to favor the proposed “sparse-random phase” over the prevalent “uniform-synchronized” model at least in some situations.
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These observations are also a reflection of the well-known fact that the EEG inverse problem is ill-posed, with an infinite number of cortical activation patterns leading to the same scalp electric potential distribution. Assumptions regarding the low complexity of the generator morphology are routinely made to interpret the scalp potential distribution, either visually or by means of source localization algorithms. While the resulting interpretation can be useful, the presented results remind us that the fine spatial scale of distributed generators, and the phase coherence of oscillating generators cannot be determined uniquely from scalp measurements.
Acknowledgments
This work was supported by Canadian Institutes of Health Research grants MOP-102710 and FDN143208.
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ACCEPTED MANUSCRIPT Figure 1: Examples of random spatial activation patterns with different spatial scales and coverage densities. Each square represents an area of 5x5 cm of flattened cortex, with light gray indicating active regions and black indicating inactive regions. The spatial scale is related to the typical size of the active/inactive regions.
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Figure 2: Examples of cortical generators and their associated scalp electric potential distributions and intracerebral EEG signals. The maximum scalp potential is lower for generators with random spatial activation profile than for constant profile, but the shape of the distribution is similar. A: Cortical region of 100 cm2 geodesic surface area, located below scalp electrode F3. The white dots indicate the position of the contacts of four depth electrodes. B and C: Examples of random spatial activation profiles with 50% coverage of the cortical region, and a spatial scale of 4 mm, i.e. the activation between points on the cortex more than 4 mm apart is uncorrelated. The color shows the relative current density, constant over all active regions and equal in the three panels. D, E, and F: Scalp electric potential distribution produced by the generators shown in panels A, B, and C. Note that the amplitude is slightly lower in panels E and F, but the shape of the distribution is quite similar in the three cases. G, H, and I: EEG signals that would be observed in scalp channels F3-C3 and F7-T7, and in the intracerebral bipolar channels recording in the neighborhood of the neocortex (higher numbers indicate outer contacts). The scale of the scalp channels is magnified 5 times with respect to the intracerebral channels. The scale is the same in the three panels. The time scale is intentionally undefined to indicate that the signals could represent activity of different frequency. Note that very different intracerebral patterns can be associated to essentially the same scalp distribution. Observe that even in the case of a constant cortical activation profile, the signal in intracerebral channels can have different amplitude and polarity depending on the relative position of the contacts with respect to the cortex.
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Figure 3: Examples of cortical generators and their associated scalp electric potential distributions and intracerebral EEG signals. If the strength of the generators is allowed to vary by a factor lower than two, the distribution on the scalp is almost identical for constant and random activation profiles. A: Cortical region of 100 cm2 geodesic surface area, located below scalp electrode F3. The white dots indicate the position of the contacts of four depth electrodes. B and C: Examples of random spatial activation profiles with 50 % coverage of the cortical region, and a spatial scale of 4 mm, i.e. the activation between points on the cortex more than 4 mm apart is uncorrelated. The color indicates the relative current density, which was scaled to get the same amplitude for the scalp distributions. D, E, and F: Scalp electric potential distribution produced by the generators shown in panels A, B, and C. Note that the distribution is almost identical in the three cases. G, H, and I: EEG signals that would be observed in scalp channels F3-C3 and F7-T7, and in the intracerebral bipolar channels recording in the neighborhood of the neocortex (higher numbers indicate outer contacts). The scale of the scalp channels is magnified 5 times with respect to the intracerebral channels. The scale is the same in the three panels. The time scale is intentionally undefined to indicate that the signals could represent activity of different frequency. Note that very different intracerebral patterns can be associated to essentially the same scalp distribution. Figure 4: Example of sleep spindles measured with simultaneous scalp EEG and intracerebral EEG. The different panels show three very similar sleep spindles clearly visible in the scalp, and their intracranial correlates in the neocortical contacts of four depth electrodes. Underlined segments correspond to spindles marked by an expert scorer. The scale of the scalp channels is magnified 5 times with respect to the intracerebral channels. The scale is the same in the three panels. Different intracerebral patterns are visible in each case, as in panels G-I of Figures 2 and 3, suggesting that the cortical spatial activation
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Figure 5: Quantification of differences between scalp electric potential distributions using the Normalized Relative Difference Measure (NRDM). In all the panels each point corresponds to the median of 1000 realizations and the bars extend from the 5th to the 95th percentiles. A NRDM value of 0 represents “no difference”; the maximum value of NRDM is 2. A: Difference between scalp distribution produced by a patch of cortex with constant current density and by generators with random spatial profiles covering different percentages of the total area, and with different spatial scale (i.e. spatial correlation length of the active regions). Random generators located in a 100 cm2 cortical region below electrode F3. Note that lower spatial coverage implies larger differences in the scalp distribution. However the variability is very high indicating that some random spatial activation profiles will generate a scalp distribution similar to that of a constant activation profile. B: NRDM values that can be expected for normally distributed additive noise at different Signal to Noise Ratios. An SNR as low as 26 dB can be expected in clinical situations. This indicates that NRDM values below 0.2 are indistinguishable in practice. C: Difference between scalp distributions of constant current density generators and generators with random spatial profile with 50 % coverage and 4 mm spatial scale in 100 different locations over the cortical surface, and in cortical regions of different extents. Overall, NRDM values indicate that the scalp distributions of generators with random spatial profiles and constant profile are similar.
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Figure 6: A: Attenuation observed when averaging a finite number of sinusoidal signals of uniformly distributed random phase compared to equal phase. Each point corresponds to the median of 1000 realizations, and the bars extend from the 5th to the 95th percentiles. Results are shown for averaging with constant weights, uniformly distributed random weights between -1 and 1, and weights with distribution given by the sensitivity to generators in different parts of the cortex, observed for scalp electrode F3. An attenuation is observed for constant weights, increasing with the number of sinusoids; note that with around 20 sinusoids, the attenuation is only by a factor of about 5. No attenuation is seen for uniformly distributed random weights, but the variability is large as some weight combinations can result in an increase in amplitude (attenuation lower than 1) and others in the more expected decrease. If the weights are taken at random from a distribution equal to that of the sensitivity profile of the scalp electrode F3 shown in panel B, the situation is intermediate, with 20 generators resulting in a median attenuation by a factor of only 2. B: Sensitivity of scalp electrode F3 to generators at different points of the cortical surfaces, as determined by the solution of the EEG forward problem. Observe that due to the folding of the cortex negative and positive sensitivities can be seen. Figure 7: Example of scalp signals arising from synchronous and asynchronous generators. On the left the color scale shows the sensitivity profile of electrode C4 to different regions of the cortex. Generators in three different locations marked as A, B and C are shown. The right panels show the summation of the contributions of the 3 generators, each with an amplitude corresponding to its sensitivity for C4. The top right panel shows that in the synchronous case there is some cancellation between the contributions of generators A and B, and the bottom right panel shows three examples of the same generators with different phases (randomly selected), leading to different levels of cancellation, of the same order as in the synchronous case.
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Figure 8: Attenuation observed for generators with random phase compared to generators with equal phase. In both panels each point corresponds to the median of 4000 realizations and the bars extend from the 5th to the 95th percentiles. A: Results for different coverage of the cortical region and different degrees of spatial scale of the phase, i.e. correlation length of the spatial distribution of the phase. Results correspond to a 100 cm2 cortical region on the left frontal lobe below electrode F3, and 4 mm spatial scale of the activations. B: Results for 100 cortical regions on different locations on the left hemisphere, and different extent of the cortical area containing the generators, for spatial scale of 4 mm for the activation profile and for the phase of the generators. Median attenuation values are between 2 and 4 times, indicating that in most cases if generators with coherent phase produce a measurable scalp distribution, generators with random phase also will.
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Figure 9: Examples of cortical generators with 100 % coverage density, and their associated scalp potential distributions and intracerebral EEG signals. The maximum scalp potential is attenuated only 3 to 5 times when the generators have random phase compared to constant phase. A, B, and C: Spatial distribution of the phase of cortical generators of 100 cm2. The white dots indicate the position of the contacts of four depth electrodes. The intensity of the current density is the same in the three panels. A: Constant phase. B and C: Two examples of random phase with 4 mm spatial scale, i.e. the phase between points on the cortex more than 4 mm apart is uncorrelated. D, E, and F: Absolute value of the amplitude of the scalp electric potential distribution produced by the generators shown in panels A, B, and C. Note that the shape of the scalp distribution varies, but their amplitudes are only three to five times lower in the case of random phase compared to constant phase. G, H, and I: EEG signals that would be observed in scalp channels F3-C3 and F7-T7, and in the intracerebral bipolar channels recording in the neighborhood of the neocortex (higher numbers indicate outer contacts). The scale of the scalp channels is magnified 10 times with respect to the intracerebral channels. The scale is the same in the three panels. The time scale is intentionally undefined to indicate that the signals could represent activity of different frequencies. Figure 10: Examples of cortical generators with 50 % coverage density and their associated scalp electric potential distributions and intracerebral EEG signals. The maximum scalp potential is attenuated only 3 to 5 times when the generators have random phase compared to constant phase. A, B, and C: Spatial distribution of the phase of cortical generators of 100 cm2, with random spatial activation profiles of 4 mm spatial scale, i.e. the activation between points on the cortex more than 4 mm apart is uncorrelated. The white dots indicate the position of the contacts of four depth electrodes. The intensity of the current density is the same in the three panels. A: Constant phase. B and C: Two examples of random phase with 4 mm spatial scale. D, E, and F: Absolute value of the amplitude of the scalp electric potential distribution produced by the generators shown in panels A, B, and C. Note that the shape of the scalp distribution varies, but the amplitude are only three to five times lower in the cases of random phase compared to constant phase. G, H, and I: EEG signals that would be observed in scalp channels F3-C3 and F7-T7, and in the intracerebral bipolar channels recording in the neighborhood of the neocortex (higher numbers indicate outer contacts). The scale of the scalp channels is magnified 10 times with respect to the intracerebral channels. The scale is the same in the three panels. The time scale is intentionally undefined to indicate that the signals could represent activity of different frequency. Inline Supplementary Figure 1: Examples of electric potential distribution on the scalp for different spatial scales and coverage density. The 100 % coverage case is taken as reference in the upper right corner. The examples serve as a guide to interpret NRDM values.
ACCEPTED MANUSCRIPT Inline Supplementary Figure 2: Examples of random phase distributions with different spatial scales, compounded with different spatial activation profiles. The examples shown correspond to the parameter values shown in Figure 8. The phase and spatial activation distributions are independent. The strength of the activation on the cortex is assumed constant.
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Inline Supplementary Figure 3: Example of spindles measured with simultaneous scalp EEG and intracerebral EEG. The different panels show three sleep spindles clearly visible in the scalp, and their intracranial correlates in the neocortical contacts of four depth electrodes. The scale of the scalp channels is magnified 5 times with respect to the intracerebral channels. The scale is the same in the three panels. This figure is a zoomed version of Figure 4 to show that the intracerebral spindles are not necessarily in phase with the scalp signal.
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Inline Supplementary Figure 4: Example of HFOs measured in the neocortical contacts of four intracerebral electrodes. The scale is the same in the three panels. In most cases HFOs are visible in only one or two neighboring channels (example in the right panel), but occasionally HFOs can be observed more or less simultaneously in many channels, with varying morphology. Underlined segments correspond to HFOs marked by an expert scorer. This supports the proposed model of cortical generators with random phase and activation profile. A zoomed version of this figure is shown on Inline Supplementary Figure 5, illustrating the lack of phase synchrony between the intracranial signals.
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Inline Supplementary Figure 5: Example of HFOs measured in the neocortical contacts of four intracerebral electrodes. This figure is a zoomed version center and right panel of of Inline Supplementary Figure 4 to show that the intracerebral HFOs are not necessarily in phase in different intracerebral channels.
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