Exploring the limits of MEG spatial resolution with multipolar expansions_2

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arXiv:2210.02863v2 [physics.med-ph] 6 Mar 2023

Exploring the limits of MEG spatial resolution with multipolar expansions

Vincent Wensa,b,∗

aLN2T – Laboratoire de Neuroanatomie et Neuroimagerie translationnelles, UNI – ULB Neuroscience Institute, Universit´e libre de Bruxelles (ULB), Brussels,

Belgium

bDepartment of Translational Neuroimaging, H.U.B. – Hˆopital Erasme, Brussels, Belgium

Abstract

The advent of scalp magnetoencephalography (MEG) based on optically pumped magnetometers (OPMs) may represent a step

change in the ﬁeld of human electrophysiology. Compared to cryogenic MEG based on superconducting quantum interference

devices (SQUIDs, placed 2–4 cm above scalp), scalp MEG promises signiﬁcantly higher spatial resolution imaging but it also

comes with numerous challenges regarding how to optimally design OPM arrays. In this context, we sought to provide a systematic

description of MEG spatial resolution as a function of the number of sensors (allowing comparison of low- vs. high-density MEG),

sensor-to-brain distance (cryogenic SQUIDs vs. scalp OPM), sensor type (magnetometers vs. gradiometers; single- vs. multi-

component sensors), and signal-to-noise ratio. To that aim, we present an analytical theory based on MEG multipolar expansions

that enables, once supplemented with experimental input and simulations, quantitative assessment of the limits of MEG spatial

resolution in terms of two qualitatively distinct regimes. In the regime of asymptotically high-density MEG, we provide a math-

ematically rigorous description of how magnetic ﬁeld smoothness constraints spatial resolution to a slow, logarithmic divergence.

In the opposite regime of low-density MEG, it is sensor density that constraints spatial resolution to a faster increase following

a square-root law. The transition between these two regimes controls how MEG spatial resolution saturates as sensors approach

sources of neural activity. This two-regime model of MEG spatial resolution integrates known observations (e.g., the diﬃculty of

improving spatial resolution by increasing sensor density, the gain brought by moving sensors on scalp, or the usefulness of multi-

component sensors) and gathers them under a unifying theoretical framework that highlights the underlying physics and reveals

properties inaccessible to simulations. We propose that this framework may ﬁnd useful applications to benchmark the design of

future OPM-based scalp MEG systems.

Keywords:

High-density MEG; Magnetic ﬁeld smoothness; Magnetoencephalography; Optically pumped magnetometry; Scalp MEG.

Highlights:

•We develop a two-regime theory describing the limits of MEG spatial resolution.

•The low-density regime exhibits the advantage of multi-component MEG sensors.

•The high-density regime reveals a slow divergence as sensors are added to MEG.

•Scalp MEG exhibits saturated resolution through an interplay of the two regimes.

•This theoretical framework may be helpful to design new generation scalp MEG.

1. Introduction

The physics of electric and magnetic ﬁelds sets fundamen-

tal limits to the spatial resolution of non-invasive electrophys-

iology. As these ﬁelds spread from the brain to extra-cranial

sensors (scalp electrodes for electroencephalography, EEG;

magnetometers and gradiometers for magnetoencephalography,

MEG; see, e.g., H¨am¨al¨ainen et al., 1993), ﬁne details of neural

Abbreviations: ECoG, electrocorticography; EEG, electroencephalogra-

phy; MEG, magnetoencephalography; MRI, magnetic resonance imaging;

OPM, optically pumped magnetometer; QZFM, quantum zero-ﬁeld magne-

tometer; SQUID, superconducting quantum interference device; SNR, signal-

to-noise ratio.

∗Corresponding author. Address: Department of Translational Neuroimag-

ing, H.U.B. – Hˆopital Erasme, 808 route de Lennik, 1070 Brussels, Belgium.

E-mail address: vincent.wens@ulb.be.

current source distributions get blurred and information is lost.

This loss of spatial resolution and the accompanying decrease

in ﬁeld amplitude lie at the heart of the MEG/EEG inverse prob-

lem (H¨am¨al¨ainen et al., 1993) and thus cannot be overcome

fully by technological developments (Tarantola, 2006). Still,

our ability to harvest smaller and smaller details of brain elec-

trophysiological signals improves as technology evolves.

The development of scalp MEG based on optically pumped

magnetometers (OPMs) may lead to major advances in this

regard (Boto et al., 2018). By avoiding the heavy cryogen-

ics needed for MEG systems based on superconducting quan-

tum interference devices (SQUIDs), this technology allows to

place magnetometers closer to the scalp (from about 2–4 cm

above scalp for SQUIDs to about 5 mm for OPMs) and thus

leads to substantial improvements in signal focality and am-

plitude (Boto et al., 2016; Feys et al., 2022; Iivanainen et al.,

Article published in NeuroImage 270 (2023) 119953. DOI: 10.1016/j.neuroimage.2023.119953. Open access under CC BY-NC-ND license.

Exploring the limits of MEG spatial resolution with multipolar expansions V. Wens (2023)

2017, 2020). The ensuing reﬁnement in MEG data remains

limited in practice due to the relatively low number of mag-

netometers in current wearable OPM systems (up to 50 in

Hill et al., 2020; Boto et al., 2021) and their sensitivity to envi-

ronmental noise (Boto et al., 2018; Seymour et al., 2022). The

situation is evolving rapidly though, as denser OPM arrays

(Boto et al., 2016; Iivanainen et al., 2017), new types of OPM

sensors (Labyt et al., 2019; Borna et al., 2020; Nardelli et al.,

2020; Brookes et al., 2021), and background ﬁeld cancellation

techniques (Holmes et al., 2018, 2019; Iivanainen et al., 2019;

Mellor et al., 2022) are being invented. In this exciting context,

one important question to consider is how much detail of the

neuromagnetic ﬁeld can be harvested by MEG sensor arrays if

we could place as many sensors as we wanted on these arrays.

In other words, what are the limits of MEG spatial resolution,

given a system design (i.e., sensor coverage, density, sensor-to-

brain distance, ﬁeld sensitivity, and noise level)? This question

is admittedly abstract, since practical constraints such as sensor

size (about 2 cm2scalp contact area for state-of-the-art OPMs)

or cost restrict the number of available sensors in current MEG

arrays. That said, its answer could have an important impact

on the development of future OPM systems as sensors become

smaller and cheaper. Characterising the limits of MEG spa-

tial resolution quantitatively and systematically would allow to

assess how many sensors are ideally needed to map neuromag-

netic ﬁelds as precisely as possible, and how this number is

aﬀected by system design.

This type of question was already asked in the early days

of the whole-brain-covering SQUID array technology. In their

seminal study, Ahonen et al. (1993) used a two-dimensional

version of Nyquist’s sampling theorem to estimate the dis-

tribution of radial magnetic sensors needed to image dipolar

magnetic ﬁelds faithfully without aliasing. This information

was crucial for the development of modern multi-SQUID sys-

tems. More recent approaches in the context of OPM develop-

ments focused instead on simulated models of MEG signals.

Forward modeling (i.e., the explicit numerical evaluation of

ﬁeld propagation from neural current sources to sensors) was

used extensively to estimate the resolution gain expected when

passing from SQUIDs to OPMs (i.e., with similar design but

placed on scalp; see Boto et al., 2016; Iivanainen et al., 2017;

Tierney et al., 2020). Multipolar expansions (Jackson, 1998;

Zangwill, 2012) provide another modeling technique that is ide-

ally suited to investigate spatial resolution as they decompose

MEG data in terms of angular frequency, i.e., a measure of spa-

tial scale on sensor topographies. This decomposition underlies

signal-space separation, a preprocessing technique of SQUID

signals that allows to suppress both focal sensor noise at high

angular frequency and widespread long-distance environmental

magnetic interferences at low angular frequency (Taulu et al.,

2004, 2005). Tierney et al. (2022) explored OPM spatial sam-

pling with simulations built from MEG multipolar expansions.

However, simulation-based approaches are impractical to han-

dle the case of asymptotically dense sensor arrays needed to

assess the limits of MEG spatial resolution.

Here, we expand upon the multipolar expansion technique

and provide a systematic framework for MEG spatial resolution

that encompasses its limits. We use an analytical description of

neuromagnetic ﬁeld smoothness in asymptotically high-density

MEG with hemispherical geometry to characterize spatial res-

olution in terms of the highest angular frequency accessible to

the array. Given that magnetic ﬁeld spread exerts a smoothing

on extra-cranial neuromagnetic topographies, we hypothesized

that MEG spatial resolution would converge to a deﬁnite limit

controlled by ﬁeld smoothness once sensor density gets large

enough. Our speciﬁc goal was thus to measure this limit, assess

how many sensors are needed to reach it, and examine the eﬀect

of key parameters such as sensor type, sensor-to-brain distance,

or signal-to-noise ratio (SNR). We further used simulations to

investigate the opposite regime of low sensor density where the

asymptotic theory breaks down.

2. Theory

We consider a multi-channel MEG system composed of sen-

sors surrounding the head as illustrated in Fig. 1. In this section,

we present the results of a theoretical analysis of MEG spatial

resolution in the asymptotic limit where a large number of sen-

sors are distributed homogeneously on a hemispherical array

(shown red in Fig. 1). This allows us to describe explicitly a

measure of spatial resolution as a function of sensor type, array-

to-brain distance, and SNR. The detailed developments leading

to these results are relegated to two appendices; Appendix A

gathers useful background and minor results on the machinery

of MEG multipolar expansions, and Appendix B develops our

original asymptotic analysis of MEG spatial resolution. The

theory is supplemented with experimental data and numerical

simulations in Sections 3 and 4, where we also explore the

regime of low sensor density outside the domain of validity of

the asymptotic theory. Further intuition and practical conclu-

sions are discussed in Section 5.

We start by describing the general framework of the theory

and formulate explicitly its assumptions.

2.1. Multipolar expansion of multi-channel MEG signals

Observation model. For our purposes, a MEG array consists

of a number Nof sensor locations where one or several com-

ponents of the magnetic ﬁeld or its gradient are measured. We

assume that these locations can be parameterised by the angular

position Ωin a suitable spherical coordinate frame centered on

the subject’s brain (e.g., the polar angle θand the azimuthal an-

gle ϕshown in Fig. 1). We focus mainly on spherically shaped

arrays where the radial coordinate r=Rarray is constant (Fig. 1),

but angle-dependent radial coordinates will be allowed when

we consider a realistic MEG geometry (see Sections 3 and 4).

Sensor measurements b(Ω) may be related to point ﬁeld val-

ues φ(Ω) (i.e., components of the magnetic ﬁeld or its gradient

at the center of the sensor) and intrinsic sensor noise ε(Ω) via

the observation model

b(Ω)=φ(Ω)+ε(Ω),(1)

which typically holds to a good approximation in MEG sys-

tems. We allow for multimodal setups where a number Mof

Exploring the limits of MEG spatial resolution with multipolar expansions V. Wens (2023)

Rbrain

Rarray

Figure 1: Geometric arrangement. This illustration shows a subject’s head

inside a hemispherical array of radius Rarray centered on their brain (red), along

with the N=102 sensor locations of the Neuromag MEG sensor array (black

dots). The smallest concentric sphere enclosing the brain (blue) deﬁnes the

anatomical brain radius Rbrain. The spherical coordinate system (r, θ, ϕ) used in

this paper is also indicated.

diﬀerent sensors sit at the same place, so all symbols in Eq. (1)

represent M-vectors. For example, M=1 for CTF systems

consisting of N=275 axial gradiometers and for OPM arrays

composed of single-axis radial magnetometers; M=3 for Neu-

romag systems consisting of N=102 chipsets (Fig. 1, black

dots) of one radial magnetometer and two planar gradiometers

(see, e.g., Hari and Puce, 2017) and for arrays of tri-axis OPMs

(Brookes et al., 2021).

Spatial whiteness of intrinsic sensor noise. We also assume

that sensor noise is homogeneous and uncorrelated, so its co-

variance across all array locations (i.e., gathering the Nvectors

ε(Ω) in a single NM-vector) takes the form

cov(ε)=σ2

εI,(2)

with Ithe NM ×NM identity matrix. Equation (2) should

be amended in multimodal setups that mix magnetometers and

gradiometers (since their noise levels σεdo not carry the same

physical units), but the ensuing changes are not essential so we

keep it unmodiﬁed for notational simplicity.

Multipolar expansion. Since neuromagnetic activity is probed

outside the head and works in a quasi-static regime, extra-

cranial ﬁeld values φ(Ω) may be subjected to an interior

multipolar expansion of the form (Taulu et al., 2004, 2005;

Tierney et al., 2022)

φ(Ω)=X

ℓ,m

aℓ,mS(Ω|ℓ, m).(3)

The M-vectors S(Ω|ℓ, m) denote the vectorial spherical harmon-

ics (Hill, 1954) indexed by integers ℓ≥0 and −ℓ≤m≤ℓ.

See Appendix A.1 (Tables A.1 and A.2) for detailed expres-

sions in cases of interest. The series (3) corresponds to a spec-

tral decomposition of the neuromagnetic topographies in terms

of angular frequency k=√ℓ(ℓ+1)/Rarray, so ℓindexes an-

gular frequency (Jackson, 1998; Zangwill, 2012). We use in

this work the inverse of the radial coordinate rrelative to the

brain sphere radius Rbrain (see Fig. 1) as expansion parameter

(Appendix A.1). In this way, all multipole moment coeﬃcients

aℓ,mshare the same physical unit [T ·m] and may be compared

numerically. This allows to formulate the following hypothesis

that is fundamental to our analysis of MEG spatial resolution.

Maximum-entropy hypothesis. We assume that all multipole

moments are uncorrelated and of equal variance, i.e.,

cov(a)=σ2

aI(4)

using formal notations where the coeﬃcients aℓ,mare gathered

into an inﬁnite column vector aand where Idenotes an inﬁnite

square identity matrix. This corresponds to a situation of “max-

imum entropy” where brain activity is spatially unstructured

and involves all spatial scales equally, from microscopic (e.g.,

single-channel synaptic currents) to macroscopic (i.e., whole-

brain network) levels. That is both unphysical and biologically

unrealistic, but nevertheless useful for exploring the limits of

MEG spatial resolution. Extra-cranial measurements are at best

sensitive to the mean activity of neural populations, but the as-

sumption (4) also includes undetectable microscopic and other

non-physiological electrical source conﬁgurations, leading to

an overestimation of MEG spatial resolution. This overestima-

tion is illustrated with experimental data in Section 4.

A solution to this important caveat is to abandon a direct

physiological interpretation of the two parameters of the MEG

multipolar expansion model, i.e., the brain sphere radius Rbrain

and the multipole amplitude σa. Instead, we propose to treat

them as eﬀective parameters of the theory to be assessed em-

pirically from data. According to the hypothesis (4), a brain

sphere with radius Rbrain estimated na¨ıvely from anatomy (blue

sphere in Fig. 1) would include highly localized neural activity

right under (or even slightly above) the brain convexity beneath

the scalp. Such conﬁguration must be associated with a focal

ﬁeld topography and thus high MEG spatial resolution, but it

might not be representative of the experimental data at hand. In

turn, this might lead to an underestimation of the multipole am-

plitude parameter σa, which can be determined from the SNR

estimate

SNR =1

NM Tr hcov(ε)−1cov(b)i(5)

of MEG recordings (1) via the relation (Appendix A.2)

σ2

=NM (SNR −1)

Tr(S S†)·(6)

Here, Sis a formal matrix with an inﬁnite number of columns

indexed by (ℓ, m), each column gathering the NM elements of

the M-vectors S(Ω|ℓ, m) at the Nsensor locations of the MEG

Exploring the limits of MEG spatial resolution with multipolar expansions V. Wens (2023)

array. We describe in Section 3 how to combine anatomical

brain images and MEG recordings in order to determine func-

tional estimates of Rbrain and σa/σεand obtain physiologically

meaningful MEG multipolar expansions.

Spatial resolution from multipolar expansions. The vectorial

spherical harmonics S(Ω|ℓ, m) in Eq. (3) measure the sensitivity

of MEG sensors to neuromagnetic ﬁelds with deﬁnite angular

frequency ℓ. Sensitivity decreases exponentially fast for highly

focal neuromagnetic topographies characterized by large values

of ℓ(Appendix A.1). This exponential suppression embodies

the physical smoothing process that neuromagnetic ﬁelds un-

dergo as they propagate from brain sources to sensors. On the

other hand, intrinsic sensor noise contributes equally at all the

spatial scales sampled by the MEG array; this is embodied by

the spatial whiteness assumption (2). It is the interaction of

these two features that inherently limits the sensitivity of MEG

data to focal brain activity. Measurement noise is typically

subdominant at low angular frequency but overshadows focal

neuromagnetic activity at high angular frequency. Eﬀectively,

noise should cut oﬀthe expansion (3) at a critical value ℓ=ℓ∗

where this cross-over occurs. This idea is the basis of signal-

space separation (Taulu et al., 2004, 2005). We leverage it here

and seek to measure MEG spatial resolution using the critical

value ℓ∗, since it corresponds to the smallest spatial scale that is

experimentally accessible.

Our main goal is to determine explicitly this spatial reso-

lution index ℓ∗. Quite amazingly, this turns out possible for

hemispherically shaped MEG arrays in the limit N→ ∞ corre-

sponding to an inﬁnitely dense, homogeneous sensor coverage

(Fig. 1). The usefulness of considering this situation inaccessi-

ble to both experiment and simulations is that it allows precisely

to assess the limits of MEG spatial resolution and how they de-

pend on sensor type, array-to-brain distance, and SNR.

Signal-space dimension. In situations where the validity of the

asymptotic theory is not settled, we will resort to signal-space

dimension as proxy measure of spatial resolution. We deﬁne it

here as the number νof degrees of freedom contained in brain

MEG signals and estimated according to

ν=#neigenvalues λ2

uof S S†with λ2

u> σ2

ε/σ2

ao.(7)

This corresponds to the number of linearly independent neu-

romagnetic topographies (i.e., eigenvectors of the N M ×N M

matrix S S†) whose contribution (measured by their eigenvalue

λ2

u) exceeds noise level (σ2

ε/σ2

a) and thus is experimentally

detectable (Appendix A.2). These topographies span what is

known as the signal space (Taulu et al., 2004, 2005).

The signal-space dimension νassesses the information con-

tent of MEG data rather than their spatial resolution per se.

It must be commensurate to spatial resolution since access to

more focal details should increase the number of detectable to-

pographies. Yet, like any other complexity metric (may they be

linear dimensions or non-linear, information-theoretic capaci-

ties), it turns out to mix spatial resolution and other geometric

factors of the MEG array. This is demonstrated below.

2.2. Asymptotic regime of high-density MEG

Spatial resolution index in the large-N limit. We present here

our main theoretical result about asymptotically high-density

MEG arrays with hemispherical geometry and homogeneously

distributed sensors (Fig. 1). Mathematical analysis of the limit

N→ ∞ developed from Appendix B.1 to Appendix B.3 de-

termines the spatial resolution index as

ℓ∗=

log N

4πr2+deg(P)

σ2

Plog N

2 log r

2 log r+O log log N

log N!.(8)

This result enables the quantitative measurement of MEG spa-

tial resolution as a function of the number Nof sensors, the

sensor-to-brain distance r=Rarray/Rbrain (i.e., the radius of the

hemispherical array relative to that of the brain sphere), and the

multipole SNR parameter σa/σε. It also depends on the type

of sensors composing the MEG array through a polynomial P

that is identiﬁed in Appendix B.1 (Table B.1).

Properties of high-density MEG spatial resolution. Let us de-

scribe the eﬀect of diﬀerent MEG array characteristics dis-

closed by Eq. (8). See Appendix B.4 for more details.

(i) Sensor density. The spatial resolution index ℓ∗exhibits a

logarithmic divergence as Ngrows indeﬁnitely. In other

words, the limits of spatial resolution increase without

bound (albeit very slowly) as MEG arrays become denser.

This observation contradicts our initial expectation that

spatial resolution would converge towards a deﬁnite limit

controlled by magnetic ﬁeld smoothness. Rather, it is the

extreme slowness of this divergence that corresponds to

the constraints imposed by ﬁeld smoothness.

(ii) Sensor-to-brain distance. The rate at which the spatial

resolution diverges turns out to be controlled by the pa-

rameter rand not by any other MEG characteristics. The

dependence in other characteristics is milder because both

sensor type and SNR only contribute through sub-leading

corrections that are small compared to the leading diver-

gence. This means that the limits of spatial resolution are

mostly modulated by the sensor-to-brain distance. In fact,

ℓ∗appears to increase without bound as the sensor array

approaches the brain surface (r→1), i.e., spatial reso-

lution improves drastically as the MEG array approaches

the brain. Nevertheless, this divergence is, in a sense, only

an artifact as the asymptotic theory breaks down before

sensors reach the brain (see Section 4).

(iii) Sensor type. The magnetometric or gradiometric nature of

MEG sensors makes a sub-leading contribution to ℓ∗that

is nevertheless numerically signiﬁcant, because it also ex-

hibits a divergence as Ngrows indeﬁnitely (although an

even slower one). It turns out that this contribution is

twice larger for gradiometers, so we conclude that gradio-

metric arrays exhibit moderately higher limits of spatial

resolution than magnetometric arrays (at similarly large

number N of sensors). On the other hand, the number

of recorded components or their orientation only have a

minute impact as their contribution is either ﬁnite or neg-

ligibly small at large N.

Exploring the limits of MEG spatial resolution with multipolar expansions V. Wens (2023)

(iv) Multipole SNR. Likewise, the SNR makes a subtle, ﬁnite

contribution that is negligible.

Signal-space dimension in the large-N limit. We further

demonstrate in Appendix B.1 that the signal-space dimension

νmay be expressed in terms of the spatial resolution index ℓ∗

by merely counting the number of vectorial spherical harmon-

ics whose contribution to MEG signals exceeds noise level, i.e.,

for which ℓ≤ℓ∗. A straightforward count (as done in, e.g.,

Taulu et al., 2005; Tierney et al., 2022) suggests a value

νS=(ℓ∗+1)2(9)

but that is not quite right. In fact, this relation is only valid

for a hypothetical MEG array that covers a complete sphere S

enclosing the brain (notwithstanding that this would be nonsen-

sical from the experimental standpoint); this is emphasized by

the subscript attached to the symbol νin Eq. (9). A proper

analysis of MEG multipolar expansions on a hemisphere H

(Appendix A.4) reveals instead that

νH=1

2(ℓ∗+1)(ℓ∗+2) .(10)

This is approximately twice smaller, which reﬂects the halving

of sensor coverage compared to the whole sphere. The depen-

dence in sensor coverage demonstrates the diﬀerence between

the spatial resolution index (which is the same for spherical and

hemispherical MEG; see Appendix B.1) and complexity met-

rics such as signal-space dimension (see also Appendix B.4).

3. Methods

We describe numerical and experimental methods to estimate

the parameters of MEG multipolar expansions (rand σa/σε),

examine the domain of validity for our asymptotic theory (i.e.,

how large the number Nof sensors must be to ensure the quan-

titative accuracy of Eq. 8), explore what happens at low sen-

sor density outside this domain of validity, and ﬁnally measure

quantitatively the limits of MEG spatial resolution.

3.1. Numerical evaluation of signal-space dimension

At the core of our numerical experiments is an estimation

of signal-space dimension that works whatever the MEG ar-

ray (e.g., hemispherical or Neuromag geometries illustrated in

Fig. 1) and whatever the number Nof sensors (as long as it is

not so large that simulations become untractable).

The N M ×NM matrix S S†appearing in Eq. (7) gathers

M×Mblocks of the form P∞

ℓ=0Pℓ

m=−ℓS(Ω|ℓ, m)S(Ω′|ℓ, m)†,

where Ω,Ω′run over the Nsensor locations. The inﬁnite sum

over ℓwas evaluated by computing terms at successive val-

ues ℓ=0,1,2,... and adding them iteratively until numeri-

cal convergence (which is guaranteed). The vectorial spheri-

cal harmonics S(Ω|ℓ, m) were evaluated at angular (θ, ϕ) and

radial (r) coordinates corresponding to sensor locations of the

MEG arrays described below and for diﬀerent sensor types (ra-

dial and tri-axis magnetometers, axial and planar gradiometers,

see Appendix A.1). Summation was performed over the ﬁrst

hundred terms (0 ≤ℓ≤100) and then continued iteratively un-

til the last term to add got small enough; as precise criterion,

we required that the squared Frobenius norm of the current, ℓth

term, relative to that of the partial sum over all ℓ−1 previous

terms, reach below 10−5. The signal-space dimension (7) was

then evaluated by diagonalizing the partial sum and counting

the number of eigenvalues above threshold σ2

ε/σ2

3.2. Anatomical MEG expansion parameters

We considered MEG resting-state data of 14 healthy adult

subjects used in previous studies (Coquelet et al., 2020, 2022),

to which we refer for details. Brieﬂy, MEG signals were ac-

quired at rest (5 min, 0.1–330 Hz analog bandpass, 1 kHz sam-

pling rate) using a Neuromag Vectorview system (MEGIN Oy,

Helsinki, Finland) and denoised using signal-space separation

(Maxﬁlter v2.2 with default parameters ℓin =8 and ℓout =3,

MEGIN; Taulu et al., 2004, 2005) and independent component

analysis (Hyv¨arinen and Oja, 2000). We used these data to ex-

tract geometric information needed to construct the Smatrices

(Appendix A.1) and functional information related to the SNR.

The Neuromag MEG array is composed of sensors located at

N=102 locations (Fig. 1) comprising one radial magnetome-

ter and two orthogonal planar gradiometers. We used individ-

ual brain magnetic resonance images (MRIs) co-registered with

the MEG array to deﬁne subject-speciﬁc spherical coordinates

of each sensor location. The coordinate origin was set at the

centre of the sphere ﬁtted to the vertices of the scalp surface

obtained after tissue segmentation (Freesurfer, Martinos Center

for Biomedical Imaging, Massachussetts, USA; Fischl, 2012).

This allowed to assign radial (distance from origin) and angular

coordinates to each sensor. The array radius Rarray was deﬁned

as the root-mean-square of all 102 radial coordinates, and the

anatomical brain sphere of radius Rbrain was determined as the

smallest sphere centered on the origin that encloses the inner

skull surface (Fig. 1, blue). The ratio Rarray/Rbrain determined

the “anatomical” estimate of the expansion parameter rfor the

Neuromag MEG array. To illustrate the impact of array-to-brain

distance, we also considered a virtual OPM array placed 6.5

mm above scalp and thereby obtained an anatomical estimate

of rcorresponding to scalp MEG. The 6.5-mm height corre-

sponds to the center location of the alkali vapour cell in Gen-2

QZFM sensors (QuSpin Inc., Colorado, USA) placed directly

on scalp.

The SNR associated with Neuromag MEG recordings at rest

was estimated for magnetometers (N=102, M=1) and planar

gradiometers (N=102, M=2) separately according to Eq. (5),

with the NM ×N M data covariance cov(b) extracted from the

resting-state recordings and the noise covariance cov(ε), from

empty-room recordings. The noise covariance was regularized

prior to inversion by adding 10% of the mean sensor variance

to its diagonal. Combining this SNR measure with the com-

putation of the corresponding S S†matrix (based on the above

geometric information and on Section 3.1) and with Eq. (6), we

could then estimate the multipole SNR parameter σa/σε.

The MEG multipolar expansion models constructed in this

way will be referred to as “anatomical MEG” as they are in-

ferred from the actual brain size of subjects.

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arXiv:2210.02863v2[physics.med-ph]6Mar2023ExploringthelimitsofMEGspatialresolutionwithmultipolarexpansionsVincentWensa,b,∗aLN2T–LaboratoiredeNeuroanatomieetNeuroimagerietranslationnelles,UNI–ULBNeuroscienceInstitute,Universit´elibredeBruxelles(ULB),Brussels,BelgiumbDepartmentofTranslationalNeuroimag...

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Exploring the limits of MEG spatial resolution with multipolar expansions_2.pdf

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Exploring the limits of MEG spatial resolution with multipolar expansions_2

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