a review on the importance of volume currents john c. mosher biological and quantum physics group...

A Review on the A Review on the Importance of Volume Importance of Volume

CurrentsCurrents

John C. MosherJohn C. MosherBiological and Quantum Physics Biological and Quantum Physics

GroupGroup

Los Alamos National LaboratoryLos Alamos National Laboratory

Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117

AcknowledgementsAcknowledgements• Cite as: Mosher JC, “A Review on the Importance of Volume

Currents,” (invited presentation), 14th International Conference on Biomagnetism, Boston, Massachusetts, August 2004, available as Los Alamos Technical Report LA-UR-04-6117.

• This work was supported by the National Institutes of Health under grant R01-EB002010, and by Los Alamos National Laboratory, operated by the University of California for the United States Department of Energy, under Contract W-7405-ENG-36.


AbstractAbstractGiven an elemental current dipole inside the brain, the

forward problem is the calculation of the external scalp potential or the magnetic field; by superposition, any more complicated distribution of primary current can be found by integration or summation of the basic solution. Although the solutions have been derived over the last four decades under a variety of situation, some users remain uncertain about the effects of the volume currents in the models. The confusion may arise in part because most forward models have been reworked to make the volume currents implicit, rather than explicit. We review the general development of the forward solution, including our discovery of an early 1971 paper missed by the MEG community that elegantly yields the general solution. We discuss the principal computational issues in boundary element methods (BEMs) in accurately accounting for these volume currents, and how it impacts both MEG and EEG models. We also review the “myth” of the silent radial dipole, reviewing classic and recent work that shows that radial dipoles are generally measurable in MEG data under realistic conditions.


OutlineOutline• Basic Assumptions, and the definition Basic Assumptions, and the definition

of Primary and Volume currentsof Primary and Volume currents

• Simple MEG solution and the possible Simple MEG solution and the possible confusion about volume currentsconfusion about volume currents

• Historical review of the development Historical review of the development of the general solutions for EEG and of the general solutions for EEG and MEGMEG

• The “myth” of the “silent” radial dipoleThe “myth” of the “silent” radial dipole


Excellent Mathematical Excellent Mathematical Reference:Reference:

• Jukka Sarvas, 1987, Physics in Jukka Sarvas, 1987, Physics in Medicine and Biology.Medicine and Biology.


Sarvas References:Sarvas References:


Basic AssumptionsBasic Assumptions• Source region is non-magneticSource region is non-magnetic

• Currents are quasi-staticCurrents are quasi-static– Electric field is gradient of a scalarElectric field is gradient of a scalar– Curl of magnetic field is the currentCurl of magnetic field is the current– Divergence of magnetic field is zeroDivergence of magnetic field is zero

• Define static magnetic field as the Define static magnetic field as the curl of a “vector potential” A(r), curl of a “vector potential” A(r), yieldingyielding

''

)'()(

head

rrr

rjrA d


Magnetic Vector PotentialMagnetic Vector Potential

• Integrate the Integrate the totaltotal current density current density flowing in the head, flowing in the head, divided by its divided by its distance to the distance to the observation.observation.

''

)'()(

head

rrr

rjrA d

)'(rj

)(rA

BrainStorm


Biot-Savart LawBiot-Savart Law')'(

'

1)(

head

drrjrr

rA

head3

head

''

')'(

')'()'

1()(

drrr

rrrj

drrjrr

rB

Vector potential

CURL yields magnetic field

But the Biot-Savart Law is expressed in total But the Biot-Savart Law is expressed in total current, we need the solution in terms of the current, we need the solution in terms of the neural source generators.neural source generators.


Primary vs. Secondary Primary vs. Secondary CurrentsCurrents

• Picture Picture primaryprimary currentcurrent as a small as a small battery inside the battery inside the brain.brain.

• Secondary or volume Secondary or volume currents are the currents are the gradient currents to gradient currents to “complete the “complete the circuit.”circuit.”

• Boundaries shape Boundaries shape the volume currents.the volume currents.

BrainStorm


Primary Neural SourcesPrimary Neural Sources

• Primary currents are Primary currents are produced by current produced by current flow in apical flow in apical dendrites in cortical dendrites in cortical pyramidal neurons.pyramidal neurons.

• Millions of EPSPs Millions of EPSPs summed over ~ten summed over ~ten milliseconds.milliseconds.

• ““Macrocellular” vs. Macrocellular” vs. “microcellular.”“microcellular.”

Ramon y Cajal 1888 from Hamalainen et al. 1993 Reviews of Modern Physics


Two Types of CurrentTwo Types of Current

• Volume currents flow simply due to Volume currents flow simply due to the presence of a the presence of a macroscopicmacroscopic voltage gradient in conducting voltage gradient in conducting mediummedium

• Simply define Primary Current as Simply define Primary Current as “not volume”“not volume”

)'()'( rrj vv

)'()'()'( rjrjrj vp


Unbounded SolutionsUnbounded Solutions

')'('

1)(

head

drrjrr

rA pp

')'()'

1()(

head

drrjrr

rB pp

')'()'

1()(

head

drrjrr

rv pp

Assume an unbounded homogeneous region:

CURL: Homogeneous magnetic field

DIVERGENCE: Homogeneous electric potential


Numerical ExampleNumerical Example• Consider a dipole 7 cm up from an Consider a dipole 7 cm up from an

origin, and observation points origin, and observation points arranged 12 cm from the origin.arranged 12 cm from the origin.

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

0

0.02

0.04

0.06

0.08

0.1

0.12

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

-0.1

-0.05

0

0.05

0.1


Primary Dipole Magnetic FieldsPrimary Dipole Magnetic Fields

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

-0.1

-0.05

0

0.05

0.1


Unbounded RegionsUnbounded Regions

• Volume currents Volume currents do notdo not contribute contribute to the potential or magnetic field in to the potential or magnetic field in infiniteinfinite homogeneous regions homogeneous regions

• Volume currents only contribute Volume currents only contribute when bounded regions are nearbywhen bounded regions are nearby


Bounded RegionsBounded Regions

• Given primary current, what is the Given primary current, what is the magnetic field?magnetic field?

• MEG general solution includes the general MEG general solution includes the general solution of EEG surface potentials.solution of EEG surface potentials.


Boundary Effects (cf. Sarvas)Boundary Effects (cf. Sarvas)


Fictitious CurrentsFictitious Currents• Standard vector identities allow us to Standard vector identities allow us to

delete the true 3-D volume currents delete the true 3-D volume currents and replace them with and replace them with fictitiousfictitious 2-D 2-D currents only on the boundaries, currents only on the boundaries, normally orientednormally oriented

True physical True physical currentscurrents

All surface current elements All surface current elements discontinuousdiscontinuous


Radial Field Outside of a Radial Field Outside of a SphereSphere• In a sphere, then all fictitious In a sphere, then all fictitious

currents are radial.currents are radial.

• If a sensing coil is oriented If a sensing coil is oriented radiallyradially outside of the a outside of the a perfectperfect sphere, then sphere, then nonenone of the fictitious currents are of the fictitious currents are visiblevisible

Radial fictitious currents are Radial fictitious currents are

Unobservable by a radial sensorUnobservable by a radial sensor


Radial Field StrengthRadial Field Strength

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

-0.1

-0.05

0

0.05

0.1


The Simple MEG SolutionThe Simple MEG Solution• Biot-Savart Law using All currents:Biot-Savart Law using All currents:

• Spherical Case, Radial Direction:Spherical Case, Radial Direction:

• Looks like the Biot-Savart Law, but only Looks like the Biot-Savart Law, but only involves the primary currentinvolves the primary current

head

30 '

'

')'(

4)( r

rr

rrrjrB d

head

30 '

'

')'(

4)( r

rr

rrrj

rrB

rd

rrp


The ConfusionThe Confusion• We can also write the spherical solution We can also write the spherical solution

for the radial measurement as:for the radial measurement as:

• But this often leads novices to the But this often leads novices to the conclusion that the volume currents are conclusion that the volume currents are unimportant. For non-radial unimportant. For non-radial measurements in the orientation “o”, measurements in the orientation “o”, they attempt:they attempt:

)()( rBr

rBr p

rrrm

)(rBor pm


Slightly Non-radial Slightly Non-radial MeasurementsMeasurements• Six degrees from radial in the y-directionSix degrees from radial in the y-direction

Incorrect: Primary Currents OnlyIncorrect: Primary Currents Only Correct: Primary and Volume CurrentsCorrect: Primary and Volume Currents


Non-radial Field Outside of a Non-radial Field Outside of a SphereSphere• In non-radial directions, the fictitious In non-radial directions, the fictitious

currents are visible and must be currents are visible and must be included in the calculationincluded in the calculation

Non-radial Non-radial orientationorientation

Sensor and Sensor and dipole in same dipole in same orientationorientation

Primary Primary current does current does NOT NOT contribute!contribute!


All Sensors in the X-directionAll Sensors in the X-direction• Sensors cannot see the primary Sensors cannot see the primary

current, only the volume currentscurrent, only the volume currents


General Solution ApproachGeneral Solution Approach• Specify an elemental primary current Specify an elemental primary current

source, calculate the infinite source, calculate the infinite homogeneous potential, then solve homogeneous potential, then solve the Fredholm Integral of the Second the Fredholm Integral of the Second Kind for all boundary potentials (cf Kind for all boundary potentials (cf Sarvas):Sarvas):

• Using this “EEG” solution, plug into Using this “EEG” solution, plug into “Geselowitz (1970) equation for MEG:“Geselowitz (1970) equation for MEG:


Classical Solution ApproachClassical Solution Approach

• First Edition 1962First Edition 1962


Vector Spherical HarmonicsVector Spherical Harmonics


Grynszpan and Geselowitz Grynszpan and Geselowitz 19731973

• Formalized the lead-field work of Baule and McFee Formalized the lead-field work of Baule and McFee (1965,1970), using vector spherical harmonics(1965,1970), using vector spherical harmonics


The Simple The Simple AcknowledgementAcknowledgement

• 1975 Second Edition1975 Second Edition


The FootnoteThe Footnote

• The clue, quite overlooked, that a The clue, quite overlooked, that a simple solution existed.simple solution existed.


Bronzan, Am. Jrnl. Phys. 1971Bronzan, Am. Jrnl. Phys. 1971


Bronzan’s Solution (1971)Bronzan’s Solution (1971)

• Magnetic scalar, given in terms of the Magnetic scalar, given in terms of the total current. In the spherical boundaries total current. In the spherical boundaries case, only primary currents can case, only primary currents can contributecontribute

• Magnetic field is simply the gradient.Magnetic field is simply the gradient.

volume rr rrrrrr

rrrJr

2

volume

p

rr rrrrrr

rrrJr

2


Further MEG DevelopmentFurther MEG Development• Bronzan’s solution is still today not widely Bronzan’s solution is still today not widely

cited, and has only recently been cited in cited, and has only recently been cited in MEG literature (Jerbi et al. 2003 PMB).MEG literature (Jerbi et al. 2003 PMB).

• As the MEG community began As the MEG community began experimental measurements in the 1980s, experimental measurements in the 1980s, a solution was sought for the non-radial a solution was sought for the non-radial field outside of a sphere.field outside of a sphere.

1984 Biomag in Vancouver1984 Biomag in Vancouver


Ambiguity of MEG DataAmbiguity of MEG Data

• Consider again the physical case. A Consider again the physical case. A radially-oriented sensor cannot radially-oriented sensor cannot distinguish between the following cases:distinguish between the following cases:

Volume Volume currents currents are are invisibleinvisible

Radial Radial line line currents currents are are invisibleinvisible


General Solution for the SphereGeneral Solution for the Sphere• Sum the magnetic field for the two Sum the magnetic field for the two

radial lines and the tangential cross radial lines and the tangential cross elements. Take the limit as the elements. Take the limit as the tangential element is made small, tangential element is made small, yielding:yielding:

• Unfortunately, while correct, the formula Unfortunately, while correct, the formula contained a singularity “0/0” condition contained a singularity “0/0” condition at certain observation points.at certain observation points.


Correct InsightCorrect Insight• Their insight was correct, however, Their insight was correct, however,

that we did that we did NOTNOT need to calculate need to calculate the volume currents the volume currents explicitlyexplicitly::


Sarvas SolutionSarvas Solution• 1987, Jukka Sarvas correctly 1987, Jukka Sarvas correctly

exploited the observations of exploited the observations of Ilmoniemi et al, yielding a concise Ilmoniemi et al, yielding a concise closed-form linear algebra solution closed-form linear algebra solution (independent of Bronzan’s (independent of Bronzan’s development):development):


Bronzan-Sarvas Model Bronzan-Sarvas Model • 1971 Bronzan solved the general 1971 Bronzan solved the general

magnetic scalar solution. No magnetic scalar solution. No specialization to primary currents specialization to primary currents and spherical geometry.and spherical geometry.

• 1987 Sarvas independently 1987 Sarvas independently addressed the specific spherical MEG addressed the specific spherical MEG case and provided the explicit case and provided the explicit gradient solution for the magnetic gradient solution for the magnetic field.field.


Full Magnetic FieldFull Magnetic Field

Primary Current onlyPrimary Current only Primary plus Volume CurrentsPrimary plus Volume Currents


The Elegant PhantomThe Elegant PhantomReturning to the Ilmoniemi et al Returning to the Ilmoniemi et al

solution. Their insight led them to solution. Their insight led them to develop a phascinating phantom. develop a phascinating phantom. These “triangular These “triangular magnetic magnetic dipoles” are dipoles” are experimentally experimentally indistinguishable indistinguishable from a current from a current dipole in a dipole in a perfect sphere of perfect sphere of conducting conducting solution.solution.


Spatial AmbiguitySpatial Ambiguity• This phantom experimentally This phantom experimentally

emphasizesemphasizes– (1) the importance of the volume (1) the importance of the volume

currents, which are now embodied in the currents, which are now embodied in the radial lines, and radial lines, and

– (2) the complete spatial ambiguity (2) the complete spatial ambiguity between two different current between two different current configurations.configurations.


The Silent Radial DipoleThe Silent Radial Dipole• Return to the fictitious currents Return to the fictitious currents

modelmodel

No No measurable measurable signal, primary signal, primary and fictitious and fictitious are all radial.are all radial.

Indeed, there is NO external Indeed, there is NO external magnetic field in any magnetic field in any direction. The field from the direction. The field from the volume currents has exactly volume currents has exactly cancelled the field from the cancelled the field from the primary current. primary current.


““Big” – “Big” Equals ZeroBig” – “Big” Equals Zero• Radially-oriented current dipole Radially-oriented current dipole

generates itself a substantial generates itself a substantial homogeneous field.homogeneous field.

• Volume currents exactly negate this Volume currents exactly negate this field everywhere.field everywhere.


The “Myth” of the Radial The “Myth” of the Radial DipoleDipole• Two critical conditions:Two critical conditions:

– The dipole must be The dipole must be perfectly radialperfectly radial..– The head must be The head must be perfectly sphericalperfectly spherical..

• Re Radial: Hillebrand and Barnes Re Radial: Hillebrand and Barnes (Neuroimage 2002) recently found (Neuroimage 2002) recently found less the 5% of the cortical surface is less the 5% of the cortical surface is within 15 degrees radial. within 15 degrees radial.


Nearly Radial Dipole in a Nearly Radial Dipole in a SphereSphere• Dipole six degrees in x-direction from Dipole six degrees in x-direction from

radial, sensors radially oriented.radial, sensors radially oriented.

Strength Strength already 10% already 10% of that of the of that of the tangential tangential dipole.dipole.

At 15 degrees, At 15 degrees, strength is strength is 25% of that of 25% of that of the tangential the tangential dipole.dipole.


The “Myth” of the Spherical The “Myth” of the Spherical Head Head • A much older and often overlooked A much older and often overlooked result of Grynszpan and Geselowitz result of Grynszpan and Geselowitz (1973) examines a (1973) examines a slight slight perturbation perturbation of the spherical head.of the spherical head.

• Let the minor axis of a prolate spheroid Let the minor axis of a prolate spheroid be be 99.5%99.5% that of the major axis. that of the major axis.

• The maximum external magnetic field The maximum external magnetic field for a radial source is now for a radial source is now 10%10% of that of that for a tangential source.for a tangential source.

• Effectively, the radial dipole in a Effectively, the radial dipole in a perfect sphere has been rotated to perfect sphere has been rotated to 66 degreesdegrees..


SummarySummary• Primary current source generates volume Primary current source generates volume

currents.currents.– If no primary current, then no volume current.If no primary current, then no volume current.– But can have closed loop primary currents that But can have closed loop primary currents that

generate no volume current.generate no volume current.

• The volume currents create differences in The volume currents create differences in potentials on the scalp surface -> EEG.potentials on the scalp surface -> EEG.– Silent EEG sources include those with no Silent EEG sources include those with no

volume currents.volume currents.

• In general, both the primary current and In general, both the primary current and the volume currents contribute to the the volume currents contribute to the magnetic field.magnetic field.– Must first solve the EEG forward model before Must first solve the EEG forward model before

solving the MEG forward model.solving the MEG forward model.


Summary continuedSummary continued• In the special case of spherical In the special case of spherical

geometry and radial MEG geometry and radial MEG measurements, then volume currents measurements, then volume currents do not contribute to the measurement.do not contribute to the measurement.– Only occurs in simulation, even a six degree Only occurs in simulation, even a six degree

offset from radial causes appreciable offset from radial causes appreciable volume current signals in the sensor.volume current signals in the sensor.

• In the special case of spherical In the special case of spherical geometry and a radial source, then geometry and a radial source, then volume currents exactly cancel the volume currents exactly cancel the primary signal, such that NO external primary signal, such that NO external magnetic field exists.magnetic field exists.– Only occurs in simulation, since head must Only occurs in simulation, since head must

be perfectly spherical.be perfectly spherical.


Summary continuedSummary continued• In the special case of spherical In the special case of spherical

geometry, then the tangential geometry, then the tangential magnetic field components may be magnetic field components may be calculated directly from the radial calculated directly from the radial magnetic field components.magnetic field components.– Therefore do not need explicitly to solve Therefore do not need explicitly to solve

EEG problem first.EEG problem first.– First solved in general by Bronzan 1971 First solved in general by Bronzan 1971

using magnetic scalars, but result using magnetic scalars, but result remains obscure.remains obscure.

– Investigated by Ilmoniemi et. al in 1984, Investigated by Ilmoniemi et. al in 1984, resulting in elegant “dry” phantom.resulting in elegant “dry” phantom.

– Solved explicitly by Sarvas 1987 for the Solved explicitly by Sarvas 1987 for the MEG case.MEG case.

a review on the importance of volume currents john c. mosher biological and quantum physics group...

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