1. A Review on the Importance of Volume Currents John C. Mosher Biological and Quantum Physics Group Los Alamos National Laboratory

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

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

4. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Outline Basic Assumptions, and the definition of Primary and Volume currents Basic Assumptions, and the definition of Primary and Volume currents Simple MEG solution and the possible confusion about volume currents Simple MEG solution and the possible confusion about volume currents Historical review of the development of the general solutions for EEG and MEG Historical review of the development of the general solutions for EEG and MEG The “myth” of the “silent” radial dipole The “myth” of the “silent” radial dipole

5. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Excellent Mathematical Reference: Jukka Sarvas, 1987, Physics in Medicine and Biology. Jukka Sarvas, 1987, Physics in Medicine and Biology.

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

7. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Basic Assumptions Source region is non-magnetic Source region is non-magnetic Currents are quasi-static Currents are quasi-static –Electric field is gradient of a scalar –Curl of magnetic field is the current –Divergence of magnetic field is zero Define static magnetic field as the curl of a “vector potential” A(r), yielding Define static magnetic field as the curl of a “vector potential” A(r), yielding

8. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Magnetic Vector Potential Integrate the total current density flowing in the head, divided by its distance to the observation. Integrate the total current density flowing in the head, divided by its distance to the observation. BrainStorm

9. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Biot-Savart Law Vector potential CURL yields magnetic field But the Biot-Savart Law is expressed in total current, we need the solution in terms of the neural source generators.

10. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Primary vs. Secondary Currents Picture primary current as a small battery inside the brain. Picture primary current as a small battery inside the brain. Secondary or volume currents are the gradient currents to “complete the circuit.” Secondary or volume currents are the gradient currents to “complete the circuit.” Boundaries shape the volume currents. Boundaries shape the volume currents. BrainStorm

11. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Primary Neural Sources Primary currents are produced by current flow in apical dendrites in cortical pyramidal neurons. Primary currents are produced by current flow in apical dendrites in cortical pyramidal neurons. Millions of EPSPs summed over ~ten milliseconds. Millions of EPSPs summed over ~ten milliseconds. “Macrocellular” vs. “microcellular.” “Macrocellular” vs. “microcellular.” Ramon y Cajal 1888 from Hamalainen et al. 1993 Reviews of Modern Physics

12. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Two Types of Current Volume currents flow simply due to the presence of a macroscopic voltage gradient in conducting medium Volume currents flow simply due to the presence of a macroscopic voltage gradient in conducting medium Simply define Primary Current as “not volume” Simply define Primary Current as “not volume”

13. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Unbounded Solutions Assume an unbounded homogeneous region: CURL: Homogeneous magnetic field DIVERGENCE: Homogeneous electric potential

14. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Numerical Example Consider a dipole 7 cm up from an origin, and observation points arranged 12 cm from the origin. Consider a dipole 7 cm up from an origin, and observation points arranged 12 cm from the origin.

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

16. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Unbounded Regions Volume currents do not contribute to the potential or magnetic field in infinite homogeneous regions Volume currents do not contribute to the potential or magnetic field in infinite homogeneous regions Volume currents only contribute when bounded regions are nearby Volume currents only contribute when bounded regions are nearby

17. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Bounded Regions Given primary current, what is the magnetic field? Given primary current, what is the magnetic field? MEG general solution includes the general solution of EEG surface potentials. MEG general solution includes the general solution of EEG surface potentials.

18. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Boundary Effects (cf. Sarvas)

19. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Fictitious Currents Standard vector identities allow us to delete the true 3-D volume currents and replace them with fictitious 2-D currents only on the boundaries, normally oriented Standard vector identities allow us to delete the true 3-D volume currents and replace them with fictitious 2-D currents only on the boundaries, normally oriented True physical currents All surface current elements discontinuous

20. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Radial Field Outside of a Sphere In a sphere, then all fictitious currents are radial. In a sphere, then all fictitious currents are radial. If a sensing coil is oriented radially outside of the a perfect sphere, then none of the fictitious currents are visible If a sensing coil is oriented radially outside of the a perfect sphere, then none of the fictitious currents are visible Radial fictitious currents are Unobservable by a radial sensor

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

22. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 The 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 involves the primary current Looks like the Biot-Savart Law, but only involves the primary current

23. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 The Confusion We can also write the spherical solution for the radial measurement as: We can also write the spherical solution for the radial measurement as: But this often leads novices to the conclusion that the volume currents are unimportant. For non-radial measurements in the orientation “o”, they attempt: But this often leads novices to the conclusion that the volume currents are unimportant. For non-radial measurements in the orientation “o”, they attempt:

24. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Slightly Non-radial Measurements Six degrees from radial in the y-direction Six degrees from radial in the y-direction Incorrect: Primary Currents Only Correct: Primary and Volume Currents

25. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Non-radial Field Outside of a Sphere In non-radial directions, the fictitious currents are visible and must be included in the calculation In non-radial directions, the fictitious currents are visible and must be included in the calculation Non-radial orientation Sensor and dipole in same orientation Primary current does NOT contribute!

26. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 All Sensors in the X-direction Sensors cannot see the primary current, only the volume currents Sensors cannot see the primary current, only the volume currents

27. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 General Solution Approach Specify an elemental primary current source, calculate the infinite homogeneous potential, then solve the Fredholm Integral of the Second Kind for all boundary potentials (cf Sarvas): Specify an elemental primary current source, calculate the infinite homogeneous potential, then solve the Fredholm Integral of the Second Kind for all boundary potentials (cf Sarvas): Using this “EEG” solution, plug into “Geselowitz (1970) equation for MEG: Using this “EEG” solution, plug into “Geselowitz (1970) equation for MEG:

28. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Classical Solution Approach First Edition 1962 First Edition 1962

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

30. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Grynszpan and Geselowitz 1973 Formalized the lead-field work of Baule and McFee (1965,1970), using vector spherical harmonics Formalized the lead-field work of Baule and McFee (1965,1970), using vector spherical harmonics

31. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 The Simple Acknowledgement 1975 Second Edition 1975 Second Edition

32. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 The Footnote The clue, quite overlooked, that a simple solution existed. The clue, quite overlooked, that a simple solution existed.

33. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Bronzan, Am. Jrnl. Phys. 1971

34. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Bronzan’s Solution (1971) Magnetic scalar, given in terms of the total current. In the spherical boundaries case, only primary currents can contribute Magnetic scalar, given in terms of the total current. In the spherical boundaries case, only primary currents can contribute Magnetic field is simply the gradient. Magnetic field is simply the gradient.

35. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Further MEG Development Bronzan’s solution is still today not widely cited, and has only recently been cited in MEG literature (Jerbi et al. 2003 PMB). Bronzan’s solution is still today not widely cited, and has only recently been cited in MEG literature (Jerbi et al. 2003 PMB). As the MEG community began experimental measurements in the 1980s, a solution was sought for the non-radial field outside of a sphere. As the MEG community began experimental measurements in the 1980s, a solution was sought for the non-radial field outside of a sphere. 1984 Biomag in Vancouver

36. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Ambiguity of MEG Data Consider again the physical case. A radially-oriented sensor cannot distinguish between the following cases: Consider again the physical case. A radially-oriented sensor cannot distinguish between the following cases: Volume currents are invisible Radial line currents are invisible

37. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 General Solution for the Sphere Sum the magnetic field for the two radial lines and the tangential cross elements. Take the limit as the tangential element is made small, yielding: Sum the magnetic field for the two radial lines and the tangential cross elements. Take the limit as the tangential element is made small, yielding: Unfortunately, while correct, the formula contained a singularity “0/0” condition at certain observation points. Unfortunately, while correct, the formula contained a singularity “0/0” condition at certain observation points.

38. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Correct Insight Their insight was correct, however, that we did NOT need to calculate the volume currents explicitly: Their insight was correct, however, that we did NOT need to calculate the volume currents explicitly:

39. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Sarvas Solution 1987, Jukka Sarvas correctly exploited the observations of Ilmoniemi et al, yielding a concise closed-form linear algebra solution (independent of Bronzan’s development): 1987, Jukka Sarvas correctly exploited the observations of Ilmoniemi et al, yielding a concise closed-form linear algebra solution (independent of Bronzan’s development):

40. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Bronzan-Sarvas Model 1971 Bronzan solved the general magnetic scalar solution. No specialization to primary currents and spherical geometry. 1971 Bronzan solved the general magnetic scalar solution. No specialization to primary currents and spherical geometry. 1987 Sarvas independently addressed the specific spherical MEG case and provided the explicit gradient solution for the magnetic field. 1987 Sarvas independently addressed the specific spherical MEG case and provided the explicit gradient solution for the magnetic field.

41. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Full Magnetic Field Primary Current only Primary plus Volume Currents

42. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 The Elegant Phantom Returning to the Ilmoniemi et al solution. Their insight led them to develop a phascinating phantom. These “triangular magnetic dipoles” are experimentally indistinguishable from a current dipole in a perfect sphere of conducting solution.

43. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Spatial Ambiguity This phantom experimentally emphasizes This phantom experimentally emphasizes –(1) the importance of the volume currents, which are now embodied in the radial lines, and –(2) the complete spatial ambiguity between two different current configurations.

44. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 The Silent Radial Dipole Return to the fictitious currents model Return to the fictitious currents model No measurable signal, primary and fictitious are all radial. Indeed, there is NO external magnetic field in any direction. The field from the volume currents has exactly cancelled the field from the primary current.

45. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 “Big” – “Big” Equals Zero Radially-oriented current dipole generates itself a substantial homogeneous field. Radially-oriented current dipole generates itself a substantial homogeneous field. Volume currents exactly negate this field everywhere. Volume currents exactly negate this field everywhere.

46. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 The “Myth” of the Radial Dipole Two critical conditions: Two critical conditions: –The dipole must be perfectly radial. –The head must be perfectly spherical. Re Radial: Hillebrand and Barnes (Neuroimage 2002) recently found less the 5% of the cortical surface is within 15 degrees radial. Re Radial: Hillebrand and Barnes (Neuroimage 2002) recently found less the 5% of the cortical surface is within 15 degrees radial.

47. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Nearly Radial Dipole in a Sphere Dipole six degrees in x-direction from radial, sensors radially oriented. Dipole six degrees in x-direction from radial, sensors radially oriented. Strength already 10% of that of the tangential dipole. At 15 degrees, strength is 25% of that of the tangential dipole.

48. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 The “Myth” of the Spherical Head A much older and often overlooked result of Grynszpan and Geselowitz (1973) examines a slight perturbation of the spherical head. A much older and often overlooked result of Grynszpan and Geselowitz (1973) examines a slight perturbation of the spherical head. Let the minor axis of a prolate spheroid be 99.5% that of the major axis. Let the minor axis of a prolate spheroid be 99.5% that of the major axis. The maximum external magnetic field for a radial source is now 10% of that for a tangential source. The maximum external magnetic field for a radial source is now 10% of that for a tangential source. Effectively, the radial dipole in a perfect sphere has been rotated to 6 degrees. Effectively, the radial dipole in a perfect sphere has been rotated to 6 degrees.

49. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Summary Primary current source generates volume currents. Primary current source generates volume currents. –If no primary current, then no volume current. –But can have closed loop primary currents that generate no volume current. The volume currents create differences in potentials on the scalp surface -> EEG. The volume currents create differences in potentials on the scalp surface -> EEG. –Silent EEG sources include those with no volume currents. In general, both the primary current and the volume currents contribute to the magnetic field. In general, both the primary current and the volume currents contribute to the magnetic field. –Must first solve the EEG forward model before solving the MEG forward model.

50. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Summary continued In the special case of spherical geometry and radial MEG measurements, then volume currents do not contribute to the measurement. In the special case of spherical geometry and radial MEG measurements, then volume currents do not contribute to the measurement. –Only occurs in simulation, even a six degree offset from radial causes appreciable volume current signals in the sensor. In the special case of spherical geometry and a radial source, then volume currents exactly cancel the primary signal, such that NO external magnetic field exists. In the special case of spherical geometry and a radial source, then volume currents exactly cancel the primary signal, such that NO external magnetic field exists. –Only occurs in simulation, since head must be perfectly spherical.

51. Dr. John C. Mosher, Los Alamos National Laboratory, Biomag 2004 Presentation, August 2004, LANL Technical Report # LA-UR-04-6117 Summary continued In the special case of spherical geometry, then the tangential magnetic field components may be calculated directly from the radial magnetic field components. In the special case of spherical geometry, then the tangential magnetic field components may be calculated directly from the radial magnetic field components. –Therefore do not need explicitly to solve EEG problem first. –First solved in general by Bronzan 1971 using magnetic scalars, but result remains obscure. –Investigated by Ilmoniemi et. al in 1984, resulting in elegant “dry” phantom. –Solved explicitly by Sarvas 1987 for the MEG case.