Brain Topography, Volume 7, Number 2, ! 994 151

DSL and MUSIC Under Model Misspecification and Noise-Conditions

Zhi Zhang* and Don L. Jewett*

Summary: The concepts of both the traditional DSL (Dipole Source Localization) and MUSIC (MUltiple Signal Classification) are explained using simple vectors. DSL and MUSIC differ in the way the weighting functions are found. In both DSL and MUSIC, the fitted generator magnitudes are found by projecting the recorded potential map onto the fitted weighting functions, and the model transfers the weighting functions into generator locations/orientations. Both DSL and MUSIC will cause errors in the fitted dipole parameters when there is model misspecification, noise, or both. Therefore, an accurate head model is essential for either method to give reliable results.

Key words: Vectors; Least square fit; MUltiple Signal Classification; Dipole source localization; Dipole parameters; Model misspecification.

Introduction Current dipoles have been used as equivalent brain

electric generators to explain recorded evoked-potential maps. A commonly-used method, DSL (Dipole Source Local izat ion) , local izes d ipoles by m i n i m i z i n g the squared potential differences be tween the recorded- potential map and the map due to the fitting dipoles. With this method, various kinds of constraints on the fitting-dipole parameters can be applied. One of the frequently-used constraints is that the fitting dipoles must have FLO (Fixed Location and Orientation) over time. We have shown that, even wi th the FLO constraint, DSL results can have insidious errors in the fit dipole parameters, and the magni tude of the errors can be large (Zhang et al. 1994). Recently, MUSIC (MUltiple Signal Classification) has been used as an alternative method to fit dipoles (Mosher et al. 1992). This method uses con- cepts totally different from that of the DSL, and its per- formance is not yet well known to most of the users of dipole modeling.

Here we will use simple vectors to illustrate the prin- ciple of the DSL. Following the same line of reasoning,

*Abratech Corporation, Research Division, Sausalito, California, U.S.A.

Accepted for publication: August 30, 1994. This research was supported by grant R01DC00328 from the Nation-

al Institute on Deafness and Other Commttoication Diseases. We thank Dan Fletcher for his suggestions on the manuscript.

Correspondence and reprint requests should be ad dressed to Dr. Zhi Zhang, Abratech Corporation, Research Division, 475 Gate Five Rd., Suite 255, Sausalito, California 94965, U.S.A.

Copyright �9 1994 Human Sciences Press, Inc.

the principle of MUSIC will be described. The explana- tion will focus on one and two simultaneously-active dipoles, so the explanation can be visualized in a 3- dimensional space. The problems related to the techni- que used to f ind the global m i n i m u m wil l not be considered here, we simply assume the global m in im um is found by the computation.

This paper is intended for the non-mathematical ly oriented readers, all equations are writ ten using only scalars or vectors, no matrices are used.

M-space (Magnitude-space) The spatio-temporal map generated by brain gener-

ators is usually described by an Ntime x Nchn matrix Vobs (or Vfit, see later), where Ntime and ~Nchn are the number of time-points and the number of recording-channels, respectively, in recording the potential map. Assuming the generating dipoles have FLO, the spatio-temporal map can also be conveniently described in a space whose number of dimensions equals the number of generating dipoles. We call this space the M-space. The M-space allows the DSL and MUSIC processes to be explained using vectors. Note that three different Magnitude- spaces will be used to explain DSL and MUSIC: M-space is used for the generators in the brain (noise-free), M- space is used for the fitting dipoles, and M'-space is used for the so called signal-space of the potential map (maybe noise-contaminated). These are the same as in Zhang and Jewett (1994).

Assuming noise-free (noise will be dealt later), the potential Vobs, 1 generated by a dipole at a given record- ing-channel can be written as

152 Zhang ancl Jewett

robs, 1 = ml wl (1)

where ml is the dipole magnitude, wl is a complicated function of both the recording-channel and the dipole location/orientation, and it is the potential generated by a unit-magnitude dipole, so is called the weighting func- tion. When the dipole changes its location and /o r orien- tation, or the r eco rd ing -channe l is changed, the numerical value of the weighting function will change. But, when both the recording-channel and the dipole location/orientation are fixed, Wl becomes a constant. The observed potential Vobs,1 is proportional to ml; when the magnitude varies over time, v0bs,1 varies in the same way, as

Vobs, 1 ( t ) = ml( t ) Wl (t = tl, t2 , . . . , tNt~,~ ) (2)

Similar relat ionship holds w h e n potentials are measured across many channels, the only difference is that for each recording-channel, the weighting function will have different numerical values, thus

VobsA (i,t ) = ml (t ) w t (i ) (i = 1,2 ..... Nchn; t = tl, t2 ..... tN~me)

(3)

where i denotes recording-channels, and t denotes time. Thus, Vobs,1 (i,t) is the potential generated by the dipole across the i-th recording-channel at time t.

The potentials generated by many dipoles obey the linear superposition law. If there are Ngen dipoles, then the total potential recorded across the i-th recording- channel, at time t, will be

Nx~n /qg~

robs (i,t ) = ~ Vobs, j (i,t) = ~ mj (t ) wj ( i ) j= l j= l

(4)

The above expression can also be rewritten in vector form (bold letters denote vectors)

Vob ( t )

Ngen

= E m (t)wj j=l

(5)

where

robs (t ) = {Vobs (1, t ), Vobs (2, t ),. �9 robs (Nchn, t )},

wj = {wj (1), wj ( 2 ) , . . . , wj (Nchn)}

are both vectors. Specifically, Vobs (t) is the single time- point potential map at time t. wj is the collection of the j-th generator's weighting functions across all the chan-

nels. All the potentials across channels over time [i.e., robs

(t) (t=t 1, t~..., tNti~ ) or rob s (i,t) (i=1,2 ..... Nchn; t=tl, h,o..,

tNt~ )] form a map Vobs (capital letters denote matrices).

Equation 5 shows that,cobs (t) over time (i.e., the Vobs map) is simply a linear combination of Ngen v e c t o r s wj (]=1,2 ..... Ngen ) so, the maximum number of dimensions of this Vobs map is Ngen, no matter how many time-points and how many recording-channels are used in recording the Vobs map. The Ngen vectors wj form an Ngen-dimen- sional space. The robs (t), at any time t, can be shown as a point within this space or a vector from the origin towards this point. The projection of this vector on w i is the j-th dipole magnitude at time t, mj (t). Therefore, this space is called the M-space (Magnitude-space). It is not easy to visualize a high dimensional space, so we will show this in a two-dimensional case.

When the Vobs map is generated by two dipoles (at FLO), Ngen =2, the M-space becomes a two-dimensional M-plane (Zhang and Jewett 1.994). The Vobs map, shown on the M-plane, is some curve (figure la). Each point on the curve represents a time-point t, its coordinates (ml (t), m 2 (t)) - the projections along the two axes of the M-plane - are the magnitudes of the two generating dfpoles aL that time-point. The Lmit vectors along the two axes are the two dipoles" weighting functions wl and w2. The weighting function w or w(i) (i=1,2,._, Nchn), as stated early, are the potentials at all ~he channels generated by a unit-magnitude dipole at the given location/orienta- tion.

To demonstrate the relationship between the M-plane and the dipole magnitudes, the two dipole magnitude waveforms ml (t) and m2 (t) over time - the projections of the Vobs map shown in figure la along the two axes - are shown in figure lb.

The statement that Vobs is two dimensional is true whether the two generators' weighting functions wl and w2 are known. When wl and w2 are not known, the M-plane can still be found either pictorially as was done with the 3-Channel Lissajous' Trajectory (Jewett et al. 1987), or mathematically using the SVD (Singular Value Decomposition). However, finding the M-plane is dif- ferent from finding w 1 and w2 (the correct dipole weight- ing functions). This is because a given plane can be uniquely defined by any two intersecting lines on the plane. Vectors other than wl and w2 on the M-plane can also be used to define the M-plm~e. For example, when SVD is applied to the Vobs map of figure la, two or- thogonal axes will be given to define the same M-plane. K the Vobs of figure la is projected on to the axes given by SVD, the projection will be different from those shown in figure lb (the correct dipole magnitudes), i.e., the derived magnitudes from the SVD axes will be wrong. Therefore,

DSL and MUSIC 153

rn2

2.0 T : L

i �9 m ct] D 1.5 i o

\ \ 0 5 10 15 20

?rn2 /

Figure 1. The concept of the M-plane. a) Regardless of the number of recording channels, a potential map Vobs generated by two dipoles each at fixed location and orientation can be shown on a plane as some curve. The plane is called the M-plane (M stands for Magnitude). The axes of the M-plane (marked as ml and m2) are chosen in the directions of Wl and w2, respectively. The two vectors Wl and w2 are treated as unit vectors when the Vobs is projected along the two axes. b) The dipole mag- ni tude waveforms corresponding to the Vobs are recovered by projecting the Vobs map onto the two axes of the M-plane. The point marked "t" in a) is identified by a dashed line in b). c) As the DSL described here is concerned,the Vobsmap (the curve shown in both a) and c)) can be replaced by two points (two vectors vl and V2). The two points may not be on the curve.

M-plane Isin~i

Figure 2, Vector view of DSL and MUSIC under two-dipole cases, a) The M-plane and the M-plane. The two gener- ator weighting functions wl and w2 define the M-plane. The vl and v2, representing a potential map Vobs generated by the two generators, are also on the M-plane (or define the same M-plane). The two fitting generators' weighting functions ~1 and w2 define the__M-plane. Usual- ly, M-plane and M-plane do not coincide with each other. b) DSL tries to find two vectors ~1 and ~ in the fitting model such that the sum of the squared distances be- tween vl and v2 to the M-plane defined by ~1 and w~2 is a minimum, c) Front view of b). d) MUSIC tries to find individual vectorsw in the fitting model such that the angle between the found vector ~ and the M-plane (defined both by wl and w2 and by vl and v2) is a minimum, e) Front view of d), In both DSL and MUSIC, after the fit of the ~'s, the dipole magnitudes are found by projecting the Vobs map on to the w's, and the fitting model transfers each ~ into a dipole location/orientation.

154 Zhang and Jewett

in order to find the correct dipole magnitudes, the correct dipole weighting functions need to be found. This is done by using a model in both DSL and MUSIC.

DSL (Dipole Source Localization) Methods In the commonly-used DSL, the dipoles are fit by

minimizing the following SE (Square Error)

Nchn tNtime

sE:2 2 i=1 t= t 1

IVobs (i, t ) - vfi t (i, t )]2

(6)

where Nch n is the total number of recording-channels, and Ntime is the total number of time-points, robs (Lt) is the observed (recorded) potential across the i-th record- ing-channel at time t, and vfit (i,t) is the potential on the i-th channel generated by all the fitting dipoles at time t within the fitting model.

The fit can be done in various kinds of ways. In the moving dipole modeling (Cuffin 1985; He and Musha 1992), the dipoles fit at each time-point are assumed to be independen t f rom any other time-points. This is equivalent to the single time-point modeling at each time-point (Henderson et al. 1977; Kavanagh et al. 1978; Lehmann et al. 1982), no advantage of using the multiple time-points is gained. In this case, we found that the errors involved in the fit dipole parameters can be large when more than one dipole is active and there is model misspecification (Zhang and Jewett 1993).

The advantage of using multiple time-points is gained by applying proper constraints to the fitting dipole parameters over time. One of the commonly-used con- straints is that the fitting dipoles must have FLO over time, this constraint assumes the brain generator loca- tions and orientations are fixed over time (Scherg and van Cramon 1985a,b; Franssen et al. 1992). Other constraints can also be applied, such as to the fitting dipole mag- nitude waveshapes (Turetsky et al. 1990).

In the following, we will only describe the DSL mini- mizing the SE of equation 6 under the FLO constraint. That is, we will assume each generating dipole is at a fixed location and fixed orientation over time, while the magnitude is allowed to vary freely. The correct number of dipoles is specified with the FLO constraint in the fitting. This is one of the few cases where MUSIC works (see later).

Two-dipole cases

To simplify the vector view of our problem, we will borrow some results developed in our previous work (Zhang et al. 1994). The reader interested in the proofs of these assertions may find them in (Zhang et al. 1994).

When the Vobs map generated by two dipoles at FLO is fitby two dipoles, under constraint FLO, as far as the DSL of equation 6 is concerned, the Vobs curve shown on the M-plane can be replaced by (i,e., simplified to) only two (virtual) time-points (Zhang et al. 1994) with the coor- dinates of the points on the M-plane being:

(1, B cos0) and (0, B sin0) (7)

Where A (=cos 0) and B are two integrals of the original dipole magnitude waveforms, ml(t) and m2(t). "A" describes the overlap between ml (t) and m2 (t) over time. For example, A=0 when there is no overlap at all between the two waveforms, while A=I when the two waveforms are proportional to each other at all time-points. "B" describes the magnitude of the second dipole relative to the first one. With the Vobs curve being replaced by the two virtual time-points on the M-plane, the same dipole location/orientation parameters will be fit by equation 6 as if the Vobs curve was used. Changes in the generator waveforms can be reflected in the values of A and B used in equation 7. All possible generating dipole waveform combinations, as far as the DSL is concerned, can each be located on an AB-plane (-1 <_ A _< +1,0 _< B _< +o0) (Zhang et al. 1994) as one point.

After the simplification of equation 7, equation 6 can be written as

•gen 1:2

SE=E E i=1 t=z !

IVobs (i, t ) - vfit (i, t )12

(8)

where ~:1 and ~2 are the two virtual time-points given by equation 7. So

Vobs ( / , Zl ) = Wl ( i ) + B cos{} W 2 ( i )

robs (i, T2 ) = 0 + B sin0 w2 (i) (9)

or in vector form,

robs ( ' r = W l + B c o s 0 w 2 -- v 1

Vobs(Z2) = B s i n 0 w 2 = v 2 (10)

Shown in figure lc, Vl and v2 are the vectors specified by (1,B cos 0) and (0,B sin 0) on the M-plane, respectively. Note that vl and v2 are not necessarily on the Vobs curve.

Denote the fitting dipole magnitudes and the weight- ing functions as m__l (t), m 2 (t), w I (i) and W__ 2 (i), respective- ly,

vf t ( i , t ) = m l ( t ) w l ( i ) + m__2(t)w__2(i) (11)

DSL and MUSIC 155

or in vector form,

vfit ( t) = m i ( t ) W l + m__2(t)w__2 (12)

where wl and w_ 2 are the weighting functions of the two fitting dipoles written in vector form. The two vectors w 1 and __w 2 define another plane (call it M-plane, the plane due to the fitting dipoles). Note that the M-plane can be different from the M-plane, i.e., the two planes intersect at a non-zero angle (figure 2a). Substituting equations 9-12 into equation 8,

Nchn

SE = s Iwl(i )+ B cos (} w2(i )- m__1('r wl(i )- m2('Cl) w2 (i)12 /--1

Nchn 2

y" [ B sin 0 w2 (i) - ml (z2) wl (i) - m_2 (ca) w__2(i )] i= 1 (13)

or in vector form

SE = I vl - ml ( 'n) W__l - m2 (,1) w_2 12

+ I v2 - m l ( 'r W l - m2 ( 'r W__2 1 2 (14)

The four fitting dipole magnitudes __ml (z) and m2 ('r (z = ~1, ~2) are constants to be linear fit in order to min imize the SE. Af ter t hey are fit, i Vl - ml (Zl) Wl - m_2 (zl) w__2 I is the distance from

vector Vl to the (fitting) __M-plane, so,

[ V1 -- m__l ( ' c l ) W---1 -- m__2 ( 'r w__2 [ = Vl sin[z1

where v 1 is the length of Vl, and (l I is the angle between vector Vl and the M-plane (figure 2b-c). Similarly,

I V 2 - m l ( z 2 ) W 1 - m 2 ( z 2 ) ~ [ = V 2 SinO. 2

is the distance from vector v 2 to the M-plane. And equa- tion 14 becomes

SE = (v I s i n ( I I )2 + (v 2 since2 )2 (15)

Equation 15 is the equation we will use to interpret the way DSL works. This equation is the same as the original equation 6 (under Ngen =2) except expressed in a different form. Here v I and v2 represent the Vobs map, for a given Fobs map, they are fixed vectors given by equation 10. During the fitting, the DSL tries different dipole loca- tions/orientations in the fitting model to minimize the SE. Changing the fitting dipole locations/orientations changes the weighting functions w 1 and w2, which in turn changes the orientation of the M-piane. Changing the orientation of the M-plane changes the two angles ct I

and c~ 2 [equation 15, figure 2b-c]. Therefore, the DSL alters c~1 and or2 by changing the orientation of the M- plane until the sum of the squared distances between the two (constant) vectors to the M-plane (equation 15) is minimized. After the minimum is found, the fitted dipole magnitudes are given by projecting the Vobs map onto __w 1 and w2. The fitting model transfers 14/1 and w2, which define the orientation of the M-plane, into dipole locations/orientations. The fitting model also limits the orientation of the M-plane to varying only within a cer- tain range. A good model should ensure that when SE is at its minimum, not only are or1 and ot 2 small (i.e., the angle between the M-plane and the M-plane is small), but also the angles between Wl and w 1, and w_ 2 and w2, are both small, because only when the Vob s map is projected on to wl and w2 (the correct dipole weighting functions), will the derived dipole magnitude waveforms be correct. For example, when SVD is applied to the Vobs map, SVD will find the correct M-plane but not the correct w's. Therefore SVD will not decompose the Vob s map into correct dipole magnitudes. Furthermore, there is no loca- t ion/ orientation attached to the SVD components, hence no generator location/orientat ion will be provided directly by SVD.

If and only if the fitting model is correctly specified and there is no noise, then at the global minimum of the SE the following are true: the correct locations/orienta- tions are found; the M-plane coincides with the M-plane; Vl and v2 are completely projected on to the M-plane; ctl = 0t2 - 0; SE=0; and the angles between _w I and wl, and __w 2 and w2, are both zero (__w 1 = w 1, __w 2 = W2) , with the result that when Vobs is projected on to _ww I and w2, the correct dipole magnitude waveforms will be derived.

When there is model misspecification, however, the following will be true: the M-plane will generally be different from the M-plane (i.e., the angle between the two planes will be non-zero) even when SE is at its minimum; and Vl and v2 cannot be completely projected on to the M-plane (i.e., ct 1 r 0, cz 2 r 0) (figure 2b-c). The consequences of this are that, first, wl and w2 are dif- ferent from Wl and w2, respectively. So, when Vobs is projected on to__w 1 and w2, the derived dipole magnitude waveforms will be wrong. Second, when another two vectors on the M-plane, different from vl and v2, are used (i.e., the generating dipoles changed their magnitude waveforms only, neither locations nor orientations!), in order to minimize the SE, the orientation of the M-plane may have to be altered. For example, assuming SE is at its minimum for two given vectors vl and v2 (figure 2b-c). Now, if Vl is magnified by a factor of 2, becoming 2vl, and v 2 is unchanged, then, as far as the SE is concerned, 2Vl now has a larger weight than the original Vl, so, if possible, the M_-plane will tend to move closer to 2vl (i.e., to decrease CXl). Changing the M-plane is done by chang-

156 Zhang and Jewett

ing __w I and w2, which in turn is done by changing the fitting dipole location and orientation in the fitting model. Thus, the fit dipole location and orientation change as A and B (thus Vl and v2) change, although the generators did not change their locations and orienta- tions. Also, since __wl and w2 change when A and B change, the derived magnitude waveforms will also change. The results of Zhang et al. (1994) showed that the change in fit dipole parameters due to the change in v 1 and v 2 can be significantly large. Note that this will not happen if the model is correct.

Single-dipole cases

Under single-dipole cases, the Vobs map is one-dimen- sional. Setting w2 and -----2 to zero in equation 14, we can write the SE expression under the one-dipole case as

SE = ! vl-__ml_wl 12 = Iwl-m__lwl 12 (16a)

After the SE is minimized by fitting ml , equation 16a becomes

SE = (w I sin ~3) 2 (16b)

where o~ 3 is the angle between w I and Wl. So, under single-dipole cases, the DSL tries to find a vector direc- tion (W_l) so that the angle (o@ between w I and w 1 is a minimum. The same as in the two-dipole cases, the dipole magnitude waveform is fit by projecting the Vobs (now ml Wl) on to Wl. The fitting model transfers wl into the fitted dipole location and orientation. Equation 16b shows that the same direction (Wl) will be fit to different Vobs maps generated by the same dipole with different waveforms ml(t), because the SE expression depends only on Wl, and not on ml (t). Therefore, the same dipole location and orientation will be fit when ml(t ) changes.

Mult ip le-dipole cases

The multiple-dipole cases can be understood in the same way as the two-dipole cases, except the M-plane now becomes a higher dimensional M-space. Under multiple-dipole cases, the SE of equation 6 after the linear parameters are fit can be written as

SE = (vj sin S/2 j= l

(17)

where Nfit fitting dipoles are to fit the Vobs map generated by Ngen generating dipoles. (The number of fitting dipoles Nit can be different from Ngen, but equation 17 is still correct. The number of generators is misspecified if Nilt ~e Ngen). The vj (j =l,2,...,Ngen) are Ngen vectors after

the simplification with coordinates within the Ngen- dimensional M-space given in the Appendix of Zhang et al. (1994), and vj is the length of vj. The Nil t fitting dipole weighting functions__w) (j=1,2 ..... Nil't) form an Nilt-dimen- sional M-space. After v] is projected onto the M-space, the remaining unexplained vector length is vj sin ~j. Also, ~j is called the "angle" between vector vy and the M-space. The DSL tries different fitting dipole loca- tions/orientations to minimize the SE of equation 17- the sum of the squared distances between the Ngen vectors v] and the M-space - by altering the "orientation" of the M-space, and thus the cz's. Following the same argument as under the two-dipole cases, under model misspecifica- tion, when the v's are changed (i.e., only the magnitude waveforms of the generating dipoles changed), different o~'s, hence different ~ j ' s may be fit, thus, different dipole parameters will be found.

Noise

As soon as noise is involved, the explanation becomes more complicated. This is because the noise-con- taminated potential map can only be viewed in a high dimensional space, while most people are familiar only with 3-dimensional space. However, based on the un- derstanding of the multiple-dipole cases, the noise-cases can also be understood.

Assuming Vobs is a noise-contaminated spatio-tem- poral map, regardless of the number of generating dipoles, using SVD, equation 6 can always be written as

Xch•

j=l

(18)

Compare equation 18 with equation 17, the only dif- ference is that in equation 18 the summation is from j=l to j=Nchn instead of from j=l to j=Ngen. This is because any potential map Vobs can be thought of as generated by Nchn "virtual generators" each has FLO (i.e, the number of dimensions of any potential map will not exceed the number of recording-channels, Nchn). In equation 18, the vj 's are vectors derived when Vobs is decomposed by the SVD (see de Munck 1990 and Zhang and Jewett 1994).

Notice that although the summation is from j=l to j=Nchn in equation 18, the number of fitting dipoles will be much smaller than Nchn, because it is the correct number of generators determined by whatever means that should be specified. For example, if it is believed that the Vobs map is generated by only two dipoles, then only two dipoles should be used to fit the map.

In the following, we will examine how noise affects the fitted dipole parameters by assuming the noise-con- taminated Vobs map is generated by two dipoles, and

DSL and MUSIC 157

fitted by two dipoles. If there is no noise, the Vobs will be a curve on the

M-plane. The DSL fitting is done by finding an M-plane such that the sum of the squared distances between two vectors vl and v2 to the __M-plane is a minimum (equation 15). When the model is correct, the M-plane found will coincide -with the M-plane. The effect of noise on the Vobs is that the noise-contaminated Vobs will no longer be a curve on the M-plane. Some points of the Vobs may still be on the M-plane. But some points are off the plane, the distance between the points to the M-plane can be quite small or large, depending the noise structure and the signal to noise ratio. The many data points (Ntime) can be replaced by (no more than) Nchn points (equation 18). Still, some points are on the M-plane, and some are not. The fit dipole parameters are found by minimizing the sum of the squared distances between these vectors and the M-plane (equation 18). It is unlikely that the M-plane found will coincide with the correct M-plane, even if the model is correct. Therefore, errors will occur in the fitted parameters regardless of the model used. When the noise changes, it is as if the "virtual generators" change their m a g n i t u d e wave fo rms , so d i f ferent dipole parameters may be found on different runs (i.e., different samples).

The DSL fitting of dipole parameters under conditions of noise may be done in a different way: by trying to find the estimated M-plane first, then do the fitting as under two-dipole cases (de Munck 1990; Masher et al. 1992) (equation 15). Because of the noise, the estimated M- plane found (call it the M'-plane, this is the signal-space in Masher et al. 1992) may be different from the M-plane. After the M'-plane is found, two vectors on the M'-plane will be used in place of vl and v2 of equation 15 for fitting the dipole parameters. Because a wrong M-plane (the M'-plane) is used as the correct M-plane, no matter how good the model is, the fitted results will contain errors.

MUSIC (MUltiple Slgnal Classification) MUSIC is a different method for fitting dipoles

(Masher et al. 1992). Assume a Vobs map is generated by Ngen generating dipoles over time, all having an FLO, then Vob s spans an Ngen-dimensional M-space (MUSIC will not work if the number of dimensions is less than Ngen). Now, if the part of the Vobs generated by any one of the Ngen dipoles is completely extracted from the Vob s , the remaining Vobs map will be one dimension less [i.e., (Ngen-1)-dimension]. Put it in a different way, the part of the Vobs map generated by any of the Ngen dipoles is within the Ngen-dimensional M-space. So if one dipole is used to scan all the locations/orientations over the cor- rect model, and at each location/orientation the dipole's weighting function_w is projected on to the M-space (not

M-space!), then when the dipole is at any of the Ngen correct locations/orientations, the projection will be complete (i.e;, the "angle" between the currently-tested weighting function and the M-space will be zero), there- fore the correct dipole locations/orientations are iden- tified (remember __w is no more than a potential map generated by the dipole). After all the Ngen weighting functions are found, the dipole magnitude waveforms are given by projecting the Vobs map on to these Ngen weighting functions.

But if there is model misspecification, the "angle" be- tween w and the M-space will generally be non-zero for all the locations/orientations within the model. In this condition, MUSIC finds the locations/orientations cor- responding to the "angle" minima as the best fit dipole parameters. As mentioned under DSL, when the weight- ing functions are wrong, the fitted parameters will be in error.

Two-dipole cases

The MUSIC fit can be written as

MSE = j 2 (19)

w - k I w 1 - k 2 w = sin2131 w

where MSE stands for MUSIC SE. kl and k2 are constants to be linear fit in order to minimize MSE. The w_ is the fitting dipole weighting function, and 131 is the angle between_w and the M-plane (figure 2d-e). Thus, MUSIC is trying to alter the orientation of the vector w, so as to minimize the angle I31. Under two-dipole cases, two such __w's should be found.

Equation 19 indicates that the fit dipole parameters under MUSIC is only a function of the M-plane (not Wl and w2 or other vectors on the M-plane). Thus, for a given M-plane the same dipole location/orientation will be fit. When different waveforms are used for the two generating dipoles to generate the Vobs maps, the same M-plane is defined, therefore the same dipole loca- t i o n / o r i e n t a t i o n will be fit. So, u n d e r mode l misspecification (without noise), the location and orien- tation MulGenErr (Multiple-Generator Error, see Zhang et aL 1994) under MUSIC will be constants when plotted over the AB-ptane.

Single-dipole cases

In a single-dipole case, equation 19 becomes

w - kl Wl 2 (20) MSE = -- = sin2132

w

MUSIC finds a vector w in the fitting model such that

158 Zhang and Jewett

4

Figure 3. The DSL fit dipole parameters vary as the generating dipoles change their magnitude waveforms. The Location MulGenErrs, LE, for one dipole of a cortical pair under a 3-shell sphere to 1-shell sphere model misspeciflcation are shown. The variation of the LE over the AB-plane indicates that different dipole locations will be fit depending the generating dipole magnitudes.

the angle ~2 between w__ and Wl is a minimum. This is the same as DSL under a single-dipole case whether modelis correctly specified or misspecified (e.g., DSL and MUSIC will find the same dipole parameters).

Mult ip le-dipole cases

In multiple-dipole cases, MUSIC minimizes

MSE =

w - ~ - j=l

w

2 (21)

= sin2 ~3

where kj (j = 1, 2, ..., Ngen) are constants to be linear fit in order to minimize the MSE. Equation 21 can be inter- preted as that MUSIC tries to find the w's in the fitting model such that the "angle" ([33) between each of these w__'s and the M-space (formed by the w's) is at a minimum.

It can be seen in equation 21 that the MUSIC fitted dipole locations/orientations are independent from each other, i.e., the fit location/orientation for one dipole will not be affected by the fit locations/orientations for the other dipoles. This is because they are each compared to the M-space separately.

Noise

Since MUSIC compares each w with the M-space, even under noise conditions MUSIC must find :the estimated M-space (the M'-space) first. This is usually done by decomposing the Vobs by SVD (see Mosher et al. 1992; Zhang and Jewett 1994). After the M'-space is deter- mined, the following MSE is minimized

N~n J ,~

kjw9 _ /=1 = sin2 ~4 MSE =1 w J

2 (22)

where w'j (j: l , 2 ..... Ngen) is a set of Ngen vectors defining the M'-space.

The noise effect on MUSIC is that the M'-space derived from SVD will be different from the M-space. By treating the M'-space as the correct M-space, even a correct model is used, the fitted dipole parameters may contain errors.

Comparison between DSL and MUSIC The differences between DSL and MUSIC are in the

way and the quantity they minimize. DSL projects the given potential map Vobs (or the vectors vj's) onto the fitted M-space, and minimizes the sum of the squared distances between the vectors (vj's) and the __M-space by changing the orientation of the M-space (equations 15- 18). While MUSIC projects the fitted weighting function w onto the M-space (or M'-space) derived from the given Vobs map, and minimizes the angle between w and the M-space (or M'-space) by testing a large set of w's in the fitting model (equations 19-22). In DSL, the non-chang- ing part is the Vobs map (or the vj's), the part that changes during the iterative search for the lowest SE is the __M, space. In MUSIC, the non-changing part is the M'space (or the M'-space), while the fitting __w changes over the locations/orientations studied (see figure 2).

When MUSIC works, it is identical to the DSL only under noise-free and i) under single-dipole cases with or without model misspecification or ii) under multiple- dipole cases without model misspecification. Under multiple-dipole cases with model misspecification or when noise is present, the two are generally different. One of the obvious differences is that, the DSL fit dipole locations/orientations are functions of the generating dipoles' waveforms, while the MUSIC fit dipole loca- tions/orientations are constants for different generating dipole waveforms.

For example, shown in figure 3 are the Location Mul- GenErrs (LE) for one dipole of a cortical pair over the AB-plane. The MulGenErrs are defined in Zhang and Jewett (1993) and Zhang et aL (1994). Basically, LE is the

DSL and MUSIC 159

30

2O

Ld -J

10

a @

@ Dipole1 ,MUSIC 0 Dipole2,MUSIC

Dipole1 ,OSL23 Dipole2,DSL23

~ A �9

0 0

30

~20 03

Ld (D

5 10 15 20 25 30 35 40 45 �9 b pair

0 5 I 0 15 20 25 30 35 40 45 pair

Figure 4. a) The LEs (Location MulGenErrs) and b) OEs (Orientation MulGenErrs) for the 47 dipole pair of Zhang and Jewett (1993) under the 3-shell to 1-shell sphere misspecification. Circles indicate results from MUSIC, Diamonds indicate results from DSL when equation 23 is applied. MUSIC give results generally better than DSL if MUSIC can separate the two dipoles. Few large errors occur under MUSIC because MUSIC could not separate the two dipoles (i.e., only one angle minimum is found although the Vobs is twO dimensional).

fitted location difference in the fitting model for the same dipole under single-dipole active and multiple-dipole active conditions. Clearly, when there is no model misspecification (and noise-free), the same location should be found for the same dipole under both condi- tions, hence the LE should be zero. However, when there is model misspecification, LE will generally not be zero. Now go back to our example. The noise-free Vobs maps were generated in a 3-shell sphere (Zhang et al. 1994), while the fits were done in a 1-shell sphere. It is clear that the LE varies over the AB-plane significantly, indicating that the fitted dipole location changes significantly when the generating dipoles changed their magnitudes. Dif- ferent points on the AB-plane represent different Vobs maps, thus, using different points on the AB-plane is equivalent to using different vectors v 1 and v 2 in equation 14 (figure 2a-b). When the angle between the two vectors

are small or their lengths are significantly different from each other (e.g., at I At ~ 1), the M-plane defined by the two vectors is not stable. A small change in the two vectors may lead to insidious changes in the fit dipole parameters. Therefore, a reasonable condition for the DSL to give stable fit dipole parameters (and hopefully with small MulGenErrs) would be to use the Vobs map at a point in the AB-space corresponding to

vi" vj = 0 and vi = vj ( i ~ j = 1 , 2 , . . . , Ngen ) (23)

Shown in figure 4 are the Location and Orientation MulGenErrs (LE and OE), under the 3-shell to 1-shell sphere misspecification, for the 47 dipole pairs studied by Zhang and Jewett (1993). For each dipole pair, a point on the AB-plane corresponding to equation 23 was used in the DSL to fit the dipole parameters. The errors shown in figure 4 (diamonds) are generally small compared with the MulGenErrs in Zhang and Jewett (1993) and Zhang et al. (1994). Note that equation 23 is only a broad condi- tion, the fit dipole parameters also depend on the fitting model, there can be other (A,B) points on the AB-plane where the MulGenErrs are even smaller than those shown in figure 4 for a given dipole pair. However, equation 23, when desired, can be applied in real applica- tion with the help of the SVD.

Although the DSL fit dipole parameters vary over the AB-plane, MUSIC will fit the same dipole locations and orientations over the AB-plane, but the fit results may be wrong. The MUSIC fit MulGenErrs for the same 47 d ipole pairs u n d e r the 3-shell to 1-shell sphere misspecification are also shown in figure 4 (circles) for comparison. The MulGenErrs are small for most pairs.

Both the DSL and MUSIC fit dipole parameters will also be affected by w's, the weighting functions of the generating dipoles. For example, under Ngen = 2, when wl and w2 are strongly correlated (i.e., the angle between w I and w 2 is very small. This will happen when the two dipoles are close to each other), the fit results (for both DSL and MUSIC) will either contain large errors or be non-separable in location. The fit results for these dipole pairs are also unstable. When waveforms with large 1A I values were used for such dipole pairs to form the Vobs maps, and MUSIC and DSL (even under equation 23) were used to do the fitting, the errors for these dipoles increased (not shown). This is because the IvI-plane is so unstable under such conditions, even a small roundoff error in the SVD components (even when noise-free!) may change its orientation. For real evoked potential maps, whether or not the DSL (under equation 23) or MUSIC will improve the fit results over the DSL of equa- tion 18 depends on whether SVD can separate the noise f rom the real signals so that the M-space can be recovered.

160 Zhan 9 and Jewett

8 4o

I 0

i.G20 __1 @

V7 @

@ MUSIC q'~ DSL23 [] DSL1 s

L_ L: ._

0 0

40

o~

~ 2 0 ._J

0 0

Figure 5.

5 10 15 20 25 30 35 40 45 50 Pun @

b

II

P

5 10 15 20 25 30 35 4Q 45 50 r u n

The LEs for both a) the first and b) the second dipoles of the pair of figure 3 when different EEG noise was added. The 1-shell sphere was used in both the forward and the inverse calculations, so there is no model misspecification. A Vobs map was formed by using figure l b as the two generating dipoles' magnitudes. 50 dif- ferent EEG noise maps were added to the noise-free map to form 50 noise-contaminated maps, each with signal to noise ratio (power) of 16. DSL were applied to each of the 50 maps (equation 18), and the LEs are shown (squares). MUSIC is also applied to each map with results shown in circles. DSL was also applied in a different way, i.e., equa- tion 23 was applied, the results are shown in diamonds. It can be seen that generally DSL with equation 23 and MUSIC are better than DSL without equation 23. Again, some large errors occur when MUSIC cannot separate the two dipoles due to noise.

To see the noise effect on the DSL and MUSIC, shown in figure 5 are the Location MulGenErrs for both dipoles of the same pair of figure 3. Except that, for figure 5, a 1-shell sphere is used in both generating the noise-free Vobs map and in the fitting, so there is no model misspecification. Only noise is added. 50 different EEG noise maps were added to the same noise-free Vobs map, with the same SNR (Signal to Noise Ratio) of 16 in power. It can be seen that the noise effect cannot be described as a function of SNR only, the noise effect also depends on

the noise structure. For both the DSL and MUSIC, some noise maps result less error than other noise maps.

MUSIC works only under limited conditions (Mosher et al. 1992; Scherg 1992). We mentioned that it will not work when the number of dimensions of the Vobs map is less than Ngen, the number of generating dipoles. To see why, assume Vobs is one-dimensional while Nge~=2. Since Vob s is generated by two dipoles, this one-dimen- sional vector v (representing the Vobs map) is neither in the direction of wl nor in the direction of w2. MUSIC will find a vector ~ closest to this vector v, but what are needed are two vectors, one is closest to wl and the other is closest to w2. Thus, even if a correct model is specified as the inverse model, wrong parameters are likely to be fit by MUSIC.

DSL can still be used to fit dipoles under the situaUon mentioned above, but the fit parameters may contain large errors. The moving-dipole modeling is an example. Still assuming Ngen=2, when only one time-point is used in the DSL, the Vobs is simplified to only one vector vl (0=0 in equation 10), in the direction of neither Wl nor w2. DSL tries to find an M-plane closest (making a smallest angle) to this vector vl. When the model is correct, the final fit M-plane should be the M-plane, hence the correct weighting functions and the correct parameters can be fotmd. However, when the model is misspecified, fitting the M-plane by constraining it to ~wo vectors v l and v2 (equation 10) is usually better than to only one vector, as two vectors, especially two orthogonal vectors with equal magnitudes (equation 23), would make both the M-plane and the fit M-plane stable.

Some of the advantages of MUSIC over DSL include that, in MUSIC the fitting process is a l~near scalming process. The time required for MUSIC to fit the dipoles is independent of the number of generating dipoles. After the scanning over a region in the fitting model is done, the MSE (or 1/MSE) can be plotted as contours. While in DSL, finding the global minimum is always a problem. The time needed to fit the dipoles in DSL increases significantly when the number of generating dipoles increases.

Conclusion Although both DSL and MUSIC can find the correct

dipole parameters if there is no model misspecification and no noise, there will be errors in both the DSL and MUSIC fitted dipole parameters when there is either model misspecification, or noise, or both. The fitted dipole parameters are completely determined by the fitted weighting functions (the w's); the projection of the recorded Vobs map onto the fitted weighting functions ~ ' s gives the fitted magnitudes, and the model transfers the w's into the fitted dipole locations and orientations.

DSL and MUSIC 161

If the w's are wrong, the fitted magni tude waveforms will be wrong. If the model is wrong, then even if the w's are correct, the model m a y transfer them into wrong dipole loca t ions /or ien ta t ions . Thus, an accurate model is desirable whether DSL or MUSIC is used.

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