Brain Topography, Volume 6, Number 4, 1994 283

Insidious Errors in Dipole Parameters Due to Shell Model Misspecification Using Multiple Time-Points

Zhi Zhang*, Don L. Jewett*, and Gilbert Goodwill*

Summary: Insidious errors in dipole modeling due to shell model misspecification in a spherical model were examined analyzing multiple time-points using the constraints of a commonly-used DSL (Dipole Source Localization) method. The computer simulation examined the differences in the fit dipole parameters for the same generator under two circumstances: 1) when computed as a single dipole active alone, and 2) when computed as a member of a simultaneously-active dipole pair. The computations were done using a simplification by which the dipole parameters computed from multiple time-points can be correctly assessed by computing dipole parameters at only two virtual time-points. Using multiple time-points in the DSL generally resulted in less error than if only a single time-point was used. However, how much improvement can be achieved by using multiple time-points, as compared with a single time-point, is a function of many factors, such as the location and orientation of the dipoles, and the relative magnitudes and overlap of the waveforms (i.e., time-varying magnitudes) of the dipoles, as well as the model used in the fitting. Further, it was shown that it is incorrect to assume that a multiple-time-point DSL will compute a zero magnitude for generators during quiescent intervals. Additionally, it was shown that a "correction" to reduce error for one pair of waveforms will not be applicable to other waveforms. Also, even if location errors are eliminated, magnitude and orientation errors can still be shown to be present. Finally, iterative reduction of the least-square error between the observed and predicted surface maps leads to increasing errors in dipole parameters. We conclude that a DSL with model misspecification can contain insidious (undetectable) errors.

Key words: Dipole source localization; Multiple time-points; Evoked response; Model misspecification; Dipole parameters; Ary correction.

Introduction

Curren t dipoles have been sugges ted as reasonable mode l s to expla in electr ical ly-recorded bra in activity (Bishop 1949; Brazier 1949; Sholl 1956; Landau 1967; Kavanagh et al. 1976; Jewett et al. 1990). To localize these dipoles f rom a m a p p i n g of surface (scalp) potentials, a ma thema t i ca l l y tractable, s implif ied mode l is usual ly used even though the mode l does not incorporate all the f e a t u r e s of t h e h u m a n h e a d (i .e. , t he m o d e l is misspecif ied to some degree). The success of a given mode l ing effort is often eva lua ted b y compar ing the surface potent ia l m a p pred ic ted f rom the mode l wi th the recorded potent ia l map ; a close cor respondence be tween the two m a p s is t aken as evidence that the analysis is "correct", i.e., the analysis accurately describes the loca- tion, orientat ion, and m a g n i t u d e of the equivalent dipole g e n e r a t o r s w h o s e s u m m e d p o t e n t i a l s c r e a t e d the r e c o r d e d su r face p o t e n t i a l m a p . W h e n e v e r a DSL

*Abratech Corporation, Research Division, Sausalito, CA, USA. Accepted for publication: December 23, 1993. We thank Dr. George Fein for helpful suggestions concerning the

manuscript. This research was supported by grant R01DC00328 from the National Institute on Deafness and Other Communication Diseases.

Correspondence and reprint requests should be addresed to Zhi Zhang, Ph.D., Research Division, Abratech Corporation, 475 Gate Five Road, Suite 255, Sausalito, California 94965, U.S.A.

Copyright �9 1994 Human Sciences Press, Inc.

(Dipole Source Localization) is used, there is a p r e s u m p - tion that if two observed potent ia l m a p s can each be individual ly fit by different single dipoles, then the al- gebraic sum of these two potent ia l m a p s will be fit b y the same two dipoles. This is the a s sumpt ion of l inear super- position. However , unde r mode l misspecification, this p r e s u m p t i o n is not correct, and we have s h o w n that the error unde r some single t ime-point cases could be large (Zhang and Jewett 1993).

To overcome some of the difficulties of the single t ime-point DSL, a spa t io- tempora l dipole m o d e l has been p r o p o s e d and used (Scherg 1984; Scherg a n d Von C r a m o n 1985, 1986; Scherg 1989). This mode l utilizes mul t ip le t ime-poin ts u n d e r the a s s u m p t i o n that the recorded potent ials are due to a few dipole generators , each of which has a fixed location and fixed orientat ion over the mul t ip le t ime-point interval. Addi t ional con- straints m a y also be used (Turetsky et al. 1990). H o w - ever, to our knowledge, the accuracy of such DSLs has not yet been evaluated.

The error due to DSL mode l misspecif icat ion can be separa ted into two parts , the SingGenErr (Single Gener- a tor Error) a n d the MulGenEr r (Mult iple Gene ra to r Error). SingGenErr is well known, and it is the par t which can be e l iminated b y an Ary- type correct ion (Ary et al. 1981), if such a correct ion is available. MulGenErr is ins id ious a n d occurs w h e n mul t ip l e gene ra to r s are

284 Zhang et al.

Table I. The c o n c e p t of the MulGenErr (Multiple Gener- ator-Error), d represents all the parameters for a dipole, so d = (r, 0, ~, m(t)) [Similarly, d = Cr, ~, ~, m(t)), d '= ~', _0', .~', m'(t))], r is d ipole locat ion. (0, ~) is dipole orientation. (Angles are def ined in the figure 1 legend.) m(t) is d ipole magn i tude over t ime, and is also cal led d ipole waveform in this paper. MulGenErr is the di f ference be tween d and d' (e.g., be tween d l and c11, _d_2 and d2, but not be tween

d_l and d_2). SingGenErr (Single Generator-Erro0 is the dif- ference be tween d and d. An Ary-type correct ion (if avai lable) can correct the SingGenErr, but not the Mul- GenErr, so d' r d (third co lumn in table).

# in GEN -> fit in FIT -> in GEN after Ary- type correction

1 dl _dl dl 2 d2 ~. d2

3 dl + d2 d'l + d2 dl + d'2

simultaneously active, it cannot be eliminated by the Ary-type correction. It is this MulGenErr we address both in Zhang and Jewett (1993) and here. "Error" and "MulGenErr" will be used interchangeably to mean Mul- GenErr.

In this paper, we examine the MulGenErr of a DSL analyzing multiple time-points with the constraints FLO (the fit dipoles have Fixed Location and Orientation over time). The computations presented here illustrate the magnitude of the errors that may occur in the multiple time-point case. We emphasize that the MulGenErrs reported here are due only to the model misspecification of approximating a true 3-shell sphere with a 1-shell sphere. Since we do not test other possible model misspecifications (e.g., head shape), this analysis may underestimate the amount of misspecification inherent when approximating the complexities of a real head with a 1-shell model.

We also introduce in this paper a new approach for analysis of multiple-time-point map (under FLO), in which we show that although waveshape in terms of bo th ove r l ap and m a g n i t u d e will affect d ipole parameters, the effects can be simplified for Ngen dipoles to Ngen "virtual time-points". This result has the implica- tion that increasing the number of real time-points used in a DSL does not necessarily improve the result.

Table I outlines the procedures. The potentials at the electrodes on GEN (a 3-shell sphere model mimicking the real head), due to a dipole with parameters dl [at a given location (rl) and orientation (101, ~1) with time-varying magnitude ml(t)] were calculated yielding a time-vary- ing surface potential map equivalent to potentials that might be observed from scalp recordings. This map was then analyzed on FIT (the 1-shell sphere DSL model), so

as to derive the best-fit underlying dipole parameters dl: Similarly, d2 was derived from the map computed f rom d2 (table I, rows #1 and #2). The time-varying surface potential map due to two simultaneously-active, spatial- ly-distinct dipoles, dl and d2, was computed by summing the maps generated by dl and d2 separately, This com- bined map was then f i t by two simultaneously-active

dipoles , d' 1 and _~. We r epo r t tha t g e n e r a l l y

d_ 1 ~ d 1 and d~ ~ d2. The differences are the MulGen- Errs. We show that the magnitude of the MulGenErrs is a function of the dt and d2 waveforms and present a f r amework for examining this error for arbi t rary wave forms.

Methods All the experimental settings are the same as our pre-

vious single time-point study (e.g., same programs, same GEN and FIT models, same electrode locations); for ad- ditional details on the methods, see Zhang and Jewett (1993). The only differences in the present paper are 1) here multiple time-points are used, and 2) only a subset of dipoles and dipole pairs of Zhang and Jewett (1993) are studied.

In this paper, lower case letters are used for dipole parameters in GEN. The fit dipole parameters in FIT are underlined. The fit parameters under multiple-active dipole conditions are marked with a prime symbol. Sub- scripts are used to distinguish different dipoles (see table I). Bold face letters denote vectors. In the appendix, corresponding capital letters are used to denote matrices.

C o m p u t a t i o n

The dipoles were fit by minimizing the LSE (Least Square Error) of the differences between Vob s and v fi t in the same way as in Zhang and Jewett (1993):

LSE

Ntime Nchn

= 2 ; s t = l i = l

[ Vobs (i,t) - vf i t (i , t) ]2

(I)

where Nchn is the number of channels, Vobs(i,t) is the forward computed potential on GEN surface and vfit(i,t) the fit potential on FIT surface, across the i-th channel at time t.

The MulGenErr is broken down into three parts as in Zhang and Jewett (1993): LE (Location Error), OE (Orien- tation Error), and ME(t) (Magnitude function Error). ME(t) is expressed as a normalized percentage:

m'(t) - m(t) (2)

ME(t) (%) - m(t) x 100%

insidious Errors in Multiple Time-Point DSL 285

ME is a function of time, while OE and LE are not, because the constraints FLO force the dipoles to have fixed locations and orientations, while the magnitudes vary freely over time. Because ME is a function of time, we did not compute it, instead, a time-independent mag- nitude error matrix 19I was computed, from which the ME may be computed (see later).

To see how much the fit _vfi t map due to the fit dipole(s) differed from the forward-computed Vobs map between models, a normalized LSE, NLSE (see Zhang and Jewett

1993) was computed. NLSE(%) -- ~ x 100%. RV is the Residual Variance (Scherg and van Cramon 1985). The smaller the NLSE, the better the vfi t fits Vobs in the least square sense. A perfect fit gives a zero NLSE.

Simpl i f i ca t ion

In a computer simulation, using multiple time-points is much more complicated than using only a single time- point. In generating the Vobs map for a dipole pair, different waveforms ml(t ) and m2(t), numbers of time- points, or time intervals can be used, each of which will result in a different time-varying Vobs map. Theoretically there are an infinite number of possible waveforms to analyze for each fixed dipole pair. Furthermore, in localizing dipoles, the required computing time increases as the number of time-points increases. We have shown in the appendix that, as far as fitting the locations and orientations under the constraints FLO is concerned, fit- ting any number (Nfit) of dipoles in FIT to a potential map generated by two fixed-location-and-orientation dipoles, d I and d2, in GEN, with arbitrary waveforms, ml(t ) and m2(t) (t=l, 2 ..... Ntime), is equivalent to fitting the same number (Nfit) of dipoles to a potential map generated by the same two dipoles, d I and d 2 , at only two virtual time-points, "r and %2, with the magnitudes given by R in A.11 as

mml('r m2('~1)" 1('r m2('r /10 B

(3)

Where A and B are defined in A.10. "~1 and "~2 are not simply two of the original multiple time-points. Instead, they are constructed from the values of A and B, which, in turn, are related to the original waveforms ml(t) and m2(t) by A.10.

After the fixed locations and the fixed orientations are found for the Nfit fitting dipoles using the two time- points, a time-independent Ngen x Nfit magnitude error matrix ~ can be computed by A.13 using the single- dipole fitting results of A.14. In this paper, 1C/I is com- puted only for Ngen=Nfit=2, so, l~I is 2 x 2. After M is

known for a certain dipole pair at a given A and B, both the fit waveforms, m'(t)s, and the ME(t)s for different ml(t) and m2(t) (if these waveforms yield the same A and B values) can be easily reconstructed by using A.12 and eq.(2). Specifically, the relationship between ME and M is

m2(t) MEI(t) = gll + m ~ g21

ml(t) ME2(t) = Ix22 + m ~ ~t12

(4)

Note that symbol g is used for M's components. This simplification has a major advantage for inves-

tigating errors in dipole modeling. By determining the errors that can exist for combinations of A and B that encompass the "AB"-plane (-1_< A_< 1, 0_< B<+oo), errors that can occur for all combinations of ml(t) and m2(t) can be addressed, without having to specify ml(t) and m2(t). An explanation of the AB-plane and the magnitude error matrix will aid in interpretation of the resutts. The A expression is called the "overlap function", and the B expression the "relative magnitude", of the activity of the two dipoles (see A.10 for exact meanings). Assuming two waveforms ml(t) and m2(t) are used for a dipole pair in generating the Vobs map, and their values of A and B computed by A.10 are a and b, respectively, then some of the properties of the AB-plane can be explained: If the point (A,B)=(a,b) on the AB-plane is used to form the "two virtual time-points", "1:1 and ~2, then the DSL will fit the same locations and orientations to the fit dipoles (having a fixed location and orientation over time) whether the two time-points, "~1 and 1:2 (eq.(3)), or the multiple time-points, ml(t) and m2(t), are used. So study- ing different points on the AB-plane is equivalent to analysis of different ml(t) and m2(t) combinations. When forming "~1 and ~2 (to be used by the DSL) using (a, [3b) instead of (a, b) (moving in the AB-plane along A=con- stant) is equivalent to replacing m2(t ) by [~m2(t) (i.e., keeping ml(t) unchanged, while scaling m2(t) by a factor of 13 over time). Using point (%a,b) instead of (a,b) is equivalent to shifting m2(t) along the time axis while keeping the waveshapes unchanged. If ml(t) and m2(t) are time-separated (i.e., at any time-point, only one dipole is active), then a=0. Note that there are an infinite number of time-overlap waveform combinations that can also locate on a=0 (along with non-overlapping waveforms). If a=l, the two waveshapes must be com- pletely overlapped (e.g., when ml(t)/m2(t) is a constant over time), which is identical to using a single time-point in the DSL. If a= -1, it is also equivalent to a single time-point, but with m2(t) replaced by -m2(t). Note that any ml(t) and m2(t) combination can be located on the

286 Zhang et ai.

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Figure 1. Dipoles used in the computer simulation, The +z-axis points to the '"vertex". The +x-axis is towards the reader. The solid circle and two dotted circles indicate the three shells, while the dashed lines and circle are to facilitate visualization. The dipole numbers are shown around their locations (marked with *) separated by com- mas if more than one is at the same location. Dipoles #1-#5 are radial with orientation outward. The rest of the dipoles (#6-#12) are neither radial nor tangential, their orientations are: (0,~) = (0,30)6, (0,45)7, (0,80)8, (0,80)9, (0,20)10, (65.5,47.4)11 and (0,20)12, respectively, where 0 is the azimuth angle, between +X axis and the projection of the dipole moment onto the XY plane, a positive angle being toward the +Y axis; ~ is the elevation angle between the dipole moment and the +Z axis; in degrees. Dipoles #10- #12 are not on the YZ plane. The locations (x,y,z) in cm are (1.42, 2.224, -3.684) for #10 and #11 and (-1.034, 2.224, -2.627) for #12. The sphere radius is R0=10 (cm). 65 electrodes were distributed on the sphere surface at 7>0. The electrode at (R0,0,0) was used as the reference electrode for both GEN and FIT.

AB-plane by A.10. Additionally, different ml(t) and m2(t) combinations can be located at a same point on the AB-plane, and if so they will have the same LE, OE and I9I, but different ME(t).

In the error matrix l~I, the non-zero components ~tgf (g,f = 1, 2) are caused by the model misspecification distor- tion of the Vobs map (see eq.(4)). Specifically, Ixff (f=1,2) is the part of MEf(t) caused by the LE and OE in the fit of the f-th dipole itself, in the pair. This is because under a cond i t ion of mul t ip ly -ac t ive d ipoles and mode l misspecification, the locations and orientation of the f-th fitting dipole will be driven away from its starting "cor- rect" (see later) location and orientation. When this hap- pens, the fit magnitude will be different from that if it

Table il. Dipole pairs studied (This is a subset of the pairs studied in the single time-point case (Zhang and Jewett !993)). Pairs are grouped into 4 categories according to their locations and orientations. Pair numbers are given under each category, starting from the smallest to the largest. The two dipole numbers formed the pair are shown together with an ampersand sign. Within paren- theses, the separation of the two dipoles in either degrees or centimeters is shown. In addition, I I means the two dipole orientations are parallel.

Categoryl Pair Number I Dipo les forming the pair Cortical - Cortical

I

II

III ]

IV

Radial- Radial (01-06)

Others (07-16)

2&3(15~ 5&4(15~ 2&4(35~

5&2(50~ 2&1(60~ 5&1(110 ~ 3&6(15~ 2&7(15~ 5&8(15~ 4&9(15~ 7&6(15~ 9&8(15~ 2&8(35~ 4&6(35~ 6&8(35 ~ 1&9(110 ~

Brainstem - Brainstem (17-18) / 11&12(1.1cm); 10&12(1.1cm, I I)

Cortical - Brainstem (19-24) 2&12(9cm); 6&12(9cm); 2&11(10cm);

2&10(10cm); 6&ll(10cm); 6&10(10era

were at the "correct" location and orientation, with a resulting error. Similarly, Ftgf (gcf) represents the part of MEf(t) caused by the other generating dipole's-map. This

part of the MEf is ~tgf multiplied by mg(t) Hence this mf(t) "

part of MEf(t) is a function of time if gCf, and is caused by the g-th dipole in the pair onlyo Put it in a different way, ttll and ~t22 do not distort the waveshapes, while ~t12 and ~21 do. From the definition of 19I [mainly shown in A.13], it can be seen that -100% < ~tff <+oo (f=l, 2), while ~gf (g;a f) can have any value. ~tgf < 0 means that the effect of the g-th dipole on the f-th dipole magnitude is to add a component opposite to the f-th dipole orientation. The smaller the absolute value of the 19I components, the better the magnitude fit.

Simulation

Simulations were performed for 12 dipoles as il- lustrated in figure 1. The dipole locations were chosen to approximate both cortical and brainstem locations. 24 dipole pairs, categorized as Cortical-Cortical, Brainstem- Brainstem and Cortical-Brainstem, were studied (table II). Dipole orientations and separations were varied across the dipole pairs within each category as anatomi- cally appropriate (i.e., Brainstem-Brainstem pairs were kept close together).

insidious Errors in Multipie Time-Point DSL 287

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+ h : : : ! i ~ a0

........ i" i ...... i ............

Figure Z 3D plots of LE over the AB-plane for some dipoles: a) #2 in pair 01. b) #2 in pair 03. c) #6 in pair 24. d) #10 in pair 24. In all the figures (a-f), the left horizontal axes is A, the right is B, the vertical axes is the LE. In this and all the 3D LE, OE and I~ plots, the errors have been computed at every line intersection. The lines between intersections are linear interpolations. The point corresponding to point (0,1) of the AB-plane is marked on all 3D plots. The values at (0,1) are a) 2%, b) 1%, c) 5%, and d) 9%. e) A small region around a peak near (0.87,0.36) in figure 2c (marked with a black triangle) was re-plotted using many points to show a smaller-scale variation of the plot. f) Similarly, the cliff shown in 2b was re-plotted using many points.

In doing all the computer simulations, a point within the AB-plane was picked which determined the mag- nitudes at the two virtual time-points for the dipole pair chosen to generate the Vobs map in GEN. The Vobs map was then fit by two dipoles within FIT. After the fit was done, LE, OE and 19I were computed. This process was repeated systematically across the AB-plane. It took ap- proximately 5 hours of computation on an IBM 6000 model 530 to compute the 28x56 points on the AB-plane for each dipole pair.

For the single-dipole fits, both single and multiple time-points will fit the same locations and orientations. These fit locations and orientations are used as "correct" ones for all the multiple-dipole fittings, so as to guarantee that the errors found were not due to inadequacies in the Simplex algorithm (see Zhang and Jewett 1993).

Results

Given a pair of dipoles, each dipole can have its LE, OE and two l~I components displayed as a surface over the AB-plane. Note that in order to give clear views of all

results displayed on the AB-plane, the scales and axis used for the 3D plots must vary. But in each figure the orientation of the axis is the same.

(a) LE (Location Error) and OE (Orientation Error) as functions of (A,B).

A large region of the AB-plane was studied first. For each dipole in a pair, the LE and OE were plotted as a function of A and B. 3D LE plots for dipole #2 in pairs 01 and 03 and the two dipoles in pair 24 are shown in figures 2a-d. The LE plots for the two dipoles in pairs 01 and 03 are similar to each other, so only one dipole from each pair is shown. It can be seen from figures 2a and 2b that the LEs for dipole #2 differ, depending on which other dipole it is paired with. Both plots have a flat region approximately around (A,B)=(0,1) with errors that are smaller than in the rest of the plane (about 2% and 1%, respectively). At (0,1), the two waveforms, Ml(t) and M2(t), have no overlap, and the "relative magnitude" implies that the magnitudes are equal. Note, however, that the flat region is not necessarily centered on (0, 1), and that the size, shape and exact location of the flat

288 Zhang et ai.

.ao

~30

Figure 3. 3D plots of OE over the AB-plane for the two dipoles in pair 24: a) #6, b) #10. The left horizontal axes is A, right is B, vertical is OE. The OEs at (0,1) are 2 ~ for both a) and b).

region differs for different dipole pairs, as can be seen in figures 2a-d. The shape of the LE plots of figures 2c and 2d are clearly different from figures 2a and 2b. There are no "typical" errors in this complex model. Figures 2a-d also show that multiple time-point cases (between A-1 and A=-I) generally have smaller MulGenErrs than their corresponding single time-point cases (on A=+I at the same B value) for the same dipole of the same pair. Generally, when the points on the AB-plane move away from the flat region, errors increase, even on A=0 (i.e., no overlap).

On the 3D plots, there are regions where LE as a function of A and B is rough, like a mountain range. To see how LE varies with a small change of A and B in the rough region, the LEs were computed for many points within a small region around a peak on figure 2c (marked with a black triangle) near (-0.78, 0.36). The results are shown in figure 2e, showing that the 3D LE plot is not smooth even within that small region, and LE is sensitive to a small change in A and B. In computing figure 2e, the peak of figure 2c was located between the grids (com- putations were done only on the grids), and the peak does

not appear on figure 2e (its location is at the center of figure 2e). Figure 2e showed other peaks which did not appear on figure 2c, and some of them are higher than the highest one on figure 2c (note the different scales on figures 2c and 2e). Part of these rough regions on these 3D plots can be explained by the weak dipole effect (Amir 1994), such as at B>>I or B<<I. However, there can be high peaks even near B=I. It is likely these rough regions are due to the interactions between the potential maps generated by the two individual dipoles.

On figure 2b, there is a cliff starting at (A,B)=(1,1) ending at about (-0.5,0.1). Similar to figure 2e, a small region around the cliff was re-computed using many points and shown in figure 2f. The cliff remained there with structure almost the same as in figure 2b.

With respect to OE, the shapes of most of the 3D plots are similar to the corresponding LE plots; only the two plots for dipole pair 24 are shown in figures 3a-b. Note again that the errors differ with position on the AB-plane, i.e., with changes in waveforms.

To show the magnitudes of the errors across all dipole pairs, plots of LE versus A are shown in figure 4 for B=1/4, 1/2, 1, 2, 4 for all the 24 pairs (48 dipoles) where the dipole in the numerator is the one first listed in the pair (table II). These plots are analogous to the mag- nitude ratio change in the single time-point study (Zhang and Jewett 1993). Similarly, figure 5 shows errors for OEs.

(i) The relationship between LE, OE and B, A. B is the "relative magnitude", thus, on B=2, the second

dipole's "magnitude" in the pair is twice as large as the first one's. Figures 4 and 5 show that for most of the dipole pairs, errors (both LE and OE) generally increase when B moves away from 1, but in different ways (i.e. the error plots are different on B=2 and B=1/2). By examin- ing the dipole pairs (table I) it can be seen that in the Cortical-Brainstem group, the cortical dipole is always the first one in the pair. Larger errors at B<I than at B>I (e.g. figure 4 at B=0.25 and B=4; figures 2c-d) indicates that when a brainstem dipole and a cortical dipole are simultaneously active, then if the cortical dipole is stronger than the brainstem dipole, the localization of the two dipoles has a greater error in the fit parameters than if the brainstem dipole is stronger. The resulting errors in the fit dipole parameters were much larger for the brainstem dipole than for the cortical dipole (e.g., figures 2c-d and figure 6 - to be discussed later). This is because a cortical- brainstem dipole pair with B=I will produce a surface potential map in which the potentials are predominantly from the cortical one.

A is the "overlap function", the smaller the absolute A, the less the overlap between the two waveforms. Figures 4 and 5 show that generally the errors decrease when A

insidious Errors in MultipLe Time-Point DSL 289

Figure 4.

s

p

LE as function of A for all the 24 dipole pairs (48 dipoles) under B=1/4, 1/2, 1, 2 and 4. All data are com- pressed on one 3D plot to save space, but different sym- bols are used for different B values.

moves to zero from both sides. However, even at A=0, the errors are not all zero. Although one cannot tell from the figures how the errors vary with A for each dipole, the actual data showed that the errors do not always decrease with decreasing I A l, which can also be seen from figures 2a-d.

To see how the errors for an individual dipole vary with A, the same LEs as those shown in figure 4 for a few dipoles are plotted in figure 6. For some dipoles, using multiple time-points reduced the LE significantly, e.g. the LEs were close to zero within almost the entire range of 1A I <1 (as in dipole #11 paired with #12, figure 6), while for some pairs, multiple time-points improved the fit parameters only slightly, resulting in still significant er- rors in the fit parameters (e.g., pairs 02, 19-22, 24). It can also be seen that the smallest errors may not necessarily occur at A=0.

(ii) The ability of the DSL to separate simultaneously-ac- tive dipoles.

From all the LE plots, it is hard to tell whether two simultaneously-active dipoles can be spatially separated by the DSL. By examining the fit parameters for B=1/4, 1/2, 1, 2 and 4 for all dipole pairs, it was found that, at each point computed on the plane, including those when A=0, as many as 4 pairs were not separated. The non- separated pairs were those which were closely spaced,

-~/~ a

Figure 5. OE as function of A for all the 24 dipole pairs under B=1/4, 1/2, 1,2 and 4.

Figure 6. LE as function of A under different B values for some dipoles: #10 in pair 24; #2 in pair 03; #6 in pair 24 and #11 in pair 17. The information shown here is included in figure 2.

290 Zhang et al.

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W

�9 ~ I 0

2 0 .......... ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b

L-- E~

3D v

LU O

1 2 0

8 0

�9 0 i � 9 - * * ; ; � 9 $ t

i o l i i i l l l l i t li . . . . . - -0 .5 0 1.5 1

A ( B = 1)

Figure 7. LE (a) and OE (b) as function of A at B=] for all the 24 pairs (48 dipoles) when the potential map generated by two dipoles was fit by three dipoles. The error for the third dipole is not shown,

and were generally those with dipole orientations in parallel (mainly pairs 02, 11 and 12). When A<0, pairs 01, 08 and 18 were also likely tobe non-separable. Although using multiple time-points resulted in large errors (not shown) in locating two parallel brainstem dipoles (pair 18), two non-parallel brainstem dipoles (pair 17) were located correctly most of the time (LE for one dipole in the pair is shown in figure 6). The data also showed that for certain dipole pairs (mainly pairs 07, 20, 22 and 24), using a single time-point solution (e.g., A=-I), the pro- gram was unable to localize the two dipoles reasonably, as one of them was fit close to the sphere surface, result- ing in a large LE. This also happened in some multiple time-point cases, as shown in figure 4 (at B=1/4 and 1/2).

(iii) LE and OE when the number of active dipoles is misspecified.

We now turn to the question of LE and OE induced when 3 dipoles are used to fit maps created by only 2 dipoles. This is to mimic the situation in a real DSL, where the exact number of active generators may not be known. This is the only test where an incorrect number of dipoles was specified in the fitting. The resulting LE and OE are shown in figures 7a and 7b for all the 24 pairs (48 dipoles) on B=I. The starting parameters for the first two dipoles were put at the "correct" ones as before, while the third dipole was put at the center of the sphere pointing towards the +z axis. All parameters were then computed for all three dipoles. Although the third dipole was put at the center of the sphere, it sometimes ended at a location closer to one of the two "correct" locations than the first two. The errors shown in figures 7a and 7b were computed from the two dipoles whose fit locations were closest to the "correct" locations. Comparing figure 7 with figures 4 and 5 (at B=I), it can be seen that, for most dipole pairs, using 3 dipoles to fit the Vobs map generated by only 2 dipoles led to larger errors than if the correct number of dipoles was used. The results also revealed (not shown) that the third dipole magnitude in some cases was large and that the LSE was always made smaller by the presence of the superfluous dipole. Since the superfluous dipole might theoretically have com- puted to have zero magnitude (with a result then identi- cal to that if only two dipoles were used in the fitting), it is clear that each such superfluous dipole makes the "fit" better (i.e., the LSE becomes smaller) with some alteration in the parameters of the other two dipoles. This problem occurs mainly when there is a residual error, such as occurs with model misspecification, but is less likely to occur if there is no model misspecification (see discus- sion).

The number of generators may be misspecified in the opposite way. E.g., one dipole is used to fit a map generated by two dipoles. In this case, even if the model is correct, the fit dipole parameters will obviously be wrong (may correspond to neither of the two generators). The fit parameters will also be a function of both A and B.

(b) M as functions of (A,B). In order to describe the magnitude errors, the error

matrix M was computed and plotted in a similar way to that for LE and OE. When the two dipole locations cannot be separated by the inverse program or one of them was fit close to the surface of the 1-shell sphere (such as occurred when the two dipoles in a pair in generating the Vobs map were close to each other, or the "relative magnitude" was far from one, or I A I was close

x

to one), the fit magnitudes I mgfl (g,f= 1, 2) in A.12 can

Insidious Errors in Multiple Time-Point DSL 291

a i!!i ~ . ilii

0j

C #

5

sH

0j , 1 ~

Figure 8. 3D plots of the 4 components of I~1 over the AB-plane for dipole pair 24. a) p11, b) #22, C) #12, d) p21. The left horizontal axes is B, right is A. The first dipole is #6 and the second is #10. The values at (0,1) are a) -20%, b) -18%, c) -2% and d) 45%.

200

150

100

50

0

-50' . . . . . . . . . . . . . . . . . . -1 -0.5 0 0.5 1

A ( B = 1)

Figure 9, The four components of I~1 for dipole pair 24 as function of A under B=I.

be extremely large, as can the magnitudes of the l~I com- ponents, I ~,gf [. Hence, in all the plots for 19I, any value greater than 200% was set to 200%, and less than -200% set to -200%.

Since 19I has four components for a dipole pair at each point on the AB-plane, only a few plots are included in this paper. 3D plots for each of the four components of 19I are shown in figures 8a-d for dipole pair 24.

(i) Effect of ~t on ME. Figures 8a and 8b show that [111 and [122 vary in dif-

ferent ways over the AB-plane. ~tll is -10% or less over the plotting region, while ~t22 varies between positive and negative values. This means that although [111 and P-22 do not alter the actual waveshapes, the waveform for dipole #6 was always decreased by ~t11, while the waveform for dipole #10 could either be decreased or magnified by [122 depending on their location on the AB-plane. So the effects of [111 and [122 on the two waveforms are not the same, i.e., the two waveforms are not scaled by the same factor. Thus, it is the difference be tween gll and [122 that is impor t an t for given waveshapes.

In contrast, the absolute values of ~t12 and [121 are important, because they alter the actual waveshapes fit for the two dipoles. Figures 8c and 8d show that while [.t12 varies from +50% and above, [121 varies between-15% to 15% (close to zero within a large part of the plot region). This means that the first dipole (#6) will have a much stronger effect on the second (#10) waveform than the second on the first. So the waveshape of dipole #10 will have much larger distortion than that of dipole #6.

In figure 9, the four components of l~I on B=I for dipole pair 24 were plotted, which are simply cross-sec-

292 Zhang et al.

- " ~legtTitude ) 4

Figure 10. I~1 components as function of A for all the 24 dipole pairs under B=1/4, 1/2, 1,2 and 4. Any component value larger than 200% was set to 200%, and less than -200% was set to -200%. The 4 components under the same B are plotted together with different symbols in the same figure. Solid and empty symbols were used for distin- guishability (e.g., both solid and empty diamonds repre- sent I~ 1 ] ) .

tions of the 3D plots shown in figure 8. The plot showed again that, at A close to +1, the error become large, and the best value of A (under which the absolute values of the/xs would be small) is not A=0.

The error matrix components for all the 24 dipole pairs were plotted along lines B=1/4, 1/2, 1, 2, 4 in figure 10. Ix11, ~t22, ~t12 and/x21, for all the pairs were plotted together for the same B. Figure 10 shows that the majority of the values are close to zero. However, there still are values far from zero, even when A is close to zero. Comparing the plots for different B values in figure 10, it canbe seen that for B=I the data points are closer to zero than for B~I. This is because the majority of the pairs are cortical pairs and brainstem pairs. When B=I, the two dipoles have similar contributions to the surface potential map. Note the scales used in the plots are -200% to +200%, so al- though some points look close to zero, their actual values can still be as large as 10% or 20%.

(ii) The upper limit of the DSL fitting a simultaneously- active dipole pair.

Generally speaking, it is true that using multiple time- points with constraints FLO will provide better results

20

V

15

10

5

0

- 5 . . . . . . . i , . . . . . . 0 8 16 24

Pair #

Figure 11. I~1 components for all the 24 pairs when the two dipoles were forced to stay at the "correct"' location and orientation determined under the single t ime-point and single-dipole cases (Zhang and Jewett 1993). Note the axes are different from those in Figures 9 and 10.

than if only a single time-point is used in fitting simul- taneously-active dipoles. However, the magnitude of the improvement depends on many factors, such as the loca- tion, orientation, waveforms of the true generators, and the model used in the fitting. For instance, the multiple time-point fit results can be very good (as for pair 17), or can only be slightly better than if a single time-point was used (e.g., pair 11). To give a bottom line, the best parameters that can be obtained by a fit to multiple time-points would be no location and orientation errors for both fit dipoles. In this case, there would still be magnitude errors as shown by the components of the error matrix for all the 24 dipole pairs (figure 11). For figure 11, the two fitting dipoles were forced to stay at the "correct" locations and orientations. No better results than those shown in figure 11 can be achieved by using multiple time-points with constraints FLO. However, as shown in most of the figures of this paper, LE and OE were non-zero in most of the cases, and, in fact, they can be large. So the results using multiple time-points will generally have errors in both location and orientation, and the absolute values of the error matrix would generally be much larger than those shown in figure 11.

(iii) Locating some real waveforms on the AB-plane. To show where an actual waveshape combination will

locate on the AB-plane, the values of A and B for some ml(t) and m2(t) combinations were computed. In each case, the second waveform [m2(t)] was constructed by

!nsidious Errors in Multiple Time-Point DSL 293

1

"~ ~ -=-sin (0, ~) 0.5 (0, [ ~\~ /~-wave E (0, 4) g

o

- 0 . 5 . . . . . . . . . . . .

- 1 ' 0 90 180 270 360

At (degrees) Figure 12, The A as function of Af for the 4 different waveform combinations given in equation (5),

expanding the first waveform [ml(t)] to mimic temporal dispersion and then shifted in latency to mimic sequen- tial activation of neural generators; the values of the waveforms were zero at all other times during the inter- val. Numerically,

ml(t) = {m0 (t)

ma(t) = {km[0.9(t - At)]

for 0 < t < T for all other t

for At< t < (At + T/0.9) for all other t

(5.1)

And m(t) was chosen as one of the following four:

1) m(t)=Sin(t), T= n (5.2)

2) m(t)=Sin(t), T=2~ (5.3)

3) m(t)= the first half of wave E in figure 6 of Jewett and Deupree (1989), the time interval was treated as n. T=

(5.4)

4) re(t)= the complete wave G in figure 6 of Jewett and Deupree (1989), the time interval to the first zero wave was treated as n. T extends over the entire waveform

( 5 . 5 )

The sine waves given by equations (5.2) and (5.3) are mathemat ica l ly easy to construct, and a decaying sinusoidal waveshape has been used in a DSL to fit recorded human evoked potentials (Turetsky et al. 1990).

Waveshapes (5.4) and (5.5) were exper imenta l ly recorded potentials from long (5.4) and short (5.5) straight nerves.

Substituting equation (5.1) into A.10 shows that B depends only on k [in equation (5.1)] while A depends on the actual waveshapes. Thus, we concentrate first on the effect on A of various latency shifts, At. In figure 12, A as a function of At is shown for the 4 different waveform combinations given by equations (5.1)-(5.5). When At varies from 0 to T, the As for these different combinations are all within -0.5 to 1.0. Obviously, these four different combinations can give the same A for values of A > -0.5. Figure 12 shows that the At-A plots for (5.2) and (5.4) are similar, indicating that when either (5.2) or (5.4) is used in generating the Vobs map, similar dipole parameters (except waveshapes) will be fit. The same applies to (5.3) and (5.5).

To locate these waveform combinations on the AB- plane, we need to know not only the value of A, but also

the value of B. When k= k0 = ~]0.9 [in m2(t) of equation (5.1)], B=I for all the four different combinations. The errors for B=I are shown in figures 2-10 assuming that these waveforms were for dipoles in the specified loca- tions and orientations. When k=klk0, the ml(t) and m2(t) combination moves to B=kl on the AB-plane.

To give an example of what the ME will be for a certain waveform combination and to further illustrate that dif- ferent waveform combinations can locate on the same point on the AB-plane (and thus have the same LE, OE and 19I), equation !5.3) was used to generate the Vobs map (at k=k0, At=88.7 and 366 ), and then both ml(t), m2(t),

s J

and the fit magnitudes ml(t) and m__2(t), were plotted for dipole pair 24 at two different time-shifts: in figure 13a at At=88.7 and in figure 13b at At=366 ~ At both At=88.7 ~ and At=366 ~ the waveform combination, ml (t) and m2 (t),

o 1-20 451 locates on (0,1) on the AB-plane, l~I(Yo) = The -2 -18 "

l~I components showed that dipole #6 h a d a constant magnitude error of-20% over time in its fit waveform due to its location and orientation errors, while the errors of #6 affected the waveform of dipole #10 significantly (45%) and this part of the error varied over time. Similar- ly, dipole #10 had a constant of-18% magnitude error due to its location and orientation errors, yet the effect of these errors on #6 was only -2%. As shown in both figures 13a and 13b, the fit waveform for dipole #6 was almost proportional to its true waveform, while the fit waveform for dipole #10 was quite different from its true waveform, since there was fit "activity" for dipole #10 during times when the "true" #10 was inactive (i.e., zero). Figures 13a and 13b show clearly that when only one dipole is active (the other being zero at that time) that the DSL attempts to fit two dipoles, thus creating errors ha both dipole

294 Zhang et al.

1

i 0.5

. m 0

-0.5

-1 ' 0

1

b 0.5

,'t=~ c 0 O)

-0.5

-1

I-True Mag. O,p. I I-True Mag. Dip. #10 I-- Fitted Mag. Dip. #6 I

0.5 1 1.5 2 2.5

0 1 2 3 4

t/=

Figure 13. When ml(t)=Sin(t) and m2(t)= Sin[q0.9(t-At)] (T=2~) were used as the waveforms for dipole ~pair 24, and the fit magnitudes are shown in a) for At=88.7 and b) for At>360 ~ (=366). Both a) and b) correspond to the same location (0,1) on the AB-plane, so have the same LE, OE and I~.

magnitudes. So, these results show that it cannot be assumed that a DSL with constraints FLO will give cor- rect (zero error) magnitudes for quiescent generators.

The two waveform combinations shown in figures 13a and 13b are physically different, as the two waveforms shown in figure 13a are partly overlapped over time, while the combinations shown in figure 13b are not over- lapped at all over time. However, as far as the DSL is concerned, the two combinations are "identical", as they locate on the same point on the AB-plane, with the same resulting LE, OE and 19I: the LEs are 5% for dipole #6 and 10% for dipole #10, the OEs are 2 ~ for both #6 and #10.

(c) The relationship between LE and NLSE. In the single time-point study (Zhang and Jewett

1993), we showed that decreasing LSE increased LE. The same is true for multiple time-points as indicated by the non-zero LEs in the plots. To demonstrate in a different way that a small LSE is generally inversely correlated with MulGenErr(!), the residual NLSEs for dipole pair 03 are shown in figure 14 (the LEs for dipole #2 of pair 03

Figure 14. The NLSE as a function of A and B for dipole pair 03. (The LE for dipole #2 of pair 03 is shown in figure 2b.) The NLSE is generally small over the AB-plane, and it be- comes even smaller when the MulGenErr increases. Thus the NLSE cannot be used as a measure of how accurate the fit dipole parameters are.

are shown in figure 2b). It can be seen that, first, the NLSE is generally very small (<6%, corresponding to <0.36% of the Residual Variance). Second, over the AB-plane, small NLSE (or LSE) corresponds to worse fit results! The NLSE becomes smaller around the boundary of the AB- plane, and that is the place where large MulGenErrs occur (figure 2b). By using only the NLSE as the criterion, one would accept as better, the result containing the larger MulGenErr! We again conclude (Zhang and Jewett 1993) that a small LSE does not mean the dipole parameters are correct.

Discussion The inherent non-linearity of the dipole model and its

sensitivity to model assumptions is such that when mul- tiple dipoles are active and the model is misspecified, misallocation of signal variance between the multiple dipoles takes place. Under model misspecification, when one fitting dipole is used to fit the potential map generated by one generating dipole (a single-dipole map), it is unlikely that the fitting dipole will explain 100% of the map. When two dipoles are to fit a map generated by two generating dipoles (a summed map, equivalent to the sum of two single-dipole maps), even if the two fitting dipoles are forced to stay at the "correct" locations and orientations, the part of the unexplained

Insidious Errors in Multiple Time-Point DSL 295

single-dipole map will now be partly explained by the other fitting dipole, introducing MulGenErrs in the fit magnitude for both fitting dipoles (figure 11). If the locations and orientations of the fitting dipoles are al- lowed to vary, there may well be other locations and orientations at which the two fitting dipoles would "ex- plain" the summed map better than at the "correct" loca- tions and orientations, i.e., give a smaller NLSE. Thus the two fitting dipoles are driven to new, incorrect locations and orientations by the DSL algorithm.

In some cases the Simplex algorithm can be trapped in a local minimum. However, we believe that our results were not affected by such a problem for the following reasons: 1) For the single-dipole cases (Zhang and Jewett 1993), the fit results were always the same when different starting parameters for the fitting dipole were used to fit the same Vobs map generated by a single dipole. Further, the fit results agreed with those predicted by the "Ary- correction". Therefore, for single dipoles, our results are unlikely to be trapped in a local minimum. 2) When two dipoles were used, the parameters found under single- dipole cases were chosen as the starting parameters. It is possible that some of the results reported by the Simplex may not be at a global minimum (e.g., Fender 1987) [this is also pointed out in Zhang and Jewett (1993)], such as in the rough regions. But the non-zero MulGenErr indi- cates that the starting parameters at the "correct location" are not a minimum..

All the computer simulations in this study were done in a manner similar to those in Zhang and Jewett (1993). A large part of the discussion in that paper also applies here. For example: 1) The errors shown in the figures are unlikely to have been caused by the small errors in the single dipole locations and orientations (due to the computer's accuracy) used as the starting values for com- puting the two simultaneous dipole parameters. 2) After an "Ary-type" correction (which we did not apply), there will be no errors in the parameters when only a single dipole is used in the forward and inverse calculations (i.e., SingGenErr is removed, table I). But, for two simul- taneously active dipoles, the correction will work if and only if the MulGenErrs (LE, OE and ~I) are all zero, which did not occur for most dipole pairs. Since the "Ary-type" correction multiplies both the dipole location and mag- nitude computed in the 1-shell sphere by a factor of approximately 1.5, after the correction, LE will be about 1.5 times larger. While the errors in orientation and magnitude (OE, ME) would remain as reported. So even after applying an "Ary-type" correction, the errors in most of the dipole parameters may still be too large to be acceptable when more than one dipole is active.

Almost all the computer simulations were done on the AB-plane without concern for an actual waveform com- bination used in generating the Vobs maps. This is be-

cause all the different waveform combinations can be located on the AB-plane. If a particular waveform com- bination is of interest, then, by using A.10, the cor- responding point on the AB-plane for this waveform combination can be found, and all of the error plots at that point on the AB-plane apply to that combination of waveforms. Note that abnormal waveshapes obtained in clinical studies may have different A and B coefficients than normal waveshapes, and hence different errors; so a "correction" derived for normal waveforms may not apply to abnormal ones (especially in "rouglh" regions).

The AB-plane covers all the waveform combinations used to generate the Vobs map, and it also includes the single time-point cases, so the information shown in the single time-point study (Zhang and Jewett 1993) is also included in the figures of this study. The boundaries of the AB-plane (i.e., on A=+I; B=0; l/B=0) correspond to the single time-points (and moving dipole model). The boundary lines B=0 and l/B=0, although not shown on any of the figures, can be computed, and each cor- responds to one point only; on B=0, the potential map is generated by the first dipole only, on l/B=0, the map is generated by the second dipole only. The whole region within these boundaries corresponds to the multiple time-points (no matter how many time-points are used).

The points on A=0 on the AB-plane are of special interest, since they correspond to a non-time-overlap waveform combination (i.e, at any time-point only one dipole is active). Although it might seem that this should give the best fit dipole parameters even when fitting two dipoles simultaneously, this is not necessarily true (e.g., figures 13a-b) because the number of active dipoles is also misspecified, i.e., at any time only one dipole is active, while two dipoles are used in the fitting. Note that although A=0 are the points on the AB-plane on which non-overlapping waveforms fall, A=0 also are the points for some waveform combinations which do have time overlap; the case shown in figure 13a is one example.

At times the experimenter may not know whether a dipole is fixed in orientation over time or "rotates" at a fixed location. This can be addressed by removing the constraint that the dipole orientation be constant over time. Alternatively, three orthogonal fixed-orientation dipoles, with the same location can be fit to the data. The AB-plane remains valid if rotating dipoles are used in the dipole estimation procedure. Also, the same analysis can be done when generators are not dipoles (e.g., quad- ruples, dipole sheets, dipole lines). Thus, the concept of MulGenErr applies to other generators as well.

Conclusions

The errors reported here are due only to model misspecification, which can occur either when the model

296 Zhang et aL

cannot be precisely described (e.g., from a real head to a spherical model) or when the real model can be precisely described but is too complicated (e.g., from a 3-shell sphere to a 1-shell sphere as we did here). Model misspecifications may include shape (e.g, head shape to sphere) and conductivity (e.g, shell conductivities, num- ber of shells, shell thicknesses, local conductivities). The magnitude of the errors depends upon many factors including the degree of model misspecification, the dipole locations and orientations, the "relative mag- nitude" and the "overlap function", etc., in complex ways we have not tried to categorize. Thus, we warn readers not to assume that specific results here can be applied to any other cases, e.g., different locations and orientations, different conductivity ratios, or different shapes.

In this study, a 3-shell sphere and a 1-shell sphere were used to demonstrate the MulGenErrs. This does not mean that there will be no MulGenErr when a 3-shell sphere is used to analyze real evoked potentials while there will be when a 1-shell sphere is used. What we want to emphasize is that MulGenErr may occur with unpredictable magnitude under any model misspecifica- tion. With computer simulations we can show the errors, but it is impossible to demonstrate this with a real poten- tial map. We mentioned the Ary-type correction, be- cause most DSL users are familiar with that, and the Ary-type correction is used in some DSL analyses. Under model misspecification, which is true for almost all cur- rently-used DSLs, fit dipole parameters are obviously wrong if no Ary-type correction is applied as d ~ d (table I). However, our results show that even if an Ary-type correction is applied, the fit results may still be wrong when multiple dipoles are simultaneously active, but such errors are insidious.

With model misspecification, the errors of this report are generally inverse to the LSE. Thus, a low LSE cannot, and should not, be used as evidence for the accuracy of a given analysis (figure 14).

The promise of DSLs in accurately separating the ac- tivity of anatomically or physiologically distinct gener- ators that are contributing simultaneously to surface potential maps will not be fulfilled until these problems have been adequately addressed. Alternatively, DSLs containing insidious MulGenErrs should be viewed as purely an empirical means of data reduction lacking accuracy with respect to underlying dipole parameters, but such an approach should be tempered by the realiza- tion that, even in the same subject, changes of relative activity of simultaneously-active generators (e.g., by a change in stimuli, or by disease) can cause movement on the AB-plane, with a correspondent change in the errors in derived dipole parameters that is not predictable from the change in generator activity.

Appendix - Simplification of multiple time-point DSL under noise-free computer simulation

We present here the mathematical proof of the simplification used in the main text. We first follow the same procedure as in Mosher et al. (1992), then, take advantage of the knowledge available from the computer simulation and reach our goal of testing the DSL under conditions of model misspecification, multiple time- points, and multiple (simultaneously-active) generators.

Constraint FLO (Fixed dipole Location and Orienta- tion) is assumed in the derivation. But the simplification equally applies for the constraint FL (Fixed dipole Loca- tion, i.e., rotating dipoles), because one FL dipole is equivalent to three FLO dipoles at the same location (so long as their orientations are not all in the same plane). Matrices are denoted by capital letters, vectors are in bold face. Estimated variables are underlined, prime symbols are used when more than one dipole is active. Nchn is the number of recording channels. Ntime is the number of time-points. Ngen is the number of FLO dipoles in generating Mob s. (Note, the simplification has nothing to do with Nfit, the number of fitting dipoles.)

Using matrix notation, the LSE (Least Square Error) of equation (1) becomes

LSE = I I Mob s - V fit I 12 A.1

where both Mob s and V fit are Ntime x Nch n matrices, representing the OBServed and the FITted potential maps, respectively. The relationship between the surface potential map Vma p (both Mob s and V fit) and the dipole parameters are governed by

Vma p = M W T A.2

M represents the dipole magnitudes. The model (in- cluding electrode locations) and dipole locations and orientations are reflected in the weighting function W. Substituting A.2 for V fit into A.1

LSE = I I Vob s - M' w'TI 12 A.3

,Note that, we assume more than one dipole is fit, so M ~V T was used in A.3. The same analysis holds when only one dipole is active, simply replace M" W 'T by ~ T . Finding the best-fit M" for any given W' leads to

LSE = J I Vobs _P' I 12 A.4

where P' is an Nchn x Nchn projection matrix, and it is related to W" (see Mosher et al. 1992).

So far, no simplification is introduced, only the matrix

Insidious Errors in Multiple Time-Point DSL 297

forms are altered from equation to equation. In de Munck (1989) and Mosher et al. (1992), SVD (Singular Value Decomposition) was applied to Vobs as

Mob s = U ~ V T A.5

where U has the same size as Vobs but with orthonormal columns, ~ is Nchn x Nchn diagonal, and V is Nchn x Nchn orthonormal. The simplification is seen when A.5 is substituted back into A.4, where the orthonormal U is dropped as it does not affect the LSE, so

LSE = !I ~ V T P'II2 A.6

The effect is that the Ntime x Nchn matrix Vob s is replaced by ~V T which is Nchn x Nch_w possibly with a lot of zero elements. For example, if only. 2 dipoles are active, then only the first two rows of ~V T are non-zero.

This kind of simplification could have been used in our computer simulation, but the drawback would be that, each time when different waveforms are used, we have to form a new multiple time-point Vobs, and then simplify the map using A.5. To avoid forming the multiple time- point Vobs, substitute A.2 for Vob s into A.4

LSE= I IMWTp'I I2 A.7

Here M is Ntime x Ngen, and W is Nchn x Ngen. Notice that the magnitude matrix M can be factored into M = QR by the Gram-Schmidt process. The columns of Q are orthonormal, and R is Ngen x Ngen upper triangular. When M is replaced by QR in A.7, Q can be dropped for the same reason as U can in A.6, therefore, the LSE for our computer simulation becomes

LSE = l iR wTp'112 A.8

So, when W is known, M can be replaced by R (or Vobs replaced by RwT). R is the simplified form of the multi- ple time-point magnitude M. However, this is still no more than what de Munck and Mosher have done. We still need to form the M, then decompose it to get R. We wish to form R without forming M first. M and R are related by

(MTM)ij = (RTR)ij = mi,mj (i,j=l, 2 ..... Ngen) A.9

where m = { m(1), m(2), ..., m(Ntime) } Define:

Aij = mi -mj / ( Imi l Ira j I) = coseij (i~j) Bi= Imil / Imll A.10

the first few components of R are given (by the Gram- Schmidt process) as

B2cos012 B3cos013 ...~

0 B2sin012 n cos023--cos012 c0s013 ... D3

sin012 0 0 B3S-([cos(012- 013) -cos023][cos023-cos(012+ 013)]) ""

sin012 . , .

A.11

In forming R, all its components were normalized by I m 1 l, because scaling by a common factor will not affect the fit dipole location/orientation parameters.

A.11 shows that R is a function of only the As and the Bs of A.10. A is the cosine angle between two magnitude vectors (in the time domain), and is closely related to the so called correlation coefficient. But A will be the correla- tion coefficient only if the means of both ml and m2 are zero. To avoid confusion, in this paper A is called "over- lap function". B is the ratio of the lengths of two vectors, and is called "relative magnitude".

When Ngen =1, R = (1). So, the fit non-linear dipole parameters will be the same no matter how many time- points are involved in Vobs. W h e n Ngen =2, R = 1 B2cose12

(0 Basine12) The fit dipole Parameters will be functions

of both A12 and B2 (in the main text, they are simply called A and B), and either the parameters or the errors can be plotted as surfaces over the plane spanned by A12 and B 2. In this sense, the fit dipole parameters can be thought of as a two-dimensional function, or varying within a two- dimensional space. Generally, when the number of gen- erators is Ngen , the dimension of this space (spanned by

3 K T

(Ngen the As and Bs) is DIM (Ngen) = ("~gen + 1)

2 - 1). With this understanding, we can compute the function values systematically over this DIM(Ngen)-dimensional AB-space without being concerned the actual magnitude M.

Compare A.7 with A.8, the original magnitudes, M, can have many (N0_me) time-points, while its simplified form, R, has no more than Ngen "virtual" time-points. Therefore, a multiple time-point DSL (under FLO) can always be simplified to no more than Ngen time- points.

After the non-linear location/orientation parameters are fit by A.8, the magnitudes can be linearly fit as

M' = M M'w = M W T _.W_W' (~vr__ 'T W_') -1 A.12

M'w is Ngen x Nfi t. It defines the magnitude linear contribution from the Ngen generators to the Nfit fitting dipoles. The components ofM w can each be plotted over the AB-space. In this paper, instead, l~'I, the MulGenErr contained in M w are plotted.

298 Zhang et al.

For simplicity, assume Ngen = Nfit,

l~I = M s l M w - I A.13

where I is the identi ty matrix, Ms is Ngen x Ngen diagonal wi th its diagonal element (__Ms)ii the fit magni tude when Vobs, generated by the i-th generator alone with uni t magnitude, is fit by one dipole, i.e., the fit is done by minimizing LSEi = I wi - ( ~ ) i i wil 2, [w = {w(1), w(2) ..... w(Nchn)}] wi th the result that

(M___s)ii = w i �9 w i / (wi �9 wi) A.14

Therefore the w's in A.13 may be different from the__ws in A.12.

Returning to the magni tude error matrix 19I. Its com- ponents, denoted by ggf in the text, represent the mag- ni tude error in the f-th fit dipole due to the g-th generator. 19I can be extended to Ngen x Nfit with Ngen ~ Nfit by adding zeros and infinities appropriately.

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