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BioMed Central Page 1 of 12 (page number not for citation purposes) Journal of NeuroEngineering and Rehabilitation Open Access Research Time and frequency domain methods for quantifying common modulation of motor unit firing patterns Lance J Myers* 1 , Zeynep Erim 1,2 and Madeleine M Lowery 1,2 Address: 1 Rehabilitation Institute of Chicago, Sensory Motor Performance Program, 345 East Superior St, Chicago, Illinois, 60611, USA and 2 Department of Physical Medicine and Rehabilitation, Feinberg School of Medicine, Northwestern University, Chicago, Illinois, USA Email: Lance J Myers* - [email protected]; Zeynep Erim - [email protected]; Madeleine M Lowery - m- [email protected] * Corresponding author coherencecommon drivemotor unit dischargedescending drive Abstract Background: In investigations of the human motor system, two approaches are generally employed toward the identification of common modulating drives from motor unit recordings. One is a frequency domain method and uses the coherence function to determine the degree of linear correlation between each frequency component of the signals. The other is a time domain method that has been developed to determine the strength of low frequency common modulations between motor unit spike trains, often referred to in the literature as 'common drive'. Methods: The relationships between these methods are systematically explored using both mathematical and experimental procedures. A mathematical derivation is presented that shows the theoretical relationship between both time and frequency domain techniques. Multiple recordings from concurrent activities of pairs of motor units are studied and linear regressions are performed between time and frequency domain estimates (for different time domain window sizes) to assess their equivalence. Results: Analytically, it may be demonstrated that under the theoretical condition of a narrowband point frequency, the two relations are equivalent. However practical situations deviate from this ideal condition. The correlation between the two techniques varies with time domain moving average window length and for window lengths of 200 ms, 400 ms and 800 ms, the r 2 regression statistics (p < 0.05) are 0.56, 0.81 and 0.80 respectively. Conclusions: Although theoretically equivalent and experimentally well correlated there are a number of minor discrepancies between the two techniques that are explored. The time domain technique is preferred for short data segments and is better able to quantify the strength of a broad band drive into a single index. The frequency domain measures are more encompassing, providing a complete description of all oscillatory inputs and are better suited to quantifying narrow ranges of descending input into a single index. In general the physiological question at hand should dictate which technique is best suited. Published: 14 October 2004 Journal of NeuroEngineering and Rehabilitation 2004, 1:2 doi:10.1186/1743-0003-1-2 Received: 30 August 2004 Accepted: 14 October 2004 This article is available from: http://www.jneuroengrehab.com/content/1/1/2 © 2004 Myers et al; licensee BioMed Central Ltd. This is an open-access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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BioMed Central

Journal of NeuroEngineering and Rehabilitation

ss

Open AcceResearchTime and frequency domain methods for quantifying common modulation of motor unit firing patternsLance J Myers*1, Zeynep Erim1,2 and Madeleine M Lowery1,2

Address: 1Rehabilitation Institute of Chicago, Sensory Motor Performance Program, 345 East Superior St, Chicago, Illinois, 60611, USA and 2Department of Physical Medicine and Rehabilitation, Feinberg School of Medicine, Northwestern University, Chicago, Illinois, USA

Email: Lance J Myers* - [email protected]; Zeynep Erim - [email protected]; Madeleine M Lowery - [email protected]

* Corresponding author

coherencecommon drivemotor unit dischargedescending drive

AbstractBackground: In investigations of the human motor system, two approaches are generallyemployed toward the identification of common modulating drives from motor unit recordings.One is a frequency domain method and uses the coherence function to determine the degree oflinear correlation between each frequency component of the signals. The other is a time domainmethod that has been developed to determine the strength of low frequency common modulationsbetween motor unit spike trains, often referred to in the literature as 'common drive'.

Methods: The relationships between these methods are systematically explored using bothmathematical and experimental procedures. A mathematical derivation is presented that shows thetheoretical relationship between both time and frequency domain techniques. Multiple recordingsfrom concurrent activities of pairs of motor units are studied and linear regressions are performedbetween time and frequency domain estimates (for different time domain window sizes) to assesstheir equivalence.

Results: Analytically, it may be demonstrated that under the theoretical condition of a narrowbandpoint frequency, the two relations are equivalent. However practical situations deviate from thisideal condition. The correlation between the two techniques varies with time domain movingaverage window length and for window lengths of 200 ms, 400 ms and 800 ms, the r2 regressionstatistics (p < 0.05) are 0.56, 0.81 and 0.80 respectively.

Conclusions: Although theoretically equivalent and experimentally well correlated there are anumber of minor discrepancies between the two techniques that are explored. The time domaintechnique is preferred for short data segments and is better able to quantify the strength of a broadband drive into a single index. The frequency domain measures are more encompassing, providinga complete description of all oscillatory inputs and are better suited to quantifying narrow rangesof descending input into a single index. In general the physiological question at hand should dictatewhich technique is best suited.

Published: 14 October 2004

Journal of NeuroEngineering and Rehabilitation 2004, 1:2 doi:10.1186/1743-0003-1-2

Received: 30 August 2004Accepted: 14 October 2004

This article is available from: http://www.jneuroengrehab.com/content/1/1/2

© 2004 Myers et al; licensee BioMed Central Ltd. This is an open-access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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IntroductionCommon oscillations in neurophysiological activity in thehuman motor system have been well documented. Duringvoluntary muscle contraction, the human central nervoussystem drives motor neurons at a range of frequencieswhich cause common modulations in the firings of theseneurons. These drives are reviewed in [1] and [2] wherethey are summarized into four broad frequency ranges: (1)A low frequency drive at around 1–3 Hz (2) A neurogeniccomponent of physiological tremor that occurs between 5–12 Hz and is likely to have both spinal and supraspinalcomponents. (3) A corticospinal drive in the beta (15–30Hz) range (4) A corticospinal drive in the low gamma (30–60 Hz) range, that increases in importance with strongercontractions and is called the Piper rhythm.

There are two distinct approaches toward the identifica-tion of these drives. The majority of the literature hasexamined common modulation to motor units using fre-quency domain methods. This methodology was firstintroduced by Rosenberg and colleagues [3] and appliedby Farmer and colleagues [4] who used coherence analysisto identify both a significant low frequency and beta-bandassociation between motor unit firings in the 1–12 Hzand 15–30 Hz frequency ranges respectively. Subse-quently, coherence analysis has become an establishedtechnique to study bivariate motor system measurementsand a number of works have used this to investigate corti-comuscular interactions [5-8,1]; tremor [9]; aging [10];oscillatory drive in Parkinson's disease [11,12]; dystonia[13]; stroke [14]; and cortical myoclonus [15].

A separate body of literature has focused specifically onthe low frequency common drive. This drive was firstidentified by De Luca and colleagues [16] who demon-strated that the firing rates of concurrently active motorunits (MUs) were modulated in a highly interdependentmanner. They low-pass filtered the impulse trains corre-sponding to MU firing times to obtain the time-varyingmean firing rates which they high-pass filtered at 0.75 Hz.They then performed a time domain cross-correlationanalysis between pairs of zero-mean signals representingthe fluctuations in mean firing rates. Peaks occurring nearthe zero lag location in the normalized cross correlationsimplied that those firings rates were essentially simultane-ously modulated with virtually no time delay. This phe-nomenon was termed 'common drive' to indicate acommon excitation to the motor neuron pool that resultsin concurrent fluctuations in the firing rates of motorunits from the same pool. A number of subsequent stud-ies have utilized this technique to investigate the relation-ship of this drive to handedness [17-19]; differentproprioceptive conditions [20]; exercise [21]; task and dis-ease [22]; and aging [23]. These works have establishedthis time domain method as an accepted means of quan-

tifying the common low frequency modulation of MUfirings.

In a recent review [1], it was suggested that the low fre-quency common drive first identified by De Luca and col-leagues [16] using time domain methods is essentially thesame low frequency drive as detected by Farmer and col-leagues [4] using frequency domain methods. In thispaper we explore the relationship between the two tech-niques using mathematical and experimental approaches.

Analytic methodsFrequency domain methods: CoherenceThe coherence between two zero-mean stationary randomprocesses x1 (t) and x2 (t), at frequency f, is defined as:

where (f) is the cross spectral density and (f)

and (f) are the auto spectral density functions of x1

(t) and x2 (t) respectively. The coherence function is acomplex quantity and its squared magnitude provides abounded measure of linear association between the twoseries, taking on a value of 1 for a perfect linear relation-ship and a value of 0 to indicate that the series are uncor-related. In practice, we are often limited to a single time-limited realization of each random process and hence it isnecessary to estimate the magnitude squared coherence,

, by windowing the time series to

obtain multiple sections as follows:

where * denotes complex conjugation, N is the number ofdata segments employed and X1n (f) and X2n (f) are the dis-crete Fourier transforms of the nth data segments of x1 (t)and x2 (t). This estimate is biased and its probability den-sity function for non-weighted and non-overlapping win-dows has been analytically determined [24]. This may beused to derive the value of the estimated coherence, witha particular probability of occurrence, α, that would beobtained when the true value is zero. Any value exceedingthis level is considered to be unlikely to be a false indica-tion of coherence with (α × 100) % confidence. This con-fidence level is given by [24,3]

Eα = 1 - (1 - α)1/(N-1) (3)

γ x xx x

x x x x

fS f

S f S f1 2

1 2

1 1 2 2

1 21( ) =

( )( ) ( )

/( )

Sx x1 2Sx x1 1

Sx x2 2

C f fx x x x1 2 1 2

2( ) = ( )γ

ˆ ( )

*

C f

X f X f

X f X fx x

nn

N

n

n nn

N

n

N1 2

11

2

2

12

22

11

2( ) =( ) ( )

( ) ( )

=

==

∑∑

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The resolution of the coherence estimate is determinedfrom the inverse of the length of the windowed sections,i.e., for a 2 s window, the coherence resolution will be 0.5Hz. Figure 1 depicts a typical coherence plot computed forthe spike trains of two MU's and the associated 95% con-fidence level. The coherence plots reveal the bandwidthand values of significant coherence for the given resolu-tion. Results from coherence analyses are usually quanti-fied in terms of either the peak value and its frequency orthe frequency range of significant coherence. In Figure 1there is significant coherence between 0.5 and 3.5 Hz andthe peak value of coherence is 0.46 and occurs at 1.5 Hz.

Time domain methods: Cross correlationThe cross correlation between two zero-mean stationaryrandom processes x1 (t) and x2 (t) is defined as:

where E [·] is the estimation operator. Assuming ergodic-ity, for single time-limited realizations of each randomprocess, this is determined using the integral:

where * denotes complex conjugation and τ is the time lagbetween the signals. The Fourier transform of the cross

Example of a magnitude squared coherence plotFigure 1Example of a magnitude squared coherence plot. Magnitude coherence between two motor unit spike trains recorded from the FDI muscle. The dashed horizontal line indicates significance at the 95% confidence level. Significant coherence occurs between 0.5 and 3.5 Hz with the peak coherence of 0.46 occurring at 1.5 Hz.

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency (Hz)

Coh

eren

ce m

agni

tude

R E x t x tx x1 2 1 2 4τ τ( ) = +[ ( ) ( )] ( )

R x t x t dtx x1 2 1 2 5( ) ( ) ( ) ( )*τ τ= +−∞∞

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correlation function, defines the cross-spectrum, (f).

Cross correlation functions are unbounded measures andare typically normalized by the values of the autocorrela-tions at zero lag to bound the estimate between -1 and 1.The autocorrelation functions are the time domain equiv-alent of the auto power spectra and their value at zero lagrepresents the total energy in the signal. The normalizedand bounded measure is known as the cross correlation

coefficient, , which provides a measure of the lin-

ear association between the two signals at a given time lagand is given by:

The original method employed by De Luca and colleagues[16] represents the time series as a binary pulse train withones corresponding to the firing times of the MUs andzeros comprising the remainder of the signal. A movingaverage window is then used to smooth these binary sig-nals, which is analogous to filtering the time-series with alow-pass filter. These smoothed signals are depicted in fig-ure 2a. A high-pass filter is then used to remove the meanbias of the signal as 'shown in figure 2b. The filtered sig-nals are subsequently cross correlated and an indexobtained from the peak value of the normalized cross cor-relation function within a specified lag window. Here weterm this index the time domain common drivecoefficient.

Figure 2c depicts the normalized cross correlation func-tion for the same two motor unit spike trains used in Fig-ure 1 obtained using a moving average Hanning windowof 400 ms duration and high pass filtering at 0.75 Hz. Thefunction is displayed for lags up to ± 400 ms and the peakof the signal indicated at a lag of 3.5 ms. The time domaincommon drive coefficient is measured as 0.75.

Relationship between cross correlation and coherenceThe cross correlation coefficient is related to coherenceusing a similar analysis to Gardner [25] as follows:

We begin with the real and stationary signals x1 (t) and x2(t), where x2 (t) = αx1 (t - τ0) + n (t) is a scaled and time-delayed version of x1 (t), with additive uncorrelated, zeromean noise, n (t). The cross-correlation function is givenby

since the cross correlation with the noise is zero every-where. The cross spectrum is given by

Assume that the signals y1 (t) and y2 (t) result from passingthe signals x1 (t) and x2 (t) respectively through a tunablenarrowband bandpass filter with transfer functiondenoted by H (f):

where and ∆ are the center frequency and bandwidth ofthe ideal bandpass filter. The cross-correlation functionbetween filtered signals y1 (t) and y2 (t) is given by:

where (f) is the cross power spectral density func-

tion of y1 (t) and y2 (t).

The cross power spectrum may be written in terms of its

magnitude and phase, .

Since for stationary, real signals, the autocorrelation is real

and even and hence, (f) is real, the phase of the cross

spectrum is given by (equation 8):

Thus replacing (f) with |H (f)|2 (f) in equation

10 we get

since is real for real x1 (t) and x2 (t).

Similarly for the autocorrelation functions we get

and

Thus as ∆ → 0 we obtain the expression for the normal-ized cross correlation function as:

Sx x1 2

ρ τx x1 2( )

ρ ττ

x xx x

x x x x

R

R R1 21 2

1 1 2 20 0

6( )( )

( ) ( )( )=

R Rx x x x1 2 1 1 0 7( ) ( ) ( )τ α τ τ= −

S f S f ex x x xi f

1 2 1 102 8( ) ( ) ( )= −α π τ

H ff f

( ),

,

/( )=

− ≤

1

0

29

otherwise

f

R S f e dfy y y yi f

1 2 1 2

2 10( ) ( ) ( )τ π τ=−∞∞

∫Sy y1 2

S f S f ex x x xi fx x

1 2 1 21 2( ) ( )

( )= − θ

Sx x1 1

θ π τx x f f1 2

2 110( ) ( )=

Sy y1 2Sx x1 2

R H f S f e df

S f e

y y x xi f f

x xi

1 2 1 20

1 2

2 2 20

2

12( ) ( ) ( ) ( )

( )

[ ]

[

τ π τ π τ=

=

−∞∫

ππ τ π τ

π τ τ

f ff

f

x x

df

S f f

−−+

∫≅ −

22

2

0

0

1 22 13

]/

/

( ) cos ( ) ( )

∆∆

Ry y1 2( )τ

R S f fx x x x1 1 1 12 14( ) ( ) cos ( )τ π τ≅ ∆

R S fx x x x1 1 1 10 15( ) ( ) ( )≅ ∆

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Example of the construction of a low frequency common drive plotFigure 2Example of the construction of a low frequency common drive plot. A low-pass, moving average Hanning window filter of length 400 ms was applied to two motor unit spike trains recorded from the FDI muscle. (a) A 5 s epoch of the time-varying smoothed firing rates; (b) the high-pass filtered version of the smoothed firing rates shown in (a); and (c) the low frequency common drive coefficient function between two motor unit spike trains. This results in an effective pass band of 0–5 Hz. The peak of the signal is 0.75 and occurs at a lag of 3.5 ms.

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The peak of the cross-correlation function occurs at thetime delay, τ = τ0. Thus

Thus we see that the peak of the normalized cross-correla-tion function between two signals after ideally bandpassfiltering to contain a single frequency, is identical to themagnitude of the coherence function of the original signalsat the frequency of the filter. The phase of the coherencefunction is the same as the phase of the cross-spectrumand provides the time delay.

For a less ideal filter that spans several frequencies therelationship is less precise and may be derived as follows:

Let W (f) be the new filter transfer function and thus thenormalized cross-correlation function is:

where f1 and f2 are the cut-off frequencies of the filter. Thuswhen multiple frequencies are present, this may bethought of as taking the weighted summation of the cross-correlation functions at each frequency present and nor-malizing this by the product of the weighted summationsof the autocorrelations across all frequencies present. Themore narrow band the filter used, the more similar thetime domain correlation and frequency domain magni-tude coherence measures. As the filter encompasses agreater range of frequencies, measures from the two meth-ods will increasingly deviate.

The low frequency time domain method employed by DeLuca and colleagues [16] utilized a moving Hanning win-dow as a low pass filter. The cut-off frequency of the filteris dependent on the time constant of the filter which istypically 400 ms [16,21] but values up to 0.95 s have alsobeen used [26]. However different window lengths willmodify the relationship between this time domain meas-ure and the coherence function.

The effect of varying window length may be illustrated byobtaining an expression for the filter transfer function.The equation for the Hanning window is given as:

where τ is the length of the window. The discrete Fouriertransform of this is given as (Kay, 1988):

where

Figure 3 depicts the transfer function power spectrum (|W(f)|2) for τ = 200, 400 and 800 ms. The figure clearly dem-onstrates that as the length of the analysis windowdecreases, the bandwidth of the filter increases. Thereforethe only information that can be ascertained with shorterwindows is that the frequency of the common modulatinginput lies somewhere within the frequency range specifiedby the window. Longer windows result in a better correla-tion with coherence values at lower single frequencies(close to zero), while shorter windows lump into a singlevalue a weighted expression of the coherence values in thefrequency range which they span.

Experimental methodsIn this section we demonstrate the relationship betweentime and frequency domain based methods to estimatethe common modulating drive using empirical data. Themethods are applied to data collected during isometriccontractions of the First Dorsal Interosseous muscle at20% of maximal effort. Two contractions where the activ-ities of 4 and 5 MUs were identified yielded a total of 16pairs of coactive MUs. The periods of concurrent activityof these MU pairs ranged between 30 s to 1 minute andwere further divided into pairs of non-overlapping 10 sintervals resulting in a total of 50 pairs of 10 s long spiketrains. Each method was applied to these spike train pairsand the correlation between the results yielded by the twomethods were investigated as discussed below.

The time domain method was used to estimate low fre-quency common drive according to the method describedby De Luca and colleagues [16]. Three different timedomain estimates were formed by smoothing the spiketrains using Hanning windows of length 200, 400 and800 ms respectively. These smoothed firing rate signalswere then digitally high pass filtered with a low frequencycut-off of 0.75 Hz using a third order Butterworth filter toremove the mean bias discharge rates. The cross correla-tion coefficient function of these high pass filtered records

lim ( )( ) cos[ ( )]

[ ( ) ( )] /∆→

0

0

1 21 2

1 2

1 1 2 2

2ρ τ

π τ τy y

x x

x x x x

S f f

S f S f(( )16

ρ τ γy yx x

x x x xx x

S f

S f S ff

1 2

1 2

1 1 2 2

1 20 1 217( )

( )

[ ( ) ( )]( ) ( )

/= =

ˆ ( )( ) ( )

( ) ( )

( )

ρ τπ τ τ

y y

x xi f df

f

f

x xf

W f S f e

W f S f df1 2

1 20

1

2

1 11

2 2

2=

−∫ff

x xf

fW f S f df2

2 21

2 218

∫ ∫ ( ) ( )( )

w tt

t( )

cos( )=

+

< ≤12

12

0

019

πτ

τelsewhere

W f W f W f W fR R R( ) ( ) ( )= −

+ + +

14

1 12

14

120

τ τ

W f ef

fRj f( )

sinsin

( )= − 2 21π τ π τπ

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was then obtained and the peak value of this functionwithin ± 50 ms of the zero time lag was recorded andtermed the time domain common drive coefficient.

The coherence analyses were performed in a similar man-ner to the procedure of Rosenberg and colleagues [3] forpoint process data. The spike trains were represented asbinary pulse trains with ones corresponding to the firingtimes of the MU's and zeros comprising the remainder ofthe signal. Fourier transforms of these trains wereobtained for each appropriately windowed section and

then averaged according to equation (2). However whereRosenberg and colleagues [3] do not use overlapping ortapered data windows, we used overlapping, tapered Han-ning windows of 2048 ms to optimize the variance andbias of the estimate. With any non-parametric spectralestimation technique, there is a trade-off between the var-iance and both the bias and resolution of the estimation.A window size of 2048 ms, gives a frequency resolution of0.49 Hz, which is adequate to discriminate frequencies forour purposes. However, when analyzing 10 s of data using2048 ms non-overlapping windows, only 5 different

Magnitude squared spectra of Hanning window filtersFigure 3Magnitude squared spectra of Hanning window filters. Magnitude spectra of the transfer functions of Hanning window filters for three different time constants, τ = 200 ms (dotted line), τ = 400 ms (dashed line) and τ = 800 ms (solid line). As the time constant increases the bandwidth of the filter decreases and its magnitude increases.

0 2 4 6 8 10 120

1

2

3

4

5

6

7

8x 10

−3

Frequency (Hz)

Am

plitu

de (

V)

200ms400ms800ms

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records are available and this small number of records willincrease the variance of the estimate. Furthermore rectan-gular windows introduce an estimation bias due to theeffect of their sidelobes. These concerns may be reducedby using the Welch periodogram method which usestapered windows (to reduce spectral leakage and thereforethe estimation bias) and overlapping windows (toincrease the total number of windows and hence reducethe variance). The minimum variance for this method isobtained using an overlap of 62.5% [27]. The frequencycorresponding to the first zero-crossing of the Hanning fil-ter was obtained according to equation (20) and the peakvalue of the coherence in the range between 0.75 Hz andthis frequency was recorded.

A linear regression between the time domain commondrive coefficients and corresponding frequency domainpeak coherence values was performed to determinewhether a linear relationship between the two indicesexisted. The regression r2 values are reported at a signifi-cance level of p < 0.05.

Results and discussionFigure 4a,4b,4c displays the regression between the lowfrequency time domain common drive coefficients forHanning windows of length 200, 400 and 800 ms andpeak low frequency coherence. All regressions are signifi-cant at p < 0.05 and the r2 statistics are 0.56, 0.81 and 0.80respectively. A unitary slope line is displayed in the figureand this describes the theoretical relationship between thetwo indices. These results indicate that for the larger 400ms and 800 ms windows, the time domain method ismore closely correlated with the coherence estimate, withthe 400 ms window yielding a marginally better fit. Thedata for the smaller 200 ms window exhibits a consistentbias, with the coherence estimate larger than the timedomain common drive estimate, whilst the 400 ms and800 ms windowed data are more evenly distributedaround the unitary slope line, indicating less bias. Thereare a number of possible factors that could contribute tothe observed mismatches between the two methods.

As demonstrated in Figure 2, the cross-correlation peakcan occur at lags slightly different than zero. A time delayor misalignment has been shown to introduce a bias intothe coherence estimate that is proportional to the delayand coherence magnitude and inversely proportional tothe FFT epoch duration [24]. However for delays of theorder of magnitude of ± 50 ms and FFT lengths of approx-imately 2 s, this type of bias is very small and unlikely toaccount for the observed differences between the time andfrequency domain estimates.

The use of a short duration window in the time domainmethod results in the inclusion of multiple frequencies in

the time domain correlation estimation according toequation (18). The bandwidths of descending oscillatorydrives may be variable. Thus when the descending driveoccupies a narrow bandwidth and the time domain win-dow includes a greater range of frequencies than thisbandwidth, this will bias the time domain estimate to belower than the peak coherence value as is the case in figure4a. Alternatively should the drive span a broader band-width, the time domain measure would encompass all thecorrelated frequencies into a single value and would thusbe different than the value obtained from any single peakcoherence frequency. This idea is illustrated in Figure 5where a typical coherence plot is displayed. Superimposedon this are vertical lines representing various moving aver-age filter cut-off frequencies. The 0.75 Hz high pass cut-offfrequency is also displayed. Thus from the figure we seethat in this case the coherence occupies a fairly broadbandwidth from 1–5 Hz, peaking at 1.5 Hz. The cut-offfrequency of the 200 ms filter is approximately 10 Hz andthus the time domain estimate will include coherence val-ues at all these frequencies which would make it signifi-cantly different from the peak coherence. The 400 ms and800 ms windows would better correlate with the peakcoherence frequencies and the 400 ms window wouldprovide a better overall index encompassing the full band-width of the drive. However, if the middle peak at 8 Hzwere stronger and actually the main peak, the wider timewindows would miss it altogether. This emphasizes theimportance of a priori knowledge in choosing the appro-priate time windows in the time-domain based method.Therefore in summary, the time domain measure is moreeffective in quantifying a range of frequencies into a singleindex and the peak coherence estimate is better at repre-senting the coherence at any single frequency.

Coherence estimates are typically formed from datarecords of around 1–5 minutes in length [4,28,29] orfrom pooled coherence measures of repeated trial meas-urements [30]. This increases the number of non-overlap-ping windows in the calculation, thereby reducing thevariance of the coherence estimate. Non-overlapping, rec-tangular windows are traditionally preferred due to theclear relationship with significance levels. Overlapping,tapered windows will allow coherence to be estimatedfrom shorter data segments and parametric techniques, inparticular multivariate autoregressive (MAR) methods aresuggested for the analysis of very short duration data seg-ments [31]. When using short records of data (<5 s), thecoherence estimates are likely to be significantly biased.However, the time domain method is more robust forsuch short data lengths and would therefore be preferredin these situations.

The time domain method uses a high pass filter to removethe mean bias from the smoothed signals, whereas the

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Regression plots for low frequency common drive time versus frequency domain techniquesFigure 4Regression plots for low frequency common drive time versus frequency domain techniques. Regression plots for low fre-quency common drive time versus frequency domain techniques. Three different moving average Hanning windows were used to low pass filter the time series for the time domain method. The time constants for the filters are as follows: (a) τ = 200 ms, (b) τ = 400 ms, (c) τ = 800 ms. All regressions are significant at p < 0.05 and the r2 statistics are (a) 0.56, (b) 0.81 and (c) 0.80. The unitary slop line is indicated in the figures as a dashed line and represents the ideal mathematical relationship.

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frequency domain coherence method simply subtracts themean component of the signal prior to forming the esti-mate. Although similar, these two methods are not iden-tical and may further explain some of the variationbetween the time and frequency domain techniques. Afurther possibility is to employ a low order polynomialdetrending technique instead of high pass filtering or sub-tracting the mean. In general, a visual examination of thesmoothed firing rate signals would indicate whether thiswould be necessary.

It is straight forward to quantify any time delay using thetime domain technique. Although this is also possiblewith the frequency domain technique, this delay informa-tion is incorporated in the phase of the estimate and istherefore 2π periodic and would thus yield the same resultfor integer multiples of delay. For significant coherencepresent over a band of frequencies, Mima and colleagues[32] suggest a constant phase shift plus constant timedelay regression model to compute time delays fromcoherence estimates. However for narrow band

Magnitude squared coherence between two motor unit spike trains recorded from the FDI muscleFigure 5Magnitude squared coherence between two motor unit spike trains recorded from the FDI muscle. Magnitude coherence between two motor unit spike trains recorded from the FDI muscle. The vertical dotted lines from left to right represent the cut-off frequencies of the 0.75 high-pass filter, and the 800 ms, 400 ms and 200 ms moving average low-pass filters. The peak coherence occurs at 1.5 Hz.

0 2 4 6 8 100

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Frequency (Hz)

Coh

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HP filter 800ms filter 400ms filter 200ms filter

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descending drives, the time domain technique provides aclearer estimate of any time delay.

It should be noted that the time domain technique maybe generalized to cover any arbitrary frequency. Thiswould necessitate bandpass filtering the signals to thefrequency range of interest, removing the mean trendsand then evaluating the cross-correlation function.Although this requires a priori knowledge of the drivebandwidth, this method would then be able to provide asingle index to quantify a particular bandwidth.

ConclusionsTwo separate bodies of literature offer techniques to esti-mate band limited common oscillatory input to motorneurons. These techniques are based in either the time orin the frequency domain. Indices derived from both meas-urement techniques are well correlated with each otherand in the theoretical limit, the techniques are shown tobe mathematically equivalent. However, for practical pur-poses there are a number of minor discrepancies whichmay favour the use of one particular method for a givenapplication.

The time domain method offers greater resolution in time(the latency of the correlations are easily revealed) at theexpense of the requirement of a priori knowledge of thebandwidth of the common modulating drive. On theother hand the frequency domain technique reveals infor-mation regarding the frequency distribution of the com-mon modulating drives but it is more difficult to obtainaccurate estimates of the coherence with short signallengths as well as of the exact delay.

Time domain methods of estimation are preferred forshort data segments and are well suited to quantifying thestrength of a broad band drive into a single index. Thisproves useful in quantitative, comparitive analyses of thecommon behavior of MUs such as statistical tests thatinvestigate the effects of aging or disease. Frequencydomain measures tend to be more encompassing as theyprovide a complete description of all common oscillatoryinputs. This facilitates qualitative analysis of distributionof coherence across frequencies and hence leads to a bet-ter understanding of the nature of the common inputs.They are well suited for estimation from large data seg-ments, that may be assumed to be stationary, and are bet-ter able to quantify narrow ranges of descending inputsinto a single index. Thus the selection of one techniqueover another should be dictated by the nature of the phys-iological question to be addressed.

AcknowledgementsThe authors gratefully acknowledge the financial support of the Falk Medi-cal Research Trust.

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