## Experimental Study

To investigate the role of the cerebellum and basal ganglia in sensorimotor synchronization, we conducted a study in which patients with cerebellar lesions or Parkinson's disease were asked to tap along with an auditory metronome. Unlike previous studies of tapping with these patient groups, we included only a paced phase, and the number of taps per trial (200) was considerably larger than in previous work. These long runs were essential for examining how the participants adjusted their behavior based on an error signal generated through the comparison of their own performance and the metronome signals. While previous studies have focused on component processes that are assumed to operate during unpaced tapping, the focus here was on the ability of these patients to use error correction during paced tapping.

We based our analysis on a linear error correction model (Vorberg and Wing, 1996). To separate the influence of the phase correction process from noise coming from the internal clock component and noise arising at stages of the motor implementation, we express the asynchrony on tap k + 1 relative to the perceived asyn-chronies on the last two taps (see Figure 19.1):

Ak +1 = Ak - aA'k - pAVi + (Tk - P) + (Mk_i - Mk) (19.1)

While this formulation includes a second-order term based on the asynchrony preceding the adjusted tap by two taps, the first-order (AR1) forms of the models proved to be sufficient with most of the data sets. With a few exceptions, discussed in the results section, the estimates for p were close to zero. Thus, we will limit our focus to the first-order model.

Because the perceived asynchrony A! is the sumof themeasuredasynchrony A and the difference of the perceptual delays for the perception of the motor event (the tap), Fk, and auditory signal, Sk, Equation (19.1) becomes

Ak+i = (1 - a)Ak - a(Fk - Sk) + (Tk - P) + (M+ - Mk) (19.2)

We can solve for a stationary solution of the expected asynchrony by taking expected values, yielding

a where the bar denotes the average or expected value.

Thus the expected asynchrony is a direct function of the difference between mean perceptual delays (Aschersleben and Prinz, 1995), the deviation of the mean i nterval of the clock from the pacing interval, and the error correction parameter a. The stochastic properties of this and related models have been described extensively i n several publications, along with different methods for estimating these parameters (Pressing, 1998; Pressing and Jolley-Rogers, 1997; Schulze and Vorberg, 2002; Vorberg and Schulze, 2002; Vorberg and Wing, 1996). To estimate the parameters, w e established the equivalence between the first-order synchronization model and an ARMA(1, 1) process of the asynchronies (see Appendix). This approach, based on earlier work by Vorberg and Wing (1996), allows the estimation of the error correction parameter a, as well as of the motor and central variances, with standard estimation methods. In addition, second-order (AR2) models were also fit to the data, and the relative adequacy of the AR1 form of the model was tested against AR2 optional formulations.

Although the equations are linear, their application to real data requires considerable care for three main reasons. First, it is necessary to circumnavigate certain parameter ranges that exhibit near indeterminacy of a solution (Pressing, 1998; Schulze and Vorberg, 2002; Vorberg and Schulze, 2002). This effect was kept to a minimum by excluding trials in which the autocovariance function showed no significant deviation from zero for lags one through five since the model fit for these data would fall in the area of near indeterminacy. We also acquired converging results using the method of bins (Pressing, 1998). This method yields a more robust estimate of the error correction parameter, but requires that the size of the motor delay variance be specified a priori.

Second, the validity of the estimates is related to the number of intervals produced on each trail. Without error correction, stable estimates of clock and motor variability can be obtained with short runs of 20 to 40 taps (Wing and Kristofferson, 1973). However, such lengths are completely inadequate when error correction is involved. An order of magnitude longer is essential (at least 200 taps, depending on the consistency of the performer's control). Our run lengths were chosen with this in mind.

Third, the data analysis presumes that the same control process is used over the course of the run, a phenomenon that is described as stationarity. If a run is markedly nonstationary, then the estimation technique may exhibit significant biases. This is not a significant problem with younger control groups, but patient groups are selected precisely due to problems in their control processes, and nonstationarity may be more likely with them (Wilson et al., 2002). This issue can be handled in part by comparing parameter estimates in different sections of each run and discarding runs that are inconsistent. Runs with a notably poor model fit are also likely to be nonstationary.

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