- Email: [email protected]
Decoupling tremor and volitional force control factors in analyses of motor task performance
Computational Statistics & Data Analysis 37 (2001) 363–372 www.elsevier.com/locate/csda
Decoupling tremor and volitional force control factors in analyses of motor task performance T. Eakina; b;∗ , K. Francisa; c , W. Spirdusoa; c a
Institute of Gerontology, The University of Texas at Austin, Austin, TX 78712, USA Academic Computing and Instructional Technology Services, The University of Texas at Austin, Austin, TX 78712, USA c Department of Kinesiology and Health Education, The University of Texas at Austin, Austin, TX 78712, USA b
Abstract A frequency domain -ltering protocol is described for separating force contributions from tremor and -ne motor control when both factors are present during performance of a goal-oriented task. This method partitions the frequency domain after Fourier transformation of time series data. Amplitudes within a range are divided by or substituted with analogous values from a de-ned task function to obtain representations of the contribution of a particular factor or that remaining in the absence of the factor. Filtered data can then be used for independent characterization of each factor with respect to c 2001 Elsevier Science B.V. All rights reserved. frequency power spectra or time series properties. Keywords: Motor control; Tremor; Force regulation; Decoupling
1. Introduction Motor control tasks extending over a time interval provide useful information in the study of biomechanics, neurophysiology, kinesiology, and other related areas. Analysis of performance of such tasks can give insight into mechanisms, inspire or validate motor behavior models, provide clinical classi-cation standards, and serve as diagnostic criteria. Motor control tasks can also be used to monitor longitudinal changes in abilities during aging, response to intervention, or progression of pathologies. ∗ Corresponding author. E-mail address: [email protected] (T. Eakin).
c 2001 Elsevier Science B.V. All rights reserved. 0167-9473/01/$ - see front matter PII: S 0 1 6 7 - 9 4 7 3 ( 0 1 ) 0 0 0 1 9 - 6
364
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
Hand function, de-ned to a large extent by dexterity and -ne motor control of the thumb and -ngers, is an area of study in which analysis of motor task performance can be of signi-cant importance (Spirduso and Choi, 1993). Various aspects of function can be assessed, but one of the most sensitive is the measurement of -ne motor control associated with digits, including coordination during tasks requiring accuracy in movement or precision in the regulation of isometric force. Regulation of force during task-driven intentional maneuvering is the primary factor that a?ects performance, but this can be impacted or even dominated in some cases by the concurrent presence of tremor—a phenomenon of the nervous system, manifested as an involuntary rhythmic activity in a body part that can be of physiological or of neuropathological origin. Even though hand function plays a critical role in negotiating tasks of daily living, and even though tremor exists to a noticeable degree among many in the elderly population (Elble, 1998), no assessment methodology has yet been developed that quanti-es -ne motor control of the hands, tremor, and the way that they interact. Although quantitative methods for the analysis of tremor alone have been developed (Deuschl et al., 1995; Riviere et al., 1997; O’Suilleabhain and Matsumoto, 1998; Forssberg et al., 2000), the lack of a quantitative method that can assess accurately and objectively the characteristics of tremor and -ne motor control when both factors are manifested simultaneously has been an impediment in the understanding of hand control and tremor, and in the comparison of the results of various treatments and interventions from one study to another. Here we demonstrate a new approach to analyzing each of the underlying components of motor control separately from each other, using examples from a goal-oriented task setting.
2. Contributing factors of motor output The aggregate manifestation of underlying components of a kinematic variable is usually observable directly and is recorded as a time series to serve as the basis for analysis of task performance. As a generalization that is applicable to a broad range of motor control tasks and data acquisition systems, of which our focused interest on isometric force regulation is an example, consider a single time dependent variable, g(t), sampled over a total task duration time T . Suppose that the objective of the task is to create a speci-ed pro-le of the dependent variable as a function of the independent variable. In our generalized prototype we can call this task objective f(t). The subject performing the task can be considered as a “black box,” having an unknown infrastructure that expresses itself during task performance as determined by any de-ciencies or pathologies related to motor performance in the form of a transfer function h(t). It is this transfer function that, through interaction with the input task function f(t), produces the output function g(t), which are the data that are directly observed and recorded. This interaction can be expressed mathematically as a time domain convolution f(t) ∗ h(t) = g(t)
(1)
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
365
which, for continuous or analog data is equivalent to g(t) = f( − t)h() d
(2)
or, for discrete or digital data is equivalent to g(t) = f(t − i)h(i):
(3)
i
Because the task objective f(t) is de-ned by its design and the output g(t) is directly recorded, it is the transfer function h(t) attributable to the task performer which is the unknown factor of Eq. (1). In the time domain, this factor can be extracted by deconvolution but such a process may be quite complicated. The usual procedure is to recast this equation into the frequency domain via a Fourier transform, de-ned by 1 X (!) = √ e−i!t x(t) dt; (4) 2 where the equivalent of time domain convolution is a simple multiplication and likewise the equivalent of time domain deconvolution is a simple division. Transformation of Eq. (1) to the frequency domain thus gives F(!)H (!) = G(!):
(5)
From Eq. (5) the transfer function H (!) attributable to the task performer in the frequency domain is the ratio of the transformed output with respect to the transformed task function, i.e., G(!)=F(!). This contribution from the task performer is thus expressed as complex-valued amplitudes of the frequency spectrum, which can be partitioned into non-overlapping segments Hk that span the entire spectrum. For example, a two region partition would be composed of H1 covering the domain 0 6 ! ¡ !1 and H2 covering the domain !1 6 ! ¡ ∞. The contribution of each of these partitioned regions, decoupled from the contributions of all the others, can subsequently be obtained with an inverse transformation back to the time domain 1 hk (t) = √ ei!t Hk (!) d!: (6) 2 Conversely, this partitioning method can generate a representation of task performance in the absence of a particular factor hk . Such hypothetical data are obtained by substituting the corresponding complex amplitudes of the task function F(!) for those of the aggregate function G(!) at all frequencies within the range characterizing the factor. The substitution is equivalent to dividing the aggregate amplitude by the factor amplitude since, by Eq. (5), the task function F(!) is the ratio G(!)=H (!). A modi-ed frequency domain function Gk (!) is created by such substitution, representing task performance in the absence of the factor Hk . The corresponding time series for force values in the absence of factor hk can then be generated with an inverse Fourier transform of Gk (!) back to the time domain 1 gk (t) = √ ei!t Gk (!) d!: (7) 2
366
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
3. Application of the decoupling method The decoupling concept is conveniently demonstrated with a speci-c example. Force magnitude data were collected over the duration of an isometric task performed by a patient diagnosed with Parkinson’s disease on instrumentation designed and constructed for this purpose: the Manual Force Quanti-cation System (MFQS), which will be described in more detail in a subsequent paper (Spirduso et al., in preparation). The objective of the task is to trace a diagonal line on a computer generated template with a cursor by applying or releasing isometric force (in a range from 0.98 to 3.43 N) at two transducers using the thumb and the index -nger in a pinching position. The force signals from the thumb transducer control the x-axis movement of the cursor and the signals from the index -nger transducer control the y-axis movement. Perfect neuromuscular coordination of the thumb and index -nger ◦ while tracing a 45 line template would result in equal and simultaneous productions of force by the two digits. Force–time plots recorded simultaneously by the thumb and index -nger transducers during the particular trial chosen for illustration are shown in Fig. 1. The patient’s tremor as well as the deviation from perfect force regulation, especially by the index -nger, are clearly observed. The frequency power
Fig. 1. Force–time plots for the thumb and index -nger during performance of a tracing task, showing aggregate manifestation of tremor and -ne motor control regulation.
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
367
Fig. 2. Frequency domain power spectra for the thumb and index -nger data displayed in Fig. 1. The range containing the tremor signal is similar for the two digits.
spectra of each of these, obtained from Fourier transforms over N samplings, is Gi (!)Gi∗ (!) ; (8) N where the asterisk superscript indicates a complex conjugate, as shown in Fig. 2. For this and subsequent signal processing procedures, the Fast Fourier Transform (FFT) and related algorithms as implemented in Matlab 4.2 (Mathworks, Inc) for Sun Microsystem’s Solaris 2.6 operating system were used. As can be seen in this -gure, the frequency range spanned by the tremor signal is similar for the two digits, thumb and index -nger. The function f(t) describing the task is dependent on the temporal strategy being used with respect to task performance and will be related to the total time duration T and the magnitude of the force di?erence over the tracing template L, i.e., the di?erence in force exerted at the beginning and at the end of the task. Three example possibilities are (1) performing the task by applying force at a uniform velocity (L=T ); (2) applying force with a uniform deceleration (−2L=T 2 ); and (3) applying force with a uniform acceleration (2L=T 2 ). These constant values for the respective variables will cover the same time span T in traversing the same force di?erential L as was observed in the actual recording of g(t). The corresponding Pi (!) =
368
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
Fig. 3. Force–time plots for isolated isometric force components, extracted from index -nger data displayed in Fig. 1. The frequency range used for tremor was 5 –7 Hz. (a) Tremor component, (b) low frequency intentional force control component.
power spectra for these task paradigms are almost the same, with di?erences at each particular frequency being 14 or more orders of magnitude smaller than the amplitudes themselves. Although the subject performing the task was not given explicit instructions concerning temporal strategy, task instructions were to complete the tracing as quickly as possible while maintaining geometrical accuracy. The relative insensitivity of the task function frequency power spectrum in terms of the velocity, acceleration, or deceleration being kept constant would indicate that similar strategies with smooth monotonically increasing force values would also be very similar. Thus, for demonstrative purposes, consider the task function f(t) to be that of constant velocity over the task duration time, i.e., (L=T )t. Assuming this task function for the experimental data shown in Fig. 1, the tremor component can be decoupled to obtain the Fourier transform F(!) needed for use in Eq. (5). The tremor, isolated from task performance, that was exhibited by the index -nger in the range 5 –7 Hz typical for Parkinson tremor, is shown in Fig. 3a using the same scaling as in Fig. 1. Similarly, the pro-le of force control regulation by the index -nger, isolated from the task performance, exhibited in the frequency range below 5 Hz is shown in Fig. 3b, again using the same scaling as in Fig. 1. In addition to extracting these components we can see what the actual performance would have been in their respective
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
369
Fig. 4. Force–time plots of the experimental index -nger data in Fig. 1 from which a component of isometric force has been removed by the decoupling procedure. The displays represent task performance at constant velocity of increase in force application had the particular component been the only control dysfunction. (a) Tremor during task performance with intentional force control contribution removed, (b) task performance with tremor contribution removed.
absences using Eq. (7). Thus, substituting F(!) for G(!) wherever ! is less than the tremor range beginning at 5 Hz and at frequencies greater than the 7 Hz upper limit for the Parkinson’s tremor range, followed by inverse Fourier transform, gives the representations of performance in the absence of any -ne motor control dysfunction. A force–time plot derived from the experimental index -nger data in Fig. 1 showing such a hypothetical performance is presented in Fig. 4a. Conversely, substituting F(!) for G(!) where ! lies above the 5 Hz lower limit of the Parkinson’s tremor range, followed by inverse Fourier transform, gives the representations of performance with the frequencies associated with tremor removed. A force–time plot showing such a hypothetical performance by the index -nger is presented in Fig. 4b. When making frequency domain divisions and substitutions as discussed above in the illustrative examples it is important to be aware of the fact that the FFT algorithm produces a vector where complex conjugates of elements in the lower half subsequent to the -rst are assigned to related positions in the upper half. Thus, any division or substitution manipulation involving element x ¿ 1 in an FFT vector of length N must also have a corresponding manipulation applied to element (N + 2 − x).
370
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
4. Limitations The methodology presented here for separating force control components was developed speci-cally for use in situations similar to the example given above, in which the task is isometric and de-ned within the context of the Manual Force Quanti-cation System instrumentation, and in which the tremor perturbations have frequencies typical of those manifested in Parkinson’s disease. Although the procedure can also be applied more generally in the analyses of motoric data where concurrent volitional and involuntary components are present, several factors and limitations must be considered. These include the number of data samplings during the task, the nature of the task and variables being measured, and the nature of the concurrent tremor. The number and acquisition rate of sampling data points during the task have to be suLcient for the FFT algorithm to accommodate the entire output frequency range of interest. Because that algorithm only considers frequencies at equally spaced intervals up to the Nyquist frequency, a sampling rate more than twice the highest output frequency of interest is needed. The data acquisition duration, i.e., the number of samplings during the task, needs to be suLcient to include multiple samplings within a period of any oscillatory component and suLcient to provide the desired output frequency resolution. For example, a sampling over n seconds in the time domain will result in a resolution of 1=n Hz in the frequency domain. It is also important that the sampling data vector be long enough such that any continuous-valued task function not be distorted substantially by a -nite digital representation in the frequency domain after FFT transformation. The applicability of the decoupling method is contingent upon the task function being well-de-ned over the entire frequency spectrum and upon perturbations of di?erent origins being separable in the frequency domain. The task objectives must thus be speci-ed both in a spatial sense and in a temporal sense so that the variable values associated with the task itself can be determined explicitly at any particular sampling time. This need can be addressed in the task design itself, but there may be diLculty in terms of presenting instructions to subjects doing the task. Subjects may also have less sensory input cues for observing temporal requirements than for observing geometrical requirements. The major obstacle to using the decoupling method is probably the uncertainty introduced when the frequency ranges of perturbations from di?erent origins are not disjoint. Although the underlying mechanisms of volitional motor control are not understood completely, the frequency output for routine high precision tasks seems to be at the low end of the spectrum, less than 4 Hz. Tremors also have underlying mechanisms that are not completely understood but generally involve output at higher frequency ranges. For example, Parkinson’s disease tremor typically has a frequency in the 5 –7 Hz range. Essential tremor has a range a bit higher on the frequency scale, but it still overlaps somewhat with the higher end of the Parkinson’s disease tremor range (Deuschl, 1999). Physiological tremor, on the other hand, has yet a higher frequency range which does not signi-cantly overlap with the ranges of neuropathological tremors. It is not likely that Parkinson tremor and essential tremor would
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
371
be manifested simultaneously, but there are certain caveats concerning overlapping frequency ranges that are related to the possible presence of higher frequency output of volitional origin. We assume that the voluntary component of task performance is executed so as to generate an output as close to the theoretical objective as possible. There may be situational circumstances that a?ect this component though, which may introduce higher frequency output in ranges characteristic of various tremors. This could be caused by an attempt at recovery from a large geometric displacement away from the ideal, or from the negotiation of a complex maneuver if there is a complicated element in the task design. A sharp angular change in task geometry might introduce sudden starting and stopping accelerations and decelerations, for example. Another source of caution pointed out by a reviewer is the evidence for an 8–10 Hz pulsatile output associated with slow -nger movements (Valbo and Wessberg, 1993). Whether this is considered to be an element of volitional movement or is considered as separate involuntary movement, it would in either case introduce a problem if tremor were simultaneously present. 5. Conclusion Although the example illustrates interesting aspects related to the clinical status of the Parkinson’s disease patient who performed the task, the focus here is on the quantitative methodology of processing the recorded data. A separate case study which addresses interpretation of time series data in biological and clinical contexts will be presented in a later paper (Francis et al., in preparation). The quantitative concern is the development of a way to segregate the contributions of independent force regulation factors manifested in a goal-oriented task such that they can be analyzed independently of each other. Previously, researchers have studied task paradigms in which postural or rest tremor is the only contributing factor in movement (Gantert et al., 1992; Timmer et al., 1993). In those open ended passive tasks, the time series of kinematic variables, primarily positional acceleration, were shown to be amenable to various methods of time series analysis that could distinguish stochastic from deterministic behavior and characterize the dimensionality of the underlying oscillator. Conversely, there have been, previously, quantitative studies of performance using task paradigms in which intentional -ne motor control is required and in circumstances where tremor is not normally expected to be present (Bronte-Stewart et al., 2000) providing characterization of interdigit coordination in terms of force regulation. However, there have not been reports of quantitative analysis of task performance where time series recordings are confounded by the presence of both voluntary -ne motor control output and tremor. A process for quantitative analysis of time series pro-les containing contributions from both of these is needed, particularly in cases where each may be signi-cant, e.g., goal-oriented task performance by elderly patients with both neuropathological tremor and age-related impairment of -ne motor control. The method described here, in which factors are separated in the frequency domain, is a straight forward way of approaching this need on a quantitative basis, although
372
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
further statistical based methods will be needed for detailed inferential analyses. It uses standard principles of engineering mechanics and eliminates errors that would be introduced by interpolations with curve smoothing to get an underlying envelope or by approximations that would be present in least squares curve -tting of linear superposition of parametric functions. Acknowledgements We wish to thank Professor R. Barr for his consulting assistance with signal processing issues, the Reeves Rehabilitation Clinic at the University of Texas Health Science Center at San Antonio for providing access to patients with Parkinson’s disease, the patients themselves, and Dr. P. New for supervising and facilitating the acquisition of the experimental data. We also appreciate the helpful comments of reviewers, in particular their suggestions for ways in which the limitations inherent in the methodology could be addressed. References Bronte-Stewart, H., Ding, L., Alexander, C., Zhou, Y., Moore, G., 2000. Quantitative digitography (QDG): a sensitive measure of digital motor control in idiopathic Parkinson’s disease. Mov. Disord. 15, 36 – 47. Deuschl, G., 1999. Neurophysiological tests for the assessment of tremors. In: Stern, G. (Ed.), Parkinson’s Disease: Advances in Neurology, Vol. 80. Lippincott, Williams, & Wilkins, Philadelphia, PA. Deuschl, G., Lauk, M., Timmer, J., 1995. Tremor classi-cation and tremor time series analysis. Chaos 5, 48–51. Elble, R., 1998. Tremor in ostensibly normal elderly people. Mov. Disord. 13, 457– 464. Forssberg, H., Ingvarsson, P., Iwasaki, N., Johansson, R., Gordon, A., 2000. Action tremor during object manipulation in Parkinson’s disease. Mov. Disord. 15, 244 –254. Francis, K., Eakin, T., Spirduso, W., manuscript in preparation. Gantert, C., Honerkamp, J., Timmer, J., 1992. Analyzing the dynamics of hand tremor time series. Biol. Cybernet. 66, 479– 484. O’Suilleabhain, P., Matsumoto, J., 1998. Time-frequency analysis of tremors. Brain 121, 2127–2134. Riviere, C., Reich, S., Thakor, N., 1997. Adaptive Fourier modeling for quanti-cation of tremor. J. Neurosci. Methods 74, 77–87. Spirduso, W.W., Choi, J., 1993. Age and practice e?ects on force control of the thumb and index -nger in precision pinch and bilateral coordination. In: Stelmach, G.E., Homberg, V. (Eds.), Sensorimotor Impairment in the Elderly. Kluwer Academic Publishers, The Netherlands. Spirduso, W., Francis, K., Eakin, T., Stanford C., manuscript in preparation. Timmer, J., Gantert, C., Deuschl, G., Honerkamp, J., 1993. Characteristics of hand tremor time series. Biol. Cybernet. 70, 75– 80. S Valbo, A.B., Wessberg, J., 1993. Organization of motor output in slow -nger movements in man. J. Physiol. London 469, 673– 691.
Decoupling tremor and volitional force control factors in analyses of motor task performance T. Eakina; b;∗ , K. Francisa; c , W. Spirdusoa; c a
Institute of Gerontology, The University of Texas at Austin, Austin, TX 78712, USA Academic Computing and Instructional Technology Services, The University of Texas at Austin, Austin, TX 78712, USA c Department of Kinesiology and Health Education, The University of Texas at Austin, Austin, TX 78712, USA b
Abstract A frequency domain -ltering protocol is described for separating force contributions from tremor and -ne motor control when both factors are present during performance of a goal-oriented task. This method partitions the frequency domain after Fourier transformation of time series data. Amplitudes within a range are divided by or substituted with analogous values from a de-ned task function to obtain representations of the contribution of a particular factor or that remaining in the absence of the factor. Filtered data can then be used for independent characterization of each factor with respect to c 2001 Elsevier Science B.V. All rights reserved. frequency power spectra or time series properties. Keywords: Motor control; Tremor; Force regulation; Decoupling
1. Introduction Motor control tasks extending over a time interval provide useful information in the study of biomechanics, neurophysiology, kinesiology, and other related areas. Analysis of performance of such tasks can give insight into mechanisms, inspire or validate motor behavior models, provide clinical classi-cation standards, and serve as diagnostic criteria. Motor control tasks can also be used to monitor longitudinal changes in abilities during aging, response to intervention, or progression of pathologies. ∗ Corresponding author. E-mail address: [email protected] (T. Eakin).
c 2001 Elsevier Science B.V. All rights reserved. 0167-9473/01/$ - see front matter PII: S 0 1 6 7 - 9 4 7 3 ( 0 1 ) 0 0 0 1 9 - 6
364
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
Hand function, de-ned to a large extent by dexterity and -ne motor control of the thumb and -ngers, is an area of study in which analysis of motor task performance can be of signi-cant importance (Spirduso and Choi, 1993). Various aspects of function can be assessed, but one of the most sensitive is the measurement of -ne motor control associated with digits, including coordination during tasks requiring accuracy in movement or precision in the regulation of isometric force. Regulation of force during task-driven intentional maneuvering is the primary factor that a?ects performance, but this can be impacted or even dominated in some cases by the concurrent presence of tremor—a phenomenon of the nervous system, manifested as an involuntary rhythmic activity in a body part that can be of physiological or of neuropathological origin. Even though hand function plays a critical role in negotiating tasks of daily living, and even though tremor exists to a noticeable degree among many in the elderly population (Elble, 1998), no assessment methodology has yet been developed that quanti-es -ne motor control of the hands, tremor, and the way that they interact. Although quantitative methods for the analysis of tremor alone have been developed (Deuschl et al., 1995; Riviere et al., 1997; O’Suilleabhain and Matsumoto, 1998; Forssberg et al., 2000), the lack of a quantitative method that can assess accurately and objectively the characteristics of tremor and -ne motor control when both factors are manifested simultaneously has been an impediment in the understanding of hand control and tremor, and in the comparison of the results of various treatments and interventions from one study to another. Here we demonstrate a new approach to analyzing each of the underlying components of motor control separately from each other, using examples from a goal-oriented task setting.
2. Contributing factors of motor output The aggregate manifestation of underlying components of a kinematic variable is usually observable directly and is recorded as a time series to serve as the basis for analysis of task performance. As a generalization that is applicable to a broad range of motor control tasks and data acquisition systems, of which our focused interest on isometric force regulation is an example, consider a single time dependent variable, g(t), sampled over a total task duration time T . Suppose that the objective of the task is to create a speci-ed pro-le of the dependent variable as a function of the independent variable. In our generalized prototype we can call this task objective f(t). The subject performing the task can be considered as a “black box,” having an unknown infrastructure that expresses itself during task performance as determined by any de-ciencies or pathologies related to motor performance in the form of a transfer function h(t). It is this transfer function that, through interaction with the input task function f(t), produces the output function g(t), which are the data that are directly observed and recorded. This interaction can be expressed mathematically as a time domain convolution f(t) ∗ h(t) = g(t)
(1)
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
365
which, for continuous or analog data is equivalent to g(t) = f( − t)h() d
(2)
or, for discrete or digital data is equivalent to g(t) = f(t − i)h(i):
(3)
i
Because the task objective f(t) is de-ned by its design and the output g(t) is directly recorded, it is the transfer function h(t) attributable to the task performer which is the unknown factor of Eq. (1). In the time domain, this factor can be extracted by deconvolution but such a process may be quite complicated. The usual procedure is to recast this equation into the frequency domain via a Fourier transform, de-ned by 1 X (!) = √ e−i!t x(t) dt; (4) 2 where the equivalent of time domain convolution is a simple multiplication and likewise the equivalent of time domain deconvolution is a simple division. Transformation of Eq. (1) to the frequency domain thus gives F(!)H (!) = G(!):
(5)
From Eq. (5) the transfer function H (!) attributable to the task performer in the frequency domain is the ratio of the transformed output with respect to the transformed task function, i.e., G(!)=F(!). This contribution from the task performer is thus expressed as complex-valued amplitudes of the frequency spectrum, which can be partitioned into non-overlapping segments Hk that span the entire spectrum. For example, a two region partition would be composed of H1 covering the domain 0 6 ! ¡ !1 and H2 covering the domain !1 6 ! ¡ ∞. The contribution of each of these partitioned regions, decoupled from the contributions of all the others, can subsequently be obtained with an inverse transformation back to the time domain 1 hk (t) = √ ei!t Hk (!) d!: (6) 2 Conversely, this partitioning method can generate a representation of task performance in the absence of a particular factor hk . Such hypothetical data are obtained by substituting the corresponding complex amplitudes of the task function F(!) for those of the aggregate function G(!) at all frequencies within the range characterizing the factor. The substitution is equivalent to dividing the aggregate amplitude by the factor amplitude since, by Eq. (5), the task function F(!) is the ratio G(!)=H (!). A modi-ed frequency domain function Gk (!) is created by such substitution, representing task performance in the absence of the factor Hk . The corresponding time series for force values in the absence of factor hk can then be generated with an inverse Fourier transform of Gk (!) back to the time domain 1 gk (t) = √ ei!t Gk (!) d!: (7) 2
366
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
3. Application of the decoupling method The decoupling concept is conveniently demonstrated with a speci-c example. Force magnitude data were collected over the duration of an isometric task performed by a patient diagnosed with Parkinson’s disease on instrumentation designed and constructed for this purpose: the Manual Force Quanti-cation System (MFQS), which will be described in more detail in a subsequent paper (Spirduso et al., in preparation). The objective of the task is to trace a diagonal line on a computer generated template with a cursor by applying or releasing isometric force (in a range from 0.98 to 3.43 N) at two transducers using the thumb and the index -nger in a pinching position. The force signals from the thumb transducer control the x-axis movement of the cursor and the signals from the index -nger transducer control the y-axis movement. Perfect neuromuscular coordination of the thumb and index -nger ◦ while tracing a 45 line template would result in equal and simultaneous productions of force by the two digits. Force–time plots recorded simultaneously by the thumb and index -nger transducers during the particular trial chosen for illustration are shown in Fig. 1. The patient’s tremor as well as the deviation from perfect force regulation, especially by the index -nger, are clearly observed. The frequency power
Fig. 1. Force–time plots for the thumb and index -nger during performance of a tracing task, showing aggregate manifestation of tremor and -ne motor control regulation.
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
367
Fig. 2. Frequency domain power spectra for the thumb and index -nger data displayed in Fig. 1. The range containing the tremor signal is similar for the two digits.
spectra of each of these, obtained from Fourier transforms over N samplings, is Gi (!)Gi∗ (!) ; (8) N where the asterisk superscript indicates a complex conjugate, as shown in Fig. 2. For this and subsequent signal processing procedures, the Fast Fourier Transform (FFT) and related algorithms as implemented in Matlab 4.2 (Mathworks, Inc) for Sun Microsystem’s Solaris 2.6 operating system were used. As can be seen in this -gure, the frequency range spanned by the tremor signal is similar for the two digits, thumb and index -nger. The function f(t) describing the task is dependent on the temporal strategy being used with respect to task performance and will be related to the total time duration T and the magnitude of the force di?erence over the tracing template L, i.e., the di?erence in force exerted at the beginning and at the end of the task. Three example possibilities are (1) performing the task by applying force at a uniform velocity (L=T ); (2) applying force with a uniform deceleration (−2L=T 2 ); and (3) applying force with a uniform acceleration (2L=T 2 ). These constant values for the respective variables will cover the same time span T in traversing the same force di?erential L as was observed in the actual recording of g(t). The corresponding Pi (!) =
368
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
Fig. 3. Force–time plots for isolated isometric force components, extracted from index -nger data displayed in Fig. 1. The frequency range used for tremor was 5 –7 Hz. (a) Tremor component, (b) low frequency intentional force control component.
power spectra for these task paradigms are almost the same, with di?erences at each particular frequency being 14 or more orders of magnitude smaller than the amplitudes themselves. Although the subject performing the task was not given explicit instructions concerning temporal strategy, task instructions were to complete the tracing as quickly as possible while maintaining geometrical accuracy. The relative insensitivity of the task function frequency power spectrum in terms of the velocity, acceleration, or deceleration being kept constant would indicate that similar strategies with smooth monotonically increasing force values would also be very similar. Thus, for demonstrative purposes, consider the task function f(t) to be that of constant velocity over the task duration time, i.e., (L=T )t. Assuming this task function for the experimental data shown in Fig. 1, the tremor component can be decoupled to obtain the Fourier transform F(!) needed for use in Eq. (5). The tremor, isolated from task performance, that was exhibited by the index -nger in the range 5 –7 Hz typical for Parkinson tremor, is shown in Fig. 3a using the same scaling as in Fig. 1. Similarly, the pro-le of force control regulation by the index -nger, isolated from the task performance, exhibited in the frequency range below 5 Hz is shown in Fig. 3b, again using the same scaling as in Fig. 1. In addition to extracting these components we can see what the actual performance would have been in their respective
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
369
Fig. 4. Force–time plots of the experimental index -nger data in Fig. 1 from which a component of isometric force has been removed by the decoupling procedure. The displays represent task performance at constant velocity of increase in force application had the particular component been the only control dysfunction. (a) Tremor during task performance with intentional force control contribution removed, (b) task performance with tremor contribution removed.
absences using Eq. (7). Thus, substituting F(!) for G(!) wherever ! is less than the tremor range beginning at 5 Hz and at frequencies greater than the 7 Hz upper limit for the Parkinson’s tremor range, followed by inverse Fourier transform, gives the representations of performance in the absence of any -ne motor control dysfunction. A force–time plot derived from the experimental index -nger data in Fig. 1 showing such a hypothetical performance is presented in Fig. 4a. Conversely, substituting F(!) for G(!) where ! lies above the 5 Hz lower limit of the Parkinson’s tremor range, followed by inverse Fourier transform, gives the representations of performance with the frequencies associated with tremor removed. A force–time plot showing such a hypothetical performance by the index -nger is presented in Fig. 4b. When making frequency domain divisions and substitutions as discussed above in the illustrative examples it is important to be aware of the fact that the FFT algorithm produces a vector where complex conjugates of elements in the lower half subsequent to the -rst are assigned to related positions in the upper half. Thus, any division or substitution manipulation involving element x ¿ 1 in an FFT vector of length N must also have a corresponding manipulation applied to element (N + 2 − x).
370
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
4. Limitations The methodology presented here for separating force control components was developed speci-cally for use in situations similar to the example given above, in which the task is isometric and de-ned within the context of the Manual Force Quanti-cation System instrumentation, and in which the tremor perturbations have frequencies typical of those manifested in Parkinson’s disease. Although the procedure can also be applied more generally in the analyses of motoric data where concurrent volitional and involuntary components are present, several factors and limitations must be considered. These include the number of data samplings during the task, the nature of the task and variables being measured, and the nature of the concurrent tremor. The number and acquisition rate of sampling data points during the task have to be suLcient for the FFT algorithm to accommodate the entire output frequency range of interest. Because that algorithm only considers frequencies at equally spaced intervals up to the Nyquist frequency, a sampling rate more than twice the highest output frequency of interest is needed. The data acquisition duration, i.e., the number of samplings during the task, needs to be suLcient to include multiple samplings within a period of any oscillatory component and suLcient to provide the desired output frequency resolution. For example, a sampling over n seconds in the time domain will result in a resolution of 1=n Hz in the frequency domain. It is also important that the sampling data vector be long enough such that any continuous-valued task function not be distorted substantially by a -nite digital representation in the frequency domain after FFT transformation. The applicability of the decoupling method is contingent upon the task function being well-de-ned over the entire frequency spectrum and upon perturbations of di?erent origins being separable in the frequency domain. The task objectives must thus be speci-ed both in a spatial sense and in a temporal sense so that the variable values associated with the task itself can be determined explicitly at any particular sampling time. This need can be addressed in the task design itself, but there may be diLculty in terms of presenting instructions to subjects doing the task. Subjects may also have less sensory input cues for observing temporal requirements than for observing geometrical requirements. The major obstacle to using the decoupling method is probably the uncertainty introduced when the frequency ranges of perturbations from di?erent origins are not disjoint. Although the underlying mechanisms of volitional motor control are not understood completely, the frequency output for routine high precision tasks seems to be at the low end of the spectrum, less than 4 Hz. Tremors also have underlying mechanisms that are not completely understood but generally involve output at higher frequency ranges. For example, Parkinson’s disease tremor typically has a frequency in the 5 –7 Hz range. Essential tremor has a range a bit higher on the frequency scale, but it still overlaps somewhat with the higher end of the Parkinson’s disease tremor range (Deuschl, 1999). Physiological tremor, on the other hand, has yet a higher frequency range which does not signi-cantly overlap with the ranges of neuropathological tremors. It is not likely that Parkinson tremor and essential tremor would
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
371
be manifested simultaneously, but there are certain caveats concerning overlapping frequency ranges that are related to the possible presence of higher frequency output of volitional origin. We assume that the voluntary component of task performance is executed so as to generate an output as close to the theoretical objective as possible. There may be situational circumstances that a?ect this component though, which may introduce higher frequency output in ranges characteristic of various tremors. This could be caused by an attempt at recovery from a large geometric displacement away from the ideal, or from the negotiation of a complex maneuver if there is a complicated element in the task design. A sharp angular change in task geometry might introduce sudden starting and stopping accelerations and decelerations, for example. Another source of caution pointed out by a reviewer is the evidence for an 8–10 Hz pulsatile output associated with slow -nger movements (Valbo and Wessberg, 1993). Whether this is considered to be an element of volitional movement or is considered as separate involuntary movement, it would in either case introduce a problem if tremor were simultaneously present. 5. Conclusion Although the example illustrates interesting aspects related to the clinical status of the Parkinson’s disease patient who performed the task, the focus here is on the quantitative methodology of processing the recorded data. A separate case study which addresses interpretation of time series data in biological and clinical contexts will be presented in a later paper (Francis et al., in preparation). The quantitative concern is the development of a way to segregate the contributions of independent force regulation factors manifested in a goal-oriented task such that they can be analyzed independently of each other. Previously, researchers have studied task paradigms in which postural or rest tremor is the only contributing factor in movement (Gantert et al., 1992; Timmer et al., 1993). In those open ended passive tasks, the time series of kinematic variables, primarily positional acceleration, were shown to be amenable to various methods of time series analysis that could distinguish stochastic from deterministic behavior and characterize the dimensionality of the underlying oscillator. Conversely, there have been, previously, quantitative studies of performance using task paradigms in which intentional -ne motor control is required and in circumstances where tremor is not normally expected to be present (Bronte-Stewart et al., 2000) providing characterization of interdigit coordination in terms of force regulation. However, there have not been reports of quantitative analysis of task performance where time series recordings are confounded by the presence of both voluntary -ne motor control output and tremor. A process for quantitative analysis of time series pro-les containing contributions from both of these is needed, particularly in cases where each may be signi-cant, e.g., goal-oriented task performance by elderly patients with both neuropathological tremor and age-related impairment of -ne motor control. The method described here, in which factors are separated in the frequency domain, is a straight forward way of approaching this need on a quantitative basis, although
372
T. Eakin et al. / Computational Statistics & Data Analysis 37 (2001) 363–372
further statistical based methods will be needed for detailed inferential analyses. It uses standard principles of engineering mechanics and eliminates errors that would be introduced by interpolations with curve smoothing to get an underlying envelope or by approximations that would be present in least squares curve -tting of linear superposition of parametric functions. Acknowledgements We wish to thank Professor R. Barr for his consulting assistance with signal processing issues, the Reeves Rehabilitation Clinic at the University of Texas Health Science Center at San Antonio for providing access to patients with Parkinson’s disease, the patients themselves, and Dr. P. New for supervising and facilitating the acquisition of the experimental data. We also appreciate the helpful comments of reviewers, in particular their suggestions for ways in which the limitations inherent in the methodology could be addressed. References Bronte-Stewart, H., Ding, L., Alexander, C., Zhou, Y., Moore, G., 2000. Quantitative digitography (QDG): a sensitive measure of digital motor control in idiopathic Parkinson’s disease. Mov. Disord. 15, 36 – 47. Deuschl, G., 1999. Neurophysiological tests for the assessment of tremors. In: Stern, G. (Ed.), Parkinson’s Disease: Advances in Neurology, Vol. 80. Lippincott, Williams, & Wilkins, Philadelphia, PA. Deuschl, G., Lauk, M., Timmer, J., 1995. Tremor classi-cation and tremor time series analysis. Chaos 5, 48–51. Elble, R., 1998. Tremor in ostensibly normal elderly people. Mov. Disord. 13, 457– 464. Forssberg, H., Ingvarsson, P., Iwasaki, N., Johansson, R., Gordon, A., 2000. Action tremor during object manipulation in Parkinson’s disease. Mov. Disord. 15, 244 –254. Francis, K., Eakin, T., Spirduso, W., manuscript in preparation. Gantert, C., Honerkamp, J., Timmer, J., 1992. Analyzing the dynamics of hand tremor time series. Biol. Cybernet. 66, 479– 484. O’Suilleabhain, P., Matsumoto, J., 1998. Time-frequency analysis of tremors. Brain 121, 2127–2134. Riviere, C., Reich, S., Thakor, N., 1997. Adaptive Fourier modeling for quanti-cation of tremor. J. Neurosci. Methods 74, 77–87. Spirduso, W.W., Choi, J., 1993. Age and practice e?ects on force control of the thumb and index -nger in precision pinch and bilateral coordination. In: Stelmach, G.E., Homberg, V. (Eds.), Sensorimotor Impairment in the Elderly. Kluwer Academic Publishers, The Netherlands. Spirduso, W., Francis, K., Eakin, T., Stanford C., manuscript in preparation. Timmer, J., Gantert, C., Deuschl, G., Honerkamp, J., 1993. Characteristics of hand tremor time series. Biol. Cybernet. 70, 75– 80. S Valbo, A.B., Wessberg, J., 1993. Organization of motor output in slow -nger movements in man. J. Physiol. London 469, 673– 691.