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Tracing tremor: Neural correlates of essential tremor and its treatment

Buijink, A.W.G.

Publication date2016Document VersionFinal published version

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Citation for published version (APA):Buijink, A. W. G. (2016). Tracing tremor: Neural correlates of essential tremor and itstreatment.

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Chapter 8 Propranolol in essential tremor affects

motor control in a pattern fitting increased Renshaw inhibition

Submitted as

Propranolol in essential tremor affects motor control in a pattern fitting increased Renshaw inhibition.

AWG Buijink, W Mugge, S Sharifi, NM Maurits, AC Schouten, LJ Bour, AF van Rootselaar.

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Abstract

Background

The tremorolytic site of action of propranolol in essential tremor (ET) is unknown. It could be located peripherally, in the muscle compartments, or centrally. Understanding its mechanism of action will aid in the development of more effective treatments.

Methods

To assess the effect of propranolol on motor control, 16 propranolol sensitive ET patients were tested on and off propranolol. Using a wrist manipulator, ramp-and-hold stretches were applied. M1 and M2 reflex responses were quantified. Continuous wrist perturbations were applied during a passive and two resist tasks (with/without external damping). Closed-loop system identification determined the frequency response, which was fitted with a neuromuscular model, including delayed position and velocity feedback to represent afferent feedback from muscle spindles.

Results

Repeated measures ANOVA revealed significantly decreased afferent feedback and increased intrinsic muscle viscoelasticity during the resist task in the on-medication condition. There was a significant effect of task for afferent feedback and intrinsic muscle viscoelasticity, reflecting varying levels of co-contraction.

Conclusions

Our results suggest increased phasic co-contraction and decreased reflexive activity in ET patients on propranolol.

Significance

Multiple sites of action combined can cause the tremorolytic effect of propranolol, yet only increased Renshaw cell sensitivity explains all findings simultaneously.

Propranolol and essential tremor - 151

Introduction Essential tremor (ET) is one of the most common neurological disorders, and is characterized by a progressive postural and kinetic tremor.4 Propranolol is a non-selective beta-adrenergic receptor antagonist and widely used as first choice of treatment with a level A recommendation of efficacy.9 Several randomized clinical trials show a mean decrease in tremor amplitude of about 50% (see Deuschl et al., 20119 for a review of clinical trials in essential tremor). Nonetheless, the efficacy of propranolol was discovered by coincidence and the drug was originally developed for other diseases. How the tremorolytic action of propranolol is mediated is still unknown. It is of crucial importance to understand the mechanism of action of tremor medications to provide insight into the pathophysiology of ET and possibly to aid in the development of more effective treatment. Other beta-adrenergic blockers are less effective in ET than propranolol.9,11 Propranolol, compared to other beta-adrenergic blockers, has a greater lipid solubility and hence an increased ability to cross the blood-tissue and blood-brain barrier.11,242 First, in the past, it has been suggested that the site of action of propranolol in ET is located in the deep peripheral muscle compartments, specifically at adrenergic receptors on the muscle spindles.11 These compartments have a blood-tissue barrier similar to the blood-brain barrier.242 How changes within the muscle spindles might reduce essential tremor is uncharted. Elble and colleagues suggested there is no evidence for unstable, or oscillating, mechanical reflex activity in ET,243 but propranolol itself might alter the reflex system in such a way that it alleviates tremor. Two other plausible sites of action are located within the central nervous system, considering the widespread presence of adrenergic receptors.12,244 It has been postulated that recurrent inhibition by Renshaw cells (RCs) within the spinal cord is increased by propranolol, effectively filtering 10 Hz muscle oscillations.13 Whether this hypothesis holds for other tremor disorders is unexplored. Another possible mechanism of action is through altering the properties of regions within the cerebello-thalamo-cortical network, within which it is hypothesized that pathological oscillations in ET are present.29

If there exists a site of action within the reflexive system, either peripherally within muscle compartments, or centrally in the spinal cord by affecting RCs sensitivity, this would in both cases lead to altered properties of the stretch reflex in ET patients taking propranolol.

In the current study, stretch reflexes are therefore tested in ET patients who were on and off propranolol medication. Ramp-and-hold stretches and continuous perturbations to the wrist were applied to induce stretch reflexes in a systematic way. With a combination

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of system identification and neuromuscular modeling it is possible to separate intrinsic muscular changes from alterations in reflexive contributions.51,52,245,246 In this way, effects of propranolol on specific parts of the motor system may be dissociated.

Participants & Methods

Participants

Sixteen propranolol sensitive ET patients with a positive family history were tested on and off propranolol medication. All participants were right-handed according to the Annett handedness questionnaire141 and patients were included when they fulfilled the criteria defined by the Tremor Investigation Group,5 they reported a positive subjective response to propranolol, they had a positive family history of at least one affected relative in the immediate family and an onset before the age of 65, a disease duration longer than 5 years and were aged 18 years or over. Exclusion criteria were neurological disorders besides ET and cognitive dysfunction (i.e., Mini-Mental State Examination

Propranolol and essential tremor - 153

A ground electrode was placed on the wrist joint of the contralateral hand. EMG signals were digitally filtered with a high-pass 3rd order Butterworth filter (20 Hz cut-off) to remove motion artefacts, subsequently with a notch filter (45-55 Hz 3rd order Butterworth) to remove line artifacts, rectified, and then digitally low pass filtered (80 Hz 3rd order Butterworth filter, causal filter for determining M1 onsets, non-causal filter for determining M1 and M2 responses247).

Figure 1. Schematic representation of the experimental setup. The subject is

seated in upright position with the forearm fixated allowing unimpeded movement of the wrist. The actuated handle induces wrist rotations to the subject since the axis of rotation of the handle is aligned with the wrist flexion/extension axis. Electromyography electrodes are placed on the flexor and extensor muscles (Adopted from Mugge et al., 2012248).

Ramp-and-hold stretches

Prior to the stretches, subjects were requested to exert maximum voluntary contraction (MVC); the EMG during the ramp-and-hold stretches subsequently was scaled by the EMG at MVC. With the wrist in its neutral position the subject was instructed to maintain a constant contraction at 5% of the MVC force level, in the direction opposite from the stretch perturbation. Two series of nine ramp-and-hold stretch perturbations were applied, one in the flexion, stretching the extensor muscles, and one in the extension direction, stretching the flexor muscles, both with random time intervals between the stretches (minimal interval of 2 seconds, ramp velocity 4.0 rad/s, ramp amplitude 0.14 rad).248,249 Visual feedback of the required force (line) as well as the exerted force, filtered

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at 1 Hz (moving bar) were displayed on a monitor. EMG signals obtained from the individual stretches from the flexor and extensor muscles were averaged. The short latency M1 response was determined as the area-under-the (EMG) curve (AUC) between 20 and 55 ms and the long latency M2 response was determined as the AUC between 55 and 100 ms.250 Onsets of M1 and M2 were identified as the time where the EMG passed the 3-standard deviations line of the average background EMG (determined over the 200

ms period prior to the stretch). Paired t-tests were used to assess differences in stretches in the on and off medication conditions.

Continuous force perturbations

Force perturbations were used to provoke proprioceptive reflexes and enabled the separation of intrinsic and reflexive contributions to human arm dynamics through closed-loop system identification and subsequent neuromuscular modeling.51 During this experiment, subjects had to hold the handle during a passive (“do not intervene”) and resist task (“maintain position”), while continuous pseudo-random force perturbations were applied for 20 seconds. All multisine perturbations were designed in the frequency domain by summing sines of 0.5 up to 20 Hz and optimizing crest factors.251 To facilitate the resist task, the position of the handle was shown on a display as a reference to prevent drift. To minimize possible non-linear effects, the standard deviation of the handle position was equalized and minimized for all conditions and subjects by scaling the torque perturbation magnitudes (appropriate scaling was determined during training252).The following experimental conditions were applied:

- Active, ‘stiff’, condition without external damping. During the active condition, the task instruction was to prevent displacement of the handle by resisting the force.

- Active, ‘stiff’, condition with external damping. External damping is used to provoke reflex activity.51,52 Damping was set to 0.4 Nms/rad.

- Passive condition. Here the task instruction was different from the other two: the subject was asked to relax his/her arm muscles and not to react to the

perturbation.

Each condition was repeated two times, resulting in six trials of 20 seconds each. In between the trials the subject could rest as long as he/she wanted to prevent fatigue. In the time domain, EMG signals were normalized to MVC and averaged over the total measurement time to assess the average muscle activity, a measure of the level of muscle contraction (mean normalized EMG). Closed-loop system identification quantified the

Propranolol and essential tremor - 155

admittance, the frequency domain representation of the displacement or rotation magnitude due to a force or torque; see Figure 2 for an example of a typical subject (next page). The admittance captures the dynamic motor control behavior as a whole and is commonly represented by a magnitude (the amount of displacement) and a phase (the relative timing), hence incorporating the effects of time delays.51

As a second step a basic neuromuscular model was fitted to the experimental data to obtain physiologically interpretable parameters (see Appendix A for the fit criterion and neuromuscular model). The neuromuscular model incorporated a mass-spring-damper, supplemented with time-delayed velocity and position feedback, representing reflexive contributions to motor control.52,245,253 Output parameters of the model are thus hand

inertia i (kg m2), muscle viscosity b (Nms/rad), muscle elasticity k (Nm/rad), and feedback

gains for muscle stretch kp (Nm/rad) and muscle stretch velocity kv (Nms/rad) with a

reflexive time delay td (s).52,254 Hand inertia represents the mass of the entire system, including the hands and fingers. Muscle viscosity represents the non-delayed, instantaneous damping of the muscles i.e. the resistance of the muscle against stretch velocity. Muscle elasticity represents the stiffness of the muscle by co-contraction, which is a non-linear property, i.e. the more active the muscle, the more resistance against stretch. Reflexive properties are subdivided into a position feedback gain and a velocity feedback gain, representing the responses of several afferents providing information about muscle stretch and muscle stretch velocity.51,245 Units and fit boundaries are given in Appendix B. These parameters were compared between task conditions and medication conditions, using a repeated measures ANOVA design, with a significance threshold of 0.05. Due to the explorative nature of this study, we chose not to perform correction for multiple comparisons.255

Results

Patient characteristics

Table 1 provides a full overview of demographic and clinical characteristics. Due to hardware issues, two subjects did not perform the flexion ramp-and-hold stretches (stretching the extensor muscles) and four subjects did not perform the continuous perturbation tasks, leaving 16 ET patients for the extension ramp-and-hold stretches, 14 ET patients for the flexion ramp-and-hold stretches, and 12 ET patients for the continuous perturbations task.

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Table 1. Demographic and clinical characteristics. Extension ramp-

and-hold stretches (n=16)

Flexion ramp-and-hold stretches (n=14)

Continuous perturbation tasks (n=12)

Male/female 8/8 8/6 8/4

Age in years 55.1 (16.1) 53.5 (16.0) 54.5 (18.4)

Disease duration in years

33.7 (17.7) 30.5 (17.7) 34.1 (18.4)

TRS A+B off propranolol

28.3 (11.0) 26.8 (10.8) 25.8 (11.6)

Propranolol effect TRS A+B

- 4.3 (3.0) - 4.1 (3.3) - 3.8 (2.7)

Propranolol dosage in mg

86.8 (76.2) 91.4 (84.5) 95.0 (84.7)

TRS: Tremor Rating Scale. Propranolol effect: improvement determined by difference between TRS A+B on and off propranolol medication. Results are presented in means with standard deviations between parentheses. Characteristics for the ramp-and-hold stretches are derived from all 16 ET patients performing the extension ramp-and-hold stretches.

Figure 2. Task and medication effect on admittance of a typical subject. For each of the three conditions during on and off medication measurements, the magnitude (top panel) and phase (lower panel) of the admittance are shown for the passive (solid), active (dashed) and damped (dotted) conditions. Each line represents the average of two trials per condition. It describes the rotation magnitude due to torque, as a function of frequency. At frequencies above the eigenfrequency, the second order decline of inertia is visible, with expected overlap of all conditions. The on-medication condition is depicted in black, the off-medication condition in grey. Low gain values represent higher stiffness i.e. at low frequencies the wrist is stiffer in the active condition than in the passive condition.

Propranolol and essential tremor - 157

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Ramp-and-hold stretches

There was no significant effect of medication condition for the M1 and M2 reflex

responses and M1 reflex onsets (paired sampled t-test, Table 2). There was no significant difference in maximum voluntary flexion contraction (paired sampled t-test, Table 2).

Table 2. Results ramp-and-hold stretches

Without propranolol

With propranolol

p-value*

M1 response extension [-] 3.83 (1.61) 3,42 (2.04) 0.40

M2 response extension [-] 3.91 (1.32) 4.01 (3.13) 0.85

M1 onset extension [ms] 16.34 (19.33) 16.64 (21.03) 0.97

M1 response flexion [-] 2.66 (1.15) 2.27 (0.88) 0.23

M2 response flexion [-] 2.89 (1.49) 2.32 (1.04) 0.16

M1 onset [ms] flexion 22.50 (22.76) 22.50 (17.21) 0.98

Mean MVC [Nm] 10.06 (4.86) 9.04 (5.02) 0.16

Results are presented in means with standard deviations in parentheses. MVC: maximum voluntary contraction. *paired t-test.

Continuous perturbation task

A repeated-measures ANOVA revealed significant task and medication condition effects in the fitted neuromuscular parameters. There was a significant effect of medication

condition for muscle elasticity k, afferent position feedback kp and a trend towards a

decrease of afferent velocity feedback kv in the on-propranolol condition (p = 0.056, Table 2, Figure 3). Individual post-hoc t-tests revealed no significant effect for a specific medication condition. There was a significant effect of task condition (passive vs. active

vs. active plus damped) for muscle visco-elasticity (b and k), afferent position feedback kp

and afferent velocity feedback kv. Individual post-hoc t-tests revealed the following: for b and k, a significant difference between the passive and active conditions (b on medication

Figure 3. Results for the fitted reflex parameters for the continuous perturbation task. Bars represent means, error bars represent standard error of the mean. I: hand inertia, b: muscle viscosity, k: muscle elasticity, kp: reflexive position feedback gain, kv: reflexive velocity feedback gain in, td: reflexive time delay. *: significant task effect, paired t-test, p < 0.05. ~: significant medication effect, one-way analysis of variance, p < 0.05. For exact p-values see Table 3.

Propranolol and essential tremor - 159

160 - Chapter 8

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Propranolol and essential tremor - 161

p = 0.009 and off medication p = 0.009, k on medication p = 0.005 and off medication p

= 0.007), and passive and damped conditions (b on medication p = 0.005 and off

medication p = 0.007, k on medication p < 0.001 and off medication p = 0.001). For k, an additional significant difference between the active and damped conditions was found

(off medication p = 0.022). For kp , a significant difference between the passive and

damped conditions (kp on medication p = 0.005 and off medication p = 0.007, kv on

medication p = 0.015 and off medication p = 0.070), and active and damped condition (kp on medication p = 0.030, kv on medication p = 0.032). See Table 2 for an overview of all tested variables. There were no significant interaction effects between task and medication condition. Furthermore, a repeated-measures ANOVA revealed no significant medication effect for mean normalized EMG during tasks. There was a significant task effect (flexor p < 0.001, extensor p < 0.001, Table 2). Individual post-hoc t-tests revealed significant task condition effects for the flexor muscle (passive vs. active p < 0.001 on medication, p < 0.001 off medication; passive vs. damped p = 0.001 on medication, p = 0.001 off medication; active vs. damped p = 0.008 off medication) and for the extensor muscle (passive vs. active p < 0.001 on medication, p < 0.001 off medication; passive vs. damped p < 0.001 on medication, p < 0.001 off medication; active vs. damped p = 0.025 off medication).

Discussion In the current study, we demonstrate an effect of propranolol on several physiological model parameters fitted to motor control in patients with essential tremor. Our results indicate that in the on-propranolol condition, compared to the off-propranolol

condition, there is an increase of muscle elasticity during active muscle contraction (k), a

decrease of afferent position feedback (kp) and a trend towards a decrease in afferent

velocity feedback (kv). Furthermore, we have observed changes in muscle elasticity (k),

muscle viscosity (b) and afferent position and velocity feedback (kp & kv) related to task

condition (passive vs. active vs. active and damped) as expected.252 We will interpret our findings within the framework of possible locations of sites of action of propranolol, leading to these parameter changes, and subsequently to a reduction of tremor amplitude.

Interpretation of parameter differences

Figure 4 presents a schematic overview of the neuromusculoskeletal system.245,253,256 Within the neuromusculoskeletal system, taking into account the different reflex and feedback loops, RCs play a central role in our interpretation with regard to the observed parameter changes. RCs recurrently inhibit motor neurons.13,257 Furthermore, RCs inhibit

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group Ia interneurons and antagonistic RCs.258–260 An increase in RCs sensitivity would result in the following three observed changes of 1) decreased muscle elasticity in the passive condition (not significant), 2) increased muscle elasticity due to increased antagonistic contraction in response to agonist muscle stretch in the active conditions and 3) reduced reflexes:

1. Through increased RCs activity, inhibition of the motor neuron increases, which results in less muscle activity in the passive condition and reduced MVC (did not reach significance).

2. Increased inhibition of the Ia interneuron by RCs consequently results in decreased inhibition of the antagonistic motor neuron (causing an increase in

antagonistic contraction leading to increased muscle elasticity (k) in the active, on-propranolol, conditions).

3. Increased inhibition of the agonistic motor neurons (observation 1) and reduced reciprocal inhibition on the antagonistic motor neuron from Ia interneurons (observation 2) both decrease the effectiveness of reflex responses

(kp & kv), a notion supported by a trend of decreased M1 responses upon

flexion and extension stretches in the on-propranolol condition.

An alternative explanation is that decreased reflexive position and velocity feedback could be mediated peripherally, by altering muscle spindle sensitivity.253 However, considering the additional increase in muscle elasticity, reflecting improved phasic co-contraction in the on-propranolol condition, the possibility that differences are based solely on an effect on muscle spindles is unlikely. Furthermore, in our study, we did not observe a medication effect on the M1 and M2 response after ramp-and-hold perturbations, supporting the idea that muscle spindles do not play a solitary role in mediating the effect of propranolol. Alternatively, one could postulate that presynaptic inhibition is affected by propranolol through changes in supraspinal control. Presynaptic inhibition influences Ia afferent input onto the motor neurons.261 However, if propranolol would affect presynaptic inhibition, one would solely expect a change in reflexive activity. What we observe however is that a muscle stretch not only leads to less reflexive activity, but also to increased muscle elasticity. Finally, we cannot exclude that our results could have been influenced by the fact that patients exhibited significantly less tremor in the on-propranolol condition. However, we hypothesize that an effect of tremor on the modelled

parameters would cause a decrease in muscle elasticity in the on-propranolol active conditions, as we found in the passive condition, rather than an increase in muscle ela-

Propranolol and essential tremor - 163

Figure 4. Neuromusculoskeletal model. An antagonistic muscle pair actuated in a one degree of freedom joint while being controlled by a spinal network with motor neurons (MN), group Ia interneurons (IA), Renshaw cells (RCs), inhibitory interneurons (IN), excitatory interneurons (EX) and group Ib interneurons (IB). Feedback is provided by Ia, Ib and II afferents. The locus coeruleus exerts descending inhibitory input on RCs. Adapted from Schuurmans et al., 2011.253 On the right the proposed influence of propranolol on the system is depicted. (1)RCs increase motor neuron inhibition (2) Increased inhibition of the Ia interneuron by RCs decreases inhibition of the antagonistic motor neuron, and (3) as a result of the prior two the effectiveness of the reflex response is reduced (reduced activation of the agonistic motorneuron and increased activation of the antagonistic motorneuron in response to muscle stretch).

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-sticity in the active conditions, which is what we observed in our study. A decrease in tremor severity (in the on-propranolol condition) would lead to less muscle activity, resulting in decreased muscle elasticity in the active conditions.

Possible beta-adrenergic receptors mediating Renshaw cell sensitivity

Propranolol, which is lipophilic, is significantly more effective at both reducing tremor11 and penetrating the blood-tissue barrier262 than the hydrophilic atenolol. This supports the notion that its sites of action supposedly lie also within the central nervous system or within another site separated by a blood-tissue barrier.242 In the light of our results and those of others,11–13 it seems likely that there is a central mechanism of action of propranolol in essential tremor, possibly by mediating RCs sensitivity. There is no available data on the presence of beta-receptors on the RCs themselves. However, it has been known for a long time that locus coeruleus activity suppresses RCs discharges.263 The locus coeruleus possesses adrenergic receptors.13,264,265 It is therefore conceivable that beta-blockade through propranolol suppresses locus coeruleus activity. As mentioned previously, Williams and Baker have hypothesized that propranolol increases recurrent inhibition by RCs in the spinal cord, effectively filtering 10 Hz muscle oscillations.13 They suggest that this reduction also occurs at lower frequencies, and can therefore potentially be the mechanism in which propranolol alleviates tremor in tremor disorders such as ET and Parkinson’s disease. They suggest that strengthening RCs feedback could result in a reduction of tremor amplitudes,13 which fits with the clinical benefit propranolol gives in several tremor disorders. Our results support the hypothesis by Williams and Baker for essential tremor, as well. Considering the widespread presence of β-receptors in the rest of the brain, it is necessary to consider an alternative, or additional, mechanism of action, somewhere within the cerebello-thalamo-cortical loop. Whether anti-tremorogenic effects are mediated through one specific class of beta-adrenergic receptors is unclear. All subtypes are widely distributed throughout the central nervous system, including the cerebellum, midbrain, inferior olive, thalamo-cortical neurons and superficial dorsal horn of the spinal cord.264–269 Many studies have reported pathophysiological changes in the cerebellar cortex in ET, with a possible decrease in the number and function of Purkinje cells35,270 and dysfunction of the cerebellar cortex and dentate nucleus.39,40,140 Noradrenergic projections from the locus coeruleus to the cerebellar cortex are thought to be an important modulator of fine motor control.271 Interestingly, local application of norepinephrine or stimulation of the locus coeruleus results in an inhibition of the spontaneous activity of Purkinje cells.271,272 One could speculate that adrenergic blockage

Propranolol and essential tremor - 165

of this effect increases spontaneous activity of Purkinje cells, which could have an additional tremorolytic effect. These effects are however beyond the scope of this study.

Methodological considerations

Due to the experimental setup, the on-medication condition was always during the first visit. Better task performance through learning and familiarization would cause an increase in reflexive activity, which contrasts with the observed decrease in reflexive activity in the second off-medication visit in our study. Furthermore, our study was not set up to make a definitive argument in favor of, or against, an effect of propranolol within the cerebello-thalamo-cortical system, therefore an effect of propranolol on this system cannot be excluded.

Conclusions In conclusion, our results provide novel support for a central mechanism of action of propranolol in terms of increased Renshaw inhibition, hence alleviating tremor in ET. Further understanding of the pharmacotherapeutic mechanism of propranolol could help future development of more specifically targeted therapies for ET.

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Appendices

Appendix A: Model derivation

The neuromuscular model is based on a linearized Hill-type muscle model.(van der Helm

et al., 2002) The model is briefly discussed in this appendix; a more detailed description of the model is provided in.(Schouten et al., 2003)

Note that the model is represented in the frequency domain where s denotes the Laplace operator. The admittance of the complete neuromuscular model is described by:

𝐻𝐻𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =1

𝐻𝐻𝑖𝑖 + 𝐻𝐻𝑣𝑣𝑚𝑚 + 𝐻𝐻𝑎𝑎𝑎𝑎𝑎𝑎𝐻𝐻𝑚𝑚𝑚𝑚

with Hi describing the inertial dynamics, Hact the muscle activation dynamics, Hve the

muscle visco-elasticity, Hms the muscle spindle activity. The equation was condensed using the following equations:

Inertial dynamics, where I represents the hand inertia:

𝐻𝐻𝑖𝑖(𝑠𝑠) = 1𝐼𝐼𝑠𝑠2

Muscle activation dynamics:

𝐻𝐻𝑎𝑎𝑎𝑎𝑎𝑎(𝑠𝑠) = 1

0.03𝑠𝑠 + 1

where Hact describes the activation dynamics (the process of active muscle force build-up following a neural activation signal) with an activation time constant assumed to be 30 ms.(Schouten et al., 2003)

Muscle visco-elasticity:

𝐻𝐻𝑣𝑣𝑚𝑚(𝑠𝑠) = 𝑘𝑘 + 𝑏𝑏𝑠𝑠

with k the intrinsic stiffness and b the intrinsic damping of (contracted) muscles.

Muscle spindle activity:

𝐻𝐻𝑚𝑚𝑚𝑚(𝑠𝑠) = 𝑒𝑒(−𝜏𝜏𝑑𝑑𝑚𝑚) ∗ �𝑘𝑘𝑣𝑣𝑠𝑠 + 𝑘𝑘𝑝𝑝�

The parameters kp and kv represent the gains of respectively the monosynaptic stretch and

stretch velocity feedback. A single short-latency time delay τd is used to model the neural

latency of signals travelling to the spinal cord and back to the muscle.(De Vlugt et al., 2002) Muscle spindles are often reported to have an excitatory effect on the alpha motor neuron: a positive gain results in activity that resists muscle stretch, decreasing the admittance. In the model fit negative values are allowed, and represent inhibitory effects.

Propranolol and essential tremor - 167

As a fit criterion the total of the squared logarithmic difference in admittance was minimized(Pintelon et al., 1994) over all conditions and frequencies; fitting all parameters

for each subject in one optimization, according to:

𝐸𝐸 = � 𝑚𝑚

𝑛𝑛=1

� 𝑔𝑔

𝑓𝑓=1

𝛾𝛾𝑛𝑛2(𝑓𝑓)1 + 𝑓𝑓𝑓𝑓𝑒𝑒𝑓𝑓(𝑓𝑓)

∗�log𝐻𝐻𝐹𝐹𝐹𝐹𝐹𝐹𝑛𝑛(𝑓𝑓)𝐻𝐻𝑀𝑀𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑛𝑛(𝑓𝑓)

�2

where the error is summed over all frequencies (f) with freq(1) representing the lowest

frequency, freq(m) the highest, and over all experimental conditions (n=1,…,m). HFRF represents the frequency response function from the system identification (admittance) and Hmodel the frequency response function of the model.

Appendix B: Parameter descriptions, units and fit boundaries

Parameter Description Lower bound

Upper bound

Unit

I Hand inertia 0.0001 0.006 [kg m2]

τd Reflexive time delay 0.015 0.05 [s]

ba Muscle viscosity 0.001 1 [Nms/rad]

ka Muscle stiffness 0 20 [Nm/rad]

kp Reflexive feedback gain -30 30 [Nm/rad]

kv Reflexive velocity feedback gain -20 20 [Nms/rad]