Dependence of peripheral tremor on mechanical perturbations: a modeling study

The present study scrutinizes the popular view that tremors of central origin but not those of peripheral origin are largely resistant to mechanical perturbations. We explore the effects of perturbations in a well-established model of peripheral tremor and document that (a) tremor frequency can remain unchanged when spring or weight loads are added, (b) entrainment by external drives can be limited to drives of similar frequency, and (c) resetting of tremor phase by torque pulses can remain fractional. This resistance to mechanical perturbations arises in the model because peripheral neuromuscular dynamics act as a limit-cycle oscillator which, by its very nature, will absorb moderate changes to signals and parameters. We conclude from our study that resistance to mechanical perturbations is not an exclusive property of central tremors, but rather may also be found in peripheral tremors. Other criteria are therefore needed to distinguish between different origins of tremor. Received: 8 April 1998 / Accepted in revised form: 6 October 1998     

Pathological tremors as diffusional processes

 Two types of pathological tremors, essential and Parkinsonian, are studied using dynamical systems theory. It is shown that pathological tremors can be characterized as diffusional processes. The time-scale range for the diffusional scaling law to be valid starts from about one to several tens of the mean oscillation period. This time-scale range contrasts sharply with the predictable time scale for deterministic chaos, which is usually only a small fraction of the mean oscillation period. The diffusions in pathological tremors are usually anomalous. A number of quantities are designed to characterize the diffusions in the tremor. Their relevance to potential clinical applications is discussed. It is argued that in order to discriminate between Parkinsonian and essential tremors, quantities not of purely dynamical origin may be more useful, since purely dynamical quantities emphasize more the dynamical similarities between the two types of tremors. Received: 30 May 2001 / Accepted in revised form: 6 November 2001     

Tremor modulation in patients with Parkinson’s disease compared to healthy counterparts during loaded postural holding

Through examining tremor dynamics, the study sought to investigate the effects of load characteristics upon control strategies in patients with Parkinson’s disease (PD) during postural holding. Eleven untreated patients and eleven healthy adults conducted a static pointing task with an outstretched arm, with a manipulated load of 100 g on the index finger. Oscillatory activities in the upper limb were contrasted between the unloaded and loaded conditions. The results showed that PD patients demonstrated abnormal modulation of tremor amplitude in the finger, hand, and upper arm in the opposing load condition. When the load was applied, the PD patients presented a nearly opposite pattern of tremor coupling between limb segments, contrary to the normal release and enhancement of tremor coupling in the finger–hand and hand–forearm complexes, respectively. Principal component analysis suggested that normal postural tremors could be explained by a load-dependent component that had high communality with tremors of the distal segments. In contrast, major principal components of PD tremor were invariant to load addition. Multi-segment tremors in PD were atypically organized during loaded postural holding, signifying that coordinative control of the upper limb in the patients was impaired in the absence of exploitation of a germane distal strategy against inertial perturbation.     

A comparison of resonance tuning with positive versus negative sensory feedback

We used a computational model of rhythmic movement to analyze how the connectivity of sensory feedback affects the tuning of a closed-loop neuromechanical system to the mechanical resonant frequency (ω_r). Our model includes a Matsuoka half-center oscillator for a central pattern generator (CPG) and a linear, one-degree-of-freedom system for a mechanical component. Using both an open-loop frequency response analysis and closed-loop simulations, we compared resonance tuning with four different feedback configurations as the mechanical resonant frequency, feedback gain, and mechanical damping varied. The feedback configurations consisted of two negative and two positive feedback connectivity schemes. We found that with negative feedback, resonance tuning predominantly occurred when ω_r was higher than the CPG’s endogenous frequency (ω_CPG). In contrast, with the two positive feedback configurations, resonance tuning only occurred if ω_r was lower than ω_CPG. Moreover, the differences in resonance tuning between the two positive (negative) feedback configurations increased with increasing feedback gain and with decreasing mechanical damping. Our results indicate that resonance tuning can be achieved with positive feedback. Furthermore, we have shown that the feedback configuration affects the parameter space over which the endogenous frequency of the CPG or resonant frequency the mechanical dynamics dominates the frequency of a rhythmic movement.     

Feedback-induced gain control in stochastic spiking networks

The joint influence of recurrent feedback and noise on gain control in a network of globally coupled spiking leaky integrate-and-fire neurons is studied theoretically and numerically. The context of our work is the origin of divisive versus subtractive gain control, as mixtures of these effects are seen in a variety of experimental systems. We focus on changes in the slope of the mean firing frequency-versus-input bias (f –I) curve when the gain control signal to the cells comes from the cells’ output spikes. Feedback spikes are modeled as alpha functions that produce an additive current in the current balance equation. For generality, they occur after a fixed minimum delay. We show that purely divisive gain control, i.e. changes in the slope of the f –I curve, arises naturally with this additive negative or positive feedback, due to a linearizing actions of feedback. Negative feedback alone lowers the gain, accounting in particular for gain changes in weakly electric fish upon pharmacological opening of the feedback loop as reported by Bastian (J Neurosci 6:553–562, 1986). When negative feedback is sufficiently strong it further causes oscillatory firing patterns which produce irregularities in the f –I curve. Small positive feedback alone increases the gain, but larger amounts cause abrupt jumps to higher firing frequencies. On the other hand, noise alone in open loop linearizes the f –I curve around threshold, and produces mixtures of divisive and subtractive gain control. With both noise and feedback, the combined gain control schemes produce a primarily divisive gain control shift, indicating the robustness of feedback gain control in stochastic networks. Similar results are found when the “input” parameter is the contrast of a time-varying signal rather than the bias current. Theoretical results are derived relating the slope of the f –I curve to feedback gain and noise strength. Good agreement with simulation results are found for inhibitory and excitatory feedback. Finally, divisive feedback is also found for conductance-based feedback (shunting or excitatory) with and without noise. This article is part of a special issue on Neuronal Dynamics of Sensory Coding.     

Tremor disorders. Diagnosis and management.

Tremor is commonly encountered in medical practice, but can be difficult to diagnose and manage. It is an involuntary rhythmic oscillation of a body part produced by reciprocally innervated antagonist muscles. Tremors vary in frequency and amplitude and are influenced by physiologic and psychological factors and drugs. Categorization is based on position, posture, and the movement necessary to elicit the tremor. A resting tremor occurs when the body part is in repose. A postural tremor occurs with maintained posture and kinetic tremor with movement. Various pathologic conditions are associated with tremors. Essential tremor, which is the most common, is postural and kinetic, with a frequency between 4 and 8 Hz, and involves mainly the upper extremities and head. Essential tremor responds to treatment with primidone, beta-blockers, and benzodiazepines. Parkinson''s disease causes a 4- to 6-Hz resting tremor in the arms and legs that responds to the use of anticholinergics and a combination of carbidopa and levodopa. Tremor can also be a manifestation of Wilson''s disease, lesions of the cerebellum and midbrain, peripheral neuropathy, trauma, alcohol, and conversion disorders. Treatment should be directed to the underlying condition. Stereotactic thalamotomy of thalamic stimulation is a last resort.     

Tremor and other oscillations in neuromuscular systems

A model has been analyzed which is based on recent experimental evidence concerning the properties of muscles and the sensory feedback pathways from muscles. Damped oscillations can arise in the absence of sensory feedback due to the interaction of a muscle with inertial loads. These mechanical oscillations can have a wide range of frequencies depending on the inertial and elastic loads that are attached to the muscle. Small amounts of sensory feedback will tend to reduce deviations from a steady muscle length, but larger amounts of feedback can produce oscillations. The frequency of these reflex oscillations is determined by the properties of the muscle and feedback pathway, and is rather independent of load. If the strength of the sensory feedback is sufficient, either the mechanical oscillations or the reflex oscillations or both can grow, rather than decay, with time. The growth of these oscillations is limited by saturation non-linearities in the muscle receptors and the muscle itself, so that the oscillations approach a steady amplitude and frequency. Using typical properties of muscles and spinal reflex pathways, the frequency of reflex oscillations will be within the range 8–12 Hz found for physiological tremor. With the longer latency found for supraspinal reflexes, oscillations will occur in the range 4–6 Hz which is characteristic of Parkinson's and cerebellar diseases. The role of longer latency reflexes in the generation of these tremors is discussed.     

A note on the response characteristics of the extraocular muscle system

Using records of the response of the extraocular muscle system to a step input, the frequency response characteristics of the system have been calculated. They show a progressive reduction of gain for frequencies above 30–40 cps. The high frequency tremor of the eyes observed during “steady” fixation would thus be expected to be damped down and this is in fact borne out by frequency analyses of tremor records.     

Spinal reflexes as a basis for tremor in the premammillary cat.

A maintained tremor observed in premammillary cats has been studied by varying the load attached to the soleus muscle, by electrical recording, and stimulation. The frequency and amplitude of the tremor varies widely with changes in elastic or inertial loading (springs or flywheels). These variations agree well with predictions based on the idea that the tremor is maintained by relfexes reinforcing a tendency for oscillation in a mass-spring system (which includes the elasticity of the muscle). Spinal stretch reflexes are probably responsible since a relatively fixed delay of 8.5 +/- 2.9 ms is observed between the peaks of recorded afferent and electromyographic activity. Stimulation of the lateral gastrocnemius--soleus nerve completely resets the tremor, again suggesting a reflex origin.     

Dynamic Reweighting of Three Modalities for Sensor Fusion

We simultaneously perturbed visual, vestibular and proprioceptive modalities to understand how sensory feedback is re-weighted so that overall feedback remains suited to stabilizing upright stance. Ten healthy young subjects received an 80 Hz vibratory stimulus to their bilateral Achilles tendons (stimulus turns on-off at 0.28 Hz), a ±1 mA binaural monopolar galvanic vestibular stimulus at 0.36 Hz, and a visual stimulus at 0.2 Hz during standing. The visual stimulus was presented at different amplitudes (0.2, 0.8 deg rotation about ankle axis) to measure: the change in gain (weighting) to vision, an intramodal effect; and a change in gain to vibration and galvanic vestibular stimulation, both intermodal effects. The results showed a clear intramodal visual effect, indicating a de-emphasis on vision when the amplitude of visual stimulus increased. At the same time, an intermodal visual-proprioceptive reweighting effect was observed with the addition of vibration, which is thought to change proprioceptive inputs at the ankles, forcing the nervous system to rely more on vision and vestibular modalities. Similar intermodal effects for visual-vestibular reweighting were observed, suggesting that vestibular information is not a “fixed” reference, but is dynamically adjusted in the sensor fusion process. This is the first time, to our knowledge, that the interplay between the three primary modalities for postural control has been clearly delineated, illustrating a central process that fuses these modalities for accurate estimates of self-motion.     

Beta-Adrenergic Modulation of Tremor and Corticomuscular Coherence in Humans

Coherence between the bioelectric activity of sensorimotor cortex and contralateral muscles can be observed around 20 Hz. By contrast, physiological tremor has a dominant frequency around 10 Hz. Although tremor has multiple sources, it is partly central in origin, reflecting a component of motoneuron discharge at this frequency. The motoneuron response to ∼20 Hz descending input could be altered by non-linear interactions with ∼10 Hz motoneuron firing. We investigated this further in eight healthy human subjects by testing the effects of the beta-adrenergic agents propranolol (non-selective β-antagonist) and salbutamol (β₂-agonist), which are known to alter the size of physiological tremor. Corticomuscular coherence was assessed during an auxotonic precision grip task; tremor was quantified using accelerometry during index finger extension. Experiments with propranolol used a double-blind, placebo-controlled crossover design. A single oral dose of propranolol (40 mg) significantly increased beta band (15.3–32.2 Hz) corticomuscular coherence compared with placebo, but reduced tremor in the 6.2–11.9 Hz range. Salbutamol (2.5 mg) was administered by inhalation. Whilst salbutamol significantly increased tremor amplitude as expected, it did not change corticomuscular coherence. The opposite direction of the effects of propranolol on corticomuscular coherence and tremor, and the fact that salbutamol enhances tremor but does not affect coherence, implies that the magnitude of corticomuscular coherence is little influenced by non-linear interactions with 10 Hz oscillations in motoneurons or the periphery. Instead, we suggest that propranolol and salbutamol may affect both tremor and corticomuscular coherence partly via a central site of action.     

Frequency and displacement amplitude relations for normal hand tremor.

Spectral analysis of hand tremor records obtained from normal subjects during continuous extension of the hand for 15-45 min revealed that the root-mean-square (rms) displacement amplitude of the tremor increased from control levels of about 30 mum to levels on the order of 100-1,000 times control. Associated with this increase in the displacement was a systematic decrease in the hand tremor frequency from control values of 8-9 Hz to values of 4-6 Hz. Spectral analysis of demodulated extensor EMG records indicated a consistent relation between EMG modulation amplitude at the tremor frequency and the tremor displacement amplitude for tremor records with rms displacement above about 100 mum. No consistent relation was found between these two variables for tremor records with displacements below 100 mum. Consideration of both mechanical and neural reflex effects indicated that a viscoelastic-mass mechanism primarily determined the small-amplitude (less than 100 mum) tremors, while the large displacement tremors may have involved both mechanical and neural feed back factors.     

Hippocampal rhythm generation: Gamma-related theta-frequency resonance in CA3 interneurons

 During different behavioral states different population activities are present in the hippocampal formation. These activities are not independent: sharp waves often occur together with high-frequency ripples, and gamma-frequency activity is usually superimposed on theta oscillations. There is both experimental and theoretical evidence supporting the notion that gamma oscillation is generated intrahippocampally, but there is no generally accepted view about the origin of theta waves. Precise timing of population bursts of pyramidal cells may be due to a synchronized external drive. Membrane potential oscillations recorded in the septum are unlikely to fulfill this purpose because they are not coherent enough. We investigated the prospects of an intrahippocampal mechanism supplying pyramidal cells with theta frequency periodic inhibition, by studying a model of a network of hippocampal inhibitory interneurons. As shown previously, interneurons are capable of generating synchronized gamma-frequency action potential oscillations. Exciting the neurons by periodic current injection, the system could either be entrained in an oscillation with the frequency of the inducing current or exhibit in-phase periodic changes at the frequency of single cell (and network) activity. Simulations that used spatially inhomogeneous stimulus currents showed anti-phase frequency changes across cells, which resulted in a periodic decrease in the synchrony of the network. As this periodic change in synchrony occurred in the theta frequency range, our network should be able to exhibit the theta-frequency weakening of inhibition of pyramidal cells, thus offering a possible mechanism for intrahippocampal theta generation. Received: 23 February 2000 / Accepted in revised form: 30 June 2000     

Crustacean motor pattern generator networks

Crustacean motor pattern-generating networks have played central roles in understanding the cellular and network bases of rhythmic motor patterns for over half a century. We review here the four best investigated of these systems: the stomatogastric, ventilatory, cardiac, and swimmeret systems. Generally applicable observations arising from this work include (1) neurons with active, endogenous cell properties (endogenous bursting, postinhibitory rebound, plateau potentials), (2) nonhierarchical (distributed) network synaptic connectivity patterns characterized by high levels of inter-neuronal connections, (3) nonspiking neurons and graded transmitter release, (4) multiple modulatory inputs, (5) networks that produce multiple patterns and have flexible boundaries, and (6) peripheral properties (proprioceptive feedback loops, low-frequency muscle filtering) playing an important role in motor pattern generation or expression.     

Tests of a linear model of visual-vestibular interaction using the technique of parameter estimation

The goal of this study was to test whether a superposition model of smooth-pursuit and vestibulo-ocular reflex (VOR) eye movements could account for the stability of gaze that subjects show as they view a stationary target, during head rotations at frequencies that correspond to natural movements. Horizontal smooth-pursuit and the VOR were tested using sinusoidal stimuli with frequencies in the range 1.0–3.5 Hz. During head rotation, subjects viewed a stationary target either directly or through an optical device that required eye movements to be approximately twice the amplitude of head movements in order to maintain foveal vision of the target. The gain of compensatory eye movements during viewing through the optical device was generally greater than during direct viewing or during attempted fixation of the remembered target location in darkness. This suggests that visual factors influence the response, even at high frequencies of head rotation. During viewing through the optical device, the gain of compensatory eye movements declined as a function of the frequency of head rotation (P < 0.001) but, at any particular frequency, there was no correlation with peak head velocity (P > 0.23), peak head acceleration (P > 0.22) or retinal slip speed (P > 0.22). The optimal values of parameters of smooth-pursuit and VOR components of a simple superposition model were estimated in the frequency domain, using the measured responses during head rotation, as each subject viewed the stationary target through the optical device. We then compared the model's prediction of smooth-pursuit gain and phase, at each frequency, with values obtained experimentally. Each subject's pursuit showed lower gain and greater phase lag than the model predicted. Smooth-pursuit performance did not improve significantly if the moving target was a 10 deg × 10 deg Amsler grid, or if sinusoidal oscillation of the target was superimposed on ramp motion. Further, subjects were still able to modulate the gain of compensatory eye movements during pseudo-random head perturbations, making improved predictor performance during visual-vestibular interactions unlikely. We conclude that the increase in gain of eye movements that compensate for head rotations when subjects view, rather than imagine, a stationary target cannot be adequately explained by superposition of VOR and smooth-pursuit signals. Instead, vision may affect VOR performance by determining the context of the behavior. Received: 16 June 1997 / Accepted: 5 December 1997     

Feedback control of the limbs position during voluntary rhythmic oscillation

The mechanisms that control the limbs position during rhythmic voluntary oscillations were investigated in ten subjects, who were asked to synchronise the lower peak of their hand or foot rhythmic oscillations to a metronome beat. The efficacy of the “position control” was estimated by measuring the degree of synchronisation between the metronome signal and the requested limb position and how it was affected by changing both the oscillation frequency (between 0.4 and 3.0 Hz) and the limbs inertial properties. With the limbs unloaded, the lower peak of both the hand and foot oscillations lagged the metronome beat of a slight amount that remained constant over the whole frequency range (mean phase delay −13.2° for the hand and −4.7° for the foot). The constancy was obtained by phase-advancing, at each frequency increment, the electromyogram (EMG) activation with respect of the clock beat of the amount necessary to compensate for the simultaneous increase of the lag between the EMG and the movement, produced by the limb mechanical impedance. After loading of either limb, the increase of the oscillation frequency induced larger EMG-movement delays and the anticipatory compensation became insufficient, so that the movement progressively phase-lagged the clock beat. The above results have been accurately simulated by a neural network connected to a pendulum model that shared the same mechanical properties of the moving limb. The network compares a central command (the intended position) to the actual position of the effector and acts as a closed-loop proportional, integrative and derivative controller. It is proposed that the synchronisation of rhythmic oscillations of either the hand or the foot is sustained by a feed-back control that conforms the position of each limb to that encoded in the central voluntary command.     

Corticomuscular Transmission of Tremor Signals by Propriospinal Neurons in Parkinson's Disease

Cortical oscillatory signals of single and double tremor frequencies act together to cause tremor in the peripheral limbs of patients with Parkinson''s disease (PD). But the corticospinal pathway that transmits the tremor signals has not been clarified, and how alternating bursts of antagonistic muscle activations are generated from the cortical oscillatory signals is not well understood. This paper investigates the plausible role of propriospinal neurons (PN) in C3–C4 in transmitting the cortical oscillatory signals to peripheral muscles. Kinematics data and surface electromyogram (EMG) of tremor in forearm were collected from PD patients. A PN network model was constructed based on known neurophysiological connections of PN. The cortical efferent signal of double tremor frequencies were integrated at the PN network, whose outputs drove the muscles of a virtual arm (VA) model to simulate tremor behaviors. The cortical efferent signal of single tremor frequency actuated muscle spindles. By comparing tremor data of PD patients and the results of model simulation, we examined two hypotheses regarding the corticospinal transmission of oscillatory signals in Parkinsonian tremor. Hypothesis I stated that the oscillatory cortical signals were transmitted via the mono-synaptic corticospinal pathways bypassing the PN network. The alternative hypothesis II stated that they were transmitted by way of PN multi-synaptic corticospinal pathway. Simulations indicated that without the PN network, the alternating burst patterns of antagonistic muscle EMGs could not be reliably generated, rejecting the first hypothesis. However, with the PN network, the alternating burst patterns of antagonist EMGs were naturally reproduced under all conditions of cortical oscillations. The results suggest that cortical commands of single and double tremor frequencies are further processed at PN to compute the alternating burst patterns in flexor and extensor muscles, and the neuromuscular dynamics demonstrated a frequency dependent damping on tremor, which may prevent tremor above 8 Hz to occur.     

Respiratory input impedance in anesthetized paralyzed patients

Respiratory impedance (Zrs) was measured between 0.25 and 32 Hz in seven anesthetized and paralyzed patients by applying forced oscillation of low amplitude at the inlet of the endotracheal tube. Effective respiratory resistance (Rrs; in cmH2O.l-1.s) fell sharply from 6.2 +/- 2.1 (SD) at 0.25 Hz to 2.3 +/- 0.6 at 2 Hz. From then on, Rrs decreased slightly with frequency down to 1.5 +/- 0.5 at 32 Hz. Respiratory reactance (Xrs; in cmH2O.l-1.s) was -22.2 +/- 5.9 at 0.25 Hz and reached zero at approximately 14 Hz and 2.3 +/- 0.8 at 32 Hz. Effective respiratory elastance (Ers = -2pi x frequency x Xrs; in cmH2O/1) was 34.8 +/- 9.2 at 0.25 Hz and increased markedly with frequency up to 44.2 +/- 8.6 at 2 Hz. We interpreted Zrs data in terms of a T network mechanical model. We represented the proximal branch by central airway resistance and inertance. The shunt pathway accounted for bronchial distensibility and alveolar gas compressibility. The distal branch included a Newtonian resistance component for tissues and peripheral airways and a viscoelastic component for tissues. When the viscoelastic component was represented by a Kelvin body as in the model of Bates et al. (J. Appl. Physiol. 61: 873-880, 1986), a good fit was obtained over the entire frequency range, and reasonable values of parameters were estimated. The strong frequency dependence of Rrs and Ers observed below 2 Hz in our anesthetized paralyzed patients could be mainly interpreted in terms of tissue viscoelasticity. Nevertheless, the high Ers we found with low volume excursions suggests that tissues also exhibit plasticlike properties.     

Mathematical modeling and parameter estimation of axonal cargo transport

This paper is about how cortical recurrent interactions in primary visual cortex (V1) together with feedback from extrastriate cortex can account for spectral peaks in the V1 local field potential (LFP). Recent studies showed that visual stimulation enhances the γ-band (25–90 Hz) of the LFP power spectrum in macaque V1. The height and location of the γ-band peak in the LFP spectrum were correlated with visual stimulus size. Extensive spatial summation, possibly mediated by feedback connections from extrastriate cortex and long-range horizontal connections in V1, must play a crucial role in the size dependence of the LFP. To analyze stimulus-effects on the LFP of V1 cortex, we propose a network model for the visual cortex that includes two populations of V1 neurons, excitatory and inhibitory, and also includes feedback to V1 from extrastriate cortex. The neural network model for V1 was a resonant system. The model’s resonance frequency (ResF) was in the γ-band and varied up or down in frequency depending on cortical feedback. The model’s ResF shifted downward with stimulus size, as in the real cortex, because increased size recruited more activity in extrastriate cortex and V1 thereby causing stronger feedback. The model needed to have strong local recurrent inhibition within V1 to obtain ResFs that agree with cortical data. Network resonance as a consequence of recurrent excitation and inhibition appears to be a likely explanation for γ-band peaks in the LFP power spectrum of the primary visual cortex.     

Parkinsonian tremor and simplification in network dynamics

We explore the behavior of richly connected inhibitory neural networks under parameter changes that correspond to weakening of synaptic efficacies between network units, and show that transitions from irregular to periodic dynamics are common in such systems. The weakening of these connections leads to a reduction in the number of units that effectively drive the dynamics and thus to simpler behavior. We hypothesize that the multiple interconnecting loops of the brain’s motor circuitry, which involve many inhibitory connections, exhibit such transitions. Normal physiological tremor is irregular while other forms of tremor show more regular oscillations. Tremor in Parkinson’s disease, for example, stems from weakened synaptic efficacies of dopaminergic neurons in the nigro-striatal pathway, as in our general model. The multiplicity of structures involved in the production of symptoms in Parkinson’s disease and the reversibility of symptoms by pharmacological and surgical manipulation of connection parameters suggest that such a neural network model is appropriate. Furthermore, fixed points that can occur in the network models are suggestive of akinesia in Parkinson’s disease. This model is consistent with the view that normal physiological systems can be regulated by robust and richly connected feedback networks with complex dynamics, and that loss of complexity in the feedback structure due to disease leads to more orderly behavior.