Identification of candidate mechanisms for dystonic traits from an analysis of a complete system neurological / biomechanical model

doi:10.21203/rs.3.rs-3446494/v1

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Research Article

Identification of candidate mechanisms for dystonic traits from an analysis of a complete system neurological / biomechanical model

https://doi.org/10.21203/rs.3.rs-3446494/v1

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The research goal is to construct a complete system model for human head movement integrating skeletal, muscular, biomechanical and neurological aspects; verifying the model by simulating ‘normal’ volitional movement. Thereafter the model was used to explore pathological scenarios by variations to model components some of which resulted in dystonic traits, including tremor and abnormal twisting. This enabled identification and analysis of the underlying mechanisms.

Once research commenced it was soon apparent that a purely analytical model (a collection of mutually dependent components each mathematically expressed) was not realisable nor mathematically tractable and so a computer simulation configured for the specific case of head movement (targeting the condition of cervical dystonia but in principle generic) was constructed to allow the study to proceed. The resulting model was exercised to verify a reasonable approximation to normal volitional control of head positioning. With the model operating well in this context, a series of variations were made to see if these might induce dystonic traits. Whilst most of the variations investigated did not cause significant behavioural changes, those relating to the transmission of information to the brain for proprioception were found to be very influential on the system behaviour.

Simulated behaviours of pathological interest included: movement of the head to an orientation of maximum lateral twist and locking in that state, oscillations that evidenced themselves as repetitive movement of the head and finally a ‘self healing’ capability in the case of defective proprioception which is a well established characteristic of the temporal progression of cervical dystonia.

This research identifies candidate mechanisms for the dystonic traits of tremor and excess muscle tone.

Mathematical and Theoretical Biology

Neurobiology of Disease

Cervical dystonia

Movement disorder

Analytical model

Computer simulation

Pathological mechanisms

Complete system model encompassing skeletal, muscular and neurological components
Computer simulation of the model and visualisation of results
Investigation of pathological abnormalities
Manifestation of key dystonic traits – tremors and abnormal twisting

1.1 Study aim

The literature on the pathophysiology of dystonic disorders is broad ranging covering the involvement of multiple brain nuclei along the movement generating pathways and of course the complex lower motor neuronal apparatus with its many feedback loops.

The aim of this study is to construct a complete system model incorporating the lower motor neuronal subsystem and its control mechanisms to see if the recurrency present there might be able to account for some of the aberrant movements present in cervical dystonia and indeed other focal dystonias. Specifically it investigates whether the interactions between feed forward control of the affected muscles together with proprioceptional feedback and the brains communication and control can be modelled and induced to show (via introduction of pathological modifications into the model), dystonic (involuntary) muscle contractions, tremors or postural abnormalities.[13]

There are a number of impediments to constructing a full system model on which to investigate behaviour and in particular look for candidate dystonic pathologies. Whilst many components of the model have been extensively researched and there is an abundance of detail on their behaviour and characteristics, other components (principally the brain itself) remain at a much earlier stage of understanding and so cannot be modelled in the same definitive way. As modelling the full system was a critical goal of this research, the level of abstraction was selected for each component to allow the analysis to proceed, using much higher abstraction levels where the detail is either not known or not easily quantifiable. This allowed the behaviour of the full system to be studied by accepting limitations as to accuracy of the representation – the focus being on the behavioural traits demonstrated by the conceptual model and their causalities, rather than accurate measurements that could be compared against clinical results.

1.2 The simplified model

The system overview is shown in Fig. 1 with the modelled components detailed in Fig. 2 and described here. A muscle / reciprocal muscle pairing (M1 and M2) drive the angular orientation of a skeletal bone (the horizontal hashed bar) by pulling on either end. The model has been configured for the case of lateral movements of the human head (targeting cervical dystonia) though the preconfigured model is generic. The primary model output is the angle of twist (skeletal orientation) of the bone termed Theta.

The primary driving signals come from the brain via descending cortical tracts conveying the upper motor neuron axons. The simplified model does not represent the details as to which brain areas originate the volitional control over the muscle pair but uses a simplifying abstraction of two driving signals, C1 and C2 to the two muscles via connections to the lower alpha motor neurons (AM1 and AM2) in the cervical plexus (or more generally the applicable spinal plexus). The brain model in particular has been highly simplified to be influenced by just two motivators; the desired lateral head orientation (ThetaD) and the urgency to make the turn required to reach that orientation (Priority). The model derives the control signal from the brain by selecting the required muscle contraction required to achieve ThetaD given the current perceived orientation (ThetaP) with an output signal level factoring in the Priority.

The model involves three distinct types of feedback via proprioceptors within the muscles and their tendons. In the case of muscle M1 there is feedback from the nuclear bag fibres (NB1), the nuclear chain fibres (NC1), and from the golgi tendon organ (GT1). The integration of these control signals (volitional control and proprioception feedback) into the muscle activation signal of the alpha motor neuron (AM1) occurs in the synapses in the spinal plexus.

The simulation components have the following functional responsibilities:

Table 1
Brain proprioception	This represents the brains receipt of proprioception information and its use to evaluate the perceived muscle lengths and the bone orientation.
Brain volitional muscle control	This represents the brains comparison of the perceived current orientation with the desired orientation. The urgency to make the required change in orientation (ResponseP) is factored in here to yield the appropriate output control signals.
Spinal plexus	In this component the control signals from the brain and proprioception feedback signals are incorporated into the intrafusal and extrafusal muscle control signals. It is also the relay point for proprioception signals emanating from the muscles en route to the brain.
Muscle innervation	Here the muscle control signals translate into the appropriate muscle length and tension changes. The current muscle state determines the outbound proprioception signals.
Muscle dynamics	This represents where the muscle tensions drive the skeletal bone movement. The outcome is a trace (time evolution) of the skeletal bone orientation and the muscle lengths.

2.1 The simulation model

The delay differential equation is a standard modelling approach to representing the delays in feed forward signal processing and the delays in the feedback control loop.

Let x(t) є ℝⁿ represent both the time varying input to and response from the system i.e. this is both the trajectory of the closed system state and the history of the signals constituting the internal drivers of state. In general

d x(t) / dt = f( t, x(t), x_t ) (1)

where x_t = { x(τ) : τ ≤ t } represents the trajectory of the solution in the past.

f is a functional operator from ℝ X ℝⁿ X C¹⁽ ℝ, ℝ^{n )} to ℝⁿ

C¹ denotes differentiable functions.

The simulation model uses a finite time step of 10 ms (τ) to discretise time and significantly we have a static closed system, i.e. a system that is structurally invariant over time and subject to no external influences (a system that is not evolving to improve its movement capabilities nor subject to any external agencies of change). Therefore

d x(t) / dt = f( t, x(t), x(t-τ₁), …, x(t-τ_m) ) for τ_{1 < … <} τ_m and τ_{j >} 0 ∀ j 1 … m (2)

and given no t variation and equal discrete time steps of 10 ms = τ

d x(t) / dt = f( x(t), x(t-τ), x(t-2τ), …, x(t-mτ) ) (3)

As f is an unknown complex function, the simulation utilises the simplification from linearity of response and independence in the input stimuli components to trace the evolution of x(t).

To see how this works consider an experiment that yields a relationship between a continuous input spike stimulus and yields a continuous output response, which we then discretise to make the experimental pattern amenable to computation. This is shown in Fig. 4.

δ_d(t) = d when − 1/(2d) < t < 1/(2d) otherwise 0. So ∫_−∞^∞ δ_d(t) dt = 1

ψ_d(t) = ∑ _i=0ⁱ⁼ⁿ A_i δ_d(t-iτ)

where A_m is the average value of ψ(t) (the continuous experimental response) over the m^th time step. A_i = 0 for i > n and i < 0. This approximation to the continuous experimental response has the important property that both the real world continuous response and the discretised approximation have the same time integral over any single time step (m-1/2) τ to (m + 1/2) τ.

For the purpose of simulating the system state, we need to construct the discrete response from a general discrete input as shown in Fig. 5.

f_d(t) = ∑ _i=0^i=p B_i δ_d(t-iτ) ϕ_d(t) = ∑ _i=0^i=q C_i δ_d(t-iτ)

ϕ_d(t) = ∑ _i=−∞^{i=∞ (}∑ _j=−∞^j=∞ B_jA_i−j) δ_d(t-iτ) (4)

The derivation of Eq. 4 is described in the research data. This is used extensively to drive the simulation offering the benefit of being computationally efficient.

Finally we can compare the discretised approximation of Eq. 4 to convolution on the continuous real world functions. The convolution (f*ψ)(t)

= ∫_−∞^∞ f(x) ψ(t-x) dx by definition and

≅ ∫_−∞^∞ (∑ _i=−∞^i=∞ B_i δ_d(x-iτ))( ∑ _j=−∞^j=∞ A_j δ_d(t-x-jτ)) dx

= ∑ _i=−∞^i=∞ ∑ _j=−∞^j=∞ B_iA_j ∫_−∞^∞ δ_d(x-iτ) δ_d(t-x-jτ) dx (5)

so if (f*ψ)(t) ≅ ∑ _i=−∞^i=∞ D_i δ_d(t-iτ) from Eq. 5 we obtain

D_k = ∑ _i=−∞^i=∞ ∑ _j=−∞^j=∞ B_iA_j γ(t-(i + j)τ) where t = kτ (6)

γ(t-(i + j)τ) = 1 when its argument is 0 and = 0 otherwise.

So D_k = ∑ _i=−∞^i=∞ ∑ _j=−∞^j=∞ B_iA_j γ(k-i-j)

D_k = ∑ _i=−∞^i=∞ B_iA_k−i (7)

= C_k from Eq. 4

Therefore ϕ = (f*ψ)(t) with the approximation becoming exact as τ -> 0 (8)

(General input function * Spike response function)(t) = General response function(t)

The model simulates the evolution of 24 mutually dependent signal levels over time from the prior system state of these variables, using convolution and transmission factors to determine the state at the end of each 10 ms time step.

2.2 Rotational dynamics of the head

The simplified model assumes a uniform spherical head of around 5 kg mass and with a radius of 8 cm. This gives a solid sphere moment of inertia of

I = (2/5) mr² = 0.0128 (kg m²)

for the head. Using the equation for the angular acceleration of a uniform solid sphere rotating around a central axis

τ = Iα = Fr = 0.0128 ∂²θ/∂t²

where F is the net muscle force and r the distance the force is applied from the fulcrum.

Thus F = (0.0128/0.08) ∂²θ/∂t² = 0.16 ∂²θ/∂t² (9)

Improvements on the accuracy here would require a more specific model tailored to the muscles and bones selected for analysis.

2.3 Muscle contraction

In the case of the conceptual muscle, tendon, bone geometry used in this simulation the following relationships exist between muscle length changes and head angle of turn. For a muscle contraction by δL the angle of turn θ produced is given by:

δL = ⎷(T² – R² (1-Cosθ)²) + R Sinθ – T (10)

where T is the tendon length and R is the distance from the fulcrum of the tendon attachment point. θ_max ≅ 0.585 radians is the maximum turn feasible due to constraints of the geometry.

For a muscle elongation by δL the angle of turn θ produced is given by:

δL = T + RSinθ -⎷( T² – R² (1-Cosθ)² ) (11)

and in this case θ_max = 0.643 radians. Since the two muscles are physically coupled, one side of the pair (that in contraction) will always be subject to the smaller twist limit so that becomes the system maximum turn.

Figure 7 below shows the asymmetry between contraction and extension in the antagonist muscles. For small angles of skeletal twist, extension and contraction are approximately equal but when the angle of twist becomes large and in particular nears the point of maximal twist allowed by the skeletal geometry, then significant departure occurs with extension exceeding contraction. The specifics will of course depend on the exact skeletal and muscle geometry with Fig. 7 showing the pattern only for the simplified (conceptual) case modelled.

2.4 Simulation summary and the volitional control abstraction

The brains’ assessment of the current state and its relationship to the desired orientation allows the controlling signals to the muscle pair to be constructed. These signals together with local feedback from the lower neuronal area allow the alpha motor neuron driving signals to be composed, which in turn controls the muscle tensions. From the muscle tension imbalance, rotational dynamics allows the head’s orientation to be projected (tracked) forward in time. The proprioception feedback signals are evaluated from the system state impacting on the proprioception organs.

A simplistic functional model has been used for deriving the upper neuronal signals (C1 and C2) from the brain to the antagonistic muscle pair. This involves a baseline component resulting in the resting tone of the muscles, which is balanced, plus a turning component which is unbalanced and directed towards the muscle that needs to contract to drive the head turn. Figure 7 shows the activation pattern used to compute this turning component.

3.1 Types of model variation

The goal of exploring variations in the model is to discover what changes have significant effects on the system behaviour and which have the potential to be causally linked to dystonic traits.

One class of variations has a purely local impact (e.g. setting the muscle sensitivity to alpha motor neuron innervation to a lower than normal level) but others have a brain ‘befuddling’ effect (e.g. variation to the proprioception signals received by the brain). This second type confuse the brain into believing the system state is different from what it really is and hence result in tricking it to send out the wrong activation signals (to make the perceived needed, but incorrect, state adjustment).

Dystonic symptoms are very often non-symmetric (i.e. exhibits a strong left / right imbalance in posture). This was investigated by setting up model scenarios where the variation was made on just one side.

It is also well known that dystonia can be partly remediated through the nervous systems ability to accommodate defects (e.g. if the proprioception signal to the brain is defective, then a compensation made by the brain to the baseline value of that signal may reduce the consequences to the system). The model was also configured to explore this scenario.

Results

S0.1 - Volitional movement to the desired head orientation

This trace shows a gentle approach at 15% of the maximum volitional urgency to the desired head orientation of 30 centi radians, with no overshoot and no oscillation around the target orientation. The chart shows the head turning clockwise (with Theta representing the heads angle of twist) to reach the target twist (ThetaD - the desired angle) in around 2.8 seconds. Note in particular that this chart shows the brains perception of the head orientation is always correct (ThetaP, the angle perceived by the brain using available proprioception measures, matches Theta, the true angle of twist) and the two traces overlap exactly.

S0.2 - Volitional movement with high urgency

The model has been run with the same configuration as in S0.1 but now with a high volitional urgency (80%). This trace shows a rapid approach to the desired head orientation which is achieved in around 0.45 seconds. The Theta trace then exhibits an overshoot followed by a dampened oscillation around the target twist (ThetaD).

S2.1- Head turns and locks at the maximum twist

This scenario involves a reduction in perceived Secondary 1 (the brain receives a 10% smaller Secondary1 signal than would be the case in the baseline model) and this results in ThetaP being less than Theta. M1 is the muscle in extension (DeltaL1 is positive) so Secondary 1 is the preferred input to the ThetaP calculation. As Theta hits its hard limit (58 centi radians) before ThetaP reaches ThetaD, the system gets ‘locked’ in a state of maximal twist, as the brain perceives the head twist as being always less than that desired. It can of course be unlocked by a simple volitional reduction in ThetaD. T2 remains high in the static locked state (yielding a static high muscle tone) with T1 at its resting muscle tone (rest tension).

S2.2 – Tremor due to proprioception error

This scenario is similar to S2.1 but with a smaller volitional twist sought (25 centi radians). The perceived Secondary 1 signal has a 10% reduction imposed and this results in ThetaP being less than Theta during the early part of the chart (where Secondary1 is the preferred proprioception as M1 is in extension). Over this period Theta increases rapidly with ThetaP shadowing below it, until ThetaD is reached by ThetaP, at which point the growth in Theta is slowed as the high level of M2 innervation ceases. When M2 finally stops contracting as Theta has reached its peak, then M2 proprioception becomes preferred as it enters a period of extension (but importantly Secondary 2 is not tainted in this scenario) so ThetaP correctly tracks Theta as it drops back. When ThetaP crosses ThetaD again, high M2 innervation resumes and after a short period Theta once again starts to increase. Secondary 1 resumes being the preferred proprioception returning to the original situation of ThetaP assessed to be less than Theta. An oscillation results (at approximately 0.6 Hz) as the cycle repeats itself indefinitely. The oscillation is in effect driven by a switching preference in proprioception between M1 and M2 depending on which is in extension and with a reduced Secondary 1 impacting on the perceived M1 state only.

S2.3 – Asymmetrical behaviour resulting from a proprioception deficiency

This is a similar scenario to S2.2 but with slightly less reduction to the perceived Secondary 1 level (8%) and a larger volitional twist (50 centi radians) that is reversed half way through the simulation (-50 centi radians). As in S2.2 it starts with ThetaP running below Theta as M1 is extending and the preferred M1 proprioception (Secondary 1) is depleted due to the scenario configuration. Theta clips briefly at the top of each oscillation as the maximum Theta is reached but the system does not lock. An oscillation results for the same reasons as S2.2.

Of note after the ThetaD reversal is an asymmetry between the oscillations around ThetaD positive and those around ThetaD negative. Theta never clips when ThetaD becomes negative as the driving innervation over the first stage remains high even after Theta exceeds ThetaD for a short while longer (until ThetaP exceeds ThetaD) but in the second stage the high innervation ceases as soon as Theta crosses ThetaD. The variation in Secondary1 acts to foster movement when Theta is increasing but this effect is not present when Theta is decreasing. The oscillations around ThetaD when negative also have a longer periodicity.

S2.27 – Tremor due to defective gamma level perception

Figure 13 shows the scenario being explored here. The proprioception organs (NB1 and NC1) are both regulated through gamma motor neuron signal levels (GM12 and GM11 respectively). These gamma signals are also conveyed to the brain so it can correctly interpret the proprioception coming from these organs. This scenario looks at the effect of an incorrect transmission of a gamma level to the brain, shown in the figure as 5% increase to GM11.

In this example only GM11 is elevated (+ 5%). In the M1 extending portions of the chart where Theta is increasing, M1 offers the preferred proprioception. The perception of a larger gamma bias component yields a smaller perceived stretch component and the proprioception results in a ThetaP that lies below Theta. Oscillating behaviour results (1.3 Hz) around a central value slightly above ThetaD (40 centi radians), the elevation in GM11 acting predominantly to drive Theta higher.

S2.28 – Ability of the brain to compensate for defective proprioception

This is the same scenario as S2.27 with one minor change which demonstrates that the simulation model can emulate ‘self healing’ properties of the nervous system. If we postulate the brain has the ability to detect the GM11 level is averaging 5% higher than normal and as a consequence ‘adjusts’ its baseline for the rest firing rate of GM11 by this same percentage value to compensate for the defect, we can look at the model to see what this might yield. This is achieved by a static adjustment to the rest firing rate of + 5% so we are looking at the position after full compensation has been effected. The outcome is a partial mitigation of the effect of the defective proprioception signal and results in the elimination of the oscillation in Theta with Theta converging on its correct desired value ThetaD.

S2.43 – Spontaneous tremor without volitional initiation

In this scenario the Secondary 1 level perceived by the brain is biased by -5 and ThetaD is zero. If Secondary 1 was not tainted, then this would be a static state with Theta remaining at a constant value of zero.

What results is an oscillation in Theta centring around 10 centi radians and with a frequency of 0.8 Hz. This oscillation results from the under estimation of ThetaP during the M1 extension phases when M1 is the preferred proprioception source and its reduced Secondary 1 level yields a lower perceived extension than the true state driving Theta higher. Once M1 stops extending, preferred proprioception reverts to M2 which is not tainted and so ThetaP tracks Theta exactly bringing the twist back to zero as desired.

S2.44 – Fast tremor results from a positive bias

This scenario is similar to S2.43 but with the bias in Secondary 1 reversed so it becomes + 5: however the two traces S2.43 and S2.44 look completely dissimilar. In this scenario, careful inspection of the ThetaP level shows that the + 5 bias causes an initial but very transient uplift in ThetaP well above the ThetaD of zero, due to the higher Secondary 1 level causing an inference of greater stretch in M1. This tainted ThetaP produces an M1 contraction response and this in turn promotes M2 (now in extension) to become the preferred proprioception. So the briefly high ThetaP has only a short lived downward effect on Theta which then quickly levels off as the system attempts to return to ThetaD of zero. At the point of reversal in Theta (when it has reached its minimum value and is starting to increase) M1 starts extending returning M1 to the preferred proprioception role and again causing a large uplift in ThetaP, propelling the Theta value down once again. This cycle repeats with the oscillation stabilising at around − 8 centi radians and a frequency of 3.04 Hz. The difference between ThetaP and Theta is very pronounced at this level of bias.

S2.49 - Tremor from tainted proprioception with no volitional component

This scenario is similar to S2.44 but involves a 5% uplift in the Secondary 1 level rather than the addition of a fixed bias. The chart analysis follows the same rationale as previously described for S2.44 An oscillation emerges despite the absence of any voluntary twist being sought, centred around − 3 centi radians and with a frequency of 2.8 Hz.

4.1 Candidate dystonic mechanisms

This study involved the construction of a full system neurological / biomechanical simulation for the movement of the human head. This was used to study the system impacts of a large set of scenario variations. These predominantly show that variations to proprioception organ sensitivity and neural transmission factors result in only minor changes to the overall behaviour of the system; which demonstrated a remarkable resilience. The exception however, where variations were found to have a significant effect, was in the proprioception signal levels transmitted to the brain and used to track muscle state and skeletal orientation. Exploration in this domain of variation produced many examples where dystonic behaviours occurred; primarily arising from the befuddling effect of incorrect proprioception on the cognitive process initiating and controlling movement. The brain is the key component of the system being modelled but was not varied to perform aberrantly.

Dystonic behaviours of interest included: movement of the head to an orientation of maximum lateral twist and locking in that state (produced by a deficient Secondary afferent level), oscillations that evidenced themselves as repetitive movement of the head (produced by either a defective Secondary afferent level or an in surfeit gamma level) and finally a ‘self healing’ capability in mitigation of defective proprioception, which is a well established characteristic of the temporal progression of cervical dystonia.

The simulation model provides an analytical tool with which to search for and identify the mechanisms at work and relate these causally to the behaviours observed. This research identified candidate mechanisms for the dystonic traits of tremor and excess muscle tone.

Conflicts of interest

The author has no conflicts of interest to declare.

Appendices

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Acknowledgements

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Research Data

The datasets generated during and/or analysed during the current study are available in the Mendeley repository, https://data.mendeley.com/datasets/34czjt9xz7/2

The following artefacts are supplied in support of this paper.

Model source code and model user guide (instructions for setup)
Traces (examples of simulation run outputs covering various system scenarios)
Detailed study documentation (description of the analytical model and its realisation as a simulation model. Narrative describing the experiments performed using the model and commentary on the results)

Winter, E.M., Brookes, F.B.C. Electromechanical response times and muscle elasticity in men and women. Eur J Appl Physiol 63, 124–128 (1991). https://doi.org/10.1007/BF00235181
Byrne, J. H. Synaptic Transmission in the Central Nervous System (Section 1, Chap. 6) Neuroscience Online: An Electronic Textbook for the Neurosciences | The University of Texas Medical School (tmc.edu)
Neuromuscular junctions and muscle contractions – Lumen Learning (lumenlearning.com) Module 8 – Muscle Tissue https://courses.lumenlearning.com/cuny-csi-ap-1/chapter/neuromuscular-junctions-and-muscle-contractions/
Houk, J.C. Responses of Golgi Tendon organs to forces applied to muscle tendon 1152/jn.1967.30.6.1466 Journal of Neurophysiology
Gregory, J.E., Proske, U. The responses of Golgi tendon organs to stimulation of different combinations of motor units. J Physiol. 1979;295:251–262. doi:10.1113/jphysiol.1979.sp012966
https://en.wikipedia.org/wiki/Delay_differential_equation
https://en.wikipedia.org/wiki/Nerve_conduction_velocity
Mann, Michael D. The nervous system in action – Muscle receptors http://michaeldmann.net/mann11.html
Katz B., Miledi R. The measurement of synaptic delay and the time course of acetylcholine release at the neuromuscular junction Proc. R. Soc. Lond. B.161483–495 http://doi.org/10.1098/rspb.1965.0016
Mann, Michael D. The nervous system in action – Muscle contraction http://michaeldmann.net/mann14.html
Wang, Joong-San. An analysis on muscle tone and stiffness of posterior cervical region during sling and plinth on static prone position https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5684023/pdf/jpts-29-1841.pdf
Ganguly, J., Kulshreshtha, D., Almotiri, M., Jog, M. Muscle tone physiology and abnormalities, Toxins, 13(4), 282 https://doi.org/10.3390/toxins 13040282
Elble, Roger J. Defining Dystonic Tremor, Current Neuropharmacology, 2013, 11, 48–52 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3580791
Sivapalan, K. Lower Motor System https://www.slideserve.com/cyndi/motor-system

The analytical model and its realisation in the computer simulation code is described in the accompanying research data (Dystonia model – Narrative and Notes). This model is an abstraction covering the muscle innervation response [10][1][3][9][11], the delay differential equation and convolution [6], muscle bone geometry, proprioception organs [4][5][8], head rotational dynamics, nerve conduction [2][7], reciprocal inhibition[12], and a brain proprioception and control model.
The derivation of formula 6 is detailed in the research data (Dystonia model – Narrative and Notes).
Full detail as to how each component is computed from those signals on which it has a dependency is given in the research data accompanying this paper.
Note that the head as a physical object has been simplified here to a uniform sphere with a radius of 8 cm which might appear inconsistent with the 5 cm radius assumed in the calculations relating to the twisting movement of the cranial bones (Section 2.3 and the associated analysis in the research data). The bone and its attachment to the muscles that cause twisting lies below the external surface of the head so it is not unreasonable that the external radius is larger than the internal distance (distance from the attachment point of the muscle tendon to the bone.
Details of the derivation of this equation and definitions of the constants involved are to be found in the research data.
The trace in Fig. 8 has been filtered to only show four state variables as displaying the full set produces a cluttered view.
ThetaD = 30 centi radians and Priority = 15%
If the trace file is reviewed, Primary 2 is seen to exhibit a sharp return to its active level (~ 27Hz) when the suppression caused by muscle contraction is lost (i.e. when the M2 contraction rate drops below a low threshold level).
This scenario involves no variation to the normal workings and output from the proprioception organs but imposes a 10% reduction on the signal level received from them by the brain. For clarity such variations have been called “Perceived” to emphasise that it’s the received signal at the brain that is affected.
The computation of ThetaP involves all of the Primary and Secondary afferent signal levels (plus the gamma efferent levels) but if any are silenced and so offer no proprioception information, then only the remainder will be used. These are referred to as the preferred inputs in the simulation of the brains proprioception.
The first stage is the period 50–136 time steps where ThetaP is less than ThetaD.
One unexpected feature of the trace in Fig. 14 is that over the initial period, whilst Theta is well below ThetaD and the head is turning sharply to attain that angle of twist, ThetaP is not noticeably below Theta. The reason is that over nearly all of this period ThetaP is far from ThetaD and so the simple model used to compute C1 never involves the abatement which would occur when the target angle comes into close proximity (the C1 innervation is not tapered). This in turn keeps AM1 low and GM11 follows also at a very low value. As this scenario involves a 5% uplift in GM11, over this period of the chart that uplift is negligible due to the very low GM11 level. Hence ThetaP accurately tracks Theta despite the tainted GM11 level.
A percentage change to a perceived signal (+ 5%) has a larger absolute effect when the signal level is high. In contrast adding a bias to a signal (+ 5) has the same absolute effect irrespective of the original level but makes the largest percentage difference when the signal level is low.
79 scenarios were setup in the simulation model and the outcomes are provided in the research data as trace files allowing further analysis.
The brain is processing tainted information as if it were fully correct, resulting in anomalous output. No variations to the brain proprioception processing model have been made (excepting in S2.28).

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Identification of candidate mechanisms for dystonic traits from an analysis of a complete system neurological / biomechanical model

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Abstract

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Highlights

1 Introduction

1.1 Study aim

1.2 The simplified model

2 Methods

2.1 The simulation model

2.2 Rotational dynamics of the head

2.3 Muscle contraction

2.4 Simulation summary and the volitional control abstraction

3 Model variations

3.1 Types of model variation

Conclusions

4.1 Candidate dystonic mechanisms

Declarations

References

Footnotes

Supplementary Files

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