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On-line compensation for perturbations of a reaching movement is cerebellar dependent: support for the task dependency hypothesis

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Abstract

Although the cerebellum has been shown to be critical for the acquisition and retention of adaptive modifications in certain reflex behaviors, this structure’s role in the learning of motor skills required to execute complex voluntary goal-directed movements still is unclear. This study explores this issue by analyzing the effects of inactivating the interposed and dentate cerebellar nuclei on the adaptation required to compensate for an external elastic load applied during a reaching movement. We show that cats with these nuclei inactivated can adapt to predictable perturbations of the forelimb during a goal-directed reach by including a compensatory component in the motor plan prior to movement initiation. In contrast, when comparable compensatory modifications must be triggered on-line because the perturbations are applied in randomized trials (i.e., unpredictably), such adaptive responses cannot be executed or reacquired after the interposed and dentate nuclei are inactivated. These findings provide the first demonstration of the condition-dependent nature of the cerebellum’s contribution to the learning of a specific volitional task.

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Acknowledgements

The study was supported by US National Institutes of Health (NIH) Grants NS30013 and NS21958.

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Correspondence to Yury Shimansky.

Appendix: procedure for the analysis of forelimb dynamics

Appendix: procedure for the analysis of forelimb dynamics

General ideas

At the time of our developing software for the analysis of the experimental data, to the best of our knowledge, no 3-D model of the dynamics of a three-segmented forelimb that could be used for our purposes had been described in the literature. The procedure we utilized for the derivation of the equation system describing forelimb movement dynamics in 3-D space is presented below. This procedure is based on Lagrange’s principle, according to which the equations of dynamics of a certain mechanical system are presented as \({{\partial L} \over {\partial q_{k} }} - {{d} \over {dt}} {{\partial L} \over {\partial \dot{q}_{k} }} = 0\), where L is the Lagrangian function \(L = L{\left( {q_{1} ,...,q_{n} ,\dot{q}_{1} ,...,\dot{q}_{n} } \right)}\) describing the mechanical energy of the system and q 1,...,q n are the system’s generalized coordinates. An external constraint g(q 1,q 2,...q n )=0 can be accounted for by the method of undetermined multipliers. According to this method, the Lagrangian function should include an additional term λ(tg(q 1,q 2,...q n ), where λ is the undetermined multiplier corresponding to the constraint. To calculate the corresponding variational component, the constraint equation should be presented in the differential form \({\sum\limits_j {h_{j} } }\dot{q}_{j} = 0\), or \({\sum\limits_j {h_{j} } }\delta q_{j} = 0\), where h j =dg/dq j . In case of the presence of external forces and torques, the Langrangian function L should include work done by generalized forces. Finally, Euler’s equation with regard to the generalized coordinate q k is

$$ \frac{{\partial T}} {{\partial q_{k} }} - \frac{{\partial V}} {{\partial q_{k} }} - \frac{d} {{dt}}\frac{{\partial T}} {{\partial \dot{q}_{k} }} + Q_{k} + {\sum\limits_i {\lambda _{i} h_{{ki}} } } = 0, $$
(1)

where T and V are kinetic and potential energy, respectively, Q k is the generalized force corresponding to the coordinate q k , and λ i is the undetermined multiplier corresponding to the i-th constraint.

The crucial advantage of Lagrange’s approach over the method of Newton-Euler laws of motion is that the user only has to write a formula for the Lagrangian function describing mechanical energy of the limb. The actual derivation of the differential equation system can be done automatically by a symbolic manipulation software package, such as Mathematica, Maple, or Macsyma. The latter package was used in this study not only to derive the equation system, but also to generate a C-language code for the computation of the elements of the equation system matrix with relation to generalized forces. This code then was included without any modification into a C++ program written for the data analysis. The Lagrangian function used to describe the mechanical energy of a cat forelimb is described below.

Forelimb as a mechanical system

The motion of the upper arm segment is described as a pure rotation with respect to a non-inertial coordinate system whose center is at the shoulder joint (mechanically a ball-joint). The non-inertial coordinate frame is moving parallel to the coordinate system fixed with relation to the external observer. The lower arm segment is viewed as a combination of two coaxial parts (connected at the segment’s center of mass). Wrist rotation is presented as the rotation of these parts against each other around their common longitudinal axis. Any changes in the inertia momentum of the whole lower arm segment caused by such rotation are considered insignificant and are neglected. The wrist joint is modeled as a hinge-joint representing wrist flexion. Radial deviation is neglected because of its relatively narrow range.

Constant parameters and variables

The first subscript in the name of a constant or variable related to a certain segment of the forelimb denotes the segment: a, b, c, and d corresponding to upper arm, elbow part of lower arm, wrist part of lower arm, and paw, respectively. The first subscript in the name of a constant or variable related to a certain joint denotes the joint: s, e, r, and w corresponding to shoulder, elbow, the virtual joint introduced for expressing wrist rotation (see explanations above), and wrist, respectively. The second subscript denotes the number of space coordinate (either in the observer’s coordinate system or in the system attached to the limb segment, depending on the variable).

–:

l i and l i0 are the length of the segment i (i=a, b, c, or d) and the distance from the segment’s proximal end to its center of gravity

–:

m i is the mass of the segment i

–:

g=9.81 m/s2, the gravitational acceleration constant

–:

I ij is inertia momentum of the segment i related to the j-th axis of the coordinate system attached to the segment (the third axis is the segment’s longitudinal axis; note that for the upper limb segment the center of rotation is the shoulder joint and for other two segments the center of rotation is the segment’s center of mass)

–:

ϖ ij is the angular velocity component of the segment i corresponding to the j-th axis of the coordinate system attached to the segment

–:

e ij is a unitary vector representing the j-th axis of the coordinate system attached to the segment i in the observer’s coordinate system

–:

x ij is the j-th coordinate of the i segment’s center of mass in the observer’s coordinate system

–:

v ij is the j-th coordinate of the velocity of the i segment’s center of mass in the observer’s coordinate system

–:

γ ij is the angle between the longitudinal axis of the segment i with the j-th axis of the observer’s coordinate system

–:

ϕ i , ψ i , and θ i are Eulerian angles describing the orientation of the coordinate system attached to the segment i with relation to the observer’s coordinate system

–:

α i is the angle at the joint i (i=e, w)

–:

F ij is the component j (j=α, ϕ, ψ, or θ) of an external force (i.e., torque generated by active muscle contraction, tissue passive resistance, or any external force acting on the limb) corresponding to the joint i

–:

Q ij is the Lagrangian multiplier (here playing the role of a generalized force) corresponding to the j-th constraint (j=1, 2, 3, 4, or 5) at the limb joint i (i=e or w)

Lagrangian function of forelimb dynamics

Lagrangian function for the entire forelimb can be presented as the sum L=L K+L G+L F+L C, where indices K, G, F, and C denote components corresponding to kinetic energy (done by inertial forces), gravitation, work done by external forces (i.e., active muscle forces, external perturbation forces, and forces resulting from passive resistance of tissue), and joint constraints (resulting in interaction forces). Formulas describing these components are presented in detail below.

Kinetic energy

The expression describing kinetic energy has two components: rotational L Krot and translational L Ktr . The formula for the rotational component is the same for each segment of the forelimb: \(L_{{Krot}} = {{1} \over {2}}{\sum\limits_{j = 1}^3 {I_{{ij}} \omega ^{2}_{{ij}} } }\). Note that for the upper-arm segment the center of rotation is the shoulder joint and for other two segments the center of rotation is the segment’s center of mass). The translational component for the upper-arm segment is \(L_{{Ktr}} = m_{1} l_{{10}} {\sum\limits_{j = 1}^3 {a_{{sj}} \cos {\left( {\gamma _{{1j}} } \right)}} }\), where a sj is shoulder joint acceleration in the direction of the j-th coordinate of the observer’s coordinate system. The translational component formula for the lower limb and paw segments is \( L_{{Ktr}} = {{1} \over {2}}m_{{ij}} {\sum\limits_{j = 1}^3 {v^{2}_{{ij}} } } \).

Potential energy (due to gravitational field)

For the upper-arm segment, this component is described by L G=−m 1 gl 10cos(γ3). For the other two segments, it is L G=−m 1 gz, where g=9.81 m/s2, the gravitational acceleration constant, and z is the vertical coordinate of the segment’s center of mass.

Non-inertiality of shoulder joint

The shoulder joint is moved with the body, so the system of coordinates fixed with respect to the shoulder is not inertial. This is taken into account by adding gravity-type of work: m 1 l 10(a s ·e a ), where a s is the vector of shoulder joint acceleration.

Work of external forces

Shoulder joint: L sF=F ϕa+F ψa+F θa. Elbow joint: L eF=F αe. Wrist joint: L wF=F bc)+F αw.

Work of interaction forces (due to joint constraints)

Elbow joint:

$$ \eqalign{ L_{eQ} = & \sum\limits_{i = 1}^3 Q_{ei} \left( x_{si} + l_a \gamma _{ai} - x_{bi} + l_{b0} \gamma _{bi} \right) + Q_{e4} \left( \mathbf{e}_{a1} \cdot \mathbf{e}_{b2} \right) \cr & + Q_{e5} \left( \mathbf{e}_{a3} \cdot \mathbf{e}_{b2} \right). \cr } $$

Wrist joint:

$$\eqalign { L_{wQ} = & \sum\limits_{i = 1}^3 Q_{wi} \left( x_{bi} + \left( l_{b} - l_{b0} \right) \gamma _{bi} - x_{di} + l_{d0} \gamma _{di} \right) \cr & + Q_{w4} \left( \mathbf{e}_{c1} \cdot \mathbf{e}_{d2} \right) + Q_{w5} \left( \mathbf{e}_{c3} \cdot \mathbf{e}_{d2} \right)\;. \cr }$$

The data analysis described in Introduction focuses on the following six torque components (generalized forces): shoulder rotation (F ), shoulder yaw (F ), shoulder elevation (F ), elbow flexion (F ), wrist rotation (F ), and wrist flexion (F ). It is important to note once more that any external perturbation force contributes to these torque components.

The mass m i and inertia momentum I i1 of the forelimb segments were calculated on the basis of the body mass and segment length according to the method described in (Hoy and Zernicke 1985). The inertia momentum I i2 was set equal to I i1. The third momentum related to the longitudinal axis of the segment was calculated as I i3=m i r i 2/2, assuming that the segment approximates to a cylinder with the radius r i =√[m i /(πρ i l i )], where ρ i is the average density of the i-th segment’s tissue. Density estimates for human forelimb segments (Winter 1979) were used.

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Shimansky, Y., Wang, JJ., Bauer, R.A. et al. On-line compensation for perturbations of a reaching movement is cerebellar dependent: support for the task dependency hypothesis. Exp Brain Res 155, 156–172 (2004). https://doi.org/10.1007/s00221-003-1713-0

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