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Modelling muscle spindles and afferent response

The response of the muscle spindles in mammals to muscle stretch has been thoroughly studied, and various models have been proposed. However, due to the difficulty in obtaining accurate data of the afferent and fusimotor responses during muscular movement, these models have usually been quite limited. For example, several of the earliest models account only for the afferent response, ignoring the fusimotor activity.

Mileusnic et al. (2006) model

One recent model, developed by Mileusnic et al. (2006), portrays the muscle spindle as consisting of several (typically 4 to 11) nuclear chain fibres, and two different nuclear bag fibres, connected in parallel as shown here in the figure below. The muscle fibres respond to three inputs: fascicle length, dynamic fusimotor input and static fusimotor input. The $bag_{1}$ fibre is mainly responsible for detecting dynamic fusimotor input, while the $bag_{2}$ and chain fibres are mainly responsible for detecting static fusimotor input. All fibres respond to changes in the fascicle length, and are modelled in largely the same way but with different coefficients to account for their different physiological properties. The responses of the three types of fibres are summed to generate the primary and secondary afferent activities. The primary afferent activity is affected by the response of all three types of muscle fibres, while the secondary afferent activity only depends on the $bag_{2}$ and chain fibre responses.

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Hasan (1983) model

Another comprehensive model of muscle spindles was proposed by Hasan in 1983 . This representation of muscle fibres and spindles is based closely on their physical properties. The muscle spindle is represented as two separate regions connected in series: sensory and non-sensory. The firing rate of the spindle afferent depends on the state of the two regions. The lengths of the two regions can be labelled $z(t)$ for the sensory and $y(t)$ for the non-sensory region. The tension $f(t)$ in the two regions is equal, since they are placed in series. The sensory zone can be assumed to act like a spring (equation (3)), while in the non-sensory region, tension is a non-linear function of $y(t)$ (equation (2) derived by Hasan).

$f(t)=k_{1}(y(t)-c)(1+[{\frac {y'(t)}{a}}]^{\frac {1}{3}})\qquad \qquad {\text{(2)}}$
$f(t)=k_{2}z(t)\qquad \qquad \qquad \qquad \qquad \qquad {\text{(3)}}$

The total length of the muscle spindle, x(t) is the sum of the length of the two regions (equation (4)).

$x(t)=z(t)+y(t)\qquad \qquad \qquad \qquad \qquad {\text{(4)}}$

Using this substitution and rearranging, we can derive the following expression for the length of the sensory zone (equation (5)):

$z'(t)=x'(t)-a({\frac {bz(t)-x(t)+c}{x(t)-z(t)-c}})^{3}\qquad {\text{(5)}}$

Here, parameter $a$ represents the sensitivity of the tension to to velocity in the non-sensory zone, parameter $b=(k_{1}+k_{2})/k_{1}$ and parameter $c$ determines the zero-length tension which influences the background firing rate of the afferent. The length of the sensory zone depends not only on the current length and velocity of the spindle, but on the history of the length changes.

The firing rate, $g(t)$ in Hasan's model depends on a combination of the sensory zone length and its first derivative (equation (6)), with an experimentally derived weighting.

$g(t)=z(t)+0.1z'(t)\qquad \qquad \qquad \qquad {\text{(6)}}$

Model parameters

Approximate values for the model parameters a, b and c were suggested by Hasan (1983), and differ for voluntary and passive movements. A summary of these values is presented in the table below. Type of ending Condition A (mm/s) B C (mm)

Type of ending	Condition	A (mm/s)	B	C (mm)
Primary	Passive	0.3	250	-15
Primary	Gamma - dynamic	0.1	125	-15
Primary	Gamma - static	100	100	-25
Secondary	Passive	50	50	-20

In the model, these values are assumed to be static for the duration of a movement, however this is not believed to be the case.

Internal models of limb dynamics

In addition to modelling the response of muscle spindle afferents to muscle stretch, several groups have worked on modelling the signals which are sent from the brain to the spindle efferents in order for muscles to complete specific movements. The complexity here lies in the fact that the brain must be able to adapt to unexpected changes in the dynamics of planned movements, using feedback from the spindle afferents.

Studies in this area suggest that humans achieve this using internal models, which are built through an “error-feedback-learning” process, and transform planned muscle states into the motor commands required to achieve them. To generate the motor commands for a particular reaching movement, the brain performs calculations based on the expected dynamics of the planned movement. However, any unexpected changes in these dynamics while the movement is being executed (e.g. external strain placed on the muscle) will lead to errors in expected muscle length (Gottlieb 1994, Shadmehr and Muss-Ivaldi 1994). These errors are communicated to the brain through the muscle spindle afferents, which experience a different sensory state to what is expected. The brain then reacts to these error signals with short and long latency responses, which work to minimise the error, but cannot eliminate it completely due to the delay in the system.

Studies suggest that the error can be eliminated in a subsequent attempt at the movement under the same dynamics, and this is where the “error-feedback-learning” idea comes from (Thoroughman and Shadmehr 1999). The corrections which are generated by the brain form an internal model, which maps a desired action (in kinematic coordinates) to the necessary motor commands (as torques). This internal model can be represented as a weighted combination of basis elements:

$torque=\sum w_{i}g_{i}(\theta ,\theta ',\theta ''...)$

Here each basis $g_{i}$ represents some characteristic of the muscle's sensory state, and the motor command is a “population code”. Population coding is a method of representing stimuli as the combined activity of many neurons (in contrast to rate coding). In order to use such a model, we need to know how the bases represent particular limb or muscle positions, and the neuronal firing rates associated with them. The bases can, in principle, represent every aspect of the state: position, velocity, acceleration and even higher derivatives. However, this high dimensionality makes it very difficult to derive relationships experimentally between each dimension of the bases and the firing rates.