Abstract
We are constructing a physiologically realistic model of the muscle spindle to assist in the analysis of natural sensorimotor control and to design biomimetic systems for functional electrical stimulation (FES). Our model is composed of mathematical elements that correspond closely to the anatomical components of spindles. The resulting nonlinear model is reasonably accurate in predicting published records of spindle activity during a variety of ramp, triangular and sinusoidal stretches applied under various fusimotor conditions.
1. Introduction
The
main objective of the study is to develop control systems that are applicable
for restoration of reach and grasp tasks of paralyzed arm and hand muscles by
FES. The motivation for the study is the development of modular, injectable
devices (BIONs) that can be used both to stimulate paralyzed muscles and to
provide artificial proprioceptive information for sensory feedback. We believe
that the design of FES controllers should be informed by the sensory regulation
of natural movement. Therefore, we are developing models of biological sensors,
such as Golgi tendon organs, Renshaw cells and, most importantly, muscle
spindles.
The muscle spindle must sense and accurately
encode length and velocity over a very wide range of kinematic conditions,
despite the relatively restricted dynamic range of firing rates for action potentials [7]. It does this by shifting
the relative importance of and sensitivity to length and velocity by means of
specialized fusimotor efferents (primarily gamma motoneurons) that the CNS
controls separately from the alpha motoneurons controlling muscle force.
There are three main types of intrafusal
muscle fibers within a typical muscle spindle: bag 1, bag 2 and chain
fibers. The bag 1 fiber receives dynamic fusimotor control and is primarily
responsible for velocity sensitivity of the spindle. The bag 2 fiber and chain
fibers are innervated by the the static
fusimotor control and contribute mainly to length sensitivity. The spindle is
innervated by two afferents, the primary (Ia) and the secondary afferent (II).
The primary afferent detects motion at the equatorial region of all three
intrafusal fibers (representing both length and velocity information), while
the secondary afferent is located more eccentrically on only the bag 2 and
chain fibers (sensitive primarily to length information).
1.1. Model Structure
The
muscle spindle model consists of two intrafusal fiber models, bag 1 and
combined bag 2 plus chain, reflecting
their common fusimotor drive (Figure 1a).
The three inputs to each intrafusal fiber
model are the fascicle length, the velocity and the relevant fusimotor input.
The spindle model computes two outputs similar to the biological sensors, i.e.
primary and secondary afferent. Each intrafusal fiber is modeled with the same
structure shown in Figure 1b, which is a modified version of the lumped linear
spindle model suggested by McMahon [1].which is a modified
version of the linear spindle model suggested by McMahon [8].
The
intrafusal fiber model is divided into a polar region and a central sensory
region. The sensory region is modeled as a pure elastic element (KSE),
whose strain is linearly related to afferent firing rate. The tension within
the sensory region is defined as:
T=KSE ((x-x1)-x0) (1)
Where x is
fascicle length, x1 polar
region length and x0 unloaded sensory region length.
·
The polar region
is modeled as a spring (KPE) with a parallel contractile element.
The contractile element consists of the active force generator and the damping
element. The tension within the polar region is
defined as:
T=CBx1 0.3 (x1-L) +KPE x1 +G (2)
The active-state
force generator, G, is defined
as the summation of a constant term (G0), gstatic term (G1gstatic) and gdynamic term (G2gdynamic), while damping term (B) is defined as
the weighted sum of passive damping coefficients (B0), gdynamic term (B1gdynamic) and gstatic term (-B2gstatic). C is a constant describing
experimentally observed effects of velocity on force, which are asymmetrical
for lengthening and shortening. The
model incorporates a length dependence of force production that assumes that
the intrafusal fiber is operating on the ascending limb of a force-length
relationship (x1-L) [9]. Importantly, the model
incorporates the nonlinear velocity property (power 0.3) that has been
described empirically [2].
For
each intrafusal fiber model, the equations for tension within polar and sensory
regions are combined into a simple nonlinear, first
order differential equation representing net mechanical state. Afterwards, the
primary afferent output is obtained by summing the outputs of bag 1 and bag 2
plus chain intrafusal fiber models, while secondary afferent output is obtained
only from the bag 2 plus chain intrafusal fiber models.
2. Results
The
anatomical and mathematical structure represented in Figure 1 was embodied as a
set of nested blocks in the Simulink® modeling environment. The free coefficients were initially adjusted
manually and in some cases optimized by using Levenberg-Marquardt method. The
database included the wide range of spindle afferent activity reported in the
experimental literature, which includes ramp, triangular and sinusoidal
stretches applied during different fusimotor states.
The model’s
ability to reproduce primary afferent activity during ramp stretches is shown
in Figure 2. The experimental data are from Crowe and Matthews [1], who
employed three different velocities and three different fusimotor inputs (no
input, γstatic=70pps, γdynamic=70pps). The responses of the same model to 8
mm triangular stretches employed Lennerstrand and Thoden’s data [4,5] and are
shown in Figure 3. Figure 3a deals with primary afferent firing at three
different velocities, while Figure 3b is a firing record at the same stretching
velocity but under different dynamic fusimotor inputs.