Biomedical Engineering Reference
In-Depth Information
2 Nsm
1
yields a simulated isotonic response with
approximately the same duration. For skeletal muscle data, the duration of the isotonic
experiment is approximately 400 ms, and a value for
approximately 100 ms. A value for
B
¼
1
600 Nsm
1
yields a simulated
isotonic response with approximately the same duration. It is known that fast and slow
muscle have differently shaped force velocity curves and that the fast muscle force-velocity
curve data has less curvature. Interestingly, the changes in the parameter values for
¼
B
1
B
1
as
suggested here give differently shaped force velocity curves consistent with fast (rectus
eye) and slow (skeletal) muscle.
The parameter value for viscous element
B
2
is selected by trial and error so the shape of
the simulation force-velocity curve matches the data. As the value for
B
2
is decreased from
0.5 Nsm
1
, the shape of the force velocity curves changes to a more linear shaped function.
Moreover, if the value of
B
2
falls below approximately 0.3 Nsm
1
, strong oscillations appear
in the simulations of the isotonic experiment, which are not present in the data. Thus, the
viscous element
B
2
is an essential component in the muscle model. Without it, the shape
of the force-velocity curve is linear and the time course of the isotonic experiment does
not match the characteristics of the data.
Varying the parameter values of the lever muscle model changes the eigenvalues of the
system. For instance, with
0.5 kg, the system's nominal eigenvalues (as defined with
the parameter values previously specified) are a real pole at
M
¼
30.71 and a pair of complex
conjugate poles at
283.9
j221.2
.
If the value of
B
2
is increased, three real eigenvalues
describe the system. If the value of
B
2
is decreased, a real pole and a pair of complex conju-
gate poles continue to describe the system. Changing the value of
B
1
does not change the
eigenvalue composition, but it does significantly change the value of the dominant eigen-
value from
292 with
B
1
¼
0.1 to
10 with
B
1
¼
6.
13.7.4 The 1995 Linear Homeomorphic Saccadic Eye Movement Model
The linear model of the oculomotor plant presented in Section 13.6 is based on a non-
linear oculomotor plant model by Hsu and coworkers [33] using a linearization of the
force-velocity relationship and elasticity curves. Using the linear model of muscle described
earlier in this section, it is possible to avoid the linearization and to derive a truer linear
homeomorphic saccadic eye movement model.
The linear muscle model described in this section has the static and dynamic properties
of rectus eye muscle, a model without any nonlinear elements. As presented, the model has
a nonlinear force-velocity relationship that matches eye muscle data using linear viscous
elements, and the length tension characteristics are also in good agreement with eye muscle
data within the operating range of the muscle. Some additional advantages of the linear
muscle model are that a passive elasticity is not necessary if the equilibrium point
x
e
¼
19.3
, rather than 15
, and muscle viscosity is a constant that does not depend on the
innervation stimulus level.
Figure 13.41 shows the mechanical components of the updated oculomotor plant for
horizontal eye movements, the lateral and medial rectus muscle, and the eyeball. The
agonist muscle is modeled as a parallel combination of viscosity
B
2
and series elasticity
K
se
, connected to the parallel combination of active-state tension generator
F
ag
, viscosity
element
B
1
, and length tension elastic element
K
lt
. For simplicity, agonist viscosity is set
