Using Animatlab for Neuromechanical Analysis
407
motoneuron stimulus-tension relationship such that muscle activation operates within
this range. When maximally stimulated, the activation term ST max is attenuated to F max
due to the selection of K SE and K PE .
This introduces stimulus-tension bounds of
ST max
1 + exp(S(x offset + .04))
+ y offset = ST max and
ST max
1 + exp(S(x offset + .06))
+ y offset = 0
with an x offset of −50 mV. These equations can be solved simultaneously for S and y offset
values for each muscle.
2.3 Physics Timestep Limits for Calculating Tension
Animatlab utilizes the backward Euler method to represent the tension equation in
Eq. (2.1.1) in a discrete form. At each timestep, the previous muscle state is used to
determine values at the present state. The discrete form of the tension equation is
T i =
1 −
dtK SE
B
1 +
K PE
K SE
T i−1 +
dtK SE
B
K PE (L i − L rest ) + B
L i − L i−1
dt
+ A m,i A l,i
. (2.3.1)
Algorithmic discretization of differential equations can cause asymptotic instability
depending on the selection of the timestep, dt [29]. To avoid asymptotic instability from
developing in the muscle, the coefficient multiplied by the previous-state tension, T i−1 ,
should be bounded such that
1 −
dtK SE
B
1 +
K PE
K SE
≤ 1.
(2.3.2)
This implies a trivial lower bound of
dtK SE
B
1 +
K PE
K SE
≥ 0,
(2.3.3)
which is always satisfied since all parameters are positive. Thus, a single upper bound
is necessary for constraining the physics timestep,
dt ≤
2B
K SE + K PE
.
(2.3.4)
The physics timestep boundary condition for a system of muscles is the minimum
value of Eq. (2.3.4) of any muscle in the system.
2.4 Steady State Length-Tension Stability Criteria
It is natural for muscle tension to settle to zero in instances where it is not activated
and its length is less than or equal to its resting length. However, there are valid lengthactivation combinations that are incapable of producing force specifically caused by
the way Animatlab calculates tension (red surface in Fig. 4). The inability to generate
tension at seemingly valid length-activation combinations leads to errors when the force
407
motoneuron stimulus-tension relationship such that muscle activation operates within
this range. When maximally stimulated, the activation term ST max is attenuated to F max
due to the selection of K SE and K PE .
This introduces stimulus-tension bounds of
ST max
1 + exp(S(x offset + .04))
+ y offset = ST max and
ST max
1 + exp(S(x offset + .06))
+ y offset = 0
with an x offset of −50 mV. These equations can be solved simultaneously for S and y offset
values for each muscle.
2.3 Physics Timestep Limits for Calculating Tension
Animatlab utilizes the backward Euler method to represent the tension equation in
Eq. (2.1.1) in a discrete form. At each timestep, the previous muscle state is used to
determine values at the present state. The discrete form of the tension equation is
T i =
1 −
dtK SE
B
1 +
K PE
K SE
T i−1 +
dtK SE
B
K PE (L i − L rest ) + B
L i − L i−1
dt
+ A m,i A l,i
. (2.3.1)
Algorithmic discretization of differential equations can cause asymptotic instability
depending on the selection of the timestep, dt [29]. To avoid asymptotic instability from
developing in the muscle, the coefficient multiplied by the previous-state tension, T i−1 ,
should be bounded such that
1 −
dtK SE
B
1 +
K PE
K SE
≤ 1.
(2.3.2)
This implies a trivial lower bound of
dtK SE
B
1 +
K PE
K SE
≥ 0,
(2.3.3)
which is always satisfied since all parameters are positive. Thus, a single upper bound
is necessary for constraining the physics timestep,
dt ≤
2B
K SE + K PE
.
(2.3.4)
The physics timestep boundary condition for a system of muscles is the minimum
value of Eq. (2.3.4) of any muscle in the system.
2.4 Steady State Length-Tension Stability Criteria
It is natural for muscle tension to settle to zero in instances where it is not activated
and its length is less than or equal to its resting length. However, there are valid lengthactivation combinations that are incapable of producing force specifically caused by
the way Animatlab calculates tension (red surface in Fig. 4). The inability to generate
tension at seemingly valid length-activation combinations leads to errors when the force