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M. N. Fitzpatrick et al.
threshold of the synapse [14]. The term h ∞ is another sigmoid, similar to m ∞ , but for
the sodium inactivation channel. The sigmoids are defined as
m ∞ (V ) =
1
1 + exp(S · (R − V ))
(8)
h ∞ (V ) =
1
1 + 0.5 · exp(S · V )
(9)
where S is the maximum slope of the sigmoid.
Sodium Channel Inactivation. The differential equation that governs the sodium
channel inactivation term, h, found in Eq. 4 is
˙
h =
h ∞ (U ) − h
τ h (U )
(10)
where τ h is the sodium inactivation time constant found using
τ h (V ) = τ h,max · h ∞ (V ) ·
0.5 · exp(S · V )
(11)
2.3 Interfacing the Dynamics
With the motor and neural dynamics laid out in the sections above, we must establish
coupling terms between them. Our goal is to use the neural states to specify position and
velocity commands for the motor. For the CPG to be able to control the movement of
the motor, the states of the neural dynamics are integrated in to the motor dynamics as
inputs as well as having certain states from the motor dynamics used in new feedback
pathways for the neurons.
Neuron Activation as Inputs to the Motor. To have the CPG specify the velocity of the
motor, the motor voltage input, V in , becomes a piecewise-linear function of the neuron
activation states.
V in (U 1 , U 2 ) = P · min(max(U 1 , 0), R) − min(max(U 2 , 0), R))
(12)
where U 1 and U 2 represent the activation states of neurons 1 and 2, respectively. The
term P is a gain term represented as
P =
4 · K v
1000 · R
(13)
that limits the maximum commanded speed to 4 rads/s, since this is the approximate
speed of the hip joint during walking locomotion found in [18].
Angular Position as Feedback for the Neurons. The feedback synapse conductance,
G fb , is set as a function of position θ . If neuron 1 (U 1 ) was active when θ reaches a
certain threshold, the feedback synapse conductance activates to allow a strong inhibitory
current to inhibit U 1 . With U 1 now inhibited, U 2 is allowed to escape and become the
M. N. Fitzpatrick et al.
threshold of the synapse [14]. The term h ∞ is another sigmoid, similar to m ∞ , but for
the sodium inactivation channel. The sigmoids are defined as
m ∞ (V ) =
1
1 + exp(S · (R − V ))
(8)
h ∞ (V ) =
1
1 + 0.5 · exp(S · V )
(9)
where S is the maximum slope of the sigmoid.
Sodium Channel Inactivation. The differential equation that governs the sodium
channel inactivation term, h, found in Eq. 4 is
˙
h =
h ∞ (U ) − h
τ h (U )
(10)
where τ h is the sodium inactivation time constant found using
τ h (V ) = τ h,max · h ∞ (V ) ·
0.5 · exp(S · V )
(11)
2.3 Interfacing the Dynamics
With the motor and neural dynamics laid out in the sections above, we must establish
coupling terms between them. Our goal is to use the neural states to specify position and
velocity commands for the motor. For the CPG to be able to control the movement of
the motor, the states of the neural dynamics are integrated in to the motor dynamics as
inputs as well as having certain states from the motor dynamics used in new feedback
pathways for the neurons.
Neuron Activation as Inputs to the Motor. To have the CPG specify the velocity of the
motor, the motor voltage input, V in , becomes a piecewise-linear function of the neuron
activation states.
V in (U 1 , U 2 ) = P · min(max(U 1 , 0), R) − min(max(U 2 , 0), R))
(12)
where U 1 and U 2 represent the activation states of neurons 1 and 2, respectively. The
term P is a gain term represented as
P =
4 · K v
1000 · R
(13)
that limits the maximum commanded speed to 4 rads/s, since this is the approximate
speed of the hip joint during walking locomotion found in [18].
Angular Position as Feedback for the Neurons. The feedback synapse conductance,
G fb , is set as a function of position θ . If neuron 1 (U 1 ) was active when θ reaches a
certain threshold, the feedback synapse conductance activates to allow a strong inhibitory
current to inhibit U 1 . With U 1 now inhibited, U 2 is allowed to escape and become the