Neuromechanical Model of fCO Sensory Inhibition
145
to non-spiking neurons up until a threshold value, θ. Their membrane voltages
evolve according to Eq. 1, with the added condition that:
if V = θ, then V (t) ←− E r .
(5)
When a presynaptic neuron spikes, for each of its synapses, G s is set to G max ,
then decays according to the differential equation:
τ s
dG s
dt
= −G s
(6)
where τ s is the time constant of the synapse.
To directly control a simulated limb, we added an additional non-spiking
neuron after the slow MNs as an analog for the insect’s muscle fibers. We set
the time constant of these slow muscle fiber neurons to 2000 ms so the neuron
acts as a leaky integrator of synaptic inputs similar to a muscle [15]. The neuron’s voltage is then used to command the position and velocity of a simulated,
Drosophibot-sized limb through a pair of neuromechanical adapters. The limb’s
mechanics are governed by the equation of motion:
J · ¨
θ = τ ext + τ flex − k spring · θ
(7)
where J is the moment of inertia of the limb, k spring is the stiffness of the
limb’s parallel elastic elements and τ ext,f lex are the torques determined by the
software to drive the joint at the speeds and to the positions indicated by the
adapters. The precise tuning of these adapters can be found in ref. [8].
We defined the synapse conductance strengths in the network according to
the relative values given in Fig. 11 of ref. [13] with some additional tuning to
more closely match the recorded response of each NSI to stimuli presented in
Fig. 2 of ref. [13]. Figure 2 compares the responses of the interneurons in model
to those in the animal. As the exact conversion between fCO stretch and injected
current has not been characterized, we arbitrarily chose a stimulus strength of
5 nA applied to the sensory neurons over 3.25 s. The stimulus ramps up to and
down from the hold current over a period of 0.25 s. Because Drosophibot is about
5 times larger than the stick insect, the length of the stimulus was made about
5 times longer [10].
After validating the behavior of the connections for extension, we expanded
the network to include flexion. Control of the full leg’s motion was necessary
to simulate full joint behavior and observe the effects of modulating the NSI.
However, innervation of the flexor muscle is believed to be more complicated
than that of the extensor, and the network connections for the flexor slow MN
have not been mapped in the insect. In regards to the functional responses of the
joint, B¨ assler et al. recorded the forces of the extensor and flexor tibiae muscles
in the stick insect for sinusoidal stimulus of the fCO and found that the forces
varied in nearly equal and opposite ways (Fig. 3 in ref. [2]). These results seem
to imply that the flexor networks may resemble or mirror those for extension.
As such, we elected to mirror the connections from the extensor network onto
a series of flexor NSIs, as well as adding a flexor slow MN and slow muscle
145
to non-spiking neurons up until a threshold value, θ. Their membrane voltages
evolve according to Eq. 1, with the added condition that:
if V = θ, then V (t) ←− E r .
(5)
When a presynaptic neuron spikes, for each of its synapses, G s is set to G max ,
then decays according to the differential equation:
τ s
dG s
dt
= −G s
(6)
where τ s is the time constant of the synapse.
To directly control a simulated limb, we added an additional non-spiking
neuron after the slow MNs as an analog for the insect’s muscle fibers. We set
the time constant of these slow muscle fiber neurons to 2000 ms so the neuron
acts as a leaky integrator of synaptic inputs similar to a muscle [15]. The neuron’s voltage is then used to command the position and velocity of a simulated,
Drosophibot-sized limb through a pair of neuromechanical adapters. The limb’s
mechanics are governed by the equation of motion:
J · ¨
θ = τ ext + τ flex − k spring · θ
(7)
where J is the moment of inertia of the limb, k spring is the stiffness of the
limb’s parallel elastic elements and τ ext,f lex are the torques determined by the
software to drive the joint at the speeds and to the positions indicated by the
adapters. The precise tuning of these adapters can be found in ref. [8].
We defined the synapse conductance strengths in the network according to
the relative values given in Fig. 11 of ref. [13] with some additional tuning to
more closely match the recorded response of each NSI to stimuli presented in
Fig. 2 of ref. [13]. Figure 2 compares the responses of the interneurons in model
to those in the animal. As the exact conversion between fCO stretch and injected
current has not been characterized, we arbitrarily chose a stimulus strength of
5 nA applied to the sensory neurons over 3.25 s. The stimulus ramps up to and
down from the hold current over a period of 0.25 s. Because Drosophibot is about
5 times larger than the stick insect, the length of the stimulus was made about
5 times longer [10].
After validating the behavior of the connections for extension, we expanded
the network to include flexion. Control of the full leg’s motion was necessary
to simulate full joint behavior and observe the effects of modulating the NSI.
However, innervation of the flexor muscle is believed to be more complicated
than that of the extensor, and the network connections for the flexor slow MN
have not been mapped in the insect. In regards to the functional responses of the
joint, B¨ assler et al. recorded the forces of the extensor and flexor tibiae muscles
in the stick insect for sinusoidal stimulus of the fCO and found that the forces
varied in nearly equal and opposite ways (Fig. 3 in ref. [2]). These results seem
to imply that the flexor networks may resemble or mirror those for extension.
As such, we elected to mirror the connections from the extensor network onto
a series of flexor NSIs, as well as adding a flexor slow MN and slow muscle