Abstract
In this work, we analyze a simplified, dynamical, closed-loop, neuromechanical simulation of insect joint control. We are specifically interested in two elements: (1) how slow muscle fibers may serve as temporal integrators of sensory feedback and (2) the role of common inhibitory (CI) motor neurons in resetting this integration when the commanded position changes, particularly during steady-state walking. Despite the simplicity of the model, we show that slow muscle fibers increase the accuracy of limb positioning, even for motions much shorter than the relaxation time of the fiber; this increase in accuracy is due to the slow dynamics of the fibers; the CI motor neuron plays a critical role in accelerating muscle relaxation when the limb moves to a new position; as in the animal, this architecture enables the control of the stance phase speed, independent of swing phase amplitude or duration, by changing the gain of sensory feedback to the stance phase muscles. We discuss how this relates to other models, and how it could be applied to robotic control.
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Appendices
Appendix A: Parameter values
Variable parameters Within the parameter sets shown in Table 3, the values for m and l reflect those from the legs of a cockroach as collected for a previous study. The values of \(k_\mathrm{e}\) and \(b_\mathrm{e}\) as well as \(T_\mathrm{max}\) were taken from our group’s previous model (Rubeo et al. 2017). Values for \(y_\mathrm{off}\) were calculated as described below.
Joint model As the attachment points can vary greatly between the individual muscles of a joint, and because those values are difficult to accurately quantify, we used the same value of \( r_\mathrm{a} = 1\) mm from our group’s previous model for all muscles (Rubeo et al. 2017).
Muscle model The coefficients of the muscle model, \( b = 0.1~\frac{{\text {N}\cdot \text {s}}}{{\text {m}}}\), \( k_\mathrm{pe} = 11.24~\frac{{\text {mN}}}{{\text {mm}}}\), and \( k_\mathrm{se} = 45~\frac{{\text {mN}}}{{\text {mm}}}\) were taken from our previous model (Rubeo et al. 2017) and match the values for \( k_\mathrm{pe} \) and \( k_\mathrm{se} \) reported in (Blümel et al. 2012b).
Muscle contractile element The maximum slope of the muscle activation sigmoid \(S_\mathrm{m} = 0.3\) and the curve fitting parameter \(x_\mathrm{off} = 10\) mV were selected to match our curve to the data presented in (Blümel et al. 2012a). The value of the second curve fitting parameter \( y_\mathrm{off} \) was calculated for a given value of \( T_\mathrm{max} \) to ensure that the contractile element produced zero force when the muscle membrane was at its resting potential \( U = 0 \) mV.
Muscle membrane L-glutamate is the excitatory neurotransmitter at the neuromuscular junctions of most arthropods (Shinozaki 1988), consequently our excitatory synapses were implemented with \(\varDelta E_{e} = 40\) mV, based on measurements of the excitatory post-synaptic potential induced by the application of L-glutamate to insect muscle membranes presented in (Jan and Jan 1976a, b). A physiologically plausible value of \( g_\mathrm{m} = 1~\mu \)S was selected to allow us to define all synaptic conductances relative to the conductance of the muscle membrane, as discussed in Sect. 2.5. A value of \(C_\mathrm{m} = 150\) nF was selected to achieve a membrane time constant \(\tau _\mathrm{m} = C_\mathrm{m}/g_\mathrm{m} = 150\) ms that is similar to the value for arthropod slow muscles reported in literature (relaxation time \(\approx \) 450 ms) (Iles and Pearson 1971).
CPG The commanded amplitude \( 2\theta _\mathrm{max} = 0.5 \) rad was selected to match the realistic range of motion of a cockroach FTi joint (Watson and Ritzmann 1997a, b).
Appendix B: Derivation of muscle membrane model
The Hodgkin–Huxley model defines the dynamics of a neural or (in our case) muscle membrane potential V with respect to a membrane capacitance \(C_\mathrm{m}\), and trans-membrane leakage, synaptic, and applied currents:
As there is no external current injected into the muscle membrane, we set \(I_{app} = 0\). The leakage of ions through the membrane is governed by Ohm’s Law, where \(g_\mathrm{m}\) is the membrane’s constant leak conductance (i.e., inverse of resistance) and \(E_r\) is the resting potential of the membrane:
The sum of ion flow across each neuromuscular junction defines the synaptic current:
where \(E_i\) is the synaptic potential for the ith synapse, and \(G_i\), the instantaneous conductance of the ith synapse, is defined as a function of the maximum synaptic conductance \(g_i\) and \(V_{pre,i}\), the instantaneous potential of the presynaptic neuron:
Here, \(E_{hi}\) and \(E_{lo}\) are the upper and lower thresholds of the synapse, respectively. When we require that \(E_{lo} = E_r\), and define \(R = E_{hi} - E_{lo}\) to be the operating range of the presynaptic neuron, the instantaneous conductance of the ith synapse can be written as:
This expression reduces to:
if we restrict the presynaptic voltage \(V_{pre,i} \in \left[ E_{lo},E_{hi}\right] \) or alternatively \(V_{pre,i} \in \left[ E_r,R-E_r\right] \).
To further simplify analysis, we define a new variable \(U = V - E_r\), which quantifies the voltage of the neural or muscle membrane relative to its resting potential. Note that because V does not appear outside of this derivation, we refer to U throughout the paper simply as the muscle membrane potential. In a similar manner, we define the presynaptic voltage relative to the resting potential \(U_{pre,i} = V_{pre,i} - E_r\). In terms of this variable, the instantaneous conductance of the ith synapse becomes:
where \(U_{pre,i}\) is restricted such that \(U_{pre,i} \in \left[ 0,R\right] \).
Furthermore, it is convenient to select \( R = 1 \), effectively representing the behavior of all presynaptic neurons as a fraction of their maximum activity. We call \( \hat{U}_{pre} \in \left[ 0,1 \right] \) the activation of the ith presynaptic neuron. When we also define the synaptic potential relative to the resting potential \(\varDelta E_i = E_i - E_r\) the sum of synaptic currents simplifies to the form:
and using the same variables, the leak current simplifies to:
Given that \(\dot{U} =\dot{V}\), combining the modified current terms together into the original equation yields:
For more details regarding these manipulations of the neural and synaptic models, see (Szczecinski et al. 2017b).
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Naris, M., Szczecinski, N.S. & Quinn, R.D. A neuromechanical model exploring the role of the common inhibitor motor neuron in insect locomotion. Biol Cybern 114, 23–41 (2020). https://doi.org/10.1007/s00422-019-00811-y
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DOI: https://doi.org/10.1007/s00422-019-00811-y