Design process and tools for dynamic neuromechanical models and robot controllers

Abstract

We present a serial design process with associated tools to select parameter values for a posture and locomotion controller for simulation of a robot. The controller is constructed from dynamic neuron and synapse models and simulated with the open-source neuromechanical simulator AnimatLab 2. Each joint has a central pattern generator (CPG), whose neurons possess persistent sodium channels. The CPG rhythmically inhibits motor neurons that control the servomotor’s velocity. Sensory information coordinates the joints in the leg into a cohesive stepping motion. The parameter value design process is intended to run on a desktop computer, and has three steps. First, our tool FEEDBACKDESIGN uses classical control methods to find neural and synaptic parameter values that stably and robustly control servomotor output. This method is fast, testing over 100 parameter value variations per minute. Next, our tool CPGDESIGN generates bifurcation diagrams and phase response curves for the CPG model. This reveals neural and synaptic parameter values that produce robust oscillation cycles, whose phase can be rapidly entrained to sensory feedback. It also designs the synaptic conductance of inter-joint pathways. Finally, to understand sensitivity to parameters and how descending commands affect a leg’s stepping motion, our tool SIMSCAN runs batches of neuromechanical simulations with specified parameter values, which is useful for searching the parameter space of a complicated simulation. These design tools are demonstrated on a simulation of a robot, but may be applied to neuromechanical animal models or physical robots as well.

Appendix

1.1 Neural parameters

The parameter values for the network in Fig. 1 are listed below. All neural properties are scaled to ms time scale, producing base units of mV, uS, nF, and nA. Unless otherwise noted, all neurons have the parameters $C_m = 5$, $G_m = 1$, $E_r = -60$, $G_{Na} = 0$.

HC Neurons $C_m = 5$, $G_m = 1$, $E_r = -60$, $G_{Na} = 1$, $E_{Na} = 50$, $A_h = 0.5$, $S_h = -0.046$, $E_h = -60$, $A_m = 1$, $S_m = 0.046$, $E_m = -40$.

Servo Interface $C_m = 50$, $E_r = -60 - 20 \cdot \theta _{\min } / (\theta _{\max } - \theta _{\min })$. This ensures that when the simulation starts with $\theta = 0$, the Servo Interface neuron starts at its resting potential. Any value will work, but a different value will cause unnecessary motion at startup.

Perceived Rotation Like the Servo Interface neuron, $E_r = -60 - 20 \cdot \theta _{\min } / (\theta _{\max } - \theta _{\min })$.

Rotation (slow) $C_m = 50$.

Flexion Trigger $E_r = -75$.

Rotation (slow) $C_m = 10$.

1.2 Synaptic parameters

Unless stated otherwise, all synapses have $E_{lo} = -60$ and $E_{hi} = -40$.

HC $\rightarrow$ IN. $g = 0.118$, $E_s = 300$, $E_{hi} = -20$.

IN $\rightarrow$ HC. $g = 1.041$, $E_s = -100$, $E_{hi} = -20$.

MN $\rightarrow$ Compare. Excitatory connection: $g = 1$, $E_s = -20$, $E_{lo} = -70$, $E_{hi} = -30$. Inhibitory connection: $g = 1$, $E_s = -100$, $E_{lo} = -70$, $E_{hi} = -30$.

Compare $\rightarrow$ Servo Interface. Excitatory connection: $g = 1$, $E_s = 0$. Inhibitory connection: $g = 1$, $E_s = -100$.

Perceived Rotation $\rightarrow$ Extension MN. $g = 0.5$, $E_s = -100$

Perceived Rotation $\rightarrow$ Rotation (fast/slow). $g = 0.0588$, $E_s = 300$. This parameter combination makes the steady-state voltage of the postsynaptic neuron the current voltage of the presynaptic neuron.

Rotation (fast/slow) $\rightarrow$ Velocity. $g = 0.133$, $E_s = -20$, $E_{hi} = -60$.

Rotation (fast/slow) $\rightarrow$ Velocity. $g = 0.133$, $E_s = -100$, $E_{hi} = -60$.

1.3 Mechanical parameters

The servomotors run a proportional feedback loop with an experimentally determined gain of $k =$ 15.2 Nm/rad. Damping has been determined to be $c =$ 2.5 Nms/rad. The inertia was that of one MX-64T servomotor placed 30 cm from the joint, $I = 13.5\times 10^{-3}$ kg$\cdot$m$^2$.

1.4 Computer specifications

All data were collected using a desktop computer with an Intel i5-4690K CPU running at 3.50 GHz and 8 GB of RAM. FEEDBACKDESIGN, CPGDESIGN, and SIMSCAN all parallelize their computations. Four parallel threads were used in this work.

1.5 Modulatory pathways

To simplify figures in this paper, modulatory connections were drawn as synapses onto other synapses, as shown in Fig. 11a. In practice, this was implemented as a disinhibitory pathway that changed the conductance, and thus sensitivity, of an interneuron (Fig. 11b). This is similar to GABAergic synapses, which change the effective size of the postsynaptic neuron, reducing its sensitivity to incoming currents [39]. The Modu. (Modulatory) neuron’s voltage controls the effective conductance of the connection between the Input and Output neurons between 0% when at $-60$ mV and 100% when at $-40$ mV. The plots in Fig. 11c, generated with SIMSCAN, show that the activation of the Modu. neuron directly scales the Output neuron’s activation for the same Input activations.

All neurons in Fig. 11b. have the typical set of parameters listed in Sect. 1, except the Inter. 1 neuron, whose resting potential is $E_r = -40$. The inhibitory synapses are both identical, with $G_s = 20$, $E_s = -61$, $E_{lo} = -60$, and $E_{hi} = -40$.

Fig. 11

Modulatory networks, as drawn in this paper (a), were implemented in the model as disinhibitory pathways (b). c Changing the Modu. neuron’s activation adjusts the input–output gain

1.6 FEEDBACKDESIGN

FEEDBACKDESIGN automates the analysis presented in Sects. 3.1 and 3.2. The user provides parameter values for a closed- or open-loop network of neurons, synapses, and a servomotor and limb if desired. Neurons that interface as inputs or outputs to the servomotor, and that feedback to the input neuron, are specified by the user. The equilibrium state is found by simulating the system until the energy goes to 0. If this happens, then the eigenvalues of the equilibrium point are found to ensure that the point is stable. If instead the energy diverges, the system is deemed unstable and no further analysis is conducted.

The transfer function is generated for each neuron and neuron-servomotor complex, and the open-loop transfer function of each node (i.e. neuron or servomotor) is calculated by compounding Eqs. 34 and 35 for each node along the path. The closed loop transfer function is calculated by Eq. 39. The user can also query the stability margins, which are calculated by using a Newton minimizer to find the crossing points of the gain and phase responses (Sect. 3.2). The system’s parameter values can be varied and these analyses repeated to produce plots like those in Figs. 3 and 4. FEEDBACKDESIGN can be downloaded at

+http://biorobots.case.edu/download/neural_des+

+ign_tools/FEEDBACKDESIGN.zip+.

1.7 CPGDESIGN

CPGDESIGN automates the analysis presented in Sect. 4.1. It assumes the network structure in Fig. 5. For a set of parameter values, it will calculate $\delta$, or if $\delta$ is provided, it will compute the corresponding strength of mutual inhibition. The user can simulate the dynamics, with the option to add external stimuli. When the simulation is complete, the equilibrium points will be found at every time step, using a log-bounded interior point Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimizer. The eigenvalues and eigenvectors will also be calculated. The user then has the option to animate the phase portrait for each neuron over time, and calculate the nullclines corresponding to the equilibrium states. In this way the user can gain an intuitive understanding of the CPG’s dynamics. Finally, the program facilitates studying phase response properties by calculating whether or not the output was periodic, the period of oscillation, and the rising edges of one HC’s activity. CPGDESIGN can be downloaded at

+http://biorobots.case.edu/download/neural_des+

+ign_tools/CPGDESIGN.zip+.

1.8 SIMSCAN

SIMSCAN automates the analysis presented in Sects. 4.2 and 5.2. The user makes a simulation with AnimatLab 2 and exports a “Standalone” simulation, which can be run from the command line. The user then tells SIMSCAN the directory of this file, the parameters to change and their values, and an objective function with which to process the data (in this paper, it simply extracts the speed of the “treadmill”). SIMSCAN then runs the simulations with the desired values and saves the output from the objective function. This process can easily be parallelized, and multiple objective functions can be used to measure different quantities. SIMSCAN will also produce a function handle that takes the desired parameters as inputs, and runs the simulation and computes the objective as an output. This handle can be used to optimize the parameters of the simulation to perform a specific task. SIMSCAN can be downloaded at

+http://biorobots.case.edu/download/neural_des+

+ign_tools/SIMSCAN.zip+.