Towards more biologically plausible central-pattern-generator models

Central pattern generators (CPGs) are relatively small neural networks that play a fundamental role in the control of animal locomotion. In this paper we define a method for the systematic design of CPG models able to exhibit biologically plausible gait transitions by implementing short-term synaptic plasticity mechanisms. As a case study, we focus on a simple CPG for quadruped locomotion. By applying the proposed method, three of four standard quadruped gaits were correctly reproduced by the obtained CPG model, not only in terms of the alternating sequence of the limbs but also in terms of frequency, duty cycle, and phase lags.

Figure 1

Comparison diagram of two alternatives to model a biological CPG: (left) a detailed network architecture with simple functional units [10]); (right) simplest design with biologically plausible neural and synaptic models.

Figure 2

CPG circuit of four coupled cells labeled as follows: FL and FR (fore-left and -right), HL and HR (hind-left and -right). The dashed box includes the gHCO that governs the fore limbs. The following symbols $▸, •, •$, and $•$ denote, respectively, the excitatory $E$ and inhibitory $S, F$, and $D$ synapses (see the table above). Each CPG cell controls flexor muscles regulating the swing phase of a limb, while $D$ synapses simulate the actions of the neural populations (not explicitly represented in the given model) controlling the extensor muscles. The fore and hind gHCOs are coupled through ipsilateral $F$ synapses.

Figure 3

Spatiotemporal patterns of the four mouse gaits (with the colors matching the cells in Fig. 2): bound, gallop, trot, and walk with the characteristic phase lags (as listed in Table 2) between the reference cell 1 and the other three cells of the CPG.

Figure 4

Voltage traces of the isolated burster (A1) at three different $I_c$ values: $-0.43 \left(\frac{\mu \mathrm{A}}{{\mathrm{cm}}^2}\right)$ (left panel), $-0.15 \left(\frac{\mu \mathrm{A}}{{\mathrm{cm}}^2}\right)$ (center), and 0.13 $\left(\frac{\mu \mathrm{A}}{{\mathrm{cm}}^2}\right)$ (right); the other parameters are set as listed in Appendix pp1. The threshold $V_t$ (gray horizontal line) is used to calculate the duty cycle $d$ as the ratio between the red interval and the period (red plus blue intervals).

Figure 5

Flowchart summarizing the design strategy proposed: Steps 0 through 5 focus on the steady-state dynamics of the network, while step 6 verifies the transitions between gaits.

Figure 6

Duty-cycle $d$ and bursting frequency $f$ (Hz) of the neuron model (to be compared to $d$ and $f$ in Table 2) against $I_c$ ($\mu \mathrm{A}/{\mathrm{cm}}^2$). In both panels the horizontal dashed gray lines mark the $d$ or $f$ ranges corresponding to each gait (see Table 2); the dotted rectangles mark the corresponding $I_c$ intervals for $d$ and $f$. Observe that there is no threshold separating B and G duty-cycle intervals as their ranges overlap. Colored areas highlight the $I_c$ intervals suitable (in terms of frequency and duty cycle) for each gait, as obtained from the intersection of the conditions for both features: B (blue), G (red), T (yellow), and W (green).

Figure 7

Bottom: The ratio $\psi$ between mean inhibition or excitation influx through the $S$ and $E$ synapses, plotted against the external drive $I_c$ ($\mu \mathrm{A}/{\mathrm{cm}}^2$). Top insets showing steady-state time plots of $s_j^S\left(t\right)$ at $I_c=-0.43$ (left panel), $-0.15$ (center), and 0.13 (right).

Figure 8

Color map of the phase lag ${\mathrm{\Delta }}_{14}$ over a grid of $I_c$ ($\mu \mathrm{A}/{\mathrm{cm}}^2$) and $\mathrm{\Delta }{I}_c$ ($\mu \mathrm{A}/{\mathrm{cm}}^2$) values, for fixed $g_{ij}^F$ (see Appendix for numeric values). Red dots mark ($I_c$, $\mathrm{\Delta }{I}_c$) pairs that yield ${\mathrm{\Delta }}_{14}$ values closest to the desired ones for bound, trot and walk in the proper $I_c$ intervals. Appropriate values of ${\mathrm{\Delta }}_{14}$ to model gallop are not achieved in the corresponding $I_c$ interval. Red solid lines denote the piecewise-linear function $\mathrm{\Delta }{I}_c\left(I_c\right)$ that interpolates the found ($I_c$, $\mathrm{\Delta }{I}_c$) pairs for bound, trot and walk.

Figure 9

Bifurcation diagram that shows the phase lag ${\mathrm{\Delta }}_{12}$ at steady state of the cells in the gHCO against $I_c$ ($\mu \mathrm{A}/{\mathrm{cm}}^2$). Colored rectangles highlight the $I_c$ intervals suitable for each gait determined in step 1: B (blue), G (red), T (yellow), and W (green). The vertical dashed lines indicate the parameter range where the CPG circuit becomes bistable.

Figure 10

Steady-state bound. Membrane voltage $V_i$ (mV) of each of the four neurons (colors as in Fig. 2) and synaptic activation $s_i^S$ of the efferent $S$ synapse (black lines).

Figure 11

Steady-state trot. Membrane voltage $V_i$ (mV) of each of the four neurons (colors as in Fig. 2) and synaptic activation $s_i^S$ of the efferent $S$ synapse (black lines).

Figure 12

Steady-state walk. Membrane voltage $V_i$ (mV) of each of the four neurons (colors as in Fig. 2) and synaptic activation $s_i^S$ of the efferent $S$ synapse (black lines).

Figure 13

Membrane voltage $V_i$ ($i=1$, 2, 3, 4) of the four neurons (colors as in Fig. 2) and phase lags ${\mathrm{\Delta }}_{1j}$ ($j=2$ solid line, $j=3$ dashed line, $j=4$ dotted line) during the transitions between bound and trot (a), trot and walk (b), walk and trot (c), and trot and bound (d). The transitions are obtained by varying the values of $I_c$ and $\mathrm{\Delta }{I}_c$ following the red piecewise-linear function shown in Fig. 8.

Figure 14

Fragment of the time interval where neurons 1 and 2 skip a burst after the transition from trot to walk. Membrane voltage $V_i$ (mV) of each of the four neurons (colors as in Fig. 2) and synaptic activation $s_i^S$ of the efferent S synapse (black lines).

Figure 15

Recursive flowchart that summarizes the alternative steps 4B and 5B.