Robustness, flexibility, and sensitivity in a multifunctional motor control model

Abstract

Motor systems must adapt to perturbations and changing conditions both within and outside the body. We refer to the ability of a system to maintain performance despite perturbations as "robustness," and the ability of a system to deploy alternative strategies that improve fitness as "flexibility." Different classes of pattern-generating circuits yield dynamics with differential sensitivities to perturbations and parameter variation. Depending on the task and the type of perturbation, high sensitivity can either facilitate or hinder robustness and flexibility. Here we explore the role of multiple coexisting oscillatory modes and sensory feedback in allowing multiphasic motor pattern generation to be both robust and flexible. As a concrete example, we focus on a nominal neuromechanical model of triphasic motor patterns in the feeding apparatus of the marine mollusk Aplysia californica. We find that the model can operate within two distinct oscillatory modes and that the system exhibits bistability between the two. In the "heteroclinic mode," higher sensitivity makes the system more robust to changing mechanical loads, but less robust to internal parameter variations. In the "limit cycle mode," lower sensitivity makes the system more robust to changes in internal parameter values, but less robust to changes in mechanical load. Finally, we show that overall performance on a variable feeding task is improved when the system can flexibly transition between oscillatory modes in response to the changing demands of the task. Thus, our results suggest that the interplay of sensory feedback and multiple oscillatory modes can allow motor systems to be both robust and flexible in a variable environment.

Keywords: Adaptive behavior; Aplysia; Central pattern generator; Heteroclinic channel; Limit cycle; Multistability; Sensory feedback.

Fig. 1

Dynamical architectures for motor pattern generation differ in their responsiveness to sensory input. Left At one extreme, the endogenous oscillatory dynamics generated within the nervous system (cycling arrows) drive the musculature (brain $\to$ body) and produce behavior (body $\to$ environment). Mechanical perturbations (environment $\to$ body) may have only weak effects on the central neural dynamics since they are insensitive to sensory inputs (body $\to$ brain). This architecture facilitates robustness in behaviors for which the sensitivity to internal and external perturbations should be low, but can fail during behaviors that require high sensitivity to sensory inputs. Right In contrast, in a sensory-driven pattern generator, the neural dynamics depend upon appropriately timed sensory inputs to progress through the cycle, rather than being driven by an endogenous pattern generator. This architecture can facilitate robustness in behaviors that require the timing of motor outputs to change in response to environmental conditions, since the sensitivity to sensory input is high, but can fail for behaviors where high sensitivity to sensory inputs or their absence is detrimental

Fig. 2

Phases of ingestive behavior and neural pool-muscle relationships in the model. a Ingestive behaviors in the model can be divided into three major phases, depending on the direction of movement and state (open or closed) of the grasper: protraction while open (1), protraction while closed (2), and retraction (3). The grasper (red) is moved anteriorly (to the right) by the contraction of the sheet-like protractor muscle (I2; blue). The grasper is moved posteriorly (to the left) by the contraction of the ring-like retractor muscle (I3; yellow). A section of the retractor muscles has been cut away so that the grasper is visible. The green strand is seaweed, with the arrows indicating seaweed movement. b The protractor muscle is activated by the $a_0$ and $a_1$ neural pools, and the retractor muscle is activated by the $a_2$ neural pool. The grasper is closed if the sum $a_1+a_2$ exceeds a specified threshold and open otherwise. The protraction-open phase corresponds to $a_0>\text{max}\{a_1,a_2\}$, the protraction-closed phase corresponds to $a_1>\text{max}\{a_0,a_2\}$, and the retraction-closed phase corresponds to $a_2>\text{max}\{a_0,a_1\}$. As each neural pool activates in turn, it inhibits the neural pool preceding it in the sequence. This, in combination with weak endogenous excitation in each neural pool, is sufficient to create periodic activation that drives ingestive behavior. Figure adapted from Shaw et al. (2015) (color figure online)

Fig. 3

Model of the Aplysia feeding apparatus has two coexisting stable oscillatory modes that exhibit distinct behaviors: the limit cycle mode (left) and heteroclinic mode (right). In both cases, the task is continuous-swallowing, the endogenous excitation $\mathit{\mu }={10}^{-5}$, and the mechanical load $F_{sw}=0$ is constant. The different behaviors are produced by using different initial conditions for the neural state variables: ($a_0(0)$, $a_1(0)$, $a_2(0)$) is ($1-{10}^{-9}$, ${10}^{-9}$, ${10}^{-9}$) for the heteroclinic mode and (0.2, 0.4, 0.7) for the limit cycle mode. Top panels Trajectories of the three neural state variables $a_0$, $a_1$, and $a_2$ (blue, red, and yellow, respectively), and the position of the grasper (black when open, thick dark green when closed). In the limit cycle mode, the neural dynamics are relatively insensitive to sensory feedback, so the durations of the three phases of the motor pattern are short and approximately equal. Since the muscles are slow to respond to changes in the neural variables (${\mathit{\tau }}_m\gg {\mathit{\tau }}_a$), rapid neural cycling does not allow adequate time for the muscles to develop strong forces or to fully relax, causing the antagonistic muscles to be tonically moderately activated, and thus movement is limited. In contrast, in the heteroclinic mode, the neural dynamics are sensitive to and asymmetrically slowed by proprioceptive feedback. The protraction-open ($a_0$, blue) and retraction-closed ($a_2$, yellow) phases last longer, resulting in a longer cycle period overall and greater range of motion. Bottom panels Movement of the seaweed. In the limit cycle mode, since the durations of the protraction-closed ($a_1$, red) and retraction-closed ($a_2$, yellow) phases are approximately equal, the grasper pushes seaweed out more than it pulls seaweed in each cycle, resulting in a net loss of food. In contrast, in the heteroclinic mode, the retraction-closed phase is extended, and seaweed is consumed (color figure online)

Fig. 4

Mechanism distinguishing the two oscillatory modes. The key difference is whether the neural variables collide with the boundaries. This can occur when the proprioceptive feedback $g_i(x_{\text{r}})$ overcomes the endogenous excitation $\mathit{\mu }/{\mathit{\tau }}_{\text{a}}$, such that the net input is negative. Plots show the total inputs, $g_i(x_{\text{r}})+\mathit{\mu }/{\mathit{\tau }}_{\text{a}}$, to the protraction-open ($a_0$, blue) and protraction-closed ($a_1$, red) neural pools over one complete cycle in the limit cycle mode (top) and in the heteroclinic mode (bottom). Here the beginning of the cycle is defined as the onset of the $a_0$ neural pool, and time $t=0$ corresponds to the closing of the grasper. Parameters and initial conditions as in Fig. 3. The dashed lines indicate zero net input. Bottom In the heteroclinic mode, the net input can become negative due to the proprioceptive inputs overcoming the endogenous excitation (downward arrows). If this net negative input is sufficiently strong, and the neural state variable receiving it is sufficiently close to zero, the trajectory of the neural variables will collide with the boundary $a_i=0$. Collision with the boundary stops the cycling of the neural variables, which does not resume until the grasper reaches the appropriate position, releasing the neural pool from inhibitory proprioceptive input (upward arrows). Top Such boundary collisions do not occur in the limit cycle mode. Consequently, in this mode the neural cycle progresses almost independently of the state of the biomechanical variables. Note that we did not plot the total input to the $a_{\text{2}}$ pool because it never overcomes the endogenous excitation in either oscillatory mode (color figure online)

Fig. 5

Heteroclinic mode is more common than the limit cycle mode when endogenous excitation is weak and load is large. The percentage of initial conditions sampled for the neural pool variables converging to the heteroclinic mode is plotted on the vertical axis (standard errors are approximately 0.5$\%$ and are not shown). Each line corresponds to a different value of a fixed resisting force. For each combination of $\mathit{\mu }$ and $F_{\text{sw}}$, 10,000 independent simulations were performed, each with initial conditions for the three neural state variables sampled independently and uniformly from within the unit cube. Initial conditions for the other state variables were always the same: $x_{\text{r}}(0)=0.5,u_0(0)=0$, and $u_1(0)=0$. For sufficiently small $\mathit{\mu }$, virtually all initial conditions converge to the heteroclinic mode. As $\mathit{\mu }$ increases, this proportion drops, and for sufficiently large $\mathit{\mu }$, almost all initial conditions converge to the limit cycle mode. In contrast, as $F_{\text{sw}}$ increases, this proportion increases as long as $\mathit{\mu }$ is not too great. This is explained by the mechanism of competing endogenous excitation and inhibitory sensory feedback (see Fig. 4): since larger force evokes greater sensory feedback, net input to a neural pool is more likely to become negative if $F_{\text{sw}}$ is large and $\mathit{\mu }$ is small

Fig. 6

Limit cycle mode is more robust to changes in sensory input parameters than the heteroclinic mode. Plots show the rate of seaweed intake in the continuous-swallowing task with $F_{\text{sw}}=0$ across a range of values (between $-10\%$ and $+10\%$ of their default values) for the sensory input parameters $S_i$. The blue, red, and yellow lines represent the effect of varying the input parameters to the $a_0$, $a_1$, and $a_2$ neural pools, respectively. Right The slopes of the red and blue lines are significantly steeper in the heteroclinic mode case, indicating that in this mode, the performance of the model is more strongly affected by small changes in parameter values. Left In comparison, the performance of the limit cycle mode changes much less in response to varying sensory input parameters. The limit cycle mode is more robust in the sense that it is less susceptible to potentially deleterious changes in the parameters. Here the limit cycle mode has been tuned to have comparable performance to that of the heteroclinic mode when the parameters are unperturbed (color figure online)

Fig. 7

Heteroclinic mode is more responsive to changes in mechanical load than the tuned limit cycle mode. All panels show the time courses of the neural activity variables and position of the grasper for one cycle in the continuous-swallowing task (time axes have the same scale in all panels; see Fig. 3 for color key). Right In the heteroclinic mode, the motor pattern adapts to an increase in load (from $F_{\text{sw}}=0$ to $F_{\text{sw}}=0.1$) by lengthening the durations of the protraction-open and retraction-closed phases. Left In contrast, in the limit cycle mode, the motor pattern does not change significantly in response to the same load increase (color figure online)

Fig. 8

In the heteroclinic mode the system compensates for a load increase by pulling longer and stronger during the retraction phase of the motion. Plots show the change in grasper position (top) and net muscle force applied (bottom; Eq. 10) before and after a 40% increase in load while in the heteroclinic mode ($\mathit{\mu }={10}^{-5}$). Solid green lines load condition $F_{\text{sw}}=0.05$. Dashed black lines load condition $F_{\text{sw}}=0.07$. Thick segments grasper closed on seaweed. Thin segments grasper open. Under the higher load, grasper displacement during the closed phase (and hence seaweed displacement) increases by 4%, i.e., the grasper takes in more seaweed per cycle. This is achieved by a 25% increase in the integrated net force (inward) applied to the seaweed during the grasper-closed period. At the same time the cycle time lengthens by 5%, reducing the fitness measure (length of seaweed consumed per second). The net effect on fitness is a decrease of 1%, an order of magnitude smaller than the 40% percent load increase (color figure online)

Fig. 9

Robustness to load perturbations in the heteroclinic mode is achieved through adjustments in the timing of neural activation, but not the trajectory paths of the neural variables through phase space. Panels show planar projections of the neural variables (left) and the body variables (right) for two trajectories operating in the heteroclinic mode ($\mathit{\mu }={10}^{-5}$) under different loads (as in Fig. 8). Solid green lines load condition $F_{\text{sw}}=0.05$. Dashed black lines load condition $F_{\text{sw}}=0.07$. Thick line segments grasper closed on seaweed. Thin line segments grasper open. See footnote 8 for description of projections. Arrows indicate the direction of cycling. Left In a planar projection of the neural variables, the paths of the two trajectories are indistinguishable, and only their timing may differ. Right A planar projection of the body variables shows that the heteroclinic mode responds to load by increasing retractor muscle activation (left on the x-axis) and retracting farther (down on the y-axis). Since the neural variables follow the same path, the difference in their activity driving the changes in the body variables is in the timing of transitions between the neural fixed points (color figure online)

Fig. 10

Transitions between oscillatory modes can be induced solely by load. In this example of the forage-and-feed task, $\mathit{\mu }=1.6\times {10}^{-5}$, $\mathit{\kappa }=0$, all parameters other than load are constant, and seaweed was placed in the buccal cavity for a predetermined period of time. The system starts in the limit cycle mode without seaweed, biting rapidly. As soon as the grasper closes on the force-loaded seaweed (“Load on”; $F_{\text{sw}}=0.09$), it is pulled forward by the force. One cycle later, the system transitions to the heteroclinic mode because the increased protraction triggers increased inhibitory proprioceptive feedback to the protraction-open neural pool ($a_0$, blue) that overcomes endogenous excitation (see Fig. 4), allowing the retraction-closed neural pool ($a_2$, yellow) to remain active longer during the following retraction phase. This results in greater retractor muscle force and an enhanced retraction of the grasper, allowing the system to successfully overcome the load on the seaweed and pull it inward. The enhanced retraction triggers increased inhibitory proprioceptive feedback to the protraction-closed neural pool ($a_1$, red) that again overcomes endogenous excitation, and the following protraction-open phase of the grasper movement is further enhanced. This cycling in the heteroclinic mode continues until the seaweed is removed (or consumed), after which the system transitions back to the limit cycle mode. With $\mathit{\kappa }=0$, this transition is not guaranteed to occur for all seaweed forces and is less likely to occur for weak forces. See Fig. 3 for full color key (color figure online)

Fig. 11

Fitness is maximized for the forage-and-feed task when the system flexibly switches between oscillatory modes. Left Performance of the model (seaweed intake rate, S) on the forage-and-feed task as a function of endogenous excitation ($\mathit{\mu }$) and seaweed force ($F_{\text{sw}}$), presented as a heat map (a1) and a 3D surface plot (a2). Here performance is measured in terms of mean seaweed intake over a 300-second trial, averaged over 500 trials for each set of parameter values. The strength of the seaweed-triggered inhibition is fixed at $\mathit{\kappa }=0.5$. Across nearly the entire range of forces explored, mean seaweed intake peaks at a nonzero $\mathit{\mu }$ value, forming the ridge that runs down the $F_{\text{sw}}$-axis of the plot. This ridge corresponds to parameter combinations that allow for flexible switching between oscillatory modes. In contrast, when $\mathit{\mu }$ is too small, the system never leaves the heteroclinic mode, and performance is moderate because biting is slow. Similarly, when $\mathit{\mu }$ is too large, the seaweed-triggered inhibition is too weak to induce a switch into the heteroclinic mode, and the system operates only in the limit cycle mode, which fails to swallow seaweed. Right Predicted performance of the model on the forage-and-feed task based on measurements from the continuous-swallowing task, presented as a heat map (b1) and a 3D surface plot (b2). Equation 20 (see “Appendix 3”; $L=0.5$, $\mathit{\kappa }=0.5$, $p=0.1$ to match simulations) approximates the mean seaweed intake rate and captures the key qualitative features of the numerically computed plots (left), particularly the region of enhanced performance at intermediate values of $\mathit{\mu }$. Note that the range of $\mathit{\mu }$ values over which this improvement occurs matches well with the numerical plots