Tradeoffs in the energetic value of neuromodulation in a closed-loop neuromechanical system

Abstract

Rhythmic motor behaviors controlled by neuromechanical systems, consisting of central neural circuitry, biomechanics, and sensory feedback, show efficiency in energy expenditure. The biomechanical elements (e.g., muscles) are modulated by peripheral neuromodulation which may improve their strength and speed properties. However, there are relatively few studies on neuromodulatory control of muscle function and metabolic mechanical efficiency in neuromechanical systems. To investigate the role of neuromodulation on the system's mechanical efficiency, we consider a neuromuscular model of motor patterns for feeding in the marine mollusk Aplysia californica. By incorporating muscle energetics and neuromodulatory effects into the model, we demonstrate tradeoffs in the energy efficiency of Aplysia's rhythmic swallowing behavior as a function of the level of neuromodulation. A robust efficiency optimum arises from an intermediate level of neuromodulation, and excessive neuromodulation may be inefficient and disadvantageous to an animal's metabolism. This optimum emerges from physiological constraints imposed upon serotonergic modulation trajectories on the energy efficiency landscape. Our results may lead to experimentally testable hypotheses of the role of neuromodulation in rhythmic motor control.

Keywords: Central pattern generator; Closed-loop control; Energetics; Muscle; Neuromechanics; Neuromodulation.

Figure 1:

(A) Schematic of Aplysia swallowing, consisting of three phases. During the protraction-open phase (lower right), the grasper (red), which stays open to grasp food, is protracted (moves forward) by the I2 buccal muscle (blue). Then the grasper begins closing on the food and is protracted a small distance while closed, referred to as protraction-closed phase (left middle). In the last phase, retraction phase (upper right), the I3 buccal muscle (yellow) retracts the grasper backwards to pull the food into the buccal cavity, completing the swallowing cycle. The seaweed is indicated by the green strand, with the green arrows showing how the seaweed moves within a cycle. (B) Three neural pools, with firing rates denoted by $a_0$, $a_1$, $a_2$, control the three phases. The protraction muscle I2 is driven by the neural inputs from $a_0$ and $a_1$ (blue solid line and triangle), and the retraction muscle I3 receives neural inputs from $a_2$ (yellow solid line and triangle). The grasper is closed when the sum of $a_1$ and $a_2$ exceeds a threshold and is open otherwise, as represented by the red dashed line and summation symbol. Serotonin, as a neuromodulatory substance, acts in part on the two buccal muscles (gray arrows). Figure redrawn from Shaw et al. (2015).

Figure 2:

Time courses for the default system (in the absence of neuromodulation) of (A) firing rates of neural pools ($a_0$, $a_1$, $a_2$), (B) muscle activation ($u_0$, $u_1$), (C) grasper position ($x_r$), (D) protractor muscle force ($F_{\text{musc},\text{pro}}$), retractor muscle force ($F_{\text{musc},\text{ret}}$), and total muscle force ($F_{\text{musc},\text{pro}}+F_{\text{musc},\text{ret}}$), (E) seaweed movement, and (F) energy output (Total = Work + Heat), plotted over two periods with $T=6.44\mathrm{s}$. The gray shaded regions denote the phase when the grasper is closed (protraction-closed and retraction), with duration $T_{\text{closed}}=3.03\mathrm{s}$; the white regions indicate the phase when the grasper is open (protraction-open), with duration $T_{\text{open}}=3.41\mathrm{s}$. The seaweed moves together with the grasper during the grasper-closed phase in a rate $-d{x}_r∕dt$, while it does not move when the grasper is open (compare panels (C) and (E)).

Figure 3:

Work loops of the default system, with the same parameter values as Fig. 2. Black: protractor muscle force $F_{\text{musc},\text{pro}}$. Blue: retractor muscle force $F_{\text{musc},\text{ret}}$. Red: total muscle force $F_{\text{musc}}$. Dots and stars denote the start and the end of the grasper-closed phase, respectively. Arrows mark the flow direction of the loops. The area enclosed by the $F_{\text{musc}}$ curve (red loop) indicates the net mechanical work done by the muscles during a feeding cycle, $W=0.0875\mathrm{N}\cdot \text{cm}$.

Figure 4:

Effects of the concentration of serotonin $\rho$ on muscle contraction strength $k_i$ (blue) and muscle relaxation time ${\tau }_{m,i}$ (red). The effect on the muscle contraction strength is modeled by $k_i(\rho )=0.4+3.6∕(1+e^{-3(\log \rho +7.5)})$, which is qualitatively similar to the 5-HT curve in Fig. 11 of Hurwitz et al. (2000). The effect on the muscle relaxation time follows ${\tau }_{m,i}(\rho )=2.45+0.93∕(1+e^{2.5(\log \rho +8.85)})$, approximating the series of decay durations shown in Fig. 8(A2) of Hurwitz et al. (2000) and scaled appropriately to fit our model.

Figure 5:

Effects of serotonin concentration $\rho$ on the energetic consumption and efficiency. (A): Heat output $H$ (blue), work output $W$ (black), and total energy costs $W+H$ (red). (B): Efficiency $\Phi =W∕(W+H)$. The peak efficiency is marked in green, where ${\rho }^{\ast }\approx {10}^{-7.78}\mathrm{M}$ (i.e., 16.6 nM) and ${\Phi }^{\ast }\approx 0.532$. Each pair of arrows of the same color corresponds to the systems being compared before the optimum (magenta arrows $\mathrm{①}$, see Figs. 6, 7, 8), around the optimum (gray arrows $\mathrm{②}$, see Figs. 9, 10), and after the optimum (yellow arrows $\mathrm{③}$, see Fig. 11).

Figure 6:

Comparison of grasper trajectories prior to optimal modulation. Solid: $\rho ={10}^{-9}\mathrm{M}$; dashed: $\rho ={10}^{-8.5}\mathrm{M}$ (Fig. 5B, magenta arrows $\mathrm{①}$). Same layout as in Fig. 2. Both systems are initiated at the start of their respective closed phase and plotted over one period. The gray shaded region represents the closed phase of the system with $\rho ={10}^{-9}\mathrm{M}$, and the white region represents its open phase. The vertical magenta solid line denotes the transition time of the system with $\rho ={10}^{-8.5}\mathrm{M}$ out of the closed phase. The effect of the increased neuromodulation on neural pools is delayed until the transition from closed to open (panel A, red arrow). The vertical magenta dotted line denotes the end of the open phase. Both closed and open phases are advanced due to the effect of the higher serotonin concentration on accelerating muscles’ relaxation (panel B). Another effect is manifested by the stronger muscle forces (panel D), which allows the system to pull in more seaweed (panels C and E) in even a shorter time. Hence, more work is done by the muscles (panel F).

Figure 7:

Comparison of the work loops for the modulated systems with $\rho ={10}^{-9}\mathrm{M}$ (solid) and $\rho ={10}^{-8.5}\mathrm{M}$ (dashed) (the same systems as in Fig. 6). Colors and notations as in Fig. 3. The higher concentration of serotonin increases muscle forces to induce the system to produce larger retraction and protraction movements. The net work generated by the muscles within a single movement (area inside the red loops) is increased from 0.1043 N·cm for $\rho ={10}^{-9}\mathrm{M}$ to 0.133 N·cm for $\rho ={10}^{-8.5}\mathrm{M}$.

Figure 8:

Comparison of heat production for the modulated systems with $\rho ={10}^{-9}\mathrm{M}$ (solid) and $\rho ={10}^{-8.5}\mathrm{M}$ (dashed) (the same systems as in Figs. 6, 7) The gray shaded regions and the magenta lines have the same meanings as in Fig. 6. Top panels: heat generated by lengthening contraction, consisting of (A) stable heat, (B) labile heat, and (C) shortening heat. Bottom panels: heat generated by shortening contraction, consisting of (D) stable heat, (E) labile heat, (F) shortening heat, and (G) thermoelastic heat. Note that the end points represent the total amount of each heat component after a swallowing movement. The modulation makes a large difference in the protraction phase where the I3 muscle is stretched. The higher serotonin concentration reduces the labile heat component (panel B) while increasing the heat generated by the work done on the contractile component (panel C) during the muscles’ stretch. The two changes offset each other, leading to a slight decrease in the total heat output (cf. Fig. 6F, blue curves).

Figure 9:

Comparison of grasper trajectories around optimal modulation. Solid: $\rho {=}^{10-8}\mathrm{M}$; dashed: $\rho ={10}^{-7.6}\mathrm{M}$ (Fig. 5B, gray arrows $\mathrm{②}$). Same layout as in Fig. 2. Both systems are initiated at the start of their respective closed phase and plotted over one period. The gray shaded region represents the closed phase of the system with $\rho ={10}^{-8}\mathrm{M}$, and the white region represents its open phase. The vertical magenta solid line denotes the transition time of the system with $\rho ={10}^{-7.6}\mathrm{M}$ out of the closed phase. The vertical magenta dotted line denotes the end of the open phase. The higher concentration of serotonin strengthens muscle forces (panel D), which accelerates the movement of the grasper (panel C) for a longer distance (panel E). The sensory feedback propagates the biomechanical change to the neural pools and advances the timing of neural (de)activation (panel A). The muscles’ work output is increased due to the larger forces and longer movement (panel F).

Figure 10:

Comparison of heat production for the modulated systems with $\rho {=}^{10-8}\mathrm{M}$ (solid) and$\rho ={10}^{-7.6}\mathrm{M}$ (dashed) (the same systems as in Fig. 9) The gray shaded regions and the magenta lines have the same meanings as in Fig. 9. Top panels: heat generated by lengthening contraction, consisting of (A) stable heat, (B) labile heat, and (C) shortening heat. Bottom panels: heat generated by shortening contraction, consisting of (D) stable heat, (E) labile heat, (F) shortening heat, and (G) thermoelastic heat. Increased neuromodulation significantly increases the “shortening heat” component during muscles’ stretch (panel C).

Figure 11:

Comparison of grasper trajectories above optimal modulation. Solid: $\rho ={10}^{-7}\mathrm{M}$; dashed: $\rho ={10}^{-6.5}\mathrm{M}$ (Fig. 5B, yellow arrows $\mathrm{③}$). Panels (A-F) follow the same layout as in Fig. 2; (G): heat generated during muscle stretch; (H): heat generated during muscle contraction. Both systems are initiated at the start of their respective closed phase and plotted over one period. The gray shaded region represents the closed phase of the system with $\rho ={10}^{-7}\mathrm{M}$, and the white region represents its open phase. The vertical magenta solid line denotes the transition time of the system with $\rho ={10}^{-6.5}\mathrm{M}$ out of the closed phase. The vertical magenta dotted line denotes the end of the open phase. Panels A-F show the same traces as in Fig. 2; panels G and H show the shortening heat component during lengthening contraction and shortening contraction, respectively. The insensitivity of the muscular properties (${\tau }_{m,i}$ and $k_i$) to the modulation difference results in similar neural and muscle dynamics for the two systems (panels A-E), leading to a similar amount of net work generation (panel F, black curves). The heat dynamics differs (panel F, blue curves) because muscles’ stronger stretch absorbs more work, which is converted into heat during lengthening contraction (panel G).

Figure 12:

Efficiency levels as muscle contraction strength $\mid k_i\mid$ and relaxation time ${\tau }_{m,i}$ are independently varied (in the absence of neuromodulation), with the muscle activation time ${\tau }_{m,i}^0$ fixed at (A) 2.45 s or (B) 1.5 s. Each contour line is labeled with its efficiency level. In each panel, the superimposed thick curve plots the modulation path with base values $\mid k_i^0\mid =0.4\mathrm{N}$, ${\tau }_{m,i}^0=2.45$ or 1.5 s, and serotonin concentration $\rho$ varied. The serotonin concentration starts at picomolar concentrations ($\rho ={10}^{-12}\mathrm{M}$) at the low end of each path up to almost millimolar ($\rho ={10}^{-4}\mathrm{M}$) at the high end. An optimum at intermediate concentrations for each path is observed, consistent with the energetic plateaus and neuromodulatory tradeoffs shown in Fig. 5. The solid dot on each path indicates the optimum neuromodulation point, labeled with the value of ${\rho }^{\ast }$ (in nM) and the peak efficiency ${\Phi }^{\ast }$. Note that as ${\tau }_{m,i}^0$ decreases, the contours shift to the left, indicating that the muscular time constant property has a larger effect on the efficiency pattern than the peak force property does.

Figure 13:

Effects of serotonin concentration $\rho$ on the energy efficiency with changing seaweed force $F_{\text{sw}}$. Black curve corresponds to the default system with $F_{\text{sw}}=0.01\mathrm{N}$ (cf. Fig. 5B). Similar plateaus and optima arise in response to different values of $F_{\text{sw}}$, which are relatively insensitive to the change of $F_{\text{sw}}$.