Locally and Globally Coupled Oscillators in Muscle

At an intermediate activation level, striated muscle exhibits autonomous oscillations called SPOC, in which the basic contractile units, sarcomeres, oscillate in length, and various oscillatory patterns such as traveling waves and their disrupted forms appear in a myofibril. Here we show that these patterns are reproduced by mechanically connecting in series the unit model that explains characteristics of SPOC at the single-sarcomere level. We further reduce the connected model to phase equations, revealing that the combination of local and global couplings is crucial to the emergence of these patterns.

Figure 1

Schematic representation of a myofibril and a half-sarcomere. (a) Half-sarcomeres are connected alternatively in series across the $M$ and $Z$ lines. (b) The thick and thin filaments face each other and slide past one another during contraction. (c) Cross-sectional view of the overlapping zone of the thick and the thin filaments under the relaxing conditions. The lattice spacing, $l_r(\xi )$, is that at equilibrium in the absence of force-generating cross bridges. $l_m$ is an average natural length of the myosin head thermally fluctuating without attachment to the thin filament. (d), (e), (f) Representative patterns of SPOC waves—the traveling wave (d), the disrupted traveling wave (e), and the out-of-phase synchronization (f). The corresponding movies and experimental conditions are shown in the Supplemental Material, Movie 1 and Text 2, respectively [17].

Figure 2

Examples showing typical spatiotemporal patterns of $SL$ oscillations. (a), (b), (c), and (d) show, respectively, in-phase synchronization, traveling waves, disrupted traveling waves, and out-of-phase synchronization. The values of $N$ and $K$ used in each simulation are $N=24$ and $K=0.01$, 0.18, 0.7, 3.75, respectively, and the values of the other parameters are indicated in the legend of Fig. 3. It is to be noted that the SPOC period largely depends on ${\eta }_m$ as shown in Eq. (2), that is, the larger the value of ${\eta }_m$, the longer the SPOC period (see Fig. 8A in 14 for model simulation, and Figs. 3 and 4 in 3 for experimental results). The movies are shown in the Supplemental Material, Movie 2 [17].

Figure 3

Phase diagram of the $SL$ oscillation patterns in the steady state under auxotonic conditions obtained from the connected model. Abscissa, the total number of sarcomeres, $N$. Ordinate, the ratio of the spring constant of the external spring, $K$, to the stiffness of the $Z$ (or $M$) line, $k_{MZ}$, $K{N}^3/k_{MZ}$. The color region indicates the steady state of $SL$: in-phase synchronization (white), traveling wave (blue), disrupted traveling wave (red), out-of-phase synchronization (green), nonstationary (yellow), and contraction without oscillation (black). Typical oscillation patterns corresponding to the markers (a)–(d) are shown in Fig. 2. In general, the final state depends on the initial conditions (e.g., $S{L}_i$ and its distribution), the dependence of which is indicated by the two-color circles where the proportion of each color indicates the probability of the appearance of the two states. The initial conditions used here are given by adding random numbers $\delta \in [-0.0001,0.0001]$ to a stationary solution $(\overline{P},\overline{\xi },\overline{d})=(0.18,1.08,25.1)$. When the initial randomness is increased to, for example, $\delta \in [-0.01,0.01]$, the contraction region (black) at larger $N$ values becomes the out-of-phase oscillations (green), while the other states are essentially unaltered. $c^{*}$ is $\log (-{\pi }^2{B}^{\prime }(0)/A^{\prime }(0))$. The parameters are ${\alpha }_1=19$, $F_0=20$, ${\beta }_0=20$, $k_{MZ}=0.5$, ${\eta }_d=0.0005$, $k_m=1$, $s_0=0.01$, $l_m=23$, $k_r=60$, ${\eta }_m=1.5$, $d_0=25.3$, $S{L}^{(0)}=3.6$, $l_{r0}=23$, $l_{r1}=2.53$, and $a=1$, which are nondimentionalized such that $k_m$ the force and length along the short axis ($1(\mathrm{a}.\mathrm{u}.){\mathrm{nm}}^{-1}$) and $a$ the force along the long axis (1 pN) are unity. In addition, the unit time is chosen as 1 s, and the unit lengths for the short and long axes are chosen as 1 nm (for $d$, $l_m$, and $l_r$) and $1\mu \mathrm{m}$ (for $s_0$ and $SL$), respectively. (For more details see the Supplemental Material, Table 1 [17].) The phase diagram is basically insensitive to the choice of the parameter values except that the disrupted traveling waves disappear when the parameters approach the Hopf bifurcation points.

Figure 4

Phase-coupling functions obtained from Eq. (6). The parameters and the conditions are the same as in Fig. 3. $A(\varphi )$ (red) and $B(\varphi )$ (blue) are those for the global and local couplings, respectively.