Saccadic elongations: ANLV predictions.

<p>Forces induced by saccadic elongations (same elongations shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0009595#pone-0009595-g004" target="_blank">Fig. 4</a>). Data (black), force predicted by the AQLV model (red) and force predicted by the ANLV model based on the AQLV model (green). <b>A</b>: Short elongation (1.6 mm) starting from a large initial elongation, applied to the superior rectus in m3. <b>B</b>: Intermediate elongation (3.0 mm) starting from a small initial elongation, applied to the lateral rectus in m4. <b>C</b>: Large elongation (4.0 mm) starting from an intermediate initial elongation, applied to the lateral rectus in m3. <b>D–F</b>: Here we show, for the same elongations represented in the top row, the force induced in the muscle as a function of length. Whereas the AQLV model fits the data well only for the first 0.5 mm of the elongations, the ANLV provides a good fit throughout.</p

Parameters for the <i>k_i(L)</i> functions in m3LR.

<p>Parameters for the <i>k_i(L)</i> functions in m3LR.</p

Force relaxation after saccadic elongations: ANLV predictions.

<p>Simulations of the same elongations described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0009595#pone-0009595-g002" target="_blank">Figs. 2</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0009595#pone-0009595-g006" target="_blank">6</a> using the ANLV model. In each panel in the top row we plot simulations of elongations having different amplitudes but the same final length, all done with the ANLV model based on the QLV model. In the bottom row simulations of the same elongations with the ANLV model based on the AQLV model are shown. In all cases the ANLV model performs better than either the QLV and the AQLV model. However, in this case the model based on the QLV model does a better job, as the simulations in B & C match the data much better than those in E & F. More precisely, the cross-over observed in E and F is considerably larger than the slight one observed in the data (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0009595#pone-0009595-g002" target="_blank">Fig. 2</a>).</p

Stable dynamics of spikes in solutions to a system of activator-inhibitor type

<p>Data (black) and force predicted by the AQLV model (green). The values for the parameters of the model at each step length were derived from the parameters of the generalized QLV model described above. A cubic spline interpolation (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0006480#pone-0006480-g007" target="_blank">Fig. 7C</a>) was then used to determine the value of the parameters at other lengths. Data for the superior rectus in m3. Each step has been offset in time for clarity. Note that in a logarithmic plot to carry out this operation without deforming the shape the time axis must be compressed, not shifted.</p

Parameters for the ANLV model based on the QLV model.

<p>Parameters for the ANLV model based on the QLV model.</p

Constant-speed elongations: ANLV predictions.

<p>Blue: Static length-tension relationship. Black: Force measured in muscle. Cyan: force predicted by the QLV model. Red: force predicted by the AQLV model. Green: ANLV (based on the QLV) model prediction. <b>A</b>: Stretch at 0.1 mm/s. <b>B</b>: Stretch at 1 mm/s. <b>C</b>: Stretch at 10 mm/s. <b>D</b>: Stretch at 80 mm/s. In all cases the ANLV model predicts the generated force quite well, vastly outperforming the AQLV model. Note very different ordinate scales.</p

Force relaxation after constant-speed elongations.

<p>Time course of the force decay after the completion of constant-velocity stretches. Dashed black line: Static force predicted by the length-tension relationship. Different colors indicate different stretch speeds (see key). Each panel contains data from a different muscle. Note the cross-over between 20 ms and 1 s after the end of the stretch. Before the cross-over higher stretching rates are associated with higher forces, but after the cross-over higher stretching rates are associated with lower forces. Only in the last two muscles (shown in A and B) do we have data long after the end of the stretch. But even when the record is short (panel C) the pattern is evident.</p

Force relaxation after constant-speed elongations: model predictions.

<p>Simulations of the same elongations described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0009595#pone-0009595-g003" target="_blank">Fig. 3A</a> using the QLV and AQLV models. The extensive cross-over observed in the data is not present in the simulations. <b>A</b>: An unexpected partial cross-over is produced by the QLV model. It is caused by numerical errors, and more precisely by small (less that 0.2%) differences between the integral of the elongation speed and the change in muscle length. Such differences are not surprising since these simulations were carried out using the length measurements from the experiments. In the inset we plot the results of the same simulations after manually scaling the elongation speed so that its integral matches the change in muscle length. As expected, no cross-over occurs. In both cases, the traces for the fastest and slowest elongations only converge very late. <b>B</b>: No cross-over is observed in the AQLV simulations, and convergence occurs even later than with the QLV model.</p

Parameters for the <i>k_i(L)</i> functions in m3SR.

<p>Parameters for the <i>k_i(L)</i> functions in m3SR.</p

Peri-elongation forces and generalized QLV model fits.

<p>Data (black), force predicted by the generalized QLV model using the parameters derived analytically from the E-T fit (green), and force predicted by the generalized QLV model after parameter optimization and addition of a viscous term (red). A: Data for a step at short elongations from the superior rectus in m3. B: Same forces as in A, but plotted as a function of muscle elongation rather than time. The initial rapid rise in force is due to the pure viscosity, which was not part of the original model (green trace). C & D: Same as A & B, but for a step at intermediate elongations from the lateral rectus in m3. E & F: Same as A & B, but for a step at large elongations from the lateral rectus in m4. Note how in this case the fit is not as good, as the force in panel F is convex, whereas the model predictions are always concave. Force scale is different across rows. SSQ: sum of squared residuals (fit – data).</p