Historically, nonlinearities have never been particularly popular among scientists. The main reason for this unpopularity is that while we have very sophisticated tools to analyze and describe linear systems, our toolbox for studying nonlinear systems is very light, and the few tools we do have are very hard to use. Three strategies have been employed to get around these problems. The first approach is to focus on those nonlinear behaviors that are more amenable to examination. The large popularity of the QLV model and of nonlinear superposition in general, with its corresponding focus on a single elongation, is an example of this approach. The second approach is to limit the study of the nonlinear system to a small range over which the behavior appears linear. This approach has been very successful, and there are examples of small-signal analyses in all fields of science. The third approach is to develop sophisticated linear analysis techniques that are tailored to a certain class of nonlinear systems (usually cascades of linear systems and static nonlinearities) and are capable of isolating the linear component of the underlying system. This approach is particularly popular in the analysis of neural responses. These techniques are ingenious, and their application has resulted in enormous progress in understanding natural phenomena. However, they also have a downside: they have contributed to the widely held belief that nonlinearities are inherently difficult and, correspondingly, that nonlinear systems are hard, or at least harder, to control than linear systems. For example, Anderson and colleagues, 22
while discussing the nonlinearity associated with EOM activation, concluded, “How these nonlinearities might be mitigated by, for example, recruitment to make eye-movement control simpler is an important topic for further work.” We would like instead to argue the opposite position, i.e., that proper nonlinearities might in fact lighten the burden of the neural controller.
The first reason for arguing in favor of nonlinearities is evolutionary: if nonlinearities in biological materials were detrimental, evolution would have disposed of them a long time ago. Conversely, if they are there they either serve a purpose or at least are irrelevant. A classic example of this type of specialization is represented by tendons and ligaments, which are composed of a set of collagen fibers arranged in parallel. The individual fibers behave linearly (at least within the physiological range of elongations); however, a “toe” region, where stiffness is low and increases nonlinearly, often characterizes the composite structure. This occurs because different collagen fibers have different lengths, so that as the load increases additional fibers are recruited. 32–34
Once all the fibers have been recruited, the whole structure behaves linearly. This mechanism, which helps to smooth out stress build-up and thus protect the muscle, is so important that many muscle–tendon groups operate mostly in the toe region. However, this compliance would be very detrimental for digital flexors and EOMs, whose length must be rapidly and precisely controlled: not surprisingly, evolution took care of that, and the tendons of those muscles are much stiffer.
We believe that the nonlinearities that we have discovered in EOMs have also evolved to serve a purpose: simplifying neural control. To see how, let’s consider a muscle whose viscoelastic behavior conforms to the QLV model, and thus within the first 3 mm of elongation is actually linear (but of high order). According to Dean and colleagues, 20–22
such a model actually describes the zero-load creep behavior of the whole orbit (i.e., the time course with which an eccentrically rotated eye ball passively returns to the rest position once released). Every time the eye rotates, the lengths of muscles and orbital tissues, as well as the forces that they exert, change. Once an eye rotation is over, the lengths do not change anymore, but the passive forces still do, and they do so for a long time. As all these forces are ultimately applied to the eyeball, this would inevitably lead to eye motion, i.e., eye drift. Of course, this would be unacceptable, since vision would be severely degraded. There is only one way to avoid this: the brain must generate active forces that compensate for these passive forces. Hence, the innervation to the eye muscles must keep changing long after the eye movement is over, or the eyes would keep moving. These innervational components have indeed been observed in motoneurons, 35,36
and the forces have been measured in muscles. 36,37
Generating these innervational signals would not be a trivial matter: because of superposition, the brain would need to build an internal model of the eye plant, separately for tissues and muscles, incorporating processes with long time constants. Only by doing so could it keep track of the ever-changing internal state of each muscle and tissue. With time constants of at least 40s, this would not be a small feat. Moreover, the level of accuracy required to use retinal slip (the error signal) to solve the credit assignment problem (i.e., to figure out which parameter needs to be tweaked) for such a model is extraordinary. 38
And yet the brain is capable of using the retinal slip to cancel post-saccadic drift: repeated presentation of an artificially induced post-saccadic retinal slip will, over time, induce a post-saccadic eye drift with a magnitude and time constant appropriate to cancel that retinal slip. 39,40
Moreover, this drift can be induced not only in the same or opposite direction of the antecedent saccadic eye movement, 39
but even in a direction orthogonal to it. 41
A nonlinearity such as the one we discovered in passive EOMs opens the door to a different strategy: the brain could simply issue a learned post-saccadic innervational command based on the final eye position and the direction of the antecedent movement. There would be no need to keep track of internal states of muscles and tissues, and thus there would be no credit assignment problem. It would also be very easy to dissociate the direction of the antecedent movement and that of the eye drift. Under this scheme, the floccular complex, which is required for retinal slip cancellation, 42
would be in charge of monitoring retinal slip and associating eye movements with an appropriate post-saccadic signal to be sent to the extraocular motoneurons.
One major caveat of this scheme is that it can only work if all the tissues that exert (significant) torques on the eyeball behave similarly. If any of the tissues were to behave linearly (i.e., had a long memory) the brain could not take this shortcut. Unfortunately, there is currently little evidence to support or reject this conjecture. EOM tendons are extremely stiff, 43
so they certainly do not play a significant role. However, the rest of the extraocular tissues (usually referred to as orbital tissues, OTs) are not only poorly characterized from a mechanical standpoint, but also poorly defined: it is not at all clear which tissues contribute to OT torque and which don’t. As we noted above, to the best of our knowledge the mechanical properties of the eye plant with all six EOMs fully detached have not been studied. Of course, such a procedure could not be carried out without causing damage to the conjunctiva, tenons, and other OTs, so those force contributions would have to be separately assessed. It has been shown, 44
and we have confirmed, that when the horizontal recti are detached and the eye is rotated, it does not drift back all the way to its original resting position. This seems to imply that OT forces in central gaze are very small, but once again those experiments damaged at least the conjunctiva. As mentioned above, Dean and colleagues 20–22
have argued that the whole plant can be modeled as a linear system, but their conclusion was based on the results of zero-load creep. This type of paradigm is usually avoided in material testing, because some nonlinear systems can behave linearly when the load is removed (removing the forcing function can turn some nonlinear differential equations into linear ones). Furthermore, it has been long recognized that it is extremely difficult to predict relaxation from creep in biological tissues, 25,33,45–51
and while creep can be important in ligaments, 51
which are often subjected to repetitive load histories, it is hard to imagine under what physiological condition creep would be an important orbital phenomenon. Finally, Dean and colleagues have inferred
linearity from a small set of single elongations, which were not designed to explicitly test superposition. Understandably, they thus had to introduce a set of assumptions to draw inferences from their data. However, if one or more of their assumptions were to be proven false, then their conclusions should be revisited. In light of our results, their assumption that passive EOMs are first-order linear systems seems particularly weak. Obviously, more experiments are needed to fully characterize the mechanical properties of the OTs, as well as those of active EOMs.
Here, we have focused our attention on the lack of superposition, not only because of its importance, but also because it can be easily grasped. In our experiments, we have chosen our elongation paradigms so that we could directly measure
nonlinearity (see Figs. and ). In addition to this nonlinearity, which affects the post-elongation period, and the hyper-elasticity (), which is measured at equilibrium, we found that forces during the elongation are also nonlinear: forces for larger elongations are considerably smaller than expected from a linear system given the forces induced by small elongations. 31
In conclusion, we believe that there is enough evidence to support the idea that nonlinearities are pervasive in biological tissues, including the eye orbit. Evolution has made it easier for our brain to control our body, but it has not made it easier for scientists to understand how we achieve that control.