Abstract
We continue our study of the properties and the functional role of the neuromuscular transform (NMT). The NMT is an inputoutput relation that formalizes the processes by which patterns of motor neuron firing are transformed to muscle contractions. Because the NMT acts as a dynamic, nonlinear, and modifiable filter, the transformation is complex. In the preceding paper we developed a framework for analysis of the NMT and identified with it principles by which the NMT transforms different firing patterns to contractions. The ultimate question is functional, however. In sending different firing patterns through the NMT, the nervous system is seeking to command different functional behaviors, with specific contraction requirements. To what extent do the contractions that emerge from the NMT actually satisfy those requirements? In this paper we extend our analysis to address this issue. We define representative behavioral tasks and corresponding measures of performance, for a single neuromuscular unit, for two antagonistic units, and, in a real illustration, for the accessory radula closer (ARC)–opener neuromuscular system ofAplysia. We focus on cyclical, rhythmic behaviors which reveal the underlying principles particularly clearly. We find that, although every pattern of motor neuron firing produces some state of muscle contraction, only a few patterns produce functional behavior, and even fewer produce efficient functional behavior. The functional requirements thus dictate certain patterns to the nervous system. But many desirable functional behaviors are not possible with any pattern. We examine, in particular, how rhythmic behaviors degrade and disintegrate as the nervous system attempts to speed up their cycle frequency. This happens because, with fixed properties, the NMT produces only a limited range of contraction shapes that are kinetically well matched to the firing pattern only on certain time scales. Thus the properties of the NMT constrain and restrict the production of functional behaviors. In the following paper, we see how the constraint may be alleviated and the range of functional behaviors expanded by appropriately tuning the properties of the NMT through neuromuscular plasticity and modulation.
INTRODUCTION
In the preceding paper (Brezina et al. 2000a, henceforth referred to as Paper I), we studied the complex way in which motor neuron firing patterns are transformed to muscle contractions by the neuromuscular transform (NMT). But the contractions, in themselves, are not the ultimate goal. Rather, the firing patterns are commands of the nervous system for behavior. And for integrated, functional behavior, the contractions cannot be arbitrary, but must satisfy specific requirements arising from the need to coordinate with other muscles participating in the behavior as well as set by the task to be accomplished. Such requirements are particularly stringent and obvious in cyclical, rhythmic behaviors. To what extent can the nervous system, given that it must send its commands through the NMT, actually produce contractions satisfying those requirements? In asking this question, we are motivated, for example, by the illustration shown in Fig. 1 of Paper I, where, as the nervous system increased the frequency of rhythmic contractions of two muscles, the shapes of those contractions altered, but a behavior defined by their mutual antagonism altered—degraded and then completely disintegrated—much more dramatically. This and the following paper (Brezina et al. 2000b, referred to as Paper III) are two complementary parts of our examination of the functional issue. In this paper we show that, with fixed properties, the NMT indeed significantly constrains the production of functional rhythmic behaviors. In the following paper we show, correspondingly, how physiological mechanisms that make the properties of the NMT variable alleviate the constraint.
METHODS
The basic mechanics of our approach are as in Paper I. We briefly review them here.
Input firing patterns and parameters
The firing pattern is taken to be synonymous with the waveformf(t) of firing frequency f as a function of time t. We consider a canonical set of bursting patterns (e.g., Fig. 1, bottom row) completely definable by the alternative parameter triplets (d
_{intra},d
_{inter}, f
_{intra}), (P, F, f
_{intra}), and (P, F, 〈f〉). Hered
_{intra} is the burst duration,d
_{inter} the interburst interval,f
_{intra} the intraburst firing frequency,P the cycle period, F the duty cycle, and 〈f〉 the mean (periodaveraged) firing frequency. These parameters, and so the alternative triplets, are related by the equations
NMTs
The NMT is an inputoutput relation which converts the input waveform f(t) to an output waveformc(t), of contraction amplitude c as a function of time (Fig. 1, top row). We focus on two NMTs, the real B15ARC NMT of Aplysia and a model NMT that has similar but completely known properties. The model NMT is implicitly defined by the kinetic schema
Output contractions and parameters
We consider the whole output waveform c(t) or its parameters, in particular its periodwise maximumc̄, minimum c , and mean 〈c〉. (In this paper, however, our major focus is on the behavioral performance parameters that we define inresults.) In the dynamical steady state of the system,c(t) settles to the steadystate output waveform [c(t)]_{∞}, and c̄, c , and 〈c〉 settle to its corresponding parameters c̄ _{∞}, c _{∞}, and 〈c〉_{∞}.
RESULTS
Strategy
We continue with the same analytic framework, essentially an elementary dynamical systems approach, with the same set of canonical firing patterns, and the same two illustrative NMTs, a model NMT and the real B15ARC NMT of Aplysia, as in Paper I. A brief review of the mechanics of our approach is provided inmethods. A summary list of symbols was given in Table 1 of Paper I.
In Paper I we studied how the NMT transforms different input firing patterns or waveforms f(t) to output contraction waveforms c(t), and the relationships it thus establishes between different parameters of the former and of the latter. In Paper I, we focused on such elementary output parameters as the maximum contraction c̄, the minimum contraction c , and the mean contraction 〈c〉. In this paper we will define and study an additional, more functionally relevant set of output parameters. We will devise a series of representative behavioral tasks that the nervous system might ask the muscle to perform, and with each task a measure of how well the task is being performed. Each performance measure will be defined by a singlevalued function on the contraction waveformc(t). Just like c̄, c , and 〈c〉, therefore, each performance measure will simply be another, although more elaborately defined, output parameter.
We will begin with a single neuromuscular unit, a single motor neuron controlling a single muscle. But, as we discussed in Paper I, a more complex ensemble of multiple motor neurons and muscles, carrying out a whole complex behavior, can be analyzed in much the same way, as just a more complex dynamical system. We can therefore go on to define, in the same way, a performance measure on the whole complex output: a performance measure for a whole integrated behavior.
In Paper I, we saw how the operation of the NMT can be represented geometrically as a dynamical structure in a multidimensional inputoutput space. As its input dimensions, this space has input parameters that define the set of input patterns of interest; here, as in Paper I, it will be, most generally, one of the alternative parameter triplets (d _{intra},d _{inter}, f _{intra}), (P, F, f _{intra}), or (P, F, 〈f〉) that describe our canonical set of patterns (see methods). As its output dimensions the space has the output parameters of interest. Here, therefore, it will have a single output dimension, our performance measure, m. As in Paper I, we can study the dynamical evolution of m in the inputoutput space (seeappendices). For simplicity, however, we will focus here on just one special state, the steady state. As we emphasized in Paper I, the steady state is the key element in the dynamical structure of the NMT and its physiological operation. In the dynamical steady state of the system, with our NMTs, c(t) settles for eachf(t) to a unique steadystate waveform, [c(t)]_{∞} (shown, for example, in Fig. 1). Any parameter that is a function of c(t), therefore, likewise settles to a unique steady state. c̄ settles toc̄ _{∞}, c to c _{∞}, and 〈c〉 to 〈c〉_{∞}. m settles tom _{∞}.
The principal object of our examination in this paper is thus the structure of the steadystate m _{∞} in the alternative (d _{intra},d _{inter}, f _{intra},m), (P, F,f _{intra}, m), or (P,F, 〈f〉, m) inputoutput spaces, or simply the functionsm _{∞}(d _{intra},d _{inter}, f _{intra}),m _{∞}(P, F,f _{intra}), andm _{∞}(P, F, 〈f〉). These spaces, however, are fourdimensional, with the function m _{∞}occupying a threedimensional volume. (Multiple neuromuscular units require, strictly, additional input dimensions: see Task IV: antagonistic muscle pair below.) This is unmanageable graphically. As in Paper I, most of our figures will therefore show representative three or even twodimensional sections through the complete fourdimensional spaces, obtained by setting one or two of the input parameters to constant values. This is equivalent to temporarily restricting the set of input patterns of interest from the full canonical set. In the threedimensional sections,m _{∞} appears as a twodimensional surface (e.g., Figs. 3 and 5), and in the twodimensional sections, as a onedimensional curve (Figs. 2 and 4).
Functional appropriateness of contractions
Figure 1 shows representative examples of the kind of steadystate output waveforms [c(t)]_{∞} that are produced from different canonical input patterns f(t) by our model NMT. We have already analyzed this process, and the resulting shapes of [c(t)]_{∞}, in Paper I. Here we simply point out, once more, how different members of the set of input patterns produce quite different contractions and contraction parameters: contractions that have a large tonic component (A) or are largely phasic (D), that are small (B) or large (C) in average amplitude, that have the maximum c̄ _{∞} and minimum c _{∞} close together (A andB) or far apart (D).
This is summarized more systematically in Fig.2, which is based on Fig. 9 of Paper I. The small panels on the left are twodimensional sections of the kind mentioned above, with the input in the (d _{intra}, d _{inter},f _{intra}) representation. Together, they provide an overview of the structure of the complete fourdimensional space. Each panel shows, first of all, the maximum contractionc̄ _{∞} and the minimum contraction c _{∞} (top and bottom dotted curves, respectively). As Fig. 1 suggested, we see all possible combinations of the two parameters.
Thus the system can produce a variety of contractions. But to what extent are the different contractions functionally appropriate? Functional appropriateness, clearly, is not an intrinsic property to be found within the neuromuscular system itself, but makes reference to an external goal. It may refer to intermediate goals such as how well the contractions coordinate with those of other muscles that formally are not part of the system, and ultimately always refers to a final goal: how successfully the whole neuromuscular ensemble performs a behavior that is important in the life of the animal. Different goals impose different requirements. In different behaviors that use the same neuromuscular plant, very different contractions may be required; conversely, the same contractions, produced by the same firing patterns, may have very different functional value. We can see already that not all of the contractions in Figs. 1 and 2 are likely to be functional with respect to any particular goal.
To evaluate functional appropriateness, we must therefore impose an external criterion on the system. To do this, we will define a series of simple behavioral tasks, suggested by the general consideration of a variety of behaviors. These tasks can be thought of as complete tasks to be performed by simple neuromuscular systems, or, especially in the case of the initial, more elementary tasks, as functional motifs performed by individual units within larger neuromuscular ensembles. We will then see how well the different contractions fit the requirements of each of these tasks.
Task I: movement oscillating around a single axis
Except for the special case of steady firing, our canonical firing patterns are repetitively phasic, and the most obvious task for such patterns is to produce repetitively phasic, oscillatory contractions and movement. These are the natural components of cyclical, rhythmic behaviors. The rhythmic, oscillatory movements of the feeding and respiratory organs of many animals, or of limb segments in various types of locomotion, are prime examples.
There will typically be functional requirements concerning the amplitude of the oscillatory movement as well as its location in space, relative to other body parts or external objects to be acted upon. In our first task, we will simply suppose that the larger the oscillatory movement the better, but only to the extent to which it occurs symmetrically around a given position or axis. This requirement is schematized in Figs. 1 D (in the time domain) and 2,right (a representative section through the inputoutput space), and expressed in a corresponding formula (appendix , 1), that allows us to compute, from the oscillatory contraction waveform [c(t)]_{∞}, indeed from just the two parameters c̄ _{∞}and c _{∞} that define the range of its oscillation, the functional contraction or movementm _{∞}. In this formula,m _{∞} is simply that part of the range of the oscillation that is, as required, symmetrical around the given axis (i.e., the distance between the dotted lines in Fig. 1 D, and the height of the gray area in Fig. 2, right). But at the same time, because the symmetrical part is to be maximized,m _{∞} can immediately serve as a measure of performance. It is zero whenever the oscillation does not cross the axis at all—no part of the contraction is functional—and becomes increasingly positive as the symmetrical part of the oscillation grows.
We note that this equates contraction and movement. In reality, the functionally important movement, of a whole body part for instance, will not be numerically equal or even linearly related to muscle contraction, especially that of a single muscle. The muscle will contribute to the movement through nonlinear interactions with other muscles and with rigid or elastic external structures (e.g.,Alexander 1988; Gronenberg 1996). We saw in Paper I, however, how such complications are easily accommodated in the quantitative structure of the NMT, given the requisite quantitative information in any particular case. Here, in our generic tasks, we can therefore simply take contraction and movement (and, similarly, movement and performance) to be equivalent. This implies that the variable c here designates muscle length. If it designated force or some combination of length and force (cf. Paper I), our analysis would be identical, but we would be dealing, rather, with various kinds of oscillatory squeezing or pumping motions, which are also important components of many behaviors.
It will be useful to extend our measure of performance in one respect.m _{∞}, as defined, is the steadystate functional contraction or movement per cycle: the size of each individual functional movement. And, as will be seen, there is a fundamentally inverse relationship between the size of the movement and its repetition rate. As a consequence of this, in Task I and many others, perfect functional movement—m _{∞} as large as the physical parameters of the situation allow—can always be achieved, provided that the behavior cycles slowly enough. But, in terms of performance, this is not very realistic. To be meaningful, the movements should be as large as possible, but also reasonably frequent. Even though perfect, one movement in the animal's lifetime does not constitute good performance. If a movement of a certain size is performed more, or less, often, this should be measured as better, or worse, performance. m _{∞} itself fails to take the repetition rate into account. A more realistic quantity to be maximized, in fact, is not the size of the individual movement, but the total amount of movement over some interval of time. In feeding, the ultimately important quantity is the amount of food ingested, not the size of each bite; in locomotion, it is the distance traveled, not the size of each step. To measure this kind of quantity, we can normalizem _{∞} by the cycle period P. We will focus on this more realistic performance measure,m _{∞}/P, in most tasks.
It is worth recalling why the normalization by P makes sense. P, after all, is formally a parameter of the input firing pattern, not of the output contractions, movement, or behavior. But as we saw in Paper I, with our model NMT, as with most real NMTs, the output waveform always has exactly the same period as the input waveform. The input and output are rigidly coupled in this respect. In terms of functional control by the nervous system, changes in the period of the firing pattern are necessary and sufficient for changes in the period of the contractions, movement and behavior. Therefore, when below we variously refer to the speed of the firing pattern, the repetition rate of the movement, the rhythm of the behavior, and so on, we are simply emphasizing different functional aspects of the same quantity, the parameter P.
Analysis of performance in Task I
Our basic question now is, what are the determinants of performance, of m _{∞} andm _{∞}/P? One set of determinants clearly has to do with the physical parameters of the situation: where the movement axis is located in relation to the range of possible values of c, in other words to the size of the muscle. With our model NMT, for example, c ranges from 0 to 1. Clearly, if the movement axis is set above 1, there can never be any functional movement. The largest m _{∞} can be obtained with the axis set at 0.5; it decreases as the axis is moved either up or down. All of this is quite realistic: it expresses, as well as the fact that larger muscles can give larger movements, the need to match the size of the muscle to the task. We are, however, more interested in another set of determinants: how, within these physical bounds, different levels of performance can be achieved by different firing patterns.
In each of the panels in Fig. 2, left, we have plotted, as well as c̄ _{∞} and c _{∞}, the computed performance measuresm _{∞} (gray curves) andm _{∞}/P (black curves). Together, these sections provide an overview of the complete functionsm _{∞}(d _{intra},d _{inter}, f _{intra}) andm _{∞}(d _{intra},d _{inter},f _{intra})/P for Task I.
We see immediately that, indeed, although every firing pattern produces some state of contraction of the muscle—there is always some, almost always nonzero, c̄ _{∞} and c _{∞}—this contraction is very often not functional. Only in some parts of the space do we see nonzero values of the performance measures m _{∞} andm _{∞}/P. The examples in the time domain in Fig. 1 show intuitively why this happens: here, only the contractions in D cross the movement axis and are functional, whereas those in A–C are either too small or too large. Thus, with respect to a particular task such as Task I, only a subset of firing patterns is able to produce functional movement and behavior.
We note further in Fig. 2 that, for any value of the intraburst firing frequency f _{intra}, larger values ofm _{∞} are obtained, in general, as the two temporal parameters of the firing pattern, the burst durationd _{intra} and the interburst intervald _{inter}, increase. The largestm _{∞} is obtained with the largestd _{intra} and d _{inter} (as well as the largest f _{intra}), in the top righthand corner of each 3 × 3 block of panels. This is the phenomenon that we have already mentioned, that the largest movements are those that repeat most slowly, produced by the slowest firing patterns. As will be seen in the paragraphs below, the inverse relationship between movement size and repetition rate is a direct consequence, in terms of our new output parameterm _{∞}, of one of our principal concerns in Paper I: how the inputoutput space is structured by the kinetic properties of the NMT.
We saw in Paper I that many aspects of this structuring can be understood, more easily than in the (d _{intra},d _{inter}, f _{intra}) representation, with the input in the (P, F,f _{intra}) or (P, F, 〈f〉) representation. These representations combine d _{intra} and d _{inter}into a single input dimension, that of the cycle period P, so that the key factor, the speed of the firing pattern relative to that of the NMT, is straightforwardly represented in the inputoutput space. Furthermore, P is the parameter of interest in problems like our initial motivating problem (Fig. 1 of Paper I), where we wish to see how performance is affected as the nervous system changes (perhaps for higherlevel reasons) the overall rhythm of the behavior. We will therefore, from now on, work with input in the (P, F, f _{intra}) and (P, F, 〈f〉) representations. We can always obtain equivalent results in the (d _{intra}, d _{inter},f _{intra}) representation if this should be required to answer some particular question.
Figure 3 shows threedimensional sections of m _{∞} andm _{∞}/P, for input either in the (P, F, 〈f〉) or in the (P, F, f _{intra}) representation. Here the period P and the duty cycleF are varied continuously for a single fixed value of the mean firing frequency 〈f〉 or the intraburst frequency f _{intra}. All threedimensional plots in this paper will have this form. In Fig. 3only, P and F are plotted on extended log scales to provide a broad overview of performance, in particular the limiting performance with firing patterns that are much faster, and those that are much slower, than τ, the time constant of the NMT. These are located on either side of the line marked P ≈ τ in each plot. In this and most other threedimensional plots, complete failure of performance (m _{∞} andm _{∞}/P values of zero) is indicated by pure black tone, and progressively better performance by progressively lighter tone.
Because Fig. 3 merely plots in a different way the same information that we saw in Fig. 2, we find, again, that only certain firing patterns give nonzero performance.
We can understand the details of Fig. 3 by recalling from Paper I howc̄ _{∞} and c _{∞}, the contraction parameters that delimit the performance parameters in Task I, behave under the same circumstances. We saw in Paper I how this behavior arises from the critical interaction between the speeds of the NMT and the firing pattern, which determine how fast the contraction c can react, and how much time it has available to it to react, to the changes in firing frequency that constitute the pattern.
With very slow patterns (P ≫ τ, righthand end of each plot in Fig. 3), c always has time to relax essentially completely to its true steady state c _{∞}, eitherc _{∞}(f _{intra}) orc _{∞}(0) = 0 during the burst and the interburst interval, respectively. (The contractions in Fig.1 D, for instance, come close to this.) Thereforec̄ _{∞} =c _{∞}(f _{intra}) and c _{∞} = 0. In other words, the contraction is completely phasic. [The exception throughout this analysis is the special case of steady, continuous firing (F = 1; front edge of each plot) where the contraction is likewise steady, or completely tonic,c̄ _{∞} and c _{∞}are one and the same and consequently, always,m _{∞} = 0.] When P ≫ τ, the size of the functional movement m _{∞} thus depends just on f _{intra} and can be found simply by examining the steadystatec _{∞}(f) relation in conjunction with the given physical parameters of the situation. With the model NMT that we have used here, we recall that thec _{∞}(f) relation increases monotonically from zero to saturation. So, asf _{intra} increases, m _{∞} is zero as long asc _{∞}(f _{intra}) remains smaller than the movement axis; beyond this,m _{∞} increases at double the rate ofc _{∞}(f _{intra}); finally, m _{∞} saturates whenc _{∞}(f _{intra}) either saturates or becomes double the movement axis. Becausem _{∞} depends just onf _{intra}, for any particular value off _{intra} (Fig. 3, top right) we see the same value of m _{∞} for any F (exceptF = 1) and any P ≫ τ: any slow firing pattern with that intraburst firing frequency.m _{∞} is independent of F andP in this range.
We now consider, instead of f _{intra}, the mean firing frequency 〈f〉 (Fig. 3, top left) when P ≫ τ. Because 〈f〉 =f _{intra} F, keeping 〈f〉 constant requiresf _{intra} to increase as F decreases. From the last paragraph we conclude that, as this happens,m _{∞} always eventually reaches the same saturating value—whenc _{∞}(f _{intra}) either saturates or becomes double the movement axis—with every 〈f〉. Although in this casem _{∞} depends on F, it remains independent of P.
What is the broader meaning of all this? We see that, if the cycle period is large enough relative to the time constant of the NMT, there is always some firing frequency (often, indeed, many frequencies) with which it can be combined to produce the largest possible functional movement that the physical parameters of the situation allow. With a sufficiently slow firing pattern, perfect movement can always be achieved.
But, at the same time, because the cycle period is large, the movement is not performed very often, and the total amount of movement over time, our more realistic performance measurem _{∞}/P, is very small (Fig. 3,bottom plots). Furthermore, m _{∞} has saturated and no longer varies with P in this range, so thatm _{∞}/P decreases unopposed asP increases. And even the perfect movement is, ultimately, not very large, because the individual movement cannot grow beyond the rather narrow limits imposed by the physical parameters of the situation. These limits are not easily altered.
A priori, an inverse relationship between movement size and repetition rate—a direct relationship between m _{∞} andP—suggests two opposite behavioral strategies for maximizing m _{∞}/P. One strategy is to increase m _{∞}, even at the cost of increasedP: to produce larger, although slower, movements. The above, however, shows that this strategy ultimately fails. What about the opposite strategy, of decreasing P, even at the cost of decreased m _{∞}: of producing faster, although smaller, movements?
Examining the region of intermediate (P ≈ τ) and fast (P < τ) firing patterns toward the lefthand end of the plots in Fig. 3, we see that, while many patterns fail to give any performance at all, some on the contrary give very high performance (high values of m _{∞}/P,bottom plots). Thus in Task I, at least, the second strategy is better, provided the nervous system can select one of the correct patterns. It will be seen that this is true also for other tasks. We will therefore from now on focus particularly on the intermediate and moderately fast range of patterns, and in it on the question of what happens as the nervous system progressively speeds up the pattern. This range of patterns is, in any case, most likely to be relevant in real systems.
To aid in our explanation of what happens in Task I in this range, Fig.4 shows twodimensional sections through it, like those in Fig. 2 but using the (P, F, 〈f〉) and (P, F,f _{intra}) input representations, and Fig. 5 shows threedimensional sections like those in Fig. 3 but on linear scales.
With very fast patterns (P ≪ τ), we saw in Paper I thatc has essentially no time to make any progress toward either steady state during the two phases of the pattern, but stabilizes somewhere between the two. In the limit, the contraction is steady; it no longer oscillates at all. But anywhere short of the limit, some oscillation remains (e.g., Fig. 1 A). The oscillation becomes progressively smaller—with the model NMT used here,c̄ _{∞} monotonically decreases, c _{∞} monotonically increases, so that the two converge monotonically—as the pattern speeds up toward the limit. In other words, the contraction progressively converts from a more phasic to a more tonic, and ultimately completely tonic, form. The convergence of c̄ _{∞} and c _{∞} can be seen in each panel in Fig. 4(dotted curves).
As c̄ _{∞} and c _{∞} converge, the functional movementm _{∞} decreases (gray curves in Fig. 4). This, again, is the inverse relationship between movement size and repetition rate. As the pattern, and so the movement, speeds up, the size of the movement diminishes.
The question now is, what is the limiting value to whichc̄ _{∞} and c _{∞}converge? As the three panels of Fig. 4 A illustrate, it is only if that value is precisely equal to the movement axis that the contraction continues to oscillate across the axis, so that functional movement continues to be produced, as the pattern speeds up arbitrarily close to the limit (middle panel). Only in this case doesm _{∞} remain nonzero, in Task I (although not in later, more realistic tasks) indefinitely, to the fastest patterns. If, on the other hand, the limiting value is smaller (bottom panel) or larger (top panel) than the movement axis, then m _{∞} drops at some point to zero and remains zero for all faster patterns.
As P decreases, m _{∞}/P(black curves in Fig. 4) increases as long asm _{∞} remains reasonably high. But whenm _{∞} drops to zero, so of course doesm _{∞}/P. As the pattern speeds up, therefore, performance progressively improves until a certain point, but then, in general, rapidly fails. However, in the special case where the limiting value is precisely equal to the movement axis, so thatm _{∞} never becomes zero,m _{∞}/P continues to increase to very high values for the fastest patterns (Fig. 4 A, middle panel; Fig. 3, bottom right). In essence, the inputoutput structure of the model NMT used here (specifically, the shape of the dependence of c̄ _{∞} and c _{∞}, and so m _{∞}, on P) is such that the faster the pattern can become while still maintaining m _{∞}, the higher is the value of m _{∞}/P. Thus the structure of the NMT favors the strategy of producing faster, even though smaller, movements to maximize performance. Patterns that tend to the limiting value precisely equal to the movement axis maintainm _{∞} indefinitely, give the highest performance, and so can be said to be optimal (Fig. 4 A, middle panel).
Which are the optimal patterns? In other words, which values of the other two input parameters, F andf _{intra}, or F and 〈f〉, can be combined with very smallP to give a limiting value precisely equal to the movement axis?
We showed in Paper I that, with its standard parameter values, the model NMT becomes in a certain sense linear when P ≪ τ, and in consequence its output parameters such asc̄ _{∞} and c _{∞}become “pattern independent.” By this we mean thatc̄ _{∞} and c _{∞}, as they converge to the limiting value, come to depend only on the mean density of spikes—the mean firing frequency 〈f〉—regardless of the spikes' temporal arrangement (Paper I; Brezina et al. 1997). The optimality of a pattern thus depends just on 〈f〉; for a pattern to be optimal, a necessary and sufficient condition is that it have an optimal value of 〈f〉: one that, when P ≪ τ, gives contraction equal to the movement axis. This dependence can best be appreciated, therefore, in the (P, F, 〈f〉) representation, in the twodimensional sections in Fig. 4 A, and even better in the corresponding threedimensional sections in Fig.5 A.
Because the model NMT has a monotonically increasingc _{∞}(f) relation, there is just one optimal value of 〈f〉; with the standard NMT parameters, and the movement axis set at 0.4, 〈f〉_{optimal} ≈ 2.8 (appendix , 2). Patterns with this value produce functional movement m _{∞}, and so high performance m _{∞}/P, to the smallest values of P (middle panels in Figs. 4 Aand 5 A). Patterns with smaller or larger 〈f〉 do not permit such close approach to small P and high m _{∞}/P(bottom and top panels). Figure 5 Ashows that this is true in a “patternindependent” way, across all values of the duty cycle F.
To keep 〈f〉 =f _{intra} F constant, F andf _{intra} must always vary in a precisely compensatory way. For a given period P, any decrease inF, and therefore in the intraburst durationd _{intra}, must be exactly compensated by an increase in the intraburst firing frequencyf _{intra} to maintain the same number of spikes, and so 〈f〉. Intuitively, the mechanism underlying the phenomena we have just discussed is that, when the NMT is linear, the parameters F andf _{intra} that compensate to maintain 〈f〉 at the input to the NMT also compensate to maintain contraction, movement, and performance at the output. This is the basic mechanism of “patternindependent” output (Brezina et al. 1997). Such compensation is effective even if 〈f〉 is not completely optimal. In this case, the firing pattern cannot be made as fast, and performance is not as high. Nevertheless, as can be seen in Fig.4 B, left (also Fig. 3, bottom left), if f _{intra} increases to compensate asF decreases, so maintaining 〈f〉, performance is also approximately maintained. In contrast, if f _{intra} remains fixed, 〈f〉 declines and performance fails (Fig. 4 B, right).
We conclude that, to control and maximize performance in this sort of behavioral task, the nervous system must regulate two parameters of the firing pattern, the cycle period P and the mean firing frequency 〈f〉. Significant performance is obtained only with relatively fast firing patterns, with relatively small P. At any particular small P, performance depends, to a first approximation, on 〈f〉. Performance increases asP is decreased further, provided that, at the same time, 〈f〉 is brought closer and closer to the optimal value, 〈f〉_{optimal}.
〈f〉 is already a composite input parameter: the elementary, more “physiological” parameter isf _{intra}, the frequency of firing that the nervous system must actually generate during each burst. How does the requirement for 〈f〉_{optimal}appear in terms of f _{intra}? Because 〈f〉 =f _{intra} F, 〈f〉_{optimal} corresponds to multiple optimal pairs of f _{intra} andF. In the (P, F,f _{intra}) representation in Fig. 5 B, therefore, we see that each f _{intra} ≥ 〈f〉_{optimal} corresponds to some particular value of F where alone the firing pattern can be speeded up to the smallest P, and so to high performance. Given a particular f _{intra} (dictated perhaps by other considerations, for example the limited range of frequencies at which the controlling motor neuron can actually fire) the nervous system must select the correct matching F to maximize performance; conversely, given F, it must select the correct matching f _{intra}. This illustrates in an especially dramatic way the major concept that emerges from our discussion here. When filtered through the inputoutput structure of the NMT, only certain firing patterns give high performance, or indeed any performance at all, in a behavioral task. To obtain that performance, the nervous system must select those patterns. The functional requirements of the task, ultimately, dictate these patterns to the nervous system.
We will now proceed to more complex situations and tasks. In these, many of the same phenomena that we have analyzed here will appear again, for the same reasons. We will therefore for the most part dispense with the systematic analysis referred back to the contraction level. Rather, we will use the results in these tasks to recapitulate and extend the functional ideas that we have introduced in this section.
Nonlinear NMT
In the last section we discussed the case where the NMT is linear for fast firing patterns, when P ≪ τ. How are matters altered if the NMT is not linear? In real systems, a situation of this kind may occur when multiple cellular processes with very different time constants contribute to the NMT. For instance, the response of the contractile machinery may be slow, and ratelimiting for the overall time constant of the contraction. But in addition there may be much faster—in relative terms, effectively instantaneous—processes, such as fast facilitation of transmitter release, that preprocess the spikes of the firing pattern (cf. Paper III). Because the slow contractile process is ratelimiting, the contraction will still be steady, without significant oscillation, when P ≪ τ. If the fast processes are nonlinear, however, this nonlinearity will manifest itself in “pattern dependence”: the contraction will stabilize in the steady state at different amplitudes with different firing patterns (Brezina et al. 1997). This will mean different performance in a behavioral task such as Task I.
Our model NMT is no longer linear for fast firing patterns when one of its parameters, p, is no longer equal to 1 (appendix , 2; Paper I; Brezina et al. 1997). We will set p = 3.
Figure 6 shows the results for Task I, in a format comparable to that in Fig. 5 A for the linear case. We see that, when the NMT is nonlinear, there is no longer a single value of 〈f〉_{optimal} that gives high performance for all F. Instead, what we saw before in the (P, F,f _{intra}) representation, we now see in the (P, F, 〈f〉) representation as well. Only certain matching pairs of 〈f〉 and F allow the firing pattern to be speeded up to the smallest P and high performance. Again, the nervous system must select one of the few patterns that are correct for the task. Furthermore, the fact that these patterns now differ in 〈f〉—in their number of spikes—raises the issue of the cost and efficiency of different patterns. We will pursue this issue with the next task and, in more general terms, later.
Task II: tonic contraction
Our phasic firing patterns are most obviously suitable for producing phasic, oscillatory contractions and movement, and this is the main subject of our work here. But under different circumstances the same neuromuscular system may well be called on to produce a steady, tonic contraction, to apply a prolonged steady force or to hold a body part in a fixed position for a prolonged period (cf., e.g.,Morris and Hooper 1998). We will therefore very briefly examine the performance in such a task.
The task is simply to approach the contraction waveform as closely as possible to a given steady contraction goal (Fig. 7, left). Essentially, it is the converse of Task I. Our performance measure (appendix ) assumes its highest value,m _{∞} = 1, when performance is perfect: when the contraction is steady, with no oscillation at all, at precisely the given contraction goal. In the threedimensional sections in Fig.7, A and B, this perfect performance is indicated by pure white tone, and progressively worse performance by progressively darker tone.
Steady contraction, we have seen, is obtained either with steady firing (F = 1), or with a firing pattern that is much faster than the time constant of the NMT (P ≪ τ). When the NMT is linear for fast patterns, the contraction is “pattern independent,” with the same amplitude in both cases, with steady firing or with any fast pattern whatever, provided that 〈f〉 is the same—this is the meaning of “pattern independence.” In Fig. 7 A we see, therefore, that with the model NMT with p = 1, and having set the contraction goal the same as in Task I, 〈f〉_{optimal} ≈ 2.8 again gives perfect performance, with steady firing or any fast pattern.
But when p = 3, the NMT is no longer linear, the contraction is no longer pattern independent, and there is no longer a single 〈f〉_{optimal} that gives perfect performance for any F. Rather, once again, there are multiple matching pairs of 〈f〉 andF that give high performance as the pattern speeds up (Fig.7 B).
In these pairs 〈f〉 is always smaller, and smaller by an increasing margin as F also becomes smaller, than the value that gives perfect performance with steady firing (now 〈f〉 ≈ 1.4, fromEq. A3a in appendix , 2). In other words, the “positive” pattern dependence (Brezina et al. 1997; Paper I) now exhibited by the NMT allows the same performance to be achieved with progressively fewer spikes as those spikes are grouped into progressively more extreme bursts. If spikes impose significant metabolic costs (cf., e.g., Laughlin et al. 1998), the saving in spike number could be significant. Thus it is by no means obvious that steady, tonic contraction requires steady firing. Bursting patterns, provided they are fast enough, achieve equally good performance and, if the NMT is nonlinear, may in fact do so more efficiently.
Task III: movement oscillating beyond upper and lower thresholds
One clearly unrealistic element in our Task I was the fact that arbitrarily small movements, when repeated fast enough, could still constitute good performance. In reality, very small and fast movements would simply be damped out by the inertial and elastic properties of biological tissues. (These are the kinds of complicating properties that, as we noted earlier, could be incorporated into the quantitative structure of a more realistic NMT.) But even more importantly, behaviorally meaningful movements will always have some minimum amplitude—the body part must be moved some significant distance in relation to other parts of the body or the environment—generally of the same order as the size of the muscles involved. In other words, the minimum movement is part of the requirements of the task.
We will therefore amend Task I to require the oscillatory contraction and movement to be at least of a certain size. We will split the single movement axis into distinct upper and lower thresholds, both of which the oscillation must cross in each cycle for nonzero performance. Beyond the thresholds, the larger the oscillation grows the better, to the extent that it grows symmetrically, as before (Fig.8, left; appendix c).
The performance measure m _{∞}/P for this task, using the linear model NMT with p = 1, is plotted in Fig. 8, A and B, which we can compare to Fig. 5, A and B, for Task I.
The major difference, we see, comes as the pattern speeds up toward small P. The oscillatory contraction, of course, is just the same as before; very small and fast oscillations can still be produced. But they no longer constitute functional movement. At some point as the pattern speeds up, and the oscillation—the range fromc̄ _{∞} to c _{∞}—becomes ever smaller, it ceases to span the distance between the two thresholds. At that point functional movement and performance fall to zero. But until shortly before that point, the performance measure m _{∞}/Pincreases. To maximize performance, it is still a good strategy to speed up the pattern to values of P as small as possible, provided nonzero movement can be maintained. And, if the NMT is linear for fast patterns, there is still an optimal value of 〈f〉—one where, in their convergence, the increasing c _{∞} meets the lower threshold, and the decreasing c̄ _{∞} meets the upper threshold, simultaneously as P decreases—that maintains nonzero movement to the smallest P, and the highest m _{∞}/P, that is possible in the task. Indeed, because in Fig. 8 we have separated the upper and lower thresholds symmetrically, and c̄ _{∞}and c _{∞} also converge approximately symmetrically, 〈f〉_{optimal} ≈ 2.8 as before (see Fig. 8, left, and 8 A). As before, this value translates into multiple optimal pairs off _{intra} and F (Fig. 8 B). The difference is that now, in this more realistic task, the smallestP that is possible is no longer arbitrarily small.P smaller than some value, determined by the setting of the two thresholds in conjunction with the inputoutput structure of the NMT, is no longer functional. Correspondingly, the highestm _{∞}/P that is possible is not as high. (And, because the limit of arbitrarily small P can no longer be achieved, our statements concerning the optimality of patterns are no longer exactly true, only approximately so.)
Thus, by making the requirements of the task more stringent, we have brought out more prominently several features that we already noted, or that were implicit, in Task I. The subset of firing patterns that satisfy the requirements and produce functional movement and behavior is now even more restricted. Furthermore, certain absolute restrictions have become apparent. Whereas before the firing pattern could have any speed, provided its other parameters were set to the correct matching values, now patterns faster than some speed are absolutely not functional. No values exist to which the other parameters can be set that can give nonzero performance at those faster speeds. But those faster speeds of the firing pattern, movement and behavior may be very desirable to the nervous system, perhaps for higherlevel reasons but most obviously because, as we have found, they maximize performance.
As we saw in our analysis of Task I, these restrictions are an immediate consequence of the way in which the inputoutput space is structured by the properties of the NMT. For the question of what happens as the pattern speeds up, it is the speed of the NMT, relative to the speed of the pattern, that is important. And, while the speed of the pattern varies, the speed of the NMT is fixed. As the pattern speeds up, the contraction is given less and less time to react. But it continues to react at the same fixed rate, and so covers less and less distance. When Task III, even more explicitly than Task I, nevertheless requires the contraction to cover some given distance in each cycle, at some point it becomes absolutely unable to do so.
This implies that, if the speed of the NMT could be increased, it should be possible to maintain functional movement to faster patterns, and higher performance. We will return to this idea below, and it will be our main focus in Paper III.
Task IV: antagonistic muscle pair
So far we have been considering just a single neuromuscular unit. But most real muscles work in intimate conjunction with one or more other muscles, in a multimuscle ensemble whose integrated activity generates the behavior. A common motif in such ensembles is that of the antagonistic muscle pair. We will use this arrangement in our remaining tasks.
We consider two neuromuscular units, each consisting of one muscle controlled by one motor neuron as before, arranged so that the muscles contract in opposite directions. The combined contraction (or, more properly, movement) produced by the pair is taken to be simply the sum of the two individual contractions, which themselves are not altered by their interaction (Fig. 9,left; appendix d, 1; complete waveforms can be seen in Fig. 11). Again, a more complex combination of the two contractions could easily be incorporated, if required, into the quantitative structure of the NMT.
The first task we will give to the antagonistic muscle pair is precisely the same as our original Task I. The contraction—now, however, the combined contraction, rather than the contraction of a single muscle—is required to oscillate symmetrically around a given movement axis (Fig. 9, left; appendix d,2).
With two motor neurons, the combined firing pattern, too, becomes more complex. To describe all possible canonical firing patterns of the two motor neurons, we require seven parameters (three for each motor neuron plus the relative phase of the two patterns) and so a sevendimensional input space for their geometric representation. For simplicity, we will here restrict the firing of the two motor neurons to always have equal period P and duty cycle F, and a phase of 0.5. (These restrictions are not entirely unphysiological: for instance,P may very well be the same for two motor neurons driven by the same central pattern generator.) With particular fixed values of the firing frequencies, 〈f〉 orf _{intra}, of the two motor neurons, we can plot threedimensional sections of m _{∞}/Pas before (Fig. 9, right). These are now, however, sections through an eightdimensional, rather than a fourdimensional, inputoutput space.
Each of the two neuromuscular units, separately, performs as we analyzed for Task I. The added interest comes from considering their interaction, particularly, once again, as the nervous system speeds up the firing pattern toward smaller and smaller P. Under what circumstances can functional movement be maintained as this happens, so that performance increases? For exactly the same reasons as in Task I, there exist optimal values of the mean firing frequencies 〈f〉_{1} and 〈f〉_{2} of the two motor neurons (or, if the NMT is not linear for fast patterns, optimal pairs of 〈f〉_{1} andF _{1}, and 〈f〉_{2} andF _{2}) that allow functional movement to be maintained to the smallest values of P. But what is optimal is no longer 〈f〉_{1} alone, or 〈f〉_{2} alone. Rather, there are multiple optimal combinations in which 〈f〉_{1} and 〈f〉_{2} are correctly balanced, taking the relative size or strength of the two muscles—the relative ranges of c _{1} and c _{2}— into account (appendix , 3). With equally sized muscles, for instance, and the movement axis set at 0 (a completely symmetrical situation), multiple equal pairs 〈f〉_{1} = 〈f〉_{2} are optimal (Fig.9 A). If, as is common in real systems, one of the muscles is larger or stronger than the other (Fig. 9 B), or if one of the motor neurons fires at a higher frequency than the other (Fig.9 C), the situation is no longer optimal. The two inequalities can, however, be balanced against each other (at least over a certain range) to restore optimality (Fig. 9 D). In other words, a small or weak muscle stimulated at a high frequency is functionally equivalent to a large or strong muscle stimulated at a low frequency.
Because this task is precisely the same as Task I, it raises the question whether the antagonistic muscle pair performs it better (or worse) than the single muscle. We will examine this question with the next task.
Task V: antagonistic pair compared to single muscle
Just as the preceding task was the same as Task I, this task is the same as Task III, except for the antagonistic muscle pair rather than a single muscle. The combined contraction is required to oscillate symmetrically beyond distinct lower and upper thresholds (Fig.10, left; appendix e).
The results in this task relate to those in the preceding task in precisely the same way in which the results in Task III related to those in Task I. That is to say, the parameter combinations that were optimal in the preceding task are still optimal here; those combinations are still the ones that allow functional movement to be maintained furthest, and performance to increase to the highest values, as the pattern speeds up. But it is no longer possible to reach arbitrarily fast patterns. Patterns faster than some speed are now absolutely nonfunctional.
We can now compare the performance of the antagonistic muscle pair with that of the single muscle in this task. Indeed, our earlier Task III for the single muscle was just a special case of this task when one of the muscles has zero size or strength, or is left unstimulated (appendix , 3).
We can gain insight into the relative performance of the two muscle arrangements by comparing the individual with the combined contraction waveforms in the various examples shown in Fig. 11. We make two main observations.
First, adding to the single muscle the second muscle contracting in the opposite direction broadens the physical bounds of the situation. The combined contraction can span a larger range than the single contraction, functional movements can be larger, and they can be large for a broader range of movement goals (on opposite sides of the linec = 0, for example), facilitating the performance of multiple behaviors with the same neuromuscular plant. But to exploit this broader capability, the nervous system must, once again, fire the two motor neurons in patterns that are correct, now not just each on its own but relative to each other. In particular, their relative phase must be correct, for best performance around 0.5 (the exact value will depend on the detailed quantitative structure of the NMT) so that the two muscles contract in alternation. All of this becomes significant, however, mainly with slow patterns, when the contraction has time to approach the broadened bounds; with these patterns, the performancem _{∞}/P is low.
More interestingly in view of our earlier stress on the speed of the NMT as a key factor, the combined contraction can be faster than the single contraction (see for instance the two examples in Fig. 11,bottom). It is faster by the largest amount, again, when the phase is 0.5, when the movements of the two muscles that go in the same direction—the contraction phase of one and the relaxation phase of the other—reinforce each other. (In real systems, even larger effects may be expected if, for instance, the muscle stiffness varies with activation, so that the contracting muscle can stretch the relaxing muscle.) As the phase approaches 0 or 1, where the two muscles contract and relax in synchrony, the combined contraction becomes, on the contrary, slower than the single contraction.
Furthermore, not only is the combined contraction faster, but its speed is no longer fixed, but variable. Different combinations of the firing frequencies of the two motor neurons, and in particular the different optimal pairs of 〈f〉_{1} and 〈f〉_{2}, give different speeds. As 〈f〉_{1} and 〈f〉_{2} increase while maintaining their optimal balance, so that the two individual contractions increase while maintaining the combined contraction at the same, optimal level, the speed of the combined contraction increases (see Fig. 10 B). Fundamentally, this is possible because, unlike with the single contraction, each phase of the combined contraction contains a component that is controlled by an independent input, either 〈f〉_{1} or 〈f〉_{2} (cf. appendix l of Paper I, appendix of Paper III).
As the combined contraction becomes faster, functional movement can be maintained to faster patterns, and higher performance. In Fig.10 A, this is shown for three optimal pairs of 〈f〉_{1} and 〈f〉_{2}, the first, with 〈f〉_{2} = 0, being just the singlemuscle case. As 〈f〉_{1} and 〈f〉_{2} increase while maintaining their optimal balance (top tobottom), faster patterns become functional, and performance increases. Figure 10 B compares the contraction waveforms in one particular location in the three cases to illustrate the underlying mechanism.
Thus, whereas in the preceding task all optimal combinations of 〈f〉_{1} and 〈f〉_{2} gave equivalent performance in the limit of the fastest patterns, in this task, where that limit cannot be reached, some of the combinations prove to be more optimal than others. Those combinations where the motor neurons fire at higher frequency can reach closer to the limit, and give better performance.
Cost and efficiency of patterns
There is a price to be paid for this enhanced performance and versatility, however. Not just one but two muscles are needed, and the best performance is obtained when both motor neurons fire at high frequency and both muscles contract strongly. Both neuromuscular units are working hard, but, especially as the firing pattern speeds up, much of the work is dissipated in their mutual opposition (Fig.10 B); the net movement, while it is what the task requires, is relatively small. We can see that under these circumstances the energy expended to perform the task may be large, and perhaps disproportionately large relative to that expended by other patterns that also perform the task. If this is taken into account, the performance may be relatively inefficient.
So far we have been considering the contractions, movement, and performance produced by different firing patterns in absolute terms, without regard to their energy cost. But every firing pattern has a corresponding energy cost; we can imagine a parallel “energy” NMT, with the energy cost as the output parameter. As we noted earlier, significant energy may be expended in generating the spikes of the pattern, but most, presumably, in the muscle contraction itself. If we could compute these energies, we could relate them to the performance —most simply, normalize the performance by the energy cost—and arrive at the relative efficiency of different patterns in performing a particular task.
Unfortunately, the energetics of muscle contraction are complex and only partially understood (Woledge et al. 1985). In particular, while the energy will presumably bear some relationship to the size and shape of the contraction waveform (and thus automatically to the spike density or number through the relationships described in Paper I), the relationship is not straightforward because the overall contraction waveform may conceal multiple contraction and relaxation processes with different energetics, on multiple time scales and with different pattern dependence. Muscles are energetically optimized for contraction at different speeds (see the discussion). Further complications arise at the levels of movement and behavior, because interaction with other muscles and external elastic structures often means that certain patterns of movement impose large energy penalties, or on the contrary afford significant energy savings (Alexander 1988; Ettema 1996; Full 1997; Gans et al. 1997). In a real system, the energy expenditure can, however, be empirically measured. Here, to give some idea of the use to which such information may be put, we will simply make, once more, a linear assumption, namely that the energy cost is directly proportional to the area under the contraction waveform: to the mean contraction amplitude 〈c〉_{∞}, or, when multiple muscles are involved, to the sum of their individual contraction means.
Normalizing by this energy cost will produce, for each of the plots of performance in this paper, a corresponding plot of efficiency. Each surface of m _{∞}/P in a threedimensional section, for instance, will have a corresponding surface of efficiency, with a related but quantitatively different shape. Some regions will be lowered, others raised, relative to each other. When, therefore, there are multiple firing patterns that give the same performance (the invariable case, as the plots show) they may be differentiated through their efficiency.
Figure 11, middle, shows one such efficiency plot. It is for the antagonistic muscle pair, for simplicity for Task IV rather than Task V, but for a nonoptimal situation (that presented earlier in Fig. 9 B) where the same problem arises, that performance cannot be maintained to the fastest patterns. For the purposes of discussion, we will focus on the rear region of small F, where the bursts of firing are well developed, and on the corresponding two examples of complete contraction waveforms at the top.
With firing patterns faster than some value, the muscles contract—indeed, in the limit, remain tonically contracted against each other—but produce no functional movement at all (top left). As contractions are driven to oscillate faster, the range of the oscillation diminishes, and at some point becomes too small to perform the task. But the mean contraction amplitude does not diminish (Paper I); the contraction simply converts from a more phasic to a more tonic form. The result is that considerable energy is expended even though the task is not performed, for zero efficiency.
If the pattern is not too much faster than the value where performance fails, we saw in the last section that performance may appear if both muscles are driven harder. (The precise range of speeds where this will happen will depend on the quantitative details of the NMT.) But, especially as much of the extra effort is dissipated in the opposition of the two muscles, this is a very inefficient expedient.
As the pattern slows just slightly more, performance rises dramatically, indeed, as we have seen, to its highest value. The oscillations are still small, but now they are large enough to perform the task, and as they are still fast, the performancem _{∞}/P is high. The mean contraction amplitude remains essentially unchanged (Paper I). The same energy is expended, but now it returns high performance, with high efficiency.
Finally, as the pattern becomes very slow, the oscillations become very large, but slow (top right). The performancem _{∞}/P declines. The efficiency, however, remains high. This is because the mean contraction amplitude also declines (Paper I): these large, slow oscillations expend less energy. In essence, as we have seen, as the pattern becomes slower than the time constant of the NMT, the contraction becomes more phasic, and ultimately completely phasic. The contraction follows more faithfully the bursts of firing (compare the various waveforms in Fig. 11). The bursts can therefore be used more effectively to contract the muscle only at the times required (in our simple tasks that consider only the range of the contraction, ultimately only at a single time point in each period) rather than, as with fast patterns, necessarily also throughout the rest of the period. Similarly, with the appropriate alternating phase, the contractions of the two antagonist muscles avoid each other more completely, build up less tonic cocontracture, and so are reflected more fully in the combined contraction and movement. In sum, contractions can be shaped more easily to direct more of their energy cost into productive movement and behavior.
Thus, not just the functional movement and performance, but even (at least in our simplified formulation) the energy cost and efficiency depend on the inputoutput structure of the NMT, indeed, for our simple tasks, as it is reflected in the basic output parametersc̄ _{∞}, c _{∞}, and 〈c〉_{∞} that we studied in Paper I. Furthermore, we see again that, in addition to the strictly mechanical requirements of the task, the nervous system is faced with a choice of higherorder criteria to be optimized. Is it best to produce large movements, or movements with a certain speed, or high overall performance, or high efficiency? Not all of these criteria can be satisfied simultaneously.
Task VI: ARCopener neuromuscular system of Aplysia
So far we have devised and studied relatively abstract, generic tasks for our mathematical model NMT. To what extent do our findings hold for a real NMT, such as the B15ARC NMT of Aplysia? In Paper I we saw that the two NMTs are similar in their qualitative properties. Are they similar also in their functional performance?
For a real NMT, we wish no longer to devise a generic task, but rather to deduce from the available experimental data what its real, specific task in the behaving animal is.
The accessory radula closer (ARC) muscle of Aplysiaparticipates in rhythmic consummatory feeding behaviors such as biting, swallowing, and rejection of unsuitable food (Kupfermann 1974). It is part of the buccal mass, a complex ensemble of muscles whose coordinated contractions generate various movements of a grasping mouthpart called the radula (Drushel et al. 1997; Scott et al. 1991). In an ingestive cycle (biting, swallowing), the radula protracts from the mouth, closes on food, retracts, and opens to release the food into the esophagus. In rejection, the phase relationship between the opening and closing of the radula and its forward and backward movement is altered so that the radula is closed during protraction rather than retraction, pushing material out rather than pulling it in (Morton and Chiel 1993a,b;Rosen et al. 1998; Weiss et al. 1986).
The ARC is a powerful muscle that closes the radula (Cohen et al. 1978); it acts in opposition to a compound radula opener muscle (Evans et al. 1996). The behavioral task of the antagonistic ARCopener pair is presumably, first of all, to open and close the radula in each cycle, and then to correctly coordinate that opening and closing with the other movements produced by the rest of the buccal mass, in particular the protraction and retraction of the radula.
For simplicity, we will consider here just one of the feeding behaviors, namely swallowing. (Biting and rejection would be entirely analogous, but different in the quantitative details of the task, such as the various phase relationships.) In swallowing, to a first approximation, the radula must be closed as it retracts, and open as it protracts (more precisely, see appendix ). It must switch states rapidly, within a short time window, at the transitions between protraction and retraction. In appendix , in conjunction with Fig. 12 A, we devise a precise definition of the task embodying these requirements, which allows us to compute the functional movement m _{∞}, our basic measure of how well the task is being performed. Overall, the task closely resembles our earlier Task IV for the antagonistic muscle pair, but adds to it (and to the task implicit in previous, more conceptual models of ARCopener function, such as that in Fig. 1 of Paper I) an additional layer of complexity, in that the radula is now required not merely to open and close, but also to do so with the correct timing, for functional behavior.
One important benefit of explicit modeling is that it reveals what information is still missing. In the case of the ARCopener system (as with most real systems) perhaps the most fundamental problem is that, especially in its quantitative parameters and so in our quantitative evaluation of the quality of its performance, the precise task of the system in the real behavior is not easy to deduce (see further thediscussion). Other problems are more technical. How the ARC and opener muscles interact mechanically is unknown, and, in a complex muscular organ without rigid structure such as the buccal mass, may be difficult to determine (cf. Drushel et al. 1997). Here we make, once more, the linear assumption that the individual contractions simply sum to give the combined contraction or movement. Finally, all of the NMTs involved in the task should be known. We have characterized the B15ARC NMT, but the ARC muscle is innervated by a second motor neuron, B16 (Cohen et al. 1978), that also fires and helps contract the muscle in swallowing (Cropper et al. 1990), and the opener muscle has its own motor neuron, B48 (Evans et al. 1996). These two additional NMTs have not yet been characterized. Here we simply use the B15ARC NMT to control the ARC muscle (in effect lumping the B16ARC NMT into it) as well as (with scaleddown amplitude because the opener is weaker than the ARC;Scott et al. 1991, 1997a) the opener muscle. In reality, however, the B16ARC NMT and the B48opener NMT both probably differ somewhat from the B15ARC NMT. In particular, they may have faster kinetics (Cohen et al. 1978; Scott et al. 1997b); furthermore, contraction of the opener muscle is markedly potentiated by stretch (Evans et al. 1996). We have already seen in our work here how a system with multiple NMTs, multiple motor neurons as well as multiple muscles, can be dealt with. It will be interesting, in future studies, to see whether the properties of the three NMTs in theAplysia ARCopener system differ in such a way as to contribute to better performance in its various tasks.
Using in the meantime just the B15ARC NMT, the performance in the swallowing task is presented in Figs. 1214. Here we have predicted (from the output of the NMT in response to steady input as described in Paper I) the individual contraction waveforms of the ARC and opener muscles, and hence the combined contraction or movement, for various firing patterns in the dynamical steady state of the system. Representative examples are shown in Fig.12. From the contraction waveforms, we have then computed the functional movement m _{∞}, the performance m _{∞}/P, and the efficiencym _{∞}/P(‖〈c _{1}〉_{∞}‖ + ‖〈c _{2}〉_{∞}‖). Figure13 shows threedimensional sections of the performance, for firing patterns in the (P,F, 〈f〉) and (P,F, f _{intra}) representations. Figure14 compares, for firing patterns in the physiologically most relevant (P, F,f _{intra}) representation, the functional movement, performance, and efficiency. As in our antagonisticmuscle tasks for the model NMT, we have here restricted the ARC and opener motor neurons to fire in the same canonical pattern and with a phase of 0.5; the plots in Figs. 13 and 14 are merely sections through the general eightdimensional inputoutput space.
Because this task has many elements in common with our earlier tasks, and the real NMT resembles the model NMT, we see in Figs. 1214, indeed, many of the same features that we have already discussed. We see that only a subset of firing patterns produces functional behavior, and a still smaller subset efficient functional behavior. Many patterns are completely nonfunctional. In particular, for exactly the same reasons as before (as the fixed speed of the NMT becomes increasingly unable to react to the speed of the pattern) patterns cannot maintain functional, alternating opening and closing of the radula as they speed up toward small P (e.g., Fig. 12 B, and the corresponding black dots “B” in Figs. 13 and 14). Still other patterns are rendered nonfunctional by the more stringent requirements of this task. In Fig. 12 E, for example, the radula opens and closes quite well, but it does so at the wrong times relative to its protraction and retraction.
We see again that, in general, slow patterns produce large movements, and fast patterns small movements (m _{∞}; Fig.14, left column). But fast patterns—the faster the better, provided they can maintain functional movement—become increasingly advantageous when we consider the total movement over an interval of time (m _{∞}/P; middle column), and especially when we consider the efficiency (m _{∞}/P(‖〈c _{1}〉_{∞}‖ + ‖〈c _{2}〉_{∞}‖); right column). It is striking also that, when the size of the movements is varied by changing firing frequency, their efficiency is not much affected: smaller movements, provided that they take correspondingly less effort, can be just as efficient as larger movements with more effort.
Our computation of performance in the swallowing task fixes, too, the precise phase, relative to the cycle of radula protraction and retraction, with which the ARC and opener motor neurons should fire for best performance (vertical dashed lines in Fig. 12; appendix ). It is interesting that this phase alters somewhat as the ARC and opener firing pattern varies (compare, for example, Fig. 12,A and D). As already noted, such phase relationships are merely further parameters of the overall firing pattern that the nervous system sends through the NMT to generate behavior. This observation thus underscores, once again, the fundamental concept that, when only a subset of firing patterns generates functional behavior, the nervous system must select a pattern within that subset. And if the nervous system then changes one of the parameters of the pattern, it may have to change others too in a coordinated manner so that the pattern remains within the bounds of the functional subset.
Modeling a real system allows us to test the predictions of the model in the real system. Our approach here predicts, above all, the firing patterns of the motor neurons. If we have adequately understood the NMT and the task that the system is performing, we should find that the real motor neurons fire in patterns that are within the predicted subset of efficient patterns, and certainly within the subset of functional patterns. Is this in fact the case in the AplysiaARCopener system?
Unfortunately, in vivo recordings of the firing patterns of the ARC and opener motor neurons during feeding behaviors such as swallowing are as yet fragmentary. (Definitive comparisons will in any case require more specific modeling of all of the NMTs in the system.) Some additional information can be drawn, however, from observations of the behavior itself, in particular of its cycle period P, which is also that of the underlying motor neuron firing patterns. Aplysiaconsummatory feeding behavior, although stereotyped in form, varies quite widely in its parameters, in response to changing quality of the food (e.g., Hurwitz and Susswein 1992), as well as, over the course of the perhaps hourlong meal, with the changing motivational state of the animal. Particularly characteristic is a progressive change in P. The feeding movements start slowly, with P in the tens of seconds, then rapidly accelerate in a manifestation of what has been termed foodinduced arousal, then gradually decelerate again as the animal satiates (Kupfermann 1974; Susswein et al. 1976,1978). Meanwhile, the other parameters of the firing pattern may be of the order ofF ≈ 0.5 and f _{intra} ≈ 8–12 Hz (for the B15ARC NMT) or f _{intra}≈ 15–20 Hz (for the B16ARC NMT) (Church and Lloyd 1994; Cropper et al. 1990). In Figs. 13 and 14we see that such firing patterns lie squarely in the region of functional and indeed effective patterns. In particular, performance and efficiency are highest around F = 0.5, as long asP is relatively large; as P decreases,F should decrease also. This may well happen (Cropper et al. 1990). Overall, then, there appears to be reasonable agreement between the model and the real system.
There is one clear discrepancy, however. At the height of foodinduced arousal, the real cycle period can be as short as ∼3 s (e.g., Hurwitz and Susswein 1992; Susswein et al. 1978; Weiss et al. 1986). Figure 14 suggests why such fast movements should be very desirable: they are likely to give the highest performance and especially efficiency. But it appears that our model cannot maintain function to such small values ofP. We believe that this discrepancy is explained by, and illuminates the functional reason for, the existence in the real ARCopener system, as in many other neuromuscular systems, of numerous modulators that at the behaviorally appropriate times modulate the properties of the NMT, in particular accelerate its kinetics so as to enable it to maintain function to significantly smaller values ofP. This is the subject of Paper III.
DISCUSSION
In Paper I we introduced the NMT as an inputoutput relation that transforms patterns of motor neuron firing to shapes of muscle contractions. We developed a framework for analysis of the NMT and identified with it principles of the transformation. We saw that, because the NMT is a dynamic, nonlinear, and (as Paper III will show further) modifiable filter, the transformation is complex.
However, the ultimate functional goal is not muscle contraction, but behavior. In sending different firing patterns through the NMT, the nervous system is seeking to command different behaviors. These are defined in large part by their specific, often very precise, contraction requirements. To what extent do the contractions that emerge from the NMT actually satisfy those requirements? How are changes in the firing pattern reflected, not just in altered contractions, but in altered behavior? This question naturally arises, for instance, with respect to the dramatic modulations and reconfigurations of the activity of central pattern generators and their follower motor neurons that have been observed (Dickinson 1995; HarrisWarrick et al. 1989,1992; Katz 1995; Marder and Calabrese 1996; Stein et al. 1997). Given that it must send its commands through the complex filter of the NMT, to what extent can the nervous system in fact achieve the desired behavior? And, conversely, what properties should the NMT have to make it possible?
In this paper we have extended our analysis to this functional level. We have devised and studied, with our mathematical model NMT and then with the real B15ARC NMT of Aplysia, a series of representative behavioral tasks that the nervous system might ask muscles to perform. Each task has specific, precisely defined contraction requirements. This allows us to compute an explicit measure of how well any particular contraction waveform, produced through the NMT by a particular firing pattern, satisfies those requirements: how well that pattern performs the task. In effect, we have extended the inputoutput scope of the NMT so that it transforms firing patterns not just to muscle contractions, but immediately to behavior.
Formal framework for analysis of neuromuscular function
An important aspect of our approach is that it allows a global formalization of the kind of elements that emerge only in fragmentary fashion in more intuitive models of neuromuscular function (for theAplysia ARCopener system, for instance, that ofWeiss et al. 1992, 1993). Such models are typically built around some elementary a priori functional principle, such as “larger contractions are better,” “muscles must be able to relax fully before the next contraction,” or “the contraction should faithfully follow the motor neuron firing pattern.” This principle—an implicit, elementary task—is then used to focus on representative contractions and firing patterns that clearly do, or do not, fulfill the criterion. As we saw in results, each of the three principles just mentioned, for example, can indeed be an important element in determining performance in a task. But in any realistic task, as even in our simple tasks here, it will be just one element. If we were to pursue the matter along these lines, we would conclude that overall performance in a real task usually depends on many such lowlevel principles, interacting in a complex, nonlinear fashion, and varying in relative importance with the firing pattern. Even in our simple tasks, we would find it difficult to fully explain performance, for every firing pattern, using just such lowlevel principles.
Instead, we take a more formal approach that avoids or postpones appeal to such principles. Collectively for the whole NMT—for every firing pattern, not just those given meaning by an a priori principle—we first compute the contraction waveforms, which in themselves are valuefree. We then evaluate the functional performance of each in a particular task. We do this using a formula, derived from a description of the task, which can be just as simple, or just as complex, as the task is. We thus compute, in a mechanical fashion, a complete overview of the performance for all patterns, for the whole NMT. Only then, if we wish, do we try to explain the performance (although very likely it will be only partially) through whatever intuitive principle seems appropriate. Alternatively, as we saw in results, our explicit formalization allows a more mathematical analysis.
What is the task?
Clearly, a critical requirement of our approach is a sufficiently precise understanding of the task. What exactly is the neuromuscular system trying to achieve in a behavior? What behavioral parameter, or combination of parameters, is the nervous system trying to optimize? In real behaviors, this may not be at all obvious. Depending on how broadly we conceive the system, it may be concerned with optimizing different things. For the whole animal, for example, feeding behavior is successful, presumably, if the animal remains in optimal energy balance. The purely ingestive components of the behavior may be most concerned simply with acquiring the largest amount of food, efficiently if possible, but this may be less critical if energy savings can be made elsewhere. Individual swallowing movements, in turn, may have to optimize not just the amount of food delivered, but the timing of its delivery (as in our Task VI for theAplysia ARCopener system). Another difficulty in arriving at an understanding of what is of value in a behavior is that, without systematic experimental manipulation, we have only the animal's actual behavior (the actual movements, muscle contractions, and firing patterns) to guide us. Presumably the actual behavior is functional. If we devise a task formula that predicts that the actual contractions and firing patterns are not functional, we have not understood the task correctly. But the actual behavior occupies only a very small part of the space of possible functional behaviors, and tells us nothing about the nonfunctional parts of the space. Thus we lack a good standard of comparison for seeing why the functional behavior is functional.
Here we have worked with three different, although related, measures of performance: m, the size of the individual functional movement per cycle; m/P, the total movement over an interval of time; and m/P〈c〉, a simple measure of the relative efficiency of the movements. These are all somewhat different quantities that the nervous system might optimize: really, measures of somewhat different tasks. We saw that, in general, no single firing pattern will maximize all three simultaneously (e.g., Figs. 3, 11, and 14; cf., e.g., Curtin and Woledge 1996; Rome and Lindstedt 1997). In sending firing patterns through the NMT, the nervous system must choose whether to emphasize large movements, or fast movements, or efficient movements.
NMT constrains the production of functional behavior
Our computation of the functional output of the NMT, visualized in plots like those in the figures, gives us an overview and comparison of the performance of all firing patterns in a task. The quantitative details clearly depend on the particular task and NMT, but some general conclusions stand out.
Although every pattern of motor neuron firing produces some state of muscle contraction, only a subset of the patterns produces functional and efficient behavior. Many patterns are completely nonfunctional. The more complex the task is—the more stringent its requirements are—the smaller is the subset of functional patterns.
The functional subset is different for different tasks that the neuromuscular system may perform (compare, e.g., Fig. 5 A for Task I with Fig. 7 A for Task II). Different behaviors require different firing patterns; conversely, the same pattern may have very different functional value.
All this must be taken into account by the nervous system when sending its motor commands through the NMT. To obtain functional and efficient performance in a task, it must send firing patterns that are within that task's functional subset. If it changes one of the parameters of the firing pattern, it may have to change others too in a coordinated fashion for the pattern to remain within the bounds of the functional subset.
But this may not be possible. Functional behavior with certain parameters, perhaps very desirable parameters, may not be producible through that NMT by any firing pattern.
Thus the NMT constrains the relationship between motor commands and behavior. Through the filter of the NMT, behavioral requirements dictate certain firing patterns to the nervous system; conversely, the NMT restricts the behaviors that the patterns sent out by the nervous system can produce.
Speeding up the rhythm of behavior
To illustrate our general conclusions and to guide us through the output space of the NMT, we have paid particular attention to the question (our initial motivating question in Fig. 1 of Paper I) of what happens as the nervous systems speeds up the rhythm of the behavior. This question arises in many real systems. Rhythmic behaviors, although stereotyped in form, often vary considerably in their parameters, and in particular in their cycle period, P. Most types of locomotion, for instance, can speed up or slow down manyfold (Burrows 1996; Clarac 1984; Full 1997; Gans et al. 1997; Stein et al. 1997); we noted that the same is true for the feeding movements performed by the Aplysia ARCopener system.
Particular values of P may be dictated by external considerations, such as the need to coordinate with rhythmic activity of other muscles or rhythms in the environment. In this case a particular P would be part of the requirements of the task (a more stringent task that, with the incorrect speed, the NMT might well be unable to perform at all). But there may also be internal reasons for a particular P: simply because it gives the highest performance or efficiency. Certain values of P may afford especially large energy savings through interactions with elastic tissue elements (Alexander 1988; Ettema 1996; Full 1997; Gans et al. 1997). More generally, we saw here that the structure of our two NMTs, indeed that of most real NMTs, is such that, up to a point, a typical rhythmic task aiming to maximize the total movement over time is better performed with fast, even though small, movements, rather than large but slow movements. The structure of the NMT itself favors speeding up of movements for best performance. This in fact happens in the Aplysia ARCopener system in foodinduced arousal.
At large P, the choice of the other parameters of the firing pattern is not as critical (e.g., Figs. 3, 5, and 8), but asP decreases the nervous system must control the other parameters more and more carefully. As P decreases, performance increases, but the subset of patterns able to give that performance becomes ever smaller. It becomes ever more important that, as P decreases, the other parameters change in a coordinated manner. Such careful matching and coordination of parameters is observed in real behaviors (e.g., Baldissera et al. 1998; Clarac 1984; Clemens et al. 1998; DiCaprio et al. 1997; Hooper 1997).
But P can be decreased, in real tasks, only so far. Although still smaller P might be very desirable because, if functional movement could be maintained there, it might give still higher performance, at some point performance dramatically fails. This is because, as P decreases, the movements do not simply become compressed in time, but become increasingly distorted in shape and eventually completely dysfunctional (Figs. 11 and 12; Fig. 1 of Paper I). With a particular NMT, a particular behavior is functional and efficient only over some range of P. It is interesting in this context that, while many animals are able to vary their speed of locomotion over a great range, often they do not perform the same behavior throughout. Rather, they switch between distinctly different locomotory modes or gaits in different parts of the range (Burrows 1996; Full 1997; Gans et al. 1997).
Constraint by the fixed properties of the NMT
As we saw, particularly in our detailed analysis of Task I, all this is an immediate consequence of the way in which the inputoutput space is structured by the properties of the NMT. For the question of what happens as the firing pattern speeds up, it is the characteristic speed of the NMT, relative to the speed of the pattern, that is important. While the speed of the pattern varies, the speed of the NMT is fixed. Indeed, with our two NMTs, it is fixed in a particularly restrictive way in that, as we saw in Paper I, the pointwise kinetics of the contraction take no account of the pattern at all. As the pattern speeds up, the contraction is given less and less time to react. But it continues to react at the same fixed rate, and so covers less and less distance. When the contraction is nevertheless required to cover some given distance in each cycle, at some point it becomes absolutely unable to do so.
These changes in contractions and behavior are most dramatic with pattern speeds around the characteristic speed of the NMT. On either side of this, patterns divide into two groups. At speeds much slower than the speed of the NMT, phasic patterns produce completely phasic contractions. Here the contraction does indeed “faithfully follow the motor neuron firing pattern.” Here, consequently, changes in the speed of the pattern do indeed compress or expand contractions and the behavior in time in the “ideal” periodinvariant manner with no distortion of shape (cf. Fig. 1 of Paper I). At speeds much faster than the speed of the NMT, on the other hand, all patterns produce completely tonic contractions. Phasic behavior is not possible.
In sum, with fixed properties, the NMT produces only a limited range of contraction shapes that, for the purposes of performing functional behavior, are well matched to the firing pattern only on certain time scales. Limitations imposed by fixed—especially insufficiently fast—contraction kinetics are well documented for such lowerlevel functional parameters as mechanical work and power output (Baldissera et al. 1998; Caiozzo and Baldwin 1997; Josephson 1993; Partridge 1966; Rome et al. 1988; Swoap et al. 1993). Such limitations are clearly important also in systems that are strictly outside the scope of our work here, notably the vertebrate heart in various physiological and pathological states (Katz 1992).
Matching the NMT to the behavior
Even with optimally selected firing patterns, a particular NMT, with a particular set of fixed properties, constrains behavior severely. Many behaviors are performed poorly or not at all. No matter what firing pattern the nervous system sends, for example, a slow NMT will not perform fast phasic behavior. But animals perform a vast range of behaviors. They are able to do this, in part, by employing different NMTs with very different properties.
Thus, at different muscles in an animal as well as across species, NMT speeds range over several orders of magnitude (Hoyle 1983). Some NMTs of fast spiking vertebrate skeletal muscle have characteristic speeds in the milliseconds (Rome et al. 1996; Rome and Lindstedt 1997); at the other end of the spectrum, NMTs of vertebrate smooth muscle and various nonspiking invertebrate muscles have speeds of many seconds or minutes (Hoyle 1983; Rüegg 1971). These differences reflect corresponding differences in the speeds of the various component steps of the NMT, including release of the neuromuscular transmitter and its plasticity (Atwood 1976; Atwood and Cooper 1996; Hoyle 1983) and Ca^{2+} handling in the muscle (Rome et al. 1996; Rüegg 1992).
Relating the diversity of NMT properties to the behaviors in which they are employed, it is difficult to escape the conclusion that the two are matched (Rome and Lindstedt 1997). In particular, the characteristic speed of the NMT very closely matches the speed of the behavior performed by that NMT. Fast phasic behaviors are performed by fast NMTs (see, for instance, Rome et al. 1988,1996), slow behaviors by slow NMTs.
From our work here we can see that fast behavior absolutely requires a fast NMT. But it is less clear why a slow behavior should be performed by a slow NMT. Although a slow NMT absolutely cannot perform fast behavior, a fast NMT, given a slow firing pattern, canperform a slow behavior. So, especially in view of the difficulties that arise when a slowNMT system is also asked to perform a fast behavior (see next section), why are not all NMTs fast? This question is discussed by Rome and Lindstedt (1997) in terms of parameters such as mechanical power, efficiency, and economy of force generation: lowlevel functional parameters that, while not fully explaining performance, provide important insights into it. Essentially, a tradeoff exists between the mechanics and the energetics of contractions. Fast muscles generate more power in fast behaviors, but at high energy cost; slow muscles give a better economy of force production in slow or tonic behaviors. Overall, both fast and slow muscles perform most efficiently at their particular optimal speed. Thus performing a slow behavior with fast muscles, although possible, is not efficient.
It is also worth noting the implications of such an arrangement for control. Performing a slow behavior with a fast NMT, thus in the relative speed range where the contraction faithfully follows the motor neuron firing pattern, transfers all computation to the nervous system. (Precisely what is involved in such “computation” or “selection” of patterns by the nervous system was discussed in Paper I, and will be a further issue in Paper III.) The nervous system must itself compute a firing pattern that is a perfect template for the contraction, without making use of the computational “hardwiring” in the nonlinear structure of the NMT (Paper I). As we noted in Paper I, the relatively fast, spiking vertebrate skeletal muscles, with relatively linear NMTs, go some way in this direction.
The correct specification of NMT properties has presumably been arrived at through evolution (cf. Rome and Lindstedt 1997), but during the animal's lifetime it is precisely maintained—thus underscoring its functional importance—by trophic processes that regulate, for instance, the underlying synaptic anatomy and morphology, transmitter release properties, and biochemical and contractile properties of the muscle (Atwood 1976; Atwood and Lnenicka 1987; Davis and Goodman 1998;Davis and Murphey 1994; Grinnell 1995;Hoyle 1983; Pette and Vrbová 1992). Interestingly, many of these processes (for example, those involved in specifying the speed of mammalian skeletal muscle; Pette and Vrbová 1992) are themselves under the longterm control of the motor neuron firing pattern. Thus the firing pattern can gradually change the properties of the NMT. But to produce behavior, the firing pattern must, conversely, take into account the existing properties. If the properties change, the subset of functional and efficient firing patterns that the nervous system must employ will change too. It is best, perhaps, to think of a coadaptation and coevolution of the firing patterns, NMT properties, and the behaviors that they can produce.
All this has consequences for what kind of behaviors can be produced at different stages of an animal's development and growth (e.g.,Altringham and Johnston 1990; Johnson et al. 1993), across related animal groups (Full 1997;Rome and Lindstedt 1997), as well as when the NMT is perturbed, for example, by changing temperature (Rome 1990; Stevenson and Josephson 1990; Swoap et al. 1993).
Expanding the range of behavior produced through the same NMT
Even an apparently optimal NMT, however, will still very likely present a fundamental problem. Although, as we have seen, no single behavior is likely to remain functional over an arbitrarily large range of speeds, real behaviors can nevertheless be speeded up or slowed down considerably. Furthermore, many neuromuscular systems perform multiple behaviors, often with very different intrinsic speeds: for example, the same muscles hold a limb steady against gravity, then contract rapidly to jump (compare our Tasks I and II). The problem is that a single NMT with the simple, fixed kinetic properties that we have considered so far cannot perform well over the whole range of speeds. If its speed is correctly matched to one part of the range, it will fail in another. (As we discussed above, a sufficiently fast NMT can perform over the whole range, but not well.) The solution that allows real systems to exhibit the behavioral range they do is, essentially, that the speed and other properties of real NMTs are not fixed, but variable.
It will be seen in Paper III that the variability can be viewed either as being imposed on the NMT from the outside or, equivalently, as being incorporated into the details of its structure. Our work here and in Paper I implies, for instance, that an NMT in which the pointwise contraction kinetics were made pattern dependent, in such a way that they accelerated along with the firing pattern, could produce the ideal periodindependent behavior (Fig. 1 of Paper I) not just for large, but for all P. Several solutions, employing different mechanisms but all formally along these lines, occur in real systems.
Extreme examples of such kinetically complex NMTs are found in specialized muscles that can contract phasically but can also enter a tonic, energetically efficient catch state (Hoyle 1983;Rüegg 1971; Twarog and Muneoka 1972; Wilson and Larimer 1968). But similar phenomena are observed in ordinary muscles, including vertebrate skeletal muscle (Burke et al. 1970). Some muscles exhibit spikes, which convert slow, graded contractions into faster twitches, in some parts of the NMT but not others (e.g., Evans et al. 1996). Probably most NMTs contain regions of different speeds, inherent in the electrical and contractile mechanisms of the NMT in addition to those that we will distinguish (below and in Paper III) as due to “plasticity,” that can serve to expand the range of speeds in behavior.
In vertebrate skeletal muscle (BiglandRitchie et al. 1998; Josephson 1993; Rome and Lindstedt 1997; Rome et al. 1988) as well as some invertebrate muscles (Atwood 1976; Burrows 1996; Satterlie 1993), the wellknown solution is to fractionate the range of speeds, recruiting different motor units—different NMTs, or equivalently parts of a larger overall NMT—with the correct matching speed in different parts of the range.
As we saw here, an antagonistic pairing or more complex mechanical grouping of muscles can also create a limited region of variable speed in the overall NMT.
Finally, most neuromuscular systems, including the AplysiaARCopener system, incorporate various kinds of peripheral plasticity and modulation, either intrinsic to the system or by a neuromodulator or hormone released from an external source, through which the speed and other properties of the NMT are dynamically tuned to match the changing behavior. Some of the underlying processes are in fact just fast, shortterm analogues of the processes that maintain the longterm properties of the NMT. In Paper III, we go on to examine in more detail how such tuning of the NMT alleviates the constraint on and expands the range of functional behavior.
Acknowledgments
This work was supported by National Institute of Mental Health Grants MH36730 and K05 MH01427 to K. R. Weiss and by funds from the Whitehall Foundation to V. Brezina.
Footnotes

Address for reprint requests: V. Brezina, Dept. of Physiology and Biophysics, Box 1218, Mt. Sinai School of Medicine, 1 Gustave L. Levy Place, New York, NY 10029.
 Copyright © 2000 The American Physiological Society
Appendix
Task I: movement oscillating around a single axis
1) Performance measure. In period n,c ranges from
c
_{n}
toc̄_{n}
. The part of this range that is symmetrical around axis
2) Steadystate performance of the model NMT with fast patterns. As P → 0, the contraction becomes steady, with the same c̄
_{∞} and
c
_{∞}. From appendix g,1 of Paper I, we have
Appendix
Task II: tonic contraction
Performance measure (cf. Fig. 7, left). A reasonable approximation to the deviation of the whole contraction waveform from the contraction goal
Appendix
Task III: movement oscillating beyond upper and lower thresholds
Performance measure. As in Task I (appendix
,1), but with
Appendix
Task IV: antagonistic muscle pair
1) The combined contractionc _{1}(t) +c _{2}(t) is taken to be the sum of the contractions c _{1}(t) andc _{2}(t) of the two antagonist muscles, muscle 1 and muscle 2, where c _{2} is inverted relative to c _{1} (cf. Figs. 9, left,and 11). ̅c̅ ̅1̅ ̅+̅c̅ ̅2̅ ̅, c _{2} +c _{2} and, in the steady state, [c _{1}(t) +c _{2}(t)]_{∞}, ̅c̅ ̅1̅ ̅+̅c̅ ̅2̅ ̅ _{∞} and c _{2} +c _{2} _{∞} take the place ofc̄, c , [c(t)]_{∞},c̄ _{∞}, and c _{∞}, respectively.
2) Performance measure (Fig. 9, left). Having made the above substitutions, the task is the same as Task I (appendix , 1 ). We simply use Eq.A1.
3) Optimal steadystate performance of the model NMT with fast patterns. A similar argument as in appendix ,2
, but for two muscles, leads to multiple pairs of optimal mean firing frequencies, 〈f〉_{optimal,1} and 〈f〉_{optimal,2}, satisfying the equation
Appendix
Task V: antagonistic pair compared to single muscle
Performance measure for antagonistic pair (Fig. 10,left). Having made the substitutions in appendix , 1 the task is the same as Task III (appendix ). We simply use Eqs. C1.
Appendix
Task VI: ARCopener neuromuscular system of Aplysia
Performance measure (cf. Fig. 12
A). The contractions of the ARC (muscle 1) and the opener (muscle 2) sum to a combined contraction, or openclose movement of the radula, as inappendix
, 1. Positive values of the combined contraction, above the movement axis or neutral position
The protractionretraction duty cycle is not well established, and may be variable, but 0.5 is consistent with the available data (e.g.,Drushel et al. 1997; Evans and Cropper 1998; Weiss et al. 1986). Similarly, the width of each time window is unknown, but 0.15P is reasonable. We assume that the windows are centered on the transitions between protraction and retraction: the window begins 0.075P before, and ends 0.075P after, the transition. Thus, in each period there are two identical, evenly spaced windows, in which the radula must alternately open and close.
For a simple performance measure, we take the largest overall value of the set of the four amplitudes of the combined contraction, at the beginning and end of each window, when combined as follows. Lett′ be the time since the beginning of period n, which we can take, arbitrarily, as coinciding with the beginning of the window at the transition from protraction to retraction. Note that this defines the period with respect to protraction and retraction, but leaves undefined the phase relationship with respect to them of the ARCopener firing pattern. Let that phase, the ARCprotraction phase for short, be φ. The four amplitudes are thenc_{tx}
= [c
_{1}(tx, φ) + c
_{2}(tx, φ)]_{n} where t1 =t′, t2 = t′ + 0.15P,t3 = t′ + 0.5P, andt4 = t′ + 0.65P. At t1the radula must be open, at t2 closed (at the transition from protraction to retraction, the open radula must close); att3 closed, at t4 open (at the transition from retraction to protraction, the closed radula must open). If these conditions are satisfied at the four time points t1–t4, the simple dynamical structure of the B15ARC NMT (see Paper I) in conjunction with our simple canonical firing patterns guarantees that the radula will remain open throughout protraction (although, as already mentioned, this may not be strictly required, and the real opener motor neuron fires in a perhaps more efficient noncanonical pattern), and closed throughout retraction. Expressions for the four amplitudes that incorporate these conditions (cf. Eqs. C
1a
and C
1b
in appendix
) are