INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F REFERENCES

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 altereddegraded and then completely disintegratedmuch 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

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F REFERENCES

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 waveform f(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). Here d_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 (period-averaged) firing frequency. These parameters, and so the alternative triplets, are related by the equations

(1a)

(1b)

(1c)

NMTs

The NMT is an input-output relation which converts the input waveform f(t) to an output waveform c(t), of contraction amplitude c as a function of time (Fig. 1, top row). We focus on two NMTs, the real B15-ARC NMT of Aplysia and a model NMT that has similar but completely known properties. The model NMT is implicitly defined by the kinetic schema

(2)

where 0  a(t)  1 and , , p, q are constants, or by the corresponding equations

(3)

Unless stated otherwise, we use the standard parameter values  = 1,  = 1, p = 1, and q = 3.

Output contractions and parameters

We consider the whole output waveform c(t) or its parameters, in particular its period-wise maximum , minimum , and mean c. (In this paper, however, our major focus is on the behavioral performance parameters that we define in RESULTS.) In the dynamical steady state of the system, c(t) settles to the steady-state output waveform [c(t)], and , , and c settle to its corresponding parameters , , and c.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F REFERENCES

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 B15-ARC NMT of Aplysia, as in Paper I. A brief review of the mechanics of our approach is provided in METHODS. 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 , the minimum contraction , 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 single-valued function on the contraction waveform c(t). Just like , , 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 input-output 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 input-output space (see APPENDICES). 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 each f(t) to a unique steady-state 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. settles to , to , and c to c. m settles to m.

View larger version (54K): [in this window] [in a new window]   Fig. 1. Four canonical input patterns or waveforms f(t) (bottom) and the corresponding steady-state output waveforms [c(t)] produced by the model neuromuscular transform (NMT; top). Each f(t) is completely described by the (dintra, dinter, fintra) parameter triplet below. Each [c(t)] is a plot of Eq. F3 of Paper I, with the standard parameter values (see METHODS), for the given f(t). Gray bars mark the phases when f = fintra. For explanation of the functional movement markings in D see Fig. 2, right, and Task I: movement oscillating around a single axis in RESULTS.

The principal object of our examination in this paper is thus the structure of the steady-state m in the alternative (d_intra, d_inter, f_intra, m), (P, F, f_intra, m), or (P, F, f, m) input-output spaces, or simply the functions m(d_intra, d_inter, f_intra), m(P, F, f_intra), and m(P, F, f). These spaces, however, are four-dimensional, with the function m occupying a three-dimensional 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 two-dimensional sections through the complete four-dimensional 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 three-dimensional sections, m appears as a two-dimensional surface (e.g., Figs. 3 and 5), and in the two-dimensional sections, as a one-dimensional curve (Figs. 2 and 4).

Functional appropriateness of contractions

Figure 1 shows representative examples of the kind of steady-state 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 and minimum close together (A and B) 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 two-dimensional 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 four-dimensional space. Each panel shows, first of all, the maximum contraction and the minimum contraction (top and bottom dotted curves, respectively). As Fig. 1 suggested, we see all possible combinations of the two parameters.

View larger version (29K): [in this window] [in a new window]   Fig. 2. Behavioral Task I: single neuromuscular unit, movement oscillating around a single axis. The model NMT was used. Right: definition of task. The contraction waveformhere, in the steady state, the waveform [c(t)]is required to oscillate symmetrically around a given movement axis, here c = 0.4. It carries out one complete oscillation in each cycle period P (cf. Fig. 1). The actual range of the oscillation is from the minimum to the maximum . The functional movement m is taken to be that part of this range that is symmetrical around the given axis (gray area). This can be our performance measure or a basis for it, most usefully when further normalized by P (see Task I: movement oscillating around a single axis in RESULTS). For the sake of illustration, this plot shows how all these output parameters change along one input dimension, with fintra. [Here fintra varies from 0 to 30 while dintra = 0.2, dinter = 0.4 (P = 0.6); this plot is therefore the same, except over a greater range of fintra, as the middle panel of the bottom block on the left.] Left: overview of the dependence of the output parameters on all of the input parameters, in the (dintra, dinter, fintra) representation. Layout as in Fig. 9 of Paper I. These plots show , , m, and m/P (top dotted, bottom dotted, gray and black curves, respectively; same coding as on the right) when dintra, dinter, and fintra are each varied against a background of low, intermediate, and high settings of the other 2 parameters in all possible combinations. These are therefore 2-dimensional sections through the general 4-dimensional spaces containing the functions (dintra, dinter, fintra), (dintra, dinter, fintra), m(dintra, dinter, fintra), and m(dintra, dinter, fintra)/P where 2 of the 3 input parameters are held constant. The letters A, B, C, and D mark the locations of the example waveforms A-D in Fig. 1. and were computed using Eqs. F4 and F5 of Paper I (these plots are identical to those in Fig. 9 of Paper I); m was then computed using Eq. A1.

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. 1D (in the time domain) and 2, right (a representative section through the input-output space), and expressed in a corresponding formula (APPENDIX A, 1), that allows us to compute, from the oscillatory contraction waveform [c(t)], indeed from just the two parameters and that define the range of its oscillation, the functional contraction or movement m. 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. 1D, 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 allno part of the contraction is functionaland 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 steady-state 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 movementm as large as the physical parameters of the situation allowcan 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 normalize m 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 and m/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 and , the computed performance measures m (gray curves) and m/P (black curves). Together, these sections provide an overview of the complete functions m(d_intra, d_inter, f_intra) and m(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 musclethere is always some, almost always nonzero, and this contraction is very often not functional. Only in some parts of the space do we see nonzero values of the performance measures m and m/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 of m are obtained, in general, as the two temporal parameters of the firing pattern, the burst duration d_intra and the interburst interval d_inter, increase. The largest m is obtained with the largest d_intra and d_inter (as well as the largest f_intra), in the top right-hand 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 parameter m, of one of our principal concerns in Paper I: how the input-output 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 input-output 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 higher-level 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 three-dimensional sections of m and m/P, for input either in the (P, F, f) or in the (P, F, f_intra) representation. Here the period P and the duty cycle F are varied continuously for a single fixed value of the mean firing frequency f or the intraburst frequency f_intra. All three-dimensional plots in this paper will have this form. In Fig. 3 only, 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 , the time constant of the NMT. These are located on either side of the line marked P   in each plot. In this and most other three-dimensional plots, complete failure of performance (m and m/P values of zero) is indicated by pure black tone, and progressively better performance by progressively lighter tone.

View larger version (61K): [in this window] [in a new window]   Fig. 3. Task I: 3-dimensional sections of the performance measures m and m/P, for input in the (P, F, f) and (P, F, fintra) representations, over a wide range of P and F so as to provide a broad overview of how performance depends on firing pattern. Note in particular the limiting performance with patterns much faster, or much slower, than , the time constant of the NMT. The line P   demarcating these is approximately indicated in each plot (with the standard parameter values used,   1). In this figure only, P and F are plotted on log scales. P varies continuously from 103 to 104, F varies continuously from 103 to 1, with fixed f = 10 or fintra = 10. The model NMT was used. and (not shown) were computed using Eqs. F4 and F5 of Paper I; m was then computed using Eq. A1. Here and in most other 3-dimensional sections in this paper, pure black tone indicates complete failure of performance (no functional movement at all, m = 0), and progressively lighter tone progressively better performance.

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 how and , 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  , right-hand end of each plot in Fig. 3), c always has time to relax essentially completely to its true steady state c, either c(f_intra) or c(0) = 0 during the burst and the interburst interval, respectively. (The contractions in Fig. 1D, for instance, come close to this.) Therefore  = c(f_intra) and  = 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, and are one and the same and consequently, always, m = 0.] When P  , the size of the functional movement m thus depends just on f_intra and can be found simply by examining the steady-state c(f) relation in conjunction with the given physical parameters of the situation. With the model NMT that we have used here, we recall that the c(f) relation increases monotonically from zero to saturation. So, as f_intra increases, m is zero as long as c(f_intra) remains smaller than the movement axis; beyond this, m increases at double the rate of c(f_intra); finally, m saturates when c(f_intra) either saturates or becomes double the movement axis. Because m depends just on f_intra, for any particular value of f_intra (Fig. 3, top right) we see the same value of m for any F (except F = 1) and any P  : any slow firing pattern with that intraburst firing frequency. m is independent of F and P in this range.

We now consider, instead of f_intra, the mean firing frequency f (Fig. 3, top left) when P  . Because f = f_intraF, keeping f constant requires f_intra to increase as F decreases. From the last paragraph we conclude that, as this happens, m always eventually reaches the same saturating valuewhen c(f_intra) either saturates or becomes double the movement axiswith every f. Although in this case m 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 measure m/P, is very small (Fig. 3, bottom plots). Furthermore, m has saturated and no longer varies with P in this range, so that m/P decreases unopposed as P 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 ratea direct relationship between m and Psuggests two opposite behavioral strategies for maximizing m/P. One strategy is to increase m, even at the cost of increased P: 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  ) and fast (P < ) firing patterns toward the left-hand 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 two-dimensional 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 three-dimensional sections like those in Fig. 3 but on linear scales.

View larger version (29K): [in this window] [in a new window]   Fig. 4. Task I: 2-dimensional sections of contraction and performance parameters, for input in the (P, F, f) and (P, F, fintra) representations, focusing on fast and intermediate firing patterns. Most details as in Fig. 2. The model NMT was used. A: dependence of contraction and performance parameters on the mean firing frequency f. Input in the (P, F, f) representation. Here , , m, and m/P are plotted as P varies continuously from 0 to 4, with F fixed at 0.1, for 3 different values of f, i.e., f = 1, 2.8, and 10. With f = 2.8, as P  0, m/P exceeds 1 and has been clipped. For discussion of OPTIMAL performance see Analysis of performance in Task I in RESULTS. B: dependence of contraction and performance parameters on the duty cycle F, comparing input in the (P, F, f) and (P, F, fintra) representations. Similar to A, except that f or fintra is fixed at 10, and F stepped through the values F = 0.01, 0.1, 0.5, and 0.9.  and were computed using Eqs. F4 and F5 of Paper I; m was then computed using Eq. A1.

With very fast patterns (P  ), we saw in Paper I that c 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. 1A). The oscillation becomes progressively smallerwith the model NMT used here, monotonically decreases, monotonically increases, so that the two converge monotonicallyas 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 and can be seen in each panel in Fig. 4 (dotted curves).

As and converge, the functional movement m 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 which and converge? As the three panels of Fig. 4A 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 does m 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 as m remains reasonably high. But when m drops to zero, so of course does m/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 that m never becomes zero, m/P continues to increase to very high values for the fastest patterns (Fig. 4A, middle panel; Fig. 3, bottom right). In essence, the input-output structure of the model NMT used here (specifically, the shape of the dependence of and , 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 maintain m indefinitely, give the highest performance, and so can be said to be optimal (Fig. 4A, middle panel).

Which are the optimal patterns? In other words, which values of the other two input parameters, F and f_intra, or F and f, can be combined with very small P 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  , and in consequence its output parameters such as and become "pattern independent." By this we mean that and , as they converge to the limiting value, come to depend only on the mean density of spikesthe mean firing frequency fregardless 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  , gives contraction equal to the movement axis. This dependence can best be appreciated, therefore, in the (P, F, f) representation, in the two-dimensional sections in Fig. 4A, and even better in the corresponding three-dimensional sections in Fig. 5A.

View larger version (56K): [in this window] [in a new window]   Fig. 5. Task I: 3-dimensional sections of the performance measure m/P, comparing input in the (P, F, f) and (P, F, fintra) representations, focusing on fast and intermediate firing patterns. The model NMT was used. A: plots of m/P as P varies continuously from 0 to 2, F varies continuously from 0 to 1 (see scales at bottom), and f is stepped through the values f = 1, 2.8, and 10. With f = 2.8, as P  0, m/P exceeds 1 and has been clipped. B: as in A, except that fintra is stepped through the values fintra = 3.3, 5, and 10.  and (not shown) were computed using Eqs. F4 and F5 of Paper I; m was then computed using Eq. A1.

Because the model NMT has a monotonically increasing c(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 A, 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. 4A and 5A). Patterns with smaller or larger f do not permit such close approach to small P and high m/P (bottom and top panels). Figure 5A shows that this is true in a "pattern-independent" way, across all values of the duty cycle F.

To keep f = f_intraF constant, F and f_intra must always vary in a precisely compensatory way. For a given period P, any decrease in F, and therefore in the intraburst duration d_intra, must be exactly compensated by an increase in the intraburst firing frequency f_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 and f_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 "pattern-independent" 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. 4B, left (also Fig. 3, bottom left), if f_intra increases to compensate as F decreases, so maintaining f, performance is also approximately maintained. In contrast, if f_intra remains fixed, f declines and performance fails (Fig. 4B, 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 as P 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 is f_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_intraF, f_optimal corresponds to multiple optimal pairs of f_intra and F. In the (P, F, f_intra) representation in Fig. 5B, 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 input-output 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  . 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 rate-limiting for the overall time constant of the contraction. But in addition there may be much fasterin relative terms, effectively instantaneousprocesses, such as fast facilitation of transmitter release, that preprocess the spikes of the firing pattern (cf. Paper III). Because the slow contractile process is rate-limiting, the contraction will still be steady, without significant oscillation, when P  . 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 A, 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. 5A 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 fin their number of spikesraises 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.

View larger version (22K): [in this window] [in a new window]   Fig. 6. Task I: 3-dimensional sections of the performance measure m/P, for input in the (P, F, f) representation, when the NMT is no longer linear for fast firing patterns. This is the case, with the model NMT, when p  1 (APPENDIX A, 2). Here p = 3. Otherwise as in Fig. 5A.

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 fix