Optimization of Rhythmic Behaviors by Modulation of the Neuromuscular Transform

Vladimir Brezina, Irina V. Orekhova, and Klaudiusz R. Weiss

Department of Physiology and Biophysics and Fishberg Research Center for Neurobiology, Mount Sinai School of Medicine, New York, New York 10029


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

Brezina, Vladimir, Irina V. Orekhova, and Klaudiusz R. Weiss. Optimization of Rhythmic Behaviors by Modulation of the Neuromuscular Transform. J. Neurophysiol. 83: 260-279, 2000. We conclude our study of the properties and the functional role of the neuromuscular transform (NMT). The NMT is an input-output 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 two preceding papers we developed a framework for analysis of the NMT and identified with it principles by which the NMT transforms different firing patterns to contractions. We then saw that, with fixed properties, the NMT significantly constrains the production of functional behavior. Many desirable behaviors are not possible with any firing pattern. Here we examine, theoretically as well as experimentally in the accessory radula closer (ARC) neuromuscular system of Aplysia, how this constraint is alleviated by making the properties of the NMT variable by neuromuscular plasticity and modulation. These processes dynamically tune the properties of the NMT to match the desired behavior, expanding the range of behaviors that can be produced. For specific illustration, we continue to focus on the relation between the speed of the NMT and the speed of cyclical, rhythmic behavior. Our analytic framework emphasizes the functional distinction between intrinsic plasticity or modulation of the NMT, dependent, like the contraction itself, on the motor neuron firing pattern, and extrinsic modulation, independent of it. The former is well suited to automatically optimizing the performance of a single behavior; the latter, to multiplying contraction shapes for multiple behaviors. In any case, to alleviate the constraint of the NMT, the plasticity and modulation must be peripheral. Such processes are likely to play a critical role wherever the nervous system must command, through the constraint of the NMT, a broad range of functional behaviors.


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

In the first of these papers (Brezina et al. 2000, 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). In the second paper (Brezina and Weiss 2000, referred to as Paper II), we extended our analysis to functional movements and behavior. In sending the firing patterns through the NMT, the nervous system is attempting to command behavior. But the filter of the NMT constrains which firing patterns produce functional and efficient behavior, and, even more importantly, the range of behavior that can be produced. Such constraints are particularly clear in cyclical, rhythmic behaviors. With fixed properties of the NMT, the constraints are severe. But the properties of real NMTs are not fixed. Rather, they are variable by virtue of the fact that most NMTs incorporate or are subject to various kinds of plasticity and modulation (reviewed by Bittner 1989; Calabrese 1989; Fisher et al. 1997; Hooper et al. 1999; Hoyle 1983; Worden 1998; Zucker 1989; further references in RESULTS and DISCUSSION). In this paper we examine how such mechanisms tune the properties of the NMT to match the desired behavior, alleviating the constraints imposed by the NMT to expand the range and optimize the production of functional rhythmic behaviors.


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

We continue with the approach described in detail in Papers I and II. We briefly review it 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. (For a summary list of symbols, see Table 1 of Paper I.) We consider a canonical set of bursting patterns completely definable by the alternative parameter triplets (dintra, dinter, fintra), (P, F, fintra), and (P, F, < f> ). Here dintra is the burst duration, dinter the interburst interval, fintra 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
<IT>d</IT><SUB><IT>intra</IT></SUB><IT>+</IT><IT>d</IT><SUB><IT>inter</IT></SUB><IT>=</IT><IT>P</IT> (1a)

<IT>d</IT><SUB><IT>intra</IT></SUB><IT>/</IT><IT>P</IT><IT>=</IT><IT>F</IT> (1b)

<IT>f</IT><SUB><IT>intra</IT></SUB><IT>F</IT><IT>=</IT>⟨<IT>f</IT>⟩ (1c)
In this paper we use primarily the (P, F, fintra), and to some extent the (P, F, < f> ), representation.

NMTs

The NMT is an input-output relation that converts the input waveform f(t) to an output waveform c(t), of contraction amplitude c as a function of time. 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
[1−<IT>a</IT>(<IT>t</IT>)] <LIM><OP><ARROW>⇄</ARROW></OP><LL><IT>&bgr;</IT></LL><UL><IT>&agr;</IT><IT>f<SUP>p</SUP></IT>(<IT>t</IT>)</UL></LIM> <IT>a</IT>(<IT>t</IT>)<IT>; </IT><IT>a<SUP>q</SUP></IT>(<IT>t</IT>)<IT>=</IT><IT>c</IT>(<IT>t</IT>) (2)
where 0 <=  a(t<=  1 and alpha , beta , p, q are constants, or by the corresponding equations
<FR><NU>d<IT>a</IT>(<IT>t</IT>)</NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT>=&agr;</IT><IT>f<SUP>p</SUP></IT>(<IT>t</IT>)[<IT>1−</IT><IT>a</IT>(<IT>t</IT>)]<IT>−&bgr;</IT><IT>a</IT>(<IT>t</IT>)<IT>; </IT><IT>a<SUP>q</SUP></IT>(<IT>t</IT>)<IT>=</IT><IT>c</IT>(<IT>t</IT>) (3)
In this paper we start with the standard parameter values alpha  = 1, beta  = 1, p = 1, and q = 3, then modify alpha  and beta , or more often directly the higher-level time constants of the NMT, as described in RESULTS and APPENDIX A, 1.

The B15-ARC NMT was studied experimentally as in Paper I. Motor neuron B15 was intracellularly stimulated to fire in the desired pattern; the resulting contractions of the accessory radula closer (ARC) muscle were measured under isotonic, lightly loaded conditions.

Output contractions and parameters

We consider the whole output waveform c(t) or its parameters, in particular its period-wise maximum <A><AC>c</AC><AC>&cjs1171;</AC></A>, minimum c, and mean < c> . In the dynamical steady state of the system, c(t) settles to the steady-state output waveform [c(t)]infinity , and <A><AC>c</AC><AC>&cjs1171;</AC></A>, c, and < c> settle to its corresponding parameters <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity , cinfinity , and < c> infinity .

Functional movement and performance

We consider a further output parameter, the functional movement m, or, in the steady state, minfinity . By itself or in the normalized forms minfinity /P and minfinity /P< c> infinity , this parameter provides a measure of performance and efficiency in different behavioral tasks.

Geometric and graphical representation

The operation of the NMT can be represented as a dynamical structure in a multidimensional input-output space. Here we focus on the structure of the steady state minfinity (or one of its normalized forms) primarily in the (P, F, fintra, m), and to some extent in the (P, F, < f> , m), spaces, or simply on the functions minfinity (P, F, fintra) and minfinity (P, F, < f> ). These spaces are four-dimensional, with the function minfinity occupying a three-dimensional volume. (A more complex neuromuscular system, such as the antagonistic muscle pair in Figs. 5 and 6, requires, strictly, additional input dimensions.) For graphical manageability, we show representative three-dimensional sections, obtained by setting one of the input parameters to a constant value, in which minfinity appears as a two-dimensional surface (Figs. 2-6).


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
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 Papers I and II. 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. We focused on such elementary output parameters as the maximum contraction <A><AC>c</AC><AC>&cjs1171;</AC></A>, minimum contraction c, and mean contraction < c> . We studied primarily the dynamical steady state of the system, which, as we saw, is the key element in the dynamical structure of the NMT and its physiological operation. In the steady state, c(t) settles to the steady-state output waveform [c(t)]infinity , and <A><AC>c</AC><AC>&cjs1171;</AC></A>, c, and < c> settle correspondingly to <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity , cinfinity , and < c> infinity . In Paper II, we then extended the scope of the NMT from contractions to functional movement and behavior. For a series of representative behavioral tasks, we computed from the contraction waveform a new output parameter, the functional movement m, or in the steady state minfinity , a measure of performance in the task.

Throughout, we have observed and analyzed how the input-output space is critically structured by the properties of the NMT. We have stressed, in particular, how the speed of the NMT limits the speed of functional behavior. So far, the properties of the NMT have been fixed, indeed, with our two NMTs, fixed in a very restricted way (Paper I). Here, working first with our mathematical model NMT, we will vary or modulate the properties of the NMT in certain ways that are common in real systems (see below and DISCUSSION). For example, we will modulate the NMT so as to alter the size of contractions, or alter their kinetics. We will examine how this alters the functional performance of the NMT in some of the tasks from Paper II, again particularly as the behavior accelerates.

We will then describe results of an experimental examination of such modulation of NMT properties in the real ARC muscle of Aplysia. As will be seen, modulation of the B15-ARC NMT by a number of endogenous modulators, very much like the modulation of the model NMT, is such as to significantly expand the range of speeds of functional behavior.

Effects of NMT modulation on contractions

In real systems, modulation of the NMT is usually described in terms of the effects that it has on contraction shape. The ARC and other buccal muscles of Aplysia present a typical case. Their numerous modulators can be classified, broadly, as 1) changing (increasing or decreasing) contraction amplitude, 2) accelerating the rate of contraction, and 3) accelerating the rate of relaxation (see further Modulation of the B15-ARC NMT below). In detail these can be complex, and usually not pure, effects in the real system.

Our model NMT, however, allows us to implement each of these three effects in pure form, and then, as desired, in combination. Our work in Paper I gives us equations (APPENDIX A, 1) for the whole contraction waveform [c(t)]infinity and parameters such as <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity and cinfinity explicitly in terms of tau contr and tau relax, time constants underlying the kinetics of contraction and relaxation, respectively (see APPENDIX A, 2). These time constants, as well as the amplitude of the contraction, can then be independently varied (APPENDIX A, 1). In this paper, we will restrict ourselves to just three illustrative manipulations (and their combinations): 1) we will increase contraction amplitude twofold (to decrease contraction amplitude, we can simply interchange the unmodulated and modulated contractions); 2) to accelerate the kinetics of contraction, we will decrease tau contr fivefold; 3) to accelerate the kinetics of relaxation, we will decrease tau relax fivefold. The magnitude of these changes is entirely physiological in the ARC muscle, for instance (e.g., Brezina et al. 1995).

How these three manipulations affect contraction shape can be seen in Fig. 1.



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Fig. 1. Effect of different kinds of modulation on the steady-state contraction waveform [c(t)]infinity and its parameters <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity , cinfinity , and < c> infinity . The model neuromuscular transform (NMT) was used. A and B: the contraction waveform [c(t)]infinity . Each panel shows the same unmodulated waveform (thin trace), produced by the model NMT with its standard parameter values (see METHODS) in response to the firing pattern P = 0.3, F = <FR><NU>1</NU><DE>3</DE></FR>, < f>  = 3 (or, equivalently, fintra = 9), and the corresponding waveform produced when the NMT was modulated (thick trace). The different kinds of modulation applied were as follows. A: 2-fold increase of contraction amplitude. B1: contraction accelerated: 5-fold decrease of the time constant of contraction, tau contr. B2: relaxation accelerated: 5-fold decrease of the time constant of relaxation, tau relax. B3: overall kinetics of the NMT accelerated: 5-fold decrease of both tau contr and tau relax. The waveforms were computed and the modulation was implemented as described in APPENDIX A, 1. The gray bars mark the bursts of firing when f = fintra, when the muscle contracts toward the true steady state cinfinity (fintra) with a time course reflecting tau contr, separated by the interburst intervals when f = 0, when the muscle relaxes toward the true steady state cinfinity (0) = 0 with a time course reflecting tau relax (Eq. A2; Paper I). C: dependence of the contraction parameters <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity , cinfinity , and < c> infinity , unmodulated and modulated as in B, on the cycle period P. C1, C2 and C3 correspond to B1, B2 and B3, respectively. F = <FR><NU>1</NU><DE>3</DE></FR> and < f>  = 3 or fintra = 9 as in B, and P varies from 0 to 2.

Increasing contraction amplitude (Fig. 1A) simply scales up the contraction waveform [c(t)]infinity and its parameters <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity , cinfinity , and < c> infinity , to the same extent for all firing patterns. Although this modulation does not alter kinetics, we note that over any absolute amplitude interval the contraction can rise faster than before. Even pure amplitude modulation, therefore, potentially affects the functional speed of the NMT.

The effect of decreasing tau contr and tau relax (Fig. 1B) is immediately understandable from our analysis in Paper I, where we saw how the period-wise shape of [c(t)]infinity depends on the point-wise kinetics of contraction and relaxation described by tau contr and tau relax. During each burst of firing, when f = fintra, the muscle contracts toward the true steady state cinfinity (fintra) with a time course reflecting tau contr; during each interburst interval, when f = 0, it relaxes toward the true steady state cinfinity (0) = 0 with a time course reflecting tau relax. The amplitude of [c(t)]infinity and its parameters such as <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity and cinfinity then reflects the balance of the progress in the two directions. Consequently, favoring the contraction by decreasing tau contr raises [c(t)]infinity , <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity , and (to a lesser extent) cinfinity closer to cinfinity (fintra); favoring the relaxation by decreasing tau relax lowers [c(t)]infinity , cinfinity , and <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity closer to cinfinity (0) = 0. Decreasing both tau contr and tau relax---accelerating the overall kinetics of the NMT---spreads [c(t)]infinity in both directions, raising <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity and lowering cinfinity . Thus, in general, altering the kinetics of contraction inevitably changes its amplitude too.

As Fig. 1C shows, such effects of altered kinetics are especially large for firing patterns of intermediate speed, comparable to the speed of the NMT, where the contraction makes significant progress toward but does not actually reach either true steady state in each cycle: where the contraction is partly phasic and partly tonic. With very slow patterns, which produce a phasic contraction oscillating quasi-instantaneously from one steady state to the other, altering tau contr and tau relax has little effect because it (exactly converse to the modulation in Fig. 1A) alters selectively just the approach to the steady state, not the steady state itself. Similarly for very fast patterns, which produce a tonic contraction. The contraction remains tonic, although, as we can see in Fig. 1, C1 and C2, its amplitude can change as a result of pattern dependence of the sort discussed in Paper I. [Essentially, altering in an uncompensated way just one parameter such as tau contr or tau relax gives equations that are no longer solutions of the simple differential equation (Eq. 3 in METHODS) that becomes linear for fast patterns (APPENDIX G, 1 of Paper I).]

In sum, different kinds of modulation change the input-output structure of the NMT in different and sometimes quite complex ways in different parts of the space, to greater or lesser effect (revealing, in other words, greater or lesser sensitivity of the NMT to that kind of modulation) depending on the firing pattern and the contraction parameter being considered. With respect to what we found in Paper II to be important for functional performance as rhythmic behavior accelerates, we can broadly summarize by saying that these different kinds of modulation, to different degrees, speed up the NMT in such a way that, especially over the intermediate, physiological range of firing pattern speeds, they produce larger phasic contractions for a particular pattern, or, conversely, extend phasic contractions to faster patterns. We will express this more precisely later, after we have seen how it affects performance.

Effects of NMT modulation on functional performance

We can now observe how these effects on contraction shape translate into effects on performance in some of our behavioral tasks from Paper II.

Figures 2-4 show how our three illustrative manipulations of the NMT and their combinations affect the performance measure minfinity /P---the total functional movement over time---in a typical task, Task III. We recall from Paper II that this task requires a single neuromuscular unit to produce rhythmic movement beyond distinct upper and lower thresholds. Column 1 in each figure recapitulates the unmodulated performance from Paper II. We recall our main conclusions: only a subset of firing patterns gives functional performance; to obtain that performance, the nervous system must send a pattern with parameters so matched that it is within the bounds of the subset. Performance increases as the period P of the pattern decreases---as the pattern and the behavior accelerates---provided that its other parameters, here the duty cycle F and the intraburst firing frequency fintra, are matched within ever narrower bounds. But eventually, as the pattern becomes too fast relative to the speed of the NMT, performance fails, essentially as the contraction becomes too tonic, or insufficiently phasic, for the task.



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Fig. 2. Effect of increase of contraction amplitude on the performance of Behavioral Task III from Paper II, for a single neuromuscular unit. The model NMT was used. Based on Fig. 8B of Paper II. Three-dimensional sections (see METHODS) of the performance measure minfinity /P, for firing patterns in the (P, F, fintra) representation, with P varying continuously from 0 to 2, F varying continuously from 0 to 1 (scales at bottom left), and fintra stepped through the values fintra = 3.3, 5, and 10, with the NMT unmodulated (left) and modulated by a 2-fold increase of contraction amplitude (right). For both conditions, <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity and cinfinity were computed as in APPENDIX A, 1; minfinity was then computed using Eq. C1 of Paper II. Here and in all other 3-dimensional sections in this paper, pure black tone indicates complete failure of performance (no functional movement at all, minfinity  = 0), and progressively lighter tone progressively better performance.

Because performance is just another output parameter, we find that our summary picture of how modulation of the NMT appears at the level of contractions (end of the preceding section) is valid also for performance. In Fig. 2 we see, for instance, that increase of contraction amplitude increases performance for some firing patterns, but decreases it for others. Roughly, the former are those where the contraction was too small for the task, and now is more optimal (for example, with smaller than optimal fintra: bottom pair of plots); the latter those where it was optimal, but now is too large (for example, with larger than optimal fintra and large F: front of top pair of plots). The subset of functional firing patterns does not obviously expand or contract, but it shifts its bounds. The nervous system must alter the parameters of the pattern that it sends correspondingly. This becomes increasingly critical as P decreases and the bounds of the functional subset narrow. To maintain performance at a particular small P, with a particular fintra, Fig. 2 shows that increased amplitude modulation must be accompanied by a matching decrease in F.

But is the highest performance achievable through the NMT, with any pattern, increased by the modulation? With pure modulation of contraction amplitude, this does not obviously happen. Performance increases as P decreases, and substantial increase in performance is achieved, we saw in Paper II, primarily by extending the range of functional, sufficiently phasic contractions to substantially smaller P. Because this range is limited by the speed of the NMT, that speed must be increased correspondingly. But a pure increase of contraction amplitude speeds up the NMT too little to extend phasic contractions to much smaller P. Because the contraction can rise over an absolute amplitude interval faster than before, provided the smaller P is matched with other alterations in the firing pattern, some parts of the NMT can become functionally faster, but the effect of increased amplitude modulation, alone, is small.

Phenomena of the same kind as in Fig. 2 can be seen in Fig. 3, where the kinetics of contraction have been accelerated by decreasing tau contr (column 2), the kinetics of relaxation accelerated by decreasing tau relax (column 3), or both (column 4). In some of these cases, however, we do see significant increases in the highest performance achievable through the NMT, because these manipulations do speed up the NMT so as to extend phasic contractions to substantially smaller P.



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Fig. 3. Effect of acceleration of the kinetics of the NMT on the performance of Task III. The model NMT was used. As in Fig. 2, but with the NMT first unmodulated (column 1), then with the contraction accelerated: 5-fold decrease of tau contr (column 2); the relaxation accelerated: 5-fold decrease of tau relax (column 3); and the overall kinetics of the NMT accelerated: 5-fold decrease of both tau contr and tau relax (column 4).

Decreasing tau contr, alone, increases performance relatively little. Indeed, the effect is not very different from that of increasing contraction amplitude (compare Fig. 2 and Fig. 3, columns 1 and 2). A much larger increase in performance is obtained by decreasing tau relax. This reflects an interesting asymmetry in the importance of the two time constants, and more generally of the processes of contraction and relaxation, for functional phasic contractions (APPENDIX B, and next section). In particular, it reflects the fact that while decreasing tau contr, just like increasing contraction amplitude, can give contractions whose phasic component is absolutely larger, their tonic component is also correspondingly larger: the contractions are not more phasic than before. Only decreasing tau relax, alone or as part of a more complex modulation, can give contractions that are relatively more phasic (compare Fig. 1, B1 and B2, C1 and C2; see next section).

The largest increase in performance, furthermore without complex shifts in the subset of functional patterns, is obtained by decreasing both tau contr and tau relax: by speeding up the overall kinetics of the NMT (Fig. 3, column 4). Intuitively, this reflects the fact that here the two component effects balance to leave a relatively pure, large enhancement of the phasic nature of contractions without much change in their overall amplitude (Fig. 1, B3 and C3).

As Fig. 4 shows, combining the modulation increasing contraction amplitude with that decreasing tau relax (a combination common, for instance, in the Aplysia ARC system: see Modulation of the B15-ARC NMT) also brings about a large increase in the highest performance that can be achieved. Intuitively, again, the increase and the decrease in contraction amplitude (Fig. 1, A and B2) balance to leave, simply, a more phasic contraction.



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Fig. 4. Effect of more complex, physiologically realistic modulation, increase of contraction amplitude together with accelerated relaxation, on the performance of Task III. The model NMT was used. As in Fig. 2, but with the NMT first unmodulated (left), then with a 2-fold increase of contraction amplitude and 5-fold decrease of tau relax (right).

Very similar phenomena can be seen in other tasks from Paper II. Figure 5, for instance, shows performance in Task IV, in which the combined contraction of two antagonistic muscles, here of unequal size or strength, is required to rhythmically cross a given movement axis. The kinetics of the stronger muscle, the weaker muscle, or both, have been accelerated by decreasing both tau contr and tau relax. All three manipulations, but especially the modulation of both muscles, extend the subset of functional patterns to smaller P and increase the performance achievable through the NMT.



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Fig. 5. Effect of acceleration of the kinetics of the NMT on the performance of Behavioral Task IV from Paper II, for an antagonistic pair of neuromuscular units, here with unequal size/strength. The model NMT was used. Based on Fig. 9B, right, of Paper II. Three-dimensional sections of the performance measure minfinity /P, for firing patterns in the (P, F, fintra) representation, with P1 = P2 (subscripts refer to the 2 neuromuscular units) varying from 0 to 2, F1 = F2 varying from 0 to 1, and fintra,1 = fintra,2 = 4.5. The phase between the neuromuscular units was 0.5, and, to make muscle 2 weaker than muscle 1, c2 was divided by 3 (see Paper II). The NMT was first unmodulated (plot 1), then the contraction kinetics of the stronger muscle 1 were accelerated by a 5-fold decrease of both tau contr,1 and tau relax,1 (plot 2); the kinetics of the weaker muscle 2 were accelerated by a 5-fold decrease of both tau contr,2 and tau relax,2 (plot 3); and the kinetics of both muscles (the overall kinetics of the whole NMT) were accelerated by a 5-fold decrease of all of tau contr,1, tau relax,1, tau contr,2, and tau relax,2 (plot 4). The contraction waveforms of the 2 muscles were computed as in APPENDIX A, 1 they were summed and <OVL><IT>c</IT><SUB>1</SUB> + <IT>c</IT><SUB>2</SUB></OVL>infinity and c1 + c2infinity , the parameters of the combined waveform relevant to the task, were identified numerically; minfinity was then computed using the equivalent of Eq. A1 of Paper II.

Finally, in Paper II we normalized the performance minfinity /P by the mean contraction amplitude < c> infinity to arrive at a measure of the relative efficiency of different firing patterns in producing the behavior. In Fig. 6 we see that speeding up of the NMT (in this case acceleration of the kinetics of both muscles in Task IV, as in the last panel of Fig. 5) increases, even more than the performance, the highest efficiency that can be achieved through the NMT. Examination of a representative set of contraction waveforms (top) shows that the faster NMT not only enables more phasic contractions to perform the task where they could not before, but at the same time decreases the mean contraction amplitude, a measure of the energy expended in the process (cf. Fig. 1C3). As we discussed in Paper II, the faster NMT changes the shape of the contraction so as to direct its energy more productively into rhythmic movement and behavior.



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Fig. 6. Effect of acceleration of the kinetics of the NMT on the efficiency of different firing patterns in performing Task IV. The model NMT was used. Situation and parameters as in Fig. 5, except for firing patterns in the (P, F, < f> ) representation, with < f>  = 1.5, and for P up to 10. Based on Fig. 11 of Paper II. Bottom plots show the performance measure minfinity /P normalized further by the sum of the means of the 2 individual contractions, |< c1> infinity | + |< c2> infinity |, with the NMT first unmodulated (left), then with the kinetics of both antagonistic muscles accelerated by a 5-fold decrease of all of tau contr,1, tau relax,1, tau contr,2, and tau relax,2 (right). Top plots show how the modulation alters one representative set of contraction waveforms, with the parameters P = 0.5, F = 0.05. Gray bars mark the bursts of firing when f = fintra. < c1> infinity and < c2> infinity were computed as in APPENDIX A, 1.

Phasic fraction of the contraction as a simple indicator of NMT speed

The key to extending the range of functional rhythmic behavior to faster speeds and increasing its performance and efficiency is the speed of the NMT. The aspect of "speed" is difficult to extract from the overall structure of the NMT; it is not exactly expressible by a single number even for our simple model NMT (APPENDIX A, 2) and certainly not for a real NMT such as the Aplysia B15-ARC NMT (Paper I). But, as we saw in Papers I and II and in our discussion above, the speed is broadly reflected, in a way that is immediately relevant to functional rhythmic behavior, in the extent to which the contraction is phasic, or tonic.

A single number that expresses this is the phasic fraction of the contraction, (<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity  - cinfinity )/<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity [or its complement, the tonic fraction of the contraction, cinfinity /<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity  = 1 - (<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity  - cinfinity )/<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity ]. Because this single output parameter lumps together the whole contraction waveform [c(t)]infinity , and by normalizing by <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity gives up knowledge of absolute amplitude, it cannot be a precise quantitative measure of performance in the way that minfinity is. It does, however, provide a good qualitative idea of the possibility of rhythmic behavior: the larger the phasic fraction, the better the performance of a rhythmic task can be. It is especially useful in dealing with a real NMT (in the next section, for instance, the B15-ARC NMT) where the quantitative requirements of the task may not be completely certain in any case, and where the normalization helps reconcile measurements from different preparations that may vary greatly in absolute amplitude.

Figure 7, an extension of Fig. 1, shows explicitly how acceleration of the overall kinetics of the model NMT, the fivefold decrease of both tau contr and tau relax, affects contractions produced by firing patterns of different speeds (Fig. 7A), raising <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity , lowering cinfinity (Fig. 7B), and so increasing the phasic fraction of the contraction at any particular P, or, conversely, shifting a particular phasic fraction to smaller P (Fig. 7C). The phasic fraction changes most at intermediate pattern speeds, comparable to the speed of the NMT.



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Fig. 7. Phasic fraction of the contraction. The model NMT was used. Extension of Fig. 1. A: as in Fig. 1B3, comparing the contraction waveform produced by the unmodulated NMT (thin trace) and with the kinetics of the NMT accelerated by a 5-fold decrease of both tau contr and tau relax (thick trace), for 3 representative values of P (the plot with P = 0.3 is identical to Fig. 1B3). With P = 0.01, the unmodulated waveform is completely obscured by the modulated waveform. Gray bars mark the bursts of firing when f = fintra. B: dependence of the contraction parameters <A><AC>c</AC><AC>&cjs1171;</AC></A>infinity and cinfinity , unmodulated and modulated, on P (as in Fig. 1C, left, and C3). C: phasic fraction of the contraction, unmodulated and modulated, and its dependence on P. The phasic fraction, (<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity  - cinfinity )/<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity , is represented by the height of the curve above the bottom of the plot, where the contraction has no phasic component at all, and is completely tonic. Conversely, therefore, the distance from the curve to the top of the plot, where the contraction is completely phasic, represents the tonic fraction of the contraction, cinfinity /<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity  = 1 - (<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity  - cinfinity )/<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity .

Interestingly, with the model NMT, the phasic fraction depends, of the two time constants, only on tau relax, and not at all on tau contr. Likewise with a real NMT that has similar properties, such as the B15-ARC NMT, relaxation kinetics are likely to be the primary, and contraction kinetics only a secondary, determinant of the phasic fraction (APPENDIX B, 1). Because functional performance, too, is affected much more by modulation of tau relax (Fig. 3, columns 3 and 4, and Fig. 4) than of tau contr (Fig. 3, column 2), this further validates the phasic fraction as a functionally relevant indicator of NMT speed. Pure modulation of contraction amplitude has, with the model NMT, no effect at all on the phasic fraction.

Modulation of the B15-ARC NMT

Contractions of the ARC muscle, as well as those of its antagonist, the radula opener, and other buccal muscles of Aplysia, are shaped by numerous endogenous modulators. Well studied modulators are serotonin (5-HT) and neuropeptides of the small cardioactive peptide (SCP), myomodulin (MM), buccalin (BUC), and FRF/FMRFamide families (e.g., Brezina et al. 1995; Church et al. 1993; Cropper et al. 1987, 1988, 1994; Evans et al. 1999; Fox and Lloyd 1997, 1998; Lloyd et al. 1984; Weiss et al. 1978; Whim and Lloyd 1990; reviewed by Kupfermann et al. 1997; Weiss et al. 1992, 1993).

The effect of each of these modulators is likely to be complex. Where investigated, the modulators have been found to act through multiple cellular mechanisms (e.g., Brezina et al. 1994a,b; Probst et al. 1994; Scott et al. 1997) that then underlie multiple, distinguishable components of the modulation of contraction shape (Brezina et al. 1995, 1996, and see below). The standard practice, however, is to demonstrate the effects on contraction shape using single contractions, elicited by single, brief bursts of motor neuron firing. These, too, are part of the NMT---that with short burst duration and very long interburst interval, or very large P and small F, as input---but a part that, functionally, is not very significant. In our experiments here, we have examined a more functionally relevant part of the B15-ARC NMT, contractions produced by more physiological firing patterns, focusing on how the modulation affects the phasic fraction of the contraction, our indicator of the speed of the NMT and the possibility of functional rhythmic behavior.

Figures 8-11 show the results for four representative modulators of the B15-ARC NMT. In each figure, A shows the typical effect on a single contraction, known from previous work. B then shows the effect on the steady-state contraction waveforms produced by physiological, repetitive firing patterns of different speeds. The parameters used were F = 0.4-0.5, fintra = 10-12 Hz (both fixed in any particular experiment) and P ranging from 0.5 to 10 s; these values well cover the physiological range (Paper II). C compares the unmodulated and modulated phasic fraction, plotted as a function of P.

Buccalin (BUCA; Fig. 8) is usually described as simply decreasing contraction amplitude (Fig. 8A). On the patterned contractions, however, its effect was clearly more complex. Contraction amplitude decreased (Fig. 8B), but at the same time the phasic fraction increased (Fig. 8C). This is inconsistent with a pure modulation of contraction amplitude of the kind that we studied with the model NMT. The BUC effect was largest for patterns of intermediate speed, where the contraction was partly phasic and partly tonic. Furthermore, with very slow patterns BUC appeared to have relatively little effect even on contraction amplitude. All this suggests that a significant component of BUC action amounts to an acceleration of the kinetics (perhaps primarily of the relaxation kinetics) of the NMT (compare Figs. 8B and 7A, 8C and 7C).



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Fig. 8. Modulation of the B15-ARC NMT by buccalin (BUC). A: typical effect of 1 µM BUCA on single contraction of the accessory radula closer (ARC) muscle elicited by brief firing of motor neuron B15 (see METHODS). BUC decreases the amplitude of the contraction without much effect on its kinetics. B: effect of 1 µM BUCA on the steady-state contraction waveform produced by the B15-ARC NMT (cf. Fig. 4C of Paper I) in response to 3 representative firing patterns, with F = 0.4, < f>  = 4 Hz or fintra = 10 Hz, and P = 0.5, 1, or 10 s (for P = 2 s, see Fig. 13B). Gray bars mark the bursts of firing when f = fintra. C: phasic fraction of the contraction, unmodulated and modulated, and its dependence on P.

Myomodulin C (MMC; Fig. 9) was included in these experiments because, like BUC, it can appear to have a relatively simple effect on single contractions. Through distinct mechanisms, MMs, and other modulators such as SCP and 5-HT, exert three simultaneous effects on contractions that have been described as increasing and decreasing their amplitude and accelerating their relaxation rate (Brezina et al. 1995, 1996). With MMC, at some relatively high concentration, the effects on amplitude balance out, leaving just a net acceleration of the relaxation rate of single contractions (Fig. 9A). As expected with such acceleration, MMC increased the phasic fraction of the patterned contractions (Fig. 9C), indeed very much as BUC did.



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Fig. 9. Modulation of the B15-ARC NMT by myomodulin C (MMC). Format as in Fig. 8. A: typical effect of 5 µM MMC on single contraction of the ARC muscle (from the same experiment as in B and C). At this concentration, MMC accelerates the relaxation phase of the contraction without much effect on its amplitude. B: effect of 5 µM MMC on the steady-state contraction waveform produced by the B15-ARC NMT in response to 3 representative firing patterns, with F = 0.4, < f>  = 4 Hz or fintra = 10 Hz, and P = 0.5, 1, or 3 s. C: phasic fraction of the contraction, unmodulated and modulated, and its dependence on P.

Small cardioactive peptide (SCPB; Fig. 10) exerts the three effects just mentioned, but decreases contractions only weakly: the net effect is to increase the amplitude and accelerate the relaxation rate of single contractions (Fig. 10A). With the patterned contractions, the phasic fraction increased (Fig. 10C) as with BUC and MMC. Similar effects were seen with 5-HT.



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Fig. 10. Modulation of the B15-ARC NMT by small cardioactive peptide (SCP). Format as in Fig. 8. A: typical effect of 100 nM SCPB on single contraction of the ARC muscle (from the same experiment as in B and C). SCP increases the amplitude of the contraction and accelerates its relaxation. B: effect of 100 nM SCPB on the steady-state contraction waveform produced by the B15-ARC NMT in response to 3 representative firing patterns, with F = 0.4, < f>  = 4 Hz or fintra = 10 Hz, and P = 0.5, 1, or 10 s. C: phasic fraction of the contraction, unmodulated and modulated, and its dependence on P.

Finally, myomodulin A (MMA; Fig. 11) exerts the three effects, but decreases contractions strongly: at moderately high concentrations of MMA, the net effect is to decrease the amplitude and accelerate the relaxation rate of single contractions (Fig. 11A). Again, the phasic fraction of the patterned contractions increased (Fig. 11C) as with the other modulators.



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Fig. 11. Modulation of the B15-ARC NMT by myomodulin A (MMA). Format as in Fig. 8. A: typical effect of 1 µM MMA on single contraction of the ARC muscle. At high concentrations, MMA decreases the amplitude of the contraction and accelerates its relaxation. B: effect of 10 µM MMA on the steady-state contraction waveform produced by the B15-ARC NMT in response to 3 representative firing patterns, with F = 0.4, < f>  = 4 Hz, or fintra = 10 Hz, and P = 0.5, 1, or 3 s. C: phasic fraction of the contraction, unmodulated and modulated, and its dependence on P.

Not all manipulations of the B15-ARC NMT increase the phasic fraction, however. Figure 12 shows the effect of decreased temperature, which increases contractions (cf. Vilim et al. 1996a) and slows their relaxation rate (Fig. 12A). In this case the phasic fraction of the patterned contractions decreased (Fig. 12C).



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Fig. 12. Modulation of the B15-ARC NMT by temperature. Format as in Fig. 8. A: relaxation phases of ARC-muscle contractions, produced by a very slow firing pattern (P = 36 s) in the same experiment as in B and C, at 22 and 15°C. At 15°C, contractions are larger and relax more slowly than at 22°C. B: steady-state contraction waveforms produced by the B15-ARC NMT in response to 3 representative firing patterns, with F = 0.5, < f>  = 6 Hz or fintra = 12 Hz, and P = 0.5, 1, or 8 s, at 22 and 15°C. C: phasic fraction of the contraction, unmodulated and modulated, and its dependence on P.

Several points can be made with regard to these findings. Extrapolation of the effects of modulation from one part of the NMT to another, in particular from single contractions to more physiological, patterned contractions, can be very misleading. The manifestations of the modulation are likely to be quantitatively, perhaps even qualitatively, different. With BUC, for example, the simple decrease of the amplitude of single contractions would not have predicted the increased phasic fraction of the patterned contractions. Similarly in Fig. 9, where we titrated the MMC concentration so as to leave the amplitude of single contractions unchanged (Fig. 9A), yet there were significant changes in the amplitude of the patterned contractions (Fig. 9B). Such quantitative differences were a consistent finding with all of the modulators.

Such differences can be expected for three reasons. First, we saw with the model NMT that even a pure modulatory effect will manifest itself differently in different parts of the NMT: a pure modulation of kinetics will change the amplitude and the phasic fraction of contractions produced by firing patterns of intermediate speed, but not of those produced by very slow patterns. Second, the outward manifestations of the modulation, especially on just a limited set of contractions produced by a real NMT, often leave ambiguous what variable is primarily being modulated, of the kind that we had a priori knowledge of with our model NMT and that we would need to identify for successful extrapolation to other parts of the NMT. Modulation of kinetics also appears as change in amplitude, and vice versa (Fig. 1). Because, over a fixed time interval, faster rise and larger amplitude are necessarily coupled, does the effect of, for example, SCP on single ARC contractions (Fig. 10A) reflect (in addition to a modulation of relaxation kinetics) a primary modulation of amplitude, or of contraction kinetics? (See further below.) Third, as we see already in the ARC system, real modulators are unlikely to modulate just one such variable. This is because they actually modulate some underlying physiological process (and probably several of these) such as Ca2+ entry and handling in the presynaptic terminal or in the muscle (cf. Brezina et al. 1994a,b). Such a process will inevitably influence, very likely in a complex pattern-dependent manner, both amplitude and kinetics.

In some cases, of course, there is a correlation between the modulation of the single and the patterned contractions. Thus the accelerated relaxation rate of single contractions with the MMs and SCP correlates, qualitatively, with the increased phasic fraction of the patterned contractions. (And similarly for the opposite changes with decreased temperature.) The problem is that, with a complex real NMT, it is difficult to predict when the correlation will hold.

Functionally, the most striking finding in these experiments is the increase in phasic fraction brought about by the modulators. In different cases, this may be coupled with different other effects, such as increased or decreased contraction amplitude, but it is clear that a significant component of the action of all of the modulators, even an unsuspected one such as BUC, is to increase the phasic fraction of contractions: to speed up the NMT.

In view of this and the ambiguity of interpretation of the modulation of single contractions noted above, it may be more correct to think of SCP, for instance, as primarily modulating, in addition to relaxation kinetics, not contraction amplitude but rather contraction kinetics: as accelerating the overall kinetics of the NMT. The increased amplitude of single contractions (Fig. 10A) would then simply reflect the faster contraction rate operating over the same fixed time interval. Acceleration of contraction kinetics by SCP and other modulators has indeed been emphasized in studies of other buccal muscles of Aplysia (e.g., Evans et al. 1999; Fox and Lloyd 1997, 1998). This interpretation would also explain why SCP had little effect on contraction amplitude with very slow patterns (Fig. 10B, right). (These contractions were large, but this was not limiting because much larger contractions could be produced by increasing fintra.)

Speeding up the NMT should extend the range of functional rhythmic behavior to faster speeds. Does this in fact happen? In Fig. 13 we have used the unmodulated and modulated contraction waveforms from one of the experiments just described (that with BUC in Fig. 8) to reconstruct the activity of the antagonistic ARC-opener neuromuscular system and its performance in Task VI, our realistic task from Paper II. Figure 13, A and B, shows examples at two values of P; Fig. 13C then compares the unmodulated and modulated performance minfinity /P as a function of P. Indeed, speeding up of the NMT by BUC extends opening and closing of the radula (crossing of the movement axis) to very small P (Fig. 13A, P = 1 s). And at only slightly larger P, it enables good performance of the full Task VI---opening and closing of the radula timed correctly relative to its protraction and retraction---where there was none before (Fig. 13, B and C, P = 2 s). At the same time, the mean amplitude of the contractions decreases (Fig. 13, A and B), increasing the efficiency of the behavior. The extension of performance occurs over those intermediate values of P where the largest shift of the phasic fraction occurred (cf. Fig. 8C). At large P, where performance was good before, it may conversely decrease (Fig. 13C, P = 10 s).



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Fig. 13. Effect of modulation by BUC on the performance of Behavioral Task VI, a realistic task for the antagonistic ARC-opener neuromuscular system of Aplysia, from Paper II. The unmodulated and modulated contraction waveforms from the experiment in Fig. 8, B and C, were used to reconstruct the activity of the ARC-opener system as in Fig. 12 of Paper II. The opener contractions were derived from the ARC contractions by inverting, scaling to <FR><NU>1</NU><DE>3</DE></FR> amplitude, and shifting by 0.5P. The combined contraction, or open-close movement of the radula, was then tested for performance as in APPENDIX F of Paper II. A and B: examples at 2 values of P. For the individual contraction waveforms, gray bars mark the bursts of motor neuron firing when f = f intra; for the combined contraction, they indicate the phases of protraction of the radula, fixed relative to its closure when the task is successfully performed (here only in the last of the 4 cases; see APPENDIX F and Fig. 12 of Paper II). C: unmodulated and modulated performance minfinity /P as a function of P. Vertical scales throughout are arbitrary.

We see in this example how we might work toward an eventual full quantitative accounting of the effects and the functional roles of all of the diverse modulators in the ARC-opener system. It would be premature to consider further details here, however, for three reasons, which apply also to many other neuromuscular systems (see the DISCUSSION).

First, as discussed in Paper II, the task requirements and the NMTs involved will need to be better understood.

Second, because many of the modulators are intrinsic modulators, released from the motor neurons themselves by the same firing patterns that produce the contractions, it is difficult to dissociate the modulation from the basic operation of the NMT. Indeed, although our "unmodulated" ARC contractions, here and in Paper I, were recorded in such a way as to keep modulation to a minimum (for example, we worked at room temperature, where modulator release is low; Vilim et al. 1996a), they may already be modulated to some degree. For further dissection, selective blockers of the release or effects of the modulators will be needed.

And third, the system is in reality modulated, not by fixed concentrations of single modulators, but by dynamically changing mixtures of multiple modulators. Somewhat different complements of modulators act on different buccal muscles. Indeed, even the simple assumption implicit in Fig. 13 that the ARC and opener muscles are both modulated similarly by BUC is not strictly correct. The two muscles do share modulators with similar effects, including the MMs, but BUC is apparently present just in the ARC, not in the opener muscle (Evans et al. 1999).


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

All motor commands of the nervous system pass through the filter of the NMT, whose properties significantly constrain the behaviors that can be produced. For optimal performance of a particular behavior, we saw in Paper II that the properties of the NMT must match the behavior, so that some set of motor neuron firing patterns exists that can produce the behavior, and produce it efficiently. The speed of the NMT, in particular, must match the speed of the desired behavior. But if the speed of the NMT is fixed, in a rigid way such as we studied with our two NMTs in Paper I, it will correctly match and allow behavior over only a narrow range of speeds. A slow NMT will allow slow behavior, but not fast behavior as well; a fast NMT will not allow slow behavior with any efficiency. It will not be possible to vary much the speed of a behavior, or to perform behaviors of different speeds with the same neuromuscular plant. However, as we discussed in Paper II, real systems do this routinely. They are able to do it because the speed and other properties of their NMTs are not fixed, but variable. In particular, real NMTs incorporate or are subject to various kinds of neuromuscular plasticity and modulation that dynamically tune their properties to match varying behavior.

Diversity of neuromuscular plasticity and modulation

At neuromuscular junctions of both vertebrates and invertebrates, presynaptic transmitter release exhibits multiple components of plasticity---facilitation, augmentation, potentiation, depression---on a wide range of time scales (Atwood 1976; Atwood and Cooper 1996; Atwood et al. 1989; Bittner 1989; Fisher et al. 1997; Magleby and Zengel 1982; Zucker 1989). Once released, the transmitter can feed back to modulate its own release (Bowman et al. 1988; Parnas et al. 1994; Wessler 1989). In addition to these effects intrinsically involving the contraction-mediating transmitter itself, many, perhaps most, neuromuscular junctions and muscles are modulated by a variety of other transmitters and peptides (reviewed by Adams et al. 1989; Calabrese 1989; Evans and Myers 1986; Hooper et al. 1999; Muneoka and Kobayashi 1992; Muneoka et al. 1991; Worden 1998; further references below). Such modulation is especially prominent in invertebrates, but is found even in fast vertebrate skeletal muscles (Bowman and Nott 1969; Williams and Barnes 1989). The modulators may be "intrinsic," released as cotransmitters from the motor neurons themselves, or "extrinsic," released from additional modulatory neurons or arriving as hormones (cf. Katz 1995; Kupfermann 1991). The site of modulation may be any of the steps comprising the NMT, including the release of the mediating transmitter (e.g., Atwood et al. 1989; Bittner 1989; Cropper et al. 1988; Meriney and Grinnell 1991), the response of the postsynaptic receptors (Jorge-Rivera and Marder 1997; Mulle et al. 1988), and the electrical and contractile events in the muscle (Brezina et al. 1994a,b; Erxleben et al. 1995; Probst et al. 1994). The modulation may interact with the plasticity of transmitter release (Jorge-Rivera et al. 1998; Qian and Delaney 1997) as well as, very likely in a pattern-dependent manner, with the basic properties of the NMT. Such plasticity and modulation may account for many of the complexities of the neuromuscular response, for instance neuromuscular "warm-up" (cf. Fig. 4C of Paper I), that are seen in practically all neuromuscular systems (Hoyle 1983). In RESULTS, we have already noted a number of these phenomena in our representative system, the ARC and other buccal muscles of Aplysia. These muscles exhibit both multiple components of synaptic plasticity (Cohen et al. 1978; Jordan et al. 1993) and extensive networks of intrinsic as well as extrinsic modulatory actions.

Formal framework for analysis of NMT variability

The overview above includes both the classic phenomena of pure synaptic plasticity as well as perhaps completely extrasynaptic modulation. In RESULTS, we focused on the latter, in part because it naturally led us to the simplest picture of NMT variability, as alteration imposed on a basic, unmodulated NMT by the action of an external modulator, as mimicked by its exogenous application. Synaptic plasticity, in contrast, seems to be embedded, in a much less separable way, within the structure of the NMT. But synaptic plasticity, too, can be conceptually separated from the basic structure. And conversely, our dynamical systems approach allows us to expand our representation of the NMT so that it incorporates within it the different states of modulation as well as synaptic plasticity. This more general view then shows the two types of variability as formally equivalent. Both appear as quantitative, region-specific alterations or added dimensions of the dynamical structure of a single, general NMT. If the formal alterations of the structure are similar with synaptic plasticity and modulation, their functional consequences will also be similar. Furthermore, as will be seen below, incorporation of the modulation within the structure of the NMT is not just a convenient formal device, but reflects physiological reality when the modulation, like the synaptic plasticity, depends on the motor neuron firing pattern.

We can expand the NMT along the lines discussed in Paper I. We recall that, with the dynamical systems approach, we are seeking a representation of the system in terms of a set of variables whose present values are sufficient to completely summarize its history, and predict its future.

Most generally, we add one or more dimensions orthogonal to those of the basic dynamical structure of the NMT. In Fig. 14, for example, we have added to our usual four-dimensional input-output structure one such dimension, for the modulation factor M (see APPENDIX A, 1), representing in this case the degree of acceleration of the overall kinetics of the NMT by decrease of both tau contr and tau relax. For the purposes of our later discussion, the output here is already performance in our standard Task III, but we could equally well have plotted the underlying contraction. The extra M dimension allows us to represent within the NMT itself all possible states of this modulation, including all those in Fig. 3. (With an extra dimension, of course, the section we can show is even more restricted: see Fig. 14 legend.)



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Fig. 14. Automatic tuning of the NMT and performance by intrinsic modulation. The model NMT was used. Three-dimensional section of the performance measure minfinity /P for Task III, for firing patterns in the (P, F, fintra) representation, with P varying from 0 to 2, F = 0.5, fintra = 10, and M, the modulation factor decreasing here both tau contr and tau relax (cf. APPENDIX A, 1), varying linearly from 1 to 5. In effect, this section gradually blends those in the top left and top right corners of Fig. 3, progressing from the 2-dimensional section at F = 0.5 through the former, here at M = 1 (unmodulated) at the front, to the corresponding section through the latter, here at M = 5 (fully modulated) at the rear. The arrowed trajectories (drawn arbitrarily) and points A-D are explained in the DISCUSSION.

When there are multiple independent sources of variability, multiple modulators for instance, each requires its own set of orthogonal dimensions.

The most general representation is necessary if the added variability, while interacting with the original structure of the NMT, is not in turn restricted by it: most importantly, if it is independent of the motor neuron firing pattern. This is the case with extrinsic modulation by a hormone or a modulator released from an independent modulatory neuron. Such extrinsic modulation behaves, formally, as extra input to the NMT.

But often the variability does depend on the motor neuron firing pattern, just as the contraction itself does. This is the case with intrinsic modulation and synaptic plasticity, the sources of variability that, for that very reason, seem to be more deeply embedded in the NMT. Each such source exhibits a relationship to the firing pattern that has all the properties that we studied in Paper I for the contraction itself, because the transforms from the firing pattern to the contraction and to the variability share the relevant formal properties (cf. Brezina et al. 1997). When the variability is incorporated in the expanded NMT, its pattern dependence appears in the NMT along with that of the contraction, as well as interactions between the two. (We saw a relatively simple case of such interaction in RESULTS, where some contraction shapes expressed modulation of kinetics more than others.) Quantitatively, of course, the pattern dependence of the contraction and the variability can be very different. Different physiological processes, with different dose-response shapes and time constants, are involved, allowing them to react differentially to, and be differentially controlled by, different patterns (Paper I; Brezina et al. 1997). Thus, for instance, different patterns of motor neuron firing are optimal for release of the contraction-mediating transmitter and intrinsic modulatory cotransmitters (Bartfai et al. 1988; Kupfermann 1991; Vilim et al. 1996a; Whim and Lloyd 1989, 1990). As a result of such differences in pattern dependence, some firing patterns may, for example, produce contractions without inducing long-term plasticity of the neuromuscular junction; others just the opposite (Atwood et al. 1989; Baxter and Byrne 1993; Bittner 1989).

As with the contraction itself, the pattern-dependent dynamics of the variability (and so indeed of the NMT as a whole) are likely to tend toward a point steady state. In that steady state, the variability will still alter the contraction, but in just one way. It will no longer contribute true variability, in the sense of expanding the range that the contraction can occupy. The important question is on what time scale the variability relaxes to the steady state relative to the time scale of the input patterns and contractions of interest. The faster this happens, the smaller will be the range of true variability around the basic contraction shape. In the steady state, and so, if the variability relaxes essentially instantaneously on the time scale of interest, always, the contraction will be altered in just one fixed way. In this case the NMT needs no extra dimensions at all to incorporate the variability: it is already implicit in the contraction shape itself. This is likely to be true, for example, for the faster components of synaptic plasticity (Fisher et al. 1997) on the relatively slow time scale of the patterns that we have used here with the B15-ARC NMT.

Formally, then, the different kinds of variability alter the structure of the NMT in two basic ways, distinguished not so much by the physiological mechanism but by the source of the variability: is it intrinsic to the motor neuron and so, like the contraction itself, dependent on its firing pattern, or extrinsic, independent of it? Intrinsic variability alters the shape of the contraction, but does not expand the range of shapes that can be produced by each pattern, except transiently. A steady-state surface within the structure of the NMT, of the sort we have plotted in many of our figures, is altered in shape, but remains a surface. Extrinsic variability, by contrast, does bring about such expansion, in the steady state as well as transiently, on any time scale. This is because it acts as extra input to the NMT (see APPENDIX L of Paper I.) For the same reason, expansion also occurs when there are multiple independent sources of intrinsic variability, such as multiple modulators released from separate motor neurons (see Again, issues of predictability and controllability).

Functionally, when the contraction shapes are then translated to performance, the fundamental point is that the structure of the NMT has been so altered that it can produce, and produce efficiently, a broader range of behaviors.

Modulation of the speed of the NMT

Throughout, we have focused on the illustrative aspect of speed. For optimal performance and efficiency, the speed of the NMT must match the speed of the behavior. The speed of a rhythmic behavior is set by the period of the firing pattern, P. Thus a particular speed of the NMT is optimal for a particular P. But it is not optimal for others. In particular, the performance and efficiency of rhythmic behavior falls dramatically as P decreases---as the nervous system attempts to accelerate the behavior---if the speed of the NMT remains fixed. If the NMT is accelerated correspondingly, however, functional behavior is maintained. If the NMT is accelerated in a correctly balanced way (with our model NMT, for example, by decreasing both tau contr and tau relax by the same amount), the behavior may indeed unfold much as before, just compressed in time (cf. Fig. 1 of Paper I). And, as we saw in Paper II, this compression may well bring higher performance and efficiency, not just at a particular speed, but overall.

There is good reason to believe that such modulation of speed is ubiquitous in NMTs. In the Aplysia ARC muscle, we found a significant component of kinetic acceleration in the action of every modulator that we tested, even an unsuspected one such as BUC. Similar modulation is seen in the antagonist radula opener and in other buccal muscles (Church et al. 1993; Evans et al. 1999; Fox and Lloyd 1997, 1998). In a variety of other neuromuscular systems, too, a frequent theme in the action of modulators is simultaneous acceleration of contraction kinetics (or increase of the amplitude of single contractions, which, as we saw, may well be a reflection of it) and relaxation kinetics (e.g., Evans and Myers 1986; Evans and Siegler 1982; Jorge-Rivera et al. 1998; Lingle 1981; Lloyd 1980; McPherson and Blankenship 1991; Whim and Evans 1988). These two simultaneous actions will balance out changes in amplitude and emphasize the component of kinetic acceleration (Figs. 1, B3 and C3, and 7). In nonspiking muscles, modulators often bring in spikes, which will accelerate the NMT (Erxleben et al. 1995; Jorge-Rivera and Marder 1996; Kravitz et al. 1985; Meyrand and Marder 1991; Satterlie and Norekian 1996). Virtually all neuromuscular systems exhibit components of synaptic plasticity, such as facilitation, with the right properties to accelerate the NMT. Such phenomena have prompted previous discussion of the importance of the speed of muscle contraction and relaxation for the performance of rhythmic behavior, by, for example, Evans and Siegler (1982), Whim and Evans (1988), Weiss et al. (1992, 1993), and Satterlie and Norekian (1996).

Very similar modulation is observed in the hearts of many species, including myogenic hearts that strictly are beyond the scope of this work (Buckett et al. 1990; Calabrese and Maranto 1984; Greenberg and Price 1980; Watson and Groome 1989). In the vertebrate heart, of course, the kinetics of relaxation are well-known to be critical for functional performance and are modulated to match changes in heart rate (Katz 1992; Morad and Rolett 1972).

Again, issues of predictability and controllability

We now return to some of the fundamental questions of Paper I. When the nervous system sends firing patterns through the NMT, to what extent can it predict the contractions and performance that will result? To what extent can it, through those patterns, control various desired behaviors? In Paper I we saw that the answers to these questions have to do with the size and structure of the NMT. Because variability, in general, makes the NMT larger and more complex, it complicates predictability, but at the same time expands the possibilities for control. More particularly, because the intrinsic and extrinsic variabilities alter the NMT in qualitatively different ways, they have different consequences in these respects.

As before, the problems as well as possibilities are greatest in non-steady-state situations, for instance during transitions between different behaviors. This is true even with intrinsic variability, to the extent that it expands the structure of the NMT.

Thus, in the Aplysia ARC system, the SCPs, MMs, and BUCs are all intrinsic cotransmitters of the motor neurons B15 and B16, and each motor neuron releases a different mixture of these peptides, with multiple effects on different time scales (Brezina et al. 1996; Brezina and Weiss 1998; Vilim et al. 1996b). After a few cycles of each of the firing patterns that drive the system's several distinct behaviors, such as biting, swallowing, and rejection, the whole NMT, the contraction itself as well as the modulatory effects, will reach a point steady state. But it will be a different steady state in each behavior, and, when the firing pattern then switches to a new behavior, the different variables of the NMT will relax to the new steady state on different time scales, most of them, experiments show, as slow as or slower than the time scale of the firing pattern (Brezina and Weiss 1998). During this period of transient expansion of the NMT, the nervous system will be able to predict the contractions and performance, strictly, only if it knows the values of those variables. And for smooth control of the transition, it may have to make transient adjustments to the firing pattern: relax the firing pattern more slowly, too, in coordination with the relaxation of the modulatory effects.

But eventually, with intrinsic variability, the dynamics of the NMT do collapse to a simple point steady state, just as those of the NMT without any variability do. Qualitatively, the structure of the NMT is just the same. Then the contractions and performance are, just as before, completely predictable: for each firing pattern, there is a single steady-state contraction waveform and performance value. But if it is not expanded, how can the NMT produce a broader range of functional behavior? It is able to do so because, quantitatively, it is so altered that it produces functional contractions, with high performance and efficiency, over a broader range of patterns. The nervous system controls, through the NMT, no more contraction shapes than before, but more of those that it does control are functional.

With respect to speed, we can see how this happens in Fig. 14. In this section, the general expansion of the NMT is represented by the two-dimensional base of the plot, P × M: for each value of the period P, any strength of the modulation M is possible. But if M represents intrinsic modulation, it depends on P, and in the steady state, only one value of M is possible for each P. The system cannot access the whole two-dimensional surface based on P × M, but is restricted to a one-dimensional curve through it. As P changes, so, in a fixed way, does M. Starting at point A, for example, as the nervous system decreases P to accelerate the behavior, M may depend on P in such a way that M coordinately increases. Then the system may follow some trajectory such as A right-arrow C. Along this trajectory, at every P, M is automatically adjusted so as to give the highest performance possible for that P. The speed of the NMT, in other words, does not remain fixed and optimal for just one value of P, but varies so that it is always optimally tuned to the current value of P. And we see again in Fig. 14 how this tuning extends the range of the functional behavior to smaller P, and so to much higher overall performance, than would be possible without the modulation (trajectory A right-arrow B).

Of course, M must have the correct dependence on P. A different dependence would give a different, less optimal, trajectory across the surface in Fig. 14. Increasing M too little as P decreases would give, in the extreme, the trajectory A right-arrow B; increasing M too much, the trajectory A right-arrow D. Like accelerating the NMT too little, accelerating it too much---more than is optimal for the current value of P---decreases performance and efficiency (Fig. 13C; Paper II). Intrinsic variability in real systems does indeed seem to have the correct sort of pattern dependence. For example, neuromuscular facilitation (Atwood 1976; Fisher et al. 1997; Magleby and Zengel 1982; Zucker 1989), release of intrinsic modulators (Vilim et al. 1996a; Whim and Lloyd 1989, 1990), and in some cases their actions (Jorge-Rivera et al. 1998) all tend to increase as motor neurons fire at higher frequency and in more frequent bursts.

Thus intrinsic variability builds into the structure of the NMT a tight coupling of contraction and variability so that the same motor neuron firing pattern that commands the muscle to contract simultaneously optimizes the contraction for the performance of a behavioral task. No additional input is needed. This automatic mechanism is well suited to extending the functional range of a single behavior.

But, because the coupling is tight, the system produces a set of contraction shapes that is no less restricted than before (Paper I)---in the steady state, just a single shape for each pattern. And if this shape has been optimized for one behavior, it may not be optimal for a different behavior that the nervous system may ask the system to perform. Sometimes it may even be desirable to suppress the contraction completely, freeing the motor neuron to fire in other peripheral or central tasks (Hooper et al. 1999). For multiple behaviors, what is needed is true variability that will provide independent control over---that will uncouple---multiple parameters of the contraction shape. Such expansion of the NMT is possible only with additional input.

Within limits, such expansion can occur with multiple sources of intrinsic variability, such as modulators from different motor neurons. In the Aplysia ARC system, we have previously described how the SCPs from motor neuron B15 and the MMs from motor neuron B16 uncouple, over a certain range, the size and the relaxation rate of contractions (Brezina et al. 1996; Brezina and Weiss 1997).

The fullest expansion occurs with extrinsic variability, input that can be completely independent of the motor neuron firing pattern. If M in Fig. 14 represents extrinsic modulation, the system can access the whole surface P × M. In the ARC system, such modulation is provided by 5-HT, with effects similar to SCP (Fig. 10), released from the purely modulatory metacerebral neurons (Weiss et al. 1978). Some of the benefits of the independence of the mediating and modulating inputs are well illustrated in this case, such as the fact that the latter can prepare the system before the former begin the actual behavior (Kupfermann and Weiss 1982). Extrinsic variability, much more readily than intrinsic variability, can preconfigure the system for a behavior, or for a class of related behaviors, without yet committing it to a specific behavior through a specific pattern of motor neuron firing.

Full expansion of the NMT, however, introduces the converse problem. Whereas intrinsic variability allowed too few contraction shapes, extrinsic variability allows too many. No mechanism prevents the system from reaching many parts of the input-output space that are not functional. In Fig. 14, the trajectory A right-arrow D, for example, can now be achieved, but it still is probably not desirable. In other words, some of the restrictive coupling that intrinsic variability built into the structure of the NMT, but which is absent with extrinsic variability, must be reintroduced. This can be done centrally, by coupling the inputs. Thus we find in some systems that the firing pattern of extrinsic modulatory neurons is closely coupled to that of the mediating motor neurons (Baudoux et al. 1998; Burrows and Pflüger 1995). In many systems, extrinsic modulators simultaneously modify, and so potentially couple, the central patterns and the peripheral NMT (e.g., Harris-Warrick et al. 1989, 1992; Kravitz et al. 1985; Satterlie and Norekian 1996; Weimann et al. 1997). They may modulate, too, sensory neurons that feed peripheral information back to the CNS. Indeed, the sensory neurons themselves may release central as well as peripheral modulators. The last three mechanisms may all be present in Aplysia buccal neuromuscular systems (Alexeeva et al. 1998; Cropper et al. 1996; Kupfermann et al. 1997; Weiss et al. 1978).

All these are the same fundamental issues of central-peripheral coordination and control that we discussed in Papers I and II, but now in a more complex NMT. A broader range of behavior is possible, provided that the nervous system can select the correct firing patterns. And it would seem to be correspondingly more difficult to compute these; they must, for instance, vary with the degree of modulation of the NMT (Figs. 2-5). But the patterns are not actually computed by the nervous system on the basis of complete knowledge of the structure of the NMT (Paper I). Rather, they are generated by an automatic mechanism that uses the information "hard-wired" in the structure of the NMT---the kinds of coupling that we have discussed---to send the system along just a few desirable trajectories through the input-output space.

Concluding remarks

Faced with evidence for peripheral neuromuscular modulation, a frequently asked question (for instance, with respect to the Aplysia ARC system, by Hoyle 1983, p. 468) is, why is the additional layer of modulation necessary? Why could not the basic contraction mechanisms, and the central motor neuron firing patterns, be configured so as to suffice? The answer emerges from our work here. The properties of the basic NMT are inherently so constraining that, through it, many desirable behaviors cannot be produced with any firing pattern. And because the constraints of the NMT are peripheral, they can only be alleviated by peripheral modulation.

The Aplysia ARC neuromuscular system is a particularly well studied but probably by no means atypical example. In this paper we have given some idea of the complex network of intrinsic as well as extrinsic modulation that allows this single muscle to perform with increasing speed and efficiency in food-induced arousal (Susswein et al. 1978), in multiple behaviors such as biting, swallowing, and rejection (Kupfermann 1974), and when dealing with food of varying quality (Hurwitz and Susswein 1992). As the ARC must integrate its contractions with those of the antagonist radula opener as well as other buccal muscles, very similar kinds of modulation occur in those muscles. (For example, the speed of both antagonist muscles must be modulated for optimal results: Fig. 5.) Most if not all buccal motor neurons contain intrinsic modulatory cotransmitters (Church and Lloyd 1991), and the metacerebral neurons provide widespread extrinsic modulation. In this way not just a single muscle, but the whole feeding musculature is modulated coordinately. Similar complexities of NMT modulation are being described in other neuromuscular systems (e.g., Evans and Myers 1986; Jorge-Rivera and Marder, 1996, 1997; Jorge-Rivera et al. 1998; Muneoka et al. 1991). Such modulation is likely to be found wherever the nervous system commands, through the constraints of the NMT, a broad range of functional behaviors.


    APPENDIX A
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

Modulation of the model NMT


1 We use as our basis the equations in APPENDIX F of Paper I. Specifically, for the dynamical steady state of the system, by making again the substitutions
&tgr;(<IT>f</IT>)<IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>&agr;</IT><IT>f<SUP>p</SUP></IT><IT>+&bgr;</IT></DE></FR> (A1a)
and
<IT>a</IT><SUB><IT>∞</IT></SUB>(<IT>f</IT>)<IT>=</IT><FR><NU><IT>&agr;</IT><IT>f<SUP>p</SUP></IT></NU><DE><IT>&agr;</IT><IT>f<SUP>p</SUP></IT><IT>+&bgr;</IT></DE></FR> (A1b)
and defining tau contr triple-bond  tau (fintra) and tau relax triple-bond  tau (0), Eqs. F3-F6 of Paper I can be returned to the simpler forms [e.g., in the (P, F, fintra) representation]
[<IT>c</IT>(<IT>t</IT><IT>′</IT>)]<SUB><IT>∞</IT></SUB><IT>=</IT><FENCE><AR><R><C>[<IT>a</IT><SUP><IT>q</IT></SUP><SUB><IT>intra</IT></SUB>(<IT>t</IT><IT>′</IT>)]<SUB><IT>∞</IT></SUB><IT>=</IT>{<IT>a</IT><SUB><IT>∞</IT></SUB>(<IT>f</IT><SUB><IT>intra</IT></SUB>)<IT>−</IT>[<IT>a</IT><SUB><IT>∞</IT></SUB>(<IT>f</IT><SUB><IT>intra</IT></SUB>)<IT>−</IT><UNL><IT>a</IT></UNL><SUB><IT>∞</IT></SUB>]<IT> exp</IT>(<IT>−t</IT><IT>′/&tgr;<SUB>contr</SUB></IT>)}<SUP><IT>q</IT></SUP></C></R><R><C><IT>if 0≤</IT><IT>t</IT><IT>′≤</IT><IT>PF</IT></C></R><R><C>[<IT>a</IT><SUP><IT>q</IT></SUP><SUB><IT>inter</IT></SUB>(<IT>t</IT><IT>′−</IT><IT>PF</IT>)]<SUB><IT>∞</IT></SUB><IT>=</IT>{<IT><A><AC>a</AC><AC>&cjs1171;</AC></A></IT><SUB><IT>∞</IT></SUB><IT> exp</IT>[−(<IT>t</IT><IT>′−</IT><IT>PF</IT>)<IT>/&tgr;<SUB>relax</SUB></IT>]}<SUP><IT>q</IT></SUP></C></R><R><C><IT>if </IT><IT>PF</IT><IT>≤</IT><IT>t</IT><IT>′≤</IT><IT>P</IT></C></R></AR></FENCE> (A2)

<IT><A><AC>c</AC><AC>&cjs1171;</AC></A></IT><SUB><IT>∞</IT></SUB><IT>=</IT><IT><A><AC>a</AC><AC>&cjs1171;</AC></A></IT><SUP><IT>q</IT></SUP><SUB><IT>∞</IT></SUB><IT>=</IT><FENCE><FR><NU><IT>a</IT><SUB><IT>∞</IT></SUB>(<IT>f</IT><SUB><IT>intra</IT></SUB>)[<IT>1−exp</IT>(<IT>−PF</IT><IT>/&tgr;<SUB>contr</SUB></IT>)]</NU><DE><IT>1−exp</IT>[<IT>−PF</IT><IT>/&tgr;<SUB>contr</SUB>−</IT><IT>P</IT>(<IT>1−</IT><IT>F</IT>)<IT>/&tgr;<SUB>relax</SUB></IT>]</DE></FR></FENCE><SUP><IT>q</IT></SUP> (A3)

<UNL><IT>c</IT></UNL><SUB><IT>∞</IT></SUB><IT>=</IT><UNL><IT>a</IT></UNL><SUP><IT>q</IT></SUP><SUB><IT>∞</IT></SUB><IT>=</IT><FENCE><FR><NU><IT>a</IT><SUB><IT>∞</IT></SUB>(<IT>f</IT><SUB><IT>intra</IT></SUB>)[<IT>1−exp</IT>(<IT>−PF</IT><IT>/&tgr;<SUB>contr</SUB></IT>)]<IT> exp</IT>[<IT>−P</IT>(<IT>1−</IT><IT>F</IT>)<IT>/&tgr;<SUB>relax</SUB></IT>]</NU><DE><IT>1−exp</IT>[<IT>−PF</IT><IT>/&tgr;<SUB>contr</SUB>−</IT><IT>P</IT>(<IT>1−</IT><IT>F</IT>)<IT>/&tgr;<SUB>relax</SUB></IT>]</DE></FR></FENCE><SUP><IT>q</IT></SUP> (A4)

⟨<IT>c</IT>⟩<SUB><IT>∞</IT></SUB><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>P</IT></DE></FR> <LIM><OP>∫</OP><LL><IT>0</IT></LL><UL><IT>P</IT></UL></LIM> [<IT>c</IT>(<IT>t</IT><IT>′</IT>)]<SUB><IT>∞</IT></SUB><IT>d</IT><IT>t</IT><IT>′</IT>

=<FR><NU>1</NU><DE><IT>P</IT></DE></FR> <FENCE><LIM><OP>∫</OP><LL><IT>0</IT></LL><UL><IT>PF</IT></UL></LIM> [<IT>a</IT><SUP><IT>q</IT></SUP><SUB><IT>intra</IT></SUB>(<IT>t</IT><IT>′</IT>)]<SUB><IT>∞</IT></SUB><IT>d</IT><IT>t</IT><IT>′+</IT><LIM><OP>∫</OP><LL><IT>PF</IT></LL><UL><IT>P</IT></UL></LIM> [<IT>a</IT><SUP><IT>q</IT></SUP><SUB><IT>inter</IT></SUB>(<IT>t</IT><IT>′−</IT><IT>PF</IT>)]<SUB><IT>∞</IT></SUB><IT>d</IT><IT>t</IT><IT>′</IT></FENCE> (A5)
where t' is the time since the beginning of the period.
Equations A1-A5 describe the original, unmodulated NMT. We can then apply different kinds of modulation.
To modulate contraction amplitude, we scale the original waveform [c(t')]infinity given by Eq. A2 into a new waveform [c(t')]'infinity using the transformation [c(t')]'infinity  = M[c(t')]infinity , where M is the modulation factor (in this paper M = 2). According to Eqs. A2-A4, this amounts to the transformation of the true steady state <IT>a</IT><IT>′<SUB>∞</SUB></IT>(<IT>f</IT><SUB><IT>intra</IT></SUB>)<IT>=</IT><RAD><RCD><IT>M</IT></RCD><RDX><IT>q</IT></RDX></RAD><IT>a</IT><SUB><IT>∞</IT></SUB>(<IT>f</IT><SUB><IT>intra</IT></SUB>) or c'infinity (fintra) = Mcinfinity (fintra). Also <A><AC>c</AC><AC>&cjs1171;</AC></A>'infinity  = M<A><AC>c</AC><AC>&cjs1171;</AC></A>infinity , c'infinity  = Mcinfinity , and < c> 'infinity  = M< c> infinity (cf. Fig. 1A).
To modulate the kinetics of contraction, we make the substitution tau 'contr = tau contr/M (in this paper M = 5) in Eqs. A2-A5.
To modulate the kinetics of relaxation, we make the substitution tau 'relax = tau relax/M (in this paper M = 5) in Eqs. A2-A5.
Modulating both tau contr and tau relax with the same factor M is equivalent to substituting alpha ' = alpha /M, beta ' = beta /M in Eq. A1.

2  tau contr and tau relax are, strictly, the time constants of the intermediate a, rather than of the final contraction c (Eq. A2). These are related by c = aq, where q = 3 throughout this work. Altering tau contr and tau relax alters the kinetics of c, but not in a simple linear way (cf. APPENDIX C of Paper I).


    APPENDIX B
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

Special importance of relaxation kinetics and their modulation


1 Combining Eqs. A3 and A4, we find that the phasic fraction of the contraction
(<IT><A><AC>c</AC><AC>&cjs1171;</AC></A></IT><SUB><IT>∞</IT></SUB><IT>−</IT><UNL><IT>c</IT></UNL><SUB>∞</SUB>)/<IT><A><AC>c</AC><AC>&cjs1171;</AC></A></IT><SUB><IT>∞</IT></SUB><IT>=1−</IT>{<IT>exp</IT>[<IT>−P</IT>(<IT>1−</IT><IT>F</IT>)<IT>/&tgr;<SUB>relax</SUB></IT>]}<SUP><IT>q</IT></SUP> (B1)
Thus, with the model NMT, the phasic fraction depends just on tau relax, and not at all on tau contr. This is a consequence of the fact that, with the model NMT, the (f, c) plane is a state space (APPENDIX E, 1 of Paper I) and that, when f = 0, the relaxation has an exponential, self-similar time course. In a real NMT such as the B15-ARC NMT, this is only approximately true. To the extent that the NMT departs from these requirements, modulation of contraction kinetics and of amplitude will also affect the phasic fraction.

2 From the broader perspective of neuromuscular control, we expect modulation of tau relax to be more significant than that of tau contr because tau relax is more restrictive for the shapes of contractions that can be achieved. Because tau contr varies with the intraburst firing frequency (APPENDIX A, 1), it can be altered through the firing pattern in different parts of the NMT, whereas tau relax is fixed for our canonical patterns (and many patterns in real systems) that all have an interburst firing frequency of zero.


    ACKNOWLEDGMENTS

This work was supported by National Institutes of Health Grants MH-36730 and K05 MH-01427 to K. R. Weiss and by funds from the Whitehall Foundation to V. Brezina. Some Aplysia were provided by the National Center for Research Resources National Resource for Aplysia at the University of Miami under NIH Grant RR-10294.


    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.

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Received 26 April 1999; accepted in final form 31 August 1999.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES

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