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 (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)

In this paper we use primarily the (P, F, f_intra), 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

(2)

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

(3)

In this paper we start with the standard parameter values  = 1,  = 1, p = 1, and q = 3, then modify  and , 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 , minimum c, and mean c. In the dynamical steady state of the system, c(t) settles to the steady-state output waveform [c(t)], and , c, and c settle to its corresponding parameters , c, and c.

Functional movement and performance

We consider a further output parameter, the functional movement m, or, in the steady state, m. By itself or in the normalized forms m/P and m/Pc, 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 m (or one of its normalized forms) primarily in the (P, F, f_intra, m), and to some extent in the (P, F, f, m), spaces, or simply on the functions m(P, F, f_intra) and m(P, F, f). These spaces are four-dimensional, with the function m 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 m 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 , 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)], and , c, and c settle correspondingly to , c, and c. 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 m, 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)] and parameters such as and c explicitly in terms of _contr and _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 _contr fivefold; 3) to accelerate the kinetics of relaxation, we will decrease _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.

View larger version (43K): [in this window] [in a new window]   Fig. 1. Effect of different kinds of modulation on the steady-state contraction waveform [c(t)] and its parameters , c, and c. The model neuromuscular transform (NMT) was used. A and B: the contraction waveform [c(t)]. 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 = , 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, contr. B2: relaxation accelerated: 5-fold decrease of the time constant of relaxation, relax. B3: overall kinetics of the NMT accelerated: 5-fold decrease of both contr and 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 c(fintra) with a time course reflecting contr, separated by the interburst intervals when f = 0, when the muscle relaxes toward the true steady state c(0) = 0 with a time course reflecting relax (Eq. A2; Paper I). C: dependence of the contraction parameters , c, and c, unmodulated and modulated as in B, on the cycle period P. C1, C2 and C3 correspond to B1, B2 and B3, respectively. F =  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)] and its parameters , c, and c, 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 _contr and _relax (Fig. 1B) is immediately understandable from our analysis in Paper I, where we saw how the period-wise shape of [c(t)] depends on the point-wise kinetics of contraction and relaxation described by _contr and _relax. During each burst of firing, when f = f_intra, the muscle contracts toward the true steady state c(f_intra) with a time course reflecting _contr; during each interburst interval, when f = 0, it relaxes toward the true steady state c(0) = 0 with a time course reflecting _relax. The amplitude of [c(t)] and its parameters such as and c then reflects the balance of the progress in the two directions. Consequently, favoring the contraction by decreasing _contr raises [c(t)], , and (to a lesser extent) c closer to c(f_intra); favoring the relaxation by decreasing _relax lowers [c(t)], c, and closer to c(0) = 0. Decreasing both _contr and _relaxaccelerating the overall kinetics of the NMTspreads [c(t)] in both directions, raising and lowering c. 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 _contr and _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 _contr or _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 m/Pthe total functional movement over timein 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 decreasesas the pattern and the behavior acceleratesprovided that its other parameters, here the duty cycle F and the intraburst firing frequency f_intra, 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.

View larger version (43K): [in this window] [in a new window]   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 m/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, and c were computed as in APPENDIX A, 1; m 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, m = 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 f_intra: bottom pair of plots); the latter those where it was optimal, but now is too large (for example, with larger than optimal f_intra 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 f_intra, 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 _contr (column 2), the kinetics of relaxation accelerated by decreasing _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.

View larger version (43K): [in this window] [in a new window]   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 contr (column 2); the relaxation accelerated: 5-fold decrease of relax (column 3); and the overall kinetics of the NMT accelerated: 5-fold decrease of both contr and relax (column 4).

Decreasing _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 _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 _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 _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 _contr and _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 _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.

View larger version (44K): [in this window] [in a new window]   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 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 _contr and _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.

View larger version (22K): [in this window] [in a new window]   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 m/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 contr,1 and relax,1 (plot 2); the kinetics of the weaker muscle 2 were accelerated by a 5-fold decrease of both contr,2 and 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 contr,1, relax,1, contr,2, and relax,2 (plot 4). The contraction waveforms of the 2 muscles were computed as in APPENDIX A, 1 they were summed and and c1 + c2, the parameters of the combined waveform relevant to the task, were identified numerically; m was then computed using the equivalent of Eq. A1 of Paper II.

Finally, in Paper II we normalized the performance m/P by the mean contraction amplitude c 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.

View larger version (61K): [in this window] [in a new window]   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 m/P normalized further by the sum of the means of the 2 individual contractions, |c1| + |c2|, with the NMT first unmodulated (left), then with the kinetics of both antagonistic muscles accelerated by a 5-fold decrease of all of contr,1, relax,1, contr,2, and 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 and c2 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, (  c)/ [or its complement, the tonic fraction of the contraction, c/ = 1  (  c)/]. Because this single output parameter lumps together the whole contraction waveform [c(t)], and by normalizing by gives up knowledge of absolute amplitude, it cannot be a precise quantitative measure of performance in the way that m 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 _contr and _relax, affects contractions produced by firing patterns of different speeds (Fig. 7A), raising , lowering c (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.

View larger version (37K): [in this window] [in a new window]   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 contr and 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 and c, 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, (  c)/, 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, c/ = 1  (  c)/.

Interestingly, with the model NMT, the phasic fraction depends, of the two time constants, only on _relax, and not at all on _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 _relax (Fig. 3, columns 3 and 4, and Fig. 4) than of _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 NMTthat with short burst duration and very long interburst interval, or very large P and small F, as inputbut 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, f_intra = 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 (BUC_A; 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).

View larger version (42K): [in this window] [in a new window]   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 (MM_C; 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 MM_C, 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, MM_C increased the phasic fraction of the patterned contractions (Fig. 9C), indeed very much as BUC did.

View larger version (36K): [in this window] [in a new window]   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 (SCP_B; 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 MM_C. Similar effects were seen with 5-HT.

View larger version (42K): [in this window] [in a new window]   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 (MM_A; Fig. 11) exerts the three effects, but decreases contractions strongly: at moderately high concentrations of MM_A, 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.

View larger version (36K): [in this window] [in a new window]   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).

View larger version (42K): [in this window] [in a new window]   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 MM_C 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 f_intra.)

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 m/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 VIopening and closing of the radula timed correctly relative to its protraction and retractionwhere 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).

View larger version (33K): [in this window] [in a new window]   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 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 m/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 di