INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G APPENDIX H APPENDIX I APPENDIX J APPENDIX K APPENDIX L REFERENCES

Cyclical, rhythmic behaviors such as biting, chewing and swallowing, breathing, and many different kinds of locomotion feature prominently in the behavioral repertoires of both vertebrates and invertebrates (Gray 1968; Pearson 1993; Stein et al. 1997). Such behaviors appear to unfold in an integrated and efficient manner, but this can only be if their muscular plant is successful in operating within specific constraints. Usually a number of muscles are involved, and the contractions of each muscle must be appropriate in amplitude and timing, not just in themselves but in relation to those of the other muscles. Furthermore, although the basic pattern of the behavior may be stereotyped, its parameters can often vary over a wide range to meet behavioral demands: for instance, the behavior can speed up or slow down manyfold. The muscles must remain effectively coordinated throughout. Finally, at different times the same muscles may be used in different behaviors, with different contraction and coordination requirements.

In the nervous system, such rhythmic behaviors are generated by central pattern generators (CPGs) (Harris-Warrick et al. 1992; Marder and Calabrese 1996; Pearson 1993; Stein et al. 1997). Neurons of the CPG fire cyclically in various patterns and phase relationships; the patterned activity is distributed to motor neurons (in some cases these may themselves be elements of the CPG) and to the muscles. Like the behaviors they generate, CPGs exhibit plasticity that modulates the basic pattern or even reconfigures the CPG entirely to switch between different behaviors (Dickinson 1995; Harris-Warrick et al. 1992; Katz 1995; Stein et al. 1997).

Thus a set of neurons with a certain complex pattern of activity drives a set of muscles to contract, likewise, in a certain complex, functionally meaningful pattern. It is easy to assume (especially because experimental work often focuses on just one or the other) that the linkage between these two halves of the whole system is straightforward, so that, for instance, changes seen in the pattern of activity of the CPG will automatically be read out as similar changes in the pattern of muscle contractions, and furthermore will constitute functional changes in the behavior. However, this is often not so.

Figure 1 illustrates this in the case of a pair of antagonistic muscles whose alternating contractions combine in an oscillatory movement that must meet a certain criterion for functional behavior. This illustration is in fact based on the functioning of a real system, the accessory radula closer (ARC)-opener neuromuscular system of Aplysia (Cohen et al. 1978; Evans et al. 1996; Weiss et al. 1992, 1993). The opposing contractions of the two muscles (top), driven by alternating bursts of motor neuron firing (middle), produce a combined movement (bottom), which, to be functional, must alternately cross the axis of functional movement. At slow cycle speed (left), this is possible. However, as the neural circuitry decreases the cycle period in an attempt to speed up the behavior (middle, expanded at right), the muscle contractions change shape so that the behavior does not simply become compressed in time, as one might naively expect, but becomes increasingly distorted and eventually completely dysfunctional (bottom right).

View larger version (49K): [in this window] [in a new window]   Fig. 1. Example of the problem: decreasing cycle period distorts and eventually completely eliminates functional rhythmic behavior. This figure uses some real experimental data to update the conceptual model of Weiss et al. (1992, 1993) of the functioning of the antagonistic radula closer-opener neuromuscular system of Aplysia. The radula is a handlike structure that the animal protrudes from its mouth to grasp food; the radula must alternately close to grip the food while it is being pulled in and open to release the food into the esophagus. The accessory radula closer (ARC, or I5) muscle (Cohen et al. 1978) is a larger and stronger muscle than the antagonistic radula opener muscle complex I7-I10 (Evans et al. 1996). Left column, top: actual recording of length of the ARC muscle (in a reduced preparation, however, isolated from the opener), showing contractions produced by firing of its motor neuron B15 in a slowly cycling pattern (schematized below), with burst duration dintra = 5.1 s, interburst interval dinter = 9.1 s, intraburst frequency fintra = 14 Hz, and so cycle period P = 14.2 s, duty cycle F = 0.36, and mean frequency f = 5 Hz (for definitions see Fig. 2 and Input firing patterns in RESULTS). Shown are 2 cycles after the contractions had stabilized at their steady-state shape and amplitude. The record used for the antagonist (i.e., radula opener) muscle (downward from the fully relaxed line) is, for the sake of simplicity in this conceptual illustration, just the ARC record inverted, scaled to amplitude, and half-cycle out of phase. Left, bottom: the net movement has been derived by summation of the contraction waveforms of the 2 antagonist muscles. It is assumed that functional behavior (repetitive, alternating closing and opening of the radula) requires that the movement make transitions across the zero line, marked axis of functional movement (cf. Weiss et al. 1992, 1993). Middle: as in left, but with the cycle period decreased ~9-fold (dintra = 0.58 s, dinter = 1.02 s, and so P = 1.6 s, but F, fintra and f as before). Thin traces predict the contractions and movement that would be recorded in an ideal period-invariant system: i.e., the records, left, simply compressed in time without distortion. Thick traces show the contractions of the ARC muscle that were actually produced by the faster firing pattern, and further records derived from them as before. Right: 2 cycles from the middle column at an expanded time resolution (expanded again by the same factor of ~9 by which the period was compressed). It can be seen that the decrease in cycle period has entirely eliminated functional behavior: the net movement (thick trace, bottom right) no longer makes the transitions across the zero line that close and open the radula. Although this is primarily a conceptual illustration, it is worth noting that it is not grossly inconsistent with what is known about the functioning of the intact closer-opener neuromuscular system in vivo (see Paper II). In particular, the slow cycle period used here is entirely physiological, and the fast one only slightly faster than physiological (e.g., Susswein et al. 1976, 1978; Weiss et al. 1986).

Analysis of such phenomena can be regarded as a problem in pattern dependence. How do different patterns of neuronal firing translate, through what we may call the neuromuscular transform (NMT), to muscle contractions and to functional behavior? We have recently presented some general ideas about pattern dependence in biological processes (Brezina et al. 1997). Here and in the following paper (Brezina and Weiss 2000, henceforth referred to as Paper II), we expand and apply these ideas to the neuromuscular problem in particular. We consider the problem theoretically from first principles and, whenever possible, use the Aplysia ARC-opener neuromuscular system as a real illustration. The aim is to develop a framework with which to answer such questions as: What parameters of the firing pattern are important for different parameters of contraction and functional behavior? Can it be said, even, that certain parameters of the firing pattern "code" for certain parameters of contraction and behavior? Given a firing pattern, what contractions and behavior will it produce? Conversely, given a behavioral requirement, what firing patterns are optimal? It will be seen that, with a fixed NMT, behaviors with certain parameters are not possible with any firing pattern. However, as we show in the last paper in this series (Brezina et al. 2000, referred to as Paper III), the range of possible behaviors can be greatly expanded by appropriately tuning the NMT through peripheral neuromuscular plasticity and modulation.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G APPENDIX H APPENDIX I APPENDIX J APPENDIX K APPENDIX L REFERENCES

Motor neuron-evoked contractions of the ARC muscle of Aplysia were measured as in previous studies (Brezina et al. 1995-1997). Briefly, motor neuron B15 was intracellularly stimulated to fire spikes in the desired pattern; to assure control over the pattern, each spike was elicited by a separate brief current injection. (The patterns used are described in Input firing patterns in RESULTS.) The muscle contracted isotonically against a light load; length was measured with an isotonic transducer.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G APPENDIX H APPENDIX I APPENDIX J APPENDIX K APPENDIX L REFERENCES

General strategy

We begin by considering an elementary motor unit: a single muscle controlled by a single motor neuron. Firing of the motor neuron evokes contraction of the muscle via a sequence of intermediate steps, including propagation of the action potentials into the motor neuron terminals, release of transmitter, reception and propagation of the signal in the muscle, elevation of Ca²⁺, and activation of the contractile machinery. The quantitative details of these intermediate steps are, in general, unknown. However, the first and last elements in the sequence, namely the firing of the motor neuron and the contraction of the muscle, are easily measurable, and every pattern of firing produces some observable state of contraction of the muscle. This allows us to represent the overall activity of the motor unit by a lumped input-output relation, the neuromuscular transform (NMT)

(1)

where f and c (defined more precisely below) are instantaneous measures of the input and output at any particular time t. In general, f, and therefore c, varies with t.

We now structure the input f in different temporal patterns: that is, we consider different input waveforms f(t). For rhythmic contractions, periodically repeating patterns are most relevant. Because in these papers we wish to emphasize general principles, for simplicity we will restrict our quantitative consideration to a set of canonical patterns, those describing firing of the motor neuron in regular repetitive bursts (like those in Fig. 1, for example; a precise definition follows in the next section). This is a natural set of elementary patterns, already of obvious physiological significance yet simple enough to keep the more technical aspects of computation and presentation of the results relatively straightforward. The approach and the principles that we will illustrate using these canonical patterns can be readily extended to more complex patterns.

Having selected a pattern, we feed it through the NMT. As will be seen, the important properties of the NMT are its nonlinearity and its time dependence, fundamental properties of NMTs generally. However, again for the sake of illustrative simplicity, we will focus quantitatively on just two NMTs. One is a simple mathematical model whose properties are completely defined so that their effects on the output can be fully understood. We can then compare the behavior of a real, experimental NMT, the Aplysia ARC-muscle NMT.

We now ask, when the input pattern is fed through the NMT, what output emerges? We can examine the whole contraction waveform c(t) or concentrate only on key parameters such as the peak or the mean contraction. In these papers we will be concerned primarily with the steady-state output achieved after sufficiently long repetition of the input pattern.

How does the output differ when the input is structured in different patterns? We will analyze what parameters of the input are important in determining various parameters of the output. Does the firing pattern contain a "code" by means of which particular parameters of motor neuron firing selectively control particular parameters of muscle contraction? Conversely, what information must the motor neuron, and the nervous system generally, have about the periphery to be able to control it effectively?

In Paper II, we then add the fact that the contractions must meet certain criteria for functional behavior. Every firing pattern produces some state of contraction of the muscle, but is it a functionally appropriate state? We can examine this by defining for the muscle, or for several interacting muscles, a behavioral task and a corresponding measure of how well, if at all, the task is being performed. We will define several tasks of increasing complexity for our elementary motor unit, for an antagonistic pair of two such units, and finally a realistic task for the real Aplysia ARC-opener neuromuscular system. It will be seen that relatively few firing patterns produce functional behavior, and even fewer produce efficient functional behavior. Conversely, functional behaviors with certain parameters are not possible with any firing pattern.

Finally, in Paper III, we use this foundation to examine how functional behavior may be expanded and optimized by appropriately tuning the NMT through peripheral neuromuscular plasticity and modulation.

We proceed as much as possible visually, through suitable graphical illustrations of the important concepts; the underlying mathematics and other more technical points can be found in the APPENDICES. The most frequently used symbols are summarized in Table 1. The DISCUSSION may be read independently for a nontechnical overview of the issues, results, and their biological implications. Papers II and III are organized similarly.

Input firing patterns

First, we must select a suitable time-dependent variable (for simplicity, just one) to formally describe the firing of the elementary motor neuron. We could take the continuous waveform of membrane voltage or, better, extract from it the timing of discrete spikes. For our purposes, however, it is most convenient to process the latter still further (by taking the reciprocal of the interspike interval) into a higher-order continuous variable, the "instantaneous" motor neuron firing frequency, f. The limits inherent in this discrete-to-continuous conversion will not be significantly felt in our work here. (Indeed, our work will enable us to say when those limits become significant: APPENDIX J.)

We can now consider different patterns of f. Temporal pattern is defined only over an interval of time, not at any single time point. Different patterns of f are thus expressed in different waveforms f(t): for our purposes the two will be synonymous. Our entire approach in this work is geared toward studying, as the basic unit, whole intervals of f(t), and the corresponding intervals of c(t), rather than their individual points.

Our canonical set of input waveforms, describing firing in regular repetitive bursts, is essentially the set of all waveforms that have the form sketched in the middle of Fig. 2 (more precisely, see APPENDIX A). Each such waveform is characterized by the structure of a single period, which then repeats endlessly. A single period, and therefore the whole waveform, can be completely defined by just three independent parameters. The most elementary triplet of parameters is that of the burst duration d_intra, interburst interval d_inter, and intraburst firing frequency f_intra, as indicated on the sketch in Fig. 2. (Note that the interburst frequency is always zero; patterns in which it is not are not members of the canonical set.) Another parameter triplet that is often used to describe such waveforms is that of the cycle period P, duty cycle (fraction of the period occupied by the burst) F, and mean (period-averaged) firing frequency f. These are composite parameters, derived from the elementary ones using the relations

(2a)

(2b)

(2c)

Finally, an intermediate parameter triplet of P, F, and f_intra is obtained by applying Eqs. 2a and 2b but not 2c.

View larger version (60K): [in this window] [in a new window]   Fig. 2. Input to the NMT: motor neuron firing pattern and its 3 representations. The small sketch in the middle shows a representative input waveform f(t) of the "instantaneous" firing frequency f as a function of time t. This waveform has the canonical form completely definable by 3 independent parameters: the burst duration dintra, interburst interval dinter, and intraburst frequency fintra [comprising the (dintra, dinter, fintra) representation], or alternatively the cycle period P, duty cycle F, and mean frequency f [the (P, F, f) representation], or an intermediate combination of parameters [the (P, F, fintra) representation]. The meaning of these parameters is graphically indicated on the sketch, and their interconversion is given at the bottom. Geometrically, the 3 representations can be pictured as alternative 3-dimensional input spaces (top). (Note that only a part of each infinite space can be shown, and that points where dintra = 0, P = 0 or F = 0 are by definition excluded: see APPENDIX A.) How the 3 spaces correspond to one another is illustrated by the 2 meshes. The thin mesh shows the surface given by 0 < dintra  1, 0  dinter  1, fintra = 1 in the (dintra, dinter, fintra) space and its mapping to the other 2 spaces. (Lines on this surface corresponding to P = 1 and F = 0.1, 0.3, 0.5, and 0.7 are also shown in the first 2 spaces.) Conversely, the thick mesh shows the surface given by 0 < P  1, 0 < F  1, f = 1 in the (P, F, f) space and its mapping (in the other 2 spaces the surface is truncated above fintra = 10). The rear planes where dinter = 0 or F = 1 represent steady, continuous firing.

The three parameter triplets constitute three alternative descriptions of the canonical set of waveforms. We will call them the (d_intra, d_inter, f_intra), (P, F, f), and (P, F, f_intra) representations. We now introduce a geometric point of view, which we will employ extensively throughout this work, designed to conceptually organize the set of waveforms and allow the whole set to be transformed through the NMT as a unit. Geometrically, the three representations can be pictured as alternative three-dimensional input spaces whose coordinates are the members of the triplet (Fig. 2, top). Every point in each space represents a canonical waveform, and every canonical waveform is represented in each space (see further APPENDIX A). How the spaces correspond to one another is given by Eqs. 2. Because Eqs. 2b and 2c are not linear, lines and planes in one space may correspond to curves and curved surfaces in another. The thick and thin meshes plotted in the spaces in Fig. 2 illustrate this for two particular subsets of the canonical set.

Why are we interested in three different descriptions of the same set of waveforms? It is because, by using different coordinates, they make explicit different parameters of the input pattern that may be important in different situations. For instance, the cycle period P, the parameter of interest in a problem like that in Fig. 1, is explicit in the (P, F, f) and (P, F, f_intra) representations but not in the (d_intra, d_inter, f_intra) representation. The (P, F, f) representation, but not the other two representations, makes explicit the mean firing frequency f; a horizontal plane in that space represents all canonical patterns with the same mean density of spikes, just arranged in different ways. Thus each space emphasizes different parameters of the input pattern, and, when the spaces are then transformed through the NMT, the effect of those parameters on the output is immediately apparent. Numerous examples of this will be seen below.

Clearly, more complex patterns would require spaces of more dimensions to represent them. A general representation of arbitrary input patterns would require an infinite number of dimensions. Our three-dimensional spaces are, in fact, reductions of this general space, possible because we have restricted our patterns to just a small subset with a special structure.

NMTs

The real Aplysia ARC-muscle NMT that we will use should more precisely be called the B15-ARC NMT, because it involves only one of the muscle's two motor neurons, neuron B15. We will not use the other, B16-ARC, NMT, although we will discuss generally how to analyze systems composed of multiple motor neurons as well as multiple muscles.

The B15-ARC NMT, like any other real NMT, is a priori unknown. We have no defining formula for it. Fundamentally, however, a transform is just a specification of the output that is produced for any particular input. If need be, this can be an explicit listing (APPENDIX B). We can begin to define the NMT, therefore, by correlating particular input waveforms f(t) with their corresponding output waveforms c(t). In Fig. 3A we have done this for a set of input waveforms that are also members of our canonical set (APPENDIX A), namely steady, continuous firing at various frequencies. We will show later that the information obtained with this special subset of input waveforms can be sufficient to define the NMT completely.

View larger version (61K): [in this window] [in a new window]   Fig. 3. The NMT: transform of motor neuron firing frequency f to contraction c. A: the real B15-ARC NMT. A1: contraction kinetics. Records of ARC-muscle length, c(t), when motor neuron B15 was fired steadily at the frequencies indicated [gray block; waveforms f(t) schematized below], starting with the muscle fully relaxed and continuing until the contraction reached steady state. Also shown is a representative record of the time course with which the muscle relaxed when the firing ended. Amplitude has been scaled so that c  1 for the largest contraction. These data are the same as those in Fig. 3B of Brezina et al. (1997), except that a more representative relaxation has been chosen. A2: steady-state c(f) relation from the data in A1. B: the model NMT. As in A, but for the NMT defined by Schema 3 or Eq. 4, with the standard parameter values , , p = 1 and q = 3 (except where indicated). B1: input waveforms f(t) (schematized) and corresponding solutions c(t) given by Eqs. F1. B2: c(f) relations for q = 3, q = 1 (Eq. F2), and for the ideal linear NMT (Eq. H1).

We see in Fig. 3A1 that, when motor neuron B15 begins to fire steadily, the ARC muscle contracts with a broadly sigmoidal time course, at first slowly (the contraction begins, in fact, only after some delay), then faster, and finally slower again, eventually reaching a steady-state contraction, c. All phases of the time course become faster, and c becomes greater, as the firing frequency f increases. These dependencies on f, too, are sigmoidal, with no contraction at all until some threshold value of f, and saturation at large f; the steady-state c(f) relation is shown in Fig. 3A2. When the motor neuron stops firing, the muscle relaxes with an exponential time course (Fig. 3A1). Overall, then, the B15-ARC NMT is highly nonlinear with respect to both time and the firing frequency f, and has a characteristic time scale (quite slow, of the order of seconds) that can be described by a characteristic time constant (APPENDIX C). Qualitatively, these properties are very typical of real NMTs generally. Quantitatively, in their characteristic speed in particular, NMTs of course differ considerably.

Implicit in any such description of a real NMT is a prior decision concerning the output variable, c. What, exactly, is being measured? In Fig. 3A, and elsewhere in this work where we have used the B15-ARC NMT, we measured the length of the muscle contracting isotonically against a constant light load. We could alternatively measure length under (probably more behaviorally realistic) auxotonic, variable-load conditions, or force under isometric conditions. The important qualitative properties of the NMT would most likely be the same: force measurements would probably yield a figure much like Fig. 3A. However, because the same input f would be transformed to a different output c in each case, formally we would be describing a different, although physiologically and functionally related, NMT. [In any case, throughout this work we represent active contraction produced by motor neuron firing as positive, upward movement of c(t), even though this corresponds to decrease in length.]

The other NMT that we will work with is a simple mathematical model. In this case we do know the NMT a priori. It is implicitly defined by the kinetic schema

(3)

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

(4)

Here, the input f controls the rate of a time-dependent reaction of an intermediate, a, on which the output c then depends. We found this model useful in our previous work (Brezina et al. 1997) because it can exhibit very different properties depending on the values of the parameters , , and especially p and q. In our work here, unless noted otherwise, we will use the "standard" parameter values  = 1,  = 1, p = 1, and q = 3. As can be seen in Fig. 3B, with these values the model NMT reproduces all of the qualitative properties of the real B15-ARC NMT. It, too, produces output that increases sigmoidally with both time and f, with a characteristic time scale. The primary point, however, is not to model the real NMT in the conventional sense. Rather, the model NMT, by exhibiting important properties in a simple and well-defined way, allows us to clearly analyze their effects on the output.

Output contractions

When an input waveform f(t) is fed through the NMT, an output waveform c(t) emerges. Examples with steady input were seen in Fig. 3, and a preliminary idea of the output waveforms that are produced by bursting input patterns may be gained from Figs. 4 and 5, for the real B15-ARC NMT and the model NMT, respectively. Clearly, even with simple input, the output of the NMT can be quite complex. Its analysis is facilitated, however, by our geometric point of view.

Some questions may require that we consider the whole output waveform. In the same way that we constructed geometric input spaces to collectively represent and relate to each other our input waveforms, we can construct output spaces for output waveforms. Insofar as the shapes of the output waveforms are more complex and variable, to represent them completely such spaces would need to have a correspondingly greater number of dimensions. (See, however, the next section.) A general representation of arbitrary output waveforms (as of arbitrary input waveforms) would require an infinite number of dimensions.

Fortunately, many questions concern, rather than the whole output waveform, just a small number of functionally important parameters of it. We will consider three such parameters: the peak (maximum) contraction, ; the minimum contraction, c; and the mean contraction, c. [As discussed in the next section, our output waveforms c(t) will always be periodic with the same period P as the corresponding input waveform f(t). The period P is thus the natural basic interval for our study. Each period of c(t) yields a single value of , c, and c =  _P c(t) dt.] Each of these single parameters can be represented in a reduced one-dimensional output space, that is, a line. Later, in Paper II, we will devise more complex parameters to measure how well the contractions are performing a behavioral task. These will still be single parameters, however, and so capable of being represented in one-dimensional output spaces.

In all cases, we wish to see which inputs yield which outputs, not just individually but collectively: that is, how an input space is globally transformed into an output space. But this collective specification is the NMT (more precisely, the part of it that we have chosen to focus on with our reduced input and output spaces), no longer in an implicit definition such as Schema 3 or Eq. 4 but as its explicit solution. It can be represented in a combined space that is the product of the input and output spaces. With a three-dimensional input space and a one-dimensional output spaceour canonical situationwe have a four-dimensional space to represent the NMT. As will be evident in the next section, all of this is no more than a generalization to higher dimensions of the familiar procedure of plotting an independent variable on a horizontal axis (a one-dimensional input space), a dependent variable on a vertical axis (a one-dimensional output space), and the relation of the two variables in the plane defined by the two axes (a two-dimensional product or input-output space).

Existence of mappings

Before proceeding to specific analysis, we must clarify one important issue. In what way is the NMT a transform? A transform, or, in the mathematical definition, a function or mapping, is a single-valued relation: each input maps to one, and only one, output. Of what aspects of the NMT is this true, and under what circumstances? This issue is important because a mapping provides us with a simple definition of control. If an input always maps to the same, predictable output, we may reasonably say that the input controls the output. We can apply this idea to whole waveformsoverall control of the muscle by the motor neuronor to specific parameters. As will be seen, controllability then conveniently classifies different physiological situations.

We will discuss the existence of mappings with the aid of Fig. 4. Figure 4C shows time courses of contractions produced by the real B15-ARC NMT, as in Fig. 3A but now not only with steady firing, but also four representative bursting patterns.

View larger version (55K): [in this window] [in a new window]   Fig. 4. Output of the NMT: contraction waveforms c(t) produced by the real B15-ARC NMT with different motor neuron firing patterns, illustrating approach of the system to the true steady state point-wise and to the dynamical steady state period-wise. C: records of ARC-muscle length c(t), starting with the muscle fully relaxed and continuing until the contraction reached steady state, produced by the 5 motor neuron B15 firing patterns f(t) below, with the parameters indicated at the bottom. (All 5 patterns have the same f = 6 Hz.) For the slowest bursting pattern, the dynamical steady-state contraction waveform [c(t)] and its parameters , c and c are approximately indicated. Horizontal arrows above the records in C indicate the sections plotted in A and B. A: the point-wise input-output space, the (f, c) plane, with the trajectories of the contractions in C (for clarity, only steady firing and one period of the slowest bursting pattern are shown) evolving toward the true steady state c(f) (replotted from Fig. 3A2). B: a period-wise input-output space, the (P, F, ) space, with the trajectories of the 4 bursting patterns in C evolving toward the dynamical steady state (P, F) (replotted from Fig. 10B1). See Existence of mappings in RESULTS for detailed discussion.

At this point, it is instructive to retreat for a moment from viewing waveforms in units of periods to the elementary point-wise view. The period-wise case will then be entirely analogous. Point-wise, the complete, unreduced input-output space is just two-dimensional: the (f, c) plane. It is so simple because all information about how successive points are related in waveforms has been discarded. Figure 4A plots the trajectory through the (f, c) plane of two of the contractions in Fig. 4C, with steady firing and for one period of one of the bursting patterns. (The horizontal arrows above the records in C indicate the sections plotted in A and B.)

At any time t, there is, of course, a single value of the input f, and a single value of the output c. (In other words, f and c are functions of t.) In that the NMT takes in that value of f, and puts out that value of c, it appears to be a transform (cf. APPENDIX D). However, the point of our geometric representation is to eliminate time as an explicit variable and consider all events simultaneously. Then, clearly, there is no single-valued mapping of f to c; in the (f, c) plane in Fig. 4A we see, for each plotted f, multiple values of c, corresponding to sections of the trajectory where c is changing even though f remains constant. Intuitively, Fig. 4C suggests that the value of c at any moment must reflect, as well as f, also the immediately preceding value of c. To predict c exactly requires knowledge of the complete state of the system, the pair (f, c) (precisely, see APPENDIX E, 1). Knowing just f, c can only, at best, be predicted to be in some likely range.

However, as time progresses, with constant f, we see in Fig. 4A (and Fig. 3A1) that each trajectory monotonically approaches a special state: the steady state. Thus the range in which, knowing just f, c might be expected to be, narrows with time and converges to the steady-state mapping c(f), a one-dimensional curve across the (f, c) plane in Fig. 4A (taken from Fig. 3A2). The model NMT, we know from solving Eq. 4 (see next section), approaches the steady state asymptotically as t  ; the real B15-ARC NMT, we see in Fig. 3A1, reaches it quite fast, if we consider that what is significant, functionally as well as experimentally, is only some close enough approximation (cf. APPENDIX B, and later).

We proceed similarly for the period-wise case. Fig. 4B shows a reduced three-dimensional input-output space, the (P, F, ) space with P and F as the input parameters and the peak contraction as the output, with the trajectories of the four bursting patterns in Fig. 4C. (All four patterns had the same f, allowing us to omit this fourth dimension.) Again, there is no single-valued mapping of the input (P, F) to the output ; to predict exactly requires knowledge of the complete state of the system, the triplet (P, F, ) (APPENDIX E, 2). But, with time, all trajectories converge monotonically to the steady-state mapping (P, F), the two-dimensional surface in Fig. 4B. (We will describe later how to compute this surface.)

The same behavior is found in every input-output space that we may construct along these lines. A sufficient condition is that the space have enough input dimensions to distinctly represent all of the input waveforms used: thus only (P, F) in this example, but for our whole set of canonical waveforms one of the complete triplets (d_intra, d_inter, f_intra), (P, F, f_intra), or (P, F, f). To these we can join one output dimension for a single parameter such as , c or c, several such dimensions, or even a general infinite-dimensional space for the whole output waveform c(t). With our NMTs, in each case the whole output space converges monotonically to a single steady-state pointtherefore a mappingfor each input waveform f(t) (APPENDIX E, 2).¹

With our NMTs, each output space behaves in this way because each, when joined to a sufficient input space, is in fact sufficient to distinctly represent every whole output waveform c(t) that can be produced by any canonical f(t) (APPENDIX E, 2). We do not need, after all, an infinite-dimensional or even a multidimensional space to represent every possible c(t): a one-dimensional space for a single parameter such as , c, or c is enough. In other words, for a particular f(t), a parameter such as , c, or c uniquely specifies the whole period of c(t) that yielded it (and vice versa, of course; consequently, any one of , c and c uniquely specifies the other two). Fundamentally, all of this reflects the fact that an NMT can produce only a restricted set of output waveforms (cf. APPENDIX L). Only certain shapes are possible. In our case, only one type of shape is possible, for which a one-dimensional space provides an optimalcomplete, distinct, and compactrepresentation. Any output space of more than one dimension will not provide a compact representation, in the sense that, for any f(t), it will contain points that cannot be reached with that f(t). Just as our reduced three-dimensional input spaces provide equivalent optimal representations of our restricted set of input waveforms, output spaces reduced to just one dimension such as , c or c provide equivalent optimal representations of the restricted set of output waveforms.

In each such output space, the steady-state point represents a unique steady-state output waveform, [c(t)] (Fig. 4C). Because there is a period-wise mapping to it from f(t), [c(t)] is periodic with the same period P as f(t), and each period of [c(t)] is identical. [c(t)] yields a unique value of , c and c, and of any other parameter that can be defined by a period-wise mapping from c(t), including our functional performance parameters in Paper II.

Note in Fig. 4 that, with bursting input, period-wise the system stabilizes at [c(t)], in what we may call the dynamical steady state of the system, even though point-wise it has not reached a true steady-state c. Indeed, periodic f(t) and [c(t)] correspond to a cycle in the elementary (f, c) plane (Fig. 4A). During each burst when f = f_intra, c rises toward c(f_intra), and during each interburst interval when f = 0, falls toward c(0) = 0. Although, in general, it reaches neither true steady state, how far it progresses toward them will be an important consideration in the following sections. With steady input, the two kinds of steady state necessarily coincide.

For any input, our model NMT, and apparently also the real B15-ARC NMT, has just a single steady state point-wise, and a single steady state period-wise; otherwise the mappings we have discussed would not exist. Such simple behavior is typical of many real NMTs. With some NMTs we may find, however, that for some inputs the dynamical steady state of the system consists of multiple points in the output space (alternative steady states that the system may reach from different regions of the space) or that it is itself periodic, or even that it does not have any recognizable simple structure. We have presented this section (and the associated APPENDIX E) in some detail partly to suggest how such possibilities, too, might be studied by our approach, essentially by expanding it into a more complete dynamical systems analysis. We will return to the physiological significance of such possibilities later, when we consider the implications of the ideas introduced in this section for controllability. First, however, we will examine the actual steady-state output produced by the model NMT, and by the real B15-ARC NMT.

Steady-state output of the model NMT

For any waveform f(t) as input to Eq. 4, and knowing the current state of the systemany state, not just the steady statewe can compute the corresponding output waveform c(t) (APPENDIX E). If necessary, we can do this numerically. With our canonical input waveforms, however, we can readily obtain analytic solutions (APPENDIX F). The output waveform has the general shape discussed in the last section. During each burst when f = f_intra, c moves toward c(f_intra), and during each interburst interval when f = 0, toward c(0) = 0. In general these movements are unequal so that the waveform gradually rises, or falls, over successive periods (exactly like the output of the B15-ARC NMT in Fig. 4C). In the dynamical steady state of the system, however, the two movementsrise during the burst and fall during the interburst intervalmust be equal and opposite. This requirement immediately gives us the unique steady-state waveform [c(t)] (three examples, to be discussed below, are shown in Fig. 5). Period-wise, we thus compute the dynamical steady state reached with bursting input just as, point-wise, we do the true steady state reached with steady input, by equating the appropriate opposing fluxes and converting our knowledge of the kinetics of c into knowledge of its absolute steady-state amplitude.

View larger version (24K): [in this window] [in a new window]   Fig. 5. Steady-state output waveforms [c(t)] produced by the model NMT, illustrating their appearance as the time scale of the input pattern, P, varies relative to the time constant of the NMT, . These are plots of Eq. F3 with the standard parameter values , , p = 1, q = 3. The gray bars mark the phases when f = fintra. In all cases F = , fintra = 1.5, f = 0.5. The input pattern (left) is thus identical in all cases, except on different time scales. In all cases   1 (APPENDICES C AND F). A: P = 0.01  . B: P = 1  . C: P = 100  . The 3 cases correspond to the locations A, B, C in Fig. 6A.

From [c(t)], we then obtain analytic expressions for , c, and c (APPENDIX F). We can write these expressions in terms of one or another of our alternative triplets of input parameters. To see how our set of inputs maps globally to the set of outputs, we need simply examine these expressions, or their graphic representations in the appropriate input-output spaces.

As we have discussed, these are four-dimensional spaces, in which the steady state occupies a three-dimensional volume. Four dimensions are already unmanageable graphically. We will therefore usually show representative three- or even two-dimensional sections through the complete four-dimensional space, obtained by setting one or two of the input parameters to constant values (as, in effect, we did in Fig. 4B). In a three-dimensional section the steady state appears as a two-dimensional surface, and in a two-dimensional section as a one-dimensional curve. Various such plots can be seen in Figs. 6-9.

To understand the input-output mapping, it is most convenient to begin with the input in the (P, F, f) representation. As we have already noted, this representation explicitly allows us to decompose the input into two components: f, the mean firing frequency or density of motor neuron spikes, and (P, F), the temporal arrangement of those spikes. In our previous work (Brezina et al. 1997) we reserved the term "pattern" technically just for the latter.

The output then depends on both components, filtered in different ways through their interaction with the properties of the NMT. Broadly, we can think of the output as being dependent on f according to the steady-state mapping c(f), but then modified, in a complex but predictable manner, by a factor that depends on the pattern (P, F) (as well as f). This factor, which we technically termed "pattern dependence," was the focus of our previous work. We found that, with a time-dependent NMT such as the model NMT, key determinants of pattern dependence are the time scale of the input pattern relative to that of the NMT, and the shapethe degree and kind of nonlinearityof the NMT (Brezina et al. 1997). The interaction of these elements can be envisioned intuitively as follows. The nonlinearity of the NMT only appears on time scales longer than , the time constant of the NMT. The input pattern exists on time scales shorter than P, the cycle period. Only when P > , so that the nonlinearity and the pattern overlap and interact, does pattern dependence become expressed, and modifies the output from its basic value of c(f).

This can be seen in Fig. 6, which shows sections through the mappings (P, F, f), c(P, F, f), and c(P, F, f) for one particular value of f. The shapes of these surfaces reflect, therefore, just changing pattern dependence, modifying the output up (lighter tone) or down (darker tone) from the basic value of c(f). Lower and higher values of f give similar surfaces layered, respectively, below and above the ones shown, although gradually changing in shape with f, as will be apparent in subsequent figures.

View larger version (31K): [in this window] [in a new window]   Fig. 6. Parameters , c and c of the steady-state output of the model NMT: overall view of their dependence on input parameters, in the (P, F, f) representation. These are plots over a broad range of P and F of the steady-state mappings (P, F), c(P, F), and c(P, F) given by Eqs. F4-F6 with the standard values , , p = 1, q = 3. In all cases f = 0.5; these are therefore only representative sections, at f = 0.5, through the general mappings (P, F, f), c(P, F f), and c(P, F, f). Extended log scales have been used to show a broad range of P and F and to emphasize the asymptotic behavior of , c, and c at its extremes. Higher values of , c, c are shown in lighter tone, lower values in darker tone. In B, when P   and F < 1, c  0, off scale (Eq. G2).   1 (APPENDICES C AND F). Locations A-C in A correspond approximately to the 3 waveforms [c(t)] in Fig. 5, A-C.

In the F direction, each surface begins at c(f) when F = 1 (steady input; front edge). Pattern dependence becomes expressed as F decreases, as the input pattern itself developsas the spikes are grouped into progressively more extreme burstsand engages the nonlinearity of the NMT. In the F direction, therefore, output increased above c(f) reflects what we may call "positive" pattern dependencegreater output as spikes are grouped into burstsand output decreased below c(f) "negative" pattern dependencegreater output as spikes are dispersed into steady firing. Output remaining precisely at c(f) is pattern independent: the arrangement of spikes is immaterial to the output.

We wish now to emphasize, however, the P direction. Demonstrating the critical importance of the relative time scales of the input pattern and the NMT, we see that each surface is divisible into two regions of distinct pattern dependence and therefore distinct output, above and below P  . (To emphasize this we have used extended log scales, in Fig. 6 only.) The characteristic output seen in each region can be understood by examining the limiting values that the output tends to when P   or P  , when the NMT, in effect, becomes time independent and the output ceases to vary with P. How , c, and c depend on input parameters in these two cases is shown in Figs. 7 and 8, respectively; typical shapes of [c(t)] in the two cases can be seen in Fig. 5, A and C. 

View larger version (35K): [in this window] [in a new window]   Fig. 7. Parameters , c, and c of the steady-state output of the model NMT: detailed view of their dependence on the parameters of fast input patterns. This figure examines in more detail just the leftmost end of the plots in Fig. 6, A-C, where P  . All linear scales. Equations F4-F6 with the standard values , , p =&nbs