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The Journal of Neuroscience, September 15, 2000, 20(18):6760-6772

Temporal Pattern Dependence of Neuronal Peptide Transmitter Release: Models and Experiments

Vladimir Brezina, Paul J. Church, and Klaudiusz R. Weiss

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

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS Release experiments RESULTS DISCUSSION REFERENCES

In this paper we construct, on the basis of existing experimental data, a mathematical model of firing-elicited release of peptide transmitters from motor neuron B15 in the accessory radula closer neuromuscular system of Aplysia. The model consists of a slow "mobilizing" reaction and the fast release reaction itself. Experimentally, however, it was possible to measure only the mean, heavily averaged release, lacking fast kinetic information. Considered in the conventional way, the data were insufficient to completely specify the details of the model, in particular the relative properties of the slow and the unobservable fast reaction. We illustrate here, with our model and with additional experiments, how to approach such a problem by considering another dimension of release, namely its pattern dependence. The mean release is sensitive to the temporal pattern of firing, even to pattern on time scales much faster than the time scale on which the release is averaged. The mean release varies with the time scale and magnitude of the pattern, relative to the time scale and nonlinearity of the release reactions with which the pattern interacts. The type and magnitude of pattern dependence, especially when correlated systematically over a range of patterns, can therefore yield information about the properties of the release reactions. Thus, temporal pattern can be used as a probe of the release process, even of its fast, directly unobservable components. More generally, the analysis provides insights into the possible ways in which such pattern dependence, widespread especially in neuropeptide- and hormone-releasing systems, might arise from the properties of the underlying cellular reactions.

Key words: synaptic transmission; neurotransmitter; neuropeptide; transmitter release; firing pattern; temporal pattern dependence; mathematical modeling; Aplysia

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS Release experiments RESULTS DISCUSSION REFERENCES

Release of neurotransmitters and hormones is brought about by a complex sequence of intracellular reactions with differing kinetics and stimulation dependence (for review, see Zucker, 1996; Neher, 1998; Kasai, 1999). The dynamic interplay of these reactions underlies the plasticity with which the release responds in a functionally appropriate manner to different patterns and histories of stimulation (Fisher et al., 1997; O'Donovan and Rinzel, 1997). For functional prediction of release, as well as to aid in identification of the underlying molecular machinery, it is therefore highly desirable to obtain a quantitative understanding of the properties of the release reactions and their mutual relations, such as may be embodied in a mathematical model (Magleby and Zengel, 1982; Heinemann et al., 1993; Dittman et al., 2000). Considerable progress in this direction has been made in preparations where release can be measured in "real time," with high temporal resolution, using a fast electrophysiological response to the released transmitter or a capacitance, amperometric, or optical measure of exocytosis (Angleson and Betz, 1997; Neher, 1998). As schematized in Fig. 1, these techniques allow direct observation of the detailed waveform of release (second row from bottom) that results from any pattern of stimulation (second row from top).

In many interesting preparations, however, these techniques cannot be easily applied. In this paper we deal with one such system: release of peptide transmitters from motor neuron B15 in the accessory radula closer (ARC) neuromuscular system of Aplysia. This release was measured in a series of studies by Vilim and colleagues (Vilim, 1993; Vilim et al., 1996a, 1996b); we begin here by modeling their data. Because the functional consequences of the release for the whole, intact system are of major interest (Weiss et al., 1993; Brezina et al., 1996), release was studied in the intact system, from B15 terminals lying inaccessible within the muscle. There is no fast electrophysiological response to these modulatory transmitters. The amounts released are small, and the radioimmunoassay (RIA) technique used by Vilim et al. (1996a,b) integrated the amounts over intervals of several minutes. Thus, rather than the detailed waveform of the release, Vilim et al. (1996a,b) could only measure the mean, heavily averaged release (see Fig. 1, bottom row).

Nevertheless, as we illustrate here using our model and with additional experiments, such measurements can provide considerable information about the properties of the release process. This is because the mean release, in general, is sensitive to the temporal pattern of the stimulationthe way in which a given "amount" of stimulation is arranged in time, even on time scales that may be much faster than that on which the release is averaged. For example, the pattern schematized in the right column of Figure 1 gives threefold greater mean release than the middle pattern, although both patterns deliver the same amount of stimulation (top row). Knowing the general rules that govern such pattern dependence (Brezina et al., 1997), its type and magnitude yield information about the properties of the release reactions that generated it. Temporal pattern can thus be used as a probe of the release process.

Release of transmitters and hormones is strongly pattern dependent in many systems, with important physiological consequences (Dutton and Dyball, 1979; Andersson et al., 1982; Ip and Zigmond, 1984; Cazalis et al., 1985; Bicknell, 1988; Birks and Isacoff, 1988; Peng and Horn, 1991). It is therefore of considerable interest to see how pattern dependence arises from the properties of the underlying intracellular reactions, so as to be able to understand when different types of pattern dependence can and cannot arise.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS Release experiments RESULTS DISCUSSION REFERENCES

Modeling

Here we describe important aspects of the modeling in more technical detail than could be given in the general overview in Results. The principal variables, parameters, and symbols used are summarized in Table 1.

Firing patterns

Vilim et al. (1996a,b) used regular repetitive bursting patterns of the kind shown in Figure 2A. In addition to the total stimulation length L, such patterns are completely definable by three parameters such as the burst duration d_intra, the interburst interval d_inter, and the intraburst firing frequency f_intra, or equivalently the cycle period P = d_intra + d_inter, the duty cycle D = d_intra/P, and the mean firing frequency f = f_intraD (Brezina et al., 1997, 2000). Figure 2A is drawn to scale to show the values that Vilim et al. (1996a,b) used as a standard reference pattern: d_intra = 3.5 sec, d_inter = 3.5 sec, f_intra = 12 Hz; or equivalently P = 7 sec, D = 0.5, f = 6 Hz; L = 1 hr.

The waveform of firing frequency f at time t can be conveniently expressed as:

(1)

where t mod P is the remainder after dividing P into t the largest possible integral number of times. For simplicity, we will use "f(t)" to refer, as well as to the value of f at a particular time t, to a section, or the whole waveform of such values.

The special case d_inter = 0 or D = 1 represents "unpatterned," tonic firing, expressible as:

(2)

For every patterned f(t) described by Equation 1 there exists unpatterned firing described by Equation 2 with the same mean frequency. We denote this f'(t).

Model equations

A general model with the required properties (see Basic model of the release data of Vilim et al. in Results) is given by the equations:

(3a)

(3b)

(3c)

Below we analyze and fit to the data two versions of the model, setting a priori either y = 1 (Model I) or x = 1 (Model II).

Model I: y = 1

Analysis. Model I gives essentially pattern-independent release with all of the patterns used by Vilim et al. (1996a,b) (see Pattern dependence generated by Models I and II in Results). For the purposes of reproducing the release measured by Vilim et al. (1996a,b), averaged on a time scale much longer than P, we may therefore replace every patterned f(t) given by Equation 1 with the equivalent unpatterned f'(t) given by Equation 2.

Equation 3b can then be alternatively written as:

(4)

where:

(5a)

is the steady-state value that p(t) approaches as t  L, with "dissociation constant" K_p  k_p/k_p+, and

(5b)

is the time constant of relaxation to the steady state.

We note that Equation 3a potentially allows rapid release of all S. In the data of Vilim et al. (1996a,b), however, S(t) decreases only slowly as the firing continues (see Fig. 2B). This means that p(t)^x, and p(t) itself, must be small. Hence 1  p(t)  1 in Equation 3b, and Equations 5a and 5b simplify to

(6a,b)

For f'(t) given by Equation 2, with p(0) = 0 and S(0) = S, and with y = 1 and x = 4 (determined below to be the integral value that best fits the data), Equations 3 and 4 have the analytical solutions:

(7a)

(7b)

(7c)

where a(t) = t  _p + _p[4 exp(t/_p)  3 exp(2t/_p) +  exp(3t/_p)  1/4 exp(4t/_p)].

Finally, if we define:

(8)

the total amount released up to time t, clearly:

(9)

Fitting the model to the data. Ideally, r(t) given by Equation 7a could now be fitted to the data. However, Vilim et al. (1996a,b) did not directly measure r(t), but rather outflow from the muscle, o(t), i.e., r(t) transformed by some transport function _T, which we do not know a priori. Nevertheless, Vilim et al. (1996a,b) presented two kinds of data that together provided a way past this problem.

[In the data of Vilim et al. (1996a,b), small cardioactive peptide (SCP) and buccalin (BUC) release appears indistinguishable in all respects except absolute amounts (see Figs. 2B, 3, 4) (see Basic model of the release data of Vilim et al. in Results). For fitting, we therefore always pooled SCP and BUC data or constrained the fit to yield identical parameter values, except absolute amounts.]

(1) Long L (1 hr). These data are reproduced in Figure 2B. The transport function _T can be thought of as being composed, roughly, of a "diffusional" component _D and a "bulk-flow" component _B, which slow and retard, respectively, o with respect to r. [In the interpretation of the model, the tail of o(t > L) in Fig. 2B, when according to Eq. 7a r(t > L) = 0, is a direct reflection of _T; the complete effect of _T can be seen in Fig. 5C.] After a sufficiently long time into the block of firing L, t > _p, p(t)  p, r(t) changes slowly, and _D can be assumed to be near steady state. [This is strictly true, of course, only if we have substituted the unpatterned f'(t).] Then release should directly appear as outflow, but with some fixed delay _B, the mean transit time out of the muscle by the bulk-flow process _B. Using Equation 7a, we predict:

(10)

Indeed, as Figure 2B shows, the decline of o(t > 20 min) was well fit by a single exponential with rate constant of 0.037 min¹ (time constant of ~27 min). By Equation 10, the rate constant is equal to p⁴f. Thus p(f = 6 Hz)  0.10. Then, using Equation 5a, K_p  54 sec¹.

(2) Short L (5-10 min). In these experiments the system did not have time to reach steady state; no simplifying assumption about _T can be made to extract r(t) from o(t). Thus the time course of o is not useful in this case. Instead, Vilim et al. (1996a,b) measured the total peptide outflow resulting from the whole block of firing of length L, O(L)i.e., not only the outflow during L itself, but also the tail of outflow afterward (see Fig. 8A). We define:

(11)

Experiments were performed with different patterns over fixed L = 10 min. Because with the patterns used Model I gives pattern-independent release, and Vilim et al. (1996a,b) indeed apparently observed O dependent simply on f (see Experimental test of the pattern dependence predicted by Models I and II in Results), all of these data are replotted together against f in Figure 3A. Other experiments were performed with fixed pattern but varying L (see Fig. 3B).

From Equations 7c and 9, we find:

(12)

Equation 12 (together with Equations 5) was simultaneously fitted to the data in Figure 3, A and B, using the value of K_p from the long-L data. As Figure 3, A and B, shows (solid curves), this fit, with x = 4, was excellent. A similar fitting was carried out for x = 1, 2, 3, and 5, using the appropriate equivalent of Equation 12 [differing in the exponent of p and the terms of a(L)] and appropriately recalculated K_p. However, each of these fits was inferior to that obtained with x = 4. The fits with x = 1, 2, and 3 had too little curvature; the fit with x = 5 had too much curvature (Fig. 3A, B, dashed curves).

The inset of Figure 3B extends the time axis to longer L and confirms that the best fit to the short-L data also fits the total outflow in the long-L experiments.

These fits yielded the values k_p+ = 2.04 × 10⁴, k_p = 1.10 × 10² sec¹, S_0,SCP = 542 fmol, and S_0,BUC = 200 fmol. The S₀ values are indicated in Figure 3B, inset, and discussed further below.

Model II: x = 1

Analysis. In Model II, for all of the patterns used by Vilim et al. (1996a,b), significant pattern dependence is generated only by the nonlinearity f(t)^y (see Pattern dependence generated by Models I and II in Results). For the purposes of reproducing the release measured by Vilim et al. (1996a,b), averaged on a time scale much longer than P, we may therefore replace every patterned f(t) with the equivalent unpatterned f'(t), and in Equation 3a replace f(t)^y with f'(t)^y, where  is the pattern dependence (see Eq. 21 in Results; the following shows that  is always in the steady state, as assumed in Eq. 21). Evaluating  = f(t)^y/f'(t)^y using Equations 1 and 2 and the relation f = f_intraD yields:

(13)

Thus  is independent of time, a constant in a plot such as Figure 2B, and independent of f, a constant in a plot such as Figure 4A.

With these replacements, Equations 4-6 hold as for Model I, and, with x = 1, p(0) = 0, and S(0) = S₀, Equations 3 and 4 have the analytical solutions:

(14a)

(14b)

(14c)

Equations 8, 9, and 11 apply as for Model I.

Fitting the model to the data. The same data (see Fig. 2B, and Fig. 4, replotting the data of Fig. 3) and similar strategies were used as with Model I.

(1) Fixed short L, varying f. The counterpart of Equation 12 is:

(15)

With essentially fixed L, , _p (Eq. 6b), and p  f (Eq. 6a), this yields:

(16)

where c is a constant. Equation 16 was fitted to the data in Figure 4A, in a first pass (fit not shown), yielding as the best integral value y = 3.

(2) Long L. The counterpart of Equation 10 is:

(17)

The exponential rate constant of 0.037 min¹ fitted to o(t > 20 min) in Figure 2B is now equal to pf^y. With y = 3, D = 0.5, and hence, by Equation 13,  = 4, p(f = 6 Hz)  7.14 × 10⁷. Then, by Equation 5a, K_p  8.4 × 10⁶ sec¹.

(3) All short L. Finally, fitting Equation 15 simultaneously to the data in Figure 4, A and B, with y = 3,  = 4.8 (by Eq. 13; this set of data, within which no differences in pattern dependence were apparent, had mean D = 0.46; see Experimental test of the pattern dependence predicted by Models I and II in Results), and the above K_p, yielded k_p+ = 4.04 × 10¹⁰, k_p = 3.4 × 10³ sec¹, S_0,SCP = 541 fmol, and S_0,BUC = 198 fmol. The inset of Figure 4B confirms that these values also fit the total outflow in the long-L experiments.

The S₀ values (indicated in Fig. 4B, inset) are essentially identical to those obtained for Model I. Their absolute interpretation is complicated by the fact that the SCPs and BUCs are families of multiple, almost certainly coreleased forms, all of which are not recognized equally by the RIA antibodies used (Vilim, 1993; Vilim et al., 1996a). Minimizing the error, however, the antibodies were raised and calibrated against SCP_B and BUC_A, both very abundant members of their families; BUC_A is the most common single BUC (Miller et al., 1993; Vilim et al., 1994), and SCP_B probably constitutes half of the released SCP (Lloyd, 1986; Cropper et al., 1987).

If we accept the S₀ values approximately, we can compare them with measurements of the total peptide present in the muscle. Those values are significantly larger: of the order of 3-18 pmol SCP and 4 pmol BUC (Lloyd et al., 1984; Whim and Lloyd, 1989; F. S. Vilim, unpublished observations). The motor neuron processes in the muscle thus contain additional peptide that appears nonreleasable on the time scale of the experiments (see also Whim and Lloyd, 1989; Cropper et al., 1990b). This may be peptide in vesicles that have not been made competent for release or perhaps simply are not located right at the release sites.

	Release experiments

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS Release experiments RESULTS DISCUSSION REFERENCES

Release preparation. The preparation was essentially as described by Vilim et al. (1996a). Briefly, the buccal ganglion and the ARC muscle were dissected from the animal while buccal nerve 3 was kept intact. The ganglion was pinned in a Sylgard-lined Petri dish and desheathed. Buccal nerve 3 was fitted through a slit in the dish, which was then sealed with silicone grease to separate the ganglion from the muscle. The muscle was perfused through an artery at 20 µl/min; every 2.5 min, a 50 µl drop of the perfusate formed at the outflow from the muscle and fell into an individual tube that was then processed for RIA. In the buccal ganglion, identified motor neuron B15 was impaled with two microelectrodes, one to monitor membrane potential and the other through which current was injected so as to fire the neuron in the desired pattern. All experiments were performed at 15°C.

RIA. SCP content was determined as described by Vilim et al. (1996a). Briefly, SCP_B was iodinated (¹²⁵I) using the chloramine-T method. Iodinated stocks were repurified using reverse-phase HPLC and diluted in RIA buffer containing (in mM): 154 NaCl, 10 Na₂HPO₄, 50 EDTA, 0.25 merthiolate, 1% BSA, pH 7.5, to a final activity of 10,000-15,000 cpm/100 µl. Antibodies were diluted in RIA buffer so that 100 µl bound up ~50% of the counts in 100 µl of the iodinated trace. The sample volume in the RIA reaction was 50 µl, i.e., the volume of each 2.5 min drop of ARC perfusate. The reaction was performed for 1-2 d at 4°C and terminated by addition of 2 ml of charcoal solution (10 mM Na₂HPO₄, 0.25 mM merthiolate, 0.25% activated charcoal, 0.025% 70,000 kDa dextran, pH 7.5). Samples were then centrifuged, and the supernatant, containing the bound peptide, was decanted and counted in a gamma counter. Counts were converted to SCP amounts using standard curves generated with serial dilutions of known amounts of SCP_B. [The absolute interpretation of the SCP amounts has the same uncertainties as in the data of Vilim et al. (1996a,b) in Model II above.]

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS Release experiments RESULTS DISCUSSION REFERENCES

Release of peptide cotransmitters from motor neuron B15 of Aplysia

Motor neuron B15 innervates the ARC muscle, an extensively studied muscle in the buccal mass of Aplysia that participates in the animal's feeding behavior. Like many other Aplysia motor neurons, motor neuron B15 uses both classical and peptidergic modes of neurotransmission. It releases ACh to contract the ARC muscle (Cohen et al., 1978), but it also releases peptide cotransmitters belonging to two families, the SCPs and the BUCs (Lloyd et al., 1984; Cropper et al., 1987, 1988, 1990b; Whim and Lloyd, 1989, 1990; Vilim et al., 1994, 1996a), that then modulate the ACh-induced contractions in various behaviorally appropriate ways (Weiss et al., 1993; Brezina et al., 1996). As is typical in peptidergic neurons, the peptides are contained in large dense-core vesicles (LDCVs), distributed differently within the terminal from the small synaptic vesicles (SSVs) containing ACh (Cropper et al., 1987; Kreiner et al., 1987; Vilim et al., 1996a; Karhunen et al., 1998).

In a series of studies, Vilim and colleagues (Vilim, 1993; Vilim et al., 1996a, 1996b) used RIA to directly measure the amounts of SCP and BUC appearing in perfusate of the ARC muscle (cf. Release experiments in Materials and Methods), while motor neuron B15 was stimulated to fire in various patterns for various extended lengths of time, from 5 min up to 1 hr. The outflow of the peptides from the muscle was taken as a reflection of their release from the neuron's terminals within the muscle. But because of filtering by the slow outflow (see Model I in Materials and Methods) and, more fundamentally, because the amounts measured were integrated over 2.5 min intervals [dictated by the flow rate and RIA sensitivity (Vilim et al., 1996a)], Vilim et al. were unable to measure release with any high degree of temporal resolution. Rather, they measured, essentially, the mean release averaged on a time scale of ~2.5 min. Explicitly, their data contain no faster kinetic information. Yet there are clear indirect indications that peptide release from motor neuron B15 does have faster kinetic components (see next section). The essence of the problem is thus as outlined in the introductory remarks and Figure 1.

View larger version (76K): [in this window] [in a new window]   Figure 1.   Pattern dependence of mean release as a probe of the release process. Three patterns of firing (second row from top) [all with the same mean frequency (top row)] produce, through their interaction with the properties of the release process, three different waveforms of release (second row from bottom). The detailed waveform may not be directly observable because of low temporal resolution of the available measurement techniques, which may yield only the mean, perhaps heavily averaged, release (bottom row), but as described in this paper, this can nevertheless provide considerable information about the properties of the release process. The mean release produced by patterned firing (middle and right columns) relative to that produced by "unpatterned," tonic firing at the same mean frequency (left column) defines the pattern dependence of mean release, . With the middle pattern, release is pattern independent ( = 1): the presence of the pattern does not alter the mean release from that produced simply by the same number of spikes presented unpatterned. With the right pattern, in contrast, release is pattern dependent (  1). See introductory remarks and Temporal pattern dependence in Results.

Basic model of the release data of Vilim et al.

Firing patterns

Vilim et al. (1996a,b) stimulated motor neuron B15 to fire in regular repetitive bursting patterns of the kind shown in Figure 2A. Patterns of this kind are a reasonable approximation of the natural firing patterns of B15 (Cropper et al., 1990a). Such patterns can most concisely be represented, as in Figure 2A, as patterns of the waveform of the firing frequency f as a function of time t. For short, we will refer to "the patterned waveform f(t)." In addition to the total stimulation length L, the patterned waveform f(t)the pattern itselfis completely definable by a triplet of parameters. For the purpose of discussing pattern dependence, the most suitable triplet is that of the cycle period P, the duty cycle D, and the mean firing frequency f (Brezina et al., 1997, 2000). (For further details see Firing patterns in Materials and Methods. The principal variables, parameters, and symbols used in this paper are summarized in Table 1.)

View larger version (23K): [in this window] [in a new window]   Figure 2.   Typical firing pattern of motor neuron B15 and corresponding peptide outflow from the ARC muscle from the data of Vilim et al. (1996a,b). A, Standard reference firing pattern used by Vilim et al.: burst duration dintra = 3.5 sec, interburst interval dinter = 3.5 sec, intraburst firing frequency fintra = 12 Hz; or equivalently cycle period P = 7 sec, duty cycle D = 0.5, mean firing frequency f = 6 Hz; total stimulation length L = 1 hr. B, Time course of SCP and BUC outflow when motor neuron B15 was stimulated to fire as in A ("long-L data"). Replotted from Vilim et al. (1996a), their Figure 8B1. Mean ± SEM from four experiments. Vilim et al. actually measured outflow integrated over 2.5 min intervals (Fig. 8A1) every 5 min (alternately for SCP and BUC), but this has been converted to outflow per minute. The plot was scaled correctly using the absolute amounts measured (F. S. Vilim, personal communication). The smooth curves show the best single-exponential fit to both the SCP and BUC values (with different scaling for the two peptides) in the interval 20 < t < 60 min (used for both Models I and II: see Eqs. 10 and 17 in Materials and Methods).

View this table: [in this window] [in a new window]   Table 1.   Principal variables, parameters, and symbols

Considerations for model formulation

With these firing patterns, Vilim et al. (1996a,b) obtained the data replotted in Figures 2B, 3, and 4. Figure 2B shows the time course of SCP and BUC outflow when motor neuron B15 was stimulated to fire in the particular pattern shown in Figure 2A. Measurements from many such experiments are plotted more analytically in Figures 3 and 4.

View larger version (23K): [in this window] [in a new window]   Figure 3.   Fitting of Model I to the data of Vilim et al. (1996a,b). Plotted in all cases is the total peptide outflow, O, resulting from the whole block of firing of length L, against the mean firing frequency f (with fixed L), or L (with fixed f) (see Eqs. 11 and 12 in Materials and Methods). A, O versus f, with fixed short L = 10 min. Replotted from Vilim et al. (1996b), their Figures 3B, 4B, 5B. These three figures of Vilim et al. presented data for varying dinter, fintra, and dintra, respectively; because in all cases outflow appeared to depend simply on f, the three plots have here been combined. [For both SCP and BUC, one point from each plot constitutes the groups at f  4, 5, and 6 Hz (the last consists of three points superimposed).] Each point is the mean ± SEM (often smaller than the symbol size) from four to five experiments. Vilim et al. (1996a,b) actually presented the data normalized per spike ( O/f), but this has been converted to O again. The plot was scaled correctly using the absolute amounts measured (F. S. Vilim, personal communication). The solid curves are best fits of Equation 12 with x = 4, the dashed curves with x = 1, 2, 3, and 5 (shown only for SCP), as described in Model I in Materials and Methods. B, O versus short L, with fixed f = 6 Hz. Replotted from Vilim et al. (1996a), their Figure 9B. Means ± SEM, n = 5. Details and fitting as in A. Inset, O versus all L, with fixed f = 6 Hz. Extension of the main plot of B to longer L to include the values from the experiments in Figure 2B (n = 4). The curves are simply extensions of the fits with x = 4 in the main plot.

View larger version (20K): [in this window] [in a new window]   Figure 4.   Fitting of Model II to the data of Vilim et al. (1996a,b). Same data as in Figure 3. The curves are the final best fits of Model II (Eq. 15) as described in Model II in Materials and Methods.

In these data, Vilim et al. (1996a,b) noted three principal features of outflow and so presumably release. (1) In the absence of firing, there is essentially no basal release (Fig. 2B); release increases, in a markedly supralinear fashion, with firing frequency (Figs. 3A, 4A). Temporally, release (2) increases over the first minutes of firing, then (3) decreases slowly beyond ~10 min (Fig. 2B).

To model observation (1), release could simply be made an instantaneous function of the firing frequency f. However, this would not be sufficient to reproduce observation (2), which implies that release responds slowly to changes in f. A model with only slow dependence of release on f would also be unsatisfactory, however, because release would continue for a long time [again, for minutes: see the behavior of p(t > L) (explained below) in Fig. 5A] after firing ended. Although the data of Vilim et al. (1996a,b) lack the temporal resolution to show that this does not in fact happen (indeed, the slow "tail" of release in Fig. 2B, for example, might be taken as evidence of it), there are indications that it does not. For instance, a downstream effect of the released SCP, elevation of cAMP in the muscle, decays relatively fast (with a time constant of perhaps 10 sec) after the end of firing (Whim and Lloyd, 1990), suggesting that release itself decays as fast or faster. (The slow tail in Fig. 2B is then explained as a tail of outflow: see Model I in Materials and Methods.) In other words, changes in firing are reflected relatively rapidly in downstream effects of the released peptides (which can be measured with better temporal resolution), indicating that the release has kinetic components that are considerably faster than can be resolved in the data of Vilim et al. (1996a,b). Altogether, then, the simplest model must incorporate both fast and slow dependence of release on f.

View larger version (47K): [in this window] [in a new window]   Figure 5.   Performance of the complete model: simulation of the typical experiment in Figure 2 using Model II. Right panels show the whole simulation; left panels show the first 3 min. Vertical scaling in B, C is correct for SCP. For the waveform of patterned firing f(t) shown in Figure 2A and here again in D, Equations 3, with the parameter values of Model II (see Model II in Materials and Methods), were solved numerically to obtain the probability of release p(t), the size of the releasable pool S(t), and the release r(t), shown in A-C. The gray areas in C, D, right, are the envelopes swept out by the excursions of r(t) and f(t). In C, r(t) was averaged periodwise to obtain the mean release r(t). In C, right, the outflow o(t) and its exponential fit have been reproduced from Figure 2B (for SCP), but scaled (×1.2) to match the total areas under o(t) and r(t), so that O(L) = R(L) (Eqs. 8, 11). The discrepancy between r(t) and o(t) is a measure of the function T transporting the released peptide out of the muscle (see Model I in Materials and Methods). The same simulation using Model I gave similar results, except that (1) p4 responded more rapidly to changes in f than p here; (2) consequently, p(t)4 varied more within P, and its envelope rose more rapidly at the start of the firing and fell much more rapidly at its end; (3) the rise of the envelope of p(t)4 was sigmoidal rather than exponential; (4) consequently, the envelope of r(t), too, rose sigmoidally and, after the initial lag, more rapidly than here; the same was true for r(t); and (5), consequently, there was a larger discrepancy between r(t) and o(t).

Basic model structure

Along these lines, we were able to account for all of the data with the following basic model of release. From a pool of releasable peptide of size S, the firing frequency f controls release, r, in two ways, through a slow and a fast reaction. First, r depends on a variable p, which can be interpreted as the probability of release or the actual availability for release of the peptide in S, and p varies slowly with f according to the schema:

(18)

with (relatively small) rate constants k_p+ and k_p. Second, r also depends in an instantaneous fashion on f. Altogether:

(19)

(The full set of equations is given in Model equations in Materials and Methods. The exponents x and y will be discussed below.) Although simple and relatively abstract, this model is consistent with the more elaborate models constructed in systems where release can be measured with high temporal resolution (see Discussion).

The model very naturally reproduces observation (3), without postulating any additional, hypothetical reactions, simply by assuming that the releasable pool S decreases from a fixed initial size S₀ as the peptide is releasedin other words, as depletion. Just as the slow reaction of Equation 18 might correspond to a real cellular "mobilizing" reaction (see Discussion), the modeled depletion might well correspond to real cellular depletion. However, other mechanisms, such as inactivation of the release machinery, would fit the formal properties of the modeled depletion equally well. The matters studied in this paper depend on the formal properties of the release reactions, as captured in the model, independent of their actual mechanisms. Of course, identification of those mechanisms will provide additional insights (see Discussion).

Performance of the model

How the model explains the features of the data is illustrated in Figure 5, where we ran the model in a simulation of the representative experiment in Figure 2. The right panels of Figure 5 show the whole simulation; the left panels show an expanded view of the first 3 min.

The characteristic relaxation time (time constant) of p, _p, is of the order of several minutes. (More generally, the relevant parameter is the relaxation time of p^x, _p^x, but here x = 1; see next section.) On the much shorter time scale of the firing pattern in this experiment, with cycle period P = 7 sec, p hardly reacts at all to changes in f; within P, p(t) is essentially constant (Fig. 5A). On time scales approaching _p, however, p begins to respond significantly. Consequently, after the firing starts, the envelope of p(t) rises to steady state with a slow relaxation time of, again, _p. This imparts a similar slow rise to the envelope of r(t) and to the mean release, r(t) (Fig. 5C). This rise is slow enough to be resolved in the data of Vilim et al. (1996a,b): it accounts in large part for observation (2), the slow buildup of outflow over the first minutes of firing visible in Figure 2B. [The buildup is additionally slowed by the relatively slow movement of the released peptide out of the muscle, in Figure 5C manifest in the discrepancy between r(t) and the outflow, o(t); see Model I in Materials and Methods.]

As the envelope of p(t) approaches steady state, the envelope of r(t), and the mean release r(t), peaks, then begins to fall as the pool of releasable peptide S(t), initially of size S₀, becomes depleted (Fig. 5B). Because p(t), the probability of release, is always small (Fig. 5A), this happens only very slowly, over tens of minutes. This accounts for observation (3), the gradual decline of outflow beyond ~10 min of firing visible in Figure 2B.

Finally, on the short time scale of P, r(t) is gated by f(t). In response to the bursts of firing, there are corresponding bursts of release. These cannot be resolved in the data of Vilim et al. (1996a,b); indeed, the model simply lumps all fast, unobservable components of release into one instantaneous reaction. (As the Discussion shows, however, at least some of these components must in reality be sufficiently slow to integrate the spikes into the firing frequency f.)

[The release in Fig. 5C is scaled correctly for SCP; for BUC, it is ~2.7 times smaller. This simply reflects the relative size of the releasable pool S of the two peptides. In the model, as in the data of Vilim et al. (1996a,b), SCP and BUC release is identical in all other respects (Figs. 2B, 3, 4). Indeed, all available evidence suggests that all of the peptide cotransmitters released by motor neuron B15the various forms of both SCP and BUCare packaged together in the same dense-core vesicles and obligatorily coreleased in an essentially invariant ratio (Vilim, 1993; Vilim et al., 1996a, 1996b). The release mechanism, which is our chief interest here, is thus the same for all.]

Two sites of nonlinearity: Models I and II

The model as so far discussed does not yet explain one more prominent feature of the data of Vilim et al. (1996a,b): observation (1), the fact that release depends on the firing frequency not in a linear, but in a highly supralinear fashion (Figs. 3A, 4A). To provide two possible sites of nonlinearity, the model includes the exponents x and y. (The dependence of p on f, for small p, is practically linear.) In essence, we can picture Equations 18 and 19, informally, as:

(20)

To explain the data of Vilim et al. (1996a,b), how should we distribute the required supralinearity between the slow and the fast reaction?

We created and fitted to the data two extreme versions of the basic model. In Model I, we set a priori y = 1, then found, as the integral value best fitting the data, x = 4. In other words, we made the fast reaction linear and allocated all of the supralinearity to the slow reaction. Conversely, in Model II, we set a priori x = 1, then found y = 3 to best fit the data. In other words, we made the slow reaction linear and allocated all supralinearity to the fast reaction.

As can be seen in Figures 3 (Model I) and 4 (Model II), both models provide an excellent quantitative fit to all of the data of Vilim et al. (1996a,b). (The fitting is described in detail in Model I and Model II in Materials and Methods.) Both models give the performance described in Figure 5, with differences only on fast, unobservable time scales (Fig. 5, see legend). On the basis of the data of Vilim et al. (1996a,b), the two models cannot be distinguished. However, the two models predict very different pattern dependence of release, and this can be used to discriminate between them.

Temporal pattern dependence

We apply some general ideas on temporal pattern dependence in biological reactions (Brezina et al., 1997). We can regard the firing frequency f as input, and a variable of interest X that f controls, such as here p or ultimately r, as output, of an input-output step f  X (Fig. 1). Because it is the mean output that is measured experimentally, we are interested in the pattern dependence of the mean output: how the mean amplitude of X depends on the temporal pattern of f. For each patterned waveform f(t), which produces a waveform of output X(t) with (period-averaged) mean output X(t), there exists "unpatterned," tonic firing with the same mean frequency f as f(t) (see Firing patterns in Materials and Methods), which we denote f'(t) and which produces output X'(t). We then define the pattern dependence, _fX, as:

(21)

[For simplicity, we focus immediately on the pattern dependence in the dynamical steady state of the system (Brezina et al., 1997, 2000), which corresponds well enough to the situation in the relevant data of Vilim et al. (1996a,b) as well as the new experiments in Fig. 8. See legends to Figs. 6 and 7.] The meaning of Equation 21 is indicated graphically in Figure 1. We are asking, how does the mean output differ when the same "amount" of inputhere, the same number of motor neuron spikesis presented unpatterned, and in a particular temporal pattern?

View larger version (51K): [in this window] [in a new window]   Figure 6.   Comparison of the individual pattern dependence generated by the slow and fast reactions in Models I and II. A1, A2, B1, and B2 are laid out identically. In each, the main plot shows the steady-state pattern dependence fX (see Eq. 21 in Results) generated by the reaction, f  X, for firing patterns over a wide range of cycle period P and duty cycle D (note that all scales, in this plot only, are log scales), but all with the same mean firing frequency f = 5 Hz. At the top left and top right of each of A1-B2 are two examples of the actual waveforms at the locations indicated in the m