Temporal pattern dependence of neuronal peptide transmitter release: models and experiments

Abstract

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.

Fig. 1.

Pattern dependence of mean release as a probe of the release process. Three patterns of firing (second rowfrom 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, Φ. With the middle pattern, release is pattern independent (Φ = 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 theright pattern, in contrast, release is pattern dependent (Φ ≠ 1). See introductory remarks and Temporal pattern dependence in Results.

Fig. 2.

Typical firing pattern of motor neuron B15 and corresponding peptide outflow from the ARC muscle from the data ofVilim et al. (1996a,b). A, Standard reference firing pattern used by Vilim et al.: burst duration d_intra = 3.5 sec, interburst interval d_inter = 3.5 sec, intraburst firing frequency f_intra = 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). Thesmooth 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).

Fig. 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. 11and 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 d_inter, f_intra, and d_intra, 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 curveswith 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). Thecurves are simply extensions of the fits with x = 4 in the main plot.

Fig. 4.

Fitting of Model II to the data of Vilim et al. (1996a,b). Same data as in Figure 3. The curvesare the final best fits of Model II (Eq. 15) as described in Model II in Materials and Methods.

Fig. 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 Figure2A 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 δ_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).

Fig. 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, themain plot shows the steady-state pattern dependence Φ_f→X (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 thetop left and top right of each ofA1–B2 are two examples of the actual waveforms at the locations indicated in the main plot, showing in each case three periods of the firing pattern f(t) (below, thick trace) and the corresponding output waveform [X(t)]_∞[i.e., X(t) in the dynamical steady state (Brezina et al., 1997, 2000);above, thick trace], compared with the equivalent unpatterned, tonic firing f′(t) at〈f〉 = 5 Hz and its outputX′_∞ (thin lines; where no thin lines are visible, they coincide with the thick traces). The two examples clearly have very different time scales (P = 1 sec vs P = 1000 sec), but the output is plotted on the same vertical scale, of Φ, throughout A1–B2.Finally, the top center plot in each of A1–B2shows the shape of the steady-state transformation f → X_∞. Throughout, for the instantaneous fast reaction, which in effect is always in the steady state, the specification of the steady state is superfluous. Plots were generated by a combination of numerical and analytical computations using Equations 1, 3a, b, 5a, 6a, 13 (in Materials and Methods), and 21. For detailed discussion see Pattern dependence generated by Models I and II in Results.

Fig. 7.

Overall pattern dependence of release predicted by Models I and II. A1, B1, Plots of the overall steady-state pattern dependence Φ_f→r generated by Models I and II for firing patterns over a wide range of cycle periodP and duty cycle D (all scales, in A1and B1 only, are log scales), but all with the same mean firing frequency 〈f〉 = 5 Hz. A2, B2, Expanded linear-scale view of the outlined regionof A1 and B1. The small dark gray region labeled Vilim is that containing the patterns used by Vilim et al. (1996a,b) (i.e., the points plotted in Figs. 3A, 4A). A3, B3, Comparison of the overall steady-state pattern dependence Φ_f→r predicted by Models I and II for two particular test patterns, P = 8 sec andP = 200 sec, in both cases with D = 0.25 and 〈f〉 = 5 Hz. Unpatterned, tonic firing at 〈f〉 = 5 Hz, defining Φ = 1, is also shown. The location of these two patterns and the tonic firing, color-coded dark gray, black, andwhite, respectively, is indicated by the dots inA1–B2. Plots were generated by numerical solution, for Equation 1, of Equations 3a and 3b, with S(t) = S₀, followed by application of Equation 21. Pattern dependence very similar to that computed in A3 andB3 for the steady state was obtained when the complete Models I and II (Eqs. 3) were run in simulations of the exact stimulation protocol used in the real experiments in Figure 8A1, bottom.

Fig. 8.

Experimental discrimination between Models I and II by test of the pattern dependence predicted in Figure 7. Specifically, the two patterns in Figure 7, A3 andB3, i.e., P = 8 sec and P = 200 sec, both with D = 0.25 and〈f〉 = 5 Hz, were compared. Color coding as in Figure 7. A, Representative experiment. A1,Time course of SCP outflow while motor neuron B15 was stimulated to fire as shown below, with the test pattern, in this case P = 8 sec (dark gray block), between two blocks of unpatterned, tonic firing at 〈f〉 = 5 Hz. Each bar of outflow is the amount of SCP contained in each 2.5 min drop of ARC perfusate (see Release experiments in Materials and Methods).A2, Total SCP outflow, O_∞, resulting from each block of patterned or unpatterned firing, obtained by summing the indicated bars of A1. B, Group data. Means ± SEM of plots like A2 from four experiments with each test pattern. In each experiment, the plot was first normalized by the mean of the two blocks of unpatterned outflow so as to yield, for the two test patterns, directly the pattern dependence Φ_f→r.