Ca2+ dependence of the binomial parameters p and n at the mouse neuromuscular junction

Abstract

The Ca(2+) dependence of synaptic quantal release is generally thought to be restricted to probability of vesicular release. However, some studies have suggested that the number of release sites (n) at the neuromuscular junction (NMJ) is also Ca(2+) dependent. In this study, we recorded endplate currents over a wide range of extracellular Ca(2+) concentrations and found the expected Ca(2+) dependency of release. A graphical technique was used to estimate p (probability of release) and n using standard binomial assumptions. The results suggested n was Ca(2+) dependent. The data were simulated using compound binomial statistics with variable n (Ca(2+) dependent) or fixed n (Ca(2+) independent). With fixed n, successful simulation of increasing Ca(2+) required that p increase abruptly at some sites from very low to high values. Successful simulation with variable n required the introduction of previously silent release sites (p = 0) with high values of p. Thus the success of both simulations required abrupt, large increases of p at a subset of release sites with initially low or zero p. Estimates of the time course of release obtained by deconvolving evoked endplate currents with average miniature endplate currents decreased slightly as Ca(2+) increased, thus arguing against sequential release of multiple quanta at higher Ca(2+) levels. Our results suggest that the apparent Ca(2+) dependence of n at the NMJ can be explained by an underlying Ca(2+) dependence of a spatially variable p such that p increases abruptly at a subset of sites as Ca(2+) is increased.

Fig. 1.

Measuring Var(m) at endplates allows direct estimates of p and n. A: accurate measurement of Var(m) requires that mean endplate current and miniature endplate current amplitudes are stable throughout the recording. Shown is a scatter plot of 30 EPC amplitudes from an endplate in solution containing 2.0 mM Ca²⁺. Inset: 3 examples of EPC traces. B: a scatter plot of 76 MEPC amplitudes recorded over a 1-min span. Inset: 3 examples of MEPC traces. C: superimposed EPCs recorded in 1 mM Ca²⁺ for each of 3 endplates. For each set of traces, a stimulus artifact precedes the EPC by several milliseconds. Shown at the bottom right of the EPCs for each endplate is the average MEPC from that endplate used to calculate m. The vertical scale bar represents 20 nA for EPCs and 1 nA for MEPCs. By measuring the variance of the EPC and m, it is possible to calculate Var(m). D: after taking into account the differences in MEPC amplitude between the 3 endplates, it is possible to plot Var(m) vs. m for each endplate (points 1–3 on the plot). It can be seen from this plot and the traces in C that the mechanism underlying the larger EPC amplitudes in endplates 1 and 3 differ. In endplate 1 the EPC is larger because of larger quantal amplitude, whereas in endplate 3, the larger EPC was because of larger m caused by a higher value of n with no increase in p. Each parabola represents the plot for a given n as p is increased from 0 to 1.0, assuming uniform and stationary p for all synaptic sites. Included in the plot are the parabolas for n = 20, 40, 60, 80, 100, and 120. All the parabolas start with Var(m) = 0 when m = 0 and increase to a maximum Var(m) when p = 0.5. When p reaches 1.0 for each n, Var(m) again becomes 0 as the parabola reaches maximal m. Intersecting the parabolas are straight lines representing the theoretical plot for a given p as n is increased. Error bars for m represent the SE for m for each endplate.

Fig. 2.

Ca²⁺ dependence of Var(m) plotted vs. m. Plotted is mean Var(m) vs. m for endplates in solutions containing 5 different concentrations of external Ca²⁺. The values of external Ca²⁺ used are given in Table 1 as well as the data for m and Var(m). As in Fig. 1, the parabolas represent the theoretical plot for a given n as p is increased from 0 to 1.0, and the straight lines represent theoretical plot for values of p as n is increased. Both p and n appear to increase as m is increased by increasing external Ca²⁺. Error bars for m and Var(m) were arrived at by determining a mean value of m and Var(m) for each muscle studied. The mean values from muscles were averaged, and the SE of these values is plotted.

Fig. 3.

Binomial analysis with and without Ca²⁺ dependence of n. A: plotted as open circles is Var(m) vs. m for the means of 5 simulations entailing 10 runs of 100 EPCs per simulation. The external Ca²⁺ concentration (in mM) that is being modeled for each point is shown in B. p, n, and the CV of mean p were varied as shown in the top row of B. By increasing both p and n, it was possible to generate values of m and Var(m) very similar to those shown in Fig. 2 as external Ca²⁺ was increased. Plotted as filled squares are Var(m) vs. m for simulations in which n was not Ca²⁺ dependent and was fixed at 100. Shown in the bottom row of B are distributions of p with n fixed at 100 for each of the 5 simulations. By varying only p, it was possible to generate values of m and Var(m) very similar to those shown in Fig. 2 as external Ca²⁺ was increased. However, when n is fixed, high values of CV of mean p must be used to simulate the data such that many release sites either have p < 0.1 or p > 0.9.

Fig. 4.

Ca²⁺ dependence of the time course of vesicular release. A: top: the EPC and MEPC used for deconvolution from an endplate recorded from in 1 mM Ca²⁺. Bottom: the EPC and MEPC used for deconvolution from an endplate recorded from in 5 mM Ca²⁺. The vertical scales for EPCs and MEPCs are 20 and 1 nA, respectively. B: the deconvolution waveforms from the endplate current traces shown in A. The peak rate of vesicular release is significantly higher in 5 mM Ca²⁺. The vertical scale for the deconvolution waveforms is in quanta per millisecond, and the time base is the same as in A. C: the deconvolution waveforms from B have been normalized and superimposed. Although the peak of the deconvolution waveform is similar in time to peak and width, the presence of prolonged release of vesicles after the peak is more prominent in the deconvolution waveform from the endplate in 5 mM Ca²⁺.

Fig. 5.

Prolonged release of vesicles widens the EPC but does not contribute to EPC amplitude. A: the EPC traces from Fig. 4 and the superimposed inverted deconvolution waveforms for the same endplates (gray). Prolonged release of vesicles occurs after the peak of the EPC (dotted line). B: the normalized fitted MEPCs (gray) and average EPCs (black) from the same endplates in A. The EPC recorded in 1 mM Ca²⁺ has a half-width that is only slightly wider than the fitted MEPC from the same endplate. However, the EPC recorded in 5 mM Ca²⁺ has a significantly wider half-width than the MEPC from the same endplate because of prolonged release of vesicles.