An improved test for detecting multiplicative homeostatic synaptic scaling

Abstract

Homeostatic scaling of synaptic strengths is essential for maintenance of network "gain", but also poses a risk of losing the distinctions among relative synaptic weights, which are possibly cellular correlates of memory storage. Multiplicative scaling of all synapses has been proposed as a mechanism that would preserve the relative weights among them, because they would all be proportionately adjusted. It is crucial for this hypothesis that all synapses be affected identically, but whether or not this actually occurs is difficult to determine directly. Mathematical tests for multiplicative synaptic scaling are presently carried out on distributions of miniature synaptic current amplitudes, but the accuracy of the test procedure has not been fully validated. We now show that the existence of an amplitude threshold for empirical detection of miniature synaptic currents limits the use of the most common method for detecting multiplicative changes. Our new method circumvents the problem by discarding the potentially distorting subthreshold values after computational scaling. This new method should be useful in assessing the underlying neurophysiological nature of a homeostatic synaptic scaling transformation, and therefore in evaluating its functional significance.

Figure 1. Test for multiplicative scaling of mEPSC amplitudes with a rank-order plot of amplitudes.

A. mEPSCs were recorded from CA1 pyramidal neurons in slice cultures of rat hippocampus. For induction of homeostatic plasticity of synapses, slice cultures were treated with 1–1.5 µM TTX for 3–4 days. Rank-ordered amplitudes of mEPSCs in TTX-treated cells were plotted against rank-ordered control mEPSCs. A total of 700 mEPSCs, with 100 events per cell, were collected. The straight lines are linear fits with variable (blue) or zero (green) y-intercept. B. Cumulative histograms of mEPSC amplitudes. Individual mEPSC amplitudes in TTX-treated cells were transformed with the equation y = 1.6x–4.9. The distribution of transformed values almost exactly overlaps the control distribution. The p value is from a K-S test between control and scaled TTX group. C. Cumulative histograms were obtained as in B (control and TTX plots are the same as those in B), but the TTX group was scaled with the equation y = 1.27x. The p value from a K-S test shows significant differences between control and scaled TTX groups. D. mIPSCs were obtained from control and TTX-treated slice cultures of hippocampus. The rank-order plot (D) and the cumulative histograms (E,F) were constructed in a similar manner to those in A–C. E. Cumulative histogram of mIPSC amplitudes. The amplitudes of TTX-treated mIPSCs were transformed with y = 0.55x–7.4. Again, the transformed distribution shows nearly complete overlap with the control distribution. The p value is from a K-S test between control and scaled TTX group. F. The TTX group was scaled with the equation y = 0.60x. The distributions of the control and scaled TTX groups are significantly different from each other (K-S test).

Figure 2. Schematic diagrams illustrating the inadequacy of the previous test for multiplicative scaling.

A. Hypothetical distribution curves, in which all of the mEPSC amplitudes are included without a detection threshold. The TTX-treated mEPSCs are assumed to be two-fold larger than the controls (i.e., multiplicative scaling; left). If the control population is multiplied by two (right), the scaled control curve becomes identical to the TTX curve. B. Schematic diagrams similar to A, but a detection threshold of mEPSC amplitude limits the minimum amplitude (left), as in realistic mEPSC recordings. If the suprathreshold control values (i.e., excluding the gray area) are multiplied by two (right), the minimum offset of the scaled curve (red) will be shifted by two-fold, resulting in a mismatch with the TTX curve (left graph). This suggests that a transformation of y = ax cannot be used to test for multiplicative scaling when a detection threshold is present. C. Schematic diagrams show how the exclusion of subthreshold amplitudes invalidates the rank-order method of the conventional multiplicative scaling test. Hypothetical mEPSC amplitudes distributed evenly are rank-order plotted. TTX-treated mEPSCs (open circles) are assumed to be two-fold larger (y = 2x) than control ones (filled circles). The left graph shows an ideal, full range of mEPSC amplitudes, including the subthreshold values (<10 pA; gray area). If the subthreshold data are discarded, as in realistic mEPSC recordings, the rank-order plot of TTX data will be shifted (blue arrows). A linear fit of the TTX data in the resultant rank-order plot (right graph) now gives y = 2x–8. D. Schematic illustrations similar to C, but with data distributed unevenly. The TTX data (open circles) again constitute a multiplicative scaling of the controls by a two-fold (left graph). The exclusion of subthreshold data points (gray area) will cause a shift of the remaining data by different distances (blue arrows). A linear fit of the new rank-order plot of TTX-treated data (right graph) yields an equation that loses information about the original multiplicative scaling factor. When the TTX data in the right graph were fitted with a y-intercept of 0, the equation was y = 1.45x (line not shown), again different from the original scaling function.

Figure 3. A new method of testing for the occurrence of multiplicative scaling.

A. Schematic diagrams of distributions of mEPSC amplitudes illustrate the new method of estimating a true multiplicative scaling factor. Only suprathreshold data (Suprathr.) are shown in the left graph. Multiplicative down-scaling of TTX data (from red curve to blue one) results in a small portion of data (gray) falling below the detection threshold. Because the data in the gray area are invisible in experimental recording, they are discarded for comparison with the control data. The degree of overlap is measured between the control (black curve) and the scaled TTX data (blue curve) excluding the gray area. B. Cumulative histograms of experimental data shown in Fig. 1A. To derive the distribution of scaled TTX data (blue), the TTX-treated mEPSCs were divided by 1.181, which yielded the maximum overlap between control and scaled TTX data excluding subthreshold values. The threshold is defined as the smallest detectable amplitude of control mEPSCs. A K-S test showed no significant difference between control and scaled TTX groups (p>10⁻⁴), suggesting the occurrence of multiplicative scaling. C. With other scaling factors, the degrees of overlap between control and scaled TTX data excluding subthreshold values were also examined. The degree of overlap of the histograms was quantified by confidence level (p value) of K-S test. The largest p (0.041) was obtained when mEPSC amplitudes of the TTX group were divided by 1.181. D. A test for multiplicative scaling was performed with mIPSCs in a similar manner to that in A–C. Because the mean amplitude of control mIPSCs was larger than that of TTX-treated ones, control mIPSC amplitudes were scaled down by multiplying by 0.6769. A K-S test between scaled control and TTX data (p>10⁻⁴) revealed multiplicative scaling. E. The degrees of overlap between scaled control and TTX data excluding subthreshold values were assessed with various scaling factors. The largest p (0.0025) was obtained with a scaling factor of 0.6769.

Figure 4. Tests for multiplicative scaling using two different methods and artificially generated data sets.

A. Data generation scheme. mIPSC amplitudes (n = 3,500) recorded experimentally from untreated cells were arithmetically transformed: i.e., they were multiplied by 1.5 and subtracted by 20 pA. After the transformation, 3,345 values were above the detection threshold and this set was called the “treatment group”. Randomly selected values from the experimental group made up the “control group” (n = 3,345). B–C. As in Fig. 3, the treatment group was scaled down and compared with the control group using K-S tests. When the scaling factor was 0.8335, the distribution of the scaled treated group most completely overlapped that of the control group. The p value (K-S test) shows significant difference between the control and scaled treatment groups. D–E. In the conventional test, the amplitudes of control and treatment groups were rank-ordered and plotted against each other. The treatment group was scaled back using the linear regression equation. The distributions of control and scaled treatment groups were not significantly different from each other (K-S test).