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Table A1:
Statistics of Quadratic Fit to (Average) l1-Norm of Error (top) and Hessian-Weighted Error (Bottom) as a Function of Heterogeneity for Stimulus Filter k, Postspike Filter h, and Constant b.
HeterogeneityNetwork Size234567
k R2 0.0293 0.0140 0.0347 0.0534 0.0725 0.0974 
 p-value for F-statistic versus Constant 3.9×10-42 1.8×10-20 7.1×10-50 4.9×10-77 2.2×10-105 3.9×10-143 
h R2 0.0103 0.01794 0.0606 0.0382 0.0530 0.0888 
 p-value for F-statistic versus Constant 3.6×10-15 6.7×10-26 1.4×10-87 6.7×10-55 2.3×10-76 5.2×10-130 
b R2 0.0141 0.0164 0.0357 0.0577 0.0620 0.0918 
 p-value for F-statistic versus Constant 1.6×10-20 9.1×10-24 3.3×10-51 2.1×10-83 1.2×10-89 1.3×10-134 
Heterogeneity Network Size 
k R2 0.0068 0.0019 0.0060 0.0191 0.0315 0.0401 
 p-value for F-statistic versus Constant 3.6×10-10 0.0022 4.5×10-9 1.5×10-27 3.0×10-45 1.6×10-57 
h R2 0.0014 0.0020 0.0101 0.0171 0.0181 0.0356 
 p-value for F-statistic versus Constant 0.0032 3.8×10-4 7.2×10-15 1.2×10-24 4.3×10-26 4.5×10-51 
b R2 0.0029 0.0047 0.0062 0.0222 0.0290 0.0329 
 p-value for F-statistic versus Constant 8.3×10-5 2.5×10-7 2.4×10-9 6.5×10-32 1.4×10-41 3.2×10-47 
HeterogeneityNetwork Size234567
k R2 0.0293 0.0140 0.0347 0.0534 0.0725 0.0974 
 p-value for F-statistic versus Constant 3.9×10-42 1.8×10-20 7.1×10-50 4.9×10-77 2.2×10-105 3.9×10-143 
h R2 0.0103 0.01794 0.0606 0.0382 0.0530 0.0888 
 p-value for F-statistic versus Constant 3.6×10-15 6.7×10-26 1.4×10-87 6.7×10-55 2.3×10-76 5.2×10-130 
b R2 0.0141 0.0164 0.0357 0.0577 0.0620 0.0918 
 p-value for F-statistic versus Constant 1.6×10-20 9.1×10-24 3.3×10-51 2.1×10-83 1.2×10-89 1.3×10-134 
Heterogeneity Network Size 
k R2 0.0068 0.0019 0.0060 0.0191 0.0315 0.0401 
 p-value for F-statistic versus Constant 3.6×10-10 0.0022 4.5×10-9 1.5×10-27 3.0×10-45 1.6×10-57 
h R2 0.0014 0.0020 0.0101 0.0171 0.0181 0.0356 
 p-value for F-statistic versus Constant 0.0032 3.8×10-4 7.2×10-15 1.2×10-24 4.3×10-26 4.5×10-51 
b R2 0.0029 0.0047 0.0062 0.0222 0.0290 0.0329 
 p-value for F-statistic versus Constant 8.3×10-5 2.5×10-7 2.4×10-9 6.5×10-32 1.4×10-41 3.2×10-47 

Note: The numbers in bold correspond to quadratic fits where the optimal level of heterogeneity is not in the interior.

Table A2:
Statistics of Quadratic Fit to Pearson's Correlation (with Prewhitening) as a Function of Heterogeneity for Stimulus Filter k, Postspike Filter h, and Constant b.
HeterogeneityNetwork Size234567
k R2 0.0018 0.0123 0.0076 0.0149 0.0212 0.0357 
 p-value for F-statistic versus Constant 0.003 6.0×10-18 2.2×10-11 1.4×10-21 1.6×10-30 2.9×10-51 
h R2 0.0199 0.0159 0.0166 0.0156 0.0232 0.0356 
 p-value for F-statistic versus Constant 1.1×10-28 4.6×10-23 5.3×10-24 1.4×10-22 2.4×10-33 4.4×10-51 
b R2 0.0008 0.0027 0.0032 0.0048 0.0160 0.0251 
 p-value for F-statistic versus Constant 0.0237 1.8×10-4 3.9×10-5 1.9×10-7 4.3×10-23 4.0×10-36 
HeterogeneityNetwork Size234567
k R2 0.0018 0.0123 0.0076 0.0149 0.0212 0.0357 
 p-value for F-statistic versus Constant 0.003 6.0×10-18 2.2×10-11 1.4×10-21 1.6×10-30 2.9×10-51 
h R2 0.0199 0.0159 0.0166 0.0156 0.0232 0.0356 
 p-value for F-statistic versus Constant 1.1×10-28 4.6×10-23 5.3×10-24 1.4×10-22 2.4×10-33 4.4×10-51 
b R2 0.0008 0.0027 0.0032 0.0048 0.0160 0.0251 
 p-value for F-statistic versus Constant 0.0237 1.8×10-4 3.9×10-5 1.9×10-7 4.3×10-23 4.0×10-36 

Note: The numbers in bold correspond to quadratic fits where the optimal level of heterogeneity is not in the interior.

Figure A1:

(A) For the raw stimulus sampled at 0.5ms and then differenced once to ensure stationarity, the autocorrelation functions (ACF, top) show an autoregressive process on the stimulus values, while the partial autocorrelation function (PACF, bottom) indicates a binary oscillatory pattern with some autocorrelation on the moving average. (B) Similar to panel A but for a once-differentiated stimulus chosen randomly: yt=xt-xt-1, for length 200ms. For prewhitening, chosen models were ARIMA(6,1,3), followed by ARIMA(8,1,4) and ARIMA(4,1,2) in cases where a model of the initial choice could not be constructed. (C) We systematically determined the length of the stimulus filter k and lag of the last postspike basis vector by considering 168 pairs of these values for a network with all 7 cells. We chose the pair that gave the smallest summed negative log likelihood (i.e., the largest maximum likelihood) after fitting to 10 segments of time, each of length of approximately 2sec. The magenta oval indicates our choice (180ms peak for last basis vector of h corresponds to a total lag of 240ms; see Figure 1(B); the magenta oval with dashed outline had a similar log likelihood but required more computational resources.

Figure A1:

(A) For the raw stimulus sampled at 0.5ms and then differenced once to ensure stationarity, the autocorrelation functions (ACF, top) show an autoregressive process on the stimulus values, while the partial autocorrelation function (PACF, bottom) indicates a binary oscillatory pattern with some autocorrelation on the moving average. (B) Similar to panel A but for a once-differentiated stimulus chosen randomly: yt=xt-xt-1, for length 200ms. For prewhitening, chosen models were ARIMA(6,1,3), followed by ARIMA(8,1,4) and ARIMA(4,1,2) in cases where a model of the initial choice could not be constructed. (C) We systematically determined the length of the stimulus filter k and lag of the last postspike basis vector by considering 168 pairs of these values for a network with all 7 cells. We chose the pair that gave the smallest summed negative log likelihood (i.e., the largest maximum likelihood) after fitting to 10 segments of time, each of length of approximately 2sec. The magenta oval indicates our choice (180ms peak for last basis vector of h corresponds to a total lag of 240ms; see Figure 1(B); the magenta oval with dashed outline had a similar log likelihood but required more computational resources.

Figure A2:

The l1-norm of the error as a function of all three forms of heterogeneity: stimulus filter (left column), postspike filter (middle column), and bias heterogeneity (right column). Here the network sizes ranged from N=2 (top row) and N=3 (middle row), to N=4 (bottom row); we ensured N cells were selected for each network. See section 4 for how random samples were generated. In all cases, a quadratic regression fit shows an optimal level of heterogeneity in the domain. With N=3,4 for postspike filter heterogeneity, the optimal levels are for smaller values of heterogeneity.

Figure A2:

The l1-norm of the error as a function of all three forms of heterogeneity: stimulus filter (left column), postspike filter (middle column), and bias heterogeneity (right column). Here the network sizes ranged from N=2 (top row) and N=3 (middle row), to N=4 (bottom row); we ensured N cells were selected for each network. See section 4 for how random samples were generated. In all cases, a quadratic regression fit shows an optimal level of heterogeneity in the domain. With N=3,4 for postspike filter heterogeneity, the optimal levels are for smaller values of heterogeneity.

Figure A3:

Similar to Figure 8 but with the remaining network sizes (N=5,6,7). The l1-norm of the error as a function of all three forms of heterogeneity: stimulus filter (left column), postspike filter (middle column), and bias heterogeneity (right column). Here the network sizes ranged from N=5 (top row) and N=6 (middle row), to N=7 (bottom row); we ensured N cells were selected for each network. See section 4 for how random samples were generated. In all cases, a quadratic regression fit shows an optimal level of heterogeneity in the domain.

Figure A3:

Similar to Figure 8 but with the remaining network sizes (N=5,6,7). The l1-norm of the error as a function of all three forms of heterogeneity: stimulus filter (left column), postspike filter (middle column), and bias heterogeneity (right column). Here the network sizes ranged from N=5 (top row) and N=6 (middle row), to N=7 (bottom row); we ensured N cells were selected for each network. See section 4 for how random samples were generated. In all cases, a quadratic regression fit shows an optimal level of heterogeneity in the domain.

Figure A4:

The Hessian-weighted error as a function of all three forms of heterogeneity: stimulus filter (left column), postspike filter (middle column), and bias heterogeneity (right column). Here the network sizes ranged from N=2 (top row) and N=3 (middle row), to N=4 (bottom row); we ensured N distinct cells were selected for each network. See section 4 for how random samples were generated. In almost all cases, a quadratic regression fit shows an optimal level of heterogeneity in the domain; the exceptions are for postspike filter heterogeneity for N=2,3,4 (middle column).

Figure A4:

The Hessian-weighted error as a function of all three forms of heterogeneity: stimulus filter (left column), postspike filter (middle column), and bias heterogeneity (right column). Here the network sizes ranged from N=2 (top row) and N=3 (middle row), to N=4 (bottom row); we ensured N distinct cells were selected for each network. See section 4 for how random samples were generated. In almost all cases, a quadratic regression fit shows an optimal level of heterogeneity in the domain; the exceptions are for postspike filter heterogeneity for N=2,3,4 (middle column).

Figure A5:

Similar to Figure A4 but with the remaining network sizes (N=5,6,7). The Hessian-weighted error as a function of all three forms of heterogeneity: stimulus filter (left column), postspike filter (middle column), and bias heterogeneity (right column). Here the network sizes ranged from N=5 (top row) and N=6 (middle row), to N=7 (bottom row); we ensured N distinct cells were selected for each network. See section 4 for how random samples were generated. In all but one case, a quadratic regression fit shows an optimal level of heterogeneity in the domain; the exception is with postspike filter heterogeneity for N=5 (middle column, top row).

Figure A5:

Similar to Figure A4 but with the remaining network sizes (N=5,6,7). The Hessian-weighted error as a function of all three forms of heterogeneity: stimulus filter (left column), postspike filter (middle column), and bias heterogeneity (right column). Here the network sizes ranged from N=5 (top row) and N=6 (middle row), to N=7 (bottom row); we ensured N distinct cells were selected for each network. See section 4 for how random samples were generated. In all but one case, a quadratic regression fit shows an optimal level of heterogeneity in the domain; the exception is with postspike filter heterogeneity for N=5 (middle column, top row).

Figure A6:

When using Pearson's correlation after prewhitening as a measure of decoding accuracy, there is an optimal intermediate level of heterogeneity for all types, for N=2,3,4, with only one exception: bias heterogeneity for N=2 (right column, top row).

Figure A6:

When using Pearson's correlation after prewhitening as a measure of decoding accuracy, there is an optimal intermediate level of heterogeneity for all types, for N=2,3,4, with only one exception: bias heterogeneity for N=2 (right column, top row).

Figure A7:

Similar to Figure A6 but with larger network sizes (N=5,6,7). There is an optimal level of heterogeneity in the domain in all cases except for postspike filter heterogeneity with N=5 (middle column, top row).

Figure A7:

Similar to Figure A6 but with larger network sizes (N=5,6,7). There is an optimal level of heterogeneity in the domain in all cases except for postspike filter heterogeneity with N=5 (middle column, top row).

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