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## Results: 5

1.

Four examples of neurons from two different monkeys (top two rows: monkey 1, bottom two rows: monkey 2) for which goodness of fit appears to be poor when the standard KS test is used but revealed to be good when either the numerically estimated reference distribution or the analytically corrected rescaled time

*y*are used. First Column: firing rate, second column: KS plot, third column: Differential KS plots. Blue: standard KS test. Red: KS test with numerical simulation of reference distribution. Green: KS test with analytically corrected rescaled time*y*. As with the simulated spike trains of Figure 4, the KS and Differential KS plot biases are eliminated when either the rescaled ISI distribution (*z*) is simulated using the fitted model, or the analytically corrected rescaled time*y*is used.2.

Illustration of KS plot bias induced when a homogeneous Poisson process is discretized to a homogeneous Bernoulli process. (A) KS plot for various spike per bin probabilities

*p*. Blue:*p*= 0.2, green:*p*= 0.1, red:*p*= 0.04 (40 Hz at 1 msec discretization). The rescaled times are not uniformly distributed but have positive bias at rescaled ISIs close to 0 and negative bias at rescaled ISIs close to 1. (B) Differential KS Plot:*CDF*−_{uniform}*CDF*(*z*)_{bernoulli}. Biases are easier to see if the difference between the expected CDF (uniform) and the actual CDF of the rescaled times is plotted. The colors indicate the same spike per bin probabilities*p*as in A). The horizontal dashed lines are the 95% confidence region assuming a 10 minutes of a 40 Hz Bernoulli process (24000 spikes).3.

Example of two simple KS plots demonstrating that temporal discretization induces biases even if the conditional intensity function used to calculate the rescaled times is exactly correct. CDF of the rescaled times

*z*is plotted along the x-axis versus the CDF of the uniform (reference) distribution along the y-axis. Spikes were generated from an inhomogeneous Poisson process with a maximum firing rate of 50 Hz. Thick grey dashed line: KS plot of rescaled ISIs generated by a continuous time model. Thick grey line: KS plot of rescaled ISIs calculated from the same model discretized at 5 msec resolution. The discretization was deliberately enhanced to emphasize the effect. Thin black 45 degree lines are 95% confidence bounds upon the KS plots.4.

Comparison of standard KS test, KS test using simulated rescaled ISI distribution, and KS test using the analytically corrected rescaled time. Spike trains were simulated using the same three models as in Figure 2 at fine 10

^{−10}msec temporal precision and then discretized at Δ = 1 msec resolution. Logistic GLM models were fit and used to estimate the rescaled ISI distributions (See text.) (A), (C) and (E) KS plots for inhomogeneous Bernoulli, homogeneous Bernoulli with spike history and inhomogeneous Bernoulli with spike history respectively. (B), (D) and (F) Differential KS plots for the same. Blue lines correspond to the standard KS test which plots the CDF of the rescaled time*z*versus the CDF of the uniform distribution, red lines to the numerical simulation method which plots the CDF of the rescaled time*z*versus the CDF of the numerically simulated reference distribution, and green lines to the analytical method which plots the CDF of the analytically corrected rescaled time*y*versus the CDF of the uniform distribution. The red and green lines essentially overlap in the plots. For all three spike train models, strong KS and Differential KS plot bias was eliminated when either the numerically estimated distribution or the analytical correction were used.5.

KS and Differential KS plots for 10 minute long 40 Hz mean firing rate simulated spike trains. Three continuous time models of the conditional intensity function were used for simulation. 1) inhomogeneous Poisson process 2) homogeneous Poisson with a renewal spike history process 3) inhomogeneous Poisson with a renewal spike history process. (See text) The continuously defined processes were discretized at various values Δ and used to simulate spikes. (A) 40 Hz mean inhomogeneous Bernoulli firing rate. (B) Spike history term λ

_{hist}as a function of time since the most recent spike. (C) and (D) KS and Differential KS plots for inhomogeneous Bernoulli process. Blue: Δ = 1 msec, green: Δ = 0.5 msec, red: Δ = 0.1 msec. Horizontal dashed lines are 95% confidence bounds. (E) and (F) Homogeneous Bernoulli process with spike renewal history term. (G) and (H) Inhomogeneous Bernoulli process with spike renewal history term. Note that when spike history effects are present the biases are larger both at short and long rescaled ISIs.