Estimation of Parameters

The process of estimating the parameters can be described in the following four steps.

First, a simulation was done with the Matlab code in Appendix A that led to a simulated data file. The data from the simulation of scalar timing theory, like the data from the experiment, consisted of the times of lever responses and food deliveries of each rat on each session (see Appendix B). This simulation was based on the specific values for each of the parameters that were set in Initialize_SET. The parameters were the mean and standard deviation of the clock, the mean and standard deviation of the memory storage constant, and the mean and standard deviation of the threshold.

Second, a summary measure was selected. The same data that were shown in Figure 1.3 were replotted in relative units in Figure 1.5. In this figure, response rate is shown as responses per minute divided by the maximum response rate in a given condition; time is shown as time since food was delivered, divided by the fixed interval. The temporal gradients for the three fixed-interval schedules increased in a similar manner. If the relative response rates as a function of the relative time intervals were identical at different intervals, the result would be described in terms of superposition of the functions. Such superposition implies that the mean time of a response is linearly related to the mean interval (i.e., proportionality), the standard deviation of the time of a response is linearly related to the mean interval (i.e., the scalar property), and thus the coefficient of variation (the standard deviation divided by the mean) is a constant. Superposition on a relative scale provides complete information about the temporal gradients on all other intervals based on the gradient on any single interval. The same Matlab program that was used to obtain the relative response rate gradients for the data (shown with the symbols identified in the legend) was used to obtain the relative response rate gradients for the simulation (shown as the solid line).

Third, a goodness-of-fit measure was used to compare the predictions of the theory with the observations from the data. The standard goodness-of-fit measure is m2, the percentage of variance accounted for. The sum of squared deviations of the data points from the mean of the data is the total variability; the sum of squared deviations of the data points from the predictions of the model is the unexplained variability. Thus, the explained variability is the difference between the total variability and the unexplained variability. m2 is the ratio of the explained variability to the total variability.

Fourth, modifications were made in the values of the parameters to optimize percentage of explained variability. The systematic way of conducting this search is by an exhaustive search of all combinations of parameter values at some resolution, or by a hill-climbing algorithm used for nonlinear fitting problems. The values shown in the program are those that were used for the solid line in Figure 1.5. The mean pacemaker rate was set at five pulses per second, with no variability; the mean memory constant was set at 1.0, with an estimate from the data of memory variability; the mean threshold was estimated from the data, with no threshold variability. Only two parameters were varied for the predictions shown in Figure 1.5: the standard deviation of memory was estimated to be about 0.2, and the mean threshold was estimated to be about 0.1. This accounted for about 96% of the variance.

FIGURE 1.5 Relative response rate as a function of relative time since food for fixed-interval schedules of 30, 60, and 120 sec. Relative response rate is the mean number of responses per second, divided by the maximum mean response rate; relative time is the time since stimulus onset divided by the fixed interval. These are the same data shown in Figure 1.3, plotted in relative rather than absolute rate and time. The line is based on the simulation of scalar timing theory.

Relative time since food

FIGURE 1.5 Relative response rate as a function of relative time since food for fixed-interval schedules of 30, 60, and 120 sec. Relative response rate is the mean number of responses per second, divided by the maximum mean response rate; relative time is the time since stimulus onset divided by the fixed interval. These are the same data shown in Figure 1.3, plotted in relative rather than absolute rate and time. The line is based on the simulation of scalar timing theory.

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