Simulations

The results of a set of four simulations are presented in Figure 20.3. These simulations varied the values of the gain and decay parameters, but held constant the number of nodes in the accumulator. In the low-gain conditions, the gain parameter was set to 0.05, while in the high-gain conditions, the gain parameter was set to 0.1. In the low-decay conditions, the decay parameter was set to 0.0001, and in the high-decay conditions, the decay parameter was set to 0.0002. The number of accumulator nodes was fixed at 1000 for all four conditions. Each point on each of the four functions represents the means and standard errors of 1000 runs of the model, or iterations of the j index loop in the Appendix A code. These results are model psychophysical functions that relate the passage of real time (with arbitrary scale), represented by the occurrence of pacemaker pulses, to subjective time, represented as the total number of nodes active after each pacemaker pulse.

The basic performance characteristics of the model are evident in Figure 20.3. The most obvious is the curvilinear relationship between the passage of real time, as indicated by the pulsing of the pacemaker, and the value of subjective time indexed by the mean total number of active accumulator nodes. This is the result of the countervalent influence of the gain and decay parameters. Early on in the timing process, the gain parameter is the main determinant of accumulator activity because of its higher value and the availability of a large population of nodes currently in the off state. This can be seen in Figure 20.3, where the two functions with the higher gain value rise more steeply early in the interval than the two functions with the lower gain value. As more nodes become active over time, the cumulative probability of nodes' transitioning to the off state increases, thereby decelerating the per-pulse increase in active nodes. As the number of active nodes approaches the practical asymptote defined by the total number of accumulator nodes and the decay parameter, the process reaches dynamical equilibrium. This occurs because the few nodes that offset quickly turn back on due to the fact that the gain parameter is greater than the decay parameter. This feature can be seen in Figure 20.3, where the two functions with a lower decay parameter reach a higher asymptotic number of active nodes than the pair of functions with the higher decay parameter.

-•— High Gain-High Decay —■— High Gain-Low Decay

-•— High Gain-High Decay —■— High Gain-Low Decay

FIGURE 20.3 The results of four runs of 1000 simulations using Miall's accumulator model. Each function depicts the total number of active accumulator nodes as a function of the number of pacemaker pulses, for one of the four runs. Runs differed by the value of the gain and decay parameters. The two functions with high gains are indicated with solid lines. The gain parameter was set to 0.1 in the high-gain conditions. The two functions with low gains are indicated with dashed lines and employed a gain of 0.05. The two functions with high decays are indicated with circular markers. The decay parameter was set to 0.0002 in the high-decay conditions. The two functions with low decays are indicated with square markers and employed a gain of 0.0001.

Pacemaker Pulses

FIGURE 20.3 The results of four runs of 1000 simulations using Miall's accumulator model. Each function depicts the total number of active accumulator nodes as a function of the number of pacemaker pulses, for one of the four runs. Runs differed by the value of the gain and decay parameters. The two functions with high gains are indicated with solid lines. The gain parameter was set to 0.1 in the high-gain conditions. The two functions with low gains are indicated with dashed lines and employed a gain of 0.05. The two functions with high decays are indicated with circular markers. The decay parameter was set to 0.0002 in the high-decay conditions. The two functions with low decays are indicated with square markers and employed a gain of 0.0001.

It should be noted that no attempt to fix the crossover point at any particular point in the x-axis or at any particular point in the functions' curvatures has been attempted. Rather, the figure and data presented represent a direct replication of Miall's simulations along with a doubling of the gain and decay parameters. These were the first values chosen by us for investigation. Because the x-axis is essentially timescale invariant, that is, the rate of the pacemaker is arbitrary, and because the parameter space is effectively unbounded, it follows that the model can be made to produce any set of curvatures and crossover points at any point in the sequence of pacemaker pulses.

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