## The Principles of Scalar Timing Theory

Some of the principles of scalar timing theory apply to many different procedures. In an article entitled "Origins of Scalar Timing Theory," John Gibbon provided a historical review of the theory (Gibbon, 1991). In the 1950s and 1960s various researchers reported a linear relationship between the mean latency of a response and the interval between stimulus and reinforcement, but the theoretical importance of this empirical regularity was not recognized at the time. This regularity is now referred to as proportional timing. Subsequently, a linear relationship between the standard deviation of the latency of a response and the interval between the stimulus and reinforcement was identified. This regularity is now referred to as the scalar property. Then it was found that the coefficient of variation (the ratio of the standard deviation to the mean) was approximately constant (Catania, 1970), a result that was logically implied by the linearity of the standard deviation and the mean. This regularity is now referred to as Weber's law for timing. The most general rule was superposition — the finding that the response rate as a function of time was approximately the same at all intervals. This result was clearly shown by an examination of the function relating response rate to time since reinforcement in fixed-interval schedules of reinforcement of pigeons of 3, 30, and 300 sec (Dews, 1970). This regularity, which logically implies scalar variance and Weber's law, is now referred to as timescale invariance. The principles of proportionality, scalar variance, Weber's law, and timescale invariance apply to a wide range of measures of timing.

For example, consider a two-response temporal discrimination procedure between temporal intervals between the onset and termination of a stimulus of 2 or 8 sec; one response is reinforced following a 2-sec stimulus and a different response is reinforced following an 8-sec stimulus. Unreinforced probe stimuli may be presented for intermediate durations, and the proportion of stimuli to which the rat responds with the "long" response can be the dependent variable (Church and Deluty, 1977). The time at which the rat is equally likely to press the "short" or "long" response (the point of bisection) increases approximately linearly with the geometric mean of the reinforced short and long responses (proportional timing); the standard deviation of this point of bisection increases approximately linearly with stimulus duration (scalar variability). Thus, the ratio of standard deviation to the mean, which is called the coefficient of variation, is approximately constant (Weber's law), and the psychophysical functions at all ranges approximately superimpose when the duration of stimulus is divided by the geometric mean (timescale invariance).

Similar principles apply to a single-response temporal discrimination procedure known as the peak or peak-interval procedure. A peak procedure includes two types of cycles: those with a reinforcer and those without (for an example, see Church et al., 1994). On some proportion of the cycles (normally half), food is delivered immediately after the first response following a fixed interval after stimulus onset (a discriminative fixed-interval schedule); on the other cycles, there is no food and the stimulus terminates after a long presentation of the stimulus (normally four times the duration of the fixed interval). A measure of the temporal discrimination is the time of the maximum response rate, which is called the peak time. The peak time increases linearly with the fixed interval (proportional timing), the standard deviation of the peak time also increases linearly with the fixed interval (scalar property), the coefficient of variation of the peak time is approximately constant (Weber's law), and the response rate as a function of time since stimulus onset is approximately the same at all stimulus intervals when response rate is scaled in terms of proportion of maximum response rate, and time is scaled in proportion of the fixed interval (timescale invariance).

Similar principles also apply to fixed-interval schedules of reinforcement. The time of the median response increases approximately linearly with the time of the reinforced interval (proportional timing); the standard deviation of the median response also increases approximately linearly with stimulus duration (scalar property); the coefficient of variation of the time of the median response is approximately constant (Weber's law); and at a wide range of fixed intervals, the response rate relative to the maximum response rate superposes when the time since stimulus onset is divided by the fixed interval (superposition).

Other principles of scalar timing theory are specific to some class of procedures. For example, the shape of the function relating the probability of a long response to the duration of the stimulus in the temporal bisection procedure is approximately symmetrical on a logarithmic scale of time; but the shape of the function relating the response rate to the time since stimulus onset in the peak procedure is approximately symmetrical on a linear scale of time. The time at which the rat is equally likely to press the short or long response (the point of bisection) is approximately at the geometric mean between the durations of the long and short reinforced stimuli in the two-response bisection procedure. A goal of the information-processing model was to explain such results as proportionality, the scalar rule, constant coefficient of variation, superposition, bisection at the geometric mean, and other results, such as preference for more variable intervals when the mean interval was fixed.

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