The mathematical principles of reinforcement (MPR) constitute of a set of mathematical equations set forth by Peter Killeen and his colleagues attempting to describe and predict the most fundamental aspects of behavior (Killeen & Sitomer, 2003). The three key principles of MPR, arousal, constraint, and coupling, describe how incentives motivate responding, how time constrains it, and how reinforcers become associated with specific responses, respectively. Mathematical models are provided for these basic principles in order to articulate the necessary detail of actual data.
First principle: arousal The first basic principle of MPR is arousal. Arousal refers to the activation of behavior by the presentation of incentives. An increase in activity level following repeated presentations of incentives is a fundamental aspect of conditioning. Killeen, Hanson, and Osborne (1978) proposed that adjunctive (or schedule induced) behaviors are normally occurring parts of an organism's repertoire. Delivery of incentives increases the rate of adjunctive behaviors by generating a heightened level of general activity, or arousal, in organisms. Killeen & Hanson (1978) exposed pigeons to a single daily presentation of food in the experimental chamber and measured general activity for 15 minutes after a feeding. They showed that activity level increased slightly directly following a feeding and then decreased slowly over time. The rate of decay can be described by the following function:
b ( t ) = b 1 × e − t α {\displaystyle b(t)=b_{1}\times e^{\frac {-t}{\alpha }}}
b1 = y-intercept (responses per minute) t = time in seconds since feeding
α {\displaystyle \alpha } = time constant e = base of natural logarithm The time course of the entire theoretical model of general activity is modeled by the following equation:
R = A × ( e − t C − e − t I ) {\displaystyle R=A\times (e^{\frac {-t}{C}}-e^{\frac {-t}{I}})}
A = arousal I = temporal inhibition C = competing behaviors To better conceptualize this model, imagine how rate of responding would appear with each of these processes individually. In the absence of temporal inhibition or competing responses, arousal level would remain high and response rate would be depicted as an almost horizontal line with a very small negative slope. Directly following food presentation, temporal inhibition is at its maximum level. It decreases quickly as time elapses, and response rate would be expected to increase up to the level of arousal in a short time. Competing behaviors such as goal tracking or hopper inspection are at a minimum directly after food presentation. These behaviors increase as the interval elapses, so the measure of general activity would slowly decrease. Subtracting these two curves results in the predicted level of general activity. Killeen et al. (1978) then increased the frequency of feeding from daily to every fixed-time seconds. They showed that general activity level increased substantially from the level of daily presentation. Response rate asymptotes were highest for the highest rates of reinforcement. These experiments indicate that arousal level is proportional to rate of incitement, and the asymptotic level increases with repeated presentations of incentives. The increase in activity level with repeated presentation of incentives is called cumulation of arousal. The first principle of MPR states that arousal level is proportional to rate of reinforcement, A = a r {\displaystyle A=ar} , where: A= arousal level a= specific activation r= rate of reinforcement (Killeen & Sitomer, 2003).
Second principle: constraint An obvious but often overlooked factor when analyzing response distributions is that responses are not instantaneous, but take some amount of time to emit (Killeen, 1994). These ceilings on response rate are often accounted for by competition from other responses, but less often for the fact that responses cannot always be emitted at the same rate at which they are elicited (Killeen & Sitomer, 2003). This limiting factor must be taken into account in order to correctly characterize what responding could be theoretically, and what it will be empirically. An organism may receive impulses to respond at a certain rate. At low rates of reinforcement, the elicited rate and emitted rate will approximate each other. At high rates of reinforcement, however, this elicited rate is subdued by the amount of time it takes to emit a response. Response rate, b {\displaystyle b} , is typically measured as the number of responses occurring in an epoch divided by the duration of an epoch. The reciprocal of b {\displaystyle b} gives the typical measure of the inter response (IRT), the average time from the start of one response to the start of another (Killeen & Sitomer, 2003). This is actually the cycle time rather than the time between responses. According to Killeen & Sitomer (2003), the IRT consists of two subintervals, the time required to emit a response, δ {\displaystyle \delta } plus the time between responses, τ {\displaystyle \tau } . Therefore, response rate can be measured either by dividing the number of responses by the cycle time:
b = 1 δ + τ {\displaystyle b={\frac {1}{\delta +\tau }}} , or as the number of responses divided by the actual time between responses:
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