Herrnstein's Hyperbola Research Articles

Characterizing operant hyperactivity in the Spontaneously Hypertensive Rat

BackgroundOperant hyperactivity, the emission of reinforced responses at an inordinately high rate, has been reported in children with ADHD and in the Spontaneously Hypertensive Rat (SHR), the most widely studied animal model of ADHD. The SHR emits behavior at hyperactive levels, relative to a normoactive strain, only when such behavior is seldom reinforced. Because of its dependence on rate of reinforcement, operant hyperactivity appears to be driven primarily by incentive motivation, not motoric capacity. This claim was evaluated in the present study using a novel strategy, based on the organization of behavior in bouts of reinforced responses separated by pauses.MethodMale SHR, Wistar-Kyoto (WKY) and Wistar rats (WIS) were exposed each to a multiple variable-interval schedule of sucrose reinforcement (12, 24, 48, 96, and 192 s) between post-natal days (PND) 48 and 93. Responding in each schedule was examined in two epochs, PND 58-62 and 89-93. Parameters of response-reinforcement functions (Herrnstein's hyperbola) and bout-organized behavior were estimated in each epoch.ResultsSHR emitted higher response rates than WKY and WIS, but only when rate of reinforcement was low (fewer than 2 reinforcers per minute), and particularly in the second epoch. Estimates of Herrnstein's hyperbola parameters suggested the primacy of motivational over motoric factors driving the response-rate differential. Across epochs and schedules, a more detailed analysis of response bouts by SHR revealed that these were shorter than those by WKY, but more frequent than those by WKY and WIS. Differences in bout length subsided between epochs, but differences in bout-initiation rate were exacerbated. These results were interpreted in light of robust evidence linking changes in bout-organization parameters and experimental manipulations of motivation and response-reinforcement contingency.ConclusionsOperant hyperactivity in SHR was confirmed. Although incentive motivation appears to play an important role in operant hyperactivity and motoric capacity cannot be ruled out as a factor, response-bout patterns suggest that operant hyperactivity is primarily driven by steeper delay-of-reinforcement gradients. Convergence of this conclusion with theoretical accounts of ADHD and with free-operant performance in children with ADHD supports the use of SHR as an animal model of ADHD.

The sampling distribution of k

Research on Herrnstein's single-schedule equation contains conflicting findings; some laboratories report variations in the k parameter with reinforcer value, and others report constancy. The reported variation in k typically occurs across very low reinforcer values, and constancy applies across higher values. Here, simulations were conducted assuming a wide range of reinforcer values, and the parameters of Herrnstein's equation were estimated for simulated responding. In the simulations, responses controlled by current reinforcement contingencies were added to other responses ('noise'), controlled by the experimental environment and by contingencies in effect at other times. Expected reinforcer rates were calculated by entering simulated responding into a reinforcement feedback function. These were then fitted using Herrnstein's hyperbola, and the sampling distributions of the two fitted parameters were studied. Both k and Re were underestimated by curve fitting when low-deprivation or reinforcer-quality conditions were simulated. Further simulations showed that k and Re were increasingly underestimated as the assumed noise level was increased, particularly when low-deprivation or reinforcer quality was assumed. It is concluded that reported variations in k from single schedules should not be taken to indicate that the asymptotic rate of responding depends on reinforcement parameters.

FEEDBACK FUNCTIONS, OPTIMIZATION, AND THE RELATION OF RESPONSE RATE TO REINFORCER RATE

The present experiment arranged a series of inverted U-shaped feedback functions relating reinforcer rate to response rate to test whether responding was consistent with an optimization account or with a one-to-one relation of response rate to reinforcer rate such as linear system theory's rate equation or Herrnstein's hyperbola. Reinforcer rate was arranged according to a quadratic equation with a maximum at a unique response rate. The experiment consisted of two phases, during which 6 Long Evans rats lever pressed for food. In the first phase of the experiment, the rats responded on six fixed-interval-plus-quadratic-feedback schedules, and in the second phase the rats responded on three variable-interval-plus-quadratic-feedback schedules. Responding in both phases was inconsistent with a one-to-one relation of response rate to reinforcer rate. Instead, different response rates were obtained at equivalent reinforcer rates. Responding did vary directly with the vertex of the feedback function in both phases, a finding consistent with optimization of reinforcer rate. The present results suggest that the feedback function relating reinforcer rate to response rate imposed by a reinforcement schedule can be an important determinant of behavior. Furthermore, the present experiment illustrates the benefit of arranging feedback functions to investigate assumptions about the variables that control schedule performance.

EFFECTS OF ADDING A SECOND REINFORCEMENT ALTERNATIVE: IMPLICATIONS FOR HERRNSTEIN'S INTERPRETATION OF re

Herrnstein's hyperbola describes the relation between response rate and reinforcer rate on variable-interval (VI) schedules. According to Herrnstein's (1970) interpretation, the parameter r(e) represents the reinforcer rate extraneous to the alternative to which the equation is fitted (the target alternative). The hyperbola is based on an assumption that extraneous reinforcer rate remains constant with changes in reinforcer rate on the target alternative (the constant-r(e) assumption) and that matching with no bias and perfect sensitivity occurs between response and reinforcer ratios. In the present experiment, 12 rats pressed levers for food on a series of 10 VI schedules arranged on the target alternative. Across conditions, six VI values and extinction were arranged on a second alternative. Reinforcer rate on the second alternative, r2, negatively covaried with reinforcer rate on the target alternative for five of the six VI values on the second alternative, and significant degrees of bias and undermatching occurred in response ratios. Given covariation of reinforcer rate on the second and target alternatives, the constant-r(e) assumption can be maintained only by assuming that reinforcer rate from unmeasured background sources, rb, covaries with reinforcer rate on the second alternative such that their sum, r(e), remains constant. In a single-schedule arrangement, however, r(e) equals rb and thus rb is assumed to remain constant, forcing a conceptual inconsistency between single- and concurrent-schedule arrangements. Furthermore, although an alternative formulation of the hyperbola can account for variations in bias and sensitivity, the modified equation also is based on the constant-r(e) assumption and therefore suffers from the same logical problem as the hyperbola when reinforcer rate on the second alternative covaries with reinforcer rate on the target alternative.

Fencamfamine-induced changes in the parameters of the matching law and the effect of previous exposure to the drug

The purpose of this study was to assess the effects of the psychomotor stimulant fencamfamine (FCF) on the operant response rate and on the parameters k and R(e) of Herrnstein's hyperbola. In the first experiment, rats were trained on a multiple variable interval (VI) schedule. Each rat was then given two i.p. injections of a randomly assigned FCF dose (0.0, 0.88, 1.75 and 3.5 mg/kg) at 48 h intervals. The overall response rate, the individual VI response rate, and the k and R(e) parameters of the hyperbola equation were calculated. In the second experiment, two groups of animals were trained drug-free on the same schedule. Afterwards, the animals in one group received six i.p. injections of 1.75 mg/kg FCF at 48 h intervals while subjects in the other group received vehicle injections, 15 min before each experimental session. Seven days after the last injection, a challenge dose (0.88 mg/kg) of FCF was administered and performance was measured. FCF increased overall response rate and decreased R(e) without changing k. Previous exposure to FCF did not induce sensitization of these measures. These results indicate that FCF has similar effects on k and R(e) as those produced by other indirect dopaminergic agents, and supports previous findings suggesting FCF as a potential drug of abuse.

Herrnstein's hyperbolic matching equation and behavioral pharmacology: review and critique.

Behavioral pharmacologists have enlisted Herrnstein's (1970) hyperbolic matching equation to understand the behavioral effects of drugs. Herrnstein's hyperbola describes the relation between absolute response rate and reinforcement rate. The equation has two fitted parameters. The parameter k represents the asymptotic response rate, and the parameter r(e) represents the reinforcement rate necessary to obtain half the asymptotic response rate. According to one interpretation of the equation, changes in k should reflect changes in response or motoric variables, and changes in r(e) should reflect changes in reinforcer or motivational variables, or changes in reinforcement from sources extraneous to the instrumental response. We review research that has applied Herrnstein's equation to distinguish the motoric from the motivational effects of drugs, and to identify additional independent variables responsible for drug effects, such as extraneous reinforcement. The validity of inferences about drug effects depends on the consistency of how k and r(e) respond to environmental manipulations: k should change only with response or motoric variables, and r(e) should change with reinforcer or motivational variables and with the rate of extraneous reinforcement. Empirical tests of these predictions, however, have produced inconsistent results. The review suggests that Herrnstein's theory has not fulfilled its promise of identifying the behavioral mechanisms of drug action. Modifications to the equation, known as bias and sensitivity, may explain some of these inconsistent results, and the modified equation may have utility in behavioral pharmacology.

Falsification of matching theory: changes in the asymptote of Herrnstein's hyperbola as a function of water deprivation.

Five rats pressed levers on variable-interval schedules of water reinforcement at various levels of water deprivation. In one phase of the experiment, three deprivation conditions that replicated conditions in Heyman and Monaghan (1987) were arranged, along with three less extreme deprivation conditions. In a second phase, water deprivation was arranged so that subjects were exposed to a greater range of access to water per day. Herrnstein's hyperbola described the rats' response-rate data well. The y asymptote, k, of the hyperbola appeared roughly constant over the conditions that replicated those of Heyman and Monaghan, but decreased markedly when less extreme deprivation conditions were included. In addition, k varied systematically when the second method of arranging deprivation was used. These results falsify a strong form of matching theory and confirm predictions made by linear system theory.

The effects of morphine on responding under variable-interval schedules

The study was designed to integrate rate-related principles and Herrnstein's hyperbolic matching equation (1970) as they apply to drug effects. It was also designed to assess the validity of an interpretation of Herrnstein's equation used to account for the behavioral mechanisms of drug action. Eight rats were trained to press a lever under a range of variable-interval (VI) schedules of reinforcement using sucrose solution (0.1 M) as a reinforcer. After 90 baseline sessions the effect of morphine sulfate, 3, 6, 12 and 17 mg/kg, on responding was tested. Response rates were suppressed in a dose-dependent manner after morphine administration. Herrnstein's equation, which states that responding is a function of relative reinforcement rate, provided a good description of the relationship between responding and reinforcement under most conditions. Therefore, the traditional variable used to predict the drug effect in rate-dependency functions, baseline response rate, was compared to a new variable, baseline relative reinforcement rate. Relative reinforcement rate performed as well as response rate in predicting the drug effect. In order to test if Herrnstein's hyperbola can identify the behavioral mechanisms of the action of morphine, the concentration of the sucrose solution was increased (0.32 M). The y-asymptote of Herrnstein's equation, k, increased as sucrose concentration increased. This calls into question one common interpretation of Herrnstein's equation used to identify the behavioral mechanisms of drug action.

APPLICATION OF HERRNSTEIN'S HYPERBOLA TO TIME ALLOCATION OF NATURALISTIC HUMAN BEHAVIOR MAINTAINED BY NATURALISTIC SOCIAL REINFORCEMENT

Five college students talked to an experimenter about various topics. Time spent looking at the experimenter was reinforced by verbal statements of praise and interest on five variable-interval schedules. Herrnstein's hyperbola provided a good description of the time-allocation data for 4 of the 5 subjects, and accounted for 95% of the variance of the median time-allocation data. The hyperbola provided a significantly better description of the data than a two-parameter ramp function with similar differential properties. Estimates of the asymptote, k, of the hyperbola varied among subjects from about 2 to about 15 seconds of eye contact per minute. These estimates were much smaller than the constant 60 seconds of eye contact per minute required by Herrnstein's matching theory. These results support the conclusion that Herrnstein's hyperbola describes naturalistic human behavior maintained by naturalistic social reinforcement as well as it describes the behavior of humans and nonhumans in typical laboratory preparations. The results also indicate that the hyperbolic form of the time-allocation version of Herrnstein's equation is accurate, but that the constant k requirement of matching theory may not hold.

CONFIRMATION OF LINEAR SYSTEM THEORY PREDICTION: CHANGES IN HERRNSTEIN'S k AS A FUNCTION OF CHANGES IN REINFORCER MAGNITUDE

Eight human subjects pressed a lever on a range of variable-interval schedules for 0.25 cent to 35.0 cent per reinforcement. Herrnstein's hyperbola described seven of the eight subjects' response-rate data well. For all subjects, the y-asymptote of the hyperbola increased with increasing reinforcer magnitude and its reciprocal was a linear function of the reciprocal of reinforcer magnitude. These results confirm predictions made by linear system theory; they contradict formal properties of Herrnstein's account and of six other mathematical accounts of single-alternative responding.

AN ANALYTIC COMPARISON OF HERRNSTEIN'S EQUATIONS AND A MULTIVARIATE RATE EQUATION

Herrnstein's equations are approximations of the multivariate rate equation at ordinary rates of reinforcement and responding. The rate equation is the result of a linear system analysis of variable-interval performance. Rate equation matching is more comprehensive than ordinary matching because it predicts and specifies the nature of concurrent bias, and predicts a tendency toward undermatching, which is sometimes observed in concurrent situations. The rate equation contradicts one feature of Herrnstein's hyperbola, viz., the theoretically required constancy of k. According to the rate equation, Herrnstein's k should vary directly with parameters of reinforcement such as amount or immediacy. Because of this prediction, the rate equation asserts that the conceptual framework of matching does not apply to single alternative responding. The issue of the constancy of k provides empirical grounds for distinguishing between Herrnstein's account and a linear system analysis of single alternative variable-interval responding.

A MULTIVARIATE RATE EQUATION FOR VARIABLE‐INTERVAL PERFORMANCE

A value-like parameter is introduced into a rate equation for describing variable-interval performance. The equation, derived solely from formal considerations, expresses rate of responding as a joint function of rate of reinforcement and "reinforcer power." Preliminary tests of the rate equation show that it handles univariate data as well as Herrnstein's hyperbola. In addition, a form of Herrnstein's hyperbola can be derived from the equation, and it predicts forms of matching in concurrent situations. For the multivariate case, reinforcer values scaled in concurrent situations where matching is assumed to hold are taken as determinations of reinforcer power. The multivariate rate equation is fitted to an appropriate set of data and found to provide a good description of variable-interval performance when both rate and power of reinforcement are varied. Rate and power measures completely describe reinforcement. The effects of their joint variation are not predicted and cannot be described by Herrnstein's equation.