In praise of the hyperbola

Abstract

Following a recent review by Killeen (2024) on the utility of hyperbolic functions in the experimental analysis of behavior, this article discusses recent developments of the multiplicative hyperbolic model (MHM) of reinforcer value, according to which “value” is defined as a dimensionless intervening variable derived from the multiplicative combination of separate hyperbolic functions that represent the influence of particular features of a reinforcing event (e.g. quantity, immediacy). The data reviewed here pertain to the choice behavior of rats responding for soluble reinforcers, such as sucrose solutions. Three main areas are covered: the determinants of the maximal value attainable with a particular reinforcer, the incorporation of concentration as a hyperbolic term in the definition of value, and the values of mixtures of appetitive and aversive tastants. It is argued that MHM is fundamentally a descriptive model; value is dimensionless, and its subordinate intervening variables, which define the individual hyperbolic functions, are either dimensionless or share the same scale of measurement as their respective independent variables. Therefore, it would be a mistake to identify any part of the model with any underlying biological entity. A comparison is drawn between MHM and its algebraically close analogue, classical pharmacological receptor theory.

INTRODUCTION

It is a great pleasure to make a contribution to this special issue of JEAB, honoring the contribution of Peter Killeen to the experimental analysis of behavior. As well as being a personal friend for more than 40 years, Peter made a great contribution to the behavioral research carried out by the team of which I was a member, adding his theoretical insights to our empirical findings and proffering wise advice on the presentation of those findings. His influence on the work described in this article has been immeasurable; the deficiencies of the article are, of course, mine alone.

Killeen (2024) recently published a review of the use of hyperbolic functions in the quantitative analysis of behavior. Two major themes explored in his review are the hyperbolic response‐strength equation of Herrnstein (1970, 1974) and hyperbolic models of incentive value (e.g., Mazur, 1987; Myerson and Green, 1995; Rachlin, 2006). Both themes are relevant to the work undertaken by our team in Manchester and more recently in Nottingham, UK, which I have tried to continue since my retirement in 2010. A unifying concept underlying this work is a mathematical model to which we gave the name multiplicative hyperbolic model (MHM; Ho et al., 1999). The model has undergone several changes since its inception 30‐odd years ago, but its founding principle remains more or less intact. A review of the theoretical basis of MHM, its relation to Herrnstein's hyperbolic response strength model, and its application to combinations of appetitive and aversive stimuli (e.g. punishment and conflict paradigms) has been offered by Bradshaw (2019), and some neurobehavioral applications of MHM have been reviewed by Valencia‐Torres et al. (2013) and Body et al. (2017). The present article starts with a summary of the origins of MHM and its early development and then focuses on work undertaken in the last 5 years that has examined some of the factors that determine the effectiveness of soluble food reinforcers in rats.

ORIGINS OF MHM

MHM started life as a modification of Mazur's (1987) hyperbolic model of delay discounting (hereafter, the standard hyperbolic model [SHM]), which proposes that the value of a delayed reinforcer (V _d) may be expressed as

$$V_d=\frac{V_i}{1+K.d}$$ (1)

where V _i is the value of the reinforcer when delivered immediately, d is the delay interposed between the reinforced response and the delivery of the reinforcer, and K is a delay‐discounting parameter, expressed in reciprocal time units. Making the simplifying assumption of direct proportionality between the size or quantity of the reinforcer (q) and its instantaneous value, Equation 1 may be written as

$$V_d=\frac{q}{1+K.d}$$ (2)

It is apparent from Equation 2 that V _d must be expressed in the same physical units of quantity as q, for example mg or μl.

An invaluable tool in the development of SHM, and more recently MHM, is the null equation. If two reinforcers, A and B, are compared in a free‐choice schedule and the properties of one reinforcer are systematically adjusted until the two reinforcers are selected with equal frequency, it is assumed that their values are equal: V _A = V _B. This equality may be expanded using the appropriate value equation (e.g. Equation. 1), allowing the relevant subordinate intervening variable (e.g., K) to be derived. Mazur's (1987) adjusting‐delay procedure provides the classic example of the method. Pigeons were trained to choose between two delayed reinforcers. The delay to the smaller reinforcer (B), d _B, was kept constant within each phase of the experiment, whereas the delay to the larger reinforcer (A), d _A, was varied in response to the choices made by the subject in successive blocks of trials. If A was preferred in block n, d _A was increased in block n + 1; if B was preferred, d _A was reduced in block n + 1. Training was continued until a stable indifference delay was obtained (d _A(50)). The procedure was repeated using a range of delays to reinforcer B, and plots of d _A(50) versus d _B were derived. On the assumption that V _A = V _B , Equation 2 may be used to derive Equation 3, the null, or “indifference,” equation:

$$d_{A\left(50\right)}=\frac{1}{K}.\frac{V_{\mathit{iA}}-V_{\mathit{iB}}}{V_{\mathit{iB}}}+d_B.\frac{V_{\mathit{iA}}}{V_{\mathit{iB}}}$$ (3)

Assuming that instantaneous value is proportional to reinforceer magnitude,

$$d_{A\left(50\right)}=\frac{1}{K}.\frac{q_A-q_B}{q_B}+d_B.\frac{q_A}{q_B}$$ (4)

(Mazur, 1987).

Mazur (1987) emphasized that the assumption of proportionality between V _i and q should be regarded as a convenient simplification of an unknown, although necessarily monotonic, relation between V and q (see also Mazur and Herrnstein, 1988), but as the data from his experiments were consonant with Equation 4, there was no reason for Mazur to reject this parsimonious assumption. However, subsequent findings have cast some doubt on the assumption.

Equation 4 specifies that d _A(50) is linearly related to d _B and that, provided that the relative sizes of the two reinforcers are kept constant, the indifference delay to the larger reinforcer should be impervious to variation of the absolute sizes of the reinforcers. However, Bradshaw and Szabadi (1992) obtained data that did not support this prediction. In an experiment in which rats made choices between different volumes of a sucrose reinforcer, the indifference delay to the larger volume (d _A(50)) declined when q _A and q _B were increased in the same proportion. To accommodate this finding, the authors suggested the following modification of Equation 1:

$$V=\frac{1}{1+K/i}.\frac{1}{1+Q/q},$$ (5)

where Q is a size‐sensitivity parameter expressed in the same units as q ¹ and i (immediacy) is the reciprocal of d. (This convention, adopted by Bradshaw (2019), illustrates the homologous relations between V and the two features of the reinforcing stimulus–size and immediacy (see Figure 1).) Both terms of the right‐hand side of Equation 5 are dimensionless; thus, V is also dimensionless. This confers some conceptual benefit upon MHM in that V becomes simply a number between 0 and 1 that summarizes the effects of the specified features of the reinforcer, without implying any questionable psychological interpretation of this parameter (Bradshaw, 2019; Ho et al., 1999). In this respect, V in MHM is consonant with Rachlin's (2006) assertion that value is a relative (dimensionless) quantity.

FIGURE 1.

Hyperbolic relations between value (V) and two properties of a reinforcer. Panel A: Relation between V and delay of reinforcement (d), as in Equation 2. The two curves, a and b, show the relation for two values of the delay‐discounting parameter, K. Panel B: Delay is expressed as its reciprocal (immediacy of reinforcement, i), as in Equation 5. Panel C: Relation between V and the quantity of the reinforcer (q: Equation 5), for two values of Q. Panel D: Relation between V and q in the absence (dashed line) and presence (dotted line) of a delay to reinforcement. Note that the hyperbolic relation between V and q is preserved when the hyperbolic immediacy term is introduced into the value equation. The vertical reference lines in graphs A, B, and C indicate the values of the parameters K and Q.

The incorporation of Q into the value equation (Equation 5) has obvious implications for the use of the indifference method for measuring the delay‐discounting parameter, K. Substitution of Equation 5 into the linear indifference equation (Equation 4) yields.

$$d_{A\left(50\right)}=\frac{1}{K}.\frac{Q/q_A-Q/q_B}{1+Q/q_B}+d_B.\frac{1+Q/q_A}{1+Q/q_B}$$ (6)

Figure 2 shows plots derived from Equation 6. It is apparent that a change in the rate of delay discounting (K) is reflected in a parallel shift in the linear indifference function, an increase in K resulting in a downward displacement, and a reduction of K in an upward displacement of the function (Figure 2A). In contrast, a change in size sensitivity (Q) is reflected in a change in both the slope and the intercept of the indifference function, a reduction in Q resulting in a reduced intercept and a flattened slope, and an increase of Q in an elevated intercept and a steepened slope (Figure 2B). Both patterns of change of the indifference function have been found to result from various neurobiological interventions. For example, lesions of the nucleus acumbens core resulted in a selective increase in the value of K (indicating an increase in the rate of delay discounting (Bezzina et al., 2007), whereas lesions of the subthalamic nucleus induced a reduction of the value of Q (indicating an enhancement of the value, V, of any given quantity of the food reinforcer (Bezzina, den Boon, et al., 2008). It is, of course, possible for the two patterns of effect to coexist, and indeed, destruction of the orbital prefrontal cortex (Kheramin et al., 2002) and disconnection of the orbital prefrontal cortex from the nucleus acumbens core (Bezzina, Body, et al., 2008) have been found to produce changes in the values of both K and Q (for review, see Body et al., 2017; Valencia‐Torres et al., 2013).

FIGURE 2.

Influence of K and Q on the linear indifference function for delay discounting (Equation 5). Ordinates: indifference delay to Reinforcer A (d _A(50)); abscissae: delay to Reinforcer B. Broken lines indicate the baseline condition; continuous lines show the change brought about by an intervention. Panel A: A reduction of the intercept is indicative of an increase of the rate of delay discounting (K); an increase of the intercept indicates a reduction of K. Panel B: A decrease of the slope is indicative of a reduction of Q (implying a reduction of the size of the reinforcer needed to educe a given value); an increase of the slope is indicative of an increase of Q (implying an increase of the size of the reinforcer needed to educe a given value).

Empirical support for the inclusion of the size‐sensitivity parameter Q in Equation 5 also derives from the application of an adjusting‐magnitude procedure modeled on Mazur's adjusting‐delay procedure (da Costa Araújo et al., 2010). In this procedure, the volume of a soluble reinforcer is adjusted in successive trial blocks in the same way as delay is adjusted in Mazur's procedure. Performance in the adjusting‐delay and adjusting‐magnitude procedures is associated with different patterns of regional neuronal activation in the orbital prefrontal cortex and nucleus accumbens (da Costa Araújo et al., 2010), whereas lesions of these structures induce different patterns of effect on the indifference points in the adjusting‐delay and adjusting‐magnitude procedures, reflected in different changes of the values of K and Q (da Costa Araújo et al., 2009; Valencia‐Torres et al., 2012).

RECENT EXPANSION OF MHM

This section outlines some recent attempts to extend MHM's definition of value to incorporate additional properties of reinforcers.

Different reinforcers may have different maximum values

Equation 5 defines value in terms of two features of a reinforcing stimulus—quantity and immediacy, each feature being represented by a single parameter (Q and K). If no delay is imposed on either reinforcer, then indifference between two reinforcers, A and B, is defined by the simple equation

$$\frac{1}{1+Q_A/q_{A\left(50\right)}}=\frac{1}{1+Q_B/q_B},$$ (7)

where q _A(50) is the indifference quantity of A corresponding to a specified quantity of B. Rearranging the terms gives Equations 8 and 9:

$$q_{A\left(50\right)}=q_B.\frac{Q_A}{Q_B}.$$ (8)

or

$$\frac{q_B}{q_{A\left(50\right)}}=\frac{Q_B}{Q_A}.$$ (9)

It is clear that q _A(50) should be directly proportional to q _B and that the ratio of equivalued quantities of B and A should be constant and impervious to variations in the absolute size of B (Figure 3A). However, this appears not to be the case. It has been found that the indifference ratio q _B/q _A(50) increases linearly as a function of q _B (Bradshaw, 2020). This departure from Equation 7 may be accommodated by the incorporation of a second free parameter, ε, into the quantity term of Equation 5:

$$V=\frac{1}{1+K/i}.\frac{\mathrm{\epsilon }}{1+Q/q}.$$ (10)

FIGURE 3.

Linear transformation of indifference functions. Ordinates: ratio of the quantity of Reinforcer B (q _B) to the indifference quantity of Reinforcer A (q _A(50)); abscissae: q _B. Panel A: If the same maximum value is assumed for A and B, (Equation 9), the slope of the function is zero. Panel B: If allowance is made for different maximal values of A and B (i.e. ε_A ≠ ε_B, Equation 13), the slope deviates from zero (the slope is positive if ε_A > ε_B). Panel C: Data from eight rats trained on an adjusting‐magnitude schedule in which choices were made between two solutions: A (adjusting magnitude, 0.4‐M sucrose) and B (fixed magnitude, 0.2‐M sucrose; data from Bradshaw, 2020).

This parameter, dubbed “efficacy,” which is allowed to vary between 0 and 1, is the asymptote of the second hyperbolic term in the right‐hand side of Equation 10. It expresses the maximum value of a reinforcer that may be attained by progressively increasing its quantity. The incorporation of this parameter alters the indifference function as follows. Allowing ε to differ between A and B and assuming that no delay is imposed on either reinforcer, the following indifference equation may be derived:

$$\frac{{\mathrm{\epsilon }}_A}{1+Q_A/q_{A\left(50\right)}}=\frac{{\mathrm{\epsilon }}_B}{1+Q_B/q_B}.$$ (11)

Rearrangement of the terms in Equation 11 yields the following indifference Equation 12:

$$q_{A\left(50\right)}=\frac{Q_A}{\frac{{\epsilon }_A}{{\epsilon }_B}.\left[1+\frac{Q_B}{q_B}\right]-1},$$ (12)

$\text{and}$ thus, the indifference ratio of the magnitudes of the two reinforcers is given by Equation 13:

$$\frac{q_B}{q_{A\left(50\right)}}=\frac{1}{Q_A}.\left(\frac{{\mathrm{\epsilon }}_A}{{\mathrm{\epsilon }}_B}-1\right).q_B+\frac{{\mathrm{\epsilon }}_A}{{\mathrm{\epsilon }}_B}.\frac{Q_B}{Q_A}$$ (13)

(Bradshaw, 2019; Kenakin, 1997). This equation defines a linear relation between the indifference ratio and q _B; the slope is positive if ε _A > ε _B (Figure 3B) The data shown in Figure 3C are indifference ratios (q _B/q _A(50)) corresponding to a range of fixed volumes of a sucrose solution (q _B) obtained from eight rats (Bradshaw, 2020). The data are clearly compatible with Equation 13 but not with Equation 9.

Linear indifference functions are of particular utility in the quantitative estimation of the parameters that enter into the definition of value (V). However, a practical problem with the application of Equation 13 for this purpose is the presence of four free parameters, which makes estimation of the numerical values of the parameters impracticable using a feasible number of data points. However, relative values of the parameters pertaining to one reinforcer may be estimated if arbitrary values are assigned to the parameters pertaining to the other. This is a standard practice in gustatory psychophysics; for example, the perceived sweetness of various sugars is generally quantified by comparison to the perceived sweetness of sucrose, which may be assigned a value of 1 (Antenucci and Hayes, 2015; Choi et al., 2021; Fujimaru et al., 2012; Shallenberger, 1993). Bradshaw (2024) used this approach to reanalyze the results from Bradshaw's (2020) experiment in which the indifference quantities of a 0.4‐M sucrose solution (Reinforcer A) corresponding to a series of fixed quantities of a 0.2‐M sucrose solution were determined using the adjusting‐magnitude procedure. The data are shown in Figure 3C. The values of ε and Q for the 0.2‐M sucrose solution (Reinforcer B) were estimated relative to values of 1 and 100, respectively, assigned to the 0.4‐M solution (Reinforcer A). Substituting these reference values into Equation 13 enabled the parameters pertaining to the 0.2‐M solution to be calculated from the formulae ${\mathrm{\epsilon }}_B=1/\left(100.\mathit{\text{slope}}+1\right)$ and $Q_B=100.{\mathrm{\epsilon }}_B.\mathit{\text{intercept}}.$ It was found that the value of ε _B was significantly less than 1 but the value of Q _B did not deviate significantly from 100.

Concentration may be represented as a hyperbolic term in the value equation

While the analysis offered by Bradshaw (2024) indicates that the efficacy parameter, ε, was greater for the higher concentration of sucrose than for the lower concentration, it leaves open the question of how concentration might be incorporated into a formal definition of reinforcer value. One promising possibility is that it might be represented by an additional hyperbolic term incorporated into the value equation:

$$V=\frac{\mathrm{\epsilon }}{1+Q/q}·\frac{1}{1+K/i}·\frac{\mathrm{\gamma }}{1+C/c}.$$ (14)

In Equation 14, c stands for concentration, expressed in molar units; C is a sensitivity parameter analogous to K and Q, expressed in the same units as c; and γ is a dimensionless efficacy parameter analogous to ε (Bradshaw, 2023, 2026). Evidence favoring this proposal was obtained using an adjusting‐concentration procedure (Bradshaw, 2026) in which the concentration of a sucrose reinforcer (A) was adjusted in successive blocks of trials by the simultaneous ejection of different volumes of a concentrated sucrose solution and water into the reinforcer receptacle of the operant chamber, keeping the overall volume of the reinforcer constant. The indifference concentration of A was determined for a range of concentrations of a comparison reinforcer (B, fructose). No delays to reinforcement were imposed; therefore, as the volumes of A and B were equal and invariant, indifference may be defined as in Equation 15:

$$\frac{{\mathrm{\gamma }}_A}{1+C_A/c_{A\left(50\right)}}=\frac{{\mathrm{\gamma }}_B}{1+C_B/c_B},$$ (15)

and thus,

$$\frac{c_B}{c_{A\left(50\right)}}=\frac{1}{C_A}.\left(\frac{{\mathrm{\gamma }}_A}{{\mathrm{\gamma }}_B}-1\right).c_B+\frac{{\mathrm{\gamma }}_A}{{\mathrm{\gamma }}_B}.\frac{C_B}{C_A},$$ (16)

which is algebraically identical to Equation 13. Values for γ _B and C _B can be calculated in the same manner as ε _B and Q _B, as described above. The results of Bradshaw's (2026) experiment, which are summarized in Figure 4, indicate that there was no significant deviation of the slope of the linear indifference equation (Equation 16) from zero, implying that the value of γ did not differ significantly between fructose and sucrose (γ _A, which was assigned the arbitrary value of 1). The intercept was significantly greater than 1, indicating that the value of the concentration sensitivity parameter for fructose, C _B, was considerably greater than that for sucrose (C _A, which was assigned the arbitrary value of 100), reflecting the greater reinforcing effectiveness of sucrose than fructose. This is concordant with previous findings of rats' preference for sucrose over equimolar concentrations of fructose in two‐bottle feeding tests (Sclafani & Mann, 1987).

FIGURE 4.

Application of Equation 16. Data from eight rats trained under an adjusting‐concentration schedule: Reinforcer A was sucrose (adjusting concentration); B was fructose (fixed concentrations). The indifference line lies above the line of equal value, indicating a greater relative potency of sucrose than of fructose, and the slope is not significantly different from zero, indicating approximately equal efficacies (γ) of the two saccharides (data from Bradshaw, 2026).

Different taste qualities may be represented by positive and negative values

Ho et al. (1999) suggested that aversive consequences of responding could be incorporated into MHM in the form of “negative value” (V ⁻) that might combine subtractively with the positive value (V ⁺) associated with an appetitive reinforcer, to yield the overall value (V _overall ) of the response‐contingent event:

$$V_{\mathit{\text{overall}}}=V^{+}+V^{-}.$$ (17)

The implications of Equation 17 for punishment and approach/avoidance conflict situations (Miller, 1959) have been discussed elsewhere (Bradshaw, 2019). The following discussion deals with another situation in which the notion of the summation of positive and negative value may apply—the combination of appetitive and aversive tastants. We refer to the type of combination of positive and negative values defined by Equation 17 as the additive‐combination hypothesis. It can be contrasted with an alternative hypothesis, the devaluation hypothesis, according to which the presence of the aversive substance reduces the value of the appetitive substance (V ⁺) by an effect on ε ⁺ and/or Q ⁺ (Bradshaw, 2022b).

A generally accepted dogma of many years standing was that there are four primary tastes that are detected by cells of the taste buds on the tongue's surface, which are conventionally labeled as sweet, bitter, salty, and sour (acid). To this quartet has been added a fifth taste, umami (savory; Servant and Frerot, 2022). As a general rule, mammals are attracted by sweet tastants and repelled by sour and bitter tastants. The aversive properties of sour and bitter tastants is attested by the suppressant effect of these stimuli on responding reinforced by sucrose and other appetitive tastants in two‐bottle preference tests and in the conditioned taste aversion test (Contreras et al., 1995; Harder et al., 1989; Iwasaki and Sato, 1981; Scalera, 2004; Soto et al., 2015). In the language of MHM, sweet tastants are sources of positive value, whereas bitter and sour tastants are sources of negative value. Since the overall value of a mixture of sweet and bitter tastants is, according to the additive‐combination hypothesis, defined by Equation 17, and if, for the sake of simplicity, we assume that no delays are imposed on the reinforcers and that the concentrations are invariant, this becomes

$$V_{\mathit{\text{overall}}}=\frac{{\epsilon }^{+}}{1+Q^{+}/q}+\frac{{\epsilon }^{-}}{1+Q^{-}/q}.$$ (18)

Figure 5 shows the derivation of V _overall from Equation 18. The broken lines are hyperbolic functions defined by the two terms of the right‐hand side of the equation, and the continuous line shows the overall value obtained by algebraic summation of these two functions. In the example shown in Figure 5, Q ⁺ < Q ⁻; thus, the influence of V ⁺ outweighs that of V ⁻ at lower reinforcer quantities. However, at higher quantities, when the curves approach their asymptotes (${\epsilon }_A^{+}$ and ${\epsilon }_B^{-}$), the influence of V ⁻ predominates and V _overall declines progressively. As is apparent from Equation 18, if ε _A < ε _B, V _overall may take a value less than zero, in which case the mixture becomes aversive. This is the case in the curves shown in Figure 5.

FIGURE 5.

Computation of the overall value (V _overall) of a mixture from the positive and negative values its appetitive and aversive components (V ⁺ and V ⁻), according to Equation 18. Dotted reference lines indicate the values of Q ⁺ and Q ⁻, defined as the quantities of the two components that correspond to their respective half‐maximal values. Note that at smaller quantities, the appetitive component makes a greater contribution than the aversive component to V _overall because Q ⁺ < Q ⁻; however, the influence of the aversive component predominates at higher quantities due the influence of its greater efficacy (ε⁻ > ε⁺).

Indifference functions derived from the additive‐combination hypothesis are nonlinear and in some cases nonmonotonic (as in Figure 5). If indifference is sought between a variable quantity of an appetitive substance (Reinforcer A) and a mixture of the same substance with an aversive substance (Reinforcer B), the null equation is.

$$V_{A\left(50\right)}=V_B^{+}+V_B^{-}.$$ (19)

Expanding the left‐hand side of Equation 19 and rearranging, we get Equations 20 and 21:

$$q_{A\left(50\right)}=\frac{Q^{+}.\left(V_B^{+}+V_B^{-}\right)}{{\epsilon }^{+}-\left(V_B^{+}+V_B^{-}\right)},$$ (20)

and thus,

$$\frac{q_B}{q_{A\left(50\right)}}=q_B.\frac{{\epsilon }^{+}-\left(V_B^{+}+V_B^{-}\right)}{Q^{+}.\left(V_B^{+}+V_B^{-}\right)}.$$ (21)

Although these equations do not facilitate numerical estimation of the parameters, the curves that they describe are readily distinguishable from the corresponding linear (or near‐linear²) functions provided by the devaluation hypothesis (Equations 8 and 9 if ε_A = ε_B = 1; Equations 12 and 13 if ε_A ≠ ε_B). Figure 6 shows the functions defined by the two versions of the devaluation hypothesis and by the additive‐combination hypothesis.

FIGURE 6.

Comparison of predictions of the devaluation hypothesis and the additive‐combination hypothesis of the values of mixtures of appetitive and aversive tastants. The left‐hand graphs show the predicted indifference quantities of a standard solution (q _A(50)) corresponding to a range of volumes of the mixture (q _B). The right‐hand graphs show the predicted relation between the indifference ratios of the volumes of the two solutions (q _B/q _A(50)) to q _B. Top row: According to the simpler version of the devaluation hypothesis in which only the parameter Q is purported to be affected by the presence of the aversive adulterant, q _A(50) increases linearly from the origin as a function of q _B and the indifference ratio is impervious to changes in q _B. Middle row: If the adulterant affects ε as well as Q, both q _A(50) and q _B/q _A(50) increase linearly as a function of q _B. Bottom row: The additive‐combination hypothesis predicts that q _A(50) is an inverted‐U function of q _B and that the indifference ratio is a positively accelerated function of q _B.

Figure 7 shows experimental data from two experiments in which a 0.4‐M sucrose solution was adulterated with a sour tastant (0.06‐M citric acid: upper graphs, Bradshaw, 2022a) or a bitter tastant (0.5 mM denatonium benzoate: lower graphs; Bradshaw, 2022b). In both cases, the results are consistent with predictions of the additive‐combination hypothesis rather than the devaluation hypothesis.

FIGURE 7.

Data from two experiments in which indifference between a variable volume of sucrose (0.4 M) and a range of fixed volumes of 0.4‐M sucrose mixed with an adulterant (top row: citric acid 0.06 M; bottom row: denatonium benzoate 0.5 mM) was determined using an adjusting‐magnitude schedule. Left‐hand graphs: In both cases, the indifference volume of sucrose (q _A(50)) was an inverted U‐function of the volume of the mixture (q _B). Right‐hand graphs: In both cases, the indifference ratio of volumes (q _B/q _A(50)) increased as a positively accelerated function of q _B. Data previously reported by Bradshaw (2022a, 2022b).

These findings may have implications for the central mechanisms of taste perception. Detection of tastants in the oral cavity is initiated by binding of molecules of the tastants to receptors located mainly in the taste buds of the tongue. Salty and sour tastes are detected by ionotropic receptors, the Otop1 receptor being primarily responsible for detection of acids, including citric acid, whereas sweet, bitter, and umami tastes are detected by metabotropic (G‐protein‐coupled) receptors. Sweet tastants are appreciated via T1R2/T1R3 heterodimeric receptors (Dutta Banik & Medler, 2021; Liu and Bohórquez, 2022), whereas bitter tastants stimulate T2 receptors, of which there are approximately 50 variants. Different bitter tastants act at different T2R variants; the supremely potent bitter substance denatonium benzoate acts via several receptor types, including T2R4, T2R8, T2R44, and T2R47 (Behrens and Schaefer, 2025). Taste receptor stimulation triggers a highly complex sequence of intracellular events that culminates in the generation of action potentials in the sensory nerves and then, via the chorda tympani and relays in the geniculate nucleus, the nucleus of the solitary tract and the parabrachial nucleus, to the insular (primary gustatory) cortex, and the orbital and medial prefrontal (secondary gustatory) cortices (Dutta Banik & Medler, 2021; Gutierrez et al., 2023; Jezzini et al., 2013). Tastants belonging to the five major categories activate partially overlapping sites in the gustatory cortices (Accolla et al., 2007; Peng et al., 2015). This is the basis of the so‐called labeled‐line theory of taste processing, which posits that tastes belonging to different categories retain their separate identities and hedonic values at the cortical level, even when they are presented to the tongue as a mixture (Accolla et al., 2007). It is now recognized that neither the labeled‐line theory nor its opponent, the combinatorial‐coding theory, is uniquely correct; the cortical representation of different tastes includes both specialist and generalist neurones and ensembles of neurones (Di Lorenzo, 2020). Nevertheless, it is worth noting that the results shown in Figure 7, insofar as they support the additive‐combination hypothesis rather than the devaluation hypothesis, would seem to be more consistent with the labeled‐line theory than the combinatorial‐coding theory.

IS MHM STILL USEFUL?

The results reviewed above and earlier data summarized by Bradshaw (2019) suggest that MHM “works” in the sense that it offers a coherent summary of some well‐known behavioral phenomena, defined algebraically. But the question still needs to be asked, does it really assist our attempt to understand the voluntary behavior of animals and people?

Merely conventional signs?

“What's the use of Mercator's North Poles and Equators,

Tropics, Zones, and Meridian Lines?”

So the Bellman would cry: and the crew would reply

“They are merely conventional signs! …”

Lewis Carroll, The Hunting of the Snark

In Lewis Carroll's rhyme, the Bellman and his crew are setting out on a sea voyage to an unknown land where they hope to seek out and destroy the monstrous Snark. When they embark, the Bellman presents his crew with “a large map, representing the sea, without the least vestige of land,” and the crew are greatly impressed. Of course they have a point; the North Pole, etc., are conventional signs in the sense that they have no material identity. However, as a navigational tool, the Bellman's map had its shortcomings; devoid of all conventional signs, it was “a perfect and absolute blank,” in contrast to real maps, replete with conventional signs, which have amply proved their worth in facilitating navigation and helping us to understand the layout of the material globe.

Mathematical models of behavior are composed of equations, formulae, parameters, and so on, all of them conventional signs. Are these signs indispensable tools in our search for a deeper understanding of the behavior of organisms, or should we, like the Bellman and his crew, shun them? Skinner (1950) addressed this matter in his famous critique of theories of learning. It is popularly believed that Skinner decried all theoretical models of learning and argued that experimental enquiry motivated solely by a quest for empirical data unencumbered by theoretical constructs of all kinds was the best way to advance our understanding of behavioral phenomena. But in fact, Skinner was at pains to distinguish between different types of theory and reserved his opprobrium for theories that postulated theoretical constructs that served a pseudo‐explanatory function. Thus, in the case of reductionist theories, he conceded that neurophysiological processes might serve a viable explanatory purpose if they can be measured and correlated with overt behavioral events. However, neural networks, internal clocks, and the like, are to be deprecated because rather than explaining behavior, they merely shift the burden of explanation deeper into the recesses of what Skinner mischievously dubbed the “conceptual nervous system” (Skinner, 1938). So far as theories couched in mathematical terms are concerned, Skinner did not find these objectionable, so long as they served to summarize and precisely describe observed behavior and not to offer specious explanations of behavior; however, he doubted whether behavior analysis was ready for such theories at the time when he wrote his critique.

Seventy‐five years have passed since Skinner wrote his critique of learning theories, and there can be no doubt that mathematical models have contributed greatly to the advance of our science during this period (the work of Peter Killeen being a preeminent example). But Skinner's cautions are still worthy of our attention. For example, it may be asked whether MHM, in summarizing the contribution of various features of a reinforcer with a single expression called value (V), has allowed this term to be reified and/or to acquire a spurious explanatory function. It is an axiom of MHM that V is simply a number; that is to say, it is a dimensionless intervening variable. However, the very use of the English word “value” as a label for this number invites the user to attach subjective, psychological meaning to the number. From the standpoint of MHM, this practice is to be deplored (although the author must confess to some occasional lapses of attention in this respect).

The danger of mistaking numbers for entities is not particular to behavior analysis. The history of pharmacological receptor theory provides a salutary lesson. The pioneers of classical receptor theory, including Hill (1909), Clark (1933), Gaddum (1955) and Stephenson (1956), accepted as axiomatic the principle that receptors are molecules located on the surface of cells to which ligands (drug or hormone molecules) may adhere. Under steady‐state conditions, the ratio of the rates of association and dissociation of the ligand and receptor molecules defines the “affinity” of the ligand for its receptor. The relative concentration of occupied receptors was believed to determine the magnitude of the physiological response, the maximal response (for example the maximum contractile force of a strip of smooth muscle) being attained when full receptor occupancy was achieved. The shapes and loci of empirical concentration/effect curves, which were in most cases hyperbolic, were therefore regarded as veridical reflections of the hyperbolic Langmuir isotherm that defines the mass action principle of reversible chemical reactions, and the concentration that elicits a half‐maximal response (the EC ₅₀) was accepted as a veridical measure of the affinity of the drug or hormone for its receptor (K _A, where K _A = 1/K _d, K _d being the dissociation constant). This admirably straightforward model was confuted by findings that indicated that in many tissues, the maximal response to agonists may be attained by occupation of only a fraction of the total population of receptors present in the tissues (Ariens et al., 1956; Stephenson, 1956), thus invalidating the identification of the EC ₅₀ with the affinity parameter, K _A. Attempts were made to overcome this obstacle by postulating hypothetical processes to bridge the gap between receptor occupancy and tissue response. For example, Stephenson (1956) postulated a “biological stimulus” (S) to serve this bridging function. Later, the term “transducer” became popular. Black and Leff (1983) showed mathematically that in a physiological system in which several transducers operate in sequence (A → B → C → D), if D (the tissue response) is a hyperbolic function of A (the drug concentration), then the intervening steps must also be hyperbolic (or linear, which was regarded as biologically implausible). This insight presented a more realistic model of pharmacological responses than the simple hyperbolic model that was current in the early twentieth century. However, it entails a daunting proliferation of hypothetical processes, each defined by its own conventional signs, which has led some skeptics to opine that classical pharmacological receptor theory has outlived its usefulness (Colquhoun, 1987), although some classical methods that succeed in isolating individual parameters from the morass of intervening variables, most notably the Schild regression method (Arunlakshana and Schild, 1959), remain of great value (Colquhoun, 2007).

Is there a lesson here for MHM, the hyperbolic terms of which, and their multiplicative combination, have exact parallels in classical pharmacological receptor theory? Perhaps; object lessons are always worthy of our attention. However, it is arguable that MHM has not yet outlived its usefulness. So long as it continues to fulfil the primary objective of all models—that is, to promote ordo ab chao—and we avoid the temptation to identify its conventional signs with physical realities, then it may merit consideration as one of a number of approaches that can help us to seek out new experimental methods and thereby gain a better understanding of reinforcement. Some years ago, in a wise and entertaining commentary on an ongoing dispute between the proponents of two rival theories of interval timing (Gibbon, 1991, 1999; Staddon and Higa, 1999), Peter Killeen remarked, “If you think models are about the truth, or that there is a best timing model, then you are in trouble. There is no best model, any more than there is a best car model or swimsuit model, even though each of us may have our favorites. It all depends on what you want to do with the model. … Models are go‐betweens: They go between the data and our sense of understanding” (Killeen, 1999, p. 148). It is the contention of this article that in this respect, MHM still has something to offer.

AUTHOR CONTRIBUTIONS

C. M. Bradshaw is the sole author of this article.

CONFLICT OF INTEREST STATEMENT

The author declares no conflict of interest.

ETHICS APPROVAL

No animal subjects or human participants were involved in the preparation of the article.

Bradshaw, C. M. (2026). In praise of the hyperbola. Journal of the Experimental Analysis of Behavior, 126(1), e70126. 10.1002/jeab.70126

DATA AVAILABILITY STATEMENT

No unpublished experimental data are included in this article.