Introduction

In many situations, organisms need to learn about the relationships between cues and their consequences. In the simplest case, a cue is paired with a significant outcome and can be used to predict the occurrence of that outcome. In nonhuman animal experiments, the cue is a conditioned stimulus (CS), and its relevant consequence is the unconditioned stimulus (US). If both of them were binary, there would be four possible presentations of the stimuli. At a given trial, the CS and the US can appear together, the CS can appear alone, the US can appear alone, or neither of them may be present. We call the frequencies of each of these possible combinations a, b, c, and d (see Table 1, e.g., Crump, Hannah, Allan, & Hord, 2007). Using the frequencies of each type of trial, the conditional probabilities of the US given the CS (p1), and of the US given the absence of the CS (p2), can be calculated. The value of p1 is a/(a + b), and the value of p2 is c/(c + d).

Table 1 Trial types in contingency learning
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Learning the relationship between a CS and a US is the basis of the conditioning, and many theories and models have been proposed to explain such contingency learning. In 1968, Rescorla reported a series of conditioned suppression experiments in which the difference between p1 and p2 determined whether learning was observed or not. According to Rescorla, a CS will produce an excitatory conditioned response if it is an informative predictor of an increase in the probability of the appearance of the US.

Rescorla and Wagner’s (1972) model had, among others things, the objective of explaining these results. The model is usually expressed using two formulas, \( \Delta V = \alpha {\beta_1}\left( {{\lambda_1} - \sum {{V_{{X_i}}}} } \right) \) and \( \Delta V = \alpha {\beta_2}\left( {{\lambda_2} - \sum {{V_{{X_i}}}} } \right) \). Parameters α and β are associated to the CS and US salience, respectively. The parenthesis formulation represents the difference between the asymptotic level of associative strength that a US is able to support (λ) and the total associative strength that is governed by the stimuli on the present trial (V Xi ). The first formula is applied when the US is presented in that trial, the second when it is absent. Parameters β1 and β2 represent, respectively, the salience of the US and of its absence. In a similar fashion, λ1 would be the maximum associative strength supported by the US, and λ 2 by the absence of the US.

Rescorla and Wagner (1972) ran several simulations of their model, showing that, through trial-by-trial learning, it can predict phenomena such as contingency learning and blocking (e.g., Kamin, 1969). In the years that followed, evidence of new phenomena predicted by the model, such as superconditioning (e.g., Rescorla, 2004) and overexpectation (e.g., Lattal & Nakajima, 1998), was found. Rescorla and Wagner’s model, and other models that share with it many features (e.g. Mackintosh, 1975; Pearce & Hall, 1980; Van Hamme & Wasserman, 1994) had been found relevant to not only animal associative learning but also to fields such as artificial intelligence, neuroscience, or categorization (e.g., Gluck, 1991; Luque, López, Marco-Pallares, Cámara, & Rodríguez-Fornells, in press; Schultz, 2002; Soto & Wasserman, 2010).

Many other models and rules were developed to study contingency learning. One of the most used is Δp (Allan, 1980; Jenkins & Ward, 1965). It is a normative model of covariation between events. In this case, the events would be the CS and the US. It states that Δp = p1 – p2. Positive values of Δp indicate an excitatory relationship between the CS and the US, whereas negative values indicate an inhibitory relationship. A value of 0 indicates that they are independent; that is, the presence or the absence of the CS gives no information about the occurrence of the US. As can be seen, it is equivalent to the rule proposed by Rescorla (1968). Following Δp, other rules were developed (see Allan, 1993, for a review in humans, and Hammond & Paynter, 1983, for nonhumans). Chapman and Robbins (1990) showed that the asymptotic predictions of Rescorla and Wagner’s (1972) model for a partially reinforced CS were equivalent to those of Δp if two assumptions were made. The first one was to assume that a contextual stimulus was present in every trial. The second was that the salience of the presence and absence of the US was equal. That is, β1 and β2 should have the same value.

The assumption about the values of β1 and β2 is theoretically relevant, because using different values can change the predictions of the model in some circumstances. An example of this comes from the relative validity phenomenon (Wagner, Logan, Haberlandt, & Price, 1968). Wagner et al. (1968) reinforced two compounds (AX and BX) differentially in two groups. In the first group, the correlated group, AX was reinforced 100% of the trials, whereas BX was never reinforced. In the second group, the uncorrelated group, both compounds were reinforced on 50% of the trials. In both groups, X was reinforced half of the times it was presented, and according to Δp, it should show the same level of conditioning. Contrary to this prediction, the uncorrelated group showed a higher level of X conditioning than the correlated group. This result is predicted by Rescorla and Wagner’s (1972) model only if β1 is higher than β2. When β1 and β2 have the same value, the model predicts that X should have the same associative strength in both the correlated and the uncorrelated groups. On the other hand, if β1 is higher than β2, it predicts correctly that X in the uncorrelated group should have more associative strength than in the correlated group (see Murphy, Baker, & Fouquet, 2001, for an extensive discussion). In their original work, Rescorla and Wagner used values of 0.2 for β1 and 0.1 for β2, which would lead to a correct prediction. These same values are very often used when this model is used in animal learning literature (e.g., Miller, Barnet, & Grahame, 1995).

Rescorla (1968), Rescorla and Wagner’s (1972) model, when β1 and β2 have the same value, and Δp all predict the same result: If a difference between p1 and p2 were enough to provoke excitatory conditioning, this should happen independently of the specific values of p1 and p2. For example, a group of rats trained with p1 = 0.1 and p2 = 0 would have a Δp = 0.1, and the CS should have an asymptotic associative strength of 0.1. The same would be true for a group with p1 = 0.9, p2 = 0.8, or a third that had p1 = 0.5 and p2 = 0.4. If excitatory conditioning were found in one of them, it should be found in all three.

But, what would be the prediction if β1 were higher than β2? In this situation, from a psychological point of view, trials in which the US is present would be more salient and relevant than those in which the US is absent. From the point of view of the model, because of the difference between β1 and β2, in those trials in which the US is presented, the change of the associative strength would tend to be higher than in those in which the US does not appear. As the proportion of trials in which the US is presented changes, this might lead to a change in the predictions.

To answer this question, a simulation with Rescorla and Wagner’s (1972) model was run (for details of the simulations, see Appendix 1), using values of 0.2 for β1 and 0.1 for β2. The results of the simulation for the CS and the context are presented in Fig. 1. The pattern of predictions for the conditioning of the CS with these parameters is very different to that mentioned before (context conditioning will be discussed below). If β1 is higher than β2, then the same difference of p1 – p2 leads to different asymptotic associative strengths of the CS, depending on the specific values of p1 and p2. A group trained with p1 = 0.1 and p2 = 0 is now predicted to have a higher asymptotic associative strength than those trained with p1 = 0.9 and p2 = 0.8, or p1 = 0.5 and p2 = 0.4. When p1 is 0.9, p2 should be 0.6 to predict the level of conditioning obtained in the case of p1 = 0.1 and p2 = 0. When p1 is 0.5, a p2 of 0.3 would be required to obtain an equivalent level of conditioning. Therefore, with these parameters, Rescorla and Wagner’s model predicts that as the value of p1 increases, the difference between p1 and p2 that is necessary to produce excitatory learning will increase. It should be noted that this is a robust prediction that occurs whenever β1 is higher than β2, independent of their specific values and of the values of the salience of the CS and of the context (see the Supplemental Data for additional simulations).

Fig. 1
figure 1

Predictions of asymptotic associative strength of the CS (top panel) and the context (bottom panel) by the Rescorla and Wagner’s (1972) model when β1 = 0.2 and β2 = 0.1. The color of the bars represents the value of Δp for that combination of p1 and p2

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The aim of the present study was to test that prediction using an appetitive procedure in rats. The objective of the series of experiments presented was to see whether the difference between p1 and p2 needed to produce an increase in the conditioned responding to the CS is the same for all the values of p1, or whether this difference becomes higher as the value of p1 increases. In Experiment 1a, we tested whether a group with contingency p1 = 0.1 and p2 = 0 has a higher rate of responding to the CS than a control group receiving a training in which p1 = 0.1 and p2 = 0.1. In Experiment 1b, we explored whether that same difference would be enough to produce an increase in the conditioned response when p1 and p2 have much higher values, p1 = 0.9 and p2 = 0.8 in the case of the experimental group, and p1 = 0.9 and p2 = 0.9 for its control group. In Experiment 2, a difference between p1 and p2 that was high enough to produce conditioning when p1 was 0.9 was investigated. In Experiment 3, the effect was tested at a medium contingency level not used before, p1 = 0.5. Finally, in Experiment 4, we replicated the findings of Experiments 1a and 1b in the same experiment, with a different procedure that allowed better measuring of context conditioning (see Table 2 for all the frequencies employed in the four experiments).

Table 2 Probabilities and frequencies of reinforcement
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Experiment 1

Experiment 1a

The aim of this Experiment 1 was to test the extent to which a magazine training procedure in rats was able to show contingency learning at a low contingency level. To do this, in the experimental group, p1 had a value of 0.1, and p2 of 0. Note that for all of the comparisons in this experiment, and in the following ones, a group in which the value of p2 is the same as that of p1 is used as the control. In previous studies, researchers have shown that control groups can show some change of performance (e.g., Kirkpatrick & Church, 2004; see Papini & Bitterman, 1990, for a review). In our procedure, excitatory conditioning will be understood as a differential increase in responses during the presence of the CS in the experimental groups as compared with the control group (using an elevation ratio, which will be explained later in more detail, as a dependent measure). Thus, the presentation of a salient CS might increase the arousal of rats so that the CS would have an elevation ratio over 0.5, even although no learning has been acquired. Such problems are obviated by making a comparison with the control group described. Baker, Murphy, Vallee-Tourangeau, and Mehta (2001) discussed extensively the advantages of using this kind of control instead of normative ones.

In Experiment 1a, the control group received 0.1 and 0.1 as p1 and p2, respectively. If experimental animals are able to show significantly more responding to the CS than are control animals, at low levels of contingency, then a difference between p1 and p2 of 0.1 would be enough to produce an increase in responding, and the paradigm with the parameters employed would have shown to be suitable for the investigation of other levels of contingency.

Method

Subjects

Subjects were 16 experimentally naive male Wistar rats that were about 120 days old and that had an ad libitum weight of 502 (range 397–580 g.). They were housed in cages, each of which containing four rats. They had free access to water. One hour after each session, they were fed to maintain them at 80% of their ad libitum weight. Each day, there were 12 hours of light, beginning at 8 a.m. The experiment was run during this light phase.

Apparatus

The apparatus consisted of eight identical operant chambers measuring 24 × 29 × 38 cm. The front and the back walls were made of aluminium, the side walls and the ceiling of transparent methacrylate, and the floor was composed of 0.4 cm stainless steel rods, spaced 1 cm apart. The center end of the front wall contained a recessed food magazine measuring 6 × 3.5 × 6 cm that was located 0.5 cm above the floor. Each chamber was enclosed in a sound- and light-attenuating shell. A speaker that produced a 600-Hz and 76-dB SPL tone, which was used as the CS, was mounted on the front wall, 8 cm over the food magazine. A fan included in the shell produced a background noise of 62 dB SPL. A pellet dispenser mounted behind the food magazine dispensed 45 mg pellets (type Test Diet-MLab Rodent Tablet) through a tube connecting the magazine and the dispenser. Photocells located in the magazine recorded entries into the food magazine in both presence and absence of the CS.

Procedure

Rats were randomly assigned to two groups of eight subjects each and then received two days of magazine training and 22 sessions of conditioning.

Magazine training

On Days 1 and 2, subjects received a 20-min session of magazine training during which pellets were delivered according to a variable-time 120-s schedule. Four pellets were placed in the magazine before the beginning of these sessions.

Conditioning

Conditioning began on Day 3 and continued through Day 24 (a total of 22 sessions). Conditioning sessions were designed to be an appetitive procedure equivalent to that used by Rescorla (1968). Each session lasted 960 s. The first and last 30 s had no event scheduled. The remaining 900 s were divided into 10-s. bins. In 10 of these bins, the tone used as the CS was presented for the duration of the bin. In the remaining time bins, the CS was absent. The mean interval between CS trials was 90 s, the minimum was 70 s, and the maximum was 110 s. (Bueno & Álvarez, 2001; Holland, 2000). A food pellet, the US, could be delivered during the last second of each time bin. Whenever the CS was present, the probability of this happening was p1, as set in the experiment, and in those bins in which the CS was absent, the probability was p2 (Table 2 shows the exact number of trials reinforced and nonreinforced). In Experiment 1a these values were p1 = 0.1 for both the Experimental group and the Control group. The value of p2 for the Experimental group was 0, and for Control group, 0.1. This means that for the Experimental group, on average, a US would appear once every 10 CS trials, but no USs would be presented in absence of the CS. On the other hand, the Control group would receive food, on average, once every 10 CS trials and also once every 10 no-CS trials.

The conditioned response controlled by the CS was computed as an elevation ratio. The value of this ratio was the number of magazine entries during the presentation of the CS divided by the sum of this number, and the number of magazine entries in the previous 10 s. Therefore, higher values of the ratio reflect more excitatory conditioning of the CS. To ensure that the effects found were asymptotic, only the last eight sessions were analyzed.

Data analysis

SPSS 14 was used to analyze the data. Sphericity was tested in every repeated measures ANOVA and, when needed, degrees of freedom were adjusted using the Greenhouse–Geisser correction. The level of significance used was α = 0.05.

Results

The mean elevation ratios for the two groups of Experiment 1a during the last eight sessions (from the fifteenth to twenty-second) are presented in Fig. 2. As can be seen in Fig. 2, animals in the Experimental group showed higher elevation ratio scores than controls, especially in those blocks that represent sessions 19 to 22 (blocks 3 and 4).

Fig. 2
figure 2

Mean elevation ratio of the responses to the CS of the experimental and control groups of Experiment 1a in the last eight conditioning sessions. Error bars represent the standard errors of the means

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A repeated measures ANOVA of the last eight sessions, using group and session as factors, showed a statistically significant effect of the factor group, F(1, 14) = 5.74, p < .05. There was no effect of session, F(7, 98) = 0.60, p > .05, or of interaction between both factors, F(7, 98) = 2.00, p > .05. The mean pre-CS responding of the Control group was 0.69 (SD = 1.14), and the mean of the Experimental group was 1.04 (SD = 1.97). Another ANOVA showed that there were no statistical differences in the levels of responding previous to the CS, F(1, 14) = 1.62, p > .05, indicating that groups did not differ in its CR to the context. A difference of 0.1 between p1 and p2 was enough to show more excitatory conditioning with low values of p1 than in a control group trained with no difference between the two probabilities.

Experiment 1b

In Experiment 1a, we showed that with the current procedure, a value of ∆p = 0.1 is enough to produce excitatory conditioning when p1 = 0.1. If the difference between p1 and p2 that is needed to produce excitatory conditioning is absolute and constant for every value of p1, then a group with ∆p = 0.1 having p1 = 0.9 and p2 = 0.8 should show a higher level of responding to the CS than a control group trained with p1 and p2 equal to 0.9. On the other hand, according to Rescorla and Wagner’s (1972) model with different values for β1 and β2, the difference between these two groups should be very small. To test this prediction, these two groups were used in Experiment 1b as experimental group and control group, respectively.

Method

The experimental subjects were 16 naive male Wistar rats that at the beginning of the experiment were 120 days old and had an ad libitum weight of 499 g (range 432–561 g). All of the maintenance details were the same as in Experiment 1a. The procedure used was the same as in Experiment 1a, and only the contingencies of the groups were changed (see Table 2).

Results

The mean elevation ratios for the groups of Experiment 1b are presented in Fig. 3. As can be seen in the figure, both groups showed similar levels of CR in the last eight sessions.

Fig. 3
figure 3

Mean elevation ratio of the responses to the CS of the experimental and control groups of Experiment 1b in the last eight conditioning sessions. Error bars represent the standard errors of the means

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A repeated measures ANOVA of the last eight sessions showed no statistically significant effect of the group factor, F(1, 13) = 0.01, p > .05. There was a significant effect of session, F(2.5, 32.6) = 3.12, p < .05 but no interaction between the factors, F(2.5, 32.6) = 0.89, p > .05. The mean pre-CS responding was 7.55 (SD = 4.91) in the Control group and 6.88 (SD = 4.60) in the Experimental group. No statistical differences between the groups were found in the levels of responding previous to the CS, F(1, 13) = 0.65, p > .05.

The difference between the probabilities of the US in the presence and absence of the CS, which produced an increase in responding to the CS when p1 value was low (Experiment 1a), was not enough to produce a similar effect when p1 value was high (Experiment 1b).

Experiment 2

The difference between p1 and p2 that produced an increase in conditioned responding as compared with its control when p1 was 0.1 did not produce it when p1 had a value of 0.9. Experiment 2 was designed to determine a difference between p1 and p2 that was high enough to produce more excitatory conditioning than a control when p1 = 0.9. To do this, four groups were used: a Control group (p1 = 0.9 and p2 = 0.9) and three experimental groups that had increasing values of ∆p. They were Experimental 1 (p1 = 0.9 and p2 = 0.75), Experimental 2 (p1 = 0.9 and p2 = 0.7), and Experimental 3 (p1 = 0.9 and p2 = 0.6). They had ∆p values of, respectively, 0.15, 0.2, and 0.3 (see Table 2).

Method

The experimental subjects were 32 naive male Wistar rats that were 90 days old and had an ad libitum weight of 437 g (range 378–508 g) at the beginning of the experiment. All of the maintenance details were the same as in Experiments 1a and 1b. The procedure used was the same as in previous experiments, and only the contingencies of the groups were changed.

Results

The mean elevation ratios for the groups of Experiment 2 during the last eight sessions are presented in Fig. 4. It can be seen that group Experiment 3 (p1 = 0.9 and p2 = 0.6) showed higher levels of CR than the other groups.

Fig. 4
figure 4

Mean elevation ratio of the responses to the CS of the experimental and control groups of Experiment 2 in the last eight conditioning sessions. Error bars represent the standard errors of the means

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By conducting a repeated measures ANOVA of the last eight sessions, we found a statistically significant difference among the groups, F(3, 28) = 3.71, p < .05, a significant effect of the sessions, F(4.8, 135.7) = 4.13, p < .05, and a significant interaction between these factors, F(14.5, 135.7) = 2.43, p < .05. A post hoc Tukey of the means of those last eight sessions test showed that only the difference between the Control group (p1 = 0.9 and p2 = 0.9) and the Experimental group 3 (p1 = 0.9 and p2 = 0.6) was statistically significant. The mean number of responses during the pre-CS period of the Control group and Experimental Groups 1, 2, and 3 was, respectively, 6.91 (SD = 3.34), 8.87 (SD = 4.44), 9.62 (SD = 5.34), and 6.36 (SD = 3.16). No statistical differences were found in the levels of responding previous to the CS, F(3, 28) = 1.15, p > .05.

Therefore, when p1 value was high, a difference between p1 and p2 of 0.3 was enough to produce an increase in conditioned responses to the CS, but differences of 0.2 and 0.15 were not.

Experiment 3

In Experiment 3, a similar design to those of Experiments 1 and 2 was used, but this time the value of p1 was medium, 0.5. The objective was to extend the findings of Experiments 1 and 2 to another level of p1 value, and to test the predictions of Rescorla and Wagner’s (1972) model also under those conditions. By doing so, a potential problem of Experiment 1a was solved. In Experiment 1, the experimental group p2 had a value of 0. It might be argued that this value might have a special psychological significance and be more salient than other values in the continuum of probabilities. Then, the results of Experiment 1a might not be applicable to the whole contingency space, preventing a correct test of the discussed predictions of Rescorla and Wagner’s model. In Experiment 3, therefore, we investigated the effects produced by ∆p = 0.1, but in a procedure without values of p1 or p2, that might be of special relevance. One group was trained with ∆p = 0.1 (p1 = 0.5 and p2 = 0.4); a second experimental group was trained with ∆p = 0.2 (p1 = 0.5 and p2 = 0.3). Would these show excitatory conditioning as compared with a control group with ∆p = 0 (p1 = 0.5 and p2 = 0.5)? (See Table 2).

Method

The experimental subjects were 24 naive male Wistar rats that, at the beginning of the experiment, were 90 days old and had an ad libitum weight of 430 g (range 363–506 g). All of the the maintenance details were the same as in previous experiments. The procedure used was the same as in Experiment 1a, and only the contingencies of the groups were changed.

Results

The mean elevation ratios of the groups of Experiment 3 during the last eight sessions of training are represented in Fig. 5. As can be seen in the figure, Experimental Group 2 (p1 = 0.5 and p2 = 0.3) showed the highest level of CR.

Fig. 5
figure 5

Mean elevation ratio of the responses to the CS of the experimental and control groups of Experiment 3 in the last eight conditioning sessions. Error bars represent the standard errors of the means

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A repeated measures ANOVA of the last eight sessions found statistically significant differences between the groups, F(2, 21) = 25.77, p < .05, a significant effect of the sessions, F(7, 147) = 3.64, p < .05, but no significant interaction between the factors, F(14, 147) = 1.06, p > .05. A post hoc Tukey test of the means of the eight sessions showed that the only difference that was statistically significant was between the Control group (p1 = 0.5 and p2 = 0.5) and Experimental Group 2 (p1 = 0.5 and p2 = 0.3). The mean number of responses during the pre-CS period of the Control group and Experimental Groups 1 and 2 was, respectively, 6.89 (SD = 4.19), 6.17 (SD = 3.89), and 4.72 (SD = 4.96). Another ANOVA found no statistical differences in responding previous to the CS, F(2, 21) = 2.24, p > .05.

Experiment 4

In Experiment 1a, we showed that a difference of 0.1 between p1 and p2 was enough to produce an increase in conditioned responding to the CS when p1 had a value of 0.1. This difference did not produce the same effect in Experiment 1b, where p1 was 0.9. In Experiment 4, we tried to replicate these two results in a single experiment. Also, the simulations presented in Fig. 1 show that Rescorla & Wagner’s (1972) model predicts that in each of the previous experiments, experimental and control groups should have differed in their level of context conditioning. For example, according to the predictions presented in Fig. 1, the associative strength of the context in a group with p1 = 0.1 and p2 = 0 should be zero, whereas one with p1 = 0.1 and p2 = 0.1 would be 0.2. However, this was not found, although differences in responding to the CS were detected. This could have been because of the fact that the measure of context conditioning was less sensitive than that used to assess the CR. Responding to the CS was measured as an elevation ratio, to control a possible effect of context conditioning. This kind of measure allows additional advantages, because it also reduces the influence of additional factors such as general activity of the animal. Since the index of context responding was the number of responses while the CS was absent, it did not have this feature. Because of this, a different procedure was used in this experiment, in order to try to measure responding to the context in a situation equivalent to the CS. In this procedure, every trial was signaled with a discrete stimulus. Therefore, this stimulus appeared in all trials and functioned as a contextual cue that indicated that a trial was taking place. This stimulus could appear on some of the trials alone, followed or not by the US (these trials were used to calculate p2), whereas in others, it appeared with the CS, followed by the US or not (to calculate p1). As a result of this training, these types of trials defined the probabilities assigned to each group. Thanks to this change, an elevation ratio equivalent to that of the CS could be calculated for all of the trials, those that included the CS and those that did not, allowing a better comparison between these two situations. Therefore the basic analysis used in Experiments 1–3 was also applied to the modified procedure. Also, as can be seen in Fig. 6, the predictions of Rescorla and Wagner’s model for this training are identical to those shown in Fig. 1 that were calculated for the training used in Experiments 1 to 3 (see Appendix 1).

Fig. 6
figure 6

Predictions of asymptotic associative strength of the CS (top panel) and the contextual stimulus (bottom panel) by Rescorla and Wagner’s (1972) model for a contingency training in which a contextual cue is used to mark each trial. The value of the parameters used were β1 = 0.2 and β2 = 0.1. The color of the bars represents the value of Δp for that combination of p1 and p2 (see Appendix 1 for more details)

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Four groups were used; two of them were experimental groups with the same Δp = 0.1, but with different values of p1 (p1 = 0.1 and p1 = 0.9). There were two equivalent control groups with the same p1 as the experimental groups and Δp = 0.

Method

Subjects

Subjects were 32 experimentally naive male Wistar rats about 120 days old that had an ad libitum weight of 383 g (range 347–411 g). All of the maintenance details were the same as in previous experiments.

Apparatus

The apparatus was the same as before, except for a new speaker that was added in Skinner boxes. It generated a new auditory stimulus: a 3,000-Hz and 82-dB SPL intermittent click that lasted for 10 s. This stimulus was used as a CS.

Procedure

Rats were randomly assigned to four groups of eight subjects each and then received 2 days of magazine training and 16 sessions of conditioning. The procedure used was similar to that employed by Murphy and Baker (2004). Each session was composed of trials in which only a 10-s tone was presented (no-CS trials) and trials in which the tone and a click, also 10 s long, were presented simultaneously (CS trials). Therefore, the click was used as a CS, and the tone served as a discriminative stimulus that indicated that a trial was taking place. This ensured that CS and no-CS trials were signaled in the same way.

Conditioning

Conditioning began on Day 3 and continued until Day 18. Each session lasted 60 min. In each session, 60 trials were presented, 30 with the presentation of the tone alone (non-CS trials), whereas in the other 30, the tone and a click were presented (CS trials). The mean interval between trials was 60 s, and the order of the trials was random. A food pellet, the US, could be delivered at the end of the trial. Each trial had a probability of reinforcement determined by the values of p1 and p2 of the group (Table 2 shows the exact number of reinforced and no reinforced trials). Four groups were established: Control Group 1 (p1 = 0.1 and p2 = 0.1), in which both probabilities were the same and fixed in a low value; Experimental Group 1 (p1 = 0.1 and p2 = 0.0), which was trained with a Δp = 0.1 and a low p1 value; Control Group 2 (p1 = 0.9 and p2 = 0.9), which worked as control with an equal value of probabilities fixed in a high value; and, finally, Experimental Group 2 (p1 = 0.9 and p2 = 0.8), which was a group trained with a Δp = 0.1 and a high value of p1. In each session, the click was conditioned according to the probabilities stated. In Control Group 1 (p1 = 0.1 and p2 = 0.1), 10% of tone and click (p1) and tone-alone trials (p2) were reinforced. Experimental Group 1 (p1 = 0.1 and p2 = 0.0) received a food pellet on 10% of the 30 tone and click trials (p1), but none of the 30 tone-alone trials was followed by food (p2). In Control Group 2 (p1 = 0.9 and p2 = 0.9), 90% of both types of trials was reinforced. Finally, 90% of tone and click trials and 80% of the tone trials were reinforced in Experimental Group 2 (p1 = 0.9 and p2 = 0.8). This method generated faster conditioning than that used in previous experiments, which is why the number of sessions was reduced as compared with those in previous experiments.

Data analysis

Elevation ratios, similar to those explained before, were used for the analysis. The elevation ratio of the non-CS trials, in which the tone was presented alone (ERtone), and of the CS trials, in which both the tone and the click were presented (ERclick+tone), were calculated using the magazine entries recorded during the presentation of the stimuli and those registered in a previous period of the same duration, 10 s. The dependent variable employed to analyze the conditioning data was the difference of elevation ratios of the two types of trials (DifER = ERclick+tone – ERtone). By doing so, the conditioning of the CS was estimated. The final six sessions were analyzed, to avoid possible preasymptotic effects.

Results

In Fig. 7, we present the results of the last six conditioning sessions (collapsed in blocks of two) and show that Experimental Group 1 (p1 = 0.1 and p2 = 0.0) had the highest asymptotic level of conditioned responses. To analyze the data, a repeated measures ANOVA was used. It had a within-subjects factor, session, and a between-subjects factor, group. No interaction effect between session and group, or a main session effect was found, F < 1. However, the main effect of group was significant, F(3, 28) = 18.897, p < .05. A Tukey post hoc analysis showed significant differences in their CR between Experimental Group 1 (p1 = 0.1 and p2 = 0.0) and the other three groups. No other difference was statistically significant.

Fig. 7
figure 7

Mean difference of elevation ratio of CS trials minus the elevation ratio of non-CS trials of the groups of Experiment 4 in the last six conditioning sessions (collapsed in blocks of two). Error bars represent the standard errors of the means

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To test for a possible difference in context conditioning, elevation ratios on CS and no-CS trials were analyzed separately. The results are presented in Fig. 8. Experimental Group 1 showed a higher ER on the CS trials, whereas the four groups showed a similar level of conditioning on no-CS trials. Another ANOVA was run, now with two within-subjects factors, session and trial Type (CS vs. no CS). There was a significant interaction between trial type and group factors, F(3, 28) = 23.57, p < .05. Additional tests showed that there was no significant difference among groups in mean elevation ratio on no-CS trials (F < 1), but a significant difference was present in CS trials, F(3, 28) = 3.04, p < .05. A Tukey test showed that this result was due to differences was between Experimental Group 1 (p1 = 0.1 and p2 = 0.0) and the other three groups.

Fig. 8
figure 8

Mean elevation ratios of CS and non-CS trials of the groups of Experiment 4 in the last six conditioning sessions. Error bars represent the standard errors of the means

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These results replicated those of Experiments 1a and 1b. A difference of Δp of 0.1 was enough to produce an increase in responding to the CS in a group with a low value of p1 as compared with its control group with Δp = 0.0. The same difference was not enough to produce a significant increase of responding in the experimental group, with a high value of p1. As in previous experiments, context-conditioning differences were not found.

General Discussion

The results of the present experiments have shown that the difference between p1 and p2 that is required to produce excitatory conditioning in an experimental group, as compared with a control group trained with the same p1 and p2, was not absolute, but that its value increased as the value of p1 increased. Specifically, the experiments have shown that a small difference between p1 and p2 (∆p = 0.1) can be enough to produce increased responding to the CS when p1 has a low value, 0.1 (Experiment 1a), but not when p1 has a high value, 0.9 (Experiment 1b). In this case, when p1 is 0.9, the difference has to be bigger, ∆p = 0.3, with this procedure (Experiment 2). When p1 has a value of 0.5, a difference of ∆p = 0.2 was enough to produce excitatory conditioning, whereas a difference of ∆p = 0.1 was not (Experiment 3). The results of Experiments 1a and 1b were replicated in a single experiment (Experiment 4). This result is contrary to the widely accepted idea that whenever p1 is bigger than p2, excitatory conditioning will occur (Rescorla, 1968).

A potential problem of Experiments 1b, 2, and 4 would be the existence of a ceiling effect. Given that a high magnitude of reinforcement was used, it might be possible that a maximum level of responding was reached. This, in turn, could have masked possible differences between high magnitude groups. If this had happened, it might explain to some extent the results found in those experiments, but it cannot explain the whole pattern of results obtained, and therefore does not invalidate the conclusions discussed before. Specifically, in Experiment 3, the probabilities of reinforcement were approximately half of those used in Experiment 1b; the mean response rate of all of the groups of Experiment 1b during the CS in the sessions analyzed was of 0.975 responses per second, whereas in Experiment 3, this rate was only 0.722, 25% less. It seems unlikely, therefore, that a ceiling level was reached in both experiments. But in neither experiment was a difference between p1 and p2 of 0.1 enough to produce an increase in conditioned response with a medium magnitude.

The asymptotic values predicted by some models applied to contingency learning for each particular contingency explored in these experiments are given in Table 3. Six different representative models were used. They are Rescorla and Wagner’s (1972) model and Δp (already explained); an adaptation of the Weber–Fechner rule (see below for an extended discussion); and three additional correlational models, ΔD (White, 2003), PowerPC (Cheng, 1997), and Φ (Hammond & Paynter, 1983). The algebraic expressions of those models not described before can be found in Table 4. As is true for Delta Δp, these three models are correlational and are usually described as rule-based models because they do not incorporate a trial × trial learning mechanism. Instead, they calculate their contingency index on the basis of the frequency of trials and the probabilities derived from them (see Table 4), each one proposing a different contingency rule.

Table 3 Empirical results and simulated predictions of contingency learning models
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Table 4 Asymptotic algorithms used to generate the data in Table 3
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The experimental results that were obtained are incompatible with rule-based models such as Δp, ΔD, PCi (White, 2003), and others (see Allan, 1993, for a review in humans and Hammond & Paynter, 1983, for nonhumans), but can be explained by the Rescorla and Wagner (1972) model, as long as the value of the parameter β1 is higher than the value of the parameter β2. For instance, a simple inspection of Table 3 shows that Δp predicts the same level of conditioning in the group with p1 = 0.1 and p2 = 0, as in the group with p1 = 0.9 and p2 = 0.8. However, Experiments 1a, 1b, 2, and 4 revealed a greater increase of excitatory conditioning in the former group than in the latter, as compared with their respective controls. Similarly, ΔD predicts a similar level of learning in a group with p1 = 0.5 and p2 = 0.3, and another in which p1 is 0.9 and p2 is 0.7. The first group showed significant greater levels of response than its control group, whereas the second did not (Experiments 2 and 3). Something similar happens with PowerPC and Φ predictions about these groups. Only the Rescorla and Wagner model, when β1 is higher than β2, accommodates the results of these experiments.

The effect shown in these experiments is compatible with the density effect, but it cannot be explained by this effect. The density effect (Allan, 1993) is an increase in CR due to an increase in the general probability of appearance of the outcome, but it does not imply a relative difference in discrepancy between p1 and p2. Density effect has been explained in terms of context conditioning, which could be present in present experiments, but its contribution to the kind of modulation reported here, if any, has been controlled by using elevation ratio as a measure of CR. What is more, no differences in contextual conditioning were found between groups in any of the experiments. In Experiments 1 to 3, direct responses to the context, in absence of the CS, were recorded. This might have reduced the sensitivity of this measure as compared with the CS conditioned response measure, which was an elevation ratio that takes into account the general level of activity of the animal and controls for it. In order to have the same sensitivity in both cases, in Experiment 4, we used a procedure in which contextual conditioning could be measured using an elevation ratio, making its measure equivalent to that of the CS. Even under these conditions, no difference in context conditioning was found between groups that should have differed according to the predictions of Rescorla and Wagner’s (1972) model and the rest of the models. Although this finding was not the main objective of the present study, and taking into account that additional experiments focused mainly on exploring it with different procedures and measures are necessary, so far it seems that Rescorla and Wagner’s model explains only partially the data obtained.

Another way to conceptualize contingency learning is as a discrimination between p1 and p2. When this discrimination is solved, and p1 is perceived as higher than p2, subjects would show an increase in responding as compared with a situation in which p1 and p2 are perceived as equal. Using this simple idea as theoretical framework, it is possible to analyze contingency learning in terms of theories and models that come from perception and psychophysics. This would allow us to look for some effects that are commonly found in psychophysics in contingency learning. For instance, the effect that has been described in the present experiments could be explained in terms of the Weber–Fechner law, a formal descriptive rule for discrimination and perception. According to this rule, the difference necessary to perceive two stimuli as different (∆S) is proportional to the magnitude of the stimuli (S) and is usually a constant (k), specific for each sensory modality. The generalized Weber–Fechner law (Laming, 1986) is expressed as ΔS = (S + b)k (it includes an additional parameter, b, to avoid numerical confusions between increments and absolute values of the stimuli). This generalized law is normally employed when the scale of the absolute values of the S are very close to the incremental values, as is the case in our experiments. Thus, the effects reported presently can be understood if we adopt a Weber–Fechner modulation of perception in contingency. Indeed, if contingency perception were similar to other perceptual tasks, it would be expected that the bigger the magnitude of the stimulus (in our case: the magnitude of the probability of the US in the presence of the CS), the bigger the difference necessary to perceive as different a new stimulus (in our case, to perceive as different the probability of the US in the absence of the CS). Other features of conditioning have been described using this law; they appear to be perceived according to a logarithmic function. For example, timing seems to follow the Weber–Fechner law; the scalar property of timing (Gibbon, 1991) means that longer times are perceived with greater variance than shorter times, and this variance is proportional to the duration being timed.

In fact, the generalized Weber-Fechner law accommodates all of the results described. The results of Experiments 1a and 1b showed that a difference between p1 and p2 of 0.1 produces more excitatory conditioning than a control when p1 is 0.1, but not when p1 is 0.9. When p1 is 0.9, that difference has to be of 0.3, according to the results of Experiment 2. Given these values, the parameters of the generalized Weber–Fechner law (Laming, 1986) can be calculated (see Appendix 2). This calculation leads to an estimation of parameters b having a value of 0.3 and k being 0.25. It is then possible to determine what difference between p1 and p2, according to the model, will be high enough to produce an increase in conditioning (ΔS) for a given value of p1 (S). For example, when p1 = 0.5, the model predicts that only those contingencies with Δp equal or higher than 0.2 will produce excitatory conditioning. The results of Experiment 3 are consistent with the predictions of this rule.

This psychophysical interpretation allows linking two different fields in experimental psychology: learning and perception. This can lead to new ideas and ways to analyze learning situations, and some authors have done this in the past for other aspects of learning (e.g., Allan, Hannah, Crump, & Siegel, 2008; Boneau & Cole, 1967; Hack, 1963; Hirsch, 1979; Mason et al., 2003; Nevin, 1964; Schmajuk, 1987). This research has made use of classical psychophysics to deal with a paradigmatic learning issue (contingency). Allan et al. (2008) proposed a psychophysical account of contingency data in which they made use of an account of contingency learning in human subjects based on signal detection theory. Both research programs share a similar conceptual framework based on the usefulness of putting together theories from apparently independent fields such as perception and learning.

Finally, it is hard to overstate the influence that Rescorla and Wagner’s (1972) model has had in associative learning. Both its theoretical aspects and mathematical formalization have been useful tools for researchers for 40 years now. The fact that novel predictions of this model are still being found and tested is a good proof of its longevity and strong predictive validity. Although it predicted correctly only part of the results, it is a powerful research tool and heuristic. Mathematical models of learning are essential research tools under constant development (see, e.g., Le Pelley, 2004; Stout & Miller, 2007; van Hamme & Wasserman, 1994; Wagner & Brandon, 2001).