Methodological and analytical issues of progressive ratio schedules: Definition and scaling of breakpoint

Abstract

Background:

The break point (BP) of the progressive ratio (PR) schedule of drug delivery is a well-recognized parameter in self-administration studies. Nonetheless, two problems remain unresolved: there is no rationally justified criterion for the last response at BP; the both commonly used definitions of BP as the number of deliveries or the last complete progressive ratio requirement are not the best assuming that BP is a measure of motivation.

New Method:

A criterion for the last lever press is proposed in this study using intravenous cocaine self-administration in rats. The rationale is based on the finding that long inter-press intervals have initially very low probability to occur during the self-administration phase of the session under the PR schedule. But this probability dramatically increases when inter-injection intervals increase due to high ratio requirements.

Results:

For cocaine these critical inter-press intervals were 7.5 min and longer. This novel criterion was applied to measure BP according to all four theoretically plausible definitions of BP including the new one: the higher of the two numbers of presses before or after the last delivery of the reinforcer.

Comparison with Existing Method:

The conventionally defined BP is significantly lower (by 12%) than BP defined according to the new proposed definition. The new definition of BP provides not only more accurate value of BP but now the variance of BP at different cocaine doses is homogeneous as required by many statistical tests.

Conclusion:

These new definitions of the last press and BP provide more accurate and statistically homogenous measure of BP.

1. Introduction

A progressive ratio (PR) schedule requires the subject to perform an increasing number of lever presses for the next successive presentation of a reinforcer (Hodos, 1961). The maximal number of presses resulting in the final delivery of a reinforcer Hodos called ‘breaking point’ also known as ‘break-point’ (BP) (Risner and Silcox, 1981).

Hodos (1961) proposed measuring BP by the number of responses in the last complete ratio or by the number of obtained reinforcers (Hodos and Kalman, 1963). The progression function defines the relationship between these units of measurements. If the slope of the progression function is constant (linear relationship between number of injections and number of required responses), then these two measures are invariant. However, most PR studies employ an exponential progression function in order to reach BP within one session. In this case the validity of statistical analysis may be compromised if variance of the measure is not homogenous at low and high BP values. This problem of the scaling was first addressed by Roberts and Richardson (1992). They proposed using log values of the final ratio data for comparison between groups (doses or conditions). In case of exponential progression function, it would be equivalent to comparison of the total number of reinforcers obtained (Depoortere et al., 1993). A new solution for this problem is proposed in this study.

Theoretically, there are four possible definitions of the BP summarized in Table 1.

Table 1.

Four theoretically possible definitions of the BP.

#	Definition of BP	Reference
1	Total number of reinforcers delivered before the last press	Hodos and Kalman, 1963
2	The last completed ratio followed by drug delivery	Hodos, 1961
3	Number of presses after the last delivery	This paper
4	The bigger number of presses by Def. 2 or Def. 3	This paper

The PR schedule was developed as a method of quantitative estimation of motivation to obtain a reinforcer. The appropriate measure would be the maximal amount of work (the number of responses) a subject really did even if it was not followed by reinforcer delivery. Therefore, Definitions 2 and 3 are equally valid and the highest of two measurements should represent the best estimate of this work. In addition, the same logic suggests that the definition of BP as the last ratio increment which did not result in the reinforcer delivery (Griffiths et al., 1975) would overestimate motivation because this amount of work was not actually finished. Indeed, this definition was never employed by other researchers.

All four definitions of BP include reference to the last response which in turn has not been defined. In order to know if the reinforcer delivery was the last one, the last press should be defined. Table 2 summarizes all used previously definitions of the time when the session should be terminated in drug self-administration studies under the PR schedule.

Table. 2.

The session abortion criteria and their approximate percentages of use in literature (629 original studies in animals).

#	Definition of the last injection	Reference	%
1	Inter-press interval 3-fold longer than longest interval during FR baseline	Bedford et al., 1978	0.2
2	1 h without presses	Bedford et al., 1978	-
3	Less than 3 presses over 15 min after last delivery	Winger and Woods, 1985	0.2
4	1 h with out delivery but not longer than 5 h from the beginning of the session	Roberts and Vickers, 1988	91.0
5	15 min without presses	Kantak et al., 1991*	0.2
6	20 min without delivery	Czachowski and Samson, 1999	0.5
7	30 min without delivery	Brown and Stephens, 2002	0.2
8	2 h without presses	Roberts et al., 2018	0.2
9	The first inter-press interval longer than 7.5 min	This paper	-

^*The same criterion was proposed by Hodos (1961) for food delivery. In 7.5% publications the criterion was not specified.

Bedford et al. (1978) applied two definitions to the same dataset. A comparison between BPs obtained using the fixed 1 h criterion and those obtained using the sliding criterion demonstrated no major shifts in the shape of the dose-BP functions.

Practically there is only one possibility to define the last press based on the observation that inter-press intervals (IPIs) become gradually longer. The very last interval is typically censored because it is unusually long and a researcher have to abort the session. The rationale for our new definition of the last press and BP is detailed herein.

2. Materials and methods

2.1. Cocaine self-administration training

Six male Sprague Dawley rats (from Harlan, Indianapolis, IN, initial weight 180 – 200 g and 400 – 500 g over the duration of the studies) were housed individually on a 12 h light dark cycle (lights on at 6:00 a.m.) and food and water were available ad lib. Rats were surgically implanted with an indwelling catheter into the right jugular vein under halothane anesthesia (Caine et al., 1993). Beginning six or seven days after the surgery, rats were trained to self-administer cocaine HCl using the fixed ratio 1 (FR1) schedule with 5 s time-out period (TO5 s). Training at the unit dose of 1.5 μmol/kg (0.49 mg/kg) cocaine continued until individual rats met the criterion of stable maintained self-administration. This criterion was no significant change of the mean and standard deviation of the inter-injection intervals between five consecutive sessions. At the same time the proportion of inactive lever presses was lower than 2.5% of the total presses.

Test chambers (modified chambers from Lafayette Instrument Co., Lafayette, IN) were each equipped with an active and an inactive lever. Each chamber was situated inside of a laminated wooden compartment (43 × 61 × 35 cm) that provided sound attenuation and was equipped with a house light (7-W fluorescent bulb). Syringe pumps (model PHM-100, Med Associates Inc., Georgia, VT) were situated outside of the laminated compartments. Computers controlled conditioned (light) and unconditioned (drug injection) stimuli using a program written in MedState Notation language (Med Associates, Inc., VT). The concentration of cocaine solution was 40 μmol/ml. Cocaine unit doses were controlled by the duration of pump activation. After self-administration of the initial priming (programmed escalating doses until initiation of self-administration or 2 doses fixed at 1.5 μmol/kg) and loading phases of the session the program allowed self-administration of 50 doses of 0.3 μmol/kg under FR1 and then switched to PR schedule delivering one of the following doses: 0.75, 1.5, 3.0, 6.0 or 12.0 μmol/kg. These doses were selected in pseudo random fashion between sessions. Every cocaine unit dose was presented 5 times totaling 150 sessions from 6 rats. Data from 24 sessions were excluded from analyses because animals did not show normal pattern of self-administration (6 sessions), due to technical errors (3 sessions) or because BPs in first 1–2 sessions under the PR schedule were substantially lower than after that (15 sessions).

The total number of injections at each cocaine dose delivered during self-administration sessions maintained under the FR1 schedule was preset to keep the total duration of self-administration between 2 and 3 hours. Sessions were aborted any time after the first inter-press interval longer than 30 min. This criterion is consistent with the conventional 1 hour since the last drug delivery, as lever-pressing activity during extinction was observed for approximately 30 min.

In order to maintain catheter patency, after every session the catheter was flushed with 0.2–0.5 ml of sterile saline containing 10 units/ml heparin. Catheter patency was evaluated by i.v. administration of short acting barbiturate methohexital (Brevital 6 mg/kg, 3 s injection). The catheter was considered patent if administration of Brevital produced a loss of righting reflex within 5 s after the injection was completed and the latency to the recovery of the righting reflex was greater than 100 s. A failed catheter was removed and a new catheter was implanted into the same right or into the left jugular vein. Self-administration data from 6 rats were gathered as long as 7.8 ± 0.6 months (mean ± SEM).

2.2. Progressive ratio function and BP measurements

For the sake of convenience, the relationship between the number of injections and the number of presses required to receive them was expressed in the form of the discrete function called progressive ratio function or R = f(n) where R is the ratio and n is a number of injections. The following equation for PR increments was selected (similar to that proposed by Roberts and Richardson (1992)):

$$\mathrm{Ratio}=\mathrm{truncate}\left(5·\left(\exp \left(j·\left(n–1\right)\right)–1\right)\right)$$ (1)

where the slope factor j = 0.4 generating ratios: 1, 3, 7, 12, 20, 32, 51, 78, 118, 178, 268, 403, 603, 902, 1348, 2013. In this study, the total number of delivered injections under a PR schedule according to Eq. 1 never exceeded 16. The magnitude of BP was determined according to all four definitions (see Table 1) and compared.

2.3. Materials

Cocaine HCl was obtained from Research Triangle Institute (Research Triangle Park, NC) under the National Institute on Drug Abuse Drug Supply Program. The drug was dissolved in saline containing one unit/ml of heparin (American Pharmaceutical Partners, Inc., Schaumburg, IL) and then passed through a sterilizing 0.2 μm acetate filter immediately prior to use in the self-administration studies. Methohexital sodium (Brevital, Jones Pharma, St. Louis, MO) was used as 1% (w/v, 0.5 ml/kg) solution in normal saline was prepared immediately before injections.

2.4. Statistical analysis

All statistical and visual analyses of data (Pearson linear correlation, nonlinear regression and survival analyses) were conducted using SigmaPlot software (SPSS Inc., Delaware). The distinct phases of the self-administration session were differentiated on the basis of abrupt and sustained changes in the inter-injection or inter-response intervals. Paired t-test, Wilcoxon signed rank test when distribution failed normality test, Brown-Forsythe equal variance test and two-way ANOVA were used to test for a significant difference with the significance criterion set at α = 0.05.

3. Results

3.1. Inter-press intervals

Lever-pressing behavior was analyzed and compared before and after the last injection. The overall rate of presses during the final ratio was in the range 0.19 – 0.83 presses per second. IPIs were lognormally distributed therefore all statistical analyses were conducted using their logarithmically transformed values. The nonlinear regression analysis showed that the Gaussian distribution was bimodal (Table 3). The overall rate of presses correlated with both peaks of the IPI distribution (r = 0.60 and 0.76, p < 0.001, n = 30, respectively). Two-way ANOVA demonstrated a significant difference between individual animals for the first and second peak (F_5,29 = 42.5, p < 0.001 and F_5,29 = 101.3, p < 0.001, respectively) but not between cocaine unit doses (F_4,29 = 0.238, p = 0.91 and F_4,29 = 1.321, p = 0.296), therefore data were collapsed across all unit doses.

Table 3.

Modes of the inter-press interval (IPI) distributions before and after the last injection in individual rats.

Rat #	Before the last injection		After the last injection
	Peak 1*	Peak 2	Peak 1	Peak 2
1	0.56 (0.53–0.59)	0.66 (0.60–0., 2)	0.52 (0.50–0.54)	0.75 (0.67–0.82)
2	0.61 (0.60–0.62)	1.64 (1.49–1.79)	0.58 (0.55–0.61)	1.78 (1.46–2.16)
3	0.70 (0.66–0.73)	1.40 (1.30–1.50)	0.73 (0.69–0.77)	1.45 (1.34–1.58)
4	0.64 (0.61–0.68)	1.63 (1.46–1.81)	0.89 (0.87–0.91)	1.68 (1.49–1.89)
5	1.19 (1.15–1.24)	13.35 (11.54–15.45)	1.34 (1.29–1.40)	15.98 (3.44–19.01)
6	0.52 (0.51–0.52)	0.61 (0.52–0.71)	0.54 (0.53–0.55)	0.81 (0.68–0.96)

^*Peaks represent the most frequent IPIs in seconds for both modes of the distribution. The range of standard errors is in brackets.

Comparison before and after the last injection showed that two peaks did not change significantly. Both peaks shifted by 0.1–0.3 s to the right in five rats and the second peak by 2.6 s in one rat (#5). However, the amplitude of the first peak decreased and the amplitude of the second peak increased after last injection indicating that probability of longer intervals increased in general and especially towards the end of the session (Fig. 1).

Fig. 1.

Comparison of probability density functions of IPIs before and after the last cocaine injection in a representative rat. Both distributions were lognormal and bimodal. Lines represent the best fit of non-linear regression analysis before (solid blue lines) and after (long-dashed red lines) the last injection. Longest IPIs were observed at the end of the sessions.

All longest in each session IPIs which occurred during the final completed requirement were analyzed separately. The Two-Way ANOVA showed that the maximal IPI did not depend on the cocaine unit dose or an individual rat. Therefore, the maximal IPI were collapsed across rats and doses from 126 sessions. The geometric mean ± (SD range) was 52.7 ± (22.2 – 124.9) s. The maximum IPI ever observed in this study was 351.8 s. Survival analysis showed that probability of a long IPI to occur was inversely proportional to the duration of that interval (Fig. 2). According to the 3-σ rule any IPI longer that 427.4 s might be considered as an outlier if it occurred before the last injection. We selected the IPI = 450 s (7.5 min) as a criterion for the end of the self-administration under the PR schedule phase of the session and the beginning of the remission phase. Parameters of the exponential distribution of longest IPIs were used to calculate that the IPIs of 7.5 min and longer duration will occur with probability less than 0.3% (p = 0.0029).

Fig. 2.

The probability of IPIs to occur during the last completed progression requirement as a function of the IPI duration. Symbols (⦁) represent probability of a few arbitrary selected IPI durations. Line represents the best fit of regression analysis according to exponential function: y = a · exp(−b·x), where a = 100 and b = 0.78. Star represents the IPI = 7.5 min (450 s) with calculated probability to occur equal to 0.29 %. Data collected from 126 sessions from 6 rats.

After the last injection, gradually longer IPIs occurred and eventually the session was terminated when the last IPI exceeded 30 min. The transition from lever-pressing activity during progressive ratio self-administration to extinction of lever presses was defined as the first IPI ≥ 7.5 min. Probability of this first interval to be at the same time longer than 7.5 min was determined (Fig. 3). Regression analysis showed that in 50% of all sessions the very first IPI ≥ 7.5 min was also longer than 56.5 min. The total time between the last injection and the beginning of the first IPI ≥7.5 min was proportional to the cocaine unit dose and never exceeded 31.1 min in this study.

Fig. 3.

Survival analysis of the first IPI ≥ 7.5 min which occurred after the last injection. Symbols (•) represent probability for arbitrary selected IPI durations. Line represents the best fit of regression analysis according to exponential function: y = y₀ + a · exp(−b·x), where y₀ =45.9 s, a = 81.6 and b = 0.05. Star represents the probability of 50 % that the first longer than 7.5 min IPI will be also longer than 56.5 min. Data are collected from 120 sessions. In 6 sessions experiment was accidently terminated before the first IPI = 7.5 min.

3.2. Definition of BP

Using the proposed definition of the last lever press we were able to determine BP according to all four definitions (see Tab. 1). Results are presented in Fig. 4. BP determined by Definition 3 was bigger than by Definition 2 in 57.1 % of cases. There was no statistically significant difference between BP determined by Definition 2 and 3 (Wilcoxon Signed Rank Test, p = 0.489, n = 118). BP according to the fourth definition was significantly higher than BP according to either Definition 2 or 3 (by 12.3 ± 1.2%, Wilcoxon Signed Rank Test, p < 0.001).

Fig. 4.

Relationship between the cocaine unit dose and BP (mean ± SEM) determined according to four definitions. Note that BP values measured by the number of injections are on the right y-axis.

The ratio between the highest and the lowest BP was 2.4 for Definition 4 (at cocaine doses of 12.0 and 0.75 μmol/kg, respectively). BP measured by the number of injections (Definition 1) demonstrated much smaller variance but also much smaller ratio of the highest to the smallest BP, 1.2 (Fig. 4).

3.3. Statistical considerations

Correct statistical analysis of the dose-BP function requires the data to be homogenous at different doses. BP defined as the last completed increment of progressive ratio (Definition 2) failed the Equal Variance Test (Brown-Forsythe, p < 0.05). All other measures of BP passed this test. The Two-way ANOVA showed that BP regardless of the definition significantly depended on both factors – the cocaine unit dose (F_4,117 ≥ 7.08, p < 0.001) and an individual rat (F_5,117 ≥ 9.12, p < 0.001) without interaction between these two factors (F_20,117 < 0.96, p > 0.51).

Standard deviation of BP was in the range of 40–70% of BP value at all cocaine doses. Regression analysis showed significant correlations between BP and the overall rate of lever-pressing activity (r = − 0.613, p = 0.022), the first peak (r = − 0.624, p = 0.002) and the highest correlation was with the second peak of IPI distribution (r = − 0.666, p < 0.001, n = 30). Correlation with the cocaine unit dose was also significant (r = 0.529, p < 0.001, n = 30). Therefore, two major factors determining BP were the rate of presses of an individual animal and the cocaine unit dose (Table 4).

Table 4.

Correlation between BP and the second peak of the IPI lognormal distribution.

Dose, μmol/kg	r	r2	p	n
0.75	−0.767	0.588	0.075	6
1.5	−0.740	0.548	0.095	6
3.0	−0.855	0.732	0.030	6
6.0	−0.887	0.786	0.019	6
12.0	−0.842	0.709	0.036	6

4. Discussion

Analysis of intervals between presses during the progressive ratio phase of the session revealed that they are lognormally distributed and have two modes in the ranges of 0.5 – 1.2 s and 0.6 – 13.4 respectively. The longest intervals during the last completed increment were around 30 – 85 s and never exceeded 351.8 s. After the last injection the lever-pressing behavior did not change significantly but both modes slightly shifted to the longer intervals. The probability of longer IPIs to occur after the last injection increased toward the end of sessions. The data suggest two things. First, the mean rate of presses is not an appropriate measure of lever-pressing activity during cocaine self-administration under the progressive ratio schedule because the IPI distribution is lognormal and bimodal. Second, the increasing probability of a long IPI to occur at the end of the session suggests the only theoretically plausible rationale for drawing the demarcation line between the self-administration and remission phases of the session. We hypothesized that there is an IPI of a certain duration probability of which to occur is very low during the self-administration but quickly increases in the end of the session delimiting the beginning of the remission phase.

The distribution analysis of IPIs during the last completed increment demonstrated that the IPI = 7.5 min is long enough to be considered as extremely long, i.e., the outlier according to the rule of three standard deviations. The estimated probability to observe an IPI ≥ 7.5 min was about 0.0029 (one per 344 sessions). The probability of the same event to occur within 30 min after the last injection increased to 100%. Moreover, the very first IPI longer than 7.5 min was also longer than 56.5 min in 50% of sessions indicating the abrupt change in lever-pressing behavior.

This dramatic change in behavior after the last injection is most likely related to the decrease of the cocaine concentration to a critical threshold level (Norman and Tsibulsky, 2006). It has been shown that the lever-pressing behavior correlates with the firing rate of nucleus accumbens cells and with cocaine levels suggesting that the temporal spacing of cocaine self-administration and the cessation of responding at BP are under control of the cocaine concentration (Nicola and Deadwyler, 2000).

We justified the selection of the first IPI ≥ 7.5 min as a criterion dividing the self-administration phase from the remission phase in this study. The lever press followed by the interval longer than 7.5 min was defined as the last press of self-administration under the progressive ratio schedule of cocaine delivery. The IPI = 7.5 min is not the absolute and universal criterion and should be adjusted depending on the self-administered drug, on the species and, perhaps, on the particular group of subjects. In our study the last press occurred within the 31.1 min time period after the last injection. Comparison of the proposed herein criterion with the employed in published studies suggests that the vast majority of researchers used very conservative criterion (1 h without drug delivery) keeping their conclusions valid in the light of the present study. Despite of the wide use of BP, it universally relied on visual analysis of data. The lack of a justified criterion for the last press is not optimal and could compromise reproducibility and conclusions.

BP values determined according to four BP definitions based on the proposed criterion for the last press were analyzed. Assuming the conventional interpretation of BP as a measure of motivation the most appropriate definition of BP is Definition 4. Therefore, in most PR studies employing Definition 2 the BP value was underestimated by approximately 12 %. One may anticipate that relative changes of BP do not depend on the definition and, therefore, conclusions of published studies should remain valid. However, using different BP definitions may potentially lead to different conclusions when different drugs or conditions are compared.

The second argument against Definition 2 is that dose-BP relationship determined using this Definition 2 does not satisfy the criterion of homogeneity while BP defined as the number of injections (Definition 1) or the highest number of presses before or after the last injection (Definition 4) have homogenous variances at different cocaine unit doses. However, satisfaction of the homogeneity requirement does not make the Definition 1 and Definition 4 equally good even when they produced seemingly identical dose-BP curves (Fig. 4). Assuming that BP measures how animals are motivated to obtain the next injection, only BP by the Definition 4 is linearly proportional to the amount of work/motivation. It makes comparison between doses, drugs, individuals etc more accurate and quantitative. In addition, BP measured by the number of presses is much less sensitive to the slope of PR increments.

Strong correlations between the both peaks of IPI distribution and BP suggest that the rate of lever-pressing activity is one of major independent variables determining the BP value. About 50–70 % of BP variance could be explained by the individual differences in pressing activity. It may explain why psychomotor stimulants have higher BPs than any other classes of drugs (Brady et al., 1987). This interpretation is also consistent with positive association found between the velocity of locomotion induced by novel environment with BP in mice (Olsen and Winder, 2006).

5. Conclusions

Analyses of IPIs demonstrated that their distribution is lognormal and bimodal during cocaine self-administration under the PR schedule. Before this study, to the best of our knowledge, there was no justified criterion of the last press of the self-administration phase of the session. The proposed criterion based on the critical IPI duration allows more accurate measurement of BP as the higher of the two numbers: the number of presses before or after the last injection. The rate of lever-pressing activity varies significantly between rats and has the largest contribution into the variability of BP between animals. It explains why psychostimulants are typically producing higher BP values compared with other drugs of abuse.

Highlights.

• The progressive-ratio method requires a definition of the last lever press followed by remission

• The proposed definition is based on a critical inter-press interval

• Having the new definition of the last press, breakpoint is redefined as the greater of two numbers: the last complete ratio requirement or the number of presses between the last injection and last press

• New breakpoint values satisfy the requirement of homogeneity between doses