Figure shows mean (± SEM) response rates of each strain as a function of rate of reinforcement in each epoch. Response rates of SHR are indicated by unfilled squares, WKY by filled circles, and WIS by filled triangles. Visual inspection of Figure reveals a positive correlation between response rate and rate of reinforcement in all strains. Differences in response rates between strains and across rates of reinforcement are visible in both epochs. When rates of reinforcement were low (fewer than 2 responses per minute), SHR responded at a higher rate than other strains. At higher rates of reinforcement, SHR and WKY response rates converged, and WIS response rates remained low (40-50 responses per minute). These patterns of response rate across strains and schedules were visible in epoch 1 and were magnified in epoch 2. SHR response rate increased with age regardless of rate of reinforcement, whereas for WKY age-dependent increases in response rate were more noticeable at higher rates of reinforcement, and for WIS there was virtually no change in response rate with age. To characterize these patterns of response rate, we estimated the parameters of Herrnstein's (1970) hyperbola and compared them across strains [40
Figure 2 Mean (± SEM) response rates of each strain (SHR: unfilled squares; WKY: filled circles; WIS: filled triangles) as a function of mean rate of reinforcement, in two epochs: PND 58-62 (epoch 1; top panel) and PND 89-93 (epoch 2; bottom panel). Response (more ...)
Herrnstein (1970) extended the Matching Law [52
] to describe the relation between response rate (B
) and rate of reinforcement (R
) on a single operandum. Herrnstein's rationale was that all the responses other than the target response are reinforced at an unknown rate. Such rate, however, may be estimated if it is assumed that (a) the ratio of two response rates matches the ratio of the corresponding reinforcement rates (Matching Law), and (b) the target response rate and the non-target response rate add to a constant k
. Under such assumptions,
where Re is the estimated rate of reinforcement provided by non-target responses. When reinforcement is programmed on VI schedules, R typically falls only slightly below programmed reinforcement rates and thus serves as the independent measure; B is the dependent measure; k and Re are free parameters. Equation 1 predicts that responding increases at a negative pace as reinforcement increases, with asymptote k. Re is the rate of reinforcement at which response rate reaches half of its asymptote (i.e., when R = Re, B = k /2).
Following Herrnstein's (1970) rationale [40
is often interpreted as a maximum limit on motoric performance, influenced only by response characteristics; Re
is interpreted as indexing motivation for the reinforcer, influenced only by reinforcer characteristics [39
]. A large body of evidence supports Equation 1 as an accurate characterization of response-reinforcement functions like those in Figure [53
]. The empirical support for motoric/motivational interpretations is, however, somewhat mixed [57
Parameters of Equation 1 were estimated by fitting Equation 1 to the data of each individual animal, in each epoch, using the method of least squares. Parameters k and Re were assumed constant across values of R, but could vary between rats, thus yielding 2 × 18 = 36 model parameters. Comparisons were conducted between mean estimates of each strain, henceforth referred to by the parameter and strain abbreviation: kSHR, kWKY, kWIS, ReSHR, ReWKY, and ReWIS. The curves in Figure are traces of Equation 1 using the mean estimates of k and Re for each strain.
Various constraints were imposed on model parameters to draw inferences on between-strain differences. These constraints consisted of holding constant the mean estimate of either model parameter across all, some, or none of the strains. Each particular combination of constraints constituted a hypothesis. Thus, for example, kSHR ≠ kWKY = kWIS, ReSHR= ReWKY= ReWISis the hypothesis that mean k varied between SHR and WKY, but not between WKY and WIS, and mean Re did not vary between strains. There were 15 possible constraint combinations.
Hypothesis testing was conducted separately in each epoch, using the corrected Akaike Information Criteria (AICc) [58
where n is the number of observations (n = 5 schedules × 18 rats = 90 observations in each epoch), RSS is the minimized residual sum of squares obtained from fitting a hypothesis to the data, and c is the number of free parameters in the hypothesis. c can also be computed as the degrees of freedom of the estimates of the overall means of k and Re plus 1 parameter for error variance, i.e., 2 parameters × 18 rats - the number of constraints + 1. In the preceding example, c = 36 - 3 + 1 = 34 free parameters.
Note that AICc increases with RSS and with c; therefore smaller AICc are indicative of close fit to the data and parsimony. Hypotheses with smaller AICc were favored over those with higher AICc. ΔAICci was computed as the difference between each the AICc of hypothesis i and the lowest AICc among all hypotheses (ΔAICci = AICci - AICcMIN). The hypothesis with fewest free parameters among those with ΔAICc < 4 was selected as the best description of the data. This selection was conducted separately for the 2 epochs in which data were collected.
Table shows the 5 hypotheses with the lowest ΔAICc in each epoch. The selected hypothesis for epoch 1 assumes that ReWIS= ReSHR, and all other parameters varied between strains. For epoch 2, the selected hypothesis assumes different parameters for each strain. Figure shows the mean estimates of k and Re for each epoch based on the selected hypotheses. It was inferred that kWKY >kSHR >kWIS in both epochs, which indicates that WKY had the highest asymptotic response rates, followed by the SHR and then WIS. Mean k estimates increased across epochs for all strains. In epoch 1, ReWKY>ReWIS= ReSHR; at epoch 2, ReWKY>ReWIS>ReSHR. ReSHRdoes not appear to change across epochs, whereas ReWKYand ReWISincreased.
Hypotheses of VI performance with lowest ΔAICc.
Figure 3 Mean (± SEM) estimates of Herrnstein's hyperbola parameters k (asymptotic target response rate; left panels) and Re (rate of reinforcement of non-target behavior; right panels) for each strain in epochs 1 and 2 (top and bottom panels, respectively) (more ...)
Inferences from Herrnstein's hyperbola parameters suggest that instrumental overactivity could be attributed to a higher motivation for the reinforcer, which did not decline over nearly 30 days that separated the two assessment epochs, and not to differences in motoric capacity. This analysis, however, was based on average response rates in each VI schedule, which conflate two types of intervals within the denominator: response latencies and inter-response times (IRTs). Because rodent VI performance is typically organized in bouts [42
], IRTs may be further disaggregated into between-bout and within-bout IRTs. Latencies, between- and within-bout IRTs may each depend on a distinct set of variables [45
], which may further inform the sources of SHR overactivity.
In the next two sections we examine the components of response rate in SHR, WKY, and WIS. This analysis is aimed at identifying candidate components that may account for the differences in response rate between SHR and WKY selectively at low rates of reinforcement, and between SHR and WIS at all rates of reinforcement. An AIC-based analysis appears to be best suited to address this goal, because the relation between response rate and its components is not linear (see Appendix). Without an a priori
selection of hypotheses, however, the combinatorial of parameters and factors implies a computationally intractable analysis that may ultimately select an unintelligible model [58
]. Conventional approaches, such as null-hypothesis testing, are not designed for this task: falsifying the null hypothesis that a particular component did not vary between strains in one or more schedules provides little information on the contribution of that component to differences in response rate. Therefore, the analysis presented here is qualitative; inferences drawn from this analysis should be taken as exploratory and provisional, pending empirical verification.
Figure shows mean (± SEM) median Latency 1 and Latencies 2-8 for each strain in each epoch, as a function of rate of reinforcement. Latency 1 (left panels) did not vary systematically with rate of reinforcement in either epoch or across strains in epoch 1. In epoch 2, mean median Latency 1 was longer for WKY than for the other strains, regardless of rate of reinforcement.
Figure 4 Mean (± SEM) median Latency 1 (first latency within each VI schedule; left panels) and Latency 2-8 (right panels) as a function of mean rate of reinforcement, for each strain in epochs 1 and 2 (top and bottom panels, respectively). The insets (more ...)
The right panels of Figure and their insets show that Latencies 2-8 declined with rate of reinforcement. In epoch 1, Latencies 2-8 were mostly undistinguishable between strains, with the possible exception of the longer latencies of WKY at the lowest rate of reinforcement. In epoch 2, median Latencies 2-8 of WIS were longer on average, but also more variable across rats, than those of SHR and WKY. Also in this epoch, when rate of reinforcement was less than 1 per minute, mean median Latencies 2-8 were about 1 s shorter for SHR than WKY. The slopes of rescaled Latencies 2-8 (each median latency was divided by the median latency in VI 12 s, then logged, base 2), shown in the insets, reveal a within-subject sensitivity of Latencies 2-8 to rate of reinforcement in both epochs. This sensitivity was more pronounced in WKY than in the other strains.
Inter-response times (IRTs): Bout-and-pause model
To account for the distribution of IRTs in each schedule, response rate in each VI schedule, excluding latencies, was disaggregated into bout-initiation rate (the reciprocal of the mean IRT separating response bouts) and within-bout response rate (the reciprocal of the mean IRT within bouts). This disaggregation consisted of estimating the parameters of a bi-exponential density function by fitting it to the distribution of IRTs in each VI schedule. The density function is
is the proportion of IRTs within bouts; 1/(1 - p
) is the mean bout length, measured in lever presses. w
is the response rate within bouts; b
is the rate at which bouts are initiated. Because responses take a minimum time to be produced, that minimum time (the shortest possible IRT) must be subtracted from the duration t
of every IRT [45
]. The minimum IRT recorded for every rat was 0.11 s, which was the resolution at which responses were recorded. Therefore, 0.11 s were subtracted from t
in the exponents of Equation 3, and w
was constrained to be less than or equal to 1/0.11 ≈ 9 responses per second.
Parameters of Equation 3 were estimated for each individual rat in each epoch using the method of maximum likelihood [59
]. Figure shows semi-log survival plots of IRTs in each schedule and epoch, averaged within each strain. These plots have been used previously to illustrate differences in pause and bout responding in both rats and pigeons [42
]. Often, these plots take on a "broken-stick" appearance with a steeply declining initial left limb and a more gradually declining right limb. A long initial limb on the leftmost side of the graphs indicates a high proportion of within-bout responses, p
. The slope of the left limb is the within-bout rate of responding, w
; the slope of the right limb is the rate of bout initiation, b
. The curves in Figure show that Equation 3 provided a good fit of the data, although the broken-stick pattern was most clearly visible in WKY at low rates of reinforcement.
Figure 5 Semi-log survival plots showing the mean proportion of IRTs greater than t in each schedule (symbols), strain (columns), and epoch (rows). Proportions were calculated for each rat in bins that contain, each, 1% of the IRTs; binned proportions were then (more ...)
Mean (± SEM) bout-and-pause parameter estimates for each strain at each VI schedule and epoch are shown in Figure . To compute the mean estimates of w and b, individual estimates were weighed by p and (1 - p), respectively, because confidence on w and b estimates co-varies with these weights.a The top panels of Figure show the mean estimates of p, w and b in epoch 1; the bottom panels show estimates in epoch 2. Mean bout-and-pause parameter estimates are labeled in the same way as Herrnstein's hyperbola estimates (e.g., pWKY, pSHR, pWIS).
Figure 6 Mean (± SEM) estimates of bout-and-pause parameters (Equation 3) as a function of mean rate of reinforcement, for each strain (SHR: squares; WKY: circles; WIS: triangles) in epochs 1 and 2 (top and bottom panels, respectively). In all epochs and (more ...)
Estimates of the proportion of within-bout IRTs (p) in epoch 1 are shown in the top-left panel of Figure . Estimates of pWKY were substantially higher (mean across schedules = .90) than those of pSHR (.17) and pWIS (.26). This means that WKY produced substantially longer bouts than SHR and WIS. Moreover, whereas pSHR and pWIS were relatively constant across rates of reinforcement, pWKY increased with higher rates of reinforcement, from .80 at the lowest rate to .99 at the highest rate.
Estimates of within-bout response rate (w) in epoch 1 are shown in the top-middle panel of Figure . Estimates of wWKY and wWIS increased only slightly with rate of reinforcement. At the lowest rate of reinforcement, wWKY = 0.80 and wWIS = 1.17 responses per second; at the highest rate of reinforcement, wWKY = 1.04 and wWIS = 1.31 responses per second. These trends were dwarfed by the large between-subject and between-schedule variability in estimates of wSHR. Moreover, in every schedule, wSHR >wWIS >wWKY (mean across schedules = 2.09, 1.24, and 0.96 responses per second, respectively). It is important to note, however, that estimates of wSHR and wWIS were based on 2-3 rats of each strain, because p = 0 for most of these rats in most schedules.
Estimates of bout initiation response rate (b) in the first epoch are shown in the top-right panel of Figure . Estimates of bSHR and bWIS systematically increased with rate of reinforcement. At the lowest rate of reinforcement, bSHR = 0.36 and bWIS= 0.32 responses per second; at the highest rate of reinforcement, bSHR = 0.85 and bWIS= 0.63 responses per second. Estimates of bWKYvaried as an inverted-U function of rate of reinforcement, peaking at the second highest rate of reinforcement (0.23 responses per second). In every schedule, bSHR >bWIS>bWKY(mean across schedules = 0.61, 0.49, and 0.15 responses per second, respectively).
The bottom panels of the Figure show the mean (± SEM) estimates of p, w and b in epoch 2. The bottom-left panel of Figure shows that, similar to those in the preceding epoch, estimates of pWKY in epoch 2 were very high and increased further with rate of reinforcement. Estimates of pSHR and pWIS increased between epochs in every VI schedule. The increase was particularly noticeable in pSHR; on the average, estimates of pSHR more than tripled between epochs. Like in the preceding epoch, however, there were no trends in pSHR and pWIS across VI schedules comparable to those of pWKY.
Estimates of w in epoch 2 are shown in the bottom-middle panel of Figure . Estimates of wWKY and wWIS increased between epochs in every schedule, but only wWKY preserved its positive correlation with rate of reinforcement. Estimates of wSHR remained relatively high, particularly when rate of reinforcement was low. Between-subject variance in individual estimates of wSHR increased substantially between epochs, further dwarfing any differences between strains. The increase in between-subject variance was due to 2 SHR with undetermined w in epoch 1, whose individual w estimates, averaged over VI schedules in epoch 2, were 5.50 and 8.89 responses per second. Such high estimates were not obtained for any other rat of any strain.
Estimates of b in the second epoch are shown in the bottom-right panel of Figure . As in the preceding epoch, bSHR >bWIS ≥ bWKY in every VI schedule, and estimates of b also increased with rates of reinforcement, including bWKY. Estimates of bSHR and bWKY increased between epochs in every VI schedule; bWIS remained relatively unchanged.