Generalizing from the Past, Choosing the Future

Abstract

Behavior in the present depends critically on experience in similar environments in the past. Such past experience may be important in controlling behavior not because it determines the strength of a behavior, but because it allows the structure of the current environment to be detected and used. We explore a prospective-control approach to understanding simple behavior. Under this approach, order in the environment allows even simple organisms to use their personal past to respond according to the likely future. The predicted future controls behavior, and past experience forms the building blocks of the predicted future. We explore how generalization affects the use of past experience to predict and respond to the future. First, we consider how generalization across various dimensions of an event determines the degree to which the structure of the environment exerts control over behavior. Next, we explore generalization from the past to the present as the method of deciding when, where, and what to do. This prospective-control approach is measurable and testable; it builds predictions from events that have already occurred, and assumes no agency. Under this prospective-control approach, generalization is fundamental to understanding both adaptive and maladaptive behavior.

Behavior in the present depends critically on experience in similar environments in the past. Traditionally, we have assumed that past experience is important because it determines the strength of a behavior (Skinner, 1938), which in turn determines how likely the behavior is to occur. This approach has served the field well; strength from the past forms the foundation of our quantitative models (e.g., Baum, 1974; Herrnstein, 1961, 1970), theories (e.g., Nevin & Grace, 2000), and interventions. Yet sometimes past strength fails to explain present behavior (e.g., see Cowie, 2019; Cowie & Davison, 2016; Shahan, 2017, for reviews). A response that has just been reinforced, and hence strengthened, should be more likely to occur than other responses—but this is true only when that response is also likely to produce a further reinforcer. When the just-reinforced response is unlikely to produce a reinforcer, the just-reinforced response is unlikely to be repeated (e.g., Cowie, Davison, & Elliffe, 2011; Krägeloh, Davison, & Elliffe, 2005). Likewise, conditional reinforcers with a long history of pairing with a primary reinforcer will occasion the response that produced them only when they are positively correlated with the availability of more primary reinforcers for the same response. When the correlation is negative and conditional reinforcers effectively predict the availability of primary reinforcers for a different response, conditional reinforcers occasion a shift in responding away from the just conditionally reinforced response (Davison & Baum, 2006, 2010). These are but snapshots of a growing body of data that sits uncomfortably within a retrospective, response-strengthening framework (see Cowie, 2018; Cowie & Davison, 2016; Killeen & Jacobs, 2017; Shahan, 2017, for extensive reviews and discussion). Nevertheless, such data serve to highlight two problems for the field: the inadequacy of a retrospective account of control by the environment; and the potential of regularities in the environment to allow for control by likely future events.

More recent explanations of the behavior-environment transaction have highlighted the structure within the environment as the primary source of control (see Baum, 2012; Cowie, 2018, 2019; Cowie & Davison, 2016; Killeen & Jacobs, 2017; Shahan, 2017). Under this approach, the past is important because it allows the structure of the current environment to be detected, and hence for the organism to behave in accordance with likely future. Indeed, even “simple” operant behavior observed under highly controlled laboratory conditions reflects control by the likely future as extrapolated from the past (e.g., in symbolic matching to sample; for elegant demonstration of control by likely future conditions, see April, Bruce, & Galizio, 2013; Branch, Galizio, & Bruce, 2014; in concurrent choice, see Cowie, Davison, & Elliffe, 2011). Further, brain activity of “simple” (e.g., rats performing a spatial memory task; Pfeiffer & Foster, 2013) and more “complex” (e.g., monkeys performing a concurrent-choice foraging task; Shahidi, Schrater, Wright, Pitkow, & Dragoi, 2019) organisms reflects representations of the structure of the environment that facilitate planning and goal-directed behavior. Thus, the likely future controls behavior, and this future is built from past experience.

Which future the organism chooses, of course, depends on the potential of a future to satisfy an organismic state (see Killeen & Jacobs, 2017). Disposition determines which future is valuable. For example, in a maze where one route leads to food and another to water, rats will follow signposts leading to water when water-deprived, and signposts leading to food when food-deprived (e.g., Hull, 1933; Leeper, 1935). Environmental conditions that signpost food will occasion less behavior when an organism has eaten just before such conditions occur (e.g., see Nevin, Mandell, & Atak, 1983); under such conditions, signposts to food exert relatively weaker control over behavior (e.g., Bizo & White, 1995; Ward & Odum, 2006). The control of this organismic component to behavioral control is still within the environment; disposition depends heavily on an organism’s recent interaction with its environment (Cowie, 2019). Thus, whereas the structure of the environment provides a map of where the organism might go, the organismic state is the compass that orients the organism toward some routes and away from others.

In this article, we focus on how behavior depends on spatial relations and temporal relations between events, as well as on order and structure in the environment. Order in the environment allows even simple organisms to use their personal past to respond according to the likely future. We know that organisms can use their personal past in a way that extends beyond mechanistic control by patterns in one’s very recent learning history; when we surprise an organism with a question about their personal past in a way that requires generalization from some more distant past situation, the organism answers appropriately. For example, Zentall, Singer, and Stagner (2008) first trained pigeons on a symbolic matching task where the location of a response was the sample stimulus, and then on a separate task with different comparison stimuli, where color was the sample stimulus. In probe trials during the color-symbolic matching task, the comparison stimuli from the location-symbolic matching task were presented. In essence, these probe trials asked, “which key did you just peck?” in a procedure in which location was usually irrelevant. Pigeons correctly reported the just-pecked location in over 60% of trials, demonstrating use of both recent personal experience (the location just pecked) and generalization of more distant past experience (the key colors and the appropriate dimension), when neither the question nor the answer had been relevant in navigating the present conditions. Understanding how past experience is used to build prediction about the present and future is fundamental for a science of behavior.

Here, we explore how generalization affects the use of past experience to predict and respond to the future. The process of generalization is not new to behavior analysis; generalization has been crucial in translating past experience to behavior in the presence of similar stimuli (e.g., Blough, 1975; Lazareva, 2012; Lazareva, Young, & Wasserman, 2014; Spence, 1937), and in explaining the mismatch between behavior and environment when past and present conditions are identical (e.g., Davison & Nevin, 1999). Here, we examine how generalization affects the way past experience contributes to predictions about the present and tentative future. First, we consider how generalization across various dimensions of an event determines the degree to which the structure of the environment exerts control over behavior. Next, we explore generalization from the past to the present as the method of deciding when, where, and what to do.

Generalization Across Dimensions of the Near-Present

When the likely occurrence of important events changes with respect to some stimulus dimension, behavior follows these changes to some degree. Yet control is imperfect; behavior tracks changes, but tends not to follow them exactly. Consider, for example, situations in which a contingency occurs once some period of time has elapsed since a marker event (e.g., Bizo & White, 1994; Cowie, Davison, & Elliffe, 2011, 2017; Cowie, Elliffe, & Davison, 2013, 2014; Davison & Cowie, 2019; Stubbs, 1980). With experience, the allocation of responses between alternatives (choice) approximates the time-based change in reinforcer availability. Behavior comes under control of the structure of the environment to some degree, although choice typically begins to change before the change in ambient stimulus conditions signals a change in contingency, and continues to change after the contingency reversal has taken place. This approximate control by the likely future, as extrapolated from past experience, is evident when contingencies change in a session, after a fixed number of trials have occurred (Rayburn-Reeves, Molet, & Zentall, 2011), or once a fixed number of responses have been emitted (e.g., Davison & Cowie, 2019). That is, where the environment changes systematically with respect to some stimulus dimension, behavior comes under some degree of control by these changes to the extent that the stimulus dimension is discriminable. Generalization may be key to understanding why control is almost always imperfect, and to predicting the environments in which control of a suitably dispositioned organism will be strongest.

Events in the present occur in context; each event has a time relative to a marker event as well as a location in space. When an event’s occurrence on any of these dimensions relates systematically to the occurrence of other events, accurate detection of this relation between events permits the organism to predict and behave in accordance with the likely future. Generalization across the relevant dimension of an event will obscure the relation between that event and other events (those that have already occurred, and those that may follow). Behavior is controlled by the discriminated, rather than the veridical, relation between events (Davison & Nevin, 1999). When the apparent relation between events differs from the actual, behavior will match the apparent relation, and will thus appear to be under weaker control by the structure of the environment.

Consider, for example, a concurrent-choice procedure in which reinforcers are nine times as likely to be obtained for one response than the other (Cowie & Davison, 2020). A reinforcer obtained in fewer than 19 s after the preceding reinforcer is more likely to be obtained for a response to the left key, whereas a reinforcer obtained once more than 19 s has elapsed since the last reinforcer is more likely to be obtained for a response to the right key. In this situation, for a hungry pigeon the important events are the food reinforcers. Detection of the structure of this environment requires discrimination of each reinforcer’s occurrence in space (left or right) and in time elapsed since the last reinforcer. Such detection requires extended exposure to the same environmental structure; detecting a correlation requires exposure to the occurrence of reinforcers in one stimulus context (time, location) and their absence, or less frequent occurrence, in another (time, location).

In order to model the process by which the structure of the environment is detected, we might use the reinforcers across time since a marker obtained over the last 20 sessions of a condition. These reinforcers are the building blocks of the discriminated structure of the environment—and hence, of the prediction about the likely time and location of future reinforcers. Panel A of Fig. 1 shows these reinforcers, aggregated according to the time since the most recent reinforcer (in 1-s bins) and response location (left or right) at which they occur. The empty circles in Panel B of Fig. 1 shows the log (left/right) ratio of these reinforcers at each time—this also represents choice we would expect given perfect discrimination of the time and location of each reinforcer, and therefore accurate detection of the structure of the environment. The filled circles in Panel B show actual choice (the log ratio of responses made by the animal in each 1-s time bin). Choice follows the change in the likely location of reinforcers, but does not track it perfectly. Choice is less extreme than the reinforcers, and changes gradually, rather than abruptly, across time since the marker event. This difference between the actual structure of the environment (empty circles) and the behavior (filled circles) suggests some degree of generalization across the time and location of reinforcers.

Fig. 1.

The number of left- and right-key reinforcers obtained in each time bin (left panels), and the log response and reinforcer ratio (right panels). The top panels show obtained data, and the lower two panels show reinforcers after generalization processes (Equations 1 and 2) are applied. Measures are calculated using data from the last 20 sessions

The time of an event can be overestimated or underestimated. Generalization across the time of a reinforcer might therefore be modeled by redistributing some proportion of obtained reinforcers to earlier and later times, e.g., according to a Gaussian function. If the mean of the function is at the veridical time at which the reinforcers were obtained, then more often than not the time of the reinforcer is discriminated from other surrounding times. The spread of the distribution—the standard deviation—captures the extent of generalization across time. Generalization becomes wider as the duration to be discriminated increases (scalar property; Gibbon, 1977). Thus, we would expect the standard deviation of our distributions to increase according to the time at which the reinforcers actually occurred. This means that reinforcers obtained at longer times since a marker event are generalized across time more widely than those at earlier times, and that generalization of later reinforcers is more extensive than generalization of earlier reinforcers. Even with these constraints, there is flexibility in the application of the Gaussian generalization process; generalization may increase linearly (Cowie and Davison, 2016) or non-linearly (Cowie & Davison, 2020) as a function of time. Here, we adopt an ogival increase in generalization (Cowie & Davison, 2020), which permits but does not necessitate a scalar increase in generalization, and provides an excellent description of the data (see Cowie & Davison, 2020, for a model comparison). For mathematically inclined readers, the equations are outlined in Appendix 1, and a more detailed discussion can be found in Cowie and Davison (2020). Here, we focus on the concepts underlying the approach, rather than its mathematical form.

Panels C and D in Fig. 1 show the reinforcers after a Gaussian generalization process (Appendix, see Equation 1) has been applied to the obtained reinforcers in the top panel. Panel C shows the time of each reinforcer as detected by the pigeon (i.e., after temporal generalization has occurred). Panel D shows the log ratio of these reinforcers—the pattern of choice we would expect if this were the discriminated structure of the environment. This generalization process smooths the curve and brings the reinforcer differential curve closer to choice, but still the curve does not match actual choice.

Generalization must also occur across location. Because a standard operant choice experiment typically has just two response alternatives—left and right keys—we might model generalization across location in the present example by shifting some proportion of reinforcers obtained on the left at each time to the right, and some proportion of right reinforcers obtained at each time to the left (Appendix, see Equation 2). The effect of this additional source of generalization on the discriminated structure of the environment is shown in Panels E and F of Fig. 1. Generalization shifts reinforcers from left to right and right to left response locations (Panel E), and further dampens the discriminated differential (Panel F). With the addition of a constant denoting inherent bias, the reinforcers after temporal and spatial generalization (Panel F, Fig. 1) are almost indistinguishable from actual choice. Choice matches discriminated reinforcers.

Spatial and temporal generalization are dealt with differently in this example. Because the location of responses is divided into two discretely different classes, we use a point estimate of spatial generalization by redistributing proportions of reinforcers obtained at each time. Temporal generalization, in contrast, uses a more continuous (here, Gaussian) distribution. The model could be easily modified to deal with location-based generalization in the same way as it deals with temporal generalization. The spatial generalization applied here is just a snapshot of a generalization across a continuous spatial dimension; it is a special case of a general model that deals with generalization across space and time in the same way. Indeed, we know that discrimination of the occurrence of events in time and space shares many similarities (e.g., Davison & Cowie, 2019; Machado & Rodrigues, 2007; Tan, Grace, Holland, & McLean, 2007), so it is reasonable to assume that the underlying processes—for space, time, and indeed, other dimensions—are separate but similar (if not identical).

Given the similarity between temporal and spatial generalization processes, we might assume that both location-based and time-based generalization becomes more extensive and more likely as reinforcers occur further in time from the marker event. Cowie and Davison (2020) have shown that the model best describes data when this increase in generalization is ogival. An ogival increase gives the model flexibility: generalization may increase progressively, abruptly, or even remain approximately constant. Because findings of scalar timing (where the standard deviation is proportional to the mean) are not ubiquitous (e.g., see Wearden & Lejeune, 2008 and Wearden & Lejeune, 2008, for reviews), a model of time-based control must be able to capture varying degrees of scalarity.

Once the separate generalization processes have been applied to the obtained reinforcers, the ratio of apparent reinforcers at each time constitutes the organism’s prediction about what is likely to produce a reinforcer at that time, and hence its choice between responses. Although we illustrate these processes separately and sequentially in this article, the two processes are in fact applied simultaneously (Appendix, see Equation 3) when fitting the model to the data.

Thus, the meaning of now and here is subjective; generalization means that what the organism discriminates about the present situation depends on both the past and the future, and on events that have not occurred at this location at all. Control is by apparent relations between events; generalization causes differences between apparent and obtained, and hence, apparently weaker environmental control. Generalization occurs even under highly discriminable environmental conditions (e.g., Davison & Jones, 1998). We have shown that a conceptually similar model describes behavior across a wide range of tasks (e.g., Cowie et al., 2014, 2016; Cowie & Davison, 2020).

In the standard operant chamber, discriminating where an event occurred is typically the same as discriminating what that event was, because we tend to define our reinforcers according to the response that produced them—left or right—rather than according to whether they are, for example, wheat or corn. Nevertheless, a what dimension might be necessary in an experiment in which qualitatively different reinforcers may be obtained. There are many more dimensions that could be relevant, depending on the contingency (i.e., which dimensions of an event must be detected in order to detect the structure of the environment). For example, reinforcer availability might differ according to color (e.g., Rayburn-Reeves et al., 2011), flash duration (e.g., Krägeloh & Davison, 2003), or number of events (e.g., Davison & Cowie, 2019). Thus, the dimension across which generalization occurs will depend on the procedure—but the nature of the generalization processes appears similar across dimensions. Where different reinforcer conditions are associated with highly discriminable stimuli, generalization will be less than when the stimuli are confusable, or even absent. For example, we might expect minimal generalization across components signaled by color in a multiple schedule, and substantial generalization across components in the equivalent mixed schedule.

This approach assumes that generalization processes across different stimulus dimensions are similar in nature, but separate (see Cowie & Davison, 2020, for details). As a result, the model allows us to separate the relative contributions to imperfect control made by each generalization process. In any situation where reinforcers (or other currently valuable events) are multidimensional (i.e., they occur in time, space, or other stimulus dimensions), generalization across each and every dimension will contribute to systematically imperfect prediction. Indeed, the model has identified location-based generalization even in procedures in which response locations ought to be highly discriminable (see Cowie, Davison, & Elliffe, 2016, for more details). To attribute generalization to a single dimension—as is common with models of timing, for example—is unwise. Such an oversight inflates the extent of generalization across that dimension. Reinforcers occur in time and in space and in other environments, and generalization occurs across all these dimensions (see Blough, 1972).

The separate, independent parameters for each generalization process may also be used as a tool to understand how aging, impairment, or other population-specific characteristics affects generalization processes and control by the likely future. We know that many psychological impairments are associated with a deficit in prospective control and planning, and our modeling approach allows us to understand how generalization contributes to such deficits. Environmental control may not be fundamentally different for these populations; some form of generalization might obscure all order in the environment, making behavior appear not to be under control of contingencies (see, e.g., McCarthy, Corban, Legg, & Faris, 1995). Behavior resulting in overselective control by the recent past might also be understood as the result of the same sort of process. Generalization processes, and their effect on the ability to predict the future, might be crucial to understanding what we consider “abnormal” patterns of behavior.

Bringing the Past into the Present to Predict the Future

An organism has many past experiences, but not all of those are germane to navigating the present. Generalization may also be key to understanding which past experience is brought into the present to predict the future. Generalization is likely to occur from past situation(s) that most closely resemble the present, and not necessarily from more recent experience (e.g., see Estes, 1944; Reid, 1958). When a behavior has been extinguished, an environmental change that makes the current context more similar to the context in which that behavior once produced reinforcers will cause the extinguished behavior to occur (e.g., Reid, 1958). The contextual cues that promote generalization from the more distant past include the delivery of reinforcers that used to be produced by the just-extinguished behavior (e.g., Reid, 1958), a return to the training context in what that behavior used to be reinforced (e.g., Bouton, Todd, Vurbic, & Winterbauer, 2011), and the removal of a behavior that provided reinforcers only when the other behavior did not (e.g., Trask, Schepers, and Bouton, 2015). The more similar the current context to the training context, and the more different from the extinction context, the more likely the previously extinguished behavior is to return (e.g., Miranda-Dukoski, Bensemann, & Podlesnik, 2016; Podlesnik & Miranda-Dukoski, 2015; see also Bai, Cowie & Podlesnik, 2017). This generalization from more distant past to present is observed with both operant and respondent behavior (e.g., Bouton & Bolles, 1979), and appears to apply not only to stable responding, but to behavior in transition (e.g., Hunter & Rosales-Ruiz, 2019). Thus, generalization from the past to the present depends in part on the similarity of the present to past conditions.

More than one past situation may generalize to the present. When stimuli in the present have each been associated with a different past, these different pasts generalize to the present and compete for control over behavior. In many situations, generalization from multiple, different, past situations weakens control by the environmental stimuli because the contingencies in these pasts were not the same. For example, Miranda-Dukoski, Davison, and Elliffe (2014) showed that when the likely location of reinforcers changed systematically across time, control was enhanced by a single marker stimulus associated with a particular pattern of change in reinforcer location. When the same marker was associated with several different changes in likely reinforcer availability, control in their presence was attenuated. This failure of stimulus control appears to reflect generalization from multiple past situations in which the stimulus conditions were similar, but the reinforcer conditions were not.

Generalization from multiple past situations occurs when the environment contains stimuli that were relevant in more than one past situation. Cowie et al. (2017; see also Gomes-Ng, Elliffe, & Cowie, 2018) showed that when elapsed time signaled the likely location of the next reinforcer, and a red key light signaled the definite location of the next reinforcer, control was divided between time and key-light color. Choice shifted to the signaled response, in accordance with control by the key-light signal, but behavior during the signal continued to change across time in accordance with the time-based contingency. Here, the current environment included different stimuli associated with different contingencies, and thus more than one past generalized to the present.

A further generalization problem occurs when the wrong past situation is generalized to the present, or when the present is sufficiently different that it occasions no generalization. For example, Sharp, Williams, Rörnes, Lau, and Lamers (2019) discuss how environmental factors such as the presence of coffee and biscuits, or the layout of furniture in a manner similar to in one’s own home, occasion social interaction in rest homes. In the absence of these cues, environments intended to promote social interaction may attenuate generalization from appropriate past experiences. Control by more distant past situations that appear more similar to the present conditions may also contribute to habitual behavior, which appears maladaptive in light of more recent experience.

Generalization determines which past experience or experiences gets brought into the present. What gets generalized to the present depends on the present environment, and its similarity to past environments, the apparent reliability of cues in those different past experiences (e.g., Cowie et al., 2017; Gomes-Ng et al., 2018), and the amount of training of such experiences in the past. Generalization is often adaptive; it allows the organism to use past experience to solve present-situation problems. The exception is when features of the current environment occasion generalization of multiple, incompatible, or simply wrong, past experiences. Under these conditions, control is divided among different past experiences, with the apparently most reliable predictors of the future exerting the greatest control.

Conclusion

Generalization is fundamental not only because it distorts our experience of the present, and hence the apparent relation between events in time, space, and other dimensions, but also because it determines what past experiences get brought into the present to control behavior. The prospective-control approach outlined here conceptualizes behavior in terms of the structure of the environment; it uses behavioral principles and measurable events to understand apparently complex behavior that is future oriented, and in this sense, goal-directed. The organism’s tentative future is constructed from events that have already taken place; in this way, this prospective-control approach retains the rigorous, measurable qualities characteristics of a science of behavior. The degree to which behavior comes under control of the environment’s structure depends on generalization across past, present, and future stimulus conditions. This assumes, of course, that all behavior is choice behavior (Herrnstein, 1970), and that generalization determines the degree to which present choice comes under control of the current environmental conditions. Generalization across time, space, and other dimensions of an event weakens control by the current environment. Generalization from past experience to the present allows the organism to navigate the future, though not necessarily accurately. Control by the future need not place control back inside the organism; animals learn about the likely future from the experienced past, and behave accordingly. This approach implies no agency; no inner pigeon. It requires no teleology, and no control by events that have not yet taken place. The focus of our enquiry remains measurable and observable.

Under this prospective approach, control remains in the past, but in a different way from a traditional reinforcement-strengthening approach. We generalize from our past experience to discriminate the likely future, and we act in the present according to what our current environmental conditions signal about possible futures. Environments contain multiple signposts to the future (Shahan, 2010), and the ones that we follow depend on what future conditions we discriminate that they will lead to, on our current disposition (Killeen & Jacobs, 2017), and on which of these future conditions is currently valuable. The control is a pull from the likely future, and a push from the present; it is not a simple mechanistic push from the past (e.g., see Cowie, 2018, 2019; Cowie & Davison, 2016). Past experience informs and guides future behavior. To behave is to choose a future.

Appendix

The present article is intended to illustrate a conceptual approach to understanding why the environment exerts imperfect control over behavior. For these purposes, we adopt the equations used by Cowie and Davison (2020) to model the generalization of reinforcers across time and location shown in Fig. 1. To model temporal generalization, reinforcers in each time bin were redistributed across surrounding time bins according to a Gaussian function with standard deviation (s) at time t since a marker event (Panels C and D in Fig. 1):

$$s_t=s_0+\frac{a_s}{1+e^{-\left(\frac{t-X_0}{{\beta }_s}\right)}}.$$ 1

In Equation 1, the parameter a is the extent of the increase in generalization between the times at which generalization is least (s₀) and most likely (i.e., the asymptote). X₀ is the time (x-value) at which s_t is halfway between its asymptotically low and high values, and β is the slope of the function around this point (i.e., the speed with which generalization increases).

Because of the discrete nature of the two response locations in the procedure, we modeled generalization across location by shifting a proportion of reinforcers at each time m to the other alternative. The proportion of reinforcers generalized to the other location (m) at time t (Panels E and F in Fig. 1) was calculated as:

$$m_t=m_0+\frac{a_m}{1+e^{-\left(\frac{t-X_0}{{\beta }_m}\right)}}.$$ 2

The parameters in Equation 2 are the same as in Equation 1, but apply to generalization across location (m) rather than time (s). As Cowie and Davison (2020) did, we used the same X₀ parameter for both temporal (s) and spatial (m) generalization.

The discriminated reinforcers (R’) in Panels E and F of Fig. 1 are thus derived from the obtained reinforcers using the equation:

$$\mathit{log}\frac{R{\prime }_{1,t}}{R{\prime }_{2,t}}=\mathit{log}\left(\frac{{\sum }_{n=1}^{\mathit{\text{tmax}}}f\left(t,n,\mathit{\gamma n}\right)⌊\left(1-m\right)R_{1,n}+m{R}_{2,n}⌋}{{\sum }_{n=1}^{\mathit{\text{tmax}}}f\left(t,n,\mathit{\gamma n}\right)\left[\left(1-m\right)R_{2,n}+m{R}_{1,n}\right]}\right)+logc.$$ 3

In this instance, the parameters are the same as in Equations 1 and 2, and tmax is the maximum time since a marker event, dictated by the procedure itself. In the example in the present article, we displayed the effects of the two generalization processes sequentially to illustrate their separate effects on the discriminated structure of the environment. As Equation 3 shows, both processes are in fact applied simultaneously when fitting the quantitative model to the data.