Internal-Clock Models and Misguided Views of Mechanistic Explanations: A Reply to Eckard & Lattal (2020)

Abstract

Eckard and Lattal’s Perspectives on Behavior Science, 43(1), 5–19 (2020) critique of internal clock (IC) mechanisms is based on narrow concepts of clocks, of their internality, of their mechanistic nature, and of scientific explanations in general. This reply broadens these concepts to characterize all timekeeping objects—physical and otherwise—as clocks, all intrinsic properties of such objects as internal to them, and all simulatable explanations of such properties as mechanisms. Eckard and Lattal’s critique reflects a restrictive billiard-ball view of causation, in which environmental manipulations and behavioral effects are connected by a single chain of contiguous events. In contrast, this reply offers a more inclusive stochastic view of causation, in which environmental manipulations are probabilistically connected to behavioral effects. From either view of causation, computational ICs are hypothetical and unobservable, but their heuristic value and parsimony can only be appreciated from a stochastic view of causation. Billiard-ball and stochastic views have contrasting implications for potential explanations of interval timing. As illustrated by accounts of the variability in start times in fixed-interval schedules of reinforcement, of the two views of causality examined, only the stochastic account supports falsifiable predictions beyond simple replications. It is thus not surprising that the experimental analysis of behavior has progressively adopted a stochastic view of causation, and that it has reaped its benefits. This reply invites experimental behavior analysts to continue on that trajectory.

Misconstruing the Internal Clock

According to Eckard and Lattal (2020), invoking an internal clock (IC) or any other unobservable mechanism to explain temporally controlled behavior should be avoided because (1) they introduce an unnecessary intermediate link in the explanatory causal chain, and (2) they shift the focus of research from observable behavior to unobservable mechanisms. This reply argues that their admonition is unwarranted because it is based on (1) a mischaracterization of IC models and the IC framework of research on interval timing, and (2) a commitment to a notion of scientific explanation that is too constrained to accommodate essential characteristics of behavior.

The mischaracterization of IC models starts with a narrow definition of clock. Eckard and Lattal’s (2020) implicit definition is arbitrarily restricted to manufactured clocks (e.g., sundials, grandmother clocks), which makes any reference to a biological or psychological clock metaphorical at best. The IC construct is most intelligible when a broader definition of clocks, one that includes anything that can track time, is adopted. To the extent that they have timekeeping properties, certain chemical reactions (Briggs & Rauscher, 1973), some neural mechanisms (Paton & Buonomano, 2018; Tallot & Doyère, 2020), and even whole organisms, are clocks. The temporal regularity of some collateral behavior of rats and pigeons performing a temporal discrimination task may be used to track the passage of time (Fetterman, Killeen, & Hall, 1998); if properly “wound up,” rats and pigeons and humans are clocks. Internal-clock models do not necessarily treat behaving organisms as if they carried a clock—they can be treated as being clocks, that is, as having timekeeping properties.

Eckard and Lattal’s (2020) interpretation of the internal nature of ICs is similarly narrow and uncharitable. It assumes that the IC is circumscribed to a spatial location inside the organism, completely isolated from the environment. Some IC models indeed claim to identify specific neural structures responsible for timekeeping (Merchant, Harrington, & Meck, 2013), but for many others the spatial location of their material instantiation is less important. In fact, the behavioral models that Eckard and Lattal present as alternatives to IC models (Dragoi, Staddon, Palmer, & Buhusi, 2003; Jozefowiez, Staddon, & Cerutti, 2009; Killeen & Fetterman, 1988; Machado, 1997) are themselves IC models (more on this below) that have little concern with the spatial location and material implementation of the mechanisms they propose. The IC is internal (or endogenous) to the organism because—independent of its material implementation—it is a property of the organism and not of the environment, the same way that the probability of rolling a six is a property of (i.e., it is internal to) the rolled die, and not of the surface on which the die rolls. The external–internal distinction conceptually separates the environmental periodicity to which the organism is responsive (external) from the mechanism that governs such responsiveness (internal).

Mechanisms and Explanations

Although it is beyond the scope of this reply to discuss in detail what constitutes a mechanism and what makes it a good explanation, it is critical for the present argument to outline some aspects of these concepts. Mechanisms can be material things, such as the cogs and gears in a machine, but they can also be explanations—mechanistic explanations. The material and heuristic features of a mechanism are different but not exclusive: an engine can have a camshaft and a piston and also explain how fuel combustion moves a car.

Mechanistic explanations may be defined in relation to functional explanations. The former identify efficient causes to answer questions of how; the latter identify final causes to answer questions of why (Killeen, 2001; Rachlin, 2017). Genes, mutations, sexual signaling, kin recognition, etc. are mechanisms of natural selection operating at multiple levels; reproductive success and fitness are functions of natural selection that these mechanisms subserve.1 Mechanisms can be simulated; functions emerge from these simulations. Neither form of explanation is better than the other; they simply explain different aspects of the same phenomenon. The distinction between these forms of explanations may be expressed in terms of Marr’s (1982) three levels of analysis: mechanistic explanations correspond to Marr’s algorithmic and implementational levels when the invoked mechanisms are, respectively, computational and physiological; functional explanations correspond to Marr’s computational level. Internal-clock models are primarily computational mechanistic (i.e., algorithmic) explanations: they aim at explaining how a simulatable system that is also biologically and behaviorally plausible may track time in the seconds-to-minutes range. The material (neural) instantiation of IC models, although a rich field of inquiry (Tallot & Doyère, 2020), does not necessarily determine the merit of an IC model as a mechanistic explanation of temporally controlled behavior, just like the material instantiation of genes (or Mendelian “factors”) in DNA says little of their merit in explaining trait heredity.

Billiard-Ball Causation

Perhaps the most consequential of Eckard and Lattal’s (2020) mischaracterization of the IC is in regard to its purpose. To examine this mischaracterization, it is first necessary to fully unfold Eckard and Lattal’s argument. According to them, the IC is meant to fill the temporal gap between periodic environmental events and their effect on behavior. The premise of this argument is the assumption that, everything else being equal, a change in the world will immediately cause a predictable and contiguous change in the world, which, by extension, will immediately cause another predictable change, and so on. This assumption of billiard-ball causation (Fig. 1A) is at the heart of what Slife, Yanchar, and Williams (1999) call metaphysical determinism: there is a level of knowledge of the world at time t that is sufficient and necessary to predict the world at time t + 1 and, by extension, at any time in the future.

Fig. 1.

Sketch Representation of billiard-ball causation (Panel a) and stochastic causation (Panel b). Squares represent observable changes in either the environment (E) or in behavior (B); circles represent unobservable intermediate events. Observable and unobservable events in billiard-ball causation are time tagged (t, t + 1, etc.) The dashed squares in stochastic causation represent possible behavioral outcomes that are not realized in a particular instance

If an experimental manipulation at time t produces a noncontiguous behavioral effect at time t + x where x >> 1, billiard-ball causation demands that intermediate contiguous changes fill up the temporal or spatial gap between manipulation and effect at times t + 1, t + 2, . . . , t + x – 1. Eckard and Lattal (2020) embrace billiard-ball causation as the only form of causation, and therefore assume that all IC models embrace it too. What they disapprove of in mechanistic models is that these models seek to characterize the intermediate contiguous links in the causal chain and present those characterizations as explanations.

Within the framework of billiard-ball causation, Eckard and Lattal (2020) correctly point out two problems with the gap-filling goal attributed to mechanistic models. First, intermediate links lack heuristic value. To the extent that the causal chain is consistent, the relation between manipulation and effect would be consistent. Therefore, prediction and control of behavior—the aims of the experimental analysis of behavior from a radical behaviorist perspective—can be achieved through proper control of environmental variables, informed just by the functional relation between environmental and behavioral variables. Second, intermediate links are unparsimonious: they carry with them a baggage of variables and processes that are absent in simple functional relations. Moreover, because those links are neither in the environment (which serves as first link) nor in the behavior (which serves as last link), they must be inside the organism in the form of hypothetical physiological or computational mechanisms. These mechanisms are relatively unconstrained by data: their predictions can always be improved by adding a modulator to the neuronal circuit or a subroutine to the algorithm. The flexibility of these hypothetical mechanisms makes them vulnerable to unchecked growth in the number of parameters and components that constitute them. Given that intermediate links, such as those that presumably constitute the IC, explain little and complicate much, why not simply dispense with them?

Computational ICs and similar mechanistic models are indeed intermediate between environmental manipulation and behavior and are, consequently, not observable.2 Such location, however, is not necessarily meant to fill a temporal gap between manipulation and effect, because that gap assumes billiard-ball causation, and IC models do not require such an assumption. Instead, a larger class of mechanistic models, which includes IC models, in general assume a different form of causation, stochastic causation, which is closely related to what Slife et al. (1999) call metaphysical probabilism.

Stochastic Causation

In contrast to billiard-ball causation, stochastic causation is the assumption that a change in the world may cause any of a number of subsequent changes in the world, only some of which are realized. Instead of representing the effect of a manipulation as a single sequence of links in a chain, stochastic causation demands a more complex tree-like representation connecting each event with all its possible outcomes (Fig. 1B). Stochastic causation implies that behavioral predictions from environmental manipulations are best described as probability distributions of behavioral outputs. Formulating those distributions as outputs of a process requires a characterization of the intermediate links connecting the triggering environmental event with the expected behavioral effects. Without a characterization of intermediate links, predictions of behavior from environmental manipulations are limited to imprecise qualitative descriptions. Whereas the logical-mathematical connection between branches of a causal tree is critical for a quantitative account of behavior, their temporal or spatial contiguity is not necessarily relevant. The intermediate, unobservable branches of the causal tree constitute a computational IC that mechanistically explains the distribution of temporally controlled behavior.

Freestone and Balcı’s (2019) Bayesian implementation of Timberlake’s (2001) behavior systems theory illustrates the importance of the mathematical characterization of intermediate branches in a stochastic causal tree. Timberlake’s (2001) theory postulates a hierarchical organization of observable behavior that makes qualitative predictions on learning and conditioning. Freestone and Balcı’s (2019) implementation assumes a Bayesian network connecting behavioral categories across hierarchical levels (subsystems, modes, modules, and actions). In this model, the unobservable Bayesian network mediates and updates the relation between environmental contingencies and the observed distribution of actions. Thus the network operates as the branches of a stochastic causal tree that is positioned to provide mathematically precise behavioral predictions, potentially overcoming the constraints inherent to the qualitative nature of Timberlake’s (2001) theory.

To illustrate the contrast in the implications of billiard-ball and stochastic causation in the analysis of timing behavior, consider the typical performance under fixed-interval (FI) schedules of reinforcement. In FI schedules, response rate appears to be low and constant shortly after reinforcement; then, at a start time, responding increases sharply and remains high and constant until subsequent reinforcement (Church, Meck, & Gibbon, 1994; Daniels & Sanabria, 2017b; Hanson & Killeen, 1981). Start times have been modeled in multiple ways, but a fairly common characterization is that they are gamma-distributed, centered around a fixed proportion of the interreinforcer interval (Hanson & Killeen, 1981; Sanabria, Thrailkill, & Killeen, 2009). It is also well-established that, over a broad range of FI requirements, the standard deviation of start times is approximately proportional to their mean (Church et al., 1994), a regularity often referred to as scalar invariance.

How do gamma-distributed start times emerge from experiencing periodic reinforcement? From the perspective of billiard-ball causation, if the interreinforcer intervals are measured under similar circumstances, start times should be similar to one another, so the only source of variance is measurement error. From this perspective, as control over variables that affect start times is tightened (e.g., handling conditions and levels of deprivation are held constant, testing is always conducted under the same circumstances) and measurement is more precise, variation in start times should approach zero. There is no evidence, however, that error is a substantial source—never mind the only source—of variance in FI start times. Moreover, why would error increase proportionally to the FI requirement?

From the perspective of stochastic causation, at least some of the variability in temporally controlled behavior is intrinsic to the mechanism that governs that behavior. Therefore, to explain temporally controlled behavior it is necessary to formulate and test potential sources of variation that, unlike error, are internal to the behaving system. In the specific case of gamma-distributed start times, the first step is to hypothesize the simplest process that yields a scale-invariant gamma distribution. An IC based on clocked Bernoulli modules (CBMs; Killeen, Hall, Reilly, & Kettle, 2002) in Fig. 2 provides a reasonable approximation. A CBM (Fig. 2A) simply increases a counter n after a geometrically distributed pause with mean τ/π.3 At the onset of the discriminative stimulus (S^D, left side of Fig. 2B) the IC resets n and initiates a CBM; when n reaches a criterion N, an abrupt increase in response rate occurs and a second CBM continues adding to n; once the FI requirement is completed, a reinforcer is delivered, pause τ is updated proportionally to n, and the cycle starts again. At equilibrium, this learning algorithm yields start times that are approximately scale invariant and gamma distributed with shape parameter N and scale parameter τ/π, centered near a constant fraction of the FI requirement.

Fig. 2.

Flowchart Representation of an Illustrative Internal Clock (IC) Based on Clocked Bernoulli Modules (CBMs), Implemented to Track Performance in Fixed-Interval (FI) Schedules of Reinforcement. Panel A shows the CBM subroutine, in which a counter n increases after a pause τ with probability π. Panel B shows an IC that uses two concatenated CBMs. Onset of the timed discriminative stimulus (S^D, on the left) resets counter n and reinforcement flag R, and initiates the first CBM. Once n reaches criterion N, a response subroutine is initiated along with a second CBM. This CBM continues adding to n until the FI elapses, a response is emitted, and reinforcement is delivered, which sets the R flag to 1. Pause τ is then updated proportionally to n before the cycle starts again

Alternative IC models replace the CBM and feedback components in Figure 2 with functionally similar mechanisms, such as diffusion processes and decision thresholds (e.g., Balci & Simen, 2016), that yield similar distributions. Despite the diversity of IC models, they all share the computational features necessary to track changes in the periodicity of environmental events. Treisman’s (1963) prototypical model identifies such features as a pacemaker, an accumulator (n in Fig. 2), a comparator (decision rule “n = N?”), and an updatable memory (encoding n and updating τ). As mentioned before, even the models that Eckard and Lattal (2020) present as alternatives to IC models include these features, and are thus IC models themselves—even if they claim otherwise. For instance, in Dragoi et al. (2003), a “response trace” of competing activities declines in such way as to function as an accumulator, which is compared to the trace of the target activity, both of which are updated by rate of reinforcement. Both traces are internal to the timing system and neither is observable.

Falsifiable Predictions

As the FI example suggests, billiard-ball causality restricts inferences to induction. Inductive reasoning is useful when relevant data are scarce, as shown, for instance, in Ferster and Skinner’s (1957) foundational work on schedules of reinforcement. Such utility, however, has a low ceiling. The value of predicting that similar events will be observed if conditions are replicated (e.g., FI schedules maintain start times distributed in a particular way) only goes so far. This limitation is evident in Eckard and Lattal’s (2020) explanation of temporal control in FI schedules of reinforcement as a gradual shift in the nominal S^D from an effective S^Δ to an actual S^D as the interreinforcer interval elapses. Because this explanation is silent about the factors that govern the temporal location and variability of such shift, it cannot make predictions about how such factors would affect performance under different circumstances. It is, in fact, unclear whether the S^Δ-to-S^D hypothesis is even falsifiable. What kind of data would prove such hypothesis wrong?

In contrast, mechanistic IC models, by identifying hypothetical mechanisms that may govern FI performance, constitute the framework from which falsifiable predictions may be formulated. For instance, consider how an IC such as the one in Figure 2 may explain the scalar invariance of start-time distributions in FI schedules. Scalar invariance implies that the ratio of standard deviation to mean, the coefficient of variation, is constant. The model in Figure 2 predicts that the coefficient of variation of start times is approximately (1/N)^0.5. Therefore, the simplest CBM-based explanation of scalar invariance involves keeping the criterion N constant, which the model in Figure 2 does. Also, if N is constant, then for the mean start time to be proportional to the FI requirement, the mean interval between increases in n (i.e., τ/π) must be proportional to the FI requirement. The CBM-based model maintains this proportionality by updating τ proportionally to n, which in turn is proportional to the FI requirement. An interesting possibility is that τ changes proportionally with n because it tracks the rate of reinforcement (Killeen & Fetterman, 1988). Therefore, the CBM-based model suggests that increasing rate of reinforcement while keeping the FI requirement constant (e.g., increasing the reinforcer magnitude) speeds up both CBMs in Figure 2, which results in n reaching N faster, which would be expressed as shorter mean start times. Counter n would also be larger when reinforcement is delivered, which would lengthen τ and slow down the CBMs, so the model also predicts that the rate-of-reinforcement effect would be transient. Changes in performance with varying reinforcer magnitude (Ludvig, Balci, & Spetch, 2011) and FI requirement (Sanabria & Oldenburg, 2014) are consistent with these predictions.

Parsimony

Mechanistic IC models not only provide broader predictions than purely inductive strategies; they also provide more parsimonious explanations of temporally controlled behavior. If the scope of an IC model like the one in Figure 2 were restricted to stable FI performance, it would be difficult to justify its complexity, in particular when a couple of linear functions relating FI requirement to mean and standard deviation of start times would make similar predictions. The scope of most IC models, however, is not performance in a particular interval-timing task, but on any task in which performance reveals the timekeeping properties of the subject, including peak-interval, temporal-bisection, time-left, and relative-duration tasks, among others (e.g., Lejeune & Wearden, 2006). Accounting for a similar range of data using a purely inductive strategy would require a separate set of parameters for each interval-timing task. From a purely inductive perspective, there is no meaningful common ground among interval-timing tasks, so performance on each requires its own explanation. After all, the reason for a particular set of reinforcement contingencies to qualify as an interval-timing task is that it assesses the operation of a hypothetical IC.

Where is the Focus?

Eckard and Lattal’s (2020) first criticism of ICs is that they are unnecessary intermediate links in the explanatory causal chain. The chain, however, is a misleading metaphor of causality; the tree is a better one (Figure 1). The causal branches of the metaphoric tree are just as important as any other part of it, even if hidden under the ever-moving canopy. Eckard and Lattal’s (2020) second criticism of ICs is that they shift the focus of research away from observable data and toward unobservable constructs. Of course, this criticism would be valid if those unobservable constructs lacked heuristic value, which they do not—they are the branches that explain the shape of the canopy. Focusing exclusively on the data is limiting at best and misleading at worst: It is limiting because it restricts all scientific inference to induction. It is misleading because it disregards the role of instruments and theoretical assumptions in the production and selection of data, conflating measurements of an object with the object being measured. Lever presses and key pecks, response rates and start times, are not the same as the behavior that produces them. Confusing behavior with behavioral data is a form naïve operationism (Grace, 2001). To aim at behavior one must focus beyond behavioral data, conjecturing about the processes that underlie the data and the appropriate measurement categories to infer those processes.

A More Inclusive Picture

Internal-clock models—and mechanistic models in general—can be more than “descriptive aids,” “metaphors,” or “heuristics” that can “be manipulated at will to fit the scientist’s need” (Eckard & Lattal, 2000, p. 13). Nor are they limited to serve as provisional constructs laboring to graduate to observable status. They imply no category mistake (Ryle, 1949) or dualism—the IC in a timekeeping organism is no more ghostly than the probability of heads in a coin flip or the spell checker in a word processor. They are not “mental way stations” (Skinner, 1963); they are, if the analogy is valid, mental transportation networks.

The components of IC models are intermediate between environmental input and behavioral output; as computational models, they are hypothetical and not observable. They are also extremely useful for behavior analysis—the transportation network may not be visible, but its structure is testable and, for its range of use, seemingly simple. The adoption of mechanistic models requires dispensing with billiard-ball causation and acknowledging the inherent stochasticity of behavior (Maye, Hsieh, Sugihara, & Brembs, 2007; Neuringer, 1991; Sanabria & Thrailkill, 2009). In fact, the notion of stochastic causation and its implications are not entirely foreign to behavior analysis (Neuringer & Jensen, 2012; Slife et al., 1999). The experimental analysis of behavior has already benefitted from adopting stochastic mechanistic models, such as those that account for the microstructure of instrumental behavior and the factors that selectively influence the various components of that microstructure (Brackney, Cheung, & Sanabria, 2017; Brackney & Sanabria, 2015; Cheung, Neisewander, & Sanabria, 2012; Daniels & Sanabria, 2017a; Jiménez, Sanabria, & Cabrera, 2017; Shull, Gaynor, & Grimes, 2001), and those that successfully implement selectionist views on instrumental reinforcement (McDowell, 2004, 2013).

The reliability of early experimental findings appears to have discouraged the field of behavior analysis to forswear billiard-ball causation. Such reliability, however, hinges on the law of large numbers, aggregating individual responses and choices into absolute and relative response rates. These are important findings, but there is no reason to limit the field to them. A richer picture emerges once behavior is disaggregated and its statistical properties are examined. This is the picture that emerges from analyzing temporally controlled behavior within the framework of IC models. It is a picture that is consistent with the reliability of aggregated data, but that opens new windows for further exploration (Sanabria, Daniels, Gupta, & Santos, 2019). It is a picture that easily maps to neurobiological hypotheses of interval timing (e.g., Gupta et al., 2019). In short, it is a more inclusive picture, one that cannot be attained, however, if mechanistic models are kept out of the behavioral scientist’s conceptual toolbox.

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