The development of procedures to reduce anxiety and fear is important in clinical psychology. Anxiety-related disorders and problems are pervasive (Kessler et al., 2005) and have serious effects on a patient’s quality of life (e.g., Rapaport et al., 2005). One of the most effective clinical interventions is exposure therapy, designed to eliminate anxiety and fear by deliberately continuing exposure to fear-evoking stimuli (Abramowitz et al., 2019). The effectiveness of exposure therapy on maladaptive anxiety and fear has been confirmed (e.g., Watts et al., 2013), and cognitive behavior therapy includes the element of exposure therapy (Abramowitz et al., 2019). Experimentally, exposure therapy is a clinical analog for extinction procedures that have received considerable study in human and nonhuman animals, using Pavlovian fear conditioning (e.g., Craske et al., 2014; Craske et al., 2022).

In fear conditioning, a subject acquires a fear response to a neutral stimulus (conditioned stimulus [CS]) through pairings of the stimulus with an unconditioned stimulus (US). After fear conditioning, the procedure for extinction presents the CS without the US, and the conditioned response (CR) to the CS is usually diminished (Pavlov, 1927). The explanation for exposure therapy by extinction assumes that maladaptive or pathological fear and anxiety are acquired by pairings of experiences and events that are analogs to the CS and US from conditioning, and that any acquired response can be attenuated by presentations of the CS alone—that is, exposure to fear-provoking stimuli (Craske et al., 2014). This framework suggests that exposure therapy is an application of the extinction procedure in clinical settings; the effect is due to a response decrement to a CS by the extinction procedure. One advantage of this framework is that it speaks to strategies to reduce fear experienced during relapse following therapy known as recovery-from-extinction effects (McConnell & Miller, 2014).

Promoting extinction and recovery-from-extinction effects

Extinction can be promoted by various procedures, for instance simply increasing the number of extinction trials (Pavlov, 1927). Whereas with deepened extinction (Rescorla, 2000, 2006), a target CS is presented without the US together with other stimuli that elicit a similar CR. This results in a larger decrement in the CR to the target CS. On the other hand, if the target CS and other stimuli that inhibit the CR are simultaneously presented without the US in extinction, the reduction in CR to the CS can be suppressed, which refers to protection from extinction (Lovibond et al., 2000; Rescorla, 2003).

There are also four types of recovery-from-extinction effects that might be relevant for exposure therapy: renewal, spontaneous recovery, reinstatement, and rapid reacquisition (Bouton, 2002). The renewal effect is defined as the reappearance of a CR by changes to the physical context after extinction (e.g., Bouton & Bolles, 1979) and can be categorized into three types. The most common and robust type is ABA renewal, which is the reappearance of extinguished CR when testing is conducted in the acquisition context after extinction is performed in a second context (Bouton & Bolles, 1979). The second type is ABC renewal (Bouton & Bolles, 1979), which is the reappearance of CR when acquisition, extinction, and test phases are all conducted in different contexts. The third type is AAB renewal (Bouton & Ricker, 1994), which is a phenomenon in which the CR reappears when acquisition and extinction are conducted in the same context and then tested in a different context. Spontaneous recovery occurs when a CR reappears over time following extinction, which was first reported by Pavlov, 1927. Reinstatement is defined as the reappearance of an extinguished CR by the presentation of a US alone after extinction (Rescorla & Heth, 1975). Rapid reacquisition is a phenomenon in which a CR in the reacquisition phase following extinction develops more rapidly than in the initial acquisition phase or more rapidly than to a novel stimulus (Bouton & Swartzentruber, 1989). Taken together these phenomena suggest that various factors can lead to relapse following exposure therapy.

Recovery-from-extinction effects can also be diminished or eliminated. Procedures that improve the effects of extinction can decrease recovery-from-extinction effects. For example, the massive extinction (i.e., introducing many trials of extinction) delays the reacquisition rate (e.g., Bouton, 1986; Bouton & Swartzentruber, 1989). However, the literature investigating the effect of this procedure on other types of recovery-from-extinction effects showed inconsistent findings. Although some studies have indicated that AAB renewal is reduced or eliminated (Rauhut et al., 2001; Rosas et al., 2007; Tamai & Nakajima, 2000), the effect on the ABA or ABC renewal is mixed (Denniston et al., 2003; Rosas et al., 2007; Tamai & Nakajima, 2000). In compound extinction, deepened extinction (i.e., compound extinction using another exciter) decreases subsequent recovery of the CR (Culver et al., 2015; Rescorla, 2006; Thomas & Ayres, 2004), although compound extinction using another inhibitor increases its effects (Thomas & Ayres, 2004). Thus, the promoting effect of extinction can decrease subsequent recovery-from-extinction effects.

Other procedures have also been developed to decrease the recovery-from-extinction effects. For example, US presentations during the extinction procedure can decrease the recovery-from-extinction effects. This strategy can be categorized into two types. First, occasional delivery of CS–US pairings during extinction (Bouton et al., 2004; Gershman et al., 2013) is especially effective in delaying the reacquisition rate (e.g., Bouton et al., 2004). The second procedure involves the non-contingent presentation of CS and US in an explicitly unpaired manner. These procedures effectively eliminate recovery-from-extinction effects, with the effect of the latter being greater than that of the former (Bouton et al., 2004; Lipp et al., 2021; Rauhut et al., 2001; Thompson et al., 2018). Some other procedures, such as the presentation of an extinction cue (Brooks & Bouton, 1993, 1994), extinction in multiple contexts (Chelonis et al., 1999; Dunsmoor et al., 2014; Gunther et al., 1998; Laborda & Miller, 2013, although see Bouton et al., 2006), extinction of recovery-from-extinction effects (Holmes & Westbrook, 2013; Quirk, 2002; Rescorla, 2004), and extending the intertrial intervals (ITI; Urcelay et al., 2009) have been reported to decrease the recovery-from-extinction effects.

Many experiments investigating these effects have been conducted to decrease the relapse of anxious symptoms following an intervention. For example, findings regarding extinction in multiple contexts suggests that relapse following exposure therapy can be decreased by conducting interventions in many contexts or situations, supported by some clinical research (e.g., Vansteenwegen et al., 2007). The effects of other procedures have also been investigated in clinical situations and have been confirmed to be effective strategies in exposure therapy (e.g., Shin & Newman, 2018).

Can associative theories explain these phenomena?

Although strategies for exposure therapy based on Pavlovian conditioning are effective, it is difficult to provide a comprehensive theory that might explain all of these effects. Traditionally, the effects of Pavlovian conditioning have been explained using associative learning theory, which assumes that mental connections or associations develop between some events during conditioning. The Rescorla–Wagner model (Rescorla & Wagner, 1972), an influential associative model, can be used to explain fear reduction through the extinction procedure using an error correction rule. However, the Rescorla–Wagner model cannot account for many recovery-from-extinction effects without attributing some of the effects of extinction to the context. Although renewal effects are explained by assuming contextual conditioning, it does not seem that renewal effects are related to contextual conditioning (Bouton & Bolles, 1979; Bouton & King, 1983; but see Miller et al., 2020). Similarly, other major associative models also have similar problems in explaining recovery-from-extinction effects (McConnell & Miller, 2014). The attentional theory by Mackintosh (1975) and the comparator hypothesis (Miller & Matzel, 1988) present the same difficulty as the Rescorla–Wagner model because these theories postulate the elimination of the established CS–US association by extinction. Thus, it does not provide an explanation why eliminated associative strength reappears without considering contextual conditioning.

There are models that assume the establishment of an alternative association in the extinction phase, which is based on an idea originally introduced by Pavlov, 1927. Accordingly, Pavlov considered that learned content through CS–US pairings is not eliminated by experimental extinction because of findings about various manipulations that recover eliminated conditioned reflex, including spontaneous recovery. The configural model (Pearce, 1987) and the Pearce–Hall model (Pearce & Hall, 1980) assume that CS–no-US association—which functions to suppress excitatory association developed by CS–US pairings—is developed through extinction. However, both models fail to explain recovery-from-extinction effects, such as why simply moving from the extinction context would cause a recurrence of the CR (i.e., AAB renewal) without additional assumptions (McConnell & Miller, 2014).

In recent years, some computational models have been proposed to explain recovery-from-extinction effects. One of these is the latent cause model (Gershman et al., 2010; Gershman & Niv, 2012). The latent cause model assumes that animals infer causes of the US delivery from the combination of a CS and contextual stimuli; moreover, the model is formalized with Bayesian principles. Simulation and experimental data indicate that the occurrences of three type of renewal can be explained by the latent cause model (Gershman et al., 2010). However, these models explain only a part of the recovery-from-extinction effects and procedures to diminish them.

Bouton’s (1993) model has been widely accepted as an account of the extinction and recovery-from-extinction effects. Bouton’s model assumes that CR is determined by the degree of retrieval of excitatory and inhibitory associations, suggesting that CR occurs when retrieval of the excitatory association is stronger than the inhibitory association. One of the features of this model is that inhibitory association is context dependent. Thus, when a subject goes outside the extinction context, retrieval of the inhibitory association is weakened. As a result, the extinguished CR appears when the CS is presented outside the extinction context. One strength of this model is that it can explain the effect of context changes on a CR in various interference preparations, such as latent inhibition, counterconditioning, and reversal learning, as well as the extinction procedure (Bouton, 1993).

However, this model also has a limitation in that it is difficult to account for some findings of recovery-from-extinction effects (McConnell & Miller, 2014). For example, this idea cannot explain the difference between magnitude of renewal effects (ABA > ABC > AAB; Thomas et al., 2003). Furthermore, Bouton’s model cannot suggest how to manipulate procedures of exposure therapy to intensify the degree of symptom reduction with an exposure to feared stimulus, although it provides some strategies for diminishing relapse. This limitation might be caused by the fact that this model is a qualitative model, unlike other major associative models. Although this is good for handling many preparations abstractly, an explanation of how the inhibitory association changes and what strengthens it is critical for the application to exposure therapy. Solving this problem is important for improving the effect of exposure session and decreasing relapse.

Therefore, in this study, we propose an alternative model, a mathematical extension of Bouton’s model. We describe the details of our model and illustrate that the model can resolve some problems that traditional models cannot. Although our model is limited to explaining only the extinction and recovery-from-extinction effects unlike Bouton’s original model, it can provide an explanation and clinical implications for exposure therapy based on associative learning theory.

New model

Our model is a mathematical extension of Bouton’s model for extinction, which additionally incorporates the assumptions of many traditional models (Bouton, 1993; Capaldi, 1994; Laborda & Miller, 2012; Pearce, 1987; Pearce & Hall, 1980; Rescorla & Wagner, 1972). The present model postulates that excitatory and inhibitory associations are each formed by a prediction error rule, and that the retrieval strength of each association varies depending on the similarity between experienced contexts. In this model, CR is determined by the following formula:

$$\textrm{CR}= Ve\ast {S}^e+ Vi\ast {S}^i.$$
(1)

In Eq. 1, Ve is the value of the excitatory associative strength and Vi is inhibitory. Both strengths are affected by similarities, S, between a context in which the CS is presented and the contexts in which each association is developed. Se represents the similarity between the acquisition and present contexts, and Si represents the similarity between the extinction and present contexts. Similarities refer to the extent to which a subject recognizes that a context is the same as in another context (e.g., Pearce, 1987). Thus, Se refers to the extent to which a subject treats the present context as the same as the acquisition context, and Si is the extent to which a subject treats the present context is the same as the extinction context. These similarities determine the extent to which each association is retrieved within a context. If the similarity is high, the association is strongly retrieved in the context, whereas low similarity leads to weak retrieval. Thus, CR is determined by two factors: strengths in both associations and similarities between the present context and the contexts in which each association is developed.

Many studies on stimulus generalization in Pavlovian conditioning have mainly focused on the similarity between two stimuli and have indicated that the similarities are affected by the objective distance between two stimuli, such as hue and size. For example, as the physical distance between a CS and another stimulus (generalization stimulus [GS]) increases, the response to GS decreases (e.g., Guttman & Kalish, 1956). In the renewal paradigm, there is evidence that similarity between physical contexts is important (e.g., Podlesnik & Miranda-Dukoski, 2015; Thomas et al., 2003). Thus, the effect of contextual stimuli on renewal is also affected by the physical distance between the acquisition, extinction, and testing contexts. This assumption is similar to Pearce’s (1987) model, in which the degree to which the associative strength of one stimulus generalizes to another is determined by the similarity between the two stimuli. In our model, generalization occurs through a similar mechanism for retrieval strengths, which is a function of the contextual stimuli. The more similar the test context is to the context in which the association was acquired, the stronger the retrieval.

Our model assumes that Ve and Vi are developed based on simple error-correction rules. If a CS is paired with a US, ΔVet(the change in Ve on trial t) increases according to Eq. 2:

$$\Delta {Ve}_t={\alpha}^e\left(\lambda -\Sigma \left({Ve}_{t-1}\ast {S}^e\right)+\Sigma \left({Vi}_{t-1}\ast {S}^i\right)\right).$$
(2)

ΔVet is determined by the difference between λ indicating the intensity of the US (e.g., Rescorla & Wagner, 1972), the strength of Ve and Vi on trial t – 1 and the rate parameter (αe). On the other hand, if the CS is presented without a US, ΔVit, the change in Vi on trial t, changes according to Eq. 3:

$$\Delta {Vi}_t={\alpha}^i\left(0-\Sigma \left({Ve}_{t-1}\ast {S}^e\right)+\Sigma \left({Vi}_{t-1}\ast {S}^i\right)\right).$$
(3)

ΔVit is determined by the difference between λ during extinction (i.e., 0), the total strength of Ve and Vi retrieved in the context in which the CS is presented (i.e., Σ(Vet − 1 ∗ Se) + Σ(Vit − 1 ∗ Si)), and a rate parameter (αi). This assumption indicates that the upper limit of the absolute value of strength in the inhibitory association (i.e., Vi) is the strength of Ve retrieved during a nonreinforcement trial (i.e., Σ(Ve ∗ Se)), suggesting that Vi produced by the absence of the US in a context is determined by the degree that Ve is retrieved in the context.

An Important feature of our model is the initial values of similarities. The initial value of Se should differ from Si even if the two contexts are physically identical based on evidence that context change has little effect on CR before extinction (e.g., Bouton & Bolles, 1979). This suggests that regardless of the type of or distance between contexts, the initial value of Se is extremely high and larger than that of Si. The retrieval of excitation is relatively more stable than inhibition, although it is not perfectly independent of context (Bouton, 1993). This assumption implies that the acquisition context is recognized as more similar to other contexts than to the extinction context.

Another feature of our model is that the similarities change through trials under specific conditions. When one type of association (e.g., excitatory) is developed in a context, and then another type (e.g., inhibitory) is developed in another context, the similarity between both contexts corresponding to the former type is reduced. For example, when an acquisition is conducted in Context A and then extinction in Context B, the \({S}_{AB}^e\) (i.e., the extent to which the Ve developed in Context A is retrieved in Context B) is diminished through extinction trials. On the other hand, if the same type of association is developed in different contexts, the similarity between their contexts corresponding to type, increases. Thus, when the acquisition is conducted in Context A and then in Context B, \({S}_{AB}^e\ \textrm{increases}\).

This principle is based on the findings of acquired equivalence and distinctiveness effect (e.g., Honey & Hall, 1989). If two CSs (X and Y) are paired with the same outcome, generalization between both stimuli is promoted (Similarity of Outcome; Honey & Hall, 1989). In addition, if a subject receives the same type of trials regarding a CS and US (i.e., reinforcement or non-reinforcement) in two contexts, a generalization between these contextual stimuli is promoted (Honey & Watt, 1999). Based on the findings, our model assumes that when the same type of trial is conducted in multiple contexts, the similarity between their contextual stimuli increases. In contrast, when different types of trials are conducted in multiple contexts, the similarity is diminished.

We assume that this change in similarity also follows an error-correction rule, although it is not based on empirical findings. For example, when the acquisition phase takes place in Context A and then the extinction in Context B, \({S}_{AB}^e\) during extinction changes according to the following formulae:

$$\Delta {S}_{t, AB}^e={\alpha}^S\left(0-{S}_{t-1, AB}^e\right).$$
(4)

In Formula 4, Δ\({S}_{t, AB}^e\) represents the change in Se between Contexts A and B (\({S}_{AB}^e\)) on trial t. αS is a rate parameter regarding the similarities. After these two phases, when the CS is conditioned again in context C, \({S}_{A,C}^e\) increase, and \({S}_{B,C}^i\) decreases in the same manner.

$${\varDelta S}_{t, AC}^e={\alpha}^S\left(1-{\varDelta S}_{t-1, AC}^e\right)$$
(5)
$${\varDelta S}_{t, BC}^i={\alpha}^S\left(0-{\varDelta S}_{t-1, BC}^i\right).$$
(6)

In these equations, the increase and decrease of the Ss are determined by 0 or 1, which represent the lower and upper limits of the S. Thus, the previous and current types of trials are the same, and the similarity between the contexts in both phases will be changed by αS(1 − ΔSt − 1), while the previous type is different from the present type, and the similarity will be changed by αS(0 − ΔSt − 1).

Explanation of acquisition, extinction, and recovery-from-extinction effects by our model

Acquisition and insensibility of the CR to a context change following acquisition

According to our model, an increment in CR during the acquisition phase is explained by the increase in excitatory strength, and a decrease in CR during extinction is explained by the increase in inhibitory strength. Whether the extinction context is the same as the acquisition context does not affect the decrement in CR during the extinction phase (e.g., Bouton & Bolles, 1979). Our model predicts that although the CR on an initial trial in a different context (Context B) other than acquisition (Context A) is diminished by the similarity between acquisition and extinction context (e.g., Ve * \({S}_{AB}^e\)), this effect is small because Se is very high regardless of the type of context. The left panel of Fig. 1 shows the simulation of the total strength of Ve and Vi in the acquisition (1–10 trials) and extinction (11–50 trials) phases, using our model. Group AA is assumed to receive extinction in the acquisition context and Group AB in the new context. This simulation showed no substantial differences in the CR between the two groups in the first trial during the extinction phase. However, there was a slight decrease in the CR of Group AB.

Fig. 1
figure 1

A simulation of acquisition, extinction, and three renewal effects when λ = 1, αe = 0.5, αi = 0.1, \({S}_{AA}^e\) = 1, \({S}_{AB}^e\) and \({S}_{AC}^e\) = 0.9, \({S}_{BA}^i\) and \({S}_{BC}^i\) = 0.8

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The recovery-from-extinction effect

One of the strengths of the original Bouton’s model is that it can account for various recovery-from-extinction effects in a consistent way. Our model generally adapts this idea with some additional assumptions.

Renewal effects

Our model can account for the difference in the size of the three renewal effects. In the ABA renewal, the similarity between the acquisition and testing context (i.e., \({S}_{AA}^e\)) is almost 1.0, and the similarity between the extinction and testing context (i.e., \({S}_{BA}^i\)) is smaller than that of \({S}_{AA}^e\). Therefore, in testing, the retrieved inhibitory association is diminished (Vi * \({S}_{BA}^i\)), although the excitatory association remains intact (Ve * \({S}_{AA}^e\)). Moreover, because the strength of the inhibitory association is determined by the strength of the excitatory association retrieved in the extinction phase (Ve * \({S}_{AB}^e\)), the strength of the inhibitory association is less than the absolute value of the excitatory association. By combining these two effects, the CR appears during testing in the ABA renewal design.

In ABC renewal, by the same mechanisms as those of ABA renewal, the strength of the inhibitory association is smaller than that of the excitatory association. However, in the test phase, unlike the ABA renewal design, the excitatory association is also slightly diminished (i.e., Ve * \({S}_{AC}^e\)). Therefore, although the CR increases during testing, the size of the CR is smaller than with ABA renewal. In the AAB renewal design, Se and Si during testing are basically the same as those in the ABC design because testing is conducted in a new context. However, the strength of the inhibitory association in the ABC design is slightly smaller than that of the AAB design because the excitatory association retrieved during the extinction phase is Ve * \({S}_{AB}^e\) in the ABC design and Ve * \({S}_{AA}^e\) (i.e., almost Ve * 1.0) in the AAB design. Thus, when acquisition and extinction are conducted in different contexts, the strength of the inhibitory association is slightly smaller than when conducted in the same context, resulting in a smaller recovery in the AAB renewal than in the ABC renewal.

The right panel of Fig. 1 shows the quantitative simulation of the three renewal effects using this model. Group NE does not receive extinction (i.e., acquisition only), whereas other groups receive acquisition and extinction. The sequence of letters in their groups represents the contexts of acquisition, extinction, and testing. These simulations indicate some differences in the CR in the test phase of the three renewal designs. These predictions are consistent with empirical evidence (e.g., Bouton & Bolles, 1979; Thomas et al., 2003).

Spontaneous recovery

This model explains spontaneous recovery by introducing the idea of a temporal context, as in the original model. Thus, spontaneous recovery is a renewal effect that occurs through changes in the temporal context over time following the extinction phase. This explanation of spontaneous recovery is supported by many findings (e.g., Rosas & Bouton, 1998). In our model, when a CS is presented following an interval that differs from that in the previous extinction phase, the temporal context is changed from extinction, and then, the retrieval of Vi is interfered with by the decay of Si. The results of the simulation of spontaneous recovery are represented in Fig. 2. Group Immediate indicates the group received the test phase immediately following extinction. Groups Short and Long represent groups that underwent the test phase after extinction and time elapsed, and the duration between extinction and testing of Group Long is assumed to be longer than that of Group Short. In the simulation, we assumed that this additive duration in Group Long induces a large change in context than that of Group Short, which is represented as a change in similarities (Se was multiplied by 0.9 and Si was multiplied by 0.80 in Group Long). Other parameters were identical to those in the simulation in Fig. 1. This prediction is consistent with the finding that not only does spontaneous recovery occur, but the longer the time interval, the greater the degree of spontaneous recovery (Quirk, 2002).

Fig. 2
figure 2

A simulation of spontaneous recovery when Se = 0.9, Si = 0.8 in Group Short and Se = 0.855, Si = 0.64 in Group Long

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Reinstatement

Reinstatement is explained differently from the original model. According to Bouton’s model, reinstatement is explained by the idea that the US alone presentations develop the context–US association, which becomes a retrieval cue for the acquisition phase (Bouton & Nelson, 1998). It is difficult to adapt this idea in our model because it does not assume the role of context–US association. Therefore, our model assumes that the US alone presentations following extinction increase the similarity between acquisition and test context (i.e., Se) and decreases the similarity between extinction and test context (i.e., Si) because the US itself acts as a type of context. Thus, the US presentations manipulate the distance between these contexts. This idea is mainly based on a reinforcer context in instrumental conditioning (Trask & Bouton, 2016), which indicates that the delivery of reinforcer is a type of context cue and has been supported by considerable instrumental renewal and resurgence literature (e.g., Trask & Bouton, 2016). Thus, reinstatement occurs because a subject perceives the testing context to be similar to the acquisition context (i.e., increment in Se) and dissimilar to the extinction context (i.e., decrement in Si) due to the existence of the US. It can also explain the context-dependency of reinstatement. Previous studies have reported that reinstatement occurs only when the testing context and the US alone presentation context are the same (Bouton & Bolles, 1979). Since this model explains reinstatement as a result of manipulation of the distance between the US presentation context and acquisition and extinction contexts, reinstatement occurs only in the context in which a US presentation is conducted.

Rapid reacquisition

In our model, the reacquisition rate is explained by the change in similarities between the reacquisition context and the acquisition and extinction contexts. For example, when acquisition and extinction are conducted in context A, \({S}_{AA}^e\) is reduced according to the rule of similarity. Therefore, if reacquisition is conducted in context A, the reacquisition rate is delayed compared to the initial conditioning, conversely it is rapid if reacquisition is conducted in the acquisition context after extinction in a different context or a neutral context (see Fig. 3). This prediction is consistent with the experimental findings investigating the reacquisition rate using AAA, ABB, ABA, and AAB procedures (Bouton & Swartzentruber, 1989).

Fig. 3
figure 3

A simulation of the reacquisition (10 trials) following extinction. Note: The sequence of letters in the five groups represents the contexts of acquisition, extinction, and reacquisition. The parameters used were the same as those shown in Fig. 1

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Explanation of the procedure that improves the effects of extinction and prevents recovery-from-extinction effects in the model

Massive and compound extinction

Our model can also explain the effects of increasing extinction trials and other procedures for preventing recovery-from-extinction effects. Simply, an increase in the number of extinction trials strengthens Vi, which decreases not only CR at the end of the extinction phase but also recovery-from-extinction effects. However, this effect is partial because the upper limit of Vi is the absolute value of Ve * Se. Therefore, although the AAB renewal can be reduced by this procedure because Ve and Vi are almost the same in absolute value by massive extinction, other types are difficult to eliminate. Second, massive extinction decreases Se between acquisition and extinction contexts because subjects receive incompatible information with acquisition during extinction, resulting in delayed reacquisition during the subsequent reacquisition phase in the extinction context.

According to our model, compound extinction using additional excitors results in a greater increase in inhibitory association than normal extinction. This model also predicts an increment in the protection-from-extinction effect if extinction is conducted with a conditioned inhibitor. Thus, the effect of extinction increases when Σ(Vet − 1 ∗ Se) + Σ(Vit − 1 ∗ Si) is large and decreases when it is small; this effect also changes the intensity of the recovery-from-extinction effects.

US presentations during extinction phase, extinction cue, and changing physical similarity between contexts

In our model, the effect of US presentations in the extinction phase is explained by introducing the US context hypothesis as well as reinstatement. If a US is presented during extinction, Ve retrieved during extinction increases because the distance between the acquisition and extinction contexts becomes shorter than the initial value. As a result, a strong inhibitory association develops in extinction because of the increase in prediction error (i.e., increment in retrieved Ve during extinction causes strong Vi; see Renewal Effects section). Additionally, our model predicts the difference between occasional reinforcement and the presentation of an unpaired US. Occasional reinforcement is expected to increase Ve during extinction, and the CR in the test phase can be larger than the unpaired presentation or normal extinction.

This prediction by the model is illustrated in Fig. 4. The left panel represents a change in associative strength through normal extinction (Ext), occasional reinforcement (OR), and presentation of an unpaired US (UP). The right panel represents a simulation of ABA renewal. These simulations show that Group UP facilitates an extinction effect and eliminates a subsequent renewal more successfully than Group Ext, while Group OR shows a higher value of total associative strength at the extinction phase and a greater subsequent renewal. Recently, this prediction has been supported by Lipp et al. (2021), indicating that presentations of unpaired US diminish the ABA renewal, while occasional reinforcement does not sufficiently decrease the CR during extinction.

Fig. 4
figure 4

A simulation of the normal extinction procedure (Group Ext), occasional reinforcement (Group OR), and unpairing with CS and US (Group UP) procedure in extinction on the ABA renewal effect. Note: Reinforcement in Group OR and US presentation in Group UP are assumed to be conducted in 5, 15, and 25 trials

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A similar account can also apply to explain the effects of extinction cues and changes to the similarity between extinction and test contexts (e.g., Bandarian-Balooch & Neumann, 2011; Brooks & Bouton, 1993, 1994). Both procedures increase the retrieval of inhibitory association in testing by an increment in Si between extinction and testing contexts, resulting in a decrease in the recovery-from-extinction effects.

Extinction in multiple contexts

Our model assumes that this procedure extinguishes ABC renewal every time a subject moves into a neutral context. The extinction of the ABC renewal increases Vi because \(\Sigma \left( Ve\ast {S}_{A,C}^e\right)+\Sigma \left( Vi\ast {S}_{B,C}^i\right)\) is larger during the test phase than during the standard extinction (i.e., AAA procedure). Thus, reextinction of recovery-from-extinction effects increases Vi, resulting in reducing subsequent recovery (see Fig. 5). This explanation is consistent with the finding that when recovery-from-extinction effects are extinguished again, subsequent relapse is decreased compared to the first time (e.g., Rescorla, 2004). Our model can explain the effect of extending intertrial or intersession intervals (Urcelay et al., 2009) in the same way. Thus, these procedures are interpreted as performing extinction in multiple temporal contexts. Additionally, our model predicts that the effect of extinction in multiple contexts is diminished when extinction trials are limited (Thomas et al., 2009) because the effect of extinction itself is reduced.

Fig. 5
figure 5

A simulation of total associative strength in the extinction in multiple contexts (Group M) and a single context (Group S) on the ABA and ABC renewal effect. Note. The parameters used are identical to the simulation of Group ABC in Fig. 1, except that the number of extinction trials is changed to 30. Group M is assumed to move to the neutral context in 10, 20, and 30 trials

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However, this prediction does not fit with the finding that extinction in multiple physical contexts does not affect spontaneous recovery (Dunsmoor et al., 2014) because an increment in Vi predicts a decrease in all recovery-from-extinction effects. This finding might indicate that this procedure promotes generalization between contexts that Bouton’s model describes. This account suggests that extinction in multiple contexts using one contextual dimension, such as physical context, provides little generalization to another contextual dimension, such as temporal context, because both dimensions have few common elements.

Clinical implication of this model

Our model provides not only comprehensive explanations for the effects of many extinction-related exposure strategies such as expectancy violations (e.g., Craske et al., 2008) but also new implications for the intervention and quantitative predictions of these effects. The extinction-related exposure strategies are traditionally understood using an inhibitory learning approach (e.g., Craske et al., 2014). However, because the inhibitory learning approach is a conceptual framework based on multiple models of Pavlovian conditioning and findings of cognitive psychology, it is unclear which mechanism each strategy involves. Our model can resolve this issue by providing a comprehensive account of the effects of many procedures that prevent recovery-from-extinction.

Similar to the explanation by the inhibitory learning approach, our model suggests that a patient strongly learns that a stimulus is safe if the exposure to feared stimuli is conducted when the patient strongly retrieves the memory of a traumatic event. A unique feature of the model is that the degree of retrieval is determined by the similarity between contexts in which a traumatic event occurs and exposure therapy is conducted. This prediction suggests that making these contexts similar is effective for maximizing learning in therapy. In addition, it is also predicted that future relapse is unlikely to occur when the relapse is reduced in exposure therapy. As such, these procedures have been widely practiced in clinical settings (e.g., Craske et al., 2022) and our model provides an explanation of their mechanisms.

Additionally, this model can provide a new explanation for traditional exposure techniques not based on Pavlovian extinction. For example, prolonged exposure, an effective intervention for pathological problems (Powers et al., 2010), includes imaginal exposure, which uses images of traumatic stimuli or situations through recalling traumatic events, and in vivo exposure (e.g., Foa, 2011). Both techniques aim to promote emotional processing by habituation based on emotional processing theory (Foa & Kozak, 1986). However, this explanation by emotional processing theory has not been supported by empirical evidence (Craske et al., 2008). Our model might provide a new perspective on the mechanism of prolonged exposure. Imaginal exposure may be able to induce strong inhibitory learning (i.e., Vi) because it is possible to present a strongly fearful stimulus and its traumatic context by using imagery stimuli. On the other hand, in vivo exposure involves exposure to the actual fear stimulus in many contexts within daily life. Our model predicts that this procedure not only facilitates inhibitory learning through extinction in multiple contexts, but also makes inhibitory learning more likely to be retrieved in several different contexts. However, these explanations are still hypothetical and need to be tested.

Although it is unclear whether the associative perspective can deal with imaginal exposure since empirical findings, Holland and Forbes (1982) has shown that representation-mediated extinction—a procedure that repeatedly activates representation of a CS without actual presentations of the CS by using associative links—reduces an effect of flavor aversion learning (see also Holland, 1990). In human fear conditioning experiments, Agren et al. (2017) reported that imaginary extinction, a technique that requires participants to imagine the CS without stimulus presentation, is as effective as normal extinction in fear conditioning preparation. It is also reported that mental rehearsal of the exposure context before the follow-up test decreases the relapse following exposure therapy (Mystkowski et al., 2006). These findings suggest that an imaginary CS and its associated contexts have functions that are similar to direct stimuli presentation.

Lastly, the current mathematical model might represent a quantitative description and prediction of the intervention. Many models of exposure therapy cannot provide quantitative predictions for the effects, as these are qualitative or informal models. If the valid prediction of symptoms throughout sessions can be provided based on a prior assessment or change of symptoms in the initial few sessions, the utility of the exposure therapy will increase. Although it is necessary to estimate the parameters in each patient, the application of this model to clinical practice might enable such predictions (see also Fullana & Soriano-Mas, 2021; Portêlo et al., 2021).

Summary and future implications

We described that our model, a mathematical extension of Bouton’s model, can comprehensively explain many procedures that promote the effects of extinction and recovery-from-extinction effects. Traditional models cannot provide sufficient mathematical representations for these phenomena, especially the recovery-from-extinction effects. Thus, our model might be the most appropriate among associative learning models in explaining the recovery-from-extinction effects. From a clinical perspective, this model can help understand the mechanisms of exposure therapy and provide some implications for therapy.

Although our model overlaps with many of the assumptions of other associative models, it successfully provides a comprehensive explanation of various phenomena that have been difficult to clarify in these models. The most obvious point is that the current model allows for an explanation of the difference in the magnitude of the three renewal effects. Another advantage of this model is that it allows for the quantitative explanation of multiple procedures diminishing recovery-from-extinction effects. Even in the latent cause model (Gershman & Niv, 2012), which is considered the leading model in extinction, only some of the phenomena can be clarified. Thus, we believe that the most significant advantage of this model is that it quantitatively provides a comprehensive explanation for many of phenomena about the recovery-from-extinction effects.

This model has some limitations. First, our model cannot account for all of the phenomena in the extinction and recovery-from-extinction effects. For example, recovery-from-extinction effects can be diminished if the interval between acquisition and extinction is short (e.g., Chang & Maren, 2009; Myers et al., 2006) or if the CS is presented before extinction (e.g., Schiller et al., 2010). These phenomena cannot be explained using our model.

Second, our model does not describe the effects of contextual conditioning because it consider contextual stimuli as a factor that determine retrieval strengths. In many studies on recovery-from-extinction in Pavlovian conditioning, it has been suggested that the role of the context–US association is not crucial—except in some phenomena, such as reinstatement (Bouton, 1993). However, there is no doubt that the context–US association affects CR in Pavlovian conditioning (e.g., Landeira-Fernandez, 1996; Wagner & Rescorla, 1972). Although our model mainly focuses on the effect of contextual stimuli on discrete CS as retrieval cues, the effect of the context–US association on the recovery-from-extinction effects is also crucial (Miller et al., 2020). Traditional models such as the Rescorla–Wagner model or the comparator hypothesis, which assume context–US association to be a key component of Pavlovian conditioning, are superior to our model with regard to the contextual conditioning.

Finally, some assumptions about similarities among contexts are somewhat arbitrary and lack empirical evidence. The most crucial assumption in our model is the initial values of Se and Si. As explained above, many phenomena related to recovery-from-extinction effects can be explained by the difference in initial values. We attribute this discrepancy to the evolutionary aspect, but more work needs to be conducted. One hypothesis about this asymmetry in initial values of Se and Si is that the updating of S is also asymmetry. If the change rule of S is caused by the experience about other CS, then the pairings of a CS with US increase similarities between present context and all contexts, whereas other types of CS were previously reinforced, and extinction decreases the similarities. So, because reinforcement is a necessary condition for future extinction, with Pavlovian conditioning under the natural environment, increment in Se across various contexts based on reinforcement procedure should occur more often than decrement based on extinction procedure. If this hypothesis is correct, the asymmetry between Se and Si could be due to the asymmetry of learning history about Pavlovian conditioning and extinction.

Whether similarities between contexts are affected by previous experiences with other stimuli is an important question for the prediction of new phenomena. It is unclear whether these similarities are specific to each CS or are carried over to other stimuli because this model mainly focuses on the processes of a specific CS and contextual stimuli. Consider the ABA renewal procedure with two CSs (X and Y): testing of X and Y in each acquisition context after X and Y are reinforced in Contexts A and B, respectively, and then both stimuli are extinguished in Contexts B and A, respectively. If Se and Si are carried over to other stimuli, it is predicted that Si between Contexts A and B increases during the extinction phase, because subjects receive non-reinforcement in both Contexts A and B, producing little ABA renewal other than the renewal procedure using one CS. On the other hand, if these similarities are specific to a CS, the size of the ABA renewal is the same for both procedures.

From a clinical perspective, although our model has clinical implications, it is unclear whether this model can predict the effect of exposure therapy on pathological fear or anxiety. The validity of this model as a mechanism of exposure therapy must be tested.

Conclusion

In this study, we proposed an alternative model, a mathematical extension of Bouton’s model in the extinction procedure, to explain several procedures that improve the effect of extinction and prevent recovery-from-extinction effects in Pavlovian conditioning. Our model can explain many of their findings and resolve some problems that cannot be addressed by original Bouton’s model. Moreover, these phenomena are critical in exposure therapy because exposure therapy is thought to be an analog of extinction in Pavlovian conditioning. Thus, when this model is used as a mechanism of exposure therapy, many clinical implications, especially promoting the effect of exposure and prevention of relapse after the intervention, can be provided.