An originally neutral stimulus can acquire predictive properties as a conditioned stimulus (hereafter, “CS”), and thereby elicit conditioned emotional behaviors and responses (a “conditioned response”), if it is repeatedly paired with an unconditioned stimulus (“US”) (Pavlov, 1927). Conditioned responses can also generalize to stimuli that resemble the original CS (generalization stimulus; hereafter, “GS”). The study of fear generalization processes such as these has historically been undertaken using animals (Pavlov, 1927), but the research has recently been extended to include human subjects (Vervliet et al., 2004, 2005; Lissek et al., 2008, 2010; Dunsmoor et al., 2009). In investigations of the perceptual generalization of fear in humans, visual conditioned and generalization stimuli comprising rings of gradually increasing size are commonly used (Lissek et al., 2008, 2010, 2014; Lissek and Grillon, 2012; Xu et al., 2016), wherein two extreme ring sizes serve as the conditioned danger cue (CS+) and the conditioned safety cue (CS-), respectively, with the CS+ and CS- counterbalanced over the smallest and largest ring. Alternatively, visual stimuli of a particular color and shape have been used as the CS and GS (Vervliet et al., 2010; Vervliet and Geens, 2014; Ahmed and Lovibond, 2015). However, studies that use such stimuli only focus on a single physical characteristic, and therefore do not accurately reflect how we perceive our natural environment. In reality, when we perceive fearful objects or experience traumatic events, we will generalize the conditioned fear to a new object or context based on regularities that go beyond physical resemblance (Dymond et al., 2015); that is, we tend to process the stimulus as a whole or as an entire concept (Pomerantz et al., 1997; Chen, 2005). Furthermore, visual perception is sensitive to the topological properties of stimuli. In topological property, a difference in a figure’s size is not considered, but properties such as whether the stimulus is closed or open, bound or unbound, or simply connected or multiply connected are considered. Among the most commonly perceived topological properties are the connectivity of the stimulus and the number of holes it has (Chen, 1990).

In typical perceptual generalization research, GSs are interchangeable—for example, eight rings of gradually increasing size (Lissek et al., 2008). However, a key step concerning perception is the selection of which aspect of a stimulus will be used as a category standard (Mai and Lei, 2000). A variety of studies suggests that the topological properties of a stimulus play fundamental roles in the processes of perception and categorization (Chen, 1982, 1990). The topological features ensure that the GSs belonging to a category are interchangeable. In addition, given that the topological features result in two different categories, it seems fair to assume that not much (if any) generalization should be expected between the CS- and the CS+ and that a specific GS will only receive generalization from the CS that defines the category (i.e., a GS+ receives mainly generalization from the CS+ and not from the CS-, while a GS- receives mainly generalization from the CS- and not the CS+).

Topological equivalence is mainly reflected during shape changes; that is, when a stimulus smoothly changes its shape without breaking or fusing, the topological properties remain unchanged as long as the connectivity and number of holes remain the same (Mai and Lei, 2000; Zhuo et al., 2003). For example, when a triangle turns into a square or circle, because the connectivity is invariant, the topological properties remain unchanged. Similarly, a doughnut and coffee cup are considered topologically equivalent since they both have a hole in the center. Topological theory proposes that perception is based on the transformative and the invariant properties of stimuli (Chen, 1982, 1990). With regard to topological perception, compared to the processing of local geometric properties and physical characteristics of a stimulus, the processing of topological properties occurs in the early stage of visual perception (Chen, 1982). Moreover, topological properties may be regarded as an informative cue to guide attention to the target more efficiently. Specifically, topological equivalence has a significant influence in maintaining the contextual cuing effect (Ma et al., 2018). For example, if a person is bitten by a dog, the fear will be generalized to other dogs, based on topological properties or categories rather than on local properties, such as its size, shape, or the color of its hair. Moreover, if the stimuli are temporary, after the stimuli disappear, the affected person will use their short-term memory to code them. In such instances, the most important information in the perception that comprised topological properties would determine the results of coding, and thus influence perceptual categorization (An et al., 2004).

In clinical settings, patients exhibiting excessive fear generalization might have certain anxiety disorders, and such fear generalization may potentially be an etiologically significant indicator of anxiety disorders (Lissek et al., 2008). Overgeneralization of conditioned fear has been documented in panic disorder, post-traumatic stress disorder and generalized anxiety disorder (Lissek et al., 2010, 2014). Thus, establishing how to effectively inhibit this overgeneralization of fear may be key to treating such anxiety disorders. Although a former study showed that extinction of a CS+ could effectively reduce the generalization level of fear (Vervliet and Geens, 2014), in actual clinical therapy, it is usually not possible to use the original CS+ to counteract fear, especially if it is abstract, ethically forbidden or related to a physical condition. Hence, almost all of the stimuli that are used clinically to extinguish conditioned fear are GSs. However, Vervliet et al. (2004) also found that the extinction using a GS could not bring about the elimination of a conditioned fear incorporating local features. Because topological features play an important role in perception and categorization, in our study, we utilized topological features but not local features, such as the shape or the size, to investigate whether a GS with the same topological features as the CS+ could influence its fear generalization and extinction.

In the present study, we used the topological property that best fitted the perceptual process in order to assess the influence of topological equivalence on the generalization and extinction of fear. Moreover, we used GSs that were topologically, but not physically, equivalent to the CS+ in order to determine whether the elimination of the GS can effectively influence the level of fear elicited by the CS+. Based on existing evidence (An et al., 2004; Vervliet and Geens, 2014), we hypothesized that the topological equivalence of the stimuli would influence both the generalization and the extinction of fear. Because the GSs were not interchangeable and of topological equivalence with the CS, the generalization of fear could confirm the topological features play important roles in the processes of perception and categorization. Further, if fear could be successfully removed by using stimuli of topological equivalence to the CS+, this could point toward a new approach in treating conditioned fear and anxiety disorders.

In the present study, by employing a basic conditioned-fear generalization paradigm, geometrical figures with the same topological property as the CS+ were used as a GS to study the influence of topological equivalence on the generalization and extinction of conditioned fear. We found that fear was successfully generalized from the CS+ to the GS with the same topological properties. Furthermore, the shared topological property successfully reduced the return of fear in the SCRs, but not in the online expectancy ratings.

During perception, the topological properties of objects (based on, e.g., physical connectivity or number of holes) are processed prior to the local geometric properties (Chen, 2005). In our study, the GS had a white area in the middle that was equivalent to that of the CS+, and this topological equivalence effectively induced fear generalization. The only difference between the CS+ and CS- was the presence of a white circle in the middle; hence, “a white circle” was the fear-relevant feature. This finding is in line with configural theories of associative learning, which assume that stimuli acquire associations as a whole, but the generalization is determined by the proportion of unique versus common features (Pearce, 1987). In differential CS+/CS- conditioning, the CS+ acquires excitatory strength (fear-eliciting properties) and the CS- acquires inhibitory strength (safety-eliciting properties) (Vervliet and Geens, 2014; Zeng et al., 2014). Depending on the similarity of the CS+/CS-, the CS+ receives generalized inhibitory strength from the CS- and excitation strength from the CS+. Their relative strengths determine the size of the conditioned response (larger in relation to the CS+ than to the CS-). It follows that the CS+ is not a simple fear elicitor, but instead involves complex interplay between its fear- and safety-eliciting properties. Based on this theory, GSs that resemble the CS+ more than they do the CS- will receive more generalized excitation than generalized inhibition, and will thus produce a generalized fear reaction (Vervliet and Geens, 2014). In the present study, if participants remembered the local features—take counterbalancing A for example—the participants might remember a gray square and a white circle in the middle. In the generalization phase, only GS₂+ had the same local feature as CS+, in having a white circle, while GS₁+ and GS₃+ did not have the same local feature as CS+. However, GS₁+, GS₂+, and GS₃+ did have the same topological features as CS+. If the local feature played a more important role in generalization, GS₂+ might have demonstrated more generalization. However, GS₁+, GS₂+, and GS₃+ showed the same generalization tendency.

The results we obtained in the fear acquisition phase suggest that the participants might have remembered the topological properties of the dangerous stimuli rather than the local features. On the other hand, CS+ (again, taking counterbalancing A as an example) can also be deemed as a compound stimulus, which consisted of a gray square and a white circle. In a prior study, compound stimuli were used (e.g., blue triangle, yellow square) to study which local features (such as shapes or colors) are more important in fear generalization, and it was found that verbal instruction focusing on one feature could transmit information about the relative importance of an individual stimulus features in terms of fear generalization (Vervliet et al., 2010; Ahmed and Lovibond, 2015). However, in the present study, GS₁+ and GS₃+ did not have any of the same local features (e.g., a gray square and a white circle) as CS+, but shared topological properties, which further verifies the importance of topological properties with respect to fear generalization. This finding is supported by previous studies showing that topological properties are superior to local features in the process of perception (Chen, 1982, 2005; Chen and Zhou, 1997; Huang et al., 2011). Indeed, this preeminence has been demonstrated both in cognitive neuroscience studies, which have found that the human brain, particularly the left temporal area, is sensitive to the topological properties of objects (Wang et al., 2007; Zhou et al., 2008; Zhuo et al., 2003), and in phylogenetic studies, which have revealed that this topological preference exists in the visual systems of lower animals too (Chen et al., 2003; Zhu et al., 2010). Together with these previous studies, our results indicate that topological properties play an important role not only during perception but also in fear generalization.

In our study, the extinction of the fear responses to the CS+ and GS was successful in all three groups according to the SCR measures during the test phase. However, in the online expectancy ratings, only Group 1, which had eliminated the fear responses to the CS+, successfully extinguished the fear responses to both the CS+ and GS. Conversely, Groups 2 and 3, which had to extinguish fear responses to one and three GSs, respectively, still had higher fear responses to the CS+. These findings are consistent with earlier studies on perceptual generalization and the extinction of fear that posit that eradicating a different-but-similar stimulus has little effect on the fear extinction of the original CS+ (Dubin and Levis, 1975; Vervliet et al., 2004, 2005, 2006, 2010; Boddez et al., 2012; Vervliet and Geens, 2014).

In addition, the lack of a significant difference between Groups 2 and 3 indicates that the number of GSs may not influence the return of fear. In Group 2, the extinction effect of the single GS transferred to the other GS during the test phase. As noted in the introduction, topological properties play a key role in the categorization of objects (Mai and Lei, 2000). In this study, all three GSs were topologically equivalent; hence, participants might have placed them in the same category. In the real world, complex objects and situations can be represented across a variety of dimensions, with any associated fear generalization spread over arbitrary stimuli with little perceptual overlap except for category (Vervoort et al., 2014). One approach to comprehending the complexity of human fear generalization incorporates theoretical knowledge, including categorization, and the organization of conceptual knowledge (Dunsmoor and Paz, 2015). Humans can use category-level knowledge to associate the GS with an entire category (Dunsmoor and Murphy, 2015). Thus, according to conceptual generalization, the extinction of fear for one object might reduce the fear for other objects in the same category (Vervliet et al., 2010; Vervoort et al., 2014). As for the present study’s CS+, although it belonged to the same category as the GS, its presence during fear acquisition established it as a cue of fear, and therefore it was considered to be different from the other stimuli. Similarly, in real life, a stimulus with intense fear relevance, such as a fearful face, would capture our attention more automatically than would a neutral face, and so the fearful face would be categorized differently (Öhman and Mineka, 2001; Öhman et al., 2001).

Here, we found a dissociation between the expectancy ratings and the SCRs, which persisted throughout the test phase. In the online expectancy ratings, Groups 2 and 3 both demonstrated the return of fear and had higher fear ratings than did Group 1. However, the SCR data did not show a return of fear or differences among the groups. The dissociation effects observed in the present study are consistent with those reported previously (Walther et al., 2009; Balderston and Helmstetter, 2010; Klucken et al., 2013; Vervoort et al., 2014), and these findings collectively support the idea that different types of memories are formed during the same training procedure. From the perspective of dual process theory, implicit and explicit performances are dissociable from one another (Knight et al., 2003; Vervoort et al., 2014; Feng et al., 2017). Additionally, they are concerned with two distinct memory systems: the former involves a declarative memory of the learned fear association between the CS and US, while the latter involves a procedural memory pertaining to the acquisition and expression of a fear response (Knight et al., 2003; Balderston and Helmstetter, 2010; Schultz et al., 2013). The results of the present study show that, even though the participants did not feel fear at an implicit physiological level, they tended to report higher fear expectancy, which is consistent with the results of former studies (Lissek et al., 2008; Haddad et al., 2013; Xu et al., 2016). In addition, SCRs not being inclined toward a stronger generalization of fear elimination from GSs to the CS+ might also indicate the absence of the return of fear on the physiological level, compared to the expectancy ratings (Zeng et al., 2014; Xu et al., 2016; Feng et al., 2017). Nevertheless, based on our findings and former studies, we posit that the larger generalization in the expectancy ratings than in the SCRs may be a robust phenomenon. In the explicit performances, some details of the stimuli may have been ignored, with participants therefore inclined to report a higher degree of fear. It is also possible that the reports of higher fear expectancy were related to participants’ attending to and collecting information about the threat stimulus (LeDoux, 2014). From an evolutionary point of view, increased reporting of fear could help us attract the attention of others to warn them of potential danger and/or obtain help (Xu et al., 2016).

One limitation of the present study was that, based on the expectancy ratings, the GS appeared to only receive a partial generalization of fear, which in turn likely produced less extinction learning. Less extinction learning implies less extinction generalization (Vervliet et al., 2004). Since our participants did not consider the GS equally as dangerous as the original CS+, the difference in extinction generalization may be just a difference in extinction learning.

Despite the above limitation, the present study successfully highlights the importance of topological properties in the processes of human fear generalization and extinction. As enhancing perceptual processing shows promise as an effective treatment for targeting excessive fear generalization (Struyf et al., 2017), topological properties could play a key role in perceptual processing and in connecting perceptual generalization with conceptual generalization. In exposure therapy, clinicians tend to use GS since the original CS+ often cannot be used during treatment. Therefore, understanding which GS will be the most effective is essential. The findings of the present study suggest that a GS with topological equivalence to the CS+ might be better than a GS with similar local features to the CS+. However, topological properties were shown to eliminate the fear for CS+ only in SCRs; the matter of how to use topological properties to effectively decrease fear generalization in clinical contexts needs further investigation. In reality, to face the same threatening stimuli or circumstance is rare. In the study, most stimuli of fear were GS similar to the original one, and topological equivalence with the CS+ was superior. These results suggest that fear extinction could transfer to other stimuli sharing the same topological properties, affirming that topological properties might be beneficial in prospective treatments for anxiety disorders.

Conceived and designed the experiments: LX, HS, and XZ. Performed the experiments: LX, HS, XX, and JL. Analyzed the data: LX, PY, and XX. Contributed reagents/materials/analysis tools: LX, HS, PY, XZ. Contributed to grant funding: XZ.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.