3.1. Mechanistic explanation

Before evaluating whether topological explanations are mechanistic explanations, we elucidate the latter. For our purposes, we focus on conceptions of mechanistic explanation that are minimal and nontrivial. Such conceptions provide the most formidable challenges to autonomism. Hence, when we show that some topological explanations are not mechanistic, we cannot be charged with chasing ghosts. We discuss these two facets of mechanistic explanation in turn.

To begin, Glennan (Reference Glennan2017, 17) provides a “minimal” characterization of mechanisms that captures a widely held consensus among mechanists about conditions that are necessary for something to be a mechanism, even if they differ about, for example, the role of regularities, counterfactuals, and functions in mechanistic explanation:

Because this conception is minimal, the counterexamples we raise to it below apply a fortiori to more demanding conceptions of mechanisms.

Importantly, mechanists are sometimes criticized for being so vague as to trivialize the concept of mechanism (Dupré Reference Dupré2015). Since minimal conceptions are less committal than other conceptions of mechanism, they are especially susceptible to this criticism. To avoid trivializing mechanistic explanation, we appeal to claims about “entities,” “activities,” “interactions,” “organization,” and “responsibility” widely endorsed by mechanists. For instance, our arguments hinge on claims that entities and activities cannot be spatiotemporal regions determined merely by convention; that interactions cannot be mere correlations, etc. We further develop these claims as they arise in the discussion of our examples of non-mechanistic topological explanation. Should mechanists deny these requirements on entities, activities, and the like, then the burden of proof falls upon them to show that their alternative conception of mechanism is nontrivial.

Finally, mechanists typically distinguish between etiological, constitutive, and contextual mechanistic explanations (e.g., Craver Reference Craver2001). Etiological explanations cite the causal history of the explanandum; constitutive explanations cite the underlying mechanism of that explanandum; and contextual explanations cite an explanandum’s contribution to the mechanism of which it is a part. Since underlying and overlying mechanisms can be synchronic with the phenomena that they explain, neither constitutive nor contextual explanations must cite their respective explananda’s causal histories, i.e., the prior events that produced the phenomenon. Therefore, constitutive and contextual explanations are distinct from etiological explanations. Furthermore, because constitutive explanations explain a larger system in terms of its parts, while contextual mechanisms do the exact opposite, constitutive and contextual explanations are also distinct. For ease of exposition, we will first discuss constitutive mechanistic explanations. Section 4.4 discusses their etiological and contextual counterparts.

To summarize, we are trying to precisely characterize mechanistic explanations in a way that poses the stiffest challenge to our claim that some topological explanations are non-mechanistic. To that end, we think that the most defensible conception of mechanistic explanation has two features: minimality and nontriviality. Furthermore, we will first compare topological explanations to constitutive mechanistic explanations, and then turn to etiological and contextual ones.

3.2. Mechanistic interpretations of topological explanations

With a clearer conception of mechanistic explanation in hand, mechanists’ next task is to translate topological explanations’ characteristic graph-theoretic vocabulary into the language of mechanistic explanation—to provide a mechanistic interpretation of topological explanations (MITE). To that end, the preceding suggests that mechanists ought to hold that a topological model is explanatory only insofar as there exists a mechanism for which all the following conditions hold:

1. (1) Node Requirement: the topological model’s nodes denote the mechanism’s entities or activities.

2. (2) Edge Requirement: the topological model’s edges denote the interactions between the mechanism’s entities or activities.

3. (3) Responsibility Requirement: the topological model specifies how the mechanism’s entities, activities, and interactions are organizedFootnote ⁴ so as to be responsible for the phenomenon.

4. (4) Interlevel Requirement: the explanandum is at a higher level than the mechanism’s entities, activities, and interactions as described by the Node and Edge Requirements.

The first three requirements fall out of Glennan’s minimal conception of mechanism; the last, from our initial focus on constitutive explanations.

As an illustration of the MITE’s plausibility, consider the earlier example in which the small-world topology of the C. elegans nervous system explains its capacity to process information more efficiently than would be expected if its topology were either regular or random. Here, the neural system’s components are individual neurons, which are represented as nodes in the topological model. Hence, the Node Requirement is satisfied. Furthermore, the edges of the graph denote synapses or gap junctions between different neurons, so the Edge Requirement is satisfied. Third, the Responsibility Requirement is satisfied, though this requires more detailed discussion. Suppose (as we shall throughout) that “responsibility” is characterized in terms of counterfactual dependence:

In regular or random topologies, the synaptic connections between neurons will be different than they are in the actual small-world topology exhibited in C. elegans. Consequently, each of these networks describes a different potential mechanistic structure. Thus, this counterfactual shows how mechanistic differences are responsible for differences in information transfer. Finally, note that the efficiency of information transfer is a global property of the C. elegans nervous system. That system is composed of neurons connected via synapses, so mechanists appear justified in claiming that the nodes and edges of this network are at a lower level than its explanandum property. Hence, the Interlevel Requirement is satisfied. Putting this all together, this means that the example fits the MITE.Footnote ⁵

Before proceeding, we note three things. First, while other MITEs are certainly possible, this one strikes us as the most plausible. We have culled it from some of the foremost mechanists’ discussions of topological explanations (Craver Reference Craver2016; Levy and Bechtel Reference Levy and Bechtel2013, Glennan Reference Glennan2017). Indeed, our MITE also accords nicely with the widely used “Craver diagrams” in the mechanisms literature. As Figure 2 illustrates, such diagrams entail that there is a phenomenon (“S’s Ψing”) at a higher level, and a mechanism exhibiting a graph-theoretic structure at the lower level, with nodes corresponding to entities (denoted by “X _i ”) performing activities (denoted by “φ _i ”), and edges corresponding to interactions (denoted by arrows). Since no other MITE has been offered, mechanists ought to propose an alternative MITE should they chafe at the one we propose here. Second, our working definition of autonomism throughout this paper is only that some topological explanations do not fit this MITE. This is consistent with some other topological explanations fitting this MITE. Third, it suffices for our purposes if only one of the MITE’s four requirements is violated. In other words, so long as our counterexample is even “partly non-mechanistic”, autonomism (sub specie this MITE) is vindicated.

Figure 2. Craver (Reference Craver and Povich2007) diagram.

4.1. Adachi et al.’s explanation

We first discuss Adachi et al.’s explanandum and then their explanans. Adachi et al.’s dependent variable is (combined) ΔFC, “the regression slope of FC on the total number of length2-AC” patterns. ^{Footnote 6} Obviously, a better understanding of our explanandum, ΔFC, requires an understanding of both FC and length2-AC.

Begin with FC. FC networks’ edges are synchronization likelihoods (SL). Stam et al. (Reference Stam, Jones, Nolte, Breakspear and Scheltens2006, 93) provide a useful definition:

As this definition suggests, FC networks’ nodes are time series. Depending on the study, these nodes can be interpreted in multiple ways. Many FC models are interpreted so that nodes correspond to the time series of blood oxygen level-dependent (BOLD) readings for an individual voxel in functional magnetic resonance imaging (fMRI) data. A voxel is a unit of graphic information that defines a three-dimensional region in space. Voxels are essentially the result of dividing the brain region of interest into a three-dimensional grid. By contrast, Adachi et al. (Reference Adachi, Takahiro Osada, Takamitsu Watanabe, Miyamoto and Miyashita2011, 1587) opt for a less “operational” interpretation of their FC model. They do this by mapping different clusters of voxels onto 39 different anatomical regions or “areas” in the macaque cortex. Examples include visual area 4 and the mediodorsal parietal area. Consequently, their FC and AC models have the same nodes.

Indeed, without this mapping of FC nodes onto anatomical regions, the neuroscientists could not characterize their explanandum with any precision. Specifically, they focus on “length2-AC” patterns—i.e., functionally connected pairs of regions that are only anatomically connected through some third region (see Figure 3). In other words, length2-AC patterns involve two brain areas that are functionally connected (and hence are correlated) but are also known to lack any direct causal link. Hence, the explanandum, ΔFC, describes the extent to which the totality of Patterns a, b, and c contributes to the overall FC in the macaque neocortex.

Figure 3. Different anatomical connectivity patterns (length = 2) discussed by Adachi et al. (Reference Adachi, Takahiro Osada, Takamitsu Watanabe, Miyamoto and Miyashita2011). The black nodes, 1 and 2, are functionally connected by the dotted line, but are only indirectly anatomically connected via node 3 by the arrows.

With the explanandum clarified, we turn to Adachi’s explanans, which appeals to the global topological properties of the AC network. Specifically, Adachi et al. argue that the AC network’s frequency of three-node motifs (which they abbreviate as “MF3”) explains why the macaque’s ΔFC is as high as it is. In this context, a three-node motif is a triplet where the nodes denote anatomical regions and the edges denote anatomical connections. Thus, MF3 is a measure of how many of these triplets can be found in the macaque neocortex.

Adachi et al. establish this explanation by running simulations involving thousands of randomly generated networks. Some of these networks have the same frequency of three-node motifs as the macaque brain but differ with respect to other topological properties. These other properties include global clustering coefficient, modularity, and the frequency of two-node motifs. Then, each network’s ΔFC is measured. In another run of simulations, they controlled for these other topological properties while varying the frequency of three-node motifs. Models in which the frequency of three-node motifs matched the macaque neocortex vastly outperformed models matching other topological properties in accounting for ΔFC. Once again using counterfactual dependence as a working definition of “responsibility,” this suggests the following:

In other words, MF3 is “responsible” for the length2-AC patterns’ contributions to FC.

4.2. Responsibility requirement.Footnote ⁷

We now turn to our broader aim of arguing for autonomism, starting with the Responsibility Requirement, which states that topological models are only explanatory if they specify how the phenomenon counterfactually depends on how a mechanism’s entities, activities, and interactions are organized. We shall consider different mechanist proposals for how Adachi et al.’s explanation satisfies the Responsibility Requirement and show that each faces serious challenges.

The most prominent mechanist strategy holds that many topological explanations describe an abstract kind of mechanistic organization (e.g., Bechtel Reference Bechtel2009; Glennan Reference Glennan2017; Kuorikoski and Ylikoski Reference Kuorikoski and Ylikoski2013; Levy and Bechtel Reference Levy and Bechtel2013; Matthiessen Reference Matthiessen2017). This “organization strategy” prompts three replies. First, the most precise versions of this strategy simply assume that mechanistic organization can be spelled out entirely in terms of graph-theory. This effectively concedes autonomism’s main tenets. For instance, Kuorikoski and Ylikoski (Reference Kuorikoski and Ylikoski2013) define “organization” and “network structure” in terms of topological structure, as evidenced by both their examples and their acknowledgement of Watts as pioneering the “new science of networks.” However, this means that mechanistic organization just is topological structure. Since they also hold that some organization is explanatory unto itself, their position entails that some topological properties are explanatory unto themselves. This is tantamount to autonomism. Second, organization is supposed to refer to the structure of interactions between entities and activities that are constitutive of the phenomenon to be explained. Yet, in this example, the organization in question is of a system that is (partly) constituted by the phenomenon to be explained. After all, the anatomical regions that figure in each of these length2-AC patterns are parts of the anatomical network, and the latter’s frequency of three-node motifs drives the explanation. Third, different models with the same frequency of three-node motifs can posit radically different interactions between any three brain regions—radically different forms of organization—yet nevertheless predict the same FC structure.Footnote ⁸ This would mean that the specific interactions between the mechanistic components are explanatorily idle, which also violates the Responsibility Requirement.

Chastened by these problems, a mechanist might remain silent about organization, and instead insist that Adachi et al.’s explanation satisfies the Responsibility Requirement by specifying how ΔFC counterfactually depends on the macaque neocortex’s entities, activities, and interactions. However, this faces something we call the causal disconnection problem. In a nutshell, the problem is that: (a) mechanistic explanations require entities, activities, and interactions that are responsible for a phenomenon to be causally connected to that phenomenon,Footnote ⁹ yet, (b) Adachi et al.’s explanation does not require these causal connections. To see why, let us distinguish different kinds of three-node anatomical motifs included in their calculation of MF3 (see Figure 4). If Adachi et al.’s explanation is mechanistic, then the only three-node anatomical motifs that can figure in Adachi et al.’s explanation are length2-AC patternsFootnote ¹⁰ and anatomical motifs that are causally connected to a length2-AC pattern. However, Adachi et al.’s model implies that even if the only change in MF3 were to the number of three-node motifs that are causally disconnected from length2-AC patterns, ΔFC would still change. Consequently, this explanation does not satisfy the MITE’s Responsibility Requirement.

Figure 4. Examples of three kinds of three-node anatomical connectivity motifs in Adachi et al. (Reference Adachi, Takahiro Osada, Takamitsu Watanabe, Miyamoto and Miyashita2011). The dotted line in between the black nodes in the length2-AC pattern is a functional connection. The other, heavier dotted lines simply distinguish the three kinds of three-node anatomical motifs. Solid lines denote anatomical connections; arrows have been removed to indicate neutrality about the direction of causation between different anatomical regions.

Finally, mechanists might try to avoid these problems by insisting that causally disconnected three-node AC motifs are explanatorily irrelevant and only the length2-AC patterns and the AC-motifs that are causally connected to them are responsible for ΔFC. However, this proposal faces two challenges. First, mechanists would need a non-question-begging argument as to why models such as Adachi et al.’s are incorrect to treat causally disconnected motifs as explanatorily relevant. To our knowledge, no such arguments have been offered. Second, the Responsibility Requirement is not easily satisfied even if causally disconnected three-node AC motifs are excluded from the explanation. Indeed, Adachi et al. (Reference Adachi, Takahiro Osada, Takamitsu Watanabe, Miyamoto and Miyashita2011, 1589) turn to a non-mechanistic topological explanation precisely because of limitations in causal explanations of a phenomenon closely related ΔFC: why there is any functional connectivity in a length2-AC pattern. On the current mechanistic construal, length2-AC patterns’ functional connections (i.e., the edge between nodes 1 and 2 in each of the patterns in Figure 3) should be explained by some indirect causal link; paradigmatically, via a third region serving as either an intermediate cause or a common cause of the functional connection between the two regions (Patterns b and c in Figure 3.) However, the anatomical structure in Pattern b—the “two-step serial relay”—decreases FC. More importantly, Adachi et al. observe that a significant number of functional connections occur even when two functionally connected areas only share a common effect (specifically: a common efferent as represented by Pattern a.) To our knowledge, no mechanists claim that an effect can be responsible for its cause. Hence, functionally connected areas that are only anatomically connected via a common efferent (as in Pattern a) cannot satisfy the Responsibility Requirement. Consequently, even if the causal disconnection problem is bracketed, mechanisms alone cannot be responsible for ΔFC, which (to repeat) is the totality of functional connectivity for which length2-AC patterns are responsible.

In summary, we have considered three ways of trying to mechanistically interpret MF3: as describing a mechanism’s organization; as describing a mechanism’s entities, activities, and interactions while allowing for causal disconnections; and as describing a mechanism’s entities, activities, and interactions while prohibiting causal disconnections. In each case, the claim that a mechanism is responsible for ΔFC faces formidable challenges.

4.3. Interlevel requirement

Adachi et al.’s explanation also violates the Interlevel Requirement. We provide three arguments. First, constitutive mechanistic explanations require explananda to be at higher levels than the parts and activities that explain them. However, given the way that Adachi et al. interpret their FC model, both the FC and AC networks have the same brain regions as their nodes. Because the explanans and explanandum appeal to the same entities, the FC network is not at a higher level than the AC network in this explanation.Footnote ¹¹ Moreover, the anatomical regions that figure in MF3 are minimally at the same level as the anatomical regions in the length2-AC patterns that figure in ΔFC. Indeed, the explanation allows for some anatomical regions that figure in MF3 to be identical to those that figure in ΔFC.

In our estimate, this first argument understates the degree to which this explanation violates the Interlevel Requirement. This leads to our second argument. As already noted, a constitutive mechanistic explanation of ΔFC would need to appeal to the entities, activities, and interactions constitutive of (and thus at a lower level than) the anatomical regions and their axonal connections—and this explanation works in the exact opposite direction. Adachi et al. appeal to the frequency of three-node network motifs, a global property of the AC network. This network is constituted by these anatomical regions. In other words, a higher-level property is explaining the behavior of lower-level constituents. Hence, the MITE’s Interlevel Requirement has been violated.Footnote ¹²

Finally, one may argue that both ΔFC and MF3 are global properties of the macaque neocortex. If this is correct, then it is not even clear that Adachi et al.’s explanation appeals to levels at all.

4.4. Contextual and etiological explanation

This explanation’s violation of the Interlevel Requirement may prompt mechanists to reply that this merely shows it not to be a constitutive mechanistic explanation. However, it may still be either a contextual or etiological mechanistic explanation. We briefly show that such replies face significant challenges.

Craver (Reference Craver2001, 63) provides the most precise definition of contextual mechanistic explanation:

If Adachi et al.’s explanation is a contextual mechanistic one, then the system (S) is the macaque’s neocortex, the relevant system capacity (ψ) is its frequency of three-node motifs, the component X’s are the length2-AC patterns in the neocortex, and the activity/property (φ) of these patterns is their functional connectivity.

In Craver’s definition, a part’s activities are supposed to “contribute” to the system’s capacities. We take this to require, at a minimum, that S’s ψ-ing counterfactually depends on X’s φ-ing, i.e.,

This accords well with Craver’s paradigmatic examples of contextual mechanistic explanations, e.g., the heart pumping blood.Footnote ¹³ This contributes to circulation and, consonant with our suggestion, the following is true:

By analogy, if Adachi et al.’s explanation is contextually mechanistic, then the following should figure prominently:

However, a quick review of Section 4.1 shows this to be precisely the converse of the counterfactual that figures in Adachi et al.’s explanation. Nor does this counterfactual accord with their methodology: They run simulations in which they vary MF3 to see how ΔFC changes—not the other way around. Consequently, ΔFC does not contribute to MF3; this is not a contextual mechanistic explanation.

As an alternative, mechanists might claim that Adachi et al.’s explanation is an etiological mechanistic explanation. Here, Bechtel’s (Reference Bechtel2009, 557–59) account of “situated mechanisms” (or mechanistic explanations that “look up”) seems especially instructive. Bechtel’s chief example is how stimuli in the environment contribute to the mechanistic explanation of visual processing. Quite plausibly, environmental stimuli provide etiological explanations of why the mechanism for vision behaves as it does. For example, an apple and various environmental conditions (lighting, the absence of smoke and mirrors, etc.) are part of the causal history of why a person comes to see an apple. So, by analogy, if Adachi et al.’s explanation involves a situated mechanism, then MF3 is a “network environment” that is part of ΔFC’s causal history.

Like Craver’s account of contextual mechanistic explanation, situated mechanisms appeal to a higher-level mechanism or environment to explain a lower-level entity. However, unlike contextual mechanisms, situated mechanistic explanations do not require the lower-level entity to “contribute” to a higher-level mechanism in the ways we have discussed. For instance, a visual system does not need to contribute to the lighting conditions in its environment. This immediately avoids the problems raised by treating Adachi et al.’s explanation as contextually mechanistic.

Despite these initial attractions for the mechanist, other autonomists have raised challenges with identifying topological and etiological mechanistic explanations that apply here. For instance, while causes precede their effects, topological explanantia need not precede topological explananda (Huneman Reference Huneman2010, 218–19). Since MF3 does not temporally precede ΔFC, the explanation is not etiological.Footnote ¹⁴ Thus, MF3 is not an “environmental condition” in Bechtel’s sense.Footnote ¹⁵

4.5. Defending explanatoriness

At this point, we have at least shown that Adachi et al.’s topological model is not mechanistic. Assuming our arguments are sound, mechanists’ only recourse is to deny that this model is explanatory. So far as we can tell, the most plausible argument to this effect is best glossed as a methodological concern about the design of the study: Do the statistical and computational models provide sufficient evidence for ΔFC’s counterfactual dependence upon MF3? Since these statistical and computational considerations do not preclude symmetric correlations between variables, the objection implies that the study provides just as good of evidence for the following symmetric counterfactual: Had ΔFC increased, then MF3 would have increased. Since explanation is widely thought to be asymmetric, mechanists may be tempted to deny that Adachi et al.’s model is explanatory.

We will argue that this objection is inconclusive given mechanists and autonomists’ common ground. To see why, we underscore the importance of background knowledge in inferring explanations (Lipton Reference Glennan2004; Psillos Reference Psillos2007). For instance, even in well-designed experiments, background information is often required to infer the most plausible causal or mechanistic explanation. Similarly, we suggest that the following background knowledge helps to vindicate Adachi et al.’s inference:

According to this principle, there is prima facie reason to believe that MF3 explains ΔFC rather than vice versa because MF3 only traffics in anatomical connectivity, while ΔFC traffics in both anatomical and functional connectivity. One motivation for this principle is that interventions on brain regions, neurons, and synapses are easier to conceive of than interventions on voxels and synchronization likelihoods. Since interventions frequently serve as a guide to explanation, this gives AC networks their presumptive explanatory priority. That said, this priority is only prima facie; more rigorous testing and further theoretical considerations can overturn it. This background principle also accords with Adachi et al.’s reasoning. Based on their simulations and statistical analyses, they claim that MF3 “shapes” (Adachi et al. Reference Adachi, Takahiro Osada, Takamitsu Watanabe, Miyamoto and Miyashita2011, 1586, 1589–91) and “influences” (1586, 1591). ΔFC. They do not claim the converse, as the symmetric counterfactual suggests.

For mechanists who are sympathetic to AC’s explanatory priority (Craver Reference Craver2016; Povich Reference Povich2015),Footnote ¹⁶ the objection is thereby defused: While Adachi et al.’s simulations and statistical tests are not sufficient unto themselves to license an explanatory claim, they become so in conjunction with this principle. Presumably, these mechanists take AC’s explanatory priority to be a consequence of: (a) AC networks typically containing more mechanistic information than FC networks and (b) mechanistic information providing more reliable evidence for judging the plausibility of counterfactuals than information about functional connectivity. Our argument only requires AC’s explanatory priority. So, these mechanists can only reject it on pain of also rejecting (a) or (b).Footnote ¹⁷

But suppose that other mechanists would bite this bullet and reject AC’s explanatory priority. We will argue that this entails that mechanistic and topological explanations are equally (in)vulnerable to symmetry problems. If asymmetry is a requirement on all correct explanations, then this means that both mechanists and autonomists are in trouble. On the other hand, if some correct explanations are symmetric, then autonomists can learn some valuable lessons from mechanists. For instance, Craver and Bechtel (Reference Craver and Bechtel2007, 553) claim that “all of the interesting cases of interlevel causation are symmetrical: components act as they do because of factors acting on mechanisms, and mechanisms act as they do because of the activities of their lower-level components.” Indeed, Craver (Reference Craver and Huneman2013) seems to leverage this point into taking constitutive and contextual mechanistic explanations to be simply two “perspectives” on the same system. So, seemingly harmless symmetries will exist when X constitutively explains Y and Y contextually explains X. Autonomists could devise analogues to constitutive and contextual mechanistic explanations to tolerate symmetries in a similar manner. On such a view, Adachi et al. would adopt one “perspective” in which MF3 explains ΔFC, but from another perspective, ΔFC explains MF3. So, regardless of whether AC enjoys explanatory priority over FC, autonomists are no worse off than mechanists with respect to symmetry.

In summary, while Adachi et al.’s explanation satisfies the Node and Edge Requirements, this does not make it a mechanistic explanation. The reasons for this are twofold. First, only topological properties are responsible for the explanandum; not mechanistic ones. Second, the explanation does not invoke the appropriate levels required for a constitutive mechanistic explanation. Thus, we have a counterexample to the most plausible mechanistic interpretation of topological explanations (MITE). Hence, absent some other MITE, we conclude that some topological explanations are non-mechanistic. Furthermore, this explanation not only resists characterization as a constitutive mechanistic explanation, but also as a contextual and as an etiological one. This suggests that other MITEs will face steep challenges going forward.