Abstract This article examines three candidate cases of non-causal explanation in computational neuroscience. I argue that there are instances of efficient coding explanation that are strongly analogous to examples of non-causal explanation in physics and biology, as presented by Batterman (), Woodward (), and Lange (). By integrating Lange’s and Woodward’s accounts, I offer a new way to elucidate the distinction between causal and non-causal explanation, and to address concerns about the explanatory sufficiency of non-mechanistic models in neuroscience. I also use this framework to shed light on the dispute over the interpretation of dynamical models of the brain. 1 Introduction 1.1 Efficient coding explanation in computational neuroscience 1.2 Defining non-causal explanation 2 Case I: Hybrid Computation 3 Case II: The Gabor Model Revisited 4 Case III: A Dynamical Model of Prefrontal Cortex 4.1 A new explanation of context-dependent computation 4.2 Causal or non-causal? 5 Causal and Non-causal: Does the Difference Matter? 1 Introduction In recent philosophy of neuroscience and cognitive science, there has been an overriding emphasis on mechanisms and mechanistic explanation,1 and some authors have gone so far as to present the mechanistic approach as the only game in town when it comes to explaining various phenomena—from sub-cellular signalling to computation in neural circuits and person-level decision-making.2 Following Woodward ([forthcoming]), I treat mechanistic explanation as a sub-type of causal explanation, characterized by features such as reliance on decompositional methods and sensitivity to the details of implementation. A working assumption of the mechanist brand of philosophy of neuroscience has been that to explain a phenomenon is to describe the parts of the causal nexus that give rise to it. Not surprisingly, a number of authors have detected a greater degree of methodological and explanatory pluralism amongst the sciences of the mind–brain.3 On a number of occasions, it has been argued that applications of dynamical systems theory (DST) yield non-mechanistic explanatory models.4 In particular, Ross () has argued that the explanatory patterns employed in highly abstract, dynamical models of spiking neurons (Izhikevich ) are of the same form as the ‘minimal model explanations’ first described by Batterman () in the context of statistical mechanics. The published position of the mechanists has been that dynamical models are either not explanatory but merely predictive and descriptive (Kaplan and Craver ), or that when they do offer explanations, these are of a mechanistic sort (Bechtel ; Kaplan and Bechtel ). The debate over DST in neuroscience is far from settled. Yet if one does accept that dynamical models exemplify minimal model explanations, this raises the interesting question of whether such explanations are also non-causal. Robert Batterman and Collin Rice have argued that minimal model explanations in physics and biology are non-causal,5 whereas Silberstein and Chemero () hold that DST offers neuroscientists and cognitive scientists a distinctive genre of causal explanation.6 My central claim in this article is that there are indeed instances of non-causal explanation to be found within neuroscience. I aim to convince you that the cases in neuroscience are at least as strong as the familiar examples from physics and biology. Along the way, I will elucidate the distinction between causal and non-causal explanation by developing James Woodward’s proposal that non-causal explanation occurs when we are able to answer questions about non-actual scenarios (‘what-if-things-had-been-different’ or ‘w-questions’), even though those scenarios are not the consequence of hypothetical interventions. In Sections 2 and 3, I will focus on ‘efficient coding explanation’ in computational neuroscience where, I will argue, one finds numerous instances of distinctively mathematical, non-causal explanation, of the sort discussed by Lange (). Importantly, these explanations meet the mechanists’ own criterion for explanatory sufficiency when they are able to answer w-questions (Kaplan , p. 354). In Section 4, I will turn to one recent example of a dynamical model of a region of the monkey brain. Here, it is less clear that the model offers a non-causal explanation and I will argue that the issue turns on whether or not one is willing to give a realist interpretation of the model components resulting from dimensionality reduction analysis employed by the model builders. In the remainder of this section, I will say more about the relevant background, defining efficient coding explanation and presenting my preferred account of non-causal explanation. 1.1 Efficient coding explanation in computational neuroscience In a recent publication (Chirimuuta ), I argued that models in computational neuroscience often yield a distinct, non-mechanistic, pattern of explanation which I call ‘efficient coding explanation’. The term ‘computational neuroscience’ labels a broad research area that uses applied mathematics and computer science to analyse and simulate neural systems. That article responded to the work of Kaplan (), which attempted to incorporate all explanatory models of this field within the mechanistic framework. The case turns on the particular example of the Gabor model of V1 receptive fields, where a mechanistic criterion for explanatory success—the ‘models to mechanism mapping constraint’ (Kaplan , p. 347; Kaplan and Craver , p. 611)—fails and yet yet the model is still able to provide counterfactual information, thus answering ‘w-questions’ (Woodward ). How can this be?7 Well, the models in question ignore biophysical specifics in order to describe the information-processing capacity of a neuron or neuronal population. They figure in computational or information-theoretic explanations of why the neurons should behave in ways described by the model. So, while on the one hand, such receptive field models may simply be thought of as phenomenological descriptions that compactly summarize observed responses of neurons in the primary visual cortex (Kaplan , pp. 358 ff.), on the other hand, by analysis of the information-theoretic properties of the Gabor function itself, one gains an explanation of why neurons with the properties captured by the model appear at this particular stage of visual processing. In short, such models figure prominently in explanations of why a particular neural system exhibits a characteristic behaviour. Neuroscientists formulate hypotheses as to the behaviour’s role in a specific information-processing task, and then show that the observed behaviour conforms to (or is consistent with) a theoretically derived prediction about how that information could efficiently be transmitted or encoded in the system, given limited energy resources. Typically, such explanations appeal to coding principles like redundancy reduction (Barlow )—the notion that more information can be transmitted through a cable (for example, axon) of fixed bandwidth if some of the correlations between signals are removed. They do not involve decomposition of the biophysical mechanisms thought to underlie the behaviour in question; rather, they take an observed behaviour and formulate an explanatory hypothesis about its functional utility. It is worth saying a word about the notion of ‘efficiency’ in play here. A feature of this research is that neuroscientists draw on knowledge of man-made computational systems and attempt to reverse-engineer the brain, looking for the ‘principles of neural design’ (Sterling and Laughlin ). A basic fact is that information processing makes substantial demands on resources, both in terms of the material required to build a computer or nervous system, and the energetic cost of computational processing. It is assumed, reasonably, that the explanation of many features of neural systems can be derived from consideration of resource constraints—the need to achieve good computational performance in spite of a relatively small resource budget. As Sarpeshkar writes: The three physical resources that a machine uses to perform its computation are time, space, and energy. Computer scientists have traditionally treated energy as a free resource and have focused mostly on time […] and space […] However, energy cannot be treated as a free resource when we are interested in systems of vast complexity, such as the brain […] Energy has clearly been an extremely important resource in natural evolution. (Sarpeshkar , p. 1602; cf. Attwell and Laughlin ; Sterling and Laughlin ) So while it is instructive to compare efficient coding explanations to optimality explanations in biology in what follows—because both are in the business of comparing actual biological systems to theoretically optimal solutions—the field does not rely on the strong adaptationist assumption that the brain of humans, or any other animal, is somehow optimal.8 One must instead make the weaker assumption that there is some process or mechanism—evolutionary, developmental, or occurring during the life of the organism as adaptation through neural plasticity—that causes the system to tend towards the optimal solution. Efficient coding explanations typically proceed without specifying what that process is. 1.2 Defining non-causal explanation Efficient coding explanations often make fine-grained predictions about what would occur in counterfactual scenarios. This is possible because of the way that the efficiency of a computational procedure is sensitive to the nature of the particular task at hand. For example, neurons with a certain kind of receptive field structure might be the most efficient means to encode sensory information in one kind of environment, but not for another. Thus one can show how neural properties are counterfactually dependent on the evolutionary or developmental environment. For this reason, I have argued that this branch of computational neuroscience employs a proprietary kind of non-mechanistic, causal explanation. Yet if one is willing to extend the notion of a mechanism to include the whole apparatus of natural selection and ontogenesis, one might propose that computational neuroscience is still just in the business of discovering mechanisms. However, the assimilatory impulses of even the most flexible-minded mechanist would have to stop at the idea of distinctively mathematical non-causal—and non-constitutive—explanation. So a central motivation for exploring the question of whether non-causal explanations occur in neuroscience is as a way to get clearer on the limits of the mechanist framework. Woodward’s interventionist theory of causal explanation has been extremely influential in the philosophy of neuroscience, and Woodward’s proposal that explanatory sufficiency is tracked by the ability of a theory or model to address w-questions is accepted by authors such as Kaplan and Craver. The basic intuition is that ‘a successful explanation should identify conditions that are explanatorily or causally relevant to the explanandum: the relevant factors are just those that “make a difference” to the explanandum in the sense that changes in these factors lead to changes in the explanandum’ (Woodward [forthcoming], p. 5). Interestingly, Woodward (, p. 221) suggests that the ability to address w-questions may range beyond causal explanation, writing: […] the common element in many forms of explanation, both causal and noncausal, is that they must answer what-if-things-had-been-different questions. When a theory tells us how Y would change under interventions on X, we have (or have the material for constructing) a causal explanation. When a theory or derivation answers a what-if-things-had-been-different question but we cannot interpret this as an answer to a question about what would happen under an intervention, we may have a noncausal explanation of some sort. Woodward gives the example of the hypothesis that the stability of the planets is counterfactually dependent on the four-dimensional structure of space-time. What if space-time had been six-dimensional? There is no intervention associated with this question,9 but the hypothesis is that if things had been different, then planetary orbits would indeed be less stable. In this article, I follow Woodward in defining an ‘intervention’ as an idealized, unconfounded experimental manipulation of one variable that causally affects a second variable only via the causal path running between these two variables (Woodward , p. 46).10 Like various other authors,11 I believe it is useful to de-couple the counterfactualist parts of Woodward’s account of explanation from the causal, interventionist ones, and thereby develop an account of non-causal explanation. Moreover, I propose that this account be integrated with Lange’s notion of ‘distinctively mathematical explanation’ to give a clearer standard for differentiating causal from non-causal explanations than is often employed in the literature.12 Marc Lange analyses distinctively mathematical explanations of regularities and events in terms of the modal strength of mathematical facts, in comparison to ordinary causal laws. For example, Lange (, p. 488) writes: That Mother has three children and twenty-three strawberries, and that twenty-three cannot be divided evenly by three, explains why Mother failed when she tried a moment ago to distribute her strawberries evenly among her children without cutting any. This is conceptually different from any causal explanation that mentions, for instance, that the attempt made Mother hungry and frustrated (‘hangry’) and so she ended up eating two of the strawberries herself, or that one of the little darlings stole from the other, and so on. Thus I share Lange’s view that distinctively mathematical explanations employed in science are non-causal (Lange , 506). Furthermore, I propose that we bring Lange’s notion of modal strength in non-causal explanation to bear on Woodward’s idea that non-causal explanations occur when we show that there are dependencies between the explananda and explans that cannot be understood in interventionist terms. In such cases, knowledge of the dependencies does not show us how things would be in a range of counterfactual scenarios in which we perform manipulations on the explananda. Instead, we are told how things would be under certain impossible scenarios in which the laws of mathematics are altered.13 This of course assumes that counterpossible statements—counterfactuals or subjunctive conditionals with impossible antecedents—can be non-vacuously true. I invite the reader to consider that it is intuitive that the statement, ‘if thirteen were evenly divisible by three, then I could share my baker’s dozen of doughnuts equally amongst my three best friends’ is non-vacuously true, and that the statement ‘if thirteen were evenly divisible by three, then a baker’s dozen of doughnuts would be a healthy snack’ is non-trivially false; what is more, there is a recent literature on the semantics of counterpossibles that underwrites these intuitions.14 In the examples I present in the following two sections, we have a non-causal explanation that is reliant on a trade-off demonstrated in the theory of information. Such trade-offs are candidates for being brute mathematical facts—nothing could be done to make them not obtain. For this reason, we can think of the trade-offs as modally strong mathematical facts, in Lange’s sense, and as yielding information about counterfactual dependencies that go beyond interventionist interpretation, in Woodward’s sense. Like Lange (), but unlike Saatsi and Pexton (), I intend my account of non-causal explanation in neuroscience to cover explanations both of particular events and regularities, since the kinds of mathematical facts that my examples depend on do yield explanations of both sorts. In contrast to the account of non-causal explanation just outlined, Robert Batterman and Collin Rice have focused on the way that certain models in physics, biology, and economics idealize away from the causal processes that lead up to a phenomenon. Such models result in representations of the causes of phenomena that are, at best ‘caricatures’ (Batterman and Rice ). So these authors argue that the relevant notion of explanation is not a causal one, since the explanation of a phenomenon does not arise because we are provided with information about what caused it. This is how Rice (, p. 600) makes the case: When considered together, these idealizations entail that optimality models usually provide little, if any, accurate information about the actual causes, or causal mechanisms, within the model’s target system(s). In the end, the highly idealized optimality model represents mathematical relationships between constraints, tradeoffs, and the system’s equilibrium point that do not mirror any causal relationships (or processes) in the target system. Put different, optimality models fail the kind of “model-to-mechanism-mapping requirement” of causal theories. This is a less stringent notion of non-causal explanation than the one that I will employ below. Although all of my examples of distinctively mathematical explanations do abstract away from causal details, this feature is not what makes them non-causal (Lange , p. 506). While my cases of efficient coding explanations involving trade-offs do bear comparison with Rice’s discussion of optimality models in biology, I would like to emphasize an important difference, namely, that the optimality models discussed by Rice (for example, the Fisher model of the 1:1 sex ratio) posit trade-offs that could be subject to experimental intervention (for example, through engineering of a species or environment such that the cost of producing sons is not equal to the cost of producing daughters), whereas the trade-offs central to my examples are fixed in a stronger sense. This is because they are mathematical rather than empirical facts. Irvine () argues that if the equilibria discovered in optimality modelling show stability or robustness in the face of experimental interventions, then the model cannot offer causal explanation. Again, I suggest this is too loose a notion of non-causal explanation because she intends it to include models whose equilibria could shift if we were to change something fundamental about the system, such as intervening on inheritance mechanisms or introducing random fluctuations in the environment. But these are just different kinds of interventions—impractical or technically infeasible, but still within the remit of causal analysis. Irvine’s proposal does not give us a clear-cut way of deciding whether an explanation is causal or non-causal. In the next two sections of the article, I will show that it pays to uncover the use of mathematical facts in efficient coding analyses because these offer cases of non-causal explanations in neuroscience that are analogous to Lange’s examples from biology, and which can be incorporated into Woodward’s counterfactualist framework. 2 Case I: Hybrid Computation My first example is an explanation of the efficiency of the brain—thought of as a biological computational system—being due to the advantages of hybrid computation.15 Sarpeshkar’s () research aims to use insights gleaned from our understanding of computation in biological systems in order to build more efficient artificial computers. His analyses must often abstract away from the implementational details that obviously differ tremendously from biological to man-made devices, and consider the problem of computation in more purely mathematical terms. At the same time, this research is in the tradition of other efforts to apply information theory to neural systems in order to gain an understanding of why the nervous system has the anatomy and physiology that it does.16 Since the birth of modern computing and the concurrent rise of quantitative neurophysiology in the mid-twentieth century, many have compared these two very different kinds of information-processing devices and have pondered the question of whether computation in the brain is analogue or digital.17 It is known that brains are vastly more energy efficient than any digital computer. One estimate for the power (energy consumption) of the human brain is 12W (Sarpeshkar , p. 1601), whereas a supercomputer such as IBM’s 2007 ‘Roadrunner’, engaged in an equivalent number of around 1015 processing events per second (that is, 1 petaflops), runs at 2.4 MW (Komornicki et al. [unpublished]). A question posed and discussed by various neuroscientists is how to account for the impressive energy efficiency of neural tissue. Sarpeshkar () considers this question by examining the relative efficiencies of analogue and digital computational systems.18 The key characteristic of a digital system is that, by definition, one of its components (for example, a wire) can only represent 1 bit of information at a given time. This is because signals are all-or-nothing events, so the wire has only two informationally relevant states—on or off, 1 or 0—which amounts to one bit of information.19 In contrast, for an analogue system, the signal varies continuously with some physical variable of its components (for example, voltage), so any one component has the potential to represent countless bits of information at a given time (Sarpeshkar , p. 1605). For example, to transmit four bits of information, one would need four separate components (wires) if coding digitally, but the same amount of information could be transmitted on a single wire in an analogue system, so long as sixteen different physical states of the wire (for example, voltages) could be unambiguously associated with sixteen different signals.20 Note that these are not empirical claims about copper wires, axons, or any other bits of hardware; they are, if you like, ‘analytical’ statements about what we mean by representing or transmitting information in these different ways. It follows that analogue systems are far less hungry for resources—the raw materials and energy needed to build and maintain the signalling components—than are digital ones. However, the downside of analogue computation is that such systems are much more susceptible to corruption due to noise—random fluctuations in the physical states of the components—than are digital ones. To take our example of a single wire used to transmit four bits of information, it is easy to see that if there are random changes in the voltage due to motion through magnetic fields, for example, then what started out as a 2V signal could end up being received as a 2.5V one. And the more information one attempts to transmit through a single wire, the greater the problem because the difference between the physical magnitudes encoding the signals must decrease. Again, this problem is not an empirical observation of the behaviour of metal wires, but occurs for any signalling system whatever its material realization, as Figure 1 illustrates. Figure 1. View largeDownload slide Illustration of the trade-off between resource consumption and noise susceptibility. Given a fixed material resource that is subject to random fluctuations in its physical states—for example, one person’s arm—a digital code can only transmit one bit of information, signalling two different possible states, ‘cold’ and ‘not cold’. In the analogue-style code, the fixed material resource can be used to transmit an indefinite amount of information, since meaningful signals are defined across a continuous array of physical states. The angle of an arm from 0 to 90° is a physical continuum, and in this analogue-style code, six states of the world are associated with six different ranges of arm angles. However, it is clear that the more states one tries to encode, the more susceptible the system is to noise. Here, any slight wobble of the arm could lead to an error in signal transmission. Figure 1. View largeDownload slide Illustration of the trade-off between resource consumption and noise susceptibility. Given a fixed material resource that is subject to random fluctuations in its physical states—for example, one person’s arm—a digital code can only transmit one bit of information, signalling two different possible states, ‘cold’ and ‘not cold’. In the analogue-style code, the fixed material resource can be used to transmit an indefinite amount of information, since meaningful signals are defined across a continuous array of physical states. The angle of an arm from 0 to 90° is a physical continuum, and in this analogue-style code, six states of the world are associated with six different ranges of arm angles. However, it is clear that the more states one tries to encode, the more susceptible the system is to noise. Here, any slight wobble of the arm could lead to an error in signal transmission. Thus there is a rigid trade-off between economy of resource consumption and susceptibility to noise.21 This trade-off is inherent to the definitions of noise, information, and signal within the mathematical theory of information, and could not be altered through any empirical intervention. Indeed, noise is a result of the physical makeup of the components (for example, random electron motion in nano wires, or random ion flux across neuronal membranes), and one may successfully engineer less noisy components. But the trade-off still obtains: the more information one tries to send through one’s components, the more the signal will be affected by noise. Given any set of components, the more economically one tries to use those components, the more one’s signals are going to be corrupted by noise.22 The point of Sarpeshkar’s analysis is to sketch out an optimally efficient system, given this trade-off. (Sarpeshkar ) contains a number of different calculations to show, for example, the point at which a given analogue system becomes ‘useless’ due to noise accumulation. Sarpeshkar proposes hybrid computation as the best general solution to the problem of building a system that is resilient to noise (like a digital computer), but economical with resources (like an analogue computer). As he puts it, ‘hybrid computation combines the best of the analog and digital worlds to create a world that is more efficient than either’ (Sarpeshkar , p. 1636). The basic idea is to alternate between digital and analogue processing steps using digital–analogue converters. This way, one has the benefit of small chunks of efficient analogue computation, in which noise accumulates, interspersed with digital processing to ‘clean up’ the signal. The idea is illustrated in Figure 2. Figure 2. View largeDownload slide Hybrid computation. Each ‘hybrid link’ combines analogue processing with an analogue–digital–analogue (A/D/A) converter to clean up the signal. Sarpeshkar (, p. 1622) writes that ‘hybrid chains allow us to operate with the precision and complexity characteristic of digital systems, while doing efficient analog processing’ (after Sarpeshkar , Figure 5). Figure 2. View largeDownload slide Hybrid computation. Each ‘hybrid link’ combines analogue processing with an analogue–digital–analogue (A/D/A) converter to clean up the signal. Sarpeshkar (, p. 1622) writes that ‘hybrid chains allow us to operate with the precision and complexity characteristic of digital systems, while doing efficient analog processing’ (after Sarpeshkar , Figure 5). Most of Sarpeshkar’s analysis is illustrated in terms of metal wires and other artificial electrical components, but he is keen to argue that his conclusions generalize to biology (Sarpeshkar , pp. 1634, 1636). Furthermore, it is interesting to consider neuronal physiology in terms of the hybrid hypothesis: Action potentials are all-or-none discrete events that usually occur at or near the soma or axon hillock. In contrast, dendritic processing usually involves graded synaptic computation and graded nonlinear spatiotemporal processing. The inputs to the dendrites are caused by discrete events. Thus, in neuronal information processing, there is a constant alternation between spiking and nonspiking representations of information […] This alternation is reminiscent of the constant alternation between discrete and continuous representations of information in [Figure 2]. Thus, it is tempting to view a single neuron as a D/A/D.23 It is worth underlining some of the various claims that Sarpeshkar makes in the article. In the passage just quoted, he offers an interpretation of an often-observed fact of neural physiology—the flow of electrical activity from dendrites to axons, and then on to dendrites of another neuron—as an implementation of hybrid computation. It could be said that he explains this fact by appealing to the efficiency of hybrid computation. But more central to his project is the explanation of the efficiency of the brain in its entirety: the fact that a man-made super computer consumes orders of magnitude more energy than a biological brain. This is explained by showing that hybrid computation is the optimal solution to the problem of maximizing resilience to noise, while minimizing resource investment, and by showing that it is plausible that hybrid computation is implemented in biological brains. As he states in the abstract of the article: Our results suggest that it is likely that the brain computes in a hybrid fashion and that an underappreciated and important reason for the efficiency of the human brain, which consumes only 12 W, is the hybrid and distributed nature of its architecture. (Sarpeshkar , p. 1601) At the heart of Sarpeshkar’s account is the demonstration of the costs and benefits of analogue and digital computation, and the optimality of the hybrid combination. Thus, I argue, the efficiency of the brain is explained, non-causally, by its implementation of hybrid computation. In support of this claim, I will now compare Sarpeshkar’s account with some standard examples of non-causal explanation. My first point is that Sarpeshkar’s proposal answers what-if-things-had-been-different questions. According to his discussion, the efficiency and reliability of the brain are counterfactually dependent on the efficiency and reliability of hybrid computation, which is in turn dependent on a trade-off between resource consumption and reliability. If this trade-off did not occur, then a purely digital or purely analogue system could, in principle, satisfy both of the desiderata, and the physiology of brains would have been very different. At the same time, any such scenario cannot be interpreted as the result of a possible intervention because the existence of the trade-off is not an empirical fact about the properties of actual objects, but the result of our information-theoretic definitions of analogue and digital coding schemes (Figure 1). The point is not that some materials make more reliable signalling systems than others, but that given any physical system, and given a fixed resource investment, the amount of information transmitted by the system trades off against the susceptibility of the signal to be corrupted from noise. This is like Woodward’s example of the non-causal explanation of the stability of planetary orbits, except that we are clearly talking about what would obtain if the principles of information theory had been different. In other words, the information-theoretic explanation of the efficiency of the brain informs us about counterfactual (or counterpossible) scenarios in which the laws of information theory are different.24 Tinkering with information theory and working out the implications for coding systems cannot be thought of as a causal intervention. One may rightly worry that such information is not as useful as counterfactual information about the effects of causal interventions.25 Nevertheless, it is worth pointing out that all these kinds of explanations can be accommodated by Woodward’s scheme for non-causal explanation. We can also ask if this example falls within the framework of Lange’s distinctively mathematical explanation. To this end, it is worth comparing it with the famous bridges of Königsberg example. The explanation for why no one has ever managed to cross all of the bridges in the city of Königsberg (in their 1735 configuration) just once, without ever doubling back over a bridge, or resorting to flight or swimming, depends on the characterization of the set of bridges as a network of nodes and edges in graph theory. As it happens, each of the four nodes (rather than the requisite number of two nodes) is touched by an odd number of edges, and it is a modally strong mathematical fact that such a network cannot allow for the desired sequential crossings (Lange , p. 489). Likewise, one may interpret Sarpeshkar’s analysis as showing that it is a modally strong mathematical fact that hybrid computation is the optimal way to satisfy the twin constraints of efficiency and robustness in the face of noise. So just as the graph theoretic properties of the bridge layout explain the observed limitations on crossing routes, the hybrid computational properties of the biological brain explain its efficiency. It is also important to note that non-causal explanations of biological systems are focused on the problem of specifying why particular strategies are highly efficient or theoretically optimal, that is, they have a specific kind of explanandum. We can, in turn, then explain why natural selection would have settled upon these strategies. But it is important to keep the causal explanation of how a trait evolved separate from the non-causal explanation of why a certain trait may be optimal. For example, if one asks simply, ‘why should the brain operate like a hybrid computer?’, the response could be, ‘because during the evolution of the brain, there was strong selective pressure for the brain to be efficient and reliable, and hybrid computation is demonstrated theoretically to be the best way to satisfy the conflicting demands of efficiency and reliability’. Here, a causal and a non-causal explanation rub shoulders, but when arguing that distinctively mathematical, non-causal explanations take place, it is important to be specific about the explanandum so that one can isolate the non-causal ‘component’ of broader explanations. Lange (, pp. 499–500) makes a similar point in his discussion of the explanation of the hexagonal shape of honeycombs: The explanation is that it is selectively advantageous for honeybees to minimize the wax they use to build their combs—together with the mathematical fact that a hexagonal grid uses the least total perimeter in dividing a planar region into regions of equal area […] This explanation works by describing the relevant features of the selection pressures that have historically been felt by honeybees, so it is an ordinary, causal explanation, not distinctively mathematical. But suppose we narrow the explanandum to the fact that in any scheme to divide their combs into regions of equal area, honeybees would use at least the amount of wax they would use in dividing their combs into hexagons of equal area […] This fact has a distinctively mathematical explanation. However, in order to support the point that Sarpeshkar’s explanation is distinctively mathematical, we would need to say more about the details of his quantitative analysis. The description above of the trade-off and the optimality of hybrid computation is based on Sarpeshkar’s qualitative (‘intuitive’) account of these coding schemes. He also gives a mathematical description of the pros and cons of analogue and digital computation, presented as a comparison between the ‘resource precision curves’ for analogue and digital systems (Sarpeshkar , Figure 3). What these show, on the one hand, is that the relationship between resource consumption and signal-to-noise ratio for digital systems is compressive, such that digital systems can increase precision by many orders of magnitude and only incur minor increases to an already high baseline level of cost. On the other hand, we see that for analogue computation, there is an expansive relationship between resource consumption and signal-to-noise ratio, and also that the maximum precision of the system is strictly bounded by thermal noise. When precision is low, costs are extremely low; but costs increase dramatically when the analogue system is operating in a more precise regime. At a certain signal-to-noise ratio (the crossover point), the costs of analogue computation begin to outpace those of digital computation. The overall point is that analogue systems offer very cheap computation, but only if one is willing to tolerate a poor signal-to-noise ratio. The question now is why we should think of this as a mathematical fact. One reason to think that the facts summarized in the resource precision curves are straightforwardly empirical is that the points plotted in such graphs are the outputs of equations that have empirically measurable parameters, such as the length and that of transistors. And as Sarpeshkar (, p. 1615) writes, ‘The exact location of the crossover point will depend on the task, technology, and ingenuity of the analog and digital designers’. The reason why there is this contrast between cheap analogue computation and costly but precise digital computation boils down to the fact that the resource precision curves follow a logarithmic relationship for digital computation, while for analogue computation, power consumed is proportional to signal-to-noise ratio, or to S/N2, depending on the kinds of transistors used. These equations are derived from examples that cite specific parameter values and kinds of transistors, and make concrete assumptions about the noise distribution. The key question is whether the resource precision curves would take these forms given any parameter values and reasonable assumptions for analogue and digital systems, or if the results could be qualitatively different with different values and assumptions. I suspect that the qualitative form of the results would be unchanged, but since I am not in a position to offer a proof of this assertion, I concede that we have an open question about whether Sarpeshkar’s explanation is distinctively mathematical in this strict sense. In the next section, I present an example that is more clear cut on this issue. Another potential worry here is that the existence of the trade-off illustrated in Figure 1—which I have argued is a result of our information-theoretic definitions of the coding schemes—is due instead to the empirical fact that the states of all physical systems are liable to random fluctuations and hence introduce noise. In a noiseless world, we would assume, analogue computers would outperform hybrid and digital ones. Moreover, the objection goes, the trade-off would not be there in the noiseless world and so its existence is not a fact about information transmission that goes beyond any empirical facts. In response, I urge that even if the laws of nature were altered such that we have perfectly noiseless physical systems for building brains and computers, we should think of the trade-off itself as being there—even though natural selection and human engineers would not be aware of it—because of the ways that noise and information are defined theoretically. Consider an analogous scenario: In contemporary society, time spent at leisure trades off against time spent at work; the optimal solution to this trade-off is known as the ‘work-life balance’. In a future Utopia where robots do all the work, nobody needs to spend time earning a wage, so everyone spends all their time at leisure. People will not even be aware of the trade-off. But that does not alter the fact that by our definitions, time spent at leisure is not time spent earning a wage, and vice versa. In some sense, the trade-off is still there, but empirical conditions do not make it apparent. That said, it must be admitted that the empirical fact that actual physical systems are noisy is bearing substantial explanatory weight here—the trade-off is only relevant to brain physiology and computer design because of that fact. More generally, the notion of a trade-off only makes sense if we assume the presence of certain constraints. So in the final section of this article, I will consider the question of whether Sarpeshkar’s explanation is ultimately a causal one because it must refer to such empirical facts. Before moving on, it is worth noting that nothing turns on whether one uses Sarpeshkar’s analysis to explain a feature of one particular brain or a trait of brains in general. Thus, I would argue, this kind of non-causal explanation is applicable to singular states of affairs as well as regularities. 3 Case II: The Gabor Model Revisited In the previous case, we can observe a proximity between causal, evolutionary explanations and non-causal, mathematical explanations in biology and neuroscience. We see this again when we compare different efficient coding explanations of the response properties of neurons in the primary visual cortex (V1). As discussed in detail in (Chirimuuta ), the two-dimensional Gabor function (the product of a sinusoid with a Gaussian envelope) is part of the ‘standard model’ of V1. It has long been observed that neurons in this area respond selectively to the visual presentation of short, bar-like stimuli of a specific width and orientation. The parameters of the Gabor function can be adjusted to capture this selectivity pattern (the receptive field or RF), and this raises the question of why V1 neurons have receptive fields that can be fit by this equation. I previously argued that the Gabor function is central to causal, but non-mechanistic, explanations of V1 response properties. Here, I contend that there is also a non-causal part of the picture. When Gabor () first introduced the one-dimensional version of the function, it was in the context of signal engineering. Given the problem of analysing information contained in signal waves, he noted that there is a trade-off between one’s ability to accurately decode temporal and spectral (frequency) information, which is known as ‘Heisenberg-Weyl uncertainty’. The more reliably one decodes the arrival time of a signal, the less reliably one decodes its frequency, and vice versa. One cannot, at the same time, be maximally certain about both of these parameters. For example, Fourier analysis provides a very accurate analysis of the spectral composition of a wave, but temporal information is lost. Gabor proved that his function, which in effect performs a temporally localized Fourier analysis, provides the optimal balance between recovering temporal and spectral information. This, we can say, is a modally strong mathematical fact. In one of the first articles to model V1 responses with the Gabor function, John Daugman (, p. 1160) pointed out that the Gabor-like properties of V1 receptive fields could be the biological solution to the problem of jointly resolving both spectral and spatial (rather than temporal) information: […] the 2D receptive-field profiles of simple cells in mammalian visual cortex are well described by members of this optimal 2D [Gabor] filter family, and thus such visual neurons could be said to optimize the general uncertainty relations for joint 2D-spatial-2D-spectral information resolution. So there is a distinctively mathematical, non-causal explanation of why the Gabor function is optimal, which can form part of a bigger causal explanation of why V1 neurons should have evolved (or developed) their observed properties. Other neuroscientists have focused on the causal factors leading to the development of Gabor-like RFs, and such enquiry does yield experimental interventions. For example, Hyvärinen and Hoyer (, p. 2413) suggest that, ‘The reason why the CRFs [classical receptive fields] have Gabor-like shapes might thus be that these kind of CRFs are optimal for analyzing the input that the visual system typically receives’. In other words, the hypothesis is that in evolutionary or developmental time, V1 RFs have been adjusted to be most sensitive to the kind of visual information usually prevalent in the natural environment. As it happens, short edges and bar-like structures are very common in images of natural scenes, and so it seems plausible that the Gabor-like RFs are a good way to recover these kinds of stimuli. In order to test the hypothesis that in the course of development, neurons in V1 adjust their RF shapes to be most tuned to the common kinds of structures in the environment, it is possible to do causal experiments in which animals are reared in particular environments (for example, one which is full of vertical bars) to see if neurons with a preference for vertical stimuli then dominate V1. Such experiments were quite popular in the 1960s and 70s, and significant developmental effects could be shown (Blakemore and Cooper ). It should be noted that causal and non-causal explanations can be mutually supportive. One may demonstrate that a neural system bears a similarity to a theoretically optimal system, but this could just be a coincidence. If by changing the circumstances in which the system lives, one changes what counts as the optimum for the system, and the system diverges from the previously observed behaviour in a predictable way, this is good evidence that the previous behaviour was functionally significant and did not just bear a coincidental resemblance to the theoretical optimum. So any alternation between causal and non-causal explanation in neuroscience should not be taken as a troubling sign of theoretical double-think, but as a scientifically responsible strategy that profits from the insights of mathematical theorizing, while keeping theoretical speculation grounded in empirical facts. 4 Case III: A Dynamical Model of Prefrontal Cortex When addressing the challenge of understanding actual behaviours, rather than just single neurons or general computational features of the brain, neuroscientists must reckon with the daunting complexity of neural systems. Multi-electrode neurophysiology allows researchers to listen in on the activity of hundreds of neurons at a time, but even such small samples of activity from one brain area are extremely difficult to interpret functionally. Techniques of dimensionality reduction and dynamical systems analysis, imported from other branches of science, have become popular in the quest to simplify the brain. One recent study from Bill Newsome and Krishna Shenoy’s labs at Stanford employs these techniques in order to shed light on the question of how the brain makes behavioural decisions based on complex sensory stimuli. It is worth considering whether their explanation of context-dependent decision-making is causal, non-causal, or both. In order to address this question, it is first necessary to describe the study in a fair amount of detail. 4.1 A new explanation of context-dependent computation The focus of enquiry was the flexibility of humans and other primates in responding to relevant sensory stimuli, depending on contextual cues. The behavioural task targeted this ability by presenting monkeys with a visual display of hundreds of red and green moving dots. The proportion of different coloured dots would change from trial to trial, as would the predominant direction of motion. In some trials, a cue indicated to the monkeys that they should give their response based on the majority colour, and in the other trials, they were cued to respond based on the predominant direction of motion. Thus the same stimulus required different kinds of responses depending on a contextual cue. During this task, recordings were taken from populations of neurons in the prefrontal cortex (PFC), an area believed to be involved in controlling flexible, context-dependent behaviour. A number of models have been proposed to account for the neural basis of this behaviour, and one of the aims of the study was to test those models against the behavioural and neural data. The crucial step in the assessment of the models was the use of principal component analysis (PCA) as a means to simplify the neuronal population data set so that it could be represented in three-dimensional space. Mante et al. ([2013a], p. 79) interpret their three axes as corresponding to choice (which out of two responses the monkey will give), motion (the representation of information about the predominant direction of movement of the dots), and colour (the representation of information about the predominant colour of the dots). The dynamical response of the population to the stimulus on any one trial is plotted as a state-space trajectory in these three dimensions. One key finding was that whether or not the contextual cue is for a colour or motion response, the irrelevant sensory information is still represented by the neuronal population. This ruled out one popular early selection model of context-dependent computation, which assumed that the irrelevant information is filtered out before it reaches the PFC. Another key finding was that the orientation of the axes relative to the others remains fixed, regardless of which context was used. This ruled out two other models, which assumed that the choice axis lines up with the axis of relevant sensory information, leading to the contextually appropriate response (Mante et al. [2013a], p. 81). Given the failure of these existing models to account for the data, the authors go on to present their own model, which they describe as, ‘a previously unknown mechanism for selection and integration of task-relevant inputs’ (Mante et al. [2013a], p. 78). The first step is the training of an artificial network of ‘recurrently connected, nonlinear neurons’ (Mante et al. [2013a], p. 81) to perform a simulated version of the behavioural experiments, and see if the artificial ‘data’ have the same qualitative features of the real neuronal data. They then analyse the dynamics of the trained, artificial network to see if they can account for these features. In the three-dimensional state-space representation, the model population did indeed have the same qualitative features as the neuronal population: irrelevant sensory information was still represented at the population level, and the position of the choice axis remained fixed relative to the other axes (Mante et al. [2013a], p. 82). The use of the artificial network allowed them to perform a simulated experiment not possible with the real neuronal population: perturb the network into a particular state, and see what state the network relaxes into. This reveals the fixed points of the network dynamics—the states that the network always relaxes back to. The analysis showed a series of approximate fixed points—known as a line attractor—along the choice axis. Another feature of this dynamical system, referred to as the selection vector, was that perturbation of the network in the direction of the relevant stimulus dimension caused the network to relax to a point along the line attractor that was closer to a final ‘choice point’, whereas perturbation of the network in the direction of the irrelevant stimulus dimension resulted in the network just relaxing back to its initial point on the line attractor. The authors interpret this feature of the system as a ‘mechanism’ for the integration of relevant sensory information (Mante et al. [2013a], p. 83). 4.2 Causal or non-causal? It is interesting that the authors of the study repeatedly refer to these features of the network dynamics—the line attractor and the selection vector—as indicating a possible mechanism for context-dependent computation in the PFC. The important point is, however, that the ‘components’ of the posited mechanism are features revealed only by the dynamical analysis.26 This means that the notion of mechanism in play here is rather different to the one usually employed by philosophers of neuroscience, and it is not at all a foregone conclusion that the kind of explanation offered by this model is a causal-mechanistic one. In their discussion of the use of dynamical models to give causal-mechanistic explanations, Kaplan and Bechtel (, pp. 439, 443) claim (pp. 439, 443) that this is possible if the equations of the model can be interpreted as describing the activities of parts of actual neurons or circuits. This is consistent with the models to mechanism mapping (3M) criterion presented by Kaplan (, p. 347) and Kaplan and Craver (, p. 611). However, the fixed points that Mante et al. ([2013a]) characterize as explaining the context-dependent computations cannot be thought of as directly mapping on to any actual parts of a real or artificial neural network; rather, they are attractors in an abstract, low-dimensional state space. It is worth comparing the PFC model with a dynamical model more obviously amenable to the mechanistic account. Bechtel (, p. 553) presents the example of the dynamical model of circadian rhythms in Drosophila, writing that, ‘the equations are advanced […] as descriptions of the operations of specific parts of a mechanism’, and that, ‘an important part of evaluating the adequacy of a computational model is that the parts and operations it describes are those that can be discovered through traditional techniques for decomposing mechanisms’. In this case, it is straightforward to advance a model-to-mechanism mapping: the equations in the model represent such things as the rate of change in concentration of mRNA of a particular gene (per) in terms of a rate of transcription and rate of decay. A computational simulation then reveals the dynamics of the system described by the equations, and the fact that both simulation and biological system manifest the same oscillatory pattern is reason for thinking that the dynamical model of the components and operations of the cell (the set of differential equations) explains circadian rhythms. Here, the issue of how to interpret highly abstract mathematical objects, such as line attractors, does not arise. In order to advance a conventional mechanistic interpretation of the PFC model, one would have to read ‘organizational features of the target mechanism’ very liberally, such that the term refers to some non-localizable and non-decomposable features of the network, ones that can only be revealed by DST analysis, rather than components and activities that are observable without such techniques. Only in this way could the 3M criterion be satisfied. Yet this move invites the objection that a very liberal application of 3M stretches the notion of mechanistic explanation beyond recognition, and indeed usefulness. On the view of Silberstein and Chemero (, p. 967), ‘global organizational principles or features of complex systems’, which are revealed by DST analysis, ‘are not explicable in principle via localization and decomposition’, and thus should not be interpreted mechanistically. The failure of the mechanistic heuristics of localization and decomposition occurs, they argue, because many structurally diverse networks that differ in their implementational details still exhibit the same global features. Note that the same independence from implementational details is true for the artificial neural networks described in the PFC study: the modellers trained one hundred different networks, with different initial conditions and therefore different ‘synaptic’ connectivity patterns, and yet the global dynamical features were the same in each case (Mante et al. [2013b], p. 23). So is the model explanatory in another non-mechanistic, or even non-causal, sense? One way to address this question is to see how closely the study resembles the kind of modelling practices used in physics and engineering (for example, nonlinear fluid dynamics), which are often taken to be an exemplary of non-causal explanation. After all, many of the analytical tools employed in the PFC study were first developed in that context (Ott ). Batterman (, p. 23) introduced the notion of ‘type-ii why questions’ as those which require an explanation as to why a collection of micro-physically very different substances all exhibit the same macro-behaviour, for example, during phase transitions. Such explanations are provided by mathematical abstraction techniques, such as the renormalization group, which demonstrate that all of the different substances are members of the same ‘universality class’, despite their low-level differences. Ross () shows that an analogous pattern of explanation occurs when dynamical models and abstraction techniques are used in cellular neuroscience to explain why anatomically very different neurons all exhibit the same spiking behaviour, described by a ‘canonical model’. Now one could argue that a similar explanation is provided by the PFC study because the authors demonstrated that one hundred different networks trained to perform the context-dependent decision, all with different patterns of synaptic connections, converged on the same high-level dynamical properties (line attractor and selection vector), which were revealed by the abstraction techniques of PCA and the perturbation–relaxation investigation. However, one must also consider that there is a functional explanation for the different networks’ convergence on the same dynamics. The authors write that, We trained many networks (around 100) from different initializations of the weights and biases, and each time the network solved the problem in the same qualitative way. This points to the fact that the selective integration task […] placed strong constraints on the optimization process. (Mante et al. [2013b], p. 23) The implication is that the demands of the task causally shaped the development of the network, through the optimization process, in such a way that all the different networks inevitably converged on a pattern of connections that could implement the high-level dynamics and thus solve the context-dependency problem. It is an interesting question whether the non-causal explanation of universality or the causal-functional explanation of convergence on to the same behaviour is deeper or more scientifically significant in this case, or indeed whether the causal and non-causal explanations complement each other, as I argued they do in the case of the Gabor model. Lacking firm grounds to answer this question, or even clear intuitions either way, I will bracket the issue for now. As I see it, the more important explanandum that the PFC study seeks to address is how it is that any one network can perform the context-dependency task, rather than the type-ii why-question, namely, why different networks perform the task in the same way. Is the explanation here a non-causal one? Given that we still have open questions about the interpretation of dynamical models, one thing we can safely say is that the answer to this question turns on how we choose to interpret the working parts of the explanation (that is, the line attractor and selection vector). If one regards these as somehow capturing or summarizing the actual causally efficacious parts of the network (the properties of real and artificial neurons), then one would be inclined to say that the explanation is a causal one. For instance, one could say that causal properties of the network that allow it to integrate information are modelled and represented by the mathematical artifice of the line attractor, where the notion of mathematical representation without explanation is invoked (Saatsi ). On the other hand, if one finds such an interpretation far-fetched, then it is natural to say that the explanation is a non-causal one. This brings us to the point that arguably the PFC model is not explanatory at all. The neuroscientists’ account of context-dependent computation is entirely reliant on the three principal axes revealed by PCA, and the interpretation of these as indicating the ‘choice’ of the network, alongside motion and colour information. This raises the thorny question of how we should interpret the results of factor analysis. Some might accuse Mante and colleagues of committing the sin of reifying their essentially meaningless primary factors—that is, of ‘awarding physical meaning to all strong principal components’ (Gould , p. 250). In sum, we have three options on the table: the first is to say that the model offers a causal-mechanistic explanation; the second that it provides some sort of non-causal explanation; and the third position is that it is not an explanatory model. What reason is there to favour the second position? We can begin by highlighting problems with the first and third options. I am not moved by the view that the PFC model is not at all explanatory; this attitude strikes me as being tone-deaf to the scientifically compelling features of the analysis. Because of the way that their network moves through a low-dimensional space in response to ‘sensory’ input, the researchers are able to show that it is able to integrate relevant information and ignore irrelevant information, while retaining information about both kinds of stimuli, as observed in the actual neural data. The explanandum is, ‘how is this brain area able to integrate relevant sensory information?’, and by being shown specific features of the global dynamics, we are indeed provided with an explanans. If one is going to be sceptical about the meaningfulness of principal components in this case, one ought to be unpersuaded by any explanations that call upon statistical constructs, such as the greater profitability of Sunnylake Farm, compared to the otherwise identical Sunnybrook Farm, being due to the higher mean number of eggs laid per day.27 One strike against the mechanistic interpretation is that the explanation offered by the dynamical model of the PFC is not a constitutive one. That is to say, it is not an explanation of how a global phenomenon—computation—comes about because of the activities of some network components, whether spatially localized or not. Instead, the dynamical properties of the network, which do the job of explaining the computational phenomena, are themselves global- or population-level properties of the network.28 Another important point is that the model gives us no information about how we might make changes to the network in order to affect changes to its information-processing properties. It does not tell us which connections would have to be rearranged in order to make the computation no longer context-dependent, for example. However, one might ask, doesn’t the model tell us how we could intervene on the global dynamical properties of the network (rather than local connectivity patterns) in order to bring about changes in its computational properties? Isn’t it therefore a pared-down representation of some causal relationships?29 I believe that we should not interpret the counterfactual dependency between global dynamical features and computational properties as a causal relationship, even in a very minimal sense. This is because the dynamical description and the descriptions of the network’s computational properties are just two different ways of describing the same thing—the network. There is no spatio-temporal separation between putative cause and putative effect. To change the dynamical properties is just to change the information-processing capacities of the network. This brings me to say more about the second option, which I endorse: that the model offers some kind of non-causal explanation. We can think of the dynamical model as offering an illuminating perspective on the network. It makes transparent certain counterfactual relationships holding between the network’s global dynamical features and the computations that it performs—for instance, that the ability of the system to integrate sensory information for the duration of a trial is counterfactually dependent on it having a line attractor rather than a point attractor. Thus the model answers w-questions. We can say, for example, that if the selection vector were never orthogonal to the irrelevant stimulus dimension, then that information would not be ignored. This is another instance of an explanatory model providing counterfactual information that should not be interpreted as describing the outcomes of possible interventions. So if my interpretation is correct, this example fits into Woodward’s schema for non-causal explanation. Is the explanation also distinctively mathematical in Lange’s sense? Well, if it can be shown that there is a modally strong connection between the having of certain dynamical properties (one kind of mathematical description) and the ability to perform certain computations (another kind of mathematical description), then this example would also satisfy Lange’s criteria. But I leave this matter for readers with greater technical expertise than I myself have. One final point is that thinking of the model as providing explanations by offering illuminating global perspectives on a system helps shed light on why Batterman and Rice’s minimal model explanations are correctly described as non-causal ones. The point is not just, as they suggest, that minimal models like the DST one only offer caricatures of the causal dependencies (Batterman and Rice ; cf. Rice ). More strongly, when the features captured by the model are truly global, population-level ones, then no causal dependencies are being represented at all—either in an accurate or caricatured way. Instead, what the model does is reveal new global features of the system that indicate why the system also has the global, observable features that have served as our explananda. 5 Causal and Non-causal: Does the Difference Matter? In this article I have shown that there are instances in computational neuroscience of explanations that are precisely analogous to familiar examples of non-causal explanations in physics and biology. So if one is persuaded by Lange’s account of non-causal, distinctively mathematical explanation or by the non-causal extension of Woodward’s counterfactualist framework, then one should agree that computational neuroscience is just another domain in which such patterns of explanatory reasoning occur. What is interesting about this result is that it stands against the dominant mechanistic current of recent work in the philosophy of neuroscience, which seeks to interpret explanatory practice as the mapping of mechanisms: working out what cause produces which effect, in order to show how neuronal components are orchestrated to govern complex behaviours, such as navigation. Moreover, by extending Woodward’s framework to incorporate distinctively mathematical explanation, I am adapting materials frequently used by mechanists to the rather different task of analysing non-causal explanation. The clearest cases of non-causal explanation in neuroscience are efficient coding explanations that refer to information-theoretic trade-offs in order to show why it is that neural systems should employ particular computational solutions, such as hybrid computation (Section 2), or Gabor filtering (Section 3). So far, most of the discussion of non-mechanistic explanation in neuroscience has focused on DST. Yet as I show in Section 4.2, the argument that explanatory practice goes beyond the detailing of mechanisms is less clear cut in this area. The controversy turns on the question of whether or not we should interpret the mathematical structures revealed by dynamical modelling (for example, fixed points and line attractors) as representing anything in the brain that has causal powers to affect the computations performed. So does the difference between causal and non-causal explanation matter to neuroscientists and neuroengineers themselves? Perhaps not. As noted at the end of the discussion of the Gabor model, these different kinds of explanation should be seen as complementary rather than as in competition with one another (cf. Andersen [forthcoming]). Still, one might point out that if what we are interested in explaining is why some empirically realized system (like the brain) exhibits some feature, then this explanation must always have an empirical component (for example, that some optimizing selective process is operative). So one objection to my account is that the purely mathematical part of the explanation is not, by itself, really an explanation of anything empirical and that one needs the component about natural selection (or something similar) in order to arrive at an explanation of the empirically observed facts. In response, I would agree that the non-causal component is, arguably, not a stand-alone explanation of the efficiency of neural computation or the Gabor-shaped receptive fields of V1, even though researchers in theoretical neuroscience do talk of these mathematical results as explaining brain features. Such talk assumes that certain empirical background conditions obtain, such that the trade-offs are relevant to the neural systems, and that there is some developmental, evolutionary, or real-time adaptational process by which the near-optimal solution is arrived at; but such a story is bracketed in order to focus on the relevant topic of enquiry, which is the interpretation of a particular feature in terms of its utility and efficiency. So what is interesting here is that there is a division of explanatory labour: some neuroscientists will focus on the non-causal, mathematical explanation of the efficiency of a feature, while it is the job of others to find out about the aetiology of that feature. For the purposes of this article, I need not take a stand in the debate about the possibility of stand-alone non-causal explanation of empirical facts, nor answer the question of whether we can call an explanation non-causal if there is some non-causal part of the story (or causal if there is some empirical part of the explanation). But it is worth pointing out that Lange’s examples, such as the honeycomb one, invite the same objection. The task of this article is to show that there are examples of explanation in neuroscience that are of the same type as the central examples of non-causal explanation as presented in the work of Lange, Batterman, and Woodward. I hope by now to have made that case. Even if I concede that my three cases do not give us stand-alone non-causal explanations, my examples are still relevant to the ongoing debate over whether there are distinctly computational and non-mechanistic patterns of explanation in neuroscience. While authors such as Kaplan () echo Saatsi (), arguing that mathematics only has the role in neuroscience of representing physical systems via a mapping from equations to parts and processes of a mechanism, I have argued that mathematics has a stronger explanatory role that is independent of mechanism description. We arrive at a picture in that theoretical neuroscience is often focused on the non-causal parts of explanations and therefore has explanatory norms—appealing to efficient coding and mathematical trade-offs—that are distinct from the evolutionary and mechanistic branches of neuroscience. This, I believe, is sufficient for me to deflect the worry that efficient coding explanations are just another kind of evolutionary explanation, which are themselves elliptical mechanistic explanations. One issue that I have not addressed is the question of how we evaluate the quality of non-causal explanations. One of the virtues of the mechanistic account is that it lays down explicit normative criteria (Craver ). There is no space here to venture into this topic, but one obvious possibility is to evaluate non-causal explanations according to the precision and range of counterfactual or counterpossible information provided (that is, how precisely and extensively they are able to answer w-questions), in the same way that one can evaluate causal explanations. A novel possibility is to consider that non-causal explanation tends to exemplify a different explanatory virtue from the causal sort, that of unifying diverse phenomena under one general principle (Kitcher ). Though the unificatory account of explanation is less popular than it once was, it should not be ignored that all the cases I have presented do a good job of explaining why mechanistically quite diverse systems will converge on similar computational solutions—in other words, answering Batterman’s type-ii why-questions. In this era of big data neuroscience and a relative scarcity of theoretical insight, unificatory explanatory knowledge is a precious commodity. It will be instructive to see whether future developments in neuroscience will place non-causal explanations in increasingly prominent roles. 1 See, for example, (Machamer et al. ; Craver ; Bechtel ; Kaplan and Craver ; Kaplan ; Kaplan and Bechtel ; Levy ), amongst many others. 2 (Kaplan ; Piccinini and Craver ; Piccinini and Bahar ). 3 For example, (Weiskopf ; Barberis ; Serban ). 4 See, for example (Chemero and Silberstein ; Stepp and Chemero Turvey ; Silberstein and Chemero ). 5 (Batterman ; Rice ; Batterman and Rice ). 6 An equivalent debate is ongoing over the status of network models in neuroscience. Craver ([unpublished]) argues, contra Huneman (), that examples put forward of network models providing non-causal, topological explanations of neural function are either mechanistic or non-explanatory. Due to limitations of space, I will not discuss network models in this article. However, it would be an interesting exercise to apply the framework I develop below to these kinds of examples. 7 See (Chirimuuta , Section 5.1) for more detailed discussion of the points covered here. 8 Nor do optimality explanations in biology always rest on this assumption (Godfrey-Smith ). Instead, the point is to show that an observed feature has similarities with a theoretically predicted optimum, though there may be substantial departures from optimality due to structural or other constraints. Also, Wouters () gives an account of non-causal ‘design explanation’ in biology which does not depend on any claims about the optimality, or near optimality, of biological systems. 9 For one thing, if in a new act of creation God were to change the dimensionality of space-time, this could not be thought of as a possible intervention to be performed by finite beings. It would be contentious to extend an interventionist account of causation to acts of creation by infinite beings. More to the point, the theory of general relativity tells us that the counterfactual dependence of planetary stability on the geometry of space-time is not a result of any causal relationship between these two. For this reason, we cannot think of alterations in space-time that result in changes in planetary stability as interventions in Woodward’s sense; see the definition at the start of next paragraph, and see (Woodward , p. 702). 10 I take this to be uncontentious since many mechanist authors have adopted Woodward’s interventionist approach to causation, most notably Craver (). 11 See (Bokulich ; Bokulich ; Saatsi and Pexton ). 12 A new article by Baron et al. ([forthcoming]) independently hits upon this idea of using the framework of counterfactual explanation to characterize distinctively mathematical explanation. Their more formal presentation of this synthesis is a useful supplement to the examples I present below. 13 While it is thought not even God could affect such interventions. 14 See, for example, (Brogaard and Salerno ; Bjerring ). I am very grateful to an anonymous reviewer for pointing me to this literature. 15 Needless to say, my arguments do not turn on whether or not this explanation is disputed or accepted within the scientific community; rather, it serves to illustrate a pattern of explanatory reasoning. Recent work on this topic is reviewed by (Sterling and Laughlin , Chapter 10). 16 See (Cover and Thomas ) for more on information theory and (Rieke et al. ) on the application to neuroscience. 17 For example, (MacKay , p. 40, quoted by Husbands and Holland ): ‘Later in the 1940s, when I was doing my Ph.D. work, there was much talk of the brain as a computer and of the early digital computers that were just making the headlines as “electronic brains”. As an analogue computer man I felt strongly convinced that the brain, whatever it was, was not a digital computer. I didn’t think it was an analogue computer either in the conventional sense’; see also (von Neumann ; McCulloch and Pitts ; MacKay and McCulloch ). 18 To be more precise, the important contrast is between analogue and asynchronous digital or pulsatile systems. These are systems in which the signals are discrete, all-or-nothing events, but they operate in a continuous time frame. Neuronal spikes are a good example. Computer microprocessors, in contrast, send their discrete signals on a strictly clocked time schedule, and this is what is normally meant by digital computation. 19 Where information is defined in the Shannon–Weaver sense as log2(N), N being the number of possible and equally probable states of the wire. 20 Of course, in a true analogue system, the signal is a continuous function of a physical magnitude such as voltage, so the idea of associating just sixteen different signals with sixteen discrete voltages in the wire is artificial. But it serves when making the comparison in resource consumption across digital and analogue systems. 21 In my exposition, I concentrate on resource consumption in terms of number of material components required to transmit a given amount of information. The trade-off also occurs with respect to energy consumed to generate signalling states. One could increase the maximum voltage of a wire in order to create a greater ‘spacing’ between the voltages of signalling states, and thus make the system more resistant to noise corruption, but this would obviously require a bigger energy investment. 22 As a historical aside, it is worth noting that back in the 1950s, John von Neumann, one of the founders of modern digital computing, observed that one could offset the effect of noisy components in the brain by using more of them in a code with built-in redundancy, essentially highlighting this trade-off between the economy and reliability of a coding system (von Neumann ). 23 Sarpeshkar (, p. 1630) adds these words of caution: ‘However, although the firing of a spike is a discrete event, it does not imply that it encodes information about a discrete state. The information encoded by a spike is meaningful only in relation to spikes in different neurons, or in relation to earlier or later spikes in the same neuron. If these relationships are analog, then all-or-none events do not imply the encoding of discrete states.’ 24 Depending on how one understands the modal strength of information theory. It is conceivable that key concepts in information theory could have been defined differently and the theory still be consistent. So the Shannon–Weaver definitions do not have the obvious modal strength of ‘thirteen is a prime number’. That said, the important point here is that any counterfactual dependency between the properties of coding schemes and information theory cannot be understood as a causal one, just as the dependency of planetary stability on the dimensionality of space-time is not causal. 25 This worry is the topic of another article (Chirimuuta [forthcoming]). 26 Note that because the fixed points are only revealed by reverse engineering the model through perturbation and relaxation, rather than through an equivalent experiment on the real neural network (which would not be possible with current techniques), the authors are careful to present it as a possible or plausible explanation of the behaviour of the PFC, rather than a ‘how-actually’ model of this brain area. However, it can be understood as an actual explanation of the artificial neural network, so for the purpose of this section, I will just focus on the explanation of the behaviour of the model itself. As Sussillo and Barak (, p. 627) note, the operation of these recurrent neural networks is often opaque to modellers because the training of the networks does not specify how the task should be performed. Thus it is necessary to open the black box using the analytical techniques of DST. 27 As Gould (, p. 250) notes, the warning against reification of principal components is not intended as a blanket ban. Sometimes interpretation is justified by our knowledge of how the system behaves and what we are likely to be measuring. 28 See (Mante et al. [2013a], p. 79): ‘To study how the PFC population as a whole dynamically encodes the task variables underlying the monkeys’ behaviour, we represent population responses as trajectories in neural state space. Each point in state space corresponds to a unique pattern of neural activations across the population’. Also (Mante et al. [2013a], p. 78, emphasis added): ‘The mechanism reflects just two learned features of a dynamical system: an approximate line attractor and a “selection vector,” which are only defined at the level of the population’. So my point is that what these authors call a ‘mechanism’ for context-dependent computation cannot be thought of as providing a constitutive explanation. 29 That is, a very roughly sketched description of difference makers (Weisberg ; Strevens ). Acknowledgements The author thanks Marc Lange, Alex Reutlinger, and Juha Saatsi for much helpful discussion on this topic, and Bob Batterman and Jim Woodward for their encouragement, advice, and comments on the manuscript. The author is also grateful to audiences at the workshop on non-causal explanation at the University of Cologne 2013 and at the 2014 BSPS, and to members of Ken Schaffner’s Neurobiology, Reduction, and Emergence seminar at the University of Pittsburgh, for many thought-provoking questions. 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