article: Reductionism assumes that causation in the physical world occurs at the micro level, excluding the emergence of macro-level causation. We challenge this reductionist assumption by employing a principled, well-defined measure of intrinsic cause-effect power–integrated information (Φ), and showing that, according to this measure, it is possible for a macro level to “beat” the micro level. Simple systems were evaluated for Φ across different spatial and temporal scales by systematically considering all possible black boxes. These are macro elements that consist of one or more micro elements over one or more micro updates. Cause-effect power was evaluated based on the inputs and outputs of the black boxes, ignoring the internal micro elements that support their input-output function. We show how black-box elements can have more common inputs and outputs than the corresponding micro elements, revealing the emergence of high-order mechanisms and joint constraints that are not apparent at the micro level. As a consequence, a macro, black-box system can have higher Φ than its micro constituents by having more mechanisms (higher composition) that are more interconnected (higher integration). We also show that, for a given micro system, one can identify local maxima of Φ across several spatiotemporal scales. The framework is demonstrated on a simple biological system, the Boolean network model of the fission-yeast cell-cycle, for which we identify stable local maxima during the course of its simulated biological function. These local maxima correspond to macro levels of organization at which emergent cause-effect properties of physical systems come into focus, and provide a natural vantage point for scientific inquiries.
  Reductionist approaches in science usually assume that the optimal causal model of a physical system is at the finest possible scale. Coarser causal models are seen as convenient approximations due to limitations in measurement accuracy or computational power [1,2]. The reductionist view is based on the conjecture that the micro level of causal interaction is causally complete, leaving no room for additional causation at a macro level [3]. The reductionist assumption is most obvious in fields such as particle physics [4], neuroscience [5], and nanotechnology [6], but it can also be found in the social sciences [7], where researchers endeavor to ‘look inside the black box’. A case has been made for the occurrence of genuine emergence at various macro levels [8,9], such as the emergence of mind above and beyond the individual neurons (or atoms) that constitute the brain [10], and for the autonomy of the special sciences such as chemistry [11], and biology [12,13], above and beyond the underlying physics. However, arguments in favor of emergence have often been vague, or they have focused on the possibility that macro variables may have greater descriptive power than micro variables, rather than greater causal power [14,15,16]. Inspired by statistical physics, macro-level descriptions of a system are typically taken to be coarse-grainings, i. e. averages over micro elements and micro time steps. The reductionist assumption has been challenged by the introduction of explicit measures of cause-effect power, which were used to show that such coarse-grainings can indeed have greater cause-effect power at the macro level [17,18]. In simulated examples of simple logic gate systems, we coarse-grained (nearly) identical elements (‘neurons’) into groups (‘neuronal groups’) and averaged over their states. We demonstrated that, under certain conditions involving degeneracy and/or indeterminism, a macro-level system of coarse-grained elements can “beat” the micro-level system in terms of cause-effect power[17,18]. However, moving beyond statistical physics to biology, the macro elements of interest cannot be obtained by coarse-graining, because they are constituted of heterogeneous micro elements that are often compartmentalized and have highly specific functions, which would be muddled by averaging (see Box 1). For example, take the neuron, considered as the fundamental unit in much of neuroscience. Clearly, a neuron cannot be represented by a coarse-grained macro element, because it is constituted of a great diversity of specific molecules, organized in highly specific and hierarchical ways, performing highly specific functions. Indeed, it is the very specificity of the internal micro elements that makes the reductionist assumption seem inevitable in these cases: while we can treat a neuron as a black box for ease of understanding and for convenience, it would seem that its full causal power can only be captured by considering all the molecules that constitute the black box, in exquisite and specific detail [19]. Here we further challenge the reductionist assumption by generalizing the causal analysis employed for coarse-graining to black-boxing [20]: we first analyze a system of heterogeneous, specific micro elements at the micro level; then we repeat the analysis at the macro level by grouping subsets of those micro elements inside black boxes (macro elements). Black boxes are characterized exclusively by their overall input-output function [21,22]. The heterogeneous micro elements inside the black box are hidden inside a macro element, rather than averaged as with coarse-graining (Fig 1). As an example of a black box, Fig 2 shows, on the left, a simple, schematic neuron constituted of a number of specific micro elements (synapses S, cell body C, and axon hillock A) that interact internally in specific ways. On the right, the neuron is treated as a single macro element, a black box, that receives inputs (spike or no spike for each input), produces a single output (spike or no spike), and conceals its micro elements inside. In what follows, we assume that the causal power of a system is quantified by its intrinsic cause-effect power as previously defined [23,24]. While reductionism assumes implicitly that causal power resides exclusively with micro elements, we assess causal power explicitly–as intrinsic cause-effect power–and determine the spatiotemporal levels at which new cause-effect properties emerge. Such emergent cause-effect properties may include an increase in the overall intrinsic cause-effect power of the system, but also specific relationships between elements within the system (“mechanisms”) that only become apparent at the macro level. To quantify intrinsic cause-effect power and system mechanisms at the micro level and all possible black-boxed macro levels, we use the interventional and counterfactual causal framework of integrated information theory [23,24]. As a measure of intrinsic cause-effect power, integrated information (Φ) captures several aspects that are often overlooked in causal accounts [23]: the dependence of cause-effect power on the specific state the system is in (state-dependency); how cause-effect power of the system is structured (composition); whether the whole system is causally irreducible to its parts (integration); and what defines the system’s borders and grain (exclusion). These features make Φ particularly suited for assessing the cause-effect power intrinsic to a system, independent of external observers. As demonstrated through several examples, including the Boolean network model of the fission yeast cell cycle, the Φ value of systems of black-box macro elements can increase when going from finer to coarser spatiotemporal grains and lead to emergent cause-effect properties at macro scales.
  In typical usage, a black box is an object into which inputs impinge and from which outputs emerge, but its internal workings are not available for inspection [21,22]. For our purposes, a ‘black-box element’ is a physical macro element that can be manipulated, observed, and partitioned, which is constituted of several micro elements (spatial), operating over several micro time steps (temporal). To qualify as a black box, it must satisfy the following conditions: Specifically: Together, the above requirements allow to specify inputs and outputs of each black-box element, to define its macro state, to include within each black box only micro elements that are integrated and contribute to its input-output function, and to draw ‘borders’ around each black-box element that exclude any overlap with other black boxes (Figure C in S3 Text). Only systems that support local maxima of Φ, both in terms of constitution and spatiotemporal grain, are definite and have intrinsic cause-effect power. A system of elements is a local maximum if there are no ‘neighboring’ systems with a higher value of Φ. When only micro elements are considered, such as in [32], it is natural to define a neighbor as any system that differs in constitution by only a single micro element, that is, any system that can be made by either adding or removing a single element. However, to determine whether two systems at different spatiotemporal grains are neighbors, several distance measures have to be taken into account. For the present purposes, we consider three different distances between systems to establish whether two systems are neighbors in this general context. The first is the constitutional distance between two systems, which is the number of micro elements that must be added / removed from one system to transform it into the other. Next is the temporal distance between two systems, which is the difference in the number of micro updates that make up the corresponding macro updates. Finally, the spatial distance between two systems is the distance between the partitions that group micro elements into macro elements. In the current work we use the maximum matching distance between partitions [33], which is essentially the number of micro elements that must be moved from one grouping to another. If the sum of the constitutional, temporal and spatial distances between two systems is equal to 1 then those systems are neighbors, i. e. , two systems are neighbors if they differ by a single step in exactly one of the three distances. Given a set of micro elements, we evaluate all possible systems (sets of black-box elements) to determine which systems have intrinsic cause-effect power, at which spatiotemporal grain (the set of black-box elements that define the system), and what their borders are (the set of micro elements that constitute the system). Evaluating all possible sets of black-box elements includes all possible groupings of micro elements into macro elements. Then, for every grouping all possible elements of each black box are considered as its output element. Finally, cause-effect power is evaluated over all possible macro time steps of each black-box system. Note that not all micro elements must be grouped into black boxes when searching for maxima of intrinsic cause-effect power. It may be that adding a specific micro element to any black-box element within the system would in fact reduce cause-effect power. In this case, such micro elements are held fixed as background conditions of the macro system (see S3 Text).
  An intuitive example in which black-boxing may be appropriate is propagation delay–the amount of time between the output of one element and its effect on another element. Such delays are largely ignored in functional analyses and are taken to be an implicit aspect of the element of interest, i. e. , they are black-boxed. In the context of logic gates, for example, NOR logic is commonly described as a “universal” in the sense that any other logic can be built strictly from NOR gates. However, building, say, an XOR gate from NOR gates requires in fact a propagation delay as an implicit part of the circuit. In the following example, we explicitly model such propagation delays as (one or more) COPY elements that take a single input and then output the same value. Fig 3 shows the micro structure of an XOR element with a one-step propagation delay, along with the corresponding macro element, a black box with XOR logic. Consider a system of three interconnected XOR elements with a one-step propagation delay. At the micro level, this system is constituted of nine micro elements—six COPY and three XOR, which can be black-boxed over two time steps into a macro system of three interconnected XOR elements (see Fig 4). The current state of all elements is OFF. Assessing the cause-effect structure of the micro system, we find that there are only three first-order mechanisms and no high-order mechanisms. The three XOR elements each specify a mechanism with φ = 0. 5: by being in the OFF state, each XOR specifies that its two inputs must have been either (OFF, OFF) or (ON, ON) and that its outputs, the COPY elements, must be OFF in the future (Fig 4, top-right). All other sets of elements do not have cause-effect power, or are reducible, so φ = 0 (see Fig 5). Recall that from the intrinsic perspective, a set of elements must constrain both the system’s past and future irreducibly to be a mechanism for the system (see Methods). The six COPY elements, taken individually, lack any potential effect within the system: by being in the OFF state, a COPY by itself does not constrain the future state of its XOR output, which is still equally likely to be ON or OFF depending on the state of its other input (Fig 5, top). On the other hand, two COPY elements in the state (OFF, OFF) that input to the same XOR element do irreducibly constrain the system’s future states, since together they specify that the XOR element they output to will be OFF. Nonetheless, these pairs of COPY elements do not form a second-order mechanism in the system since their constraint on the system’s past state is reducible: in the OFF state, the two COPY elements taken individually already specify that their inputs must have been OFF, leaving no room for additional second-order constraints (Fig 5, bottom). The lack of either irreducible past or future constraints thus prevents the COPY elements from specifying first- or high-order mechanisms in the system. The integrated information of the micro physical system is Φ = 0. 25 (see S1 Text). The macro-level physical system with black-box elements also has three mechanisms with φ = 0. 5, but they are second-order mechanisms specified by pairs of XOR elements. By being in the state (OFF, OFF), each pair of XOR elements specifies that the past state of the entire model must have been either (OFF, OFF, OFF) or (ON, ON, ON), and that the future state of their common output must be OFF (Fig 4, bottom-right). Neither of the XOR elements in this high-order mechanism can specify these constraints on its own. Individual XOR elements lack potential effects in the system for the same reason as the individual micro COPY gates above. At the macro level, the collection of mechanisms (cause-effect structure) is more integrated than that of the micro level, with a value of Φ = 1. 875. Although the system has the same number of mechanisms and the same φ values at both the micro and the macro level, the black-boxed system has higher Φ because a system partition impacts the macro level cause-effect structure more than the micro level cause-effect structure. The black-box system “wins” by having more overlap in its mechanisms, both in terms of the elements they are composed of and the constraints they impose. The high-order mechanisms of the black-box system have overlapping constraints, with each mechanism constraining all elements within the system, whereas the first-order mechanisms of the micro system only constrain their respective COPY inputs and outputs, without overlap. A system partition at the micro level thus only affects a single micro mechanism, whereas a system partition at the black-box level affects all of the mechanisms in the system, resulting in higher integration (see S1 Text). Consequently, there is irreducible cause-effect power that emerges at this macro level of the physical system. Concealing the COPY elements inside the black boxes reveals the high-order interactions between the XOR gates over two time steps. Note also, that, while the causal analysis is state-dependent, in this example the irreducibility of micro and black-box cause-effect structures (their Φ values) and thus the relationship between levels, is equivalent for all possible system states. In a second example, we consider a larger micro system constituted of 55 elements that all implement NOR logic. By testing all possible black-boxings, we establish three local maxima of cause-effect power which reveal the organizational hierarchy of the system. Fig 6, demonstrates how a group of 11 elements implementing NOR logic can be connected in such a way to produce AND/OR logic, or MAJORITY logic at coarser spatiotemporal scales. The 55-element system is arranged into five interconnected groups of 11 elements, with each group organized according to Fig 6 so that the system exhibits different functions at different spatiotemporal scales. Each group of 11 elements receives inputs from three other groups and has a single element that outputs to three other groups (Fig 7, top left). We consider the system state in which each of the 55 NOR micro elements is ON. In the following, we focus on the cause-effect structures of the system levels shown in Fig 6: the micro physical system of NOR elements, a black-boxed system of AND/OR elements, and a black-boxed system of MAJORITY elements. These systems are shown in Fig 7 (top row) ordered according to the average spatial grain of their elements. Many other possible black-boxing schemes were also evaluated; however none of these other black-box macro systems had Φ > 0. Two representative examples of macro systems with Φ = 0 are included in Fig 7 (bottom). At the micro level, the system’s cause-effect structure consists of 55 first order mechanisms, one for each micro element with φ = 0. 239 on average, and no high-order mechanisms. The integrated information of this micro physical system is Φ = 0. 453 (see S1 Text). The macro-level AND/OR black-boxed system with an average spatial grain of 2. 75 (Fig 7, top, middle) has 20 macro elements, 15 implementing AND logic and 5 implementing OR logic, operating over two time steps. Similar to the micro level, its cause-effect structure is composed of 20 first order mechanisms (one for each black-box element) but no high-order mechanisms, with φ = 0. 112 on average. This black-boxing reduces the number of first-order mechanisms, but does not reveal high-order mechanisms or overlapping constraints, thus the macro system is no more integrated than the micro system. Moreover, this black-boxing in fact reduces the integrated information of the first-order mechanisms in the system compared to the micro level (φ values are 0. 127 lower on average), leading to lower integrated information for the system (Φ = 0. 080). The macro-level black-boxed system with an average spatial grain of 11 (Fig 7, top right) is defined by considering black-box elements implementing MAJORITY logic over four time steps. Compared to the macro level with an average spatial grain of 2. 75, this additional black-boxing step further reduces the number of elements, but increases the average φ to 0. 216 (φ values are still 0. 023 lower than the micro level on average). However, this macro system is endowed not only with first-order mechanisms, but with all possible second, third and fourth-order mechanisms. In total, its cause-effect structure includes 30 of 31 possible mechanisms from the power set of black-box elements, resulting in high integration, with Φ = 2. 333, more than the micro level. Fig 7 also shows additional black-box systems with Φ = 0. One of these black-box systems with an average spatial grain of 1. 57 has 20 black-box OR elements over two time steps and 15 micro NOR elements. A second black-box system with average spatial grain of 3. 66 has 10 black-box AND elements over two time steps and 5 black-box AND elements over four time steps. For both of these systems (and many others not shown), the integrated information is Φ = 0, because there is no common temporal scale over which all the elements in the system have effects on other elements within the system. For any specific temporal scale, there will be elements that do not causally contribute, thus the system is not integrated. In summary, this example demonstrates how evaluating cause-effect power over many different spatial and temporal scales of black boxes identifies local maxima of cause-effect power and reveals emergent cause-effect properties. For this example, the analysis reveals functional relationships between elements; local maxima of cause-effect power occur specifically at the micro level of NOR elements (average spatial grain size of 1, Φ = 0. 453), at an intermediate macro level of AND/OR elements (average spatial grain size of 2. 75, Φ = 0. 080) and at a coarser macro level of MAJORITY elements (average spatial grain of 11, Φ = 2. 333). While these spatial grains reveal emergent levels of organization at which the system exhibits intrinsic cause-effect power, which shed light on its cause-effect properties, the vast majority of systems of black-box elements, on the other hand, yield Φ = 0. As a demonstration of black-boxing in biological systems, we apply the framework to the Boolean network model of the fission-yeast cell-cycle [35]. The model consists of nine Boolean (“micro”) elements representing the state of crucial proteins expressed during cell division. Each element implements linear threshold logic, and the connections between elements are weighted, with each connection being either excitatory (+1) or inhibitory (-1) in nature (see Fig 8A). One element, “SK” only inputs to the system, receiving no feedback. This element acts as a catalyst for cell division: when it is activated while the network is in its biological attractor state, the remaining eight elements cycle through a sequence of 9 states, eventually returning to the initial attractor state (see Fig 8B). This cycle of states is called the ‘biological sequence’ of the model, and captures the specific sequence of protein expressions that occur during the cell-division cycle. Since the element SK receives no feedback from the rest of the cell-cycle network, any system that includes SK will necessarily be reducible (Φ = 0). Only when SK is fixed as a background condition can we potentially identify systems with Φ > 0. Furthermore, if we consider the remaining eight elements (excluding SK) as a system, one of the states of the biological sequence (t2, see Fig 8B) has no cause (potential past state) within the system (it is caused by the catalyst element SK which initializes cell division from outside the system). For this reason, the cause-effect structure of this system is undefined in state t2. In what follows, we refer to the cell-cycle network as the eight strongly connected elements that contain both inputs and outputs (not including SK) and its biological sequence as the eight states (t1, t3-t9) with well-defined cause-effect structures. Previous work analyzing the cause-effect structure of the cell-cycle model demonstrated that the cell-cycle network constitutes a stable local maximum of integrated information across all states of the biological sequence [32]. However, this previous work only analyzed the cell-cycle model at the micro level, considering all possible subsets of micro elements. In the current work, we extend this analysis by considering the cell-cycle network at macro spatiotemporal scales. Specifically, we consider all possible groupings of the cell-cycle network into black-box macro elements, at time scales of 2,3 and 4 micro updates (greater time scales may reveal additional local maxima and emergent cause-effect properties). There are 4140 ways to group the eight micro elements in the cell-cycle network into any number of black-box elements, and for each grouping there are on average 10 different ways to define the output elements of the black boxes. Considering three different time scales for each set of black-box elements, results in a total of 124,176 macro systems to analyze. Across all states of the biological sequence, there are 2224 macro systems with Φ > 0, an average of 278 per state, or roughly 0. 22% of all possible systems. Among the 2224 macro system with Φ > 0, we identify 33 unique local maxima (some others are duplicates due to symmetries in the network). The majority of these local maxima are transient, occurring in an average of 2. 5 out of 8 states in the biological sequence. However, 5 of the local maxima are stable over all states of the biological sequence. The micro system is one example of a stable local maximum, confirming that the results of [32] hold even when considering macro systems. The remaining four local maxima occur at macro spatiotemporal scales, one at a time scale of 3 micro updates, and the others at a time scale of four micro updates (see Fig 9). Note that the intrinsic cause-effect power of a system is state dependent, and stability across subsequent time steps is not assumed at any point in the analysis. That the cell-cycle supports stable local maxima of macro cause-effect power is a feature of this biological system that is revealed by the causal analysis, rather than a requirement imposed by the framework. Our analysis moreover reveals that one element in particular (Slp1) serves as a black box’s output in every stable local maximum. This indicates that Slp1 may play a crucial role in stabilizing and integrating the network over longer time scales during the process of cell division—a property that could not be identified from its micro level interactions [32,36].
  In the current work, we use intrinsic cause-effect power as a quantification of causal power, and demonstrate several examples of systems of black-box macro elements with higher intrinsic cause-effect power than the corresponding micro systems. To the extent that the notion of causal power is appropriately captured and quantified by intrinsic cause-effect power, our results refute the reductionist assumption that causal power resides exclusively at the micro level. The value of our characterization of cause-effect power had been previously demonstrated in a number of contexts [25,28,32], and will continue to be evaluated in the future. A limitation on the practical application of this framework is the computational demands for exhaustively evaluating intrinsic cause-effect power. Currently, cause-effect properties can only be fully explored for very small systems (< 10 micro elements; propagation delay example, cell-cycle example) or by exploiting symmetries in the system (local maxima example). Future work will extend the PyPhi software for evaluating intrinsic cause-effect power [34] by including, for example, approximations based on the connectivity matrix. However, practical applications inevitably will have to use a targeted approach and only assess the intrinsic cause-effect power of a predetermined set of macro-level systems instead of evaluating all possible black-box systems. Theoretical investigations like the current work (see below) as well as previous exploration of coarse-grained macro elements [18,37]will be crucial to define the criteria that will guide such a targeted approach. The two main requirements for high Φ are that a physical system is differentiated (many specific mechanisms) and integrated (mechanisms with overlapping constraints). Typically, whenever a lower level system is mapped into a higher macro system, there is reduced state differentiation, i. e. , the macro system has fewer elements and a smaller state space. This decrease in differentiation means fewer potential mechanisms and thus less potential integrated information [29]. In order for a macro level system to have higher cause-effect power (Φ) than a finer grained system over the same elements, the macro system must increase cause-effect power either by having more specific mechanisms, or a more integrated set of mechanisms. Degeneracy and indeterminism are two factors that influence the specificity of a mechanism. Everything else being equal, decreasing degeneracy and indeterminism leads to an increase in the cause-effect power of mechanisms within the system. In [17,18] we demonstrated that coarse-graining (averaging) micro elements into macro elements can lead to an increase in intrinsic cause-effect power that can overcome the inherent loss of differentiation in macro systems. An increase in intrinsic cause-effect power through reduction of degeneracy is also possible through black-boxing, as shown in S2 Text. The particular asset of black-boxing is that it may reveal high-order mechanisms and joint constraints between mechanisms at macro spatiotemporal scales. As demonstrated by the propagation delay example, the macro can even beat the micro level through increased integration. This may occur when elements with few potential effects are concealed within black-box elements, and micro elements with many potential effects serve as the outputs of black-box elements, resulting in a more densely interconnected set of macro elements, where groups of macro elements share common inputs and common outputs. If creating common inputs and common outputs among elements leads to additional, joint constraints on the possible past and future system states, elements may form high-order mechanisms, resulting in a more integrated cause-effect structure and higher Φ. Being a part of high-order mechanisms, or being constrained by multiple mechanisms, gives an element additional ways to contribute to the cause-effect structure; when an element contributes in multiple ways, cutting that element has a greater effect on the cause-effect structure, making the system more irreducible. Being more irreducible means having higher intrinsic cause-effect power (Φ) and may thus lead to a causally emerging macro level. This suggests that black-boxing is most beneficial when there are “causal bottlenecks” in the micro system, that is, when a micro element with a single or few outputs connects to a micro element with a single or few inputs. In such cases, it is impossible for these micro elements to contribute to high-order mechanisms, and such elements represent a “weak link” in the integration of the system. More generally, black-boxing should be particularly appropriate in systems with local modular interactions whose results are distributed across the system, such as molecular interactions within neurons in the brain, or electrical interactions within computer networks. Evaluating cause-effect power of black-box systems across many spatiotemporal scales shows that, in general, there can be several local maxima of macro cause-effect power, between which integrated information decreases or falls to zero. In Fig 7, the local maxima capture emergent functional roles of black-box macro elements, corresponding to the different descriptions of the system as sets of NOR, OR/AND, or MAJORITY elements. Importantly, even within a given spatiotemporal grain, there will generally be several local maxima corresponding to overlapping subsets of elements, such that adding or subtracting an element reduces integrated information [18,23]. These local maxima of intrinsic cause-effect power across and within levels correspond to organizational macro levels and systems having emergent cause-effect properties. These are natural levels and systems for the special sciences to investigate. A prime example is biological systems, since they contain many highly specialized components which are required to perform their function. In biology we can study the molecules within an individual cell, the interactions between networks of cells (nervous system), individual organs (liver, kidneys), whole organism (animals, humans), and communities of organisms (swarms, societies). The Boolean network model of the fission yeast cell cycle is one example of a simulated biological system which contains many heterogeneous micro elements that perform specific functions in order to accomplish cell division. Applying the black-boxing framework reveals several macro local maxima that are stable throughout the biological sequence of the network model, and highlights the role of element Slp1 in stabilizing the cycle. Note that the typical approach of studying biological systems at a particular (macro) spatiotemporal scale is precisely to treat its next-lower level components as black boxes. Here we have proposed a theoretical framework to evaluate cause-effect power and the cause-effect properties of such a black-box system. If an organizational level corresponds to a local maximum of integrated information, then there will be cause-effect properties that emerge at that level, and there is knowledge to be gained by studying the system accordingly. Finally, while local maxima reveal cause-effect properties to an investigator studying the system, the global maximum specifies the set of elements and spatiotemporal grain at which the system has most cause-effect power upon itself–from its own intrinsic perspective. According to integrated information theory, a set of elements at the spatial-temporal grain that defines the global maximum of intrinsic cause-effect power corresponds to a physical substrate of consciousness [23,24].
summary: We challenge the reductionist assumption by studying causal properties of physical systems across different spatiotemporal scales. The result is that–contrary to reductionist views–causal power can emerge at macro scales. Rather than relying on the traditional notion of coarse-grains (averages), we introduce the notion of functional black boxes that are defined based on their input-output relationship. Using a sequence of examples, our work demonstrates that black boxes are particularly well suited to capture the heterogeneous and specialized nature of components in biological systems. While the emergence of coarse-grained systems relies on increased specificity, black-boxing reveals the importance of structure and integration. Our framework is mathematically rigorous and fully general, hence applicable across many disciplines; it is particularly useful in objectively identifying informative perspectives on complex systems in the physical sciences.
keywords: action potentials
  medicine and health sciences
  statistical mechanics
  engineering and technology
  nervous system
  cell cycle and cell division
  cell processes
  membrane potential
  electrophysiology
  neuroscience
  logic circuits
  network analysis
  computer and information sciences
  animal cells
  chemistry
  physics
  cellular neuroscience
  cell biology
  anatomy
  synapses
  physiology
  neurons
  biology and life sciences
  cellular types
  physical sciences
  neurophysiology
  chemical elements
  electronics engineering
title: Black-boxing and cause-effect power