Prototypical examples of black-box and coarse-grain elements that can lead to the emergence of macro-level cause-effect power.

<p>Left: A black-box element conceals many micro elements with specific functions. Right: A coarse grain macro element averages together many homogenous micro elements that share a global function.</p

Boolean network model of the fission-yeast cell-cycle.

<p>(A) A network of 9 linear threshold elements connected by excitatory (black) and inhibitory (red) connections. The element, SK receives no input from the other elements. For this reason, we do not consider it as a part of the cell-cycle network, but rather as an external input that serves as a catalyst to initiate cell division. At the micro level, the system comprised of eight elements (in blue) is a stable local maximum of Φ for the duration of the biological sequence. (B) A sequence of 9 states that represent the cell division cycle, called the networks biological sequence.</p

A system of three interconnected XOR elements with one-step propagation delay and all elements in the OFF state.

<p>Top: At the micro level, the propagation delay is modeled by COPY elements between the XOR elements (left). The micro level has only first-order mechanisms specified by individual XOR elements (right), and Φ = 0.25. Bottom: When the elements are black-boxed over two time steps, the system is comprised of three interconnected macro elements implementing XOR logic. The macro-level system has second-order mechanisms specified by pairs of XOR elements (right) and Φ = 1.875.</p

Black-boxing and cause-effect power - Fig 2

<p>A schematic neuron considered as a number of `micro' elements (left), or as a black box (right). At the micro scale, the neuron receives inputs at its synapses (S), which are passed on to the cell body (C) and then to the axon hillock (A), which outputs to other neurons. Cause-effect power is assessed by perturbing each element (small hands) and observing the effects, while irreducibility is assessed by partitioning the elements (dashed red line). At the macro scale, there is only the black-box element (neuron) which receives three inputs and generates an output. Cause-effect power is assessed by perturbing the output of the black box (big hand) and observing its effects without constraining the constituent micro elements, however its irreducibility is still assessed by partitioning between micro elements (dashed red line).</p

A system of 55 interconnected NOR micro elements viewed at several different grain sizes, with all elements in the ON state.

<p>The micro elements form 5 interconnected groups, with each group arranged according to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006114#pcbi.1006114.g006" target="_blank">Fig 6</a>. The systems are arranged with micro elements on the far left and black-box elements of increasing spatial grain to the right. The legend on the right specifies the input-output function of each element. Each of the three systems on the top row is a local maximum of cause-effect power, corresponding to the three spatiotemporal grains shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006114#pcbi.1006114.g006" target="_blank">Fig 6</a>. Shown in the bottom row are two representative examples of the many systems with Φ = 0 at spatial grains between the local maxima.</p

Black-boxing and cause-effect power - Fig 6

<p>Left: A collection of 11 NOR elements at the micro scale. Middle: A macro scale black-boxing of these elements into four black boxes. Perturbing the inputs of these black boxes reveals that three implement AND logic and the final one OR logic, each at a time scale of two time steps. Right: A macro scale black-boxing with one element, implementing MAJORITY logic over its three inputs at a time scale of four time steps. Note that there are only three inputs, but each input arrives at two different micro elements.</p

All stable local maxima of macro cause-effect power for the cell-cycle network over the course of its biological sequence.

<p>Stable local maxima are identified at two different time scales (over 3 or 4 micro updates) and with groupings of the eight micro elements into either two or three macro elements. The output element for each black box is marked by a green outline; one common feature among all of the stable maxima is that element Slp1 acts as an output element of one black box. Note that connections between black boxes that do not originate from output elements are not shown in the figure because they do not contribute to the cause-effect structure (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006114#pcbi.1006114.s003" target="_blank">S3 Text</a>).</p

Two potential mechanisms from the propagation delay network in Fig 4 which end up being reducible.

<p>On top is a COPY element that does not specify a mechanism. By being OFF in the current state, the COPY element constrains its input to be OFF in the previous state, but it does not constrain the future state of its output element, because the state of the XOR element still completely depends on the unknown state of its other input (shown here in grey). The bottom panel is a set of COPY elements which do not specify a high-order mechanism because they do not have an irreducible cause (the red line partitions the cause in two with no loss of information). Taking each COPY element independently fully constrains the past state of its input to be OFF.</p

The structure, equation of state, and phase transitions of ammonia dihydrate polymorphs.

<p>ISIS spallation neutron source experimental report.</p> <p>Describes a first high-pressure survey of the ammonia dihydrate PT phase diagram using neutron powder diffraction.</p

Table of Mechanisms from How causal analysis can reveal autonomy in models of biological systems

Description of all irreducible mechanisms in the cause-effect structure of the cell-cycle network in the biological attracto