Fig 1.
Minimal examples of causally emergent dynamics.
In Example 1 (left) the system’s parity tends to be preserved while no interactions occur between low-level elements, which is an example of causal decoupling. In Example 2 (right) the system’s parity determines one element only, corresponding to downward causation.
Fig 2.
Diagram of causally emergent relationships.
Causally emergent features have predictive power beyond individual components. Downward causation takes place when that predictive power refers to individual elements; causal decoupling when it refers to itself or other high-order features.
Fig 3.
Integrated information decomposition (ΦID).
ΦID lattice for n = 2 time series [20], with downward () causation and causal decoupling (
) terms highlighted.
Fig 4.
Causal emergence in Conway’s Game of Life.
The system is initialised in a “particle collider” setting, and run until a stable configuration is reached after the collision. Using particle type as a supervenient feature V, we find the system meets our practical criterion for causal emergence.
Fig 5.
Causal emergence in the flocking boids model.
As the avoidance parameter is increased, the flock transitions from an attractive regime (in which all boids orbit regularly around a stable center of mass), to a repulsive one (in which boids spread across space and no flocking is visible). a) Our criterion Ψ detects causal emergence in an intermediate range of the avoidance parameter (error bars represent the standard deviation estimated over surrogate data). b) Sample trajectories of boids (grey) and their center of mass ().
Fig 6.
Causal emergence in motor behaviour of an awake macaque monkey.
a) Position of electrocorticogram (ECoG) electrodes used in the recording (in blue) overlaid on an image of the macaque’s left hemisphere (front of the brain towards the top of the page). b) Sample time series from the 64-channel ECoG recordings used, which correspond to the system of interest . c) 3D position of the macaques’s wrist, as measured by motion capture (
) and as predicted by the regression model (
), taken as a supervenient feature
. d) Our emergence criterion yields
, detecting causal emergence of the behaviour with respect to the ECoG sources. Original data and image from Ref. [34] and the Neurotycho database.