The geometry of individual and collective decision-making

A team of researchers used state-of-the-art virtual reality to show the fundamental geometrical principles that result from the inherent interplay between movement and organisms’ internal representation of space. These principles apply across scales of biological organization, from individual to collective decision-making.

The findings show that animals spontaneously compress the environment into a series of sequential binary decisions, a response that helps successful decision-making and is robust to both the amount of alternatives available and context, such as whether options are static (e.g., refuges) or movable (e.g., predators) (e.g., other animals).

The researchers explore the interplay between mobility and vectorial integration during decision-making for two or more possibilities in space using an integrated theoretical and experimental approach (using immersive virtual reality).

This leads to the conclusion that the brain splits multichoice judgments into a succession of binary decisions in space–time on a regular basis. Experiments with different insects show that they all have these bifurcations, demonstrating that there are fundamental geometric rules that are required to explain how and why animals move in the way they do.

In computational models of this process, we reveal the occurrence of spontaneous and abrupt “critical” transitions (associated with specific geometrical relationships) whereby organisms spontaneously switch from averaging vectorial information among, to suddenly excluding one among, the remaining options. This bifurcation process repeats until only one option—the one ultimately selected—remains.

From an egocentric perspective, the decision process is projected to be sequential and dependent on the geometry with regard to the targets, therefore it should be possible to visualize it directly from animal trajectories when making spatial decisions.

Geometrical principles of two-choice and three-choice decision-making. (A) Schematic of the binary decision-making experiments. This simplified representation shows that a sharp transition in the animal’s direction of travel is expected near a critical angle, θc. (B) A phase diagram describing the “critical” transition exhibited while moving from compromise to decision between two options in space. The shaded area (also in E) represents the region in parameter space where both the compromise and the decision solutions exist. (C) Density plot showing trajectories predicted by the neural model in a two-choice context. The axes represent xand y coordinates in Euclidean space. The black line (also in G) presents a piecewise phase transition function fit to the bifurcation. (D) Schematic of three-choice decision-making experiments, where the central target is on the angle bisector of the angle subtended by the other two targets. (E) A phase diagram describing the first critical transition when the individual chooses among three options. After the individual eliminates one of the outermost targets, it can decide between the two remaining options, similar to the two-choice phase diagram described in B. (F) Theoretical predictions for decision-making in a three-choice context. The dashed line (also in H) is the bisector of the angle subtended by center target and the corresponding side target on the first bifurcation point.
Decision-making for a larger number of targets. Density plots of simulated trajectories for four- (A), five- (B), six- (C), and seven-choice (D) decision-making when targets are placed equidistant and equiangular from the agent. The axes represent x and y coordinates in Euclidean space. Geometrical configurations are also varied to place the targets on the same side of the agent (A and B) or in radial symmetry (C and D).

The researchers built a simple, spatially explicit model of neural decision-making to investigate how the brain minimizes choice when faced with several spatial possibilities. They were able to make theoretical predictions and discover unifying principles of spatiotemporal computation across biological scales.

The model provides explicit predictions for animal trajectories, allowing them to determine which target is reached in the simulation. The brain model is composed of individual components, called “spins,” that collectively, as a “spin system,” represent neural activity.

Hopfield was the first to introduce spin systems into the study of neurobiology in a landmark paper that provided significant insights into principles underlying unsupervised learning and associative memory. Spin systems have long been studied in physics due to their ability to give insight into a wide range of collective phenomena from magnetic to quantum systems.

In its most basic form, a spin system is made up of things that can be in either state 0 or 1, or, in physics terms, “up” or “down.” From spin and molecular systems to neurological systems undergoing phase transitions, spin systems have continually provided significant insights into complicated collective processes.

The results from their neural decision-making model are reproduced in a second model that describes spatial decision-making at a different scale of biological organization.

See Also

Consensus decision-making in simulations of animal groups follow the same geometrical principles. Results for two- (A) and three-choice (B) decision-making in a model of animal collectives. The density plots show trajectories adopted by the centroid of the animal group for 500 replicate simulations where the groups do not split. The axes represent x and y coordinates in Euclidean space. The black lines show a piecewise phase transition function fit to the trajectories. For the three-choice case (B), the dashed line is the bisector of the angle subtended by the center target and the corresponding side target on the first bifurcation point.

Although the details of the proposed model and those of brain dynamics differ in detail, the predictions for mobility during decision-making are strikingly similar.

Thus, we find that similar principles may underlie spatial decision-making across multiple scales of biological organization. Furthermore, by presenting social interactions in a decision-making context, our zebrafish experiments elucidate the neural basis of schooling, allowing us to glean insights across three scales of biological organization—from neural dynamics to individual decisions and from individual decisions to collective movement.

The geometry of decision-making in individuals and collectives, Vivek H. Sridhar, Liang Li, Dan Gorbonos, Máté Nagy, Bianca R. Schell, Timothy Sorochkin, Nir S. Gov, Iain D. Couzin

Published: December 2021
DOI: 10.1073/pnas.2102157118

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