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The authors note that co-author JARM is a PLOS ONE Editorial Board member. This does not alter the authors' adherence to all the PLOS ONE policies on sharing data and materials.

Conceived and designed the experiments: DP PMH NEL JARM. Performed the experiments: DP. Analyzed the data: DP PMH NEL JARM. Contributed reagents/materials/analysis tools: DP PMH JARM. Wrote the paper: DP PMH TS NRF NEL JARM.

We present a dynamical systems analysis of a decision-making mechanism inspired by collective choice in house-hunting honeybee swarms, revealing the crucial role of cross-inhibitory ‘stop-signalling’ in improving the decision-making capabilities. We show that strength of cross-inhibition is a decision-parameter influencing how decisions depend both on the difference in value and on the mean value of the alternatives; this is in contrast to many previous mechanistic models of decision-making, which are typically sensitive to decision accuracy rather than the value of the option chosen. The strength of cross-inhibition determines when deadlock over similarly valued alternatives is maintained or broken, as a function of the mean value; thus, changes in cross-inhibition strength allow adaptive time-dependent decision-making strategies. Cross-inhibition also tunes the minimum difference between alternatives required for reliable discrimination, in a manner similar to Weber's law of just-noticeable difference. Finally, cross-inhibition tunes the speed-accuracy trade-off realised when differences in the values of the alternatives are sufficiently large to matter. We propose that the model, and the significant role of the values of the alternatives, may describe other decision-making systems, including intracellular regulatory circuits, and simple neural circuits, and may provide guidance in the design of decision-making algorithms for artificial systems, particularly those functioning without centralised control.

Animals constantly make decisions, yet decision-making mechanisms and their evolution are still poorly understood in many cases. Recent years have seen a convergence of several research fields aiming to improve our understanding of general decision-making principles. Behavioural ecologists have argued for the need to combine the traditional study of animal behaviour through the lens of optimality arguments

Some researchers have noted the parallels between these apparently disparate fields, by observing that the interaction patterns of neurons in brain circuits and animals in groups appear to be very similar

In this paper we present a comprehensive analysis of our previous empirically-motivated model of decision-making by house-hunting honeybees swarms

Here, we show further aspects of value-sensitive decision-making that arise from cross-inhibitory stop-signalling. We analyse a model whose decision-dynamics are characterised by fast attraction to a one-dimensional decision manifold, followed by slower time-evolution along this manifold. We leverage a time-scale separation to reveal how the strength of cross-inhibition critically determines the decision-system response to both the difference in value and the mean value of the two alternatives. These analytic results considerably extend our previous initial analysis of this model's decision dynamics

We show that stronger cross-inhibition yields a greater minimum difference in value required for discrimination between the alternatives. When the difference in value is below this minimum, the alternatives are treated as equal or nearly equal, and the cross-inhibition determines whether or not the alternatives are of sufficiently high value to warrant breaking decision deadlock. A stronger cross-inhibition increases the minimum mean value of the alternatives above which a decision deadlock is broken and the system randomly chooses one of the alternatives. When the (nearly) equal alternatives have mean value below the minimum mean value threshold, deadlock is maintained, allowing for the arrival of information on other, possibly more valuable, alternatives.

We show that cross-inhibition strength determines the minimum detectable difference in the value of alternatives, as a function of their mean value, in a manner similar to Weber's law as arising from psychological studies. We further show that for decisions over alternatives that do differ sufficiently in quality, that the stochastic decision dynamics exhibit a speed-accuracy trade-off in decision-making that depends critically on the difference in value and mean value of the alternatives, with dependence controlled by the strength of the cross-inhibition. The speed-accuracy trade-off is qualitatively similar to the statistically-optimal trade-off of the drift-diffusion model of decision-making, although we present evidence that decision-making does not achieve optimality under the parameterisations we consider here.

The decision-making model we study is an extension of our previous empirically-motivated deterministic model

A collective decision is reached when one of the scout populations reaches a (variable) quorum threshold

Here we present analytic results on the general decision dynamics of the model. A well-established technique for studying models of binary decision-making similar to that described in Eq. 1 is to reduce the system of equations to a one-dimensional description of the decision dynamics (

When the accumulator for alternative

depends on

Thus, analysing the stochastic decision dynamics along the stable one-dimensional manifold will give a good understanding of the decision-making properties of the system as a whole. This is particularly relevant because the reduced dynamics resemble classical models of binary decision-making. For example, the general one-dimensional stochastic differential equation

where

Bogacz

Our previous analysis showed that the decision-making model of Eq. 1 with

A critical cross-inhibition level

The three-alternative model simulated here is a simple extension of the two-alternative model of Eq. 1, as described in section S.2 of

If, however, for the same rate of cross-inhibition

The preceding analysis assumes an evolutionarily hard-wired level of cross-inhibition, but further sophistication is possible if one considers what might happen to our hypothetical decision-maker, considering two equal but low value alternatives, if it waits too long. Any decision-maker has finite time and resources available to make decisions; in the case of a honeybee swarm members have finite energy reserves, since they load up with honey before swarming and do not resume foraging until the swarm has found a suitable nest site

The decision dynamics of the model are sensitive not only to the value of the available alternatives but also to the absolute

(Left) Bifurcation set as a function of

As

For larger values of

In

where

According to parameterisation of the decision problem and decision-maker, the dynamics include

The saddle-node bifurcation of

In

Other authors have previously presented similar bifurcation results in different contexts for different models. For example

As noted above, several classical models of decision-making, including the DDM and the (un)stable O-U process, are described using equations of stochastic motion on a line. The separation of timescales result presented above demonstrates that the decision dynamics converge rapidly to a line, along which they slowly diffuse. Of particular interest in decision-making models are speed-accuracy trade-offs

When two attractors for alternatives of different values exist, however, the presence of the unstable saddle-node can improve error rate without compromising reaction time (see

Although motivated by and presented in terms of decision-making by house-hunting honeybee swarms, our model exhibits a number of beneficial decision-making qualities that we might expect other organisms to exhibit. At the heart of our analysis is the observation that, in a choice, an animal is typically rewarded by the value of the chosen alternative, rather than whether or not it chose the best. In particular the model decision-maker displays a sensitivity to the absolute as well as the relative value of the alternatives under consideration; this enables it to wait for information on better alternatives to arise when considering equally poor alternatives, but to spontaneously choose one equal alternative at random when both are good enough relative to a crucial decision-making parameter, the rate of stop-signalling, or cross-inhibition,

Having suggested that our model might describe adaptive decision-making in general, what are the prospects for finding similar decision-making networks in other species? The form of the model equations is that of chemical reaction kinetics, in which interactions between chemical species are described by `mass action' terms. Therefore, there is the potential for intra-cellular regulatory networks to implement these decision-dynamics quite easily, for example in deciding for which of a number of available substances to activate the associated metabolic pathway. Evidence that single-cells can, for example, implement Bayesian-estimation through intra-cellular signalling

Another obvious class of decision models that invite comparison are those developed to describe neural networks for decision-making in simple perceptual decision tasks, such as those that take place in the primate visual cortex. A variety of accumulator models have been studied for their ability to fit experimental data, as well as implement optimal decision strategies (

Many accumulator models, like the classic DDM, also struggle with the correct choice when presented with zero net evidence, such as when choosing between two stimuli of equal average magnitude, and thus cross decision thresholds only through random drift. When choice of either alternative would result in an equal reward, such behaviour is clearly sub-optimal. Proposals to deal with this include implementing `urgency signals' or collapsing decision thresholds over time

Our non-linear model differs from the linear formulation of accumulator models underlying many analyses (

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Matlab code for stochastic simulation models.

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Locations of fixed-points, and simulated stochastic trajectories, as a function of varying stop-signal level (equal alternatives).

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Locations of fixed-points, and simulated stochastic trajectories, as a function of varying stop-signal level (unequal alternatives).

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Locations of fixed-points, and simulated stochastic trajectories, as a function of varying difference in quality of alternatives.

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Hysteretic effect as result of smoothly varying difference in quality of intervals repeatedly over a fixed interval.

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Further information on analytic and simulation results.

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We thank Jonathan Cohen, Jochen Ditterich, Phil Holmes, Jeremy Niven, Peter Swain and David Sumpter for discussion and comments.