Probabilistic vs. non-probabilistic approaches to the neurobiology of perceptual decision-making

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Optimal binary perceptual decision making requires accumulation of evidence in the form of a probability distribution that specifies the probability of the choices being correct given the evidence so far. Reward rates can then be maximized by stopping the accumulation when the confidence about either option reaches a threshold. Behavioral and neuronal evidence suggests that humans and animals follow such a probabilitistic decision strategy, although its neural implementation has yet to be fully characterized. Here we show that that diffusion decision models and attractor network models provide an approximation to the optimal strategy only under certain circumstances. In particular, neither model type is sufficiently flexible to encode the reliability of both the momentary and the accumulated evidence, which is a pre-requisite to accumulate evidence of time-varying reliability. Probabilistic population codes, by contrast, can encode these quantities and, as a consequence, have the potential to implement the optimal strategy accurately.

Highlights

► Optimal decision making requires accumulation of evidence to bound in confidence. ► Behavioral/neuronal evidence suggests humans and animals follow such strategy. ► Drift-diffusion and attractor network models approximate this strategy. ► Probabilistic population codes have potential to implement strategy precisely.

Introduction

Efficient decision-making requires inferring the state of the world from uncertain or ambiguous evidence [1]. Little evidence results in inaccurate decisions, such that it is in the decision maker's best interest to boost her confidence by accumulating evidence over time and, if possible, across cues before committing to a decision. Thus, it is essential for the decision maker to perform decisions in two stages: first, she accumulates evidence to reach a certain level of confidence, and – once this level is reached – commits to her decision (Figure 1). A hunting eagle, for example, needs to be fairly certain about the presence of a rabbit before initiating its dive. Similarly, humans require certainty about the state of the surrounding traffic before crossing the street. In both of these cases a period of evidence accumulation is followed by acting upon this evidence.

We first give a short overview over the origins of statistically optimal, two-stage decision-making – which we will refer to as the normative strategy – as well as behavioral evidence that humans follow such a strategy. This provides us with a set of properties that decision-making models need to feature, and with respect to which we compare three types of models: diffusion models, models based on attractor dynamics, and probabilistic population codes. We show that the model based on probabilistic population codes provide a neural implementation of the normative model of decision making, while the other approaches provides various approximation to the normative approach.

Section snippets

Decision making under uncertainty

The realization that decision-making is essentially a task of probabilistic inference based on uncertain information was pioneered by Pascal in his famous wager on the benefits and losses involved in believing in God [2]. Gaining popular following, Pascal's approach was extended to all sorts of probabilistic decision problems, such as Bernoulli's well-known St. Petersburg paradox [3] that deals with wagering in games of chance. Finally, Laplace fully formalized general decision making under

Optimal accumulation of evidence

Behavioral studies have confirmed that human observers do not only take uncertainty into account, but also do so close-to-optimally according to the two-stage procedure outlined above. In the stage of evidence accumulation across time and cues, the decision maker needs to weight the momentary evidence in proportion to its reliability. Cue integration experiments that modulate the reliability of one of the cues have confirmed that humans indeed take this reliability into account [1, 10, 11],

Committing to a decision at the bound

The second stage of optimal decision-making under uncertainty is to commit to a decision once a pre-set level of certainty, called the decision bound, has been reached. Setting this bound optimally is complex, as it depends on both factors internal to the decision maker and properties of the task at hand. Thus, we first consider evidence for the presence of such a bound, and then discuss its optimality.

In reaction time tasks, decision makers are able to trade-off their decision speed with the

Diffusion decision models for 2AFC decision-making

The dominant model of decision-making for two-alternative forced choice (2AFC) tasks in psychology is the diffusion decision model [DDM [24••, 26, 41, 42]], in which a particle drifts and diffuses between two boundaries (Figure 2). Hitting either of these boundaries triggers a decision. A decision is correct if the particle hits the boundary corresponding to the mean drift rate. Incorrect choices occur due to the stochastic particle diffusion, and are less frequent for large drift rates. Thus,

Decision-making by attractor dynamics

Many neural models of decision making are based on networks with attractor dynamics. As we will argue, these models only approximate diffusion decision models and, as such, might not be optimal even when decisions are binary and the evidence is of constant reliability. Nonetheless, these models have the advantage of incorporating many biological features, such as different types of neurotransmitter receptors and distinct classes of excitatory and inhibitory neurons [47]. Their dynamics is best

Decision-making with probabilistic population codes

To recap, probabilistic decision-making requires the decision maker to at least maintain a representation of the certainty with which either option is correct throughout both stages of the decision process. Fortunately, neural population codes seem to be well suited to not only represent this certainty, but also full probability distributions over the stimulus [45••, 54, 55, 56] (Figure 4). Furthermore, as long as the neural spike variability belongs to the exponential family with linear

Conclusions

Based on behavioral and neural data of humans and animals, we have argued for following a normative approach to the modeling of decision-making. To this respect, we have pointed out weaknesses of both diffusion models as well as decision models based on attractor networks, in particular with respect to the representation of reliability and confidence. Probabilistic population codes, on the contrary satisfy requirements posed by the normative approach, and can explain both behavioral and neural

References and recommended reading

Papers of particular interest, published within the period of review, have been highlighted as:

  • • of special interest

  • •• of outstanding interest

Acknowledgements

A.P. was supported by grants from the National Science Foundation (BCS0446730), a Multidisciplinary University Research Initiative (N00014-07-1-0937) and the James McDonnell Foundation.

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