A mathematical model of digital decision-making

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Why such a coexistence of two distinct neuronal codes, one with a monotonic variation in discharge rate as a function of numerosity, the other with a tuning curve to a preferred numerosity? These results should be interpreted with caution, as these two populations of neurons have only been observed very recently, in different laboratories, in different animals and trained on different numerical tasks. However, these results fit well with a theoretical model which assumes that monotone and tuned neurons constitute two distinct stages in the extraction of an invariant representation of numerosity (Dehaene & Changeux, 1993; Verguts & Fias, 2004). According to this model, approximate numerosity can be extracted from a detailed retinal map in three successive steps: (1) retinotopic encoding of the positions occupied by objects, irrespective of their identity and size, hence with a fixed amount of activation for each object; (2) approximate summation of these activations across the whole map, by means of "accumulation neurons" whose activity level varies monotonically as a function of numerosity; (3) thresholding of this activation by neurons with increasing thresholds and strong lateral inhibition, leading to a population of neurons tuned to different numerities. Computer simulation of this model, in the form of a formal neural network, shows that at this last level, we naturally end up with a log-Gaussian encoding of numerosity. With a few adaptations, the accumulation neurons can be identified with the LIP area neurons studied by Roitman et al. while the neurons tuned to numerosity would correspond to the LIP area neurons recorded by Nieder and Miller. It should be noted that, anatomically, LIP neurons do indeed project to VIP neurons. Moreover, VIP neurons appear to respond to the entire visual field, which is compatible with the hypothesis that they receive convergent inputs from numerous retinotopic neurons in the LIP area.

Based on this log-Gaussian neural code, a mathematical model of decision making, capable of accounting for error rates and response times in various elementary numerical tasks, has also been developed (Dehaene, 2007). When the decision is taken in a fixed time, with no pressure for speed, and only the error rate needs to be modeled, signal detection theory can be applied very directly to log-Gaussian code. This model gives a good account of animal and human skills in elementary like-different or larger-smaller comparison tasks (Dehaene, 2007; Piazza et al., 2004). It can also be adapted to number naming (Izard & Dehaene, 2008) and to the addition or subtraction of two numbers, subject to a few additional assumptions about the combination of variances associated with each operand (Barth et al., 2006; Cantlon & Brannon, 2007).

When the task involves a quick decision in a limited time, a more sophisticated mathematical model is used to model response time. This model is based on the work of Mike Shadlen, who indicates that real-time decision making, based on noisy signals, relies on certain parietal and prefrontal neurons that perform an accumulation of the stochastic data that stimuli bring in favor of each possible response. This accumulation can then be described mathematically as a random walk akin to Brownian motion. The decision is made when, for one of the responses, the accumulator's random walk reaches a predetermined threshold. The corresponding response is then selected. It can be demonstrated that this thresholded statistical accumulation mechanism is an optimal real-time decision-making mechanism (Gold & Shadlen, 2002).

The analysis shows that, at least in very simple tasks such as comparing two numbers, the log-Gaussian model coupled with accumulative decision making leads to predictions that are very closely matched to the experimental data. The influence of the distance between the numbers to be compared is correctly modeled, and the model explains why the shape of this effect differs according to whether we consider the error rate or the average response time. The distribution of response times, and how this varies with the presence of an interfering task, is also explained in great detail (Sigman & Dehaene, 2005).