# To validate the probabilities in Fig 1C, we simulated the sampling state of affairs 10,000 times

When A and B are both in y and T is in z , T will have a situation on the attention axis that is greater than one when A < B, but equally often it will have a position on the attention axis that is less than 1, when B < A . Additionally, there are six equally probable arrangements when A, B, and T are all in range y . But only two of these six arrangements result in situations where T> 1 934660-93-2 manufactureron the interest axis. Underneath these problems, the all round likelihood that T will have a value much less than one on the consideration axis in Fig 1C is .583 . The capacity of the orientation of the consideration axis to flip in specific configurations implies that there will be a bias for T to have a value shifted from 1 towards the course of , even though T was drawn from the same distribution as A.This bias in the focus axis stems from the overlap amongst distributions utilized to build it and noisy estimates of the distribution implies. If the estimates A, B and T have been every single based on a big amount of samples, or personal responses from a massive variety of neurons, they would approximate the accurate implies of the distributions, B would virtually by no means lie to the appropriate of A, and the bias would be efficiently eradicated. The amount of bias in the example in Fig 1 can be precisely calculated simply because the distributions have well-described properties. To validate the chances in Fig 1C, we simulated the sampling state of affairs ten,000 times. The mean probability that T < 1 was 0.583 when A and T were randomly drawn from random uniform distributions, while the mean probability that T < A was 0.50 , matching theoretical expectations. Thus, we can account for biased sampling on the attention axis when the statistics of the distributions are known, and potentially correct for it.In practice, however, the attention axis is built on responses from many neurons that have distributions that can only be estimated. It is therefore difficult to know how much bias might enter into attention axis measurements. If the distributions of firing rates of the two response distributions that comprise the attention axis are non-overlapping, then axis inversion is avoided. But because neurons are typically weakly modulated by attention and are often driven by suboptimal sensory stimuli in neurophysiological experiments, overlapping population responses are common.Although the bias described in Fig 1 does not occur if samples are drawn from two non-overlapping distributions, a second bias can occur even when the sample distributions do not overlap and is related to the process of projecting points in multidimensional space. This bias acts in the same direction as that described above, shifting measurements towards 0 on the attention axis. Fig 2 illustrates this other source of bias using a simplified example in which the population responses are based on responses from only two neurons. All responses are noiseless except the response of neuron 2 in the attend-right condition, which is drawn from a uniform distribution.As with the simulations in Fig 1, the attention axis is constructed using only a single sample of the neurons' responses from the attend-right response distribution, A, and a single sample of the neurons' responses from the attend-left response distribution, B .