Thus, equation(11) p(θamp;m|Nk)=p(Nk|θamp;m)p(θamp;m)p(Nk)where Selleck HSP inhibitor p(θamp;m) is the distribution
of occurrences of each amplitude θamp;m (Figure 3C) and equation(12) p(Nk)=∑m=1Mp(Nk|θamp;m)p(θamp;m)where M is the number of bins. The joint probability for the estimate of the amplitude across a population of K units, given the assumption of independent neurons, is equation(13) p(θamp;m|N1,…,NK)=∏k=1Kp(θamp;m|Nk)From this distribution, we finally assume that the value of the parameter is estimated using maximum a posteriori decoding (Dayan and Abbott, 2001): equation(14) θˆamp;m(N1=n1,…,NK=nK)=argmaxθamp;mp(θamp;m. Given this decoding model for the amplitude ((6), (7), (8) and (9)), we now wish to evaluate its predictive accuracy. As we are interested in the effects of sampling from a limited number of neurons, we use a Monte Carlo resampling process to compare the results from the model to the original amplitude. We draw a random number of spikes for each of the K neurons based on Poisson statistics and a fixed value of θamp;m (Equation 10); this defines a set of K spike counts, (n1, …, nK) that are used to estimate the joint probability (Equation 13). We then use Equation 14 to estimate θˆamp;m(N1=n1,…,NK=nK). We repeat the process of draws and estimation 1000 Bortezomib mw times to form a distribution of errors δθamp;m
(K), where equation(15) δθamp;m(K)=〈〈|θˆamp;m(N1=n1,…,NK=nK)−θamp;m|〉〉where 《…》 means averaging over draws. The mean of this distribution approaches zero and the root-mean-square width defines the accuracy with which the amplitude can be reconstructed. The entire procedure is repeated as a function Phosphoprotein phosphatase of K for each value of θamp;m. Finally, we report the expected value of the accuracy as a weighted average over all amplitudes, denoted δθamp(K), where equation(16) δθamp(K)=∑m=1Mp(θamp;m)δθamp;m(K). An analogous set of procedures
holds for the accuracy of predicting the midpoint, denoted δθmid(K). All units were represented in each simulation unless otherwise specified. For simulations of populations that were larger than the number of experimentally recorded cells, every unit was duplicated into an equal number of copies so that each unit was equally represented. A final issue concerns implementation. The range of amplitudes as well as the range of midpoints are not equal for different animals. Thus, for purposes of calculation, we normalized our responses in terms of percentiles of the range of motion (Figure 3C) so that units from different animals could be averaged together. Thus, we first transform from θamp (or θmid) to percentile, noting that the percentile steps are uniform so the transformed prior probabilities p(θamp) and p(θmid) have value 1/M = 0.02, then we complete the resampling procedure in terms of percentiles, and finally transform back to absolute angles to determine δθamp(K) and δθmid(K).
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