Fig 1.
Panel a shows the general MSPRT where all the C data streams contribute to all of the N likelihoods and thus posteriors, which are then evaluated at a termination stage. Panel b only shows the effective components after all simplifications have been applied.
Fig 2.
Time course of signals in a single trial of s-MSPRT.
The trial is for an s-MSPRT using the gamma distribution, with 4 choices, under the parameterisation set ΩIV (see Methods). Panel a, shows the spike rasters of the 4 spike trains as small vertical line markers, with that of the preferred channel in red. This panel also shows the accumulated evidence yk(T) as a red line graph. Panel b shows yk(T) of all four hypotheses with the preferred hypothesis in bold red. Panel c shows the posteriors with the preferred hypothesis in black and the others in gray. The threshold is shown by the horizontal dashed line, and was chosen to give a 5% error rate.
Table 1.
Analytic expressions for ‘evidence contributions’, Li(j) for a range of distributions.
Fig 3.
The two-parameter families of pdfs (top) and their ‘evidence contributions’ Li(j) (bottom).
Panels a-d show the lognormal, gamma, inverse Gaussian, and inverse gamma pdfs respectively, for the independent variance parameter set ΩIV (see Methods). The ‘preferred’ and ‘null’ density functions (f*, f0) are in red and black respectively. The plots are for ISIs from 1 to 100 ms. For infinitesimal ISIs, the lognormal, inverse Gaussian and inverse gamma tend to zero; for the the gamma the pdf grows up to a bound as the ISI tends to zero. Panels e-f are the corresponding contributions Li(j) to the accumulated ‘evidence’ yi(T) (see Eq 9) and the separate components therein (see Table 1). Li(j) itself is shown in red, the constant term (D = L, γ, S, M) by the solid black line, and non-constant terms by dashed-grey and broken-black lines. The horizontal dashed grey line indicates 0 on the y-axis.
Fig 4.
Mean decision samples against number of choices for a range of pdfs, parameter sets, and mechanisms.
Each panel shows mean decision sample as a function of the number of choices for a range of pdfs (see legend) and for the two alternative mechanisms: s-MSPRT (solid lines) and u-MSPRT (solid circles). Panels a and b are for the parameter sets ΩIV and ΩFV respectively (see Methods). In the case of ΩFV, the gamma and exponential distributions are identical and so not reported separately. All data points are the mean of 950 correct, out of 1000 total trials (see text for inverse gamma based s-MSPRT). Error bars are omitted for clarity and are small; the standard error of the mean is typically 2% of the mean.
Fig 5.
Decision sample in s-MSPRT against mean ISI for range of pdfs and parameter sets.
Each bar shows, for the pdf indicated in the legend, the mean decision sample for N = 10 alternatives, averaged over 950 correct out of 1000 total trials. Panel a used parameter sets ΩIV, , , , panel b used ΩFV, , , (see Methods). Each group of bars relates to one parameter set with its μ0 indicated on the x − axis (ΩIV, ΩFV have μ0 = 33). For the case of ΩFV and any , the gamma and exponential distributions are identical and so not reported separately.
Fig 6.
Decision sample for u-MSPRT when the distributions in the data were not matched to those tested for.
Each bar shows the mean decision sample, for N = 10 alternatives, averaged over 950 correct out of 1000 total trials. The parameter set was ΩIV. The pale, patterned bars are for the case when the data is always sampled from an inverse gamma distribution, but inserted into mechanisms which test using the distribution indicated on the x − axis (by definition, the bars have equal height for the inverse gamma). The solid bars are for the case when the tested-for distribution matches the true distribution of ISIs, as indicated on the x − axis. Error bars are at one standard deviation.
Fig 7.
Mean decision sample for s-MSPRT against KLD.
The KLD in all cases is D(f*‖f0) (see text). Panels a, b are for decision with the parameter sets ΩIV, ΩFV, respectively, and both use N = 10. The data points are the open symbols and the dashed lines, best fit power laws (nonlinear least squares). In panel c, the data points shown in blue symbols correspond to all the decision samples in Fig 5. The solid line is the best fit power law.
Fig 8.
Hick’s and Piéron’s laws from conservation of information.
Panel a is a direct counterpart of Fig 4a. The decision samples for s-MSPRT are shown as solid lines and the predictions from Eq 27 shown as solid symbols. Panel b shows results of the virtual experiment derived from Eq 30 (blue symbols) and a best fit power law (solid black line).
Fig 9.
Fitting pdfs to an ISI histogram.
The ISIs in the grey histogram (identical in each panel) were recorded by [31] from the MT neuron with tag e093 (Table 2). The few ISIs lying farther than four standard deviations beyond the mean were not plotted for clarity. Overlaid as a solid blue line in each panel is the best-fit pdf from a set of theoretical distributions: Gaussian, gamma, inverse gamma, inverse Gaussian, lognormal and exponential
Fig 10.
Goodness of fit of selected pdfs to ISI data.
The black closed circles show the goodness of fit statistics for each of the first five data sets in Table 2. The red closed squares are for the data set from [27]. The mean for all six data sets is shown by the large open circles which also has a line plot; the number of significant fits at a level of 0.05 is noted per pdf next to such circles. Panels a and b are for the Kolomogorov-Smirnov, and Anderson-Darling tests respectively (note log y-axis in the latter). The ordering of the results from left to right preserves rank order of the mean statistic and is the same for both tests.
Table 2.
Data sets from [31] used to generate ISI distributions.
Fig 11.
Decision thresholds and their difference between s-MSPRT and u-MSPRT.
To ease visualisation, all panels show the exponential of the thresholds; ϕs = exp(θs) ϕu = exp(θu) for s- and u-MSPRT respectively. This yields positive values pertaining to the posterior (rather than negative values for the log-posterior). Panels a and b, respectively, show the lognormal and inverse Gaussian cases in Fig 4a for the parameter set ΩIV. The red lines and symbols are for ϕs, the black lines and symbols for ϕu. In panel c, the box plot labelled ‘lognormal’ shows the median and quartiles (box lines), mean (cross) and one standard deviation of the differences ϕu − ϕs in panels a and b. Other bars show similar quantities for the test distributions of the other MSPRT instantiations used to form Fig 4a. Panel d is similar to panel c, except it pertains to differences in (exponential) thresholds for results in Fig 4b, with the parameter set ΩFV.