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Probabilistic Decision Making with Spikes: From ISI Distributions to Behaviour via Information Gain.

Caballero JA, Lepora NF, Gurney KN - PLoS ONE (2015)

Bottom Line: Computational theories of decision making in the brain usually assume that sensory 'evidence' is accumulated supporting a number of hypotheses, and that the first accumulator to reach threshold triggers a decision in favour of its associated hypothesis.Thus, we find the mean information needed for a decision is constant, thereby offering an account of Hick's law (relating decision time to the number of choices).These results show the foundations for a research programme in which spike train analysis can be made the basis for predictions about behavior in multi-alternative choice tasks.

View Article: PubMed Central - PubMed

Affiliation: Dept of Psychology, University of Sheffield, Sheffield, UK; Faculty of Life Sciences, University of Manchester, Manchester, UK.

ABSTRACT
Computational theories of decision making in the brain usually assume that sensory 'evidence' is accumulated supporting a number of hypotheses, and that the first accumulator to reach threshold triggers a decision in favour of its associated hypothesis. However, the evidence is often assumed to occur as a continuous process whose origins are somewhat abstract, with no direct link to the neural signals - action potentials or 'spikes' - that must ultimately form the substrate for decision making in the brain. Here we introduce a new variant of the well-known multi-hypothesis sequential probability ratio test (MSPRT) for decision making whose evidence observations consist of the basic unit of neural signalling - the inter-spike interval (ISI) - and which is based on a new form of the likelihood function. We dub this mechanism s-MSPRT and show its precise form for a range of realistic ISI distributions with positive support. In this way we show that, at the level of spikes, the refractory period may actually facilitate shorter decision times, and that the mechanism is robust against poor choice of the hypothesized data distribution. We show that s-MSPRT performance is related to the Kullback-Leibler divergence (KLD) or information gain between ISI distributions, through which we are able to link neural signalling to psychophysical observation at the behavioural level. Thus, we find the mean information needed for a decision is constant, thereby offering an account of Hick's law (relating decision time to the number of choices). Further, the mean decision time of s-MSPRT shows a power law dependence on the KLD offering an account of Piéron's law (relating reaction time to stimulus intensity). These results show the foundations for a research programme in which spike train analysis can be made the basis for predictions about behavior in multi-alternative choice tasks.

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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.
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pone.0124787.g011: 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.

Mentions: Fig 11 shows that this condition on the thresholds is indeed met. Fig 11a and 11b show exp(θs) < exp(θu) consistently across all numbers of choices for two examples (lognormal and inverse Gaussian based s-MSPRT) for parameter set ΩIV. (The lognormal is more typical with little variation with the number of alternatives). Similar clear differences Δeθ = exp(θu) − exp(θs) exist for all distributions across both parameter sets ΩIV, ΩFV—see Fig 11c and 11d. Notice that the gamma distribution in Fig 11c and the exponential distribution in Fig 11d have the smallest Δeθ in their groups (and in each case, Δeθ is significantly less than the next smallest one at p < 0.001, two-sided t-test). This may account for the fact that decision times for the gamma distribution (Fig 4a) and the exponential distribution (Fig 4b) fail to achieve the criterion of equal decision sample as well as the others.


Probabilistic Decision Making with Spikes: From ISI Distributions to Behaviour via Information Gain.

Caballero JA, Lepora NF, Gurney KN - PLoS ONE (2015)

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.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4414410&req=5

pone.0124787.g011: 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.
Mentions: Fig 11 shows that this condition on the thresholds is indeed met. Fig 11a and 11b show exp(θs) < exp(θu) consistently across all numbers of choices for two examples (lognormal and inverse Gaussian based s-MSPRT) for parameter set ΩIV. (The lognormal is more typical with little variation with the number of alternatives). Similar clear differences Δeθ = exp(θu) − exp(θs) exist for all distributions across both parameter sets ΩIV, ΩFV—see Fig 11c and 11d. Notice that the gamma distribution in Fig 11c and the exponential distribution in Fig 11d have the smallest Δeθ in their groups (and in each case, Δeθ is significantly less than the next smallest one at p < 0.001, two-sided t-test). This may account for the fact that decision times for the gamma distribution (Fig 4a) and the exponential distribution (Fig 4b) fail to achieve the criterion of equal decision sample as well as the others.

Bottom Line: Computational theories of decision making in the brain usually assume that sensory 'evidence' is accumulated supporting a number of hypotheses, and that the first accumulator to reach threshold triggers a decision in favour of its associated hypothesis.Thus, we find the mean information needed for a decision is constant, thereby offering an account of Hick's law (relating decision time to the number of choices).These results show the foundations for a research programme in which spike train analysis can be made the basis for predictions about behavior in multi-alternative choice tasks.

View Article: PubMed Central - PubMed

Affiliation: Dept of Psychology, University of Sheffield, Sheffield, UK; Faculty of Life Sciences, University of Manchester, Manchester, UK.

ABSTRACT
Computational theories of decision making in the brain usually assume that sensory 'evidence' is accumulated supporting a number of hypotheses, and that the first accumulator to reach threshold triggers a decision in favour of its associated hypothesis. However, the evidence is often assumed to occur as a continuous process whose origins are somewhat abstract, with no direct link to the neural signals - action potentials or 'spikes' - that must ultimately form the substrate for decision making in the brain. Here we introduce a new variant of the well-known multi-hypothesis sequential probability ratio test (MSPRT) for decision making whose evidence observations consist of the basic unit of neural signalling - the inter-spike interval (ISI) - and which is based on a new form of the likelihood function. We dub this mechanism s-MSPRT and show its precise form for a range of realistic ISI distributions with positive support. In this way we show that, at the level of spikes, the refractory period may actually facilitate shorter decision times, and that the mechanism is robust against poor choice of the hypothesized data distribution. We show that s-MSPRT performance is related to the Kullback-Leibler divergence (KLD) or information gain between ISI distributions, through which we are able to link neural signalling to psychophysical observation at the behavioural level. Thus, we find the mean information needed for a decision is constant, thereby offering an account of Hick's law (relating decision time to the number of choices). Further, the mean decision time of s-MSPRT shows a power law dependence on the KLD offering an account of Piéron's law (relating reaction time to stimulus intensity). These results show the foundations for a research programme in which spike train analysis can be made the basis for predictions about behavior in multi-alternative choice tasks.

Show MeSH
Related in: MedlinePlus