Limits...
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.

Show MeSH

Related in: MedlinePlus

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 ‘’ 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.
© Copyright Policy
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4414410&req=5

pone.0124787.g003: 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 ‘’ 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.

Mentions: However, there are other terms in Li(j) which depend on xi(j) and which can make a substantial contribution to the development of yi(T). This is demonstrated in Fig 3 for a particular set of biologically plausible pdfs and parameterisations, ΩIV, described in the Methods. Fig 3a–3d, shows the pdfs specified by ΩIV. In the corresponding panels e-h below, Li(j) is shown per pdf as a function of xi(j) (solid red line) as well as the functions for its contributory terms. The term g0 is constant (solid black line), the spike count contribution and other non-linear terms are functions of xi(j) (dashed and broken gray/black lines). There is clearly a wide variation in non-constant contributions to Li(j). Most notably, the terms linear in xi(j) in the gamma and inverse Gaussian, have an apparently disproportionate effect on Li(j). However, as noted earlier in connection with the exponential distribution, when summed over spikes, such terms give an expression which is approximately gT (for some gain g). If these terms were identically equal to gT, they may be absorbed into the constant term B(T) in Eq 8, and have a effect on the posterior. Thus, assuming the terms linear in xi(j) are a good approximation to gT, they will have a very limited influence on the outcome of a decision.


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

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

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 ‘’ 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.
© Copyright Policy
Related In: Results  -  Collection

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

pone.0124787.g003: 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 ‘’ 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.
Mentions: However, there are other terms in Li(j) which depend on xi(j) and which can make a substantial contribution to the development of yi(T). This is demonstrated in Fig 3 for a particular set of biologically plausible pdfs and parameterisations, ΩIV, described in the Methods. Fig 3a–3d, shows the pdfs specified by ΩIV. In the corresponding panels e-h below, Li(j) is shown per pdf as a function of xi(j) (solid red line) as well as the functions for its contributory terms. The term g0 is constant (solid black line), the spike count contribution and other non-linear terms are functions of xi(j) (dashed and broken gray/black lines). There is clearly a wide variation in non-constant contributions to Li(j). Most notably, the terms linear in xi(j) in the gamma and inverse Gaussian, have an apparently disproportionate effect on Li(j). However, as noted earlier in connection with the exponential distribution, when summed over spikes, such terms give an expression which is approximately gT (for some gain g). If these terms were identically equal to gT, they may be absorbed into the constant term B(T) in Eq 8, and have a effect on the posterior. Thus, assuming the terms linear in xi(j) are a good approximation to gT, they will have a very limited influence on the outcome of a decision.

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