Standard integration

2017 ◽  
pp. 521-527
Author(s):  
John Bird
Keyword(s):  
Standard Integration

Variations on a theme – Euler and the logsine integral

2012 ◽  
Vol 96 (537) ◽  
pp. 451-458 ◽  
Author(s):  
Nick Lord
Keyword(s):  
Numerical Integration ◽  
Integration Techniques ◽  
Image Position ◽  
Variations On A Theme ◽  
Standard Integration

This article concerns the evaluation of the ‘logsine’ integralWe shall encounter it in several guises. Indeed, standard integration techniques used below readily show that (1) has the same value as the following integrals:En passant, it is worth noting that forms (6) and (7) are the best behaved for numerical integration.I first met the logsine integral as a callow youth in that strange hinterland of results that you may not have met at school but were not guaranteed to meet later on either.

Standard Integration Filter (SIF) State Specification and Accuracy Projections

Navigation ◽  
1986 ◽  
Vol 33 (4) ◽  
pp. 295-313 ◽  
Author(s):  
WILLIAM R. GRAHAM ◽  
GARY R. JOHNSTON
Keyword(s):  
Standard Integration

scholarly journals Impact of Correlated Neural Activity on Decision-Making Performance

2013 ◽  
Vol 25 (2) ◽  
pp. 289-327 ◽  
Author(s):  
Nicholas Cain ◽  
Eric Shea-Brown
Keyword(s):  
Decision Making ◽  
Neural Activity ◽  
Higher Order ◽  
Firing Rates ◽  
Pairwise Correlations ◽  
Neural Populations ◽  
Cell Groups ◽  
Nonlinear Computation ◽  
Standard Integration ◽  
Simultaneous Activity

Stimulus from the environment that guides behavior and informs decisions is encoded in the firing rates of neural populations. Neurons in the populations, however, do not spike independently: spike events are correlated from cell to cell. To what degree does this apparent redundancy have an impact on the accuracy with which decisions can be made and the computations required to optimally decide? We explore these questions for two illustrative models of correlation among cells. Each model is statistically identical at the level of pairwise correlations but differs in higher-order statistics that describe the simultaneous activity of larger cell groups. We find that the presence of correlations can diminish the performance attained by an ideal decision maker to either a small or large extent, depending on the nature of the higher-order correlations. Moreover, although this optimal performance can in some cases be obtained using the standard integration-to-bound operation, in others it requires a nonlinear computation on incoming spikes. Overall, we conclude that a given level of pairwise correlations, even when restricted to identical neural populations, may not always indicate redundancies that diminish decision-making performance.

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