Difficulty of singularity in population coding |
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Authors: | Amari Shun-ichi Nakahara Hiroyuki |
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Affiliation: | Laboratory for Mathematical Neuroscience, RIKEN Brain Science Institute, Wako, Saitama, 351-0198 Japan. amari@brain.riken.jp |
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Abstract: | Fisher information has been used to analyze the accuracy of neural population coding. This works well when the Fisher information does not degenerate, but when two stimuli are presented to a population of neurons, a singular structure emerges by their mutual interactions. In this case, the Fisher information matrix degenerates, and the regularity condition ensuring the Cramér-Rao paradigm of statistics is violated. An animal shows pathological behavior in such a situation. We present a novel method of statistical analysis to understand information in population coding in which algebraic singularity plays a major role. The method elucidates the nature of the pathological case by calculating the Fisher information. We then suggest that synchronous firing can resolve singularity and show a method of analyzing the binding problem in terms of the Fisher information. Our method integrates a variety of disciplines in population coding, such as nonregular statistics, Bayesian statistics, singularity in algebraic geometry, and synchronous firing, under the theme of Fisher information. |
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