An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex

David Cai, Louis Tao, Michael Shelley, David W. McLaughlin

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Abstract

A coarse-grained representation of neuronal network dynamics is developed in terms of kinetic equations, which are derived by a moment closure, directly from the original large-scale integrate-and-fire (I&F) network. This powerful kinetic theory captures the full dynamic range of neuronal networks, from the mean-driven limit (a limit such as the number of neurons N → ∞, in which the fluctuations vanish) to the fluctuation-dominated limit (such as in small N networks). Comparison with full numerical simulations of the original I&F network establishes that the reduced dynamics is very accurate and numerically efficient over all dynamic ranges. Both analytical insights and scale-up of numerical representation can be achieved by this kinetic approach. Here, the theory is illustrated by a study of the dynamical properties of networks of various architectures, including excitatory and inhibitory neurons of both simple and complex type, which exhibit rich dynamic phenomena, such as, transitions to bistability and hysteresis, even in the presence of large fluctuations. The implication for possible connections between the structure of the bifurcations and the behavior of complex cells is discussed. Finally, I&F networks and kinetic theory are used to discuss orientation selectivity of complex cells for "ring-model" architectures that characterize changes in the response of neurons located from near "orientation pinwheel centers" to far from them.

Original languageEnglish (US)
Pages (from-to)7757-7762
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Volume101
Issue number20
DOIs
StatePublished - May 18 2004

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Visual Cortex
Neurons

ASJC Scopus subject areas

  • Genetics
  • General

Cite this

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title = "An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex",
abstract = "A coarse-grained representation of neuronal network dynamics is developed in terms of kinetic equations, which are derived by a moment closure, directly from the original large-scale integrate-and-fire (I&F) network. This powerful kinetic theory captures the full dynamic range of neuronal networks, from the mean-driven limit (a limit such as the number of neurons N → ∞, in which the fluctuations vanish) to the fluctuation-dominated limit (such as in small N networks). Comparison with full numerical simulations of the original I&F network establishes that the reduced dynamics is very accurate and numerically efficient over all dynamic ranges. Both analytical insights and scale-up of numerical representation can be achieved by this kinetic approach. Here, the theory is illustrated by a study of the dynamical properties of networks of various architectures, including excitatory and inhibitory neurons of both simple and complex type, which exhibit rich dynamic phenomena, such as, transitions to bistability and hysteresis, even in the presence of large fluctuations. The implication for possible connections between the structure of the bifurcations and the behavior of complex cells is discussed. Finally, I&F networks and kinetic theory are used to discuss orientation selectivity of complex cells for {"}ring-model{"} architectures that characterize changes in the response of neurons located from near {"}orientation pinwheel centers{"} to far from them.",
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AU - Tao, Louis

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