### 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 language | English (US) |
---|---|

Pages (from-to) | 7757-7762 |

Number of pages | 6 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 101 |

Issue number | 20 |

DOIs | |

State | Published - May 18 2004 |

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### ASJC Scopus subject areas

- Genetics
- General

### Cite this

*Proceedings of the National Academy of Sciences of the United States of America*,

*101*(20), 7757-7762. https://doi.org/10.1073/pnas.0401906101

**An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex.** / Cai, David; Tao, Louis; Shelley, Michael; McLaughlin, David W.

Research output: Contribution to journal › Article

*Proceedings of the National Academy of Sciences of the United States of America*, vol. 101, no. 20, pp. 7757-7762. https://doi.org/10.1073/pnas.0401906101

}

TY - JOUR

T1 - An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex

AU - Cai, David

AU - Tao, Louis

AU - Shelley, Michael

AU - McLaughlin, David W.

PY - 2004/5/18

Y1 - 2004/5/18

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=2442679125&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2442679125&partnerID=8YFLogxK

U2 - 10.1073/pnas.0401906101

DO - 10.1073/pnas.0401906101

M3 - Article

VL - 101

SP - 7757

EP - 7762

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 20

ER -