An embedded network approach for scale-up of fluctuation-driven systems with preservation of spike information

doi:10.1073/pnas.0404062101

An embedded network approach for scale-up of fluctuation-driven systems with preservation of spike information

^*Courant Institute of Mathematical Sciences and ^§Center for Neural Science, New York University, New York, NY 10012; and ^‡Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Contributed by David W. McLaughlin, June 8, 2004

Abstract

To address computational “scale-up” issues in modeling large regions of the cortex, many coarse-graining procedures have been invoked to obtain effective descriptions of neuronal network dynamics. However, because of local averaging in space and time, these methods do not contain detailed spike information and, thus, cannot be used to investigate, e.g., cortical mechanisms that are encoded through detailed spike-timing statistics. To retain high-order statistical information of spikes, we develop a hybrid theoretical framework that embeds a subnetwork of point neurons within, and fully interacting with, a coarse-grained network of dynamical background. We use a newly developed kinetic theory for the description of the coarse-grained background, in combination with a Poisson spike reconstruction procedure to ensure that our method applies to the fluctuation-driven regime as well as to the mean-driven regime. This embedded-network approach is verified to be dynamically accurate and numerically efficient. As an example, we use this embedded representation to construct “reverse-time correlations” as spiked-triggered averages in a ring model of orientation-tuning dynamics.

Footnotes

↵ † To whom correspondence should be addressed. E-mail: cai{at}cims.nyu.edu.
Abbreviations: CG, coarse-grained; ISI, interspike interval; I&F, integrate-and-fire; RTC, reverse time correlation; PSR, Poisson spike reconstruction.
↵ ¶ Note that, in this article, we merely use the terms “simple” and “complex” to refer to this particular network architecture. Neurons in the visual cortex are classified (16) “simple” or “complex,” with simple cells responding to visual stimulation in an essentially linear fashion (for example, responding to sinusoidally modulated standing gratings at the fundamental frequency, with the magnitude of response sensitive to the spatial phase of the grating pattern), and with complex cells that respond nonlinearly (with a significant second harmonic) in a phase-insensitive manner. See refs. 14 and 15 for a model of simple and complex cells which captures these physiological features.