Abstract
In this paper, we extend a framework for constructing low-dimensional dynamical systems models of mammalian primary visual cortex to a cortical network model that incorporates the full nonlinear effects of complex cells. The procedure consists of capturing the essential dynamics in a low-dimensional subspace using empirical methods, then recasting the equations in the reduced vector space. Previously, we considered visual cortical network models consisting of only simple cells with nearly linear responses to external stimuli. Here we show that fully nonlinear effects can be incorporated by examining the dimensional reduction of an idealized ring model of V1 with both simple and complex cells. We found it expedient to divide the subspace into four separate neuronal populations: excitatory simple, excitatory complex, inhibitory simple and inhibitory complex. In order to reproduce the fluctuation-driven dynamics in this reduced space, we incorporated (1) white noises with different intensities into individual neuronal populations, and (2) firing rate estimates to capture the probability of firing due to subthreshold fluctuations. With a more accurate, fitted connectivity, our modified dimensional reduced models can reproduce the firing rates, circular variances and modulation ratios observed in the original ring model.
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Notes
Orientation selectivity for drifting grating stimuli is measured by CV. Let m k denote the time-averaged firing rate with respect to stimulus angle θ k . The angles θ k spanned the range from 0 to 180° with equally spaced intervals. CV is defined as \( \mathrm{CV}=1-\left|{\displaystyle \sum_k{m}_k}{e}^{i2{\theta}_k}\right|/{\displaystyle \sum_k{m}_k} \). CV lies in the range of [0, 1]. Neurons with bad selectivity to orientation have CVs near 1, whereas neurons with good selectivity have CVs near 0.
Modulation ratio F1/F0 is defined as \( \mathrm{F}1/\mathrm{F}0=\left|{\displaystyle \sum_{n=1}^N\overline{R_n}{e}^{- i2\pi \left( n-1\right)/ N}}\right|/{\displaystyle \sum_{n=1}^N\overline{R_n}} \), where \( \overline{R_n} \) is the cycle-averaged response to a sinusoidal drifting grating. Usually, F1/F0 is in the range from 0 to 1.
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Acknowledgments
This work was supported by the Ministry of Science and Technology of China (Basic Research Program 973 Program 2011CB809105) and the Natural Science Foundation of China (grant number 91232715). We thank Andrew Sornborger for reading and commenting on a earlier draft of this paper.
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Wang, C., Tao, L. Dimensional reduction of a V1 ring model with simple and complex cells. J Comput Neurosci 37, 481–492 (2014). https://doi.org/10.1007/s10827-014-0516-6
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DOI: https://doi.org/10.1007/s10827-014-0516-6