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Next: Complex Cells Up: The Input Layers Previous: The Retina

Simple Cells

The combined layer of simple and complex cells is practically identical to the model proposed in [42] for modeling responses of binocular cortical neurons. For simplicity, neurons in our simulation are allowed to code negative as well as positive signal values; in [42], a push-pull configuration of neurons was used instead.

The receptive fields of the simple cells are modeled by a generic Gabor function of the form

\begin{displaymath}
g(x;\omega ,\varphi )=\exp (-0.5*(x/\sigma )^{2})\cdot \cos (\omega x+\varphi )\, ,
\end{displaymath} (6)

with phase-parameter \( \varphi \) and spatial frequency \( \omega \). The filter profiles resulting from sampling this filter function at the discrete receptor positions \( x=i \) are normalized and adjusted so that no DC-component is present in the final filter coefficients.

Simple cells sample data from both left and right input streams, which we denote here by \( e^{L}_{(k,l)} \) and \( e^{R}_{(k,l)} \). As transfer function for simple cells, a squaring nonlinearity is used. With the normalized filter coefficients given by \( \tilde{g}(k;\omega ,\varphi ) \), the output of a simple cell can be written as

\begin{displaymath}
s_{(i,j)}^{a}(\varphi ,s)=\left[ \sum _{k}\tilde{g}(k-s/2;\o...
...{g}(k+s/2;\omega ,-\varphi )\cdot e^{R}_{(i-k,j)}\right] ^{2}.
\end{displaymath} (7)

Here \( s \) is a small relative shift of the centers of the two receptive fields in the left and right retina. Varying \( s \) creates the layered structure displayed in Fig. 8; in the simulations, \( s \) runs from \( -8 \) to \( +8 \) in steps of \( 1/3 \). The spatial frequency \( \omega \) is set to \( \omega =1.5 \). The phase-shift \( \varphi \) of the simple cells depends on the type of complex cell to which the simple cell is feeding data and can have the values \( -\pi /4 \), 0 and \( +\pi /4 \) (compare below).

For quadrature filtering, the output of any simple cell having a phase-shift \( \varphi \) has to be paired with another simple cell, having a phase-shift of \( \varphi +\pi /2 \). This means that for every simple cell \( s_{(i,j)}^{a}(\varphi ,s) \) with a receptive field defined by (7) there exists another, paired one, calculating

$\displaystyle s_{(i,j)}^{b}(\varphi ,s)$ $\textstyle =$ $\displaystyle \left[ \sum _{k}g(k-s/2;\omega ,\pi /2+\varphi )e^{L}_{(i-k,j)}\right.$  
  $\textstyle +$ $\displaystyle \left. \sum _{k}g(k+s/2;\omega ,\pi /2-\varphi )e^{R}_{(i-k,j)}\right] ^{2}$ (8)


next up previous
Next: Complex Cells Up: The Input Layers Previous: The Retina

2000-11-20