= Signed spatial filter model of a simple cell
{title2=$D=\sum_{i,j}K_{ij}C_{ij},\quad R=\max(0,D)$}
A <signed spatial filter model of a simple cell> assigns a weight $K_{ij}$ to each sampled location and takes its <dot product> with the <visual contrast> image $C_{ij}$. Positive weights favor bright features; negative weights favor dark features. The signed drive
$$
D=\sum_{i,j}K_{ij}C_{ij},\qquad R=\max(0,D)
$$
is a <linear function> of the image, while the <rectified linear unit> output $R$ is nonlinear. $D$ and $R$ below are arbitrary model units. To obtain a predicted <firing rate> one would need a calibrated gain, background response and possibly a different output function. The model does not generate a <spike train>; a <linear-nonlinear-Poisson cascade model> adds a stochastic event-generation assumption.
For the first diagram, every one of five rows has weights
$$
(K_{i1},K_{i2},K_{i3},K_{i4},K_{i5})=(-1/20,-1/20,1/5,-1/20,-1/20).
$$
A bright central column, with $C_{i3}=1$ and zero elsewhere, gives $D=5/5=1$. A bright middle row gives $D=-1/20-1/20+1/5-1/20-1/20=0$; a uniformly bright patch also gives $D=0$, since every row sums to zero. Move the bright column one site right: $D=5(-1/20)=-1/4$, so $R=0$. Make that shifted column dark instead: $D=1/4$ and $R=1/4$. Thus an OFF-like weight can increase output for darkness. A negative weight is a descriptive sensitivity, not evidence for a particular inhibitory <synapse>.
For a smooth example use the real, even <Gabor filter>
$$
K(x,y)=\exp\!\left(-\frac{x^2}{2\sigma_x^2}-\frac{y^2}{2\sigma_y^2}\right)\cos(qx),\qquad C_{\theta,\varphi}(x,y)=\cos\!\bigl(q(x\sin\theta-y\cos\theta)+\varphi\bigr).
$$
The <Gaussian function> localizes the weights; the <cosine> supplies alternating polarity. Choose $\sigma_x=1/2$, $\sigma_y=1$ and $q=\pi$ in arbitrary spatial units. The stripes' <spatial frequency> is $q/(2\pi)=1/2$ cycles per unit. Their axis makes <angle> $\theta$ with horizontal, while brightness changes across the perpendicular axis. $\varphi$ is their spatial <phase>. Normalize the full-plane <integral> of $KC_{\theta,\varphi}$ by its value at $\theta=\pi/2$, $\varphi=0$. The <Gaussian integral> and the product identity for <cosines> give
$$
D(\theta,\varphi)=A(\theta)\cos\varphi,\qquad R(\theta,\varphi)=\max(0,D(\theta,\varphi)),
$$
where
$$
A(\theta)=\frac{\exp\!\left[-\frac{q^2}{2}\left(\sigma_x^2(\sin\theta-1)^2+\sigma_y^2\cos^2\theta\right)\right]+\exp\!\left[-\frac{q^2}{2}\left(\sigma_x^2(\sin\theta+1)^2+\sigma_y^2\cos^2\theta\right)\right]}{1+\exp(-2\sigma_x^2q^2)}.
$$
At $\varphi=0$, horizontal stripes give $R\approx0.004159$, $60^\circ$ stripes give $R\approx0.286741$, and vertical stripes give $R=1$. At the same vertical orientation, $\varphi=\pi/2$ gives $D=R=0$, while reversing the polarity with $\varphi=\pi$ gives $D=-1$ and $R=0$. The preferred axis persists while the response changes dramatically with position. Shifting the vertical stripes by a quarter of their wavelength changes <phase> by $\pi/2$ up to sign, without rotating them.
These are two different toy filters, not discretizations of the same fitted <receptive field>. The discrete filter exactly rejects a uniform contrast patch; this <Gabor filter> need not reject uniform contrast because its weight <integral> need not vanish. Neither toy includes stimulus history, recurrent interactions, saturation or a measured response baseline. Their value is a transparent prediction: <orientation selectivity> can emerge from <spatial summation>, and it does not by itself imply tolerance to <phase>.
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