Simple Cells and Complex Cells

Simple cells vs Complex cells

	simple cells	complex cells
feature	with excitatory and inhibitory regions in RFs	with no inhibitory and excitatory regions in RFs
example	the cell responds to a horizontal slit and responds higher when the slit is positioned somewhere in RFs; similarly, the cell will respond to drifiting gratings	the cell responds to a horizontal slit anyway, the position does not matter; similarly, the cell will not respond to drifting gratings
orientation selectivity (e.g. is the slit horizontal?)	Y	Y
spatial-frequency selectivity	Y	Y
spatial-phase selectivity	Y	N (phase invariance)
model	Gabor filter	combination of simple cell responses

Simple cell model: Gabor filter (linear)

A Gabor filter can be viewed as a sinusoidal signal of particular frequency and orientation, modulated by a Gaussian wave.
Real-valued Gabor filter in the space domain:

G (x, y) = \frac{1}{2 π σ_{x} σ_{y}} e x p （ - \frac{x^{2}}{2 σ_{x}^{2}} - \frac{y^{2}}{2 σ_{y}^{2}} ） c o s (k x - Φ)

Simple cell receptive fields (e.g. in cat striate cortex) are often found to be well fit by Gabor functions.

If including the orientation angle $θ$ and phase $φ$ :

G (x, y, [φ, θ]) = \frac{1}{2 π σ_{x} σ_{y}} e x p (- \frac{x^{2}}{2 σ_{x}^{2}} - \frac{y^{2}}{2 σ_{y}^{2}}) \cos (k (x \cos (θ) + y \sin (θ)) - φ)

with $φ = k π$ , G is an even-symmetrical Gabor
with $φ = k π / 2$ , G is an odd-symmetrical Gabor

Complex cell model: Quadrature pair of simple cell models

To model the phase-invariance, the model of complex cells is the summation of responses of simple cells with phase difference $π / 2$ , which is the quadrature pair of simple cell models.
For example, Pollen & Ronner (1981) found neighboring simple cells in cat primary visual cortex with similar orientation and spatial-frequency tuning, but 90 degree phase shift.