= Solution
A <complex cell> in <primary visual cortex> typically responds to a suitably oriented bar, edge or grating over a finite receptive-field region, with less dependence on precise spatial phase and contrast polarity than a <simple cell>. Its response need not reveal separated linear ON and OFF subregions. Many cells have additional motion or binocular selectivity, so phase tolerance alone is not a complete account of every complex cell. Moving the feature beyond its <receptive field> still reduces the response.
In a <pooling model of a complex cell>, several <simple cells> with similar preferred orientation but different positions or phases feed the output. For linear subunit responses $a_m=\langle k_m,I\rangle$, a possible model is $r=f(\sum_mw_m[a_m-\theta_m]_+)$, with $w_m\geq0$. Include opposite-polarity filters if both light and dark features are to drive the cell. Rectification stops cancellation between subunits, and pooling spreads position and phase sensitivity across their coverage.
In a <quadrature energy model of a complex cell>, use two oriented filters of matching frequency but quadrature phase, and set
$$
\boxed{r=f(E),\qquad E=a_e^2+a_o^2}.
$$
For a matched grating, choose the pair so $a_e=A\cos\varphi$, $a_o=A\sin\varphi$; then $E=A^2$, independent of grating phase while retaining orientation and frequency tuning. Finite approximate filter pairs give only approximate invariance for general images. The quadrature construction underlies the primary https://persci.mit.edu/pub_pdfs/spatio85.pdf[Adelson–Bergen energy-model work]; temporal filters can extend it to motion sensitivity.
The pooling model emphasizes a circuit of rectified subunits and can flexibly cover translated positions. The energy model gives an explicit algebraic phase-invariance mechanism with quadratic contrast dependence before its output nonlinearity. They are not mutually exclusive biological alternatives: suitable subunit pooling can approximate an energy computation. Their spatial tolerance, contrast response and direction selectivity depend on the chosen filters and nonlinearities, and both require comparison with measured receptive fields.
Back to article page
