The geometry of selectivity and invariance. The three axes are three image dimensions (e.g., the values of three pixels in an image). Real images require several thousand dimensions, but we use three for simple visualization. Any point in the space corresponds to a different image. The gray surface represents a continuous subset, or manifold, of images of a particular object. If a hypothetical neural population effectively encodes this object's identity, all object images from this manifold will yield patterns of neural responses that are distinguishable from the patterns of responses induced by other sets of images. Moving along the surface of the manifold changes the image itself but maintains the ability of the neural population to discriminate the image from others. This is a direction of invariance. Moving away from, or orthogonal to, the surface of the manifold changes the image in a way that prevents the population from effectively discriminating. This is a direction of selectivity. The manifold shown here corresponds to a set of population responses that are selective for proboscis monkeys, not just for image patches with similar color and texture, but are also invariant to changes in size (near vs far) and context (face only vs face and body). Photo used with permission, courtesy of Paul Huggins (http://www.paulhugginsphotography.com/).