Whether or not you know about or care to use the mathematical coordinate systems described in the last post, it seems that our eyes and brains already make use of them when seeing and hallucinating. How this is the case will be explained here in some detail. Of course, our senses are intricately correlated with the dynamics of our brains in ways so complex that we can barely understand or even comprehend them at all, but there are still ways to reason and model (however crude they may be) the workings of such phenomenon.



Visual hallucinations are a universal human experience which occur in varying degrees in different circumstances–from when we rub or apply pressure to our eyes to more extreme instances like when we are under the influence of psychoactive drugs. What we “see” when we hallucinate is ultimately a unique subjective experience, but there does seem to be some objective similarities amongst reported hallucinations. Some of these hallucinations appear to be fixed in the visual field and do not change as we look around. Moreover, they can even be experienced when the eyes are closed, and have also been reported by people who are blind. This gives reason to believe that the source or cause of these hallucinations may be independent of what we really see and receive as sensory input from the external world, and instead originate on a deeper level within the brain itself.



Studies suggest that a certain region of the brain, the visual cortex (also known as V1), plays a primary role in the processing of visual information. There is a correspondence from what we see in the visual field with our eyes to the neural activity in parts of V1. Spatial input of light and dark on the eyes is translated to “light/dark” regions of high/low neural activity in V1. Basically, images form certain patterns on the retinas of our eyes which are then converted to related patterns on the visual cortex in our brains leading to the perception of the image.



To describe more rigorously the nature of this image translation from the eyes to the brain, we can use mathematics and coordinate systems to define a coordinate mapping between the visual field to the V1 region of the brain. Such a transformation was modeled by J. D. Cowan and G. B, Ermentrout in their 1979 paper “A Mathematical theory of Visual Hallucinations”. A simplified version capturing the essence of their model is described in what fallows.



Interpreting the visual field as a two-dimensional plane, let’s use polar coordinates to label points in the visual field. The center of the visual field is taken as the origin of the coordinate system so that a point P in the visual field is described by a pair of numbers P = ( r , a ), where r is the distance of the point P to the center of the visual field and a is the relative angle the point makes with respect to some fixed axis.



Now we need a way a describe how points in the visual field correspond to points in the V1 region of the brain, but first we need a way to label points in the V1 region. To do this, we model the V1 region as another two-dimensional flat plane. This time, let’s use the normal Cartesian coordinate system to label the points in V1 as pairs of numbers ( x , y ).



The coordinate mapping constructed here will tell us how a point in the visual field, labelled as P = ( r , a ) in polar coordinates, gets translated to the corresponding point in the V1 region of the brain labelled as ( x , y ) in Cartesian coordinates. This ( x , y ) point in V1 is given in terms of the r and a coordinates of the visual field through the relationship



( x , y ) = ( ln r , a ),



where ln r is the natural logarithm of r.



Thus, the point that is given by the polar coordinates ( r , a ) in the visual field is mapped to the corresponding point in V1 whose Cartesian coordinate has a horizontal component of ln r and a vertical component of a. This transformation can be interpreted as log-polar coordinates, and may be recognized as the complex logarithm. We can now use this coordinate transformation to describe the shapes of certain hallucinations.



Normally, we actually receive sensory input through the visual field and corresponding neural patterns are triggered in our brains which result in the perception of whatever we are seeing. The model explained here explains the existence of hallucinations by an opposite mechanism. Without even receiving real sensory input through the visual field, V1 can still be a brain region with high neural activity (perhaps more so when on certain drugs). These patterns of neural activity in V1 may be perceived as if one is actually seeing a pattern in the visual field, but they are really just hallucinations. In the 1920s, psychologist Heinrich Klüver researched himself and others while having ingested mescaline (more info here) in the form of peyote buttons and attempted to classify the visual hallucinations they experienced. The observed hallucinations manifesting themselves as geometrical patterns were classified into four types and were referred to as form constants: 1) tunnels, 2) spirals, 3) cobwebs, and 4) lattices.



The coordinate transformation just described between the visual field and the V1 region of the visual cortex is successful in explaining the occurrence of the form constants.



1) Vertical or horizontal stripes stripes of neural activity on the V1 region may look something like this:





Applying the coordinate transformation in the reverse direction, we see that vertical stripes get mapped to circles in the visual field and horizontal stripes get mapped to rays emanating out from the center of the visual field:

2) Stripes of neural activity in V1 in arbitrary directions such as these diagonals

get mapped to spirals in the visual field:



3) Combinations of stripes such as these vertical and horizontal ones



would then get mapped to patterns that resemble cobwebs:



4) Neural activity in V1 is not limited to stripes. There may be activity with certain lattice symmetries like these checkerboard or honeycomb patterns



which would get mapped to hallucinations that appear like these







This model and these examples are idealized cases, but serve to approximate what may be happening in our brains when certain hallucinations occur. Despite having empirical applications in this context, the coordinate transformation described here can be freely applied to any image resulting in transformed images that still manage to express a psychedelic aesthetic (see here for some examples).



If you ever wondered when or why you would use polar coordinates I hope this post serves as a justifiable application. Conversely, I do not necessarily justify the use of psychedelics to help understand what you are learning in your math lessons.





This post was inspired by these papers, which explain what was attempted here and more in much greater detail:

Get the Mathematica code for these animations as a CDF file here.