> Sorry if I'm Mx-splaining -- I don't know your background
Not at all! All of this is far more advanced than my knowledge on the topic, and very interesting! Thanks for sharing
That idea of representing in the highest dimensionality possible, with some constraint is also very interesting. In that case perhaps the 3D space is being represented in a higher dimensional form that makes it more convenient for some neural processing purpose (e.g. just like we use homogenous coordinates) The 2D-2D case is then just a happy coincidence where the highest representation that makes sense maps 1 to 1 with the actual data.
We should be careful to distinguish between the dimensionality of the physical space, the dimensionality of image data coming in from the retina, and the dimensionality of the navigational representation.
Going back to the image example, a 100x100 pixel image is 2D in that it can be shown on a screen, or printed on a page, or laid flat on a plane. But the (abstract) space of all possible images is #intensities_per_pixel to the 100x100 = 10000 power.
It's abstract in that each point in this space is not a location in the external world, but specifies one particular image. If you're familiar with phase spaces or configuration spaces from physics, it's like that.
The other thing is that we don't seem to have direct access to a 2d or 3d position-tracker sense. So instead, we have to build up some internal representation for navigation, based on our senses which as outlined for images are much higher-dimensional than the physically allowable positions in the world they're sensing. Robotics SLAM is one approach.
Then finally, there's the dimensionality of the neural representation itself. Let's say that your internal navigation map is represented by the firing-pattern of a population of neurons (i.e. more than one). Define a time-step, say one millisecond. For simplicity, consider each neuron to just be have an "activity" in the range 0.0 to 1.0 per time-step by e.g. counting the number of spikes emitted per time-step and dividing by some number to get things in a nice range. So now you can represent the activity of a neuron by one number per timestep. If you have a population of 100 neurons, then that's a 100 number string. But ... you can also think of it as a point in a 100 dimensional space. Each point is one particular pattern of neural activations. The entire space is all of the possible patterns of neural activations. Again, this is not a space in the sense of physical positions in the world or in the brain. It's an abstract space. But all the vector space math works.
SO we're perceiving 3D by means of retinas that take way more than 3 measurements at an instant, and maybe our brain is finding correlations so that it can represent these in a distributed way in << input_dimensions, but > world_dimensions.
Not at all! All of this is far more advanced than my knowledge on the topic, and very interesting! Thanks for sharing
That idea of representing in the highest dimensionality possible, with some constraint is also very interesting. In that case perhaps the 3D space is being represented in a higher dimensional form that makes it more convenient for some neural processing purpose (e.g. just like we use homogenous coordinates) The 2D-2D case is then just a happy coincidence where the highest representation that makes sense maps 1 to 1 with the actual data.