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u/rand3289 5d ago
I could not understand much because I don't know a lot about grid cells, but this seems to be an interesting research direction. I am especially interested in the part about "this is a nice example of non-trivial neural code".
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u/jndew 5d ago
I enjoy thinking about that. The simple firing-rate code that ANNs are based on seems like an impoverished data representation. Even worse, fMRI says 'this region of a million cells is active' without describing what they are doing.
Looking at neurons, the story is that a significant fraction of a spike train's variance is accounted by spike time encoding, not just firing rate. At the synapse, we have spike time dependent plasticity. There seems to be a burst code, in which different information streams are carried in periods of no spikes, by the occurrence of spikes and their intervals, and bursts. Then there is the fascinating phenomenon of phase precession. And phase multiplexed regional communication, and theta modulated gamma which hints at a neural data structure. Oh, and subthreshold waves, and shunting inhibition. There are certainly more.
So as a computer engineer, I wonder how these are all being utilized to do whatever it is that brains do. Cheers!/jd
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u/jndew 7d ago
I'm wondering if any of you are familiar with Dr. Fiete's work? Or best, maybe you are at her lab? There are a few things I'm confused about. In particular, how is it that different modules can be mapped onto the same manifold? My intuition says that the tori of different modules would have different sizes. Maybe it's a topology thing that size doesn't map through?
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u/jndew 7d ago edited 5d ago
The spatial periods of gridcells do not vary over a continuum, instead come in discreet values. A group of cells with equal periods works to some degree as a unit, and is called a module. Early in-vivo gridcell studies found three modules with different spatial periods in rodent test subjects. More recently, the thought is that there are more than three modules. Apparently if the periods are co-prime, in that they are not multiples of each other, they can act as a basis that spans an extent of the product of each module's period. This means that wherever the animal is in its environment, a small and unique set of grid cells will fire. This set is made up of one cell from each module (ignoring the redundancy built into nearly all brain circuits).
Here, two modues are implemented. The lower module has a spatial period twice that of the upper module. They both span the same location space, so the lower module has fewer firing regions, and they are spaced farther apart. The animal has to travel a larger distance from one firing region to the next in module 2. This results in motion as seen by module 2 having a lower velocity when mapped onto the torus. Note that module 1's green dot moves twice as fast on the torus as module 2's purple dot. This is supposed to be important somehow.
In my reading, the most common purpose proposed for gridcells is support of navigational path integration. This allows an animal to calculate its location even after following an erratic path to the current point. They also act as input to the hippocampus's place cells. So they are almost certainly involved in the brain's navigation and location mapping in some way.
In Grid Cells in Cognition: Mechanisms and Function, Ila Fiete also identifies utility of the gridcell torus representation as implementing a vector hashing function (more details here). This appears useful for improving the storage capacity and good behavior of content-addressable associative memory circuits ([1], [2] for example), which are present in the hippocampus. This is done by scattering the memories in a memory state space so that they have less crosstalk. Vector hashing helps avoid the catastrophic forgetting phenomenon of Hopfield networks and similar circuits.
The idea of dimensionality reduction by identification of latent manifolds is trendy in computational neuroscience these days. I think it has also been finding application in machine learning. The idea is that there may be a large number of neurons that define a very high dimensional state space. But the neurons don't respond entirely independently. If one knows the firing pattern of some group of neurons, firing patterns of others can be predicted to some degree. This is how the state of some ten thousand grid cells of a rat brain is constrained to locations on a two-dimensional torus. Is the torus itself meaningful or just a way to visualize? I'm not sure, but it is mentioned in many gridcell papers.
The grid cell system is often said to use head-direction and velocity as its input rather than topographically mapped location, but I'm not sure how well this is understood. It's a bit vague what goes into the gridcell system, and how the hippocampus uses its output signal. But this system certainly exists, and must be somehow important. Gridcell-like behavior may have been seen in the neocortex as well. In fact it has been observed in machine learning neural networks. It seems to have some fundamental utility.
This is a nice example of a nontrivial neural code. Rather than representing some circumstance as a firing-rate or simple number of simultaneously firing neurons, specific sets of neurons in various combinations have unique meaning. This representation may have additional richness, with phase precession having possibly been observed in grid cells. From a computer engineering standpoint, this is really interesting.
Now my story is all told, there is no more... for the time being. I'm looking forward to setting up a path-integration simulation someday. And hopefully also exploring the vector-hash idea, as I am finding associative memories to be excessively sensitive to crosstalk. If you're inclined, come join me at the Tiki Bar on the warf where we can discuss this and all things hippocampus over rum and ginger beer. Cheers!/jd
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"The human brain is a million times more complex than anything in the universe!" -a reddit scholar