Part2: Simplified implementation of Grid&Displacement cells layer

[mraptor]

Another PART2:

The major problem of the previous post was ENCODING/DECODING the POSITION/LOCATION

Possible Simplified implementation of Grid cells layer

I have an idea for Simplified implementation of Grid cells layer, may be !! You decide if I’m bit too clever for the wrong reason What is left is the easy part i.e. how to ENCODE/DECODE this thing to/from SDR ? lets start with some snippets from the paper . To form a unique representation requires multiple grid cell modules with different scales or orientations (Figures 1C,D). For illustration purposes say we have 10 grid cell modules and each module can encode 25 p…

I think I solved it

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First lets recap the requirements for Grid Cells :

1. Large representation capacity - more primes increases capacity
2. Path integration - when you get back at the same position you get the same Location
3. Locations are unique to each environment - different maps

Lets add what else we can do with RNS - residue number system :

4. Addition
5. Subtraction
6. Multiplication

Now about the solution, lets pick two primes as quick example:

In [26]: [(i,i % np.array([2,5])) for i in range(11)]                                                                                                                        
Out[26]: 
[(0, array([0, 0])),
 (1, array([1, 1])),
 (2, array([0, 2])),
 (3, array([1, 3])),
 (4, array([0, 4])),
 (5, array([1, 0])),
 (6, array([0, 1])),
 (7, array([1, 2])),
 (8, array([0, 3])),
 (9, array([1, 4])),
 (10, array([0, 0]))]

So the maximum number we can represent w/o repetition is 2*5=10

1. The more primes we pick the space grows exponentially … by the product of primes.
2. If we get back to 5 it will always be (1,0)
3. Every 10 positions are another space/map i.e. once we anchor we get the same grid just different map.

NOW finally how do we encode/decode ?

Very easy simply use Integer encoder with a range of X to X+10, voila

Again depending on the Encoder you may need different encoder for every map, so it can generate different SDRs.

Encoder to skip SP!

Using SP is expensive, I’m looking for algorithms to generate suitable SDR’s directly from the Encoder. Below is the code of one I created. Pls comment on what benefits/drawback you can see. The normal procedure is to Encode the values and then pass it trough Spatial Pooler, but it is expensive operation. What if we can encode directly and skip SP. This is a Naive algorithm I came up to do just that, here is how it works. The algorithm depends on 3 parameters. - vsize : the number of bit…

Or let say 10 maps with single Encoder with range X to X+100.

Here is also operations i.e. motor commands arrive in RNS format

In [27]: n1 = np.array([1,1])                                                                                                                                                

In [28]: n2 = np.array([0,2])                                                                                                                                                

In [29]: n5 = np.array([1,0])                                                                                                                                                

In [30]: m = np.array([2,5])                                                                                                                                                 

In [31]: (n1+n5) % m                                                                                                                                                         
Out[31]: array([0, 1])  ==> 6

In [32]: (n5-n2) % m                                                                                                                                                         
Out[32]: array([1, 3])  ==> 3

In [33]: (n5*n2) % m                                                                                                                                                         
Out[33]: array([0, 0])  ==> 10 | 0

Reversing happens by CRT or Matrix Radix Conversion…

The benefit Displacement is simply RSN Addition and Subtraction …

BTW those 3 RSN operations do not require carry bit which makes them more biologically plausible.

… and here is the kicker in RSN there is a concept of base extension , still reading but may be can use it to convert between different grid systems i.e. Reference frames transformation

And I just found this paper, which elaborates what I found in more details.

What Grid Cells Convey about Rat Location
Ila R. Fiete, Yoram Burak, and Ted Brookings