If you read the columns paper assuming that the L2/3 is doing hex-grids fits very nicely in the descriptive text and in some ways, suggests hex-grid behavior - for example: while L2/3 and L5 cells exhibit “complex” RFs (Hubel and Wiesel, 1962; Gilbert, 1977). Key properties of complex cells include RFs influenced by a wider area of sensory input and increased temporal stability (Movshon et al., 1978).
Cells which have similar classic receptive fields when presented with isolated edge-like features, diverge, and fire uniquely when the feature is part of a larger object.
To explain border ownership, researchers have proposed a layer of cells that perform “grouping” of inputs. The grouping cells are stable over time (Craft et al., 2007).
- I am certain that the function of L4 works as a timing coordinator with the interface with the thalamus for the formation of waves and local synchronization to these waves. I have supporting papers on this but that that is not the issue I am working now.
I predict that the timing below will be related to the gamma rate (40 Hz) or 25 ms:
Activations in the output layer do not require very fast inhibition. Instead, a broad inhibition within the layer is needed to maintain the sparsity of activation patterns. Experiment evidence for both fast and broad inhibition have been reported in the literature (Helmstaedter et al., 2009; Meyer et al., 2011).
Our simulations do not model inhibitory neurons as individual cells. The functions of inhibitory neurons are encoded in the activation rules of the model. A more detailed mapping to specific inhibitory neuron types is an area for future research.
Wait - what was that last bit? …
The functions of inhibitory neurons are encoded in the activation rules of the model. A more detailed mapping to specific inhibitory neuron types is an area for future research.
The biology seems to point to smaller local pools of inhibition that are triggered locally. This is not rocket science - a little stroll through related papers should lay out the scope of inter-neurons receptive fields and modulating output connections.
While I am banging on non-biological problems with this paper here is a monster one:
In this experiment, each column receives lateral input from every other column.
This is absolutely NOT how it works in biology.
These connections are the foundation of the Hex-grid formation and the model skips right over it.
What does the biology say?
From this paper in my hex-grid post:
What Proportion of Layer 3 Pyramidal Cells Receive Long-distance, Excitatory, Monosynaptic Inputs?
Our findings also suggest that most pyramidal neurons in layer 3 are targets of long-distance, horizontal projections. Specifically, low-intensity stimulation at long distances from the recorded layer 3 pyramidal cell evoked monosynaptic EPSCs in the majority (77%) of these neurons. However, this proportion is likely to be an underestimate since some long-distance axon collaterals were probably severed by slicing of the tissue blocks. Although the present study does not indicate whether horizontal projections synapse selectively onto layer 3 pyramidal neurons, our results show that these cells frequently receive this type of synaptic input
I have several more paper (including ones referenced by Numenta) that support the exclusively longer length of lateral connections. When you consider this go back to my third drawing showing a “halo” around a given mini-column in the post above.
And then consider how that drives the formation of triangular formations.
So the answer to hex-grids questions was “we considered that - go read the columns paper” and “sure it would work but why would we look at that?”
- How about - because the biology does it?
More comments on the columns paper:
Hex-Grids are strongly compatible with most of these with no changes.
Ok, I read the methods section again - it is what I saw the first time I worked through the math.
Here’s where they play footloose and fancy-free with the biology - the tangle of axons that fan out from the mini-column is what activates the inhibition cells - not some vague inhibition field in the model.
Look very hard at this related bit in Computing Cell States
As I said above: this section totally ignores the known topology of interaction between the mutual reciprocal connections and the range and activation properties of inhibition inter-neurons. When it comes to interactions between neurons I can’t stress this enough: Topology Matters. As offered in this paper the model continues the earlier simple “thousands of synapses” models and ignores the actual biology.
As described, the calculations based on this model is correct. The problem is that the model is NOT based on the known topology of the biology so it is misleading - it does NOT serve to explain the behavior of the biology. Numenta claims to be biologically inspired to explain how the brain works; they will have to do better.
The hex-grid proposal faithfully models this aspect of the biology.
I would start with modifying the model so that lateral connections are all at a biologically plausible distance from mini-column to mini-column. Your suggestion about looking at the hex-patterns that form should produce some very interesting behaviors.
I agree that Numenta has identified the inhibition function for future consideration. Why this is important is the bit I have mentioned before - the tuning of the ratio of distances of the mutual connections and inhibition range modifies the behavior of the column. At one ratio the column acts a Gabor filter - exactly what is needed for early sensory processing. Reduce the inhibition range and the column acts the hub of a hex-grid formation. I would think that this range of behaviors would be very important to the studies of cortical column theory.