Cortical Statistics

Question

I just began reading Dayan & Abbott's Theoretical Neuroscience. I'm barely even through the second page, and I'm already stuck on this paragraph:

Axons from single neurons can traverse large fractions of the brain, or, in some cases, the entire body. In the mouse brain, it has been estimated that cortical neurons typically send out a total of about 40mm of axon and have approximately 10mm of total dendritic cable in their branched dendritic trees. The axon makes an average of 180 synaptic connections with other neurons per mm of length and the dendritic tree  receives, on average, 2 synaptic inputs per micro-meter.

If I understand correctly, the axon connections are essentially output connections, and the dendritic tree connections are input connections. Since all the connections must occur between axons and dendrites only, the number of input connections must equal the number of output connections.
Doing some very basic back-of-the-envelope calculations using the provided data shows that this is wrong:

For the axons (output connections): 180 axonal connections/mm * 40 mm total axon length = 7,200 axonal connections
For the dendrites (input connections): 2000 dendritic connections/mm * 10 mm total dendritic tree length = 20,000 dendritic connections

These two numbers don't match. So there's obviously more to this than just matching the number of input to output connections.
What else could be going on?

Bryan Krause · Answer

I don't have a copy of the book on hand so I'm not sure if they say more about this, but I can make some guesses.

External connections. They are talking about local cortical connectivity but not inputs coming from outside cortex if they are just measuring cortico-cortical axons but all dendrites. Typically external connections are a much smaller fraction than the ratio discrepancy here, though.

Cell types. Cortex has lots of different cell types, and if they present an average across cells measured but don't account for their different ratios or don't measure all cells equally you wouldn't expect the ratio to match. In particular, I wonder if they're only talking about excitatory cells in a specific layer, for example. You'd expect an imbalance because those cells may be getting more inputs from other layers than they output.

Axons are probably more likely to get cut off on a histological slice. They are longer and more meandering, whereas dendrites are typically constrained to near the cell body/within the cortical column. They're more likely to underestimate local axonal connections because of this.

Overall also these numbers are quite close, within an order of magnitude. I don't see them as conflicting much. Measuring biology is messy. If you report the sources they cite for these numbers it may be more clear which of the factors I mention are most likely to contribute.

Cortical Statistics

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