PYTHON · NUMPY

The neuron is a network

See the load-bearing fact: a real neuron's dendrites each apply their own nonlinearity, so one neuron computes what a point neuron needs a whole network for.

01Challenge

Build a single dendritic neuron, solve XOR with it, then switch off its dendritic nonlinearity and watch it collapse to a point neuron.

Predict firstA single biological neuron, with its branching dendrites, computes about as much as…

02Model

A real cortical neuron is not a point. Its inputs land on dendrites, and each dendritic branch applies its own nonlinearity — driven by NMDA channels — before anything reaches the soma. So a single neuron already computes a two-layer function. Poirazi and Mel showed this in 2003; in 2021 Beniaguev, Segev and London made it quantitative, training networks to match a real layer-5 pyramidal cell and finding it took five to eight layers — and that when they removed the NMDA nonlinearity, a single layer was enough. The depth is the dendrite. Here you build the smallest version: one neuron with a handful of nonlinear dendritic subunits. It solves XOR. Then you linearize the subunits — passive dendrites — and the same neuron collapses to a plain point neuron and fails. That one switch is the whole difference between 'a neuron is a dot' and 'a neuron is a small network.'

03Guided practice

1. A dendritic neuron. Inputs fan out to subunits, each with its own nonlinearity, summed at the soma — one unit, a two-layer computation. 2. Toggle the nonlinearity. Train it on XOR with the dendritic nonlinearity on, then off. 3. The verdict. On, one neuron solves XOR; off, it collapses to a point neuron and fails. The NMDA nonlinearity is the depth — exactly the 2021 result.

04Feedback

PASS WHEN PASS when the dendritic neuron solves XOR and its linearized version fails.

If your run fails, check:

A cell errored — run them in order.
The nonlinear dendrite should solve XOR; the linear one should fail.

05Retrieve & space

Where does a real neuron's computational depth come from? (nonlinear dendritic subunits — NMDA).
How many layers did it take to match a real L5 pyramidal cell? (five to eight — Beniaguev et al. 2021).
What happens to that depth if you remove the NMDA nonlinearity? (it collapses to a single layer).

06Mastery & project

You can build a dendritic unit and show that its nonlinearity, not extra neurons, is what buys the depth.

When a unit seems too weak, ask whether the missing power belongs inside it, as a dendritic nonlinearity.

← The point neuron, and why it won Catastrophic forgetting →

PYTHON · NUMPY

The neuron is a network

Solve XOR with one dendritic neuron, then remove the nonlinearity and watch it fail.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you