Neurons talk: summation

A alone lifts the trace to $-62$ mV — depolarization $8.0$ mV, threshold gap $7.0$ mV — and decays with $\tau = 15$ ms. B alone: the same. A+B jumps to $-54$ mV, over the dashed line, and the spike stroke lands. At weights $7$ and $7$, A+B reaches only $-56$ mV. Two near-misses, still a miss.

In this model, EPSPs — the voltage bumps synapses deposit — add linearly: simultaneous inputs depolarize by $w_A + w_B$, and the cell spikes exactly when that sum covers the $15$ mV gap to threshold. Linear summation is a modeling assumption, not a property of real dendrites, which sum non-linearly — nearby inputs shunt each other, large sums saturate, and some dendrites fire local spikes of their own. The idealized version is spatial summation at its cleanest: no single input decides; the vote does.

Now collect input activity into a vector $\mathbf{x}$ (fired $= 1$, silent $= 0$) and the synaptic strengths into a weight vector $\mathbf{w}$. The cell fires when

A weighted adder behind a threshold — the dot product from the linear algebra chapters, running on membrane voltage. Borrow this unit in software and you have the artificial neuron that neural networks are built from. Real brains stack billions of cells that are richer than any weighted adder — but the adder is where their arithmetic starts.