= Spatial summation
<Spatial summation> combines inputs arriving at different <synapses>. Location matters because a <dendrite> has axial <electrical resistance> and leaky <membrane capacitance>: a local <synaptic potential> need not arrive at the <soma> with its original amplitude or time course. A <passive dendritic cable> makes this attenuation explicit. The <superposition principle> applies to fixed <electric current> inputs in a linear passive model, but adding <synaptic conductances> changes the model's coefficients. Compare voltage changes relative to the same baseline, rather than adding absolute <membrane potentials>.
\Image[/synaptic-spatial-conductance-passive-integration.png]
{title=<Spatial summation>, reduced driving force and <shunting inhibition> in passive models}
{description=Original calculations, not biological recordings. A: equal imposed <electric current> steps at the <soma> and a <dendrite> in two coupled passive compartments yield different somatic voltage traces; applying both gives their exact sum relative to rest. B: activating two excitatory <synaptic conductances> together gives less <depolarization> than adding the separate voltage responses. C: a <synaptic conductance> whose <reversal potential> equals rest suppresses the same excitation. All steps last from $10$ to $60\,\mathrm{ms}$. There are no active <ion channels>, <action potentials> or resets. The equations and parameters are under <passive spatial and conductance integration>.}
\b[A motivating experiment.] A 2004 study used $300$–$350\,\mathrm{\mu m}$ cortical slices from $18$–$28$-day-old rats at $35$–$36\,{}^\circ\mathrm C$. Somatic <whole-cell recordings> measured <EPSPs> in layer-5 <pyramidal neurons>. Two fine stimulating <electrodes> were positioned beside imaged <dendrites>; <calcium>-sensitive dye localized activation. Each site was stimulated alone, then both together, usually with paired pulses $20\,\mathrm{ms}$ apart. Comparing the combined voltage trace with the sum of individual changes tested the summation rule. Nearby inputs on one branch showed strong amplification at intermediate strengths; inputs on different branches were approximately linear.<Computational subunits in thin dendrites of pyramidal cells by Alon Polsky and others>[pp. 622–623, Figures 1–2; p. 626, Methods]{ref}
Blocking both excitatory <receptor> classes eliminated voltage and <calcium> responses, checking against direct electrical activation. <NMDA receptor> blockers eliminated or markedly reduced amplification; comparisons with inhibition intact and blocked addressed inhibitory contributions. These controls support local active integration. Passing <axons> could still recruit unintended <synapses>; somatic voltage does not measure every local <membrane potential>. The study tested subthreshold slice responses and pairs of branches, not arbitrary activity in an intact brain.<Computational subunits in thin dendrites of pyramidal cells by Alon Polsky and others>[pp. 622–624: controls; pp. 625–626: limitations]{ref}
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