[Verse 1] Picture a dendrite as leaky cable wire Resistive core with membrane that conspires To drain the voltage as it travels down R-i per length, the axial resistance crown Wrapped in capacitive membrane shell R-m and C-m cast their dampening spell Steady-state equation tells the tale V of x equals V-naught times e to the negative x over lambda scale [Chorus] Lambda equals square root of R-m over R-i Length constant determines how far signals fly Point-one to one millimeter typical range Exponential decay, voltage rearrange Thirty-seven percent at one lambda out Fourteen percent at two, signals fade without doubt Distal inputs whisper, proximal inputs shout That's the passive cable, what it's all about [Verse 2] For cylindrical dendrite with diameter d Lambda equals square root quantity R-m times d divided by four R-i Mathematics governing how potentials die But wait there's more to this temporal game Tau-m equals R-m times C-m, membrane time constant name Ten to thirty milliseconds typical duration Low-pass filter causing temporal modulation [Chorus] Lambda equals square root of R-m over R-i Length constant determines how far signals fly Point-one to one millimeter typical range Exponential decay, voltage rearrange Thirty-seven percent at one lambda out Fourteen percent at two, signals fade without doubt Distal inputs whisper, proximal inputs shout That's the passive cable, what it's all about [Bridge] EPSP arrives not as spike but filtered bump Distal signals reach soma slower, broader, slumped Conductance-based shunting makes summation sub-linear Each synapse opens to E-exc, driving force gets thinner As voltage climbs toward reversal potential zero Driving force shrinks, second synapse less heroic Two together less than sum alone Passive dendrites make this clearly known [Verse 3] Distance penalizes inputs from afar Attenuation dims each synaptic star But dendritic democracy fights back smart Distal synapses scaled up, playing their part More receptors, higher conductance compensation Normalizing impact across dendritic plantation Location marks pathway more than amplitude Spatial coding with deliberate attitude [Verse 4] Active properties add nonlinear spice Sodium channels make voltage-gated dice NMDA spikes and calcium bursts ignite Transform passive tree to computational might But still the cable equation holds its ground Foundation physics where all signals are bound Linear approximation guides our understanding Even when nonlinearities are commanding [Outro] Cable theory governs dendritic computation Passive properties shape neural conversation Remember lambda, tau-m, and shunting rule Master these concepts, you've got the tool
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