[Verse 1]
Voltage travels down the dendrite maze
But resistance makes it fade in countless ways
Lambda tells us how far signals cruise
Before they're half the strength they started to use
Length constant lambda, space decay factor
Current spreads through cytoplasm, membrane's the subtractor
Five times lambda and you've lost most of the charge
Distance kills the signal, membranes discharge
[Chorus]
Lambda for the length, tau for the time
Passive cable theory, voltage decline
Distal EPSPs arrive weak and slow
Proximal ones hit with a sharper blow
Sub-linear summation, that's the dendritic game
Voltage don't add up the simple way, driving currents strain
[Verse 2]
Tau membrane, time constant of decay
Capacitance and resistance set the way
RC circuit built into every branch
Voltage rises slow, then falls like an avalanche
Bigger tau means longer integration window
Smaller tau makes responses quick and nimble
Ten milliseconds or a hundred, depends on the cell
Time constant shapes how voltage stories tell
[Chorus]
Lambda for the length, tau for the time
Passive cable theory, voltage decline
Distal EPSPs arrive weak and slow
Proximal ones hit with a sharper blow
Sub-linear summation, that's the dendritic game
Voltage don't add up the simple way, driving currents strain
[Bridge]
Why sub-linear when dendrites compute?
Driving force decreases, that's the truth absolute
Two EPSPs don't make twice the height
Membrane saturation limits the fight
Closer to reversal, weaker the drive
Cable equation keeps the math alive
[Verse 3]
Soma sees the filtered, dampened remains
Of synaptic inputs from dendritic domains
Far-out synapses get smoothed and delayed
Close-in connections keep their edge displayed
Passive integration, no amplification
Just cable physics and membrane filtration
Tree-like computation, non-linear and wise
Sub-threshold magic that the brain utilizes
[Verse 4]
Dendrite diameter changes the game
Thick branches carry voltage with less shame
Thin twigs filter heavy, thick ones flow free
Cable equation predicts what we'll see
Branch points cause reflections, impedance mismatch
Voltage gets divided at each dendritic catch
Geometry matters in this neural art
Every fork and taper plays its part
[Chorus]
Lambda for the length, tau for the time
Passive cable theory, voltage decline
Distal EPSPs arrive weak and slow
Proximal ones hit with a sharper blow
Sub-linear summation, that's the dendritic game
Voltage don't add up the simple way, driving currents strain
[Outro]
Cable constants rule the neural wire
Distance and time, the dendrite's desire
To filter and shape each synaptic call
Passive computation, understanding it all