Lecture 6: Dendrites and Axons Cable equation Morphoelectronic transform Multi-compartment models Action potential propagation Refs: Dayan & Abbott, Ch 6, Gerstner & Kistler, sects 2.5-6; C Koch, Biophysics of Computation, Chs 2,6

Longitudinal resistance and resistivity

Longitudinal resistance

Longitudinal resistance and resistivity Longitudinal resistance Longitudinal resistivity r L ~ 1-3 k  mm 2

Longitudinal resistance and resistivity Longitudinal resistance Longitudinal resistivity r L ~ 1-3 k  mm 2

Cable equation

current balance:

Cable equation current balance: on rhs:

Cable equation current balance: on rhs:  Cable equation:

Linear cable theory Ohmic current:

Linear cable theory Ohmic current: Measure V relative to rest:

Linear cable theory Ohmic current: Measure V relative to rest: Cable equation becomes

Linear cable theory Ohmic current: Measure V relative to rest: Cable equation becomes Now define electrotonic length and membrane time constant:

Linear cable theory Ohmic current: Measure V relative to rest: Cable equation becomes Now define electrotonic length and membrane time constant: 

Linear cable theory Ohmic current: Measure V relative to rest: Cable equation becomes Now define electrotonic length and membrane time constant:  Note: cable segment of length has longitudinal resistance = transverse resistance:

Linear cable theory Ohmic current: Measure V relative to rest: Cable equation becomes Now define electrotonic length and membrane time constant:  Note: cable segment of length has longitudinal resistance = transverse resistance:

dimensionless units:

Removes,  m from equation.

dimensionless units: Removes,  m from equation. Now remove the hats:

dimensionless units: Removes,  m from equation. Now remove the hats: ( t really means t/  m, x really means x/ )

Stationary solutions No time dependence:

Stationary solutions No time dependence: Static cable equation:

Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 :

Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 :

Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 : Point injection:

Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 : Point injection: Solution:

Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 : Point injection: Solution:

Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 : Point injection: Solution: Solution for general i e :

Boundary conditions at junctions

V continuous

Boundary conditions at junctions V continuous Sum of inward currentsmust be zero at junction

Boundary conditions at junctions V continuous Sum of inward currentsmust be zero at junction closed end:

Boundary conditions at junctions V continuous Sum of inward currentsmust be zero at junction closed end: open end: V = 0

Green’s function Response to delta-function current source (in space and time)

Green’s function Response to delta-function current source (in space and time)

Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform:

Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform:

Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform Easy to solve:

Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform Easy to solve: Invert the Fourier transform:

Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform Easy to solve: Invert the Fourier transform:

Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform Easy to solve: Invert the Fourier transform: Solution for general i e (x,t) :

Pulse injection at x=0,t=0 :

u vs t at various x : x vs t max :

Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak?

Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak?

Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak? 

Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak? 

Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak?  

Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak?   Restoring ,  m :

Compare with no-leak case:

Just diffusion, no decay

Compare with no-leak case: Just diffusion, no decay Now when does it peak for a given x ?

Compare with no-leak case: Just diffusion, no decay Now when does it peak for a given x ?

Compare with no-leak case: Just diffusion, no decay Now when does it peak for a given x ? 

Compare with no-leak case: Just diffusion, no decay Now when does it peak for a given x ? 

Compare with no-leak case: Just diffusion, no decay Now when does it peak for a given x ?  Restoring ,  m :

Finite cable Method of images:

Finite cable Method of images:

Finite cable Method of images: General solution:

Finite cable Method of images: General solution:

Morphoelectronic transform

Frequency-dependent morphoelectronic transforms

Multi-compartment models

Discrete cable equations

Resistance between compartments:

Discrete cable equations Resistance between compartments: Current between compartments:

Discrete cable equations Resistance between compartments: Current between compartments: Current per unit area:

Discrete cable equations Resistance between compartments: Current between compartments: Current per unit area: 

Action potential propagation

a “reaction-diffusion equation”

Action potential propagation a “reaction-diffusion equation” Moving solutions: look for solution of the form

Action potential propagation a “reaction-diffusion equation” Moving solutions: look for solution of the form  Ordinary DE

Action potential propagation a “reaction-diffusion equation” Moving solutions: look for solution of the form  Ordinary DE

Action potential propagation a “reaction-diffusion equation” Moving solutions: look for solution of the form  Ordinary DE HH solved iteratively for s (big success of their model)

Propagation speed a/s 2 must be independent of a

Propagation speed a/s 2 must be independent of a

Propagation speed a/s 2 must be independent of a This is probably why the squid axon is so thick.

Multi-compartment model

Multicompartment calculation

Myelinated axons Nodes of Ranvier: active Na channels

Myelinated axons Nodes of Ranvier: active Na channels

Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Nodes of Ranvier: active Na channels

Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Nodes of Ranvier: active Na channels

Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Integrate up from a1 to a2 (inverse capacitances add) Nodes of Ranvier: active Na channels

Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Integrate up from a1 to a2 (inverse capacitances add) Nodes of Ranvier: active Na channels

Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Integrate up from a1 to a2 (inverse capacitances add) Negligible leakage between nodes: cable equation becomes Nodes of Ranvier: active Na channels

Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Integrate up from a1 to a2 (inverse capacitances add) Negligible leakage between nodes: cable equation becomes Diffusion constant: Nodes of Ranvier: active Na channels

How much myelinization? optimal a 1 /a 2 Find the value of y = a 1 /a 2 that maximizes D

How much myelinization? optimal a 1 /a 2 Find the value of y = a 1 /a 2 that maximizes D

How much myelinization? optimal a 1 /a 2 Find the value of y = a 1 /a 2 that maximizes D 

How much myelinization? optimal a 1 /a 2 Find the value of y = a 1 /a 2 that maximizes D  Agrees with experiment

Speed of propagation Diffusion equation with diffusion constant

Speed of propagation Diffusion equation with diffusion constant 

Speed of propagation Diffusion equation with diffusion constant   Speed of propagation proportional to a 2

Speed of propagation Diffusion equation with diffusion constant   Speed of propagation proportional to a 2 (cf a 1/2 for unmyelinated axon)