Passive Cable Properties of the Dendritic Tree

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Passive Cable Properties of the Dendritic Tree

• Baktash Babadi
• Fall 2004,
• IPM, SCS, Tehran, Iran
• baktash_at_ipm.ir

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A Schematic View of a Single Neuron
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Neuron as a Device
Output
Input
Threshold
Equilibrium Membrane Potential
Dendrites Passive Conductance
Axon Spikes (Hodgkin Huxley Eqns)
Time
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Our Approach
Basic Physiology and Anatomy
Detailed Biophysical Models of Single Neuron
Numerical Simulations
Simplified Models of Single Neuron
Network Models
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A Closed System of Particles
The Distribution of Energy among The
Particles Boltzman Distribution
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Equilibrium Potential
q
Energy Distribution
Energy inside
Concentration inside
Energy Outside
Concentration Outside
Nernst Potential
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Question

• The Goldman Equation
• Can it be derived from boltzmann distribution?

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Membrane Voltage Dynamics
?
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Membrane Equivalent Circuit (1)
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Membrane Equivalent Circuit (2)
Current I
C Q/V 1 Farad 1 Coulomb/ 1 Volt (Q
CV) dQ/dt I I C dV/dt
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Membrane Equivalent Circuit (3)
Current I
R (V-Vr)/I 1 Ohm 1 Volt/ 1 Ampere I
(V-Vr)/R
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Membrane Equivalent Circuit (5)
Current I
I C dV/dt (V-Vr)/R (CR) dV/dt -(V-Vr) IR
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Membrane Equivalent Circuit (6)
K
Cl
Na
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Dendritic Current (1)
?
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Dendritic Current (2)
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The Dendritic Current (3)
Im
ra Intracellular resistance per unit length
V (x)
V (x?x)
Il (x?x)
Il (x)
Segment resistance R ra ?x
Im Trans-membrane Current per unit length
?x
x
x?x
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Cable Equation (1)
Im
V (x)
V (x?x)
Il (x?x)
Il (x)
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Cable Equation (2)
Im
V (x)
V (x?x)
Il (x?x)
Il (x)
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Linear Cable Equation (1)
Iext
rm
cm
Vr
Im
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Cable Equation (3)
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Linear Cable Equation (2)
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Change of Variables
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Solutions of Cable Equation

• Steady State Solution
• Electrophysiology
• Synaptic Bombardment
• Insight
• Transient Solution
• Real Situation

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Steady State Solutions

• Infinite Cable
• Sealed End
• Killed End
• Clamped End

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General Solution for Linear Second Order ODE
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Infinite Cable
I
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Finite Cable

• Length of the cable l
• Dimension less length Ll / ?
• Dimension less distance variable Xx / ?

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Sealed End Cable
I
L
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Killed End Cable
I
L
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Clamped End Cable
I
L
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Transient Solution (1)
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Transient Solution (2)
Solution Green Function
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Transient Solution
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The Importance of the Green Function

• The General Solution of the cable equation is
  given by

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Modeling The Dendritic Tree