BME 6938Neurodynamics

Instructor: Dr Sachin S Talathi

The cell membrane-equivalent circuit

The bilipid layer: Capacitance

Pore Resistance

The First ODE-For XPPAuto Passive Membrane with time dependent input

current

Look up nice tutorial on using xppauto on bards webpage at http://www.math.pitt.edu/~bard/bardware/tut/start.html

XPPAuto ODE File# passive membrane with step function #current: passive.ode parameter R_M=10000, C_m=1, I_0=2, E=-70 parameter t_on=5, t_off=10,Vm=0dV/dt = (1000*(Vm-V)/R_M + I_0*f(t))/C_M V(0)=0 # define a pulse function f(t)=heav(t_off-t)*heav(t-t_on) # track the current aux ibar=f(t)*I_0 done

Comment

Define Parameters

The ODE

Initial Conditions

The Function

Aux File

End of File

The Cable Theory for Passive Cell

Assumptions:Membrane parameters are linear and independent of mem. potential (passive) ; current entering the cable flows linearly (homogeneous); resistance of extracellular medium is zero (cell immersed in homogeneous isopotential medium, the reference)

Use the mathematical frame work of linear cable theory and the elec. circuit representationof neuronal cell membrane to understand how membrane potential is affected in function of neuronal cell geometry. Important to understand concepts like synaptic integration

Kirchoff’s Current law applied to cable

Ijext

Cm: Membrane Capacitance (F)Rm: Membrane Resistance (Ohm)Ra: Axial Resistance (Ohm)Iext: External current (Amp)

The Cable Equation

CM: Specific Capacitance (F/cm2)RM: Specific Resistance (Ohm-cm2)RA: Specific Axial Resistance (Ohm-cm)iext: Current density (Amp/cm2)

The Cable Equation: Rescaling Variables

Recap

IM

IL d

The Cable Equation

IM: Membrane Current (Amp/cm2)CM: Specific Capacitance (F/cm2)RM: Specific Resistance (Ohm-cm2)RA: Specific Axial Resistance (Ohm-cm)

The Cable Equation: Steady State

Space Constant Time Constant

Longitudinal Current: Input Resistance

Il (x)

V (x+Δx)V (x)

Im

RL: Cytoplasmic Resistance per unit length (Ohm/cm)

The Cable Equation: Steady State

Green’s Function G(X): Solution to Steady State Cable Equation for

Solution to Steady State Cable Equation:

with boundary conditions:

(Infinite cable)

Steady State: Boundary Conditions

Semi-Infinite CableSemi-Infinite Cable

Finite Cable Sealed EndFinite Cable Sealed End

(closed circuit)(closed circuit)

Finite Cable Open EndFinite Cable Open End

(open circuit)(open circuit)

Finite Cable Clamped EndFinite Cable Clamped End

Cable Type Schematic Diagram Boundary Condition

Semi-Infinite Cable: Constant CurrentI0

Finite Cable: Constant Current

Length of Cable: l

Dimensionless Length:

General Solution:

Conductance of terminal end

Conductance of semi-infinite cable

Finite Cable: Sealed End

L

I0

Finite Cable: Open End

L

I0

Finite Cable: Clamped End

L

I0

Steady State Solution

Semi-Infinite CableSemi-Infinite Cable

Finite Cable Sealed EndFinite Cable Sealed End

Finite Cable Open EndFinite Cable Open End

Finite Cable Clamped EndFinite Cable Clamped End

Cable Type Solution Boundary Condition

Steady State Solution

Cable Equation: Transient Solution

Green’s Function G(X,T) for infinite cable: solution of above equation for:

With initial condition: and Boundary condition:

General Solution to Cable Equation:

Hint: Use the formula:

Ralls Model-Equivalent cylinderBasic Idea: Impedence matchingAssumptions:1.The membrane properties are identical for soma and dendritic branches. 2. Membrane properties are uniform and voltage independent3. All dendritic branches terminate at the same electrotonic length (and the tip of dendrite ends are sealed)

Class assignment: Please read sections 4.5.1.3 and 4.5.2 on your own.

Synaptic Integration

Model for current injection into neuronthrough synapse-alpha function

Use XPP AUTO to answer following Questions (Cable.ode)

1. Sketch the potential at the soma for the synaptic input at compartments 0, 5, 10, and 20. 1a.How do the peak amplitudes depend on distance? 1b. How about the time to peak? 1c.Does the peak appear to decay slower or faster for more distant inputs?1d. How does the potential scale across various compartmentsFor synaptic input at different locations on the cable