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BME 6938 Neurodynamics Instructor: Dr Sachin S Talathi

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The cell membrane-equivalent circuit The bilipid layer: Capacitance Pore Resistance

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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

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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=0 dV/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

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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 representation of neuronal cell membrane to understand how membrane potential is affected in function of neuronal cell geometry. Important to understand concepts like synaptic integration

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Kirchoffs Current law applied to cable I j ext Cm: Membrane Capacitance (F) Rm: Membrane Resistance (Ohm) Ra: Axial Resistance (Ohm) Iext: External current (Amp)

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The Cable Equation C M : Specific Capacitance (F/cm 2 ) R M : Specific Resistance (Ohm-cm 2 ) R A : Specific Axial Resistance (Ohm-cm) i ext : Current density (Amp/cm 2 )

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The Cable Equation: Rescaling Variables

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Recap IMIM ILIL d The Cable Equation I M : Membrane Current (Amp/cm 2 ) C M : Specific Capacitance (F/cm 2 ) R M : Specific Resistance (Ohm-cm 2 ) R A : Specific Axial Resistance (Ohm-cm) The Cable Equation: Steady State Space ConstantTime Constant

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Longitudinal Current: Input Resistance I l (x) V (x+Δx) V (x) ImIm R L : Cytoplasmic Resistance per unit length (Ohm/cm)

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The Cable Equation: Steady State Greens Function G(X): Solution to Steady State Cable Equation for Solution to Steady State Cable Equation: with boundary conditions: (Infinite cable)

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Steady State: Boundary Conditions Semi-Infinite Cable Semi-Infinite Cable Finite Cable Sealed End Finite Cable Sealed End (closed circuit) Finite Cable Open End Finite Cable Open End (open circuit) (open circuit) Finite Cable Clamped End Finite Cable Clamped End Cable TypeSchematic Diagram Boundary Condition

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Semi-Infinite Cable: Constant Current I0I0

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Finite Cable: Constant Current Length of Cable: l Dimensionless Length: General Solution: Conductance of terminal end Conductance of semi-infinite cable

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Finite Cable: Sealed End L I0I0

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Finite Cable: Open End L I0I0

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Finite Cable: Clamped End L I0I0

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Steady State Solution Semi-Infinite Cable Semi-Infinite Cable Finite Cable Sealed End Finite Cable Sealed End Finite Cable Open End Finite Cable Open End Finite Cable Clamped End Finite Cable Clamped End Cable TypeSolution Boundary Condition

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Steady State Solution

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Cable Equation: Transient Solution Greens 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:

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Ralls Model-Equivalent cylinder Basic Idea: Impedence matching Assumptions: 1.The membrane properties are identical for soma and dendritic branches. 2. Membrane properties are uniform and voltage independent 3. 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.

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Synaptic Integration Model for current injection into neuron through 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 compartments For synaptic input at different locations on the cable

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