Lecture 3: linearizing the HH equations HH system is 4-d, nonlinear. For some insight, linearize around a (subthreshold) resting state. (Can vary resting.

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Presentation transcript:

Lecture 3: linearizing the HH equations HH system is 4-d, nonlinear. For some insight, linearize around a (subthreshold) resting state. (Can vary resting voltage V 0 by varying constant injected current I 0.) Ref: C Koch, Biophysics of Computation, Ch 10

Full Hodgkin-Huxley model

4 coupled nonlinear differential equations

Spikes, threshold, subthreshold dynamics threshold propertyspike

Spikes, threshold, subthreshold dynamics threshold propertyspike sub- and suprathreshold regions

Linearizing the current equation: Equilibrium: V 0, I 0

Linearizing the current equation: Equilibrium: V 0, I 0 Small perturbations: 

Linearizing the current equation: Equilibrium: V 0, I 0 Small perturbations: 

Linearizing the current equation: Equilibrium: V 0, I 0 Small perturbations: 

Linearizing the current equation: Equilibrium: V 0, I 0 Small perturbations: 

Linearized equations for gating variables fromwith

Linearized equations for gating variables fromwith 

Linearized equations for gating variables fromwith  

Linearized equations for gating variables fromwith   Harmonic time dependence:

Linearized equations for gating variables fromwith   Harmonic time dependence: 

Linearized equations for gating variables fromwith   Harmonic time dependence:  solution:

Linearized equations for gating variables fromwith   Harmonic time dependence:  solution: or

So back in current equation

For sigmoidal

So back in current equation For sigmoidal

So back in current equation For sigmoidal

So back in current equation For sigmoidal like a current

So back in current equation For sigmoidal like a current i.e.

So back in current equation For sigmoidal like a current i.e. or

So back in current equation For sigmoidal like a current i.e. or equation for an RL series circuit with

Equivalent circuit component

Full linearized equation:

A(  )= 1/R(  ) = admittance

Full linearized equation: A(  )= 1/R(  ) = admittance Equivalent circuit for Na terms:

Impedance(  ) for HH squid neuron (  =2  f )

Impedance(  ) for HH squid neuron experiment: (  =2  f )

Impedance(  ) for HH squid neuron experiment: (  =2  f ) Band-pass filtering (like underdamped harmonic oscillator)

Cortical pyramidal cell (model) (log scale)

Damped oscillations Responses to different current steps: