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Lecture 3: linearizing the HH equations

This lecture focuses on the linearization of the Hodgkin-Huxley (HH) model, a 4-dimensional nonlinear system. By analyzing the system around a subthreshold resting state, insights are gained into neuronal behavior, particularly the transition between subthreshold and suprathreshold dynamics. The approach involves varying resting voltage (V0) through injected current (I0) to explore small perturbations, leading to linearized equations for gating variables. The implications on spiking, threshold properties, and impedance in neural models are discussed, providing a foundational understanding of neural excitability.

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Lecture 3: linearizing the HH equations

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  1. 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 V0 by varying constant injected current I0.) Ref: C Koch, Biophysics of Computation, Ch 10

  2. Full Hodgkin-Huxley model

  3. Full Hodgkin-Huxley model

  4. Full Hodgkin-Huxley model

  5. Full Hodgkin-Huxley model 4 coupled nonlinear differential equations

  6. Spikes, threshold, subthreshold dynamics spike threshold property

  7. Spikes, threshold, subthreshold dynamics spike threshold property sub- and suprathreshold regions

  8. Linearizing the current equation: Equilibrium: V0, I0

  9. Linearizing the current equation: Equilibrium: V0, I0 Small perturbations: 

  10. Linearizing the current equation: Equilibrium: V0, I0 Small perturbations: 

  11. Linearizing the current equation: Equilibrium: V0, I0 Small perturbations: 

  12. Linearizing the current equation: Equilibrium: V0, I0 Small perturbations: 

  13. Linearized equations for gating variables from with

  14. Linearized equations for gating variables from with 

  15. Linearized equations for gating variables from with  

  16. Linearized equations for gating variables from with   Harmonic time dependence:

  17. Linearized equations for gating variables from with   Harmonic time dependence: 

  18. Linearized equations for gating variables from with   Harmonic time dependence:  solution:

  19. Linearized equations for gating variables from with   Harmonic time dependence:  solution: or

  20. So back in current equation

  21. So back in current equation For sigmoidal

  22. So back in current equation For sigmoidal

  23. So back in current equation For sigmoidal

  24. So back in current equation For sigmoidal like a current

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

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

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

  28. Equivalent circuit component

  29. Full linearized equation:

  30. Full linearized equation:

  31. Full linearized equation: A(w)= 1/R(w) =admittance

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

  33. Impedance(w) for HH squid neuron (w=2pf)

  34. Impedance(w) for HH squid neuron experiment: (w=2pf)

  35. Impedance(w) for HH squid neuron experiment: (w=2pf) Band-pass filtering (like underdamped harmonic oscillator)

  36. Cortical pyramidal cell (model) (log scale)

  37. Damped oscillations Responses to different current steps:

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