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 V 0 by varying constant injected current I 0.) Ref: C Koch, Biophysics of Computation, Ch 10

2. Full Hodgkin-Huxley model

5. 4 coupled nonlinear differential equations

6. Spikes, threshold, subthreshold dynamics threshold propertyspike

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

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

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

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

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

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

13. Linearized equations for gating variables fromwith

14. Linearized equations for gating variables fromwith 

15. Linearized equations for gating variables fromwith  

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

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

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

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

20. So back in current equation

21. 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:

31. A(  )= 1/R(  ) = admittance

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

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

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

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

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

37. Damped oscillations Responses to different current steps: