1. Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder

2. Last Week Linear Oscillations are not robust to noise.  Biological systems encode information in nonlinear oscillations. How to tell whether a dynamical systems will exhibit nonlinear oscillations. – N-1=1 Theorem – Poincaré-Bendixon Theorem – Hopf-Bifurcation Theorem

3. This Week 1.Neurons encode information with non-linear oscillations (spike trains). 2.How do neurons generate spikes? 3. Hodgkin-Huxley Model 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem

4. 1. Stimulus Intensity is Encoded by the Frequency of a Nonlinear Oscillation (firing rate) Stimulus intensity strong stretch medium stretch light stretch Firing rate

5. Chapter 9: Action Potentials, &Limit Cycles 1.The Hodgkin-Huxley Model 2.Dynamic Properties of the HH neuron 3.Hysteresis 4.Dynamic Neuron Types 5.Resonance in Spike Generation

6. This Week 1.Neurons encode information with non-linear oscillations (spike trains). 2.How do neurons generate spikes? 3. Hodgkin-Huxley Model 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem

7. The Leading Characters: Na + and K + Session 1: Membrane Potential is determined by equilibrium between drift and diffusion

8. Neurons’ Active Properties Session 1: Passive Properties of the cell membrane Constant Permeability  Exponential Decay towards equilibrium potential Today: Active Properties: State-Dependent Responses Voltage Dependent ion channels  Spiking

9. Equivalent Electrical Circuit cell membranecapacitor concentration gradients batteries source: Kandel ER, Schwartz JH, Jessell TM 2000. Principles of Neural Science, 4th ed. McGraw-Hill, New York., chapter 7 ion channelssteerable resistors

10. Cell Membrane as a Capacitor + - in out Lipid Bilayer Capacitator = We model the lipid-bilayer as a capacitor.

11. Ion Channels as Steerable Resistors Equilibrium Potentials

12. Equivalent Circuit  Hodgkin Huxley E E E + - + - - +

13. The Dynamics of the Membrane Potential Depends on the Neuron’s State … Q: What do we have to know about the neuron’s state in order to predict the neuron’s response to a given stimulus? A: The conductances and their dynamics.

14. The conductances are voltage- dependent! Hodgkin Huxley Hodgkin, & Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology 117, 500-544 (1952).

15. How do conductances change and why? AP animation Conductances change by voltage-dependent (de)activation and (de)inactivation.

16. This Week 1.Neurons encode information with non-linear oscillations (spike trains). 2.How do neurons generate spikes? 3. Hodgkin-Huxley Model 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem

17. Voltage Gated Na-Channel The Na + channel is open if Activation and Inactivation Gate are open. Deactivated Inactivated (De)Activated x (De)Inactivated State=

18. Voltage Gated Na-Channel m: probability of one Na channel subunit to be activated Activation Gate

19. Voltage Gated K Channel 4 subunits No inactivaton gate

20. From Deactivation to Activation and Back Again Deactivated Activated

21. From Deinactivation to Inactivation and Back Again Deinactivated Inactivated The transition probabilities are voltage dependent.

22. Hodgkin-Huxley Model, Version 1 The H-H Model comprises 4 non-linear ODEs that explain the Action Potential by the voltage dependent change in the opening probability of Na + and K + channels.

23. Hodgkin-Huxley Model, Version 2 Na-activation, K-activation, and Na-inactivation converge to their voltage- dependent equilibrium values at voltage-dependent speeds.

24. Problem: HH model is too complex to analyse mathemtically Possible Solutions: 1.Numerical Simulation 2.Mathematical Simplifications 1.Fitzhugh-Nagumo: – simple, but sacrifices biophysical interpretation 2.Rinzel – retains biophysical interpretation while being analytically tractable

25. Numerical Simulation of HH Matlab Demo

26. This Week 1.Neurons encode information with non-linear oscillations (spike trains). 2.How do neurons generate spikes? 3. Hodgkin-Huxley Model 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem

27. Rinzel’s simplification of the HH model Rinzel simplified the 4-dimensional HH model into a 2-dimensional model.  We can use the mathematical tools available for the 2-dimensional case.

28. Numerical Simulation of Rinzel’s Simplification 1.Matlab Demo

29. Spike Trains are Limit Cycles Matlab Demo

30. Rinzel’Simplification, Part 2

31. Parameters of Rinzel’s Approximation

32. Rinzel’Simplification, Part 2 Solution: Simplification Rinzel’s Model

33. How does the Neuron Switch From Resting to Spiking? Resting Spiking V in dV R

34. How Stability Changes with the Input Hopf- Bifurcation!

35. Soft or Hard Hopf-Bifurcation? Let’s check this in Matlab! The HH model has a hard Hopf-Bifurcation. An Unstable Limit Cycle emerges.

36. Is there another limit cycle that is stable? Poincaré-Bendixon: + + - - + - - R Yes!

37. Two Limit Cycles Coexist. The stable limit cycle appears while the equilibrium point is still stable, but the unstable limit cycle prevents the trajectory from converging to it.

38. This Week 1.Neurons encode information with non-linear oscillations (spike trains). 2.How do neurons generate spikes? 3. Hodgkin-Huxley Model 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem

39. Hodgkin-Huxley Model Predicts Hysteresis

40. Prediction was verified experimentally Simulation (Matlab Demo) Experiment Predicted: 1965 Verified: 1980

41. This Week 1.Neurons encode information with non-linear oscillations (spike trains). 2.How do neurons generate spikes? 3. Hodgkin-Huxley Model 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem

42. Stochastic Resonance Noise can increase the neuron’s sensitivity.

43. From Squid to Man Squid Axon fires with at least 175 Hz Fast K + current Cortical Neurons can have much lower firing rates! Matlab Demo It is easy to incorporate additional channels into the HH model.

44. Dynamical Properties of Cortical Neurons Saddle-Node Bifurcation Incorporating new channels changes the dynamics.

45. Dynamic Neuron Types There are four major dynamic neuron types.

46. The extended HH model captures FS and RS neurons