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Computational Neuroeconomics and Neuroscience Spring 2011 Action Potentials and Limit Cycles Session 8 on 20.04.2011, presented by Falk Lieder 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 This Week 1. Neurons encode information with non-linear oscillations (spike trains). 2. How do neurons generate spikes? 3. Hodgkin-Huxley Model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model 1. Stimulus Intensity is Encoded by the Frequency of a Nonlinear Oscillation (firing rate) strong stretch medium stretch light stretch Firing rate Stimulus intensity This Week 1. Neurons encode information with non-linear oscillations (spike trains). 2. How do neurons generate spikes? 3. Hodgkin-Huxley Model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model The Leading Characters: Na+ and K+ Session 1: Membrane Potential is determined by equilibrium between drift and diffusion 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 Equivalent Electrical Circuit concentration gradients batteries cell membrane capacitor ion channels steerable resistors source: Kandel ER, Schwartz JH, Jessell TM 2000. Principles of Neural Science, 4th ed. McGraw-Hill, New York., chapter 7 Cell Membrane as a Capacitor out + = Capacitator - Lipid Bilayer in Capacitance C: = Charging a Capacitor: 1 = ⋅ We model the lipid-bilayer as a capacitor. Ion Channels as Steerable Resistors • Conductance : = 1 Equilibrium Potentials • Ohm’s law: (mem − Na ) = = Na ⋅ (mem − Na ) Na Equivalent Circuit Hodgkin Huxley Na = +50mV rest = −70mV E + + E - + E - K = −80mV ⋅ () = − () − () − () + () ⋅ = − − − − − − + 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. The conductances are voltagedependent! Hodgkin 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). The fraction of open channels changes with mem. Huxley How do conductances change and why? AP animation Conductances change by voltage-dependent (de)activation and (de)inactivation. This Week 1. Neurons encode information with non-linear oscillations (spike trains). 2. How do neurons generate spikes? 3. Hodgkin-Huxley Model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model Voltage Gated Na-Channel Deactivated • : Na-conductance if all sodium channels are open • h: probability of the inactivation gate to be open Inactivated (De)Activated State= x (De)Inactivated The Na+ channel is open if Activation and Inactivation Gate are open. Voltage Gated Na-Channel Activation Gate m: probability of one Na channel subunit to be activated max Na = 3 () ⋅ ℎ() ⋅ Na Voltage Gated K Channel • 4 subunits • No inactivaton gate : probability K-channel of subunit to be activated : Na-conductance if all sodium channels are open K = 4 () ⋅ Kmax From Deactivation to Activation and Back Again (V) Deactivated Activated (V) = ⋅ 1 − − ⋅ = ⋅ 1 − − ⋅ From Deinactivation to Inactivation and Back Again () Deinactivated Inactivated () ℎ = ℎ () ⋅ 1 − ℎ − ℎ () ⋅ ℎ The transition probabilities are voltage dependent. Hodgkin-Huxley Model, Version 1 1. 2. = () ⋅ 1 − − () ⋅ ℎ = ℎ () ⋅ 1 − ℎ − ℎ () ⋅ ℎ = () ⋅ 1 − − () ⋅ 3. 4. max = − ⋅ − − 3 ⋅ ℎ ⋅ Na − Na − 4 ⋅ max ⋅ − 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. Hodgkin-Huxley Model, Version 2 max 1. = − ⋅ − − 3 ⋅ ℎ ⋅ Na − Na − 2. 3. 4. 4 ⋅ max ⋅ − ⋅ = ( − ) ℎ ℎ ⋅ = (ℎ − ℎ) ⋅ = ( − ) = + () () = () + () Na-activation, K-activation, and Na-inactivation converge to their voltagedependent equilibrium values at voltage-dependent speeds. 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 Numerical Simulation of HH • Matlab Demo Hodgkin-Huxley Model, Membrane Potential 30 20 10 0 mV -10 -20 -30 -40 -50 -60 -70 10 20 30 40 ms 50 60 70 80 This Week 1. Neurons encode information with non-linear oscillations (spike trains). 2. How do neurons generate spikes? 3. Hodgkin-Huxley Model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model Rinzel’s simplification of the HH model Simplifications: 1. Na+ activation jumps to its equilibrium: = () 2. Na+ inactivation ℎ = 1 − =: Result: 1. 2. ⋅ = − 3 ⋅ ⋅ 1 − ⋅ − 4 ⋅ ⋅ − − ⋅ ( − ) + 1 = ( () − ) − Rinzel simplified the 4-dimensional HH model into a 2-dimensional model. We can use the mathematical tools available for the 2-dimensional case. Numerical Simulation of Rinzel’s Simplification 1. Matlab Demo Hodgkin-Huxley Model, Membrane Potential Rinzel Approximation to Hodgkin-Huxley 30 50 20 10 0 0 V(t) mV -10 -20 -30 -50 -40 -50 -60 -70 -100 0 2 4 6 8 10 12 Time (ms) 14 16 18 20 10 20 30 40 ms 50 60 70 80 Spike Trains are Limit Cycles 0.6 0.5 Matlab Demo 0.4 0.3 0.2 0.1 0.8 0.6 0.4 0.2 0.0 0.2 0.4 Rinzel’Simplification, Part 2 • Goal: – As simple as possible, but retain 1. ⋅ = − − + 2. Ohm’s law 3. Dependence on and • Ansatz: 1. ⋅ = − ⋅ ( − ) − ( − ) + 2. = 1 ( () − ) Parameters of Rinzel’s Approximation Phase Plane, dV/dt = 0 (red), dR/dt = 0 (blue) 1 0.9 • Isocline 0.8 – = 0.7 R = 0: − − + (− ) – looks like a cubic polynomial Fit (V) by a quadratic function of V 0.6 0.5 0.4 • Isocline 0.3 = 0: – = (V) 0.2 – looks like a line Fit (V) by 0 + 1 ⋅ 0.1 0 -100 dV dt -80 -60 -40 -20 V 0 20 40 60 Rinzel’Simplification, Part 2 • Solution: Rinzel’s Model Simplification Rinzel Approximation to Hodgkin-Huxley 60 50 40 20 0 V(t) 0 -20 -50 -40 -60 -80 0 2 4 6 8 10 12 14 16 18 20 -100 0 2 4 6 8 10 12 14 16 18 20 How does the Neuron Switch From Resting to Spiking? Spiking Resting 0.12 0.6 R 0.5 0.11 0.4 0.10 0.3 0.09 0.2 0.08 0.1 0.74 0.72 0.70 V in dV 0.68 0.66 0.8 0.6 0.4 0.2 V in dV 0.0 0.2 0.4 How Stability Changes with the Input • Matlab Demo – has complex conjugate eigenvalues. – For 0 ≤ < , the real part is negative – For = , the real part is zero – For > , the real part is positive – Critical Value = 0.07797 HopfBifurcation! Soft or Hard Hopf-Bifurcation? Let’s check this in Matlab! The HH model has a hard Hopf-Bifurcation. An Unstable Limit Cycle emerges. Is there another limit cycle that is stable? 1.0 0.8 + - + 0.6 -- Poincaré-Bendixon: R - 0.4 + Yes! 0.2 0.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 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. This Week 1. Neurons encode information with non-linear oscillations (spike trains). 2. How do neurons generate spikes? 3. Hodgkin-Huxley Model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model Hodgkin-Huxley Model Predicts Hysteresis Prediction was verified experimentally Simulation Experiment (Matlab Demo) V(t) (red) & Current Ramp (blue) 40 20 0 -20 -40 -60 -80 -100 0 50 100 150 200 250 300 Time (ms) Predicted: 1965 Verified: 1980 This Week 1. Neurons encode information with non-linear oscillations (spike trains). 2. How do neurons generate spikes? 3. Hodgkin-Huxley Model Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem 4. Hodgkin-Huxley neuron has stable limit cycle 5. Physiological Predictions of HH-model 6. Extensions of the HH-model Stochastic Resonance Noise can increase the neuron’s sensitivity. From Squid to Man • Squid Axon fires with at least 175 Hz Cortical Neurons can have much lower firing rates! Fast K+ current Matlab Demo It is easy to incorporate additional channels into the HH model. Dynamical Properties of Cortical Neurons • Saddle-Node Bifurcation Incorporating new channels changes the dynamics. Dynamic Neuron Types There are four major dynamic neuron types. The extended HH model captures FS and RS neurons = 2.1 ms = 5.6 ms