Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session...

Action Potentials and Limit Cycles

Computational Neuroeconomics and NeuroscienceSpring 2011

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 Week1. 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 cycle5. Physiological Predictions of HH-model

6. Extensions of the HH-model

Hopf-Bifircation Theorem, Poincaré-Bendixon Theorem

1. Stimulus Intensity is Encoded by the Frequency of a Nonlinear Oscillation (firing rate)

Stimulus intensity

strong stretch

medium stretch

light stretch

Firing rate

The Leading Characters: Na+ and K+

Session 1: Membrane Potential is determined by equilibrium between drift and diffusion

Neurons’ Active PropertiesSession 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

cell membrane capacitor

concentration gradients batteries

source: Kandel ER, Schwartz JH, Jessell TM 2000. Principles of Neural Science, 4th ed. McGraw-Hill, New York., chapter 7

ion channels steerable resistors

Cell Membrane as a Capacitor

Capacitance Charging a Capacitor:

+-

in

out

Lipid Bilayer Capacitator=

We model the lipid-bilayer as a capacitor.

Ion Channels as Steerable Resistors

• Conductance • Ohm’s law:

Equilibrium Potentials

Equivalent Circuit Hodgkin Huxley

𝑪𝒎 ⋅𝒅𝑽𝒅𝒕

(𝒕 )=− 𝑰𝑵𝒂(𝒕 )− 𝑰 𝑲 (𝒕 )− 𝑰 𝒍𝒆𝒂𝒌(𝒕)+ 𝑰 (𝒕 )

E E E+

-

+-

-

+

𝑪𝒎 ⋅𝒅𝑽𝒅𝒕

(𝒕 )=−𝒈𝑵𝒂 (𝒕 ) (𝑽 (𝒕 )−𝑬𝑵𝒂 )−𝒈𝑲 (𝒕 ) (𝑽 ( 𝒕 )−𝑬𝑲 )−𝒈𝑳 (𝒕 ) (𝑽 (𝒕 )−𝑬𝑳 )+ 𝑰 𝒊𝒏𝒋

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 voltage-dependent!

The fraction of open channels changes with .

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).

How do conductances change and why?

AP animation

Conductances change by voltage-dependent (de)activation and (de)inactivation.

http://www.blackwellpublishing.com/matthews/channel.html

http://www.blackwellpublishing.com/matthews/channel.html

Voltage Gated Na-Channel

• : Na-conductance if all sodium channels are open• h: probability of the inactivation gate to be open

The Na+ channel is open if Activation and Inactivation Gate are open.

Deactivated Inactivated

(De)Activated x (De)Inactivated

State=

Voltage Gated Na-Channel

m: probability of one Na channel subunit to be activated

Activation Gate

𝑔Na (𝑡 )=𝑚3(𝑡)⋅ h(𝑡)⋅𝑔Namax

Voltage Gated K Channel

• 4 subunits• No inactivaton gate

𝑔K (𝑡 )=𝑛4(𝑡)⋅𝑔Kmax

: probability K-channel of subunit to be activated: Na-conductance if all sodium channels are open

From Deactivation to Activation and Back Again

(V)

(V)

Deactivated Activated

𝑑𝑛𝑑𝑡

=𝛼𝑛 ⋅ (1−𝑛 )− 𝛽𝑛 ⋅𝑛𝑑𝑚𝑑𝑡

=𝛼𝑚 ⋅ (1−𝑚 )− 𝛽𝑚 ⋅𝑚

From Deinactivation to Inactivation and Back Again

h𝑑𝑑𝑡

=𝛼h(𝑉 )⋅ (1−h )− 𝛽h(𝑉 )⋅ h

𝛼 (𝑉 )

𝛽 (𝑉 )

Deinactivated Inactivated

The transition probabilities are voltage dependent.

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.

Hodgkin-Huxley Model, Version 2

𝑥𝑒𝑞 (𝑉 )=𝛼𝑥 (𝑉 )

𝛼𝑥 (𝑉 )+𝛽𝑥 (𝑉 )

Na-activation, K-activation, and Na-inactivation converge to their voltage-dependent equilibrium values at voltage-dependent speeds.

Problem: HH model is too complex to analyse mathemtically

• Possible Solutions:1. Numerical Simulation2. 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

10 20 30 40 50 60 70 80

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

ms

mV

Hodgkin-Huxley Model, Membrane Potential

Rinzel’s simplification of the HH model

Simplifications:1. Na+ activation jumps to its equilibrium: 2. Na+ inactivation

Result:

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

0 2 4 6 8 10 12 14 16 18 20-100

-50

0

50

Time (ms)

V(t

)

Rinzel Approximation to Hodgkin-Huxley

10 20 30 40 50 60 70 80

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

ms

mV

Hodgkin-Huxley Model, Membrane Potential

Spike Trains are Limit Cycles

Matlab Demo

0.8 0.6 0.4 0.2 0.0 0.2 0.4

0.1

0.2

0.3

0.4

0.5

0.6

Rinzel’Simplification, Part 2

• Goal:– As simple as possible, but retain

1. Ohm’s law2. Dependence on and

• Ansatz:

Parameters of Rinzel’s Approximation

• Isocline – – looks like a cubic polynomial Fit (V) by a quadratic function of V

• Isocline – (V) – looks like a line Fit (V) by

-100 -80 -60 -40 -20 0 20 40 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V

R

Phase Plane, dV/dt = 0 (red), dR/dt = 0 (blue)

Rinzel’Simplification, Part 2• Solution:

Simplification

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

0 2 4 6 8 10 12 14 16 18 20-100

-50

0

50

Time (ms)

V(t

)

Rinzel Approximation to Hodgkin-Huxley

Rinzel’s Model

flieder

The Units don't match. Correct this mistake!

How does the Neuron Switch From Resting to Spiking?

Resting Spiking

0.74 0.72 0.70 0.68 0.66

0.08

0.09

0.10

0.11

0.12

V in dV

R

0.8 0.6 0.4 0.2 0.0 0.2 0.4

0.1

0.2

0.3

0.4

0.5

0.6

V in dV

How Stability Changes with the Input

• Matlab Demo– has complex conjugate eigenvalues.– For , the real part is negative– For , the real part is zero– For , the real part is positive– Critical Value

Hopf-Bifurcation!

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?

Poincaré-Bendixon:+ +- -

+

-

-

0.8 0.6 0.4 0.2 0.0 0.2 0.4

0.0

0.2

0.4

0.6

0.8

1.0

RYes!

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.

Hodgkin-Huxley Model Predicts Hysteresis

Prediction was verified experimentally

Simulation (Matlab Demo) Experiment

Predicted: 1965 Verified: 1980

0 50 100 150 200 250 300-100

-80

-60

-40

-20

0

20

40

Time (ms)

V(t

) (r

ed

) &

Cu

rre

nt R

am

p (

blu

e)

Stochastic Resonance

Noise can increase the neuron’s sensitivity.

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.

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

𝜏 𝑅=2.1ms 𝜏 𝑅=5.6 ms

Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session...

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