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: =
𝑄
𝑉
Charging a Capacitor:
𝑑𝑉
𝑑𝑡
1
𝐶
= ⋅𝐼
We model the lipid-bilayer as a capacitor.
Ion Channels as Steerable Resistors
• Conductance 𝑔: =
1
𝑅
Equilibrium Potentials
• Ohm’s law:
(𝑉mem − 𝐸Na )
𝐼𝑁𝑎 =
= 𝑔Na ⋅ (𝑉mem − 𝐸Na )
𝑅Na
Equivalent Circuit  Hodgkin Huxley
𝐸Na = +50mV
𝑉rest = −70mV
E +
+
E
-
+
E -
𝐸K
= −80mV
𝒅𝑽
𝑪𝒎 ⋅
(𝒕) = −𝑰𝑵𝒂 (𝒕) − 𝑰𝑲 (𝒕) − 𝑰𝒍𝒆𝒂𝒌 (𝒕) + 𝑰(𝒕)
𝒅𝒕
𝒅𝑽
𝑪𝒎 ⋅
𝒕 = −𝒈𝑵𝒂 𝒕 𝑽 𝒕 − 𝑬𝑵𝒂 − 𝒈𝑲 𝒕 𝑽 𝒕 − 𝑬𝑲 − 𝒈𝑳 𝒕 𝑽 𝒕 − 𝑬𝑳 + 𝑰𝒊𝒏𝒋
𝒅𝒕
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 𝑽mem.
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
• 𝒈𝐦𝐚𝐱
𝐍𝐚 : 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
𝑔Na 𝑡 = 𝑚3 (𝑡) ⋅ ℎ(𝑡) ⋅ 𝑔Na
Voltage Gated K Channel
• 4 subunits
• No inactivaton gate
𝒏: probability K-channel of subunit to be activated
𝒈𝐦𝐚𝐱
𝑲 : Na-conductance if all sodium channels are open
𝑔K 𝑡 = 𝑛4 (𝑡) ⋅ 𝑔Kmax
From Deactivation to Activation
and Back Again
𝛼(V)
Deactivated
Activated
𝛽(V)
𝑑𝑚
= 𝛼𝑚 ⋅ 1 − 𝑚 − 𝛽𝑚 ⋅ 𝑚
𝑑𝑡
𝑑𝑛
= 𝛼𝑛 ⋅ 1 − 𝑛 − 𝛽𝑛 ⋅ 𝑛
𝑑𝑡
From Deinactivation to Inactivation
and Back Again
𝛼(𝑉)
Deinactivated
Inactivated
𝛽(𝑉)
𝑑ℎ
= 𝛼ℎ (𝑉) ⋅ 1 − ℎ − 𝛽ℎ (𝑉) ⋅ ℎ
𝑑𝑡
The transition probabilities are voltage dependent.
Hodgkin-Huxley Model, Version 1
1.
𝑑𝑉
𝐶𝑚
𝑑𝑡
2.
𝑑𝑚
= 𝛼𝑚 (𝑉) ⋅ 1 − 𝑚 − 𝛽𝑚 (𝑉) ⋅
𝑑𝑡
𝑑ℎ
= 𝛼ℎ (𝑉) ⋅ 1 − ℎ − 𝛽ℎ (𝑉) ⋅ ℎ
𝑑𝑡
𝑑𝑛
= 𝛼𝑛 (𝑉) ⋅ 1 − 𝑛 − 𝛽𝑛 (𝑉) ⋅ 𝑛
𝑑𝑡
3.
4.
max
= 𝐼𝑖𝑛𝑗 − 𝑔𝐿 ⋅ 𝑉 − 𝐸𝐿 − 𝑚3 𝑡 ⋅ ℎ 𝑡 ⋅ 𝑔Na
𝑉 − 𝐸Na −
𝑛4 𝑡 ⋅ 𝑔𝐾max ⋅ 𝑉 − 𝐸𝐾
𝑚
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
𝑑𝑉
max
1. 𝐶𝑚 𝑑𝑡 = 𝐼𝑖𝑛𝑗 − 𝑔𝐿 ⋅ 𝑉 − 𝐸𝐿 − 𝑚3 𝑡 ⋅ ℎ 𝑡 ⋅ 𝑔Na
𝑉 − 𝐸Na
−
2.
3.
4.
𝑛4 𝑡 ⋅ 𝑔𝐾max ⋅ 𝑉 − 𝐸𝐾
𝑑𝑚
𝜏𝑚 ⋅
= (𝑚𝑒𝑞 − 𝑚)
𝑑𝑡
𝑑ℎ
𝜏ℎ ⋅ = (ℎ𝑒𝑞 − ℎ)
𝑑𝑡
𝑑𝑛
𝜏𝑛 ⋅ = (𝑛𝑒𝑞 − 𝑛)
𝑑𝑡
𝛼𝑥 𝑉
𝑥𝑒𝑞 𝑉 =
𝛼𝑥 𝑉 + 𝛽𝑥 𝑉
𝛼𝑥 (𝑉)
𝜏𝑥 (𝑉) =
𝛼𝑥 (𝑉) + 𝛽𝑥 (𝑉)
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:
𝑔𝑁𝑎 𝑡 = 𝑚𝑒𝑞 (𝑉)
2. Na+ inactivation ℎ = 1 − 𝑛 =: 𝑅
Result:
1.
2.
𝑑𝑉
𝑚𝑎𝑥
𝐶 ⋅ = −𝑚𝑒𝑞 𝑉 3 ⋅ 𝑔𝑁𝑎
⋅ 1 − 𝑅 ⋅ 𝑉 − 𝐸𝑁𝑎
𝑑𝑡 𝑚𝑎𝑥
4
𝑅 ⋅ 𝑔𝐾 ⋅ 𝑉 − 𝐸𝐾 − 𝑔𝐿 ⋅ (𝑉 − 𝐸𝐿 ) + 𝐼𝑖𝑛𝑗
𝑑𝑅
1
= (𝑅𝑒𝑞 (𝑉) − 𝑅)
𝑑𝑡
𝜏𝑅
−
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
𝑑𝑉
1. 𝐶 ⋅ 𝑑𝑡 = −𝐼𝑁𝑎 − 𝐼𝐾 + 𝐼𝑖𝑛𝑗
2. Ohm’s law
3. Dependence on 𝐸𝑁𝑎 and 𝐸𝐾
• Ansatz:
𝑑𝑉
1. 𝐶 ⋅ 𝑑𝑡 = −𝑔𝑁𝑎 𝑉 ⋅ (𝑉 − 𝐸𝑁𝑎 ) − 𝑅(𝑉 − 𝐸𝐾 ) + 𝐼𝑖𝑛𝑗
2.
𝑑𝑅
𝑑𝑡
=
1
(𝑅𝑒𝑞 (𝑉)
𝜏𝑅
− 𝑅)
Parameters of Rinzel’s Approximation
Phase Plane, dV/dt = 0 (red), dR/dt = 0 (blue)
1
0.9
• Isocline
0.8
– 𝑅=
0.7
R
= 0:
−𝑔𝑁𝑎 𝑉 𝑉−𝐸𝑁𝑎 +𝐼
(𝑉−𝐸𝐾 )
– looks like a cubic polynomial
 Fit 𝑔𝑁𝑎 (V) by a quadratic
function of V
0.6
0.5
0.4
• Isocline
0.3
𝑑𝑅
𝑑𝑡
= 0:
– 𝑅 = 𝑅𝑒𝑞 (V)
0.2
– looks like a line
 Fit 𝑅𝑒𝑞 (V) by 𝑎0 + 𝑎1 ⋅ 𝑉
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
– 𝑉𝑒𝑞 𝐼 has complex conjugate eigenvalues.
– For 0 ≤ 𝐼 < 𝛽, the real part is negative
– For 𝐼 = 𝛽, the real part is zero
– For 𝐼 > 𝛽, the real part is positive
– Critical Value 𝛽 = 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
𝜏𝑅 = 2.1 ms
𝜏𝑅 = 5.6 ms