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![Page 1: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/1.jpg)
Action Potentials and Limit Cycles
Computational Neuroeconomics and NeuroscienceSpring 2011
Session 8 on 20.04.2011, presented by Falk Lieder
![Page 2: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/2.jpg)
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
![Page 3: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/3.jpg)
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
![Page 4: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/4.jpg)
1. Stimulus Intensity is Encoded by the Frequency of a Nonlinear Oscillation (firing rate)
Stimulus intensity
strong stretch
medium stretch
light stretch
Firing rate
![Page 5: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/5.jpg)
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
![Page 6: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/6.jpg)
The Leading Characters: Na+ and K+
Session 1: Membrane Potential is determined by equilibrium between drift and diffusion
![Page 7: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/7.jpg)
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
![Page 8: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/8.jpg)
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
![Page 9: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/9.jpg)
Cell Membrane as a Capacitor
Capacitance Charging a Capacitor:
+-
in
out
Lipid Bilayer Capacitator=
We model the lipid-bilayer as a capacitor.
![Page 10: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/10.jpg)
Ion Channels as Steerable Resistors
• Conductance • Ohm’s law:
Equilibrium Potentials
![Page 11: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/11.jpg)
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
![Page 12: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/12.jpg)
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.
![Page 13: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/13.jpg)
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).
![Page 14: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/14.jpg)
How do conductances change and why?
AP animation
Conductances change by voltage-dependent (de)activation and (de)inactivation.
![Page 15: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/15.jpg)
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
![Page 16: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/16.jpg)
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=
![Page 17: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/17.jpg)
Voltage Gated Na-Channel
m: probability of one Na channel subunit to be activated
Activation Gate
𝑔Na (𝑡 )=𝑚3(𝑡)⋅ h(𝑡)⋅𝑔Namax
![Page 18: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/18.jpg)
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
![Page 19: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/19.jpg)
From Deactivation to Activation and Back Again
(V)
(V)
Deactivated Activated
𝑑𝑛𝑑𝑡
=𝛼𝑛 ⋅ (1−𝑛 )− 𝛽𝑛 ⋅𝑛𝑑𝑚𝑑𝑡
=𝛼𝑚 ⋅ (1−𝑚 )− 𝛽𝑚 ⋅𝑚
![Page 20: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/20.jpg)
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.
![Page 21: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/21.jpg)
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.
![Page 22: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/22.jpg)
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.
![Page 23: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/23.jpg)
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
![Page 24: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/24.jpg)
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
![Page 25: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/25.jpg)
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
![Page 26: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/26.jpg)
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.
![Page 27: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/27.jpg)
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
![Page 28: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/28.jpg)
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
![Page 29: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/29.jpg)
Rinzel’Simplification, Part 2
• Goal:– As simple as possible, but retain
1. Ohm’s law2. Dependence on and
• Ansatz:
![Page 30: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/30.jpg)
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)
![Page 31: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/31.jpg)
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
![Page 32: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/32.jpg)
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
![Page 33: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/33.jpg)
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!
![Page 34: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/34.jpg)
Soft or Hard Hopf-Bifurcation?
Let’s check this in Matlab!The HH model has a hard Hopf-Bifurcation. An Unstable Limit Cycle emerges.
![Page 35: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/35.jpg)
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!
![Page 36: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/36.jpg)
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.
![Page 37: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/37.jpg)
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
![Page 38: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/38.jpg)
Hodgkin-Huxley Model Predicts Hysteresis
![Page 39: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/39.jpg)
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)
![Page 40: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/40.jpg)
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
![Page 41: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/41.jpg)
Stochastic Resonance
Noise can increase the neuron’s sensitivity.
![Page 42: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/42.jpg)
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.
![Page 43: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/43.jpg)
Dynamical Properties of Cortical Neurons
• Saddle-Node Bifurcation
Incorporating new channels changes the dynamics.
![Page 44: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/44.jpg)
Dynamic Neuron Types
There are four major dynamic neuron types.
![Page 45: Action Potentials and Limit Cycles Computational Neuroeconomics and Neuroscience Spring 2011 Session 8 on 20.04.2011, presented by Falk Lieder.](https://reader031.fdocuments.us/reader031/viewer/2022032702/56649cee5503460f949bb6a1/html5/thumbnails/45.jpg)
The extended HH model captures FS and RS neurons
𝜏 𝑅=2.1ms 𝜏 𝑅=5.6 ms