1. 9.8 Neural Excitability and Oscillations
Joon Shik Kim
BI study group
(c) SNU CSE Biointelligence Lab,

2. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
Quiz
The development of Hodgkin-Huxley model was preceded by detailed studies of electric current through membrane of the ( ) giant axon.
(c) SNU CSE Biointelligence Lab,

3. 9.8.1 The Hodgkin-Huxley Equations
The basic elements of the Hodgkin-Huxley model are a set of ionic currents separated into contributions from sodium ions (INa), potassium ions (IK), and the leakage of other ionic species such as chlorine (IL).
Ionic conductance: gNa, gK, and gL, where conductance is the inverse of the resistance R.
Ohm’s law,
(c) SNU CSE Biointelligence Lab,

4. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
Faraday’s law,
Resulting circuit equation,
, where I is the applied current.
(c) SNU CSE Biointelligence Lab,

5. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
Sodium channel: fast m-gate and slow h-gate
m-gate is closed and h-gate is open at rest.
m-gate opens quickly when the membrane is depolarized allowing for the entry of sodium.
The h-gate then closes slowly to block the sodium inflow.
(c) SNU CSE Biointelligence Lab,

6. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
The potassium channel is controlled by n-gate. This gate is closed at rest. It opens after a delay in response to depolarization to permit the outflow of potassium.
, where m3h may be the fraction of open sodium channel.
(c) SNU CSE Biointelligence Lab,

7. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
,where n4 is the number
of open potassium
channels.
(c) SNU CSE Biointelligence Lab,

8. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
9.8.2 The Moris-Lecar Model
FitzHugh simplified the dynamics by noting that the variables V and m change more rapidly than h and n.
FitzHugh analyzed the dynamics further inspired by in part by the van der Pol oscillator.
For large stimulating currents, firing of an action potential and limit cycle behavior appear.
Morris-Lecar model: a hybrid of the Hodgin-Huxley and FitzHugh-Nagumo approaches
(c) SNU CSE Biointelligence Lab,

9. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
, where M and N are the fractions of open Ca2+ and K+ channels.
(c) SNU CSE Biointelligence Lab,

10. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
Ca2+ system operates on a much faster time scale than the K+.
The nullclines:
(c) SNU CSE Biointelligence Lab,

11. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr

12. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
The eigenvalues λthat describe the stability properties of the linearized system are solutions of the characteristic equation, Eq. (9.54), with the coefficient b and c given by
The condition for limit cycle oscillations is that b > 0 and c > 0.
(c) SNU CSE Biointelligence Lab,

13. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
The condition that must be satisfied by the conductance parameters in other to generate oscillatory behavior:
(c) SNU CSE Biointelligence Lab,

14. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr

15. 9.8.3 Waves and Synchrony in systems of Relaxation Oscillators
(c) SNU CSE Biointelligence Lab,

16. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
Relaxation oscillators employ a method of synchronization called fast threshold modulation
We model the external input excitatory current to the ith cell in the array through a nearest-neighbor coupling term:
(c) SNU CSE Biointelligence Lab,

17. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr

18. (c) 2000-2002 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr
A sufficient condition for synchronization is that the rate of change in the slow variable before the jump is less that that following the jump. This variation in the rate of change of the slow variable gives rise to a scalloped appearance when plotting the slow variable as a function of time.
(c) SNU CSE Biointelligence Lab,