Bo Deng
University of Nebraska-Lincoln
Sept. 2004
Rinzel & Wang (1997)
Gated Currents by Electrodiffission
Ionic Pumps by Chemical Energy
• Kandel, E.R., J.H. Schwartz, and T.M. Jessell
Principles of Neural Science, 3rd ed., Elsevier, 1991.
• Zigmond, M.J., F.E. Bloom, S.C. Landis, J.L. Roberts, and L.R. Squire
Fundamental Neuroscience, Academic Press, 1999.
Circuit Model
Alan Lloyd Hodgkin
Andrew Fielding Huxley
Hodgkin, A.L. and A.F. Huxley, A quantitative description of membrane current
and its application to conduction and excitation in nerve, J. Physiol. 117(1952),
pp.500--544.
Kirchhoff Laws
- I (t)
-I (t)
Mahowald, M. and Douglas, R.
A silicon neuron. Nature, 354(1991), pp.515-518.
Electrophysiological
Model
Equivalent
Circuit
dI L
L
  RI L  V
dt
C dV  I L  I
dt
g (V  Em , I )  0
Chua, L.O., Introduction to Nonlinear Circuit Theory, McGraw-Hill, New York, 1969.
dI L
L
  RI L  V
dt
dV
C
 IL  I
dt
g (V  Em , I )  0
Balthazar van der Pol (1889-1959)
van der Pol Oscillator (R = 0)
and
FitzHugh-Nagumo Oscillator
van der Pol (1928),
FitzHugh(1961), Nagumo(1964)
Keener(1982)
g (V  Em , I )  0
dI L
L
  RI L  V
dt
dV
C
 IL  I
dt
g (V  Em , I )  0
g (V  Em , I )  0
V
IL
I
(Non-circuit) Models for Square Burster and Other Bursters
• Morris, C. and H. Lecar,
Voltage oscillations in the barnacle giant muscle fiber,
Biophysical J., 35(1981), pp.193--213.
Deng(1991)
• Hindmarsh, J.L. and R.M. Rose,
A model of neuronal bursting using three coupled first order differential
equations,
Proc. R. Soc. Lond. B. 221(1984), pp.87--102.
• Chay, T.R., Y.S. Fan, and Y.S. Lee
Bursting, spiking, chaos, fractals, and universality in biological
rhythms, Int. J. Bif. & Chaos, 5(1995), pp.595--635.
• Izhikevich, E.M
Neural excitability, spiking, and bursting,
Int. J. Bif. & Chaos, 10(2000), pp.1171--1266.
(also see his article in SIAM Review)
Disclaimer: With the exception of
the square burster and SEED simulation,
all artworks are found from the internet.