Bo Deng
University of Nebraska-Lincoln
Outlines:
 Hodgkin-Huxley Model
 Circuit Models
--- Elemental Characteristics
--- Ion Pump Dynamics
 Examples of Dynamics
--- Bursting Spikes
--- Metastability and Plasticity
--- Chaos
--- Signal Transduction
AMS Regional Meeting at KU 03-30-12
Hodgkin-Huxley Model
(1952)
Pros:
 The first system-wide model for excitable membranes.
 Mimics experimental data.
 Part of a Nobel Prize work.
 Fueled the theoretical neurosciences for the last
60 years and counting.
Hodgkin-Huxley Model
(1952)
Pros:
 The first system-wide model for excitable membranes.
 Mimics experimental data.
 Part of a Nobel Prize work.
 Fueled the theoretical neurosciences for the last
60 years and counting.
Cons:
 It is not entirely mechanistic but phenomenological.
 Different, ad hoc, models can mimic the same data.
 It is ugly.
 Fueled the theoretical neurosciences for the last
60 years and counting.
Hodgkin-Huxley Model --- Passive vs. Active Channels
I K   g K n 4 (V  VK )
dn
  n (1  n)   n n
dt
10  V
n 
100(e1V /10  1)
 n  0.125e V / 80
Hodgkin-Huxley Model
I Na   g Na m 3 h(V  VNa )
dm
  m (1  m)   m m
dt
dh
  h (1  h)   h h
dt
25  V
m 
,  m  4e V /18
2.5 V / 10
10(e
 1)
1
 h  0.07e V / 20 ,  h  3V /10
e
1
Hodgkin-Huxley Model
I L  g L (V  VL )
Hodgkin-Huxley Model
dV
IC  C
dt
C
-I (t)
The only mechanistic part
( by Kirchhoff’s Current Law)
+
Hodgkin-Huxley Model --- A Useful Clue
 Each current is an aggregate of both passive and active channels
 The change of the current depends on some sort of product of the
current and the voltage, which is the power of the channel due to
biochemical to mechanical energy conversion :
dI K 
dt
 I K V
I K   g K n 4 (V  VK )
dn
  n (1  n)   n n
dt
10  V
n 
100(e1V /10  1)
 n  0.125e V / 80
H-H Type Models for Excitable Membranes
• Morris, C. and H. Lecar,
Voltage oscillations in the barnacle giant muscle fiber,
Biophysical J., 35(1981), pp.193--213.
• 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.
Our Circuit Models
 Elemental Characteristics -- Resistor
I  gV
Our Circuit Models
 Elemental Characteristics -- Diffusor
I  f (V ) or V  h( I )
both decreasing funcitons
Our Circuit Models
 Elemental Characteristics -- Ion Pump
dA
 AV
dt
for one - way pump
Dynamics of Ion Pump as Battery Charger
K
Na 
Kirchhoff’ s Current Law :
I Na,p  I K,p  I A  I C  I ext  0
with I C  C VC '
Equivalent IV-Characteristics
--- for parallel channels
I Na,p  f Na (V )

V  vm
v 
 g NaV  d Na   tan 1
 tan 1 m 

μ

with  
g Na
v2  v1
v v
, vm  1 2
2
| g Na  d Na |
2
derived from integrating
f Na ' (V ) 
| g Na  d Na | (V  v1 )(V  v2 )
 v2  v1  | g Na  d Na |
(V  vm ) 2

 
g Na
 2 
2
Passive sodium current can be explicitly expressed as
Equivalent IV-Characteristics
--- for serial channels
Passive potassium current can be implicitly expressed as
0
A standard circuit technique to represent the hysteresis is to
turn it into a singularly perturbed equation
Equations for Ion Pump
 By Ion Pump Characteristics
 with substitution and assumption
 to get

V  vm
v 
f Na (V )  g NaV  d Na   tan 1
 tan 1 m 

μ


g Na
v2  v1
v v
, vm  1 2
2
| g Na  d Na |
2
and
hK ( I ) 
i 
1
1  1 I  im
I
  tan
 tan 1 m 
gK
dK 



i2  i1
| dK |
i i
, im  1 2
2
| gK  dK |
2
VK = hK (IK,p)
I Na = fNa (VC – ENa)
Examples of Dynamics
---------
Bursting Spikes
Chaotic Shilnikov Attractor
Metastability & Plasticity
Signal Transduction
Geometric Method of Singular Perturbation
Small Parameters:
 0 < e << 1 with ideal
hysteresis at e = 0
 both C and  have
independent time scales
Bursting Spikes
C = 0.005
Neural Chaos
C = 0.005
gNa = 1
dNa = - 1.22
v1 = - 0.8
v2 = - 0.1
ENa = 0.6
gK = 0.1515
dK = -0.1382
i1 = 0.14
i2 = 0.52
EK = - 0.7
C = 0.5
C = 0.5
 = 0.05
g = 0.18
e = 0.0005
Iin = 0
Griffith et. al. 2009
Metastability and Plasticity
Terminology:
 A transient state which behaves like a steady state is
referred to as metastable.
 A system which can switch from one metastable state
to another metastable state is referred to as plastic.
Metastability and Plasticity
Metastability and Plasticity
Metastability and Plasticity
 All plastic and metastable states are lost with only one
ion pump. I.e. when ANa = 0 or AK = 0 we have either
Is = IA or Is = -IA and the two ion pump equations are
reduced to one equation, leaving the phase space one
dimension short for the coexistence of multispike burst
or periodic orbit attractors.
 With two ion pumps, all neuronal dynamics run on
transients, which represents a paradigm shift from basing
neuronal dynamics on asymptotic properties, which can
be a pathological trap for normal physiological functions.
Saltatory Conduction along
Myelinated Axon with Multiple Nodes
Inside the cell
Outside the cell
Joint work with undergraduate and graduate students: Suzan Coons, Noah
Weis, Adrienne Amador, Tyler Takeshita, Brittney Hinds, and Kaelly Simpson
Coupled Equations for Neighboring Nodes
• Couple the nodes by
adding a linear resistor
between them

1
VC2 VC1
 dVC
 1

1
1
1
 Iext  I Na  f K VC  EK   I A 
C
R1


dt

 1
 dI


 A   I 1 V 1  g I 1 
S C
A
 dt
 1
 dI S
  I 1A VC1  g I 1A 



 dt

1
 dI Na
 VC1  E1Na  hNa  I 1Na 
e



dt

 dV 2
VC2 VC1
 2

2
2
2
C
C
  I Na  f K VC  EK   I A 

R1


dt

 dI 2
 A   I 2 V 2  g I 2 
S  C
A 
 dt

 dI 2
 S   I 2 V 2  g I 2 
A  C
A 
 dt

 dI 2
Na  V 2  E 2  h  I 2 
e
Na
Na  Na 
C

dt
The General Case for N Nodes
 This is the general
equation for the nth
node
 In and out currents are
derived in a similar
manner:

















dVCn
n1  I n  f V n  E n   I n  I n
C
 Iout
in
Na
K  C
K 
A
dt
dI An
  I Sn VCn  g I An 


dt
dI Sn
  I An VCn  g I An 


dt
n
dI Na
n h In 
e
 VCn  ENa
Na  Na 
dt
 I
 ext
n

1
Iout  V n V n1
C
 C
n

1

R

 n1
n
VC  VC
Iinn   Rn

0
if n  1
if n  1
if 1  n  N
if n  N
C=.1 pF
(x10 pF)
C=.7 pF
Transmission Speed
C=.1 pF
C=.01 pF
Closing Remarks:
 The circuit models can be further improved by dropping the
serial connectivity assumption of the passive electrical
and diffusive currents.
 Existence of chaotic attractors can be rigorously proved,
including junction-fold, Shilnikov, and canard attractors.
 Can be easily fitted to experimental data.
 Can be used to build real circuits.
• 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.
References:
 [BD] A Conceptual Circuit Model of Neuron, Journal of Integrative
Neuroscience, 8(2009), pp.255-297.
 Metastability and Plasticity of Conceptual Circuit Models of Neurons,
Journal of Integrative Neuroscience, 9(2010), pp.31-47.