circuit models of neurons
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Circuit Models of Neurons. Bo Deng University of Nebraska-Lincoln. Outlines : Hodgkin-Huxley Model Circuit Models --- Elemental Characteristics --- Ion Pump Dynamics Examples of Dynamics --- Bursting Spikes - PowerPoint PPT PresentationTRANSCRIPT
Bo DengUniversity 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
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Vn
Vn
nn
KKK
e
eV
nndtdn
VVngI
Hodgkin-Huxley Model
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NaNaNa
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Hodgkin-Huxley Model
)( LLL VVgI
Hodgkin-Huxley Model
dtdVCIC
-I (t)CThe only mechanistic part ( by Kirchhoff’s Current Law)
+
Hodgkin-Huxley Model --- A Useful Clue
VIdt
dIK
K
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• 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.
H-H Type Models for Excitable Membranes
Our Circuit Models Elemental Characteristics -- Resistor
gVI
Our Circuit Models Elemental Characteristics -- Diffusor
funcitons decreasingboth )(or )( IhVVfI
Our Circuit Models Elemental Characteristics -- Ion Pump
pumpway -onefor
AVdtdA
Dynamics of Ion Pump as Battery Charger
Na
K
' with
0:LawCurrent sKirchhoff’
C
extCApK,pNa,
CVCI
IIIII
Equivalent IV-Characteristics --- for parallel channels
Passive sodium current can be explicitly expressed as
2
Na
NaNa2
12
21NaNaNa
21
NaNa
Na12
11NaNa
NapNa,
)(||2
))((||)('
gintegratin from derived
2 ,
||2with
tantan
)(
m
m
mm
vVg
dgvvvVvVdgVf
vvvdg
gvv
μvvVdVg
VfI
Equivalent IV-Characteristics --- for serial channels
Passive potassium current can be implicitly expressed as
A standard circuit technique to represent the hysteresis is to turn it into a singularly perturbed equation
0
By Ion Pump Characteristics
with substitution and assumption
to get
Equations for Ion Pump
2 ,
||||
2
tantan11)(
and
2 ,
||2
tantan)(
21
KK
K12
11
KKK
21
NaNa
Na12
11NaNaNa
iiidg
dii
iiId
Ig
Ih
vvvdg
gvv
μvvVdVgVf
m
mm
m
mm
I Na = fNa (VC – ENa)
VK = hK (IK,p)
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
C = 0.005
Bursting Spikes
C = 0.005
C = 0.5
Neural ChaosC = 0.5 = 0.05 g = 0.18 e =
0.0005I
in = 0
gK = 0.1515
dK
= -0.1382i1 = 0.14
i2 = 0.52
EK
= - 0.7
gNa
= 1d
Na = - 1.22
v1 = - 0.8
v2 = - 0.1
ENa
= 0.6
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
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.
Metastability and Plasticity
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 2 11 1 1 1
11
1 1 1
11 1 1
11 1 1
2 2 12 2 2 2
12
2 2 2
2
C C Cext Na K KC A
AS C A
SA C A
NaNa Na NaC
C C CNa K KC A
AS C A
S
dV V VC I I f V E I RdtdI
I V IdtdI
I V IdtdI V E h IdtdV V V
C I f V E I RdtdI
I V IdtdI
g
g
e
g
2 2 2
22 2 2
A C A
NaNa Na NaC
I V IdtdI V E h I
dt
g
e
Current between the
nodes
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:
1n
n n n n n nCout inNa K KC A
nn n nAS C A
nn n nSA C A
nn n nNa
Na Na NaC
dVC I I f V E I I
dtdI
I V Idt
dII V I
dtdI
V E h Idt
g
g
e
1 1
1
1
if 1
if 1
if 1
0 if
extn n nout C C
n
n nC Cn nin
I nI V V
nR
V Vn NI R
n N
C=.1 pF C=.7 pF
(x10 pF)
Transmission SpeedC=.01 pFC=.1 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.
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.
• 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.