Date post: | 28-Dec-2015 |
Category: |
Documents |
Upload: | hugo-mckenzie |
View: | 216 times |
Download: | 1 times |
Bo DengUniversity of Nebraska-Lincoln
Topics: Circuit Basics Circuit Models of Neurons --- FitzHuge-Nagumo Equations --- Hodgkin-Huxley Model --- Our Models Examples of Dynamics --- Bursting Spikes --- Metastability and Plasticity --- Chaos --- Signal Transduction
Topics: Circuit Basics Circuit Models of Neurons --- FitzHuge-Nagumo Equations --- Hodgkin-Huxley Model --- Our Models Examples of Dynamics --- Bursting Spikes --- Metastability and Plasticity --- Chaos --- Signal Transduction
Joint work with undergraduate and graduate students: Suzan Coons, Noah Weis, Adrienne Amador, Tyler Takeshita, Brittney Hinds, and Kaelly Simpson
Circuit Basics
Q = Q(t) denotes the net positive charge at a point of a circuit. I = dQ(t)/dt defines the current through a point. V = V(t) denotes the voltage across the point.
Analysis Convention: When discussing current, we first assign a reference direction for the current I of each device. Then we have: I > 0 implies Q flows in the reference direction. I < 0 implies Q flows opposite the reference direction.
Analysis Convention: When discussing current, we first assign a reference direction for the current I of each device. Then we have: I > 0 implies Q flows in the reference direction. I < 0 implies Q flows opposite the reference direction.
Capacitors
A capacitor is a device that stores energy in an electric potential field.Q
Review of Elementary Components
Inductors
An inductor is a device that stores (kinetic) energy in a magnetic field.dI/dt
2
2
Inductor
dt
QdL
dt
dILV
Resistors
A resistor is an energy converting device. Two Types:
Linear Obeying Ohm’s Law: V=RI, where R is resistance. Equivalently, I=GV with G = 1/R the conductance.
Variable Having the IV – characteristic constrained by an equation g (V, I )=0.
I
V
g (V, I )=0
Kirchhoff’s Voltage Law
The directed sum of electrical potential differences around a circuit loop is 0. To apply this law: 1) Choose the orientation of the loop.2) Sum the voltages to zero (“+” if its current is of the same direction as the orientation and “-” if current is opposite the orientation).
Kirchhoff’s Current Law
The directed sum of the currents flowing into a point is zero. To apply this law: 1) Choose the directions of the current branches.2) Sum the currents to zero (“+” if a current points toward the point and “-” if it points away from the point).
Example By Kirchhoff’s Voltage Law
with Device Relationships
and substitution to get
or
Circuit Models of Neurons
I = F(V)
10 C
Excitable Membranes
Neuroscience: 3ed
• 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.
• 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.
Kirchhoff’s Current LawKirchhoff’s Current Law - I (t)
Hodgkin-Huxley Model
-I (t)
• 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.
• Izhikevich, E.M Neural excitability, spiking, and bursting, Int. J. Bif. & Chaos, 10(2000), pp.1171--1266. (also see his article in SIAM Review)
(Non-circuit) Models for Excitable Membranes
Our Circuit Models
By Ion Pump Characteristics
with substitution and assumption
to get
Equations for Ion Pumps
Dynamics of Ion Pump as Battery Charger
Equivalent IV-Characteristics --- for parallel sodium channels
Passive sodium current can be explicitly expressed as
Passive sodium current can be explicitly expressed as
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
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
Equivalent IV-Characteristics --- for serial potassium channels
0
Examples of Dynamics
--- Bursting Spikes --- Metastability & Plasticity --- Chaotic Shilnikov Attractor --- Signal Transduction
--- Bursting Spikes --- Metastability & Plasticity --- Chaotic Shilnikov Attractor --- Signal Transduction
Geometric Method of Singular Perturbation
Small Parameters: 0 < << 1 with ideal hysteresis at = 0 both C and have independent time scales
C = 0.005
Rinzel & Wang (1997)Rinzel & Wang (1997)
Bursting Spikes
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.
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
C = 0.005
C = 0.5
Neural ChaosC = 0.5 = 0.05 = 0.18 = 0.0005I
in = 0
gK = 0.1515
dK
= -0.1382
i1 = 0.14
i2 = 0.52
EK
= - 0.7
gNa
= 1
dNa
= - 1.22
v1 = - 0.8
v2 = - 0.1
ENa
= 0.6
Myelinated Axon with Multiple Nodes
Inside the cell
Outside the cell
Signal Transduction along Axons
Neuroscience: 3ed
Neuroscience: 3ed
Neuroscience: 3ed
Circuit Equations of Individual Node
Cext Na K KC A
A S C A
S A C A
Na Na Na NaC
dVC I I f V E I
dtI I V I
I I V I
I V E h I
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
Rdt
dII V I
dtdI
I V IdtdI
V E h Idt
dV V VC I f V E I
Rdt
dII V I
dtdI
2 2 2
22 2 2
A C A
NaNa Na NaC
I V IdtdI
V E h Idt
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
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)
C=.7 pF
Transmission Speed
C=.01 pFC=.1 pF
Closing Remarks:
The circuit models can be further improved by dropping the serial connectivity of the passive electrical and diffusive currents. Existence of chaotic attractors can be rigorously proved, including junction-fold, Shilnikov, and canard attractors.
Can be fitted to experimental data.
Can be used to form neural networks.
Closing Remarks:
The circuit models can be further improved by dropping the serial connectivity of the passive electrical and diffusive currents. Existence of chaotic attractors can be rigorously proved, including junction-fold, Shilnikov, and canard attractors.
Can be fitted to experimental data.
Can be used to form neural networks.
References: A Conceptual Circuit Model of Neuron, Journal of Integrative Neuroscience, 2009. Metastability and Plasticity of Conceptual Circuit Models of Neurons, Journal of Integrative Neuroscience, 2010.
References: A Conceptual Circuit Model of Neuron, Journal of Integrative Neuroscience, 2009. Metastability and Plasticity of Conceptual Circuit Models of Neurons, Journal of Integrative Neuroscience, 2010.