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.