Open Worm Journal Club #2

C. Elegans body wall muscles are simple actuators

Boyle & Cohen 2007

Equivalent circuit diagram

Hodgkin Huxley basics

• 1952 work by two British physiologists

• Awarded the 1963 Nobel Prize in Physiology / Medicine

• Describes how action potentials in neurons are initiated and propagated

• A set of nonlinear ODEs that approximates the electrical characteristics of excitable cells

Cell review

Hodgkin Huxley foundations

• Membrane is modeled as a capacitor

• Voltage-gated ion channels are modeled as nonlinear electrical conductances

V

QC =

GI

VR

1==

Hodgkin Huxley foundations• Capacitance is the ability of a body to hold an electrical

charge

• Also the measure of the amount of electrical energy stored for a given electric potential

• Capacitance is measured in units called the farad.

• Farad is 1 coulomb per volt

• Coulomb is the unit of electric charge

• Volt is the unit of electromotive force / electric potential difference

V

QC =

Hodgkin Huxley foundations

• Resistance is the measure of opposition to an electrical current

• Conductance is the inverse of resistance & measures how easily electricity flows along a certain path

• Resistance is measured in units of ohms.

• Conductance is measured in units of Siemens a.k.a. mhos (get it?)

• Resistance is the ratio of voltage across it to the current through it

GI

VR

1==

Hodgkin Huxley foundations

• Electric current is a stream of charged objects

• Current is measured in terms of coulombs per second, also known as Amperes (Amps)

t

QI =

Using those basics, a general form can derived

V

QC =

GI

VR

1==

ICdt

dV

dt

dQ

Cdt

dV

C

QV

V

QC

11 =⇒=⇒=⇒=

t

QI =

dt

dQ

t

QI ⇒=

VGIR

VI

GI

VR =⇒=⇒== 1

VGCdt

dV 1=

General form of the Hodgkin-Huxley Equation

[ ]∑ −−=i

iimm

m gEVCdt

dV)(

1

VGCdt

dV 1=

Capacitance of membrane

Voltage across the membrane

Reversal potential of the i-th ion channel

Conductance of the i-th ion channel

Change of voltage across membrane over time

Equivalent circuit

diagram

[ ]∑ −−=i

iimm

m gEVCdt

dV)(

1

Capacitance of membrane

Voltage across the membrane

Reversal potential of the i-th ion channel

Conductance of the i-th ion channel

Change of voltage across membrane over time

Nonlinear, voltage dependant conductances

hmgVg namna3)( =

4)( ngVg kmk =m and n are known as “activation variables”,

while h is an “inactivation variable”. All of them are functions of voltage as well

Nonlinear, voltage dependant conductances

yximi gVg χϕ=)(

General form

Maximal conductance

Activation variable (unitless, real-valued from

0-1)

Inactivation variable (unitless, real-valued from

0-1)

Constant integers

Activation variable dynamics

[ ]ϕϕτ

ϕϕ

−= ∞ )(1

)( mm VVdt

d

First derivative of an activation variable involves a time constant, a steady state value that is a function of voltage, and a function of the activation variable with respect to time. Inactivation variable follows the same form.

Activation variable dynamics

mVmVVdt

dmmmmmm )(]1)[()( βα −−=

A different looking, but equivalent form splits the activation variable into an “alpha” and a

“beta” function of voltage

Equivalent circuit diagram

Muscle cell with “arms”

Cell Body

5 arms, 10 compartments

each, passive currents

Cell body, 1 compartment, active currents

The connectional currents

Cell Body

1

n

N

noutI

NoutI

nmI

ninI

Current flowing into body

Current leaving the n-th cell

Current entering the n-th cell

Current leaking through the

membrane of the n-th cell

The connectional currents

Cell Body

1

n

N

noutI

NoutI

nmI

ninI

Current flowing into body

Current leaving the n-th cell

Current entering the n-th cell

Current leaking through the

membrane of the n-th cell

body wall muscle cells(adult worm)cell lineage shown:

Cell Body

Cell Body

Cell Body

Cell Body

Cell Body

Cell Body

Cell Body

Cell Body

Cell Body

Cell Body

Cell Body

Cell Body

Quadrant 1 Quadrant 2

Quadrants of muscle cells

Their simplified quadrants of muscle cells

Cell Body

Cell Body

Cell Body

Cell Body

Cell Body

Cell Body

iiI ,1+

iiI ,1−

Inter-muscular,

Intra-quadrant gap

junction currents. Occurs

between cell bodies

intra, II kj ≡ Inter-quadrant gap

junction currents.Occurs

between tips of muscle arms.

Quadrant 1 Quadrant 2

inter, II LR ≡

Equivalent circuit diagram