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