BME 6938
Neurodynamics
Instructor: Dr Sachin S. Talathi
Recap
• Neurons are excitable cells
• Neuronal classification
• Communication in the brain is mediated by synapses:
Electrical and Chemical
Neuronal Signaling
• Neuronal signaling is mediated by the flow of dissociated ions
across the cell membrane.
• The fundamental laws that govern the flow of these ions
through the cell membrane are:
–
–
–
–
Ficks Law of Diffusion
Ohms Law of Drift
Space charge neutrality
Einsteins Relation between diffusion and drift
Fick’s Law of Diffusion
Fick’s law relates the diffusion gradient of ions to their
concentration.
• Jdiff:Diffusion flux, measuring the amount of substance flowing across unit
area per unit time (molecules/cm2s )
• D: Diffusion coefficient ( cm2/s )
cm
3)
• [C]: Concentration of the
ions
(molecules/cm
s
2
molecules
cm 3
Ficks Law Animation
Ohms law of drift (Microscopic view)
Charged particle in the presence of external electrical field E
experience a force resulting in their drift along the E field
gradient
• Jdrift:Drift flux, measuring the amount of substance flowing across unit area
per unit time ( molecules/cm2s )
molecules
2/sV)
• mu: electrical mobility
of
charged
particle(cm
cm s
• [C]: Concentration of the substance (ions) (molecules/cm3 )
molecules
• z: Valence of ion
2
cm 3
Space charge neutrality
Biological systems are overall electrically neutral; i.e., the total
charge of cations in a given volume of biological material
equals the total charge of anions in the same volume
biological material
Einstein relation
It relates the diffusion constant (effect of motion due to
concentration gradients) of an ion to its mobility (effect of
motion due to electrical forces)
Basic idea is that the frictional resistance created by the medium
is same for ions in motion due to drift and diffusion.
Some high-school chemistry
 1 mole= Avogadro’s number (NA) of basic units (atoms,
molecules, ions…)
 Concentration is typically given in units of molar.
 1 Molar=1 mole/litre=10-3 moles/cm3
 Relation between gas constant (R) and Boltzmann’s constant
(k): R=kNA
 Faraday constant F: Magnitude of one mole of charged
particles: F=qNA
Some-algebra
Membrane capacitance of a cell membrane is around 1 microF.
Concentration of ions within and outside of a
cell is 0.5 M. Determine the fraction of free (uncompensated)
ions required on each side of a spherical cell of radius 25
micro m to produce 100mV?
Ans: ~ 0.00235%
For realistic cell dimension, from above calculations we see that
generation of 10s of mV of voltage does not violate spacecharge neutrality (~99.9% of charges are compensated)
Nernst-Plank Equation
Nernst Plank equation governs the current generated from the
flow of individual ions across the cell membrane.
Continuity Equation
The time dependent Nernst Plank Equation:
Nernst Equation
• Nernst equation is special case of NPE, where in the
membrane potential is obtained as a function of
concentration gradient across the cell membrane when the
net current flow generated by the ion is zero.
Typical scale of reversal potential values
Question:
What is the direction of flow of following ions
under normal conditions?
1.Na+
2. K+
3. Ca2+
4. Cl(Hint: Look at the chart of reversal potentials and
Nernst Equilibrium potential equation)
At 37 oC
mV
Specific Example
• Ion concentration for cat motoneuron: Vm=-70 mV
Inside mol/m3
Outside mol/m3
Na+
15
150
K+
150
5.5
Cl-
9
125
• Nernst Potential: At body temperature 37oC
VNa +
[ ]
[ ]
æ Na + ö
out
= 62log10 ç
÷÷ = 62mV
+
ç Na
è
in ø
VCl -
VK +
[ ]
[ ]
[ ]
[ ]
æ K+ ö
out
= 62log10 ç
÷÷ = -89 mV
+
ç K
è
in ø
æ Cl - ö
out
= -62log10 ç
÷÷ = -70 mV
ç Cl
è
in ø
Ion distribution and Gradient maintenance
• Active Transport:
– Flow of ions against concentration gradient.
– Requires some form of energy source
– Examples: Na+ pump
Read section 2.5.1 in Johnston’s& Wu book for more information.
• Passive Transport:
– Selective permeability to some ions results in concentration gradient
– No energy source required
– Passive distribution of ions can be determined using the Donnan
rule of equilibrium
Donnan Equilibrium Rule
• The membrane potential equals the reversal potential of all
ions that can passively permeate through the cell membrane.
• Mathematically the Donnan Rule implies:
• Have a look at Donnan Rule in works; through animaltion
developed by Larry Keeley: http://entochem.tamu.edu/GibbsDonnan/index.html
Graphical illustration of ion gradient
maintenance
Example: Application of Donnan Rule
• Consider a two compartment system separated by a
membrane that is permeable to K+ and Cl- but is not
permeable to a large anion A-. The initial concentrations on
either side of membrane are:
Ion type
I (conc in mM)
II (conc in mM)
A-
100
0
K+
150
150
Cl-
50
150
• Is the system in electrochemical equilibrium (no ion flow
across the membrane?
• If not, what direction the ions flow? And what are the final
equilibrium concentrations?
Steady state solution to NPE
• We can solve the NPE equation in steady state (ie no time
dependence for concentrations). A special case was seen
through NE, wherein in addition to the membrane being in
steady state, the membrane was in equilibrium (I=0).
Boundary conditions-Cell membrane
Permeability P: Defined as absolute magnitude of ion flux when there is a
unit concentration difference between internal and external fluids and the
concentration is linear function of distance within the membrane
(in molar units)
: Partition coefficient, which measures the drop in concentration of
species at fluid membrane interface.
Constant field assumption
• Most common assumption is charge density within the
membrane is identically zero. This assumption leads to the
following expression for V(x) with boundary conditions:
V(x=0)=Vm and V(x=l)=0
• We get the following expression for current:
where
Note: Poisson equation from Electrostatics:
Goldman Hodgkin Katz Model
• It is widely used model to predict the resting membrane
potential Vm for nerve cells
• The formula for Vm is derived as a solution NPE with constant
field assumption
• Further more it is assumed that ions flow across the cell
membrane without interacting with each other
• For membrane that is permeable to M positive monovalent
ions and N negative monovalent ions, the GHK formula for Vm
in equilibrium conditions is:
Exercise (extra credit)
Show that the resting equilibrium membrane potential Vm for a
cell membrane that is permeable to monovalent and divalent
ions is given by
Application of the GHK equation
Lets use GHK eqn to determine the contribution to membrane
potential from active ion transport mechanism’s.
•Na-pump result in flow of 3 Na+ ions across the cell membrane
for every 2K+ ions. What is the resulting equilibrium potential of
the cell of squid axon for which the concentration gradients
across the cell are:
Ion type
Inside (mM)
Outside (mM)
K+
400
20
Na+
50
550
•The permeability ratio is Pk:Pna=1:.03