BME 6938Neurodynamics

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 )• [C]: Concentration of the ions (molecules/cm3 )

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 )

• mu: electrical mobility of charged particle(cm2/sV)• [C]: Concentration of the substance (ions) (molecules/cm3 )• z: Valence of ion

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 space-charge 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

• Nernst Potential: At body temperature 37oC

Inside mol/m3 Outside mol/m3

Na+ 15 150

K+ 150 5.5

Cl- 9 125

Ion distribution and Gradient maintenance

• Active Transport: – Flow of ions against concentration gradient. – Requires some form of energy source– Examples: Na+ pumpRead 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/Gibbs-Donnan/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:

• 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?

Ion type I (conc in mM) II (conc in mM)

A- 100 0

K+ 150 150

Cl- 50 150

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:

•The permeability ratio is Pk:Pna=1:.03

Ion type Inside (mM) Outside (mM)

K+ 400 20

Na+ 50 550