BME 6938Mathematical Principles in

NeuroscienceInstructor: Dr Sachin S. Talahthi

Excitable cells in the brain: Neurons

Anatomy of a typical neuron

More Detailed View of neuron anatomy

The neuronal cell membrane

Synapses

Excitability of neuron

Crash course on neuronal signaling Neurons communicate through electrical

signaling Intracellular signaling is mediated through flow of

ions through ion channels on cell membrane (Will discuss in details soon)

Long distance cell to cell signaling is mediated through generation of action potentials that propagate along the axons (focus of neuronal modeling will be to understand the dynamical mechanism’s underlying the generation of action potential)

Communication between neurons happen at synapses by the process of neurotransmission

Conduction of nerve impulseUnmyelinated axon Myelinated axon

Animation of impulse propagation

Excitable properties of neuronal cell membrane: Intracellular signaling

Essential fundamental laws in Cellular Neurophysiology:

Ficks Law of Diffusion

Ohms Law of Drift

Space Charge Neutrality

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 ( )

D: Diffusion coefficient ( ) [C]: Concentration of the substance (ions) ( )

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 ( )

: electrical mobility of charged particle( ) [C]: Concentration of the substance (ions) ( ) 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

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/cm2. Concentration of ions within and outside of a

cell is 0.5 M. Determine the fraction of free (uncompensated) ions required to charge a spherical cell of radius 25 micro m to produce 100mV?

Ans: ~ 0.000235%

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)

Fundamental Equations of Cellular Neurophysiology Nernst-Plank Equation Goldman-Hodgkin-Katz Equation

Are derived from the fundamental laws of neurophysiology that we talked about in lecture 2 ppt.

Nernst-Plank Equation: Reversal Potential NPE describes the passive behavior of ion flow

through biological cell membrane under the influence of concentration gradient and electric field

Reversal Potential: (Nernst Equilibrium potential)

I=Current A/cm2; u=molar mobility cm2/V-sec-mol; F=Faraday constant (96480 C/mol)R=Gas constant (1.98 cal/oK-mol); C=Concentration molecules/cm3

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 Examples-Nernst Potential and the need for active mechanism 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

Gradient maintenance Active Transport:

Flow of ions against concentration gradient. Requires some form of energy source Examples: Na+ pump

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

Graphical illustration of ionic current flow

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

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

Goldman-Hodgkin-Katz Model Relates the current carried by ion’s across the

cell membrane to the transmembrane potential. Can be derived as a solution to NPE equation

under certain constraints: The cell membrane is homogeneous medium

(uniform thin glass) Electric field across the cell membrane is

constant Ion’s flow independently without interaction The flow of ion is affected by both concentration

gradient and voltage difference across the membrane

Goldman-Hodgkin-Katz Current Eqn Nonlinear I-V relationship for ionic current flow

across a cell membrane under the influence of concentration and potential gradient

(P=permeability of ion)

Goldman-Hodgkin-Katz Voltage Eqn Commonly known as the Goldman equation; is

used to determine the membrane resting potential of a cell that is permeable to several ionic species.

For membrane that is permeable to N positive ionic species and M negative ionic species:

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