BME 6938
Mathematical Principles in Neuroscience
Instructor: 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 impulse
Unmyelinated axon
Animation of impulse propagation
Myelinated axon
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.
Ion Flux = Jdiff
¶ [C ]
= -D
¶x
 Jdiff:Diffusion
flux, measuring the amount of substance flowing across
unit area per unit time ( molecules
)
cm s
D: Diffusion coefficient ( cms )
[C]: Concentration of the substance (ions) (molecules
)
cm
2


Ficks Law Animation
2
3
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
Ion Drift = Jdrift
¶V
= -z[C]m
¶x
 Jdrift:Drift
flux, measuring the amount of substance flowing across unit
area per unit time (molecules)
cm s
: electrical mobility of charged particle(
)
[C]: Concentration of the substance (ions) ( molecules
)
cm
z:Valence of ion
2



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
å z [C ] = å z [C ]
C
i
i
A
j
i
j
j
Some high-school chemistry

1 mole= Avogadro’s number (NA) of basic units (atoms,
molecules, ions…)
 Concentration
1
is typically given in units of molar.
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)
RT æ [Cout ]ö
Vm = V ( I = 0) =
lnç
÷
zF è [Cin ] ø
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
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
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:
1
m
+m
é COut
ù
é Cin-n ù
ê C +m ú = ê C -n ú
ë in û
ë Out û

1
n
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:
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?
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
uRT
P=
Fl
V=
zVF
RT
(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:
æ M
+
P
+ Ci
å
RT ç i C i
E R = Vm (I = 0) =
lnç
F ç
+
P
C
+
å
i
ç
Ci
è
[ ]
[ ]
ö
+ å PC - C ÷
out
j
j
÷
N
÷
+
P
- Cj
å
Cj
in
out ÷
ø
j
N
[ ]
j in
[ ]
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