Two common current models

1
Two common current models
Linear model
GHK model
These are the two most common current models.
Note how they both have the same reversal
potential, as they must. (Crucial fact In
electrically excitable cells gNa (or PNa) are not
constant, but are functions of voltage and time.
More on this later.)
2
Electrodiffusion deriving current models
Poisson-Nernst-Planck equations. PNP equations.
Poisson equation and electrodiffusion
Boundary conditions
Nernst-Planck equations.
3
The short-channel limit
If the channel is short, then L 0 and so l 0.
This is the Goldman-Hodgkin-Katz equation. Note
a short channel implies independence of ion
movement through the channel.
4
The long-channel limit
If the channel is long, then 1/L 0 and so 1/l
0.
This is the linear I-V curve. The independence
principle is not satisfied, so no independent
movement of ions through the channel. Not
surprising in a long channel.
5
The cell at steady state

• We need to model
• pumps and exchangers
• ionic currents
• osmotic forces

6
Osmosis
P1
P2
water Solvent (conc. c)
water
At equilibrium
Note equilibrium only. No information about the
flow.
7
A Model of Volume Control
Putting together the three components (pumps,
currents and osmosis) gives.....
8
The Pump-Leak Model
cell volume
Nai
pump rate
Note how this is a really crappy pump model
Na is pumped out. K is pumped in. So cells have
low Na and high K inside. For now we ignore
Ca2. Cl- just equilibrates passively.
9
Charge and osmotic balance

• The proteins (X) are negatively charged, with
  valence zx.
• Both inside and outside are electrically
  neutral.
• The same number of ions on each side.
• 5 equations, 5 unknowns (internal ionic
  concentrations, voltage, and volume). Just solve.

10
Steady-state solution
If the pump stops, the cell bursts, as
expected. The minimal volume gives approximately
the correct membrane potential. In a more
complicated model, one would have to consider
time dependence also. And the real story is far
more complicated.
11
Lots of interesting unsolved problems

• How do organsims adjust to dramatic environmental
  changes (T. Californicus)?
• How do plants (especially in arid regions)
  prevent dehydration in high salt environments?
  (They make proline.)
• How do plants breathe?
• How do fish (salmon) deal with both fresh and
  salt water?
• What happens to a cell and its environment when
  there is ischemia (restriction in blood supply )?

12
Active modulation of the membrane potential
electrically excitable cells

• Basic references
• 1) Keener and Sneyd, Mathematical Physiology
• 2) Dynamical systems in neuroscience the
  geometry of excitability and bursting by Eugene
  M. Izhikevich

13
Alan Lloyd Hodgkin and Andrew Huxley described
the model in 1952 to explain the ionic mechanisms
underlying the initiation and propagation of
action potentials in the squid giant axon. They
received the 1963 Nobel Prize in Physiology or
Medicine for this work.
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Resting potential

• No ions are at equilibrium, so there are
  continual background currents. At steady-state,
  the net current is zero, not the individual
  currents.
• The pumps must work continually to maintain
  these concentration differences and the cell
  integrity.
• The resting membrane potential depends on the
  model used for the ionic currents.

linear current model (long channel limit)
GHK current model (short channel limit)
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Simplifications

• In some cells (electrically excitable cells),
  the membrane potential is a far more complicated
  beast.
• To simplify modelling of these types of cells,
  it is simplest just to assume that the internal
  and external ionic concentrations are constant.
• Justification Firstly, it takes only small
  currents to get large voltage deflections, and
  thus only small numbers of ions cross the
  membrane. Secondly, the pumps work continuously
  to maintain steady concentrations inside the
  cell.
• So, in these simpler models the pump rate never
  appears explicitly, and all ionic concentrations
  are treated as known and fixed.

16
Steady-state vs instantaneous I-V curves

• So far we have discussed how the current through
  a single open channel depends in the membrane
  potential and the ionic concentrations on either
  side of the membrane.
• But in a population of channels, the total
  current is a function of the single-channel
  current, and the number of open channels.
• When V changes, both the single-channel current
  changes, as well as the proportion of open
  channels. But the first change happens almost
  instantaneously, while the second change is a lot
  slower.

I-V curve of single open channel
Proportion of open channels
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Example Na and K channels
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K channel gating
If the channel consists of two identical
subunits, each of which can be closed or open
then
S00
S01
S10
S11
19
Na channel gating
If the channel consists of multiple subunits of
two different types, m and h, each of which can
be closed or open then
the fraction of channels in state S21
activation
inactivation
20
Experimental data K conductance
If voltage is stepped up and held fixed, gK
increases to a new steady level.
four subunits
rate of rise gives tn
steady-state
time constant
Now just fit to the data.
steady state gives n8
21
Experimental data Na conductance
If voltage is stepped up and held fixed, gNa
increases and then decreases.
Four subunits. Three switch on. One switches off.
steady-state
time constant
Fit to the data is a little more complicated now,
but still easy in principle.
22
Hodgkin-Huxley equations
applied current
generic leak
activation (increases with V)
much smaller than the others
inactivation (decreases with V)
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An action potential

• gNa increases quickly, but then inactivation
  kicks in and it decreases again.
• gK increases more slowly, and only decreases
  once the voltage has decreased.
• The Na current is autocatalytic. An increase in
  V increases m, which increases the Na current,
  which increases V, etc.
• Hence, the threshold for action potential
  initiation is where the inward Na current
  exactly balances the outward K current.

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The fast phase plane I
n and h are slow, and so stay approximately at
their steady states while V and m change quickly
25
The fast phase plane II
h0 decreasing n0 increasing
As n and h change slowly, the dV/dt nullcline
moves up, ve and vs merge in a saddle-node
bifurcation, and disappear. vr is the only
remaining steady-state, and so V returns to rest.
In this analysis, we simplified the
four-dimensional phase space by taking series of
two-dimensional cross-sections, those with
various fixed values of n and h.
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The fast-slow phase plane
Take a different cross-section of the 4-d system,
by setting mm8(v), and using the useful fact
that n h 0.8 (approximately). Why? Who knows.
It just is. Thus
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Oscillations
When a current is applied across the cell
membrane, the HH equations can exhibit
oscillatory action potentials.
28
Example Biophysical characterization of
voltage-gated currents in somatic macropatches
isolated from P14 CA3 pyramidal cells at 33?C

• Jon Brown and Andy Randall
• Medical School, UoB

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Whole cell recording in slice (poor voltage
clamp)
Nucleated macropatch TypicallyCap 2 pF
Rser5 M? Rmemgt1 G? (near perfect voltage
clamp)
Massive axon
Massive dendrite
Suck Pull
Massive dendrite
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4 consecutive macropatch recordings at 33?C under
physiological ionic conditions
Outward at 32 mV
(n7)
Peak inward
Scale bars 5 ms, 200 pA
31
Tetrodotoxin blocks the fast inward current in
macropatches from CA3 interneurones
200 pA
5 ms
TTX (1 mM)
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Na current in macropatch from CA3 cell (33?C)
32
-78
-88
200 pA
5 ms
15/7/2008 cell 2
(n7)
(n7)
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Na current inactivation at 33?C in CA3 macropatch
-17
-54
-99
100 pA
1 ms
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Macropatches from CA3 pyramids have small Ca2
currents in 2 mM Ca2o at 33?C
32
-88
20 pA
10 ms
35
Ca2 current in macropatch from CA3 pyramidal
cell (33?C, Ca2o 5 mM)
32
-88
50 pA
5 ms
16/7/2008 cell 1
36
Steady state inactivation of Ca2 current in CA3
pyramidal cell (33?C, Ca2o 5 mM)
-17
-54
-99
50 pA
5 ms
16/7/2008 cell 1
37
Isolated K current in CA3 macropatch
500 pA
50 ms