Transcript “Mathematics is the Queen of the Sciences” Carl Friedrich

GnRH neurons, calcium,
and mathematical models
James Sneyd, University of Auckland
David Wen Duan, University of Auckland
Jason Chen, University of Auckland
Kiho Lee, University of Otago
Allan Herbison, University of Otago
Karl Iremonger, University of Otago
GnRH neurons
Nice colour
picture stolen
from the web
Boring greyscale
fuzzy picture stolen
from Christine
Jasoni. She needs
to polish her
Photoshop skills.
Experimental method
Glow-in-the-dark mice.
Glow-in-the-dark stuff
P.S. This slide not approved by either Allan or Kiho. Quite the reverse, actually.
Bursting and calcium
Simultaneous measurement of
membrane current and calcium from
GnRH neurons in brain slices.
This strongly suggests
Ca2+-dependent K+ channels
Spike frequency adaptation
A rise in calcium
turns the spiking off
current
10%
1 sec
calcium
Expanded view
Where does the calcium come
from?
In most of the bursts, calcium clearly continues to rise
even after the burst has ended.
So calcium is being released from internal stores. Through IP3
receptors, as it happens.
Reasonable model
K+
Bursting off
on
V
Na+
Ca2+
Ca2+
V
IPR
ER
K+
Problems
1. It doesn’t work.
2. It doesn’t explain what happens
when you block the IPR.
Blocking the IPR
From the model, you would predict that
blocking the IPR just leads to continuous
bursting, as no calcium can come out of the
IPR to open the Ca2+-sensitive K+ channel.
Surprisingly, this doesn’t happen. Why does the bursting stop?
What else doesn’t work?
What stops the bursting here?
The only model we could get
to work
Note the new names
IAHP-SK
IAHP-UCL
Turns on quickly,
turns off slowly.
Na+
Ca2+
Ca2+
IPR
ER
IK
We had to assume the
existence of a
hypothetical calciumactivated, timedependent, very slow
after-hyperpolarisation
current.
We call this IAHP-UCL.
It’s important to continue the proud
biological tradition of using
incomprehensible names with lots of
letters in them. Don’t blame me. It’s
Allan’s fault.
According to this hypothesis
Ca2+ activates IAHP-UCL channel.
Ca2+ activates IAHP-SK channel.
Switches burst off.
10%
1 sec
IAHP-UCL channel prevents
bursting.
Structure of the model
How do I know?
How do I know?
Fast
Slow
Ca influx through
voltage-gated
channels.
Voltage submodel
Based on Hodgkin-Huxley
Calcium submodel
Ca-dependent
K channels
This part of the model
gives the fast electrical
spiking.
This part of the model
gives the calcium transient
and sets the interburst
interval.
The math nerd’s view
of calcium homeostasis
cell membrane
c e = [Ca 2+ ]ER
Jserca
JIPR
c = [Ca ]
2+
dc
= J IPR (c e - c) - J serca + d (J leak - J pm )
dt
dc t
= d (J leak - J pm )
dt
Total calcium variable
Jleak
Jpm
The calcium model is essentially just a
simple conservation equation. The
change in calcium concentration is the
influx minus the efflux.
Control simulations
Model
Model detail
heavily filtered, just for fun
The model has approximately 3 spikes
per burst, and the peak of the calcium
transient is after the spiking has
finished.
Experiment
The hypothetical channel
Remember that the model assumed the existence of a really
slow, calcium-dependent, time-dependent, afterhyperpolarisation current, which we called IAHP-UCL.
Well, is it really there? After all, a model prediction is no use if
you can’t test it.
Cue Allan and Kiho …
Perforated patch, voltage-clamp
traces, showing the evoked IAHP
and its modulation by apamin and
UCL2077.
To this day, I’m not entirely sure
where Allan got the idea to use
UCL2077. I have a vague notion
that it was known to block IAHP in
hippocampal pyramidal cells, but
please don’t ask me about this.
What does the model predict?
Block IAHP-SK, get longer
bursts, spaced further apart.
Block IAHP-UCL, get faster
bursting, a bit messier.
Cue Allan and Kiho again ...
Block IAHP-SK, get longer
bursts, spaced further apart.
Don’t ask me what this is.
Block IAHP-UCL, get faster
bursting.
Exactly as predicted. This calls for a
cheer and for Allan to buy me a beer. Or
two.
And so…
• There are two Ca2+-sensitive K+ channels that modulate the
bursting.
• One is sensitive to apamin, and regulates the end of the
burst, and spike frequency adaptation.
• The other is slower and time-dependent, and regulates the
interburst period.
• In this case, the mathematical model helped show what
things to look for, and to provide a reasonable explanation for
the entire range of experimental data.
Space:
the final frontier
So spikes are not initiated
in the soma, but start at some
initiation site along the dendrite.
We call this the iSite, because
we think that is a cool name.
Dendritic calcium responses
So, no CICR in the dendrite.
How does this work?
This is the
big question
Our initial thoughts
• IP3 receptors control when the burst stops (via CICR
and activation of Ca2+-dependent K+ channels).
• So, we said that IP3 receptors had to be present in
the iSite.
• Allan disagreed.
• He told us to go back and check the model.
• We told him to go back and look for IP3 receptors.
The most
important
parameter
D=8000 mm2/ms
is the best fit.
Close electrical coupling
This is a pretty large value for the electrical diffusive coupling
and it means that the soma and the iSite are practically identical
electrically.
Unfortunately...
Allan was right... I hate it when that happens.
Lots of questions remain
• Our model requires a very particular kind of IAHP-UCL, with specified
dynamics and calcium-dependence. This needs to be tested. If I was
forced to bet, I’d say that the model is (very) unlikely to have captured the
behaviour of that channel completely.
• We predict that blockage of the IPR, with no additional effect on calcium
pumps won’t do anything very interesting. Is this true? Good question.
• The bursting is not actually a limit cycle, as far as we know. It's driven by
stochastic inputs from outside. We've done this model, as it happens, and
very little changes, so we just show the deterministic results, as they are
easier to show.
• What on earth is the iSite doing way out there? Are there multiple iSites?
Why have an iSite at all? Weird.
Reminder... excitable systems
defines the slow manifold
dv
e
= f (v) - w
dt
dw
= v - gw
dt
Because e << 1 the solution
jumps between branches of
the slow manifold (approximately).
I have to say approximately because otherwise
Martin and Vivien will tell me off. That’s because
they are real mathematicians and I’m not.
Reminder ... bursting oscillations
dv
= f1 (v,w)
dt
dw
= f 2 (v,w)
dt
dc
= eg(v,w,c)
dt
A fast system (v and w)
modulated by a slow
variable, c.
Pretend c is
constant, and
sketch the
bifurcation diagram
of the fast system,
as a function of c.
The actual solution
now jumps between
the branches of the
fast subsystem.
Transitions at the SN
and HC bifurcations.
The fast oscillations
Assume that c, ct and x are constants.
Plot the bifurcation diagram as a function
of x.
Crucial that SN2 lies to the left of HC.
Solution starts here.
Solution moves left to SN2.
Solution falls off at SN2 and heads to
the branch of stable oscillations.
Solution moves right to HC.
Solution falls off at HC and returns
to the branch of stable steady states.
Summary of the fast phase plane
Plot the bifurcation points as
functions of x and c, which are
both slow variables.
dx/dt=0
Before it can reach the steady
Start
Calcium
bottom
eventually
right.
rises
The
state,atthe
solution
hits
theso
solution
high
heads
the solution
tofalls
the off
left
hitstothe
SN2 that
bifurcation,
the
(calcium
HC
bifurcation
slightly
and
increasing)
falls off,
periodic
orbit,
and
bursting
and
eventually
tries
toreturning
get to the
steady
the
starts.
This
brings
in to
calcium
state.
starting
letting the cycle
through point,
the voltage-gated
repeat.
channels, and so calcium
increases.
Detail of the fast phase plane
Each spike gives a
jump in c. Of course,
you can’t see the
voltage spike explicitly
here, as we are not
plotting V.
The slow phase plane
Looks exactly like a FitzHughNagumo model.
dCat
=0
dt
dCai
=0
dt
This jump in Cai is caused by
the bursting, which takes Cai
over the threshold and leads
to a large increase in Cai
The bifurcation structure of apamin
Notice how the SN2 and HC
curves have been rotated by
apamin. The solution stays in the
bursting region longer, and so
there are more spikes.
Because there are more spikes,
the calcium goes up higher before
jumping across, leading to a
larger calcium transient, as the
right branch of the slow manifold
is now further away.
One can continue...
... to play this game for all the pharmacological perturbations,
but there are no real surprises.
The End
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