GnRH neurons, calcium,and mathematical models

James Sneyd, University of AucklandDavid Wen Duan, University of AucklandJason Chen, University of AucklandKiho Lee, University of OtagoAllan Herbison, University of OtagoKarl 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.

Glow-in-the-dark mice.

Experimental method

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.

calcium

current

Expanded view

Spike frequency adaptation A rise in calcium turns the spiking off

This strongly suggestsCa2+-dependent K+ channels

Where does the calcium come from?

In most of the bursts, calcium clearly continues to riseeven after the burst has ended.

So calcium is being released from internal stores. Through IP3 receptors, as it happens.

Reasonable model

ER

K+

Na+

Ca2+

IPR

V

Ca2+

K+

V

Bursting onBursting off

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

ER

IK

Na+

Ca2+

IPR

Ca2+

IAHP-SKIAHP-UCL

Turns on quickly, turns off slowly.

We had to assume the existence of a hypothetical calcium-activated, time-dependent, 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.

Note the new names

According to this hypothesis

Ca2+ activates IAHP-SK channel.Switches burst off.

Ca2+ activates IAHP-UCL channel.

IAHP-UCL channel prevents bursting.

Structure of the model

Voltage submodelBased on Hodgkin-Huxley

Calcium submodel

Fast Slow

Ca-dependentK channels

This part of the modelgives the fast electricalspiking.

This part of the modelgives the calcium transientand sets the interburstinterval.

Ca influx throughvoltage-gatedchannels.

How do I know? How do I know?

The math nerd’s viewof calcium homeostasis

JIPRJserca

Jpm

Jleak

cell membrane

Total calcium variable

The calcium model is essentially just a simple conservation equation. The change in calcium concentration is the influx minus the efflux.

Control simulations

The model has approximately 3 spikes per burst, and the peak of the calcium transient is after the spiking has finished.

Model Model detail

Experiment

heavily filtered, just for fun

The hypothetical channel

Remember that the model assumed the existence of a really slow, calcium-dependent, time-dependent, after-hyperpolarisation 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.

Block IAHP-UCL, get faster bursting.

Don’t ask me what this is.

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 initiatedin the soma, but start at someinitiation site along the dendrite.

We call this the iSite, becausewe think that is a cool name.

Dendritic calcium responses

So, no CICR in the dendrite.

How does this work?

This is thebig 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 m2/msis the best fit.

Close electrical couplingThis is a pretty large value for the electrical diffusive couplingand it means that the soma and the iSite are practically identicalelectrically.

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

Because << 1 the solutionjumps between branches ofthe slow manifold (approximately).

I have to say approximately because otherwiseMartin and Vivien will tell me off. That’s becausethey are real mathematicians and I’m not.

Reminder ... bursting oscillations

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 oscillationsAssume that c, ct and x are constants.Plot the bifurcation diagram as a functionof 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 returnsto 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

Start at bottom right. The solution heads to the left (calcium slightly increasing) and tries to get to the steady state.

Before it can reach the steady state, the solution hits the SN2 bifurcation, falls off to the periodic orbit, and bursting starts. This brings in calcium through the voltage-gated channels, and so calcium increases.

Calcium eventually rises so high that the solution hits the HC bifurcation and falls off, eventually returning to the starting point, letting the cycle repeat.

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 FitzHugh-Nagumo model.

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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