2. 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

3. GnRH neurons Nice colour picture stolen from the web Boring greyscale fuzzy picture stolen from Christine Jasoni. She needs to polish her Photoshop skills.

4. 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.

5. Bursting and calcium Simultaneous measurement of membrane current and calcium from GnRH neurons in brain slices. calcium current Expanded view Spike frequency adaptationA rise in calcium turns the spiking off This strongly suggests Ca 2+ -dependent K + channels

6. 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 IP 3 receptors, as it happens.

7. Reasonable model ER K+K+ Na + Ca 2+ IPR V Ca 2+ K+K+ V Bursting onBursting off

8. Problems 1.It doesn’t work. 2.It doesn’t explain what happens when you block the IPR.

9. 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 Ca 2+ -sensitive K + channel. Surprisingly, this doesn’t happen. Why does the bursting stop?

10. What else doesn’t work? What stops the bursting here?

11. The only model we could get to work ER IKIK Na + Ca 2+ IPR Ca 2+ I AHP-SK I AHP-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 I AHP-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

12. According to this hypothesis Ca 2+ activates I AHP-SK channel. Switches burst off. Ca 2+ activates I AHP-UCL channel. I AHP-UCL channel prevents bursting.

13. Structure of the model Voltage submodel Based on Hodgkin-Huxley Calcium submodel FastSlow 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. Ca influx through voltage-gated channels. How do I know?

14. The math nerd’s view of calcium homeostasis J IPR J serca J pm J leak 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.

15. 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

16. The hypothetical channel Remember that the model assumed the existence of a really slow, calcium-dependent, time-dependent, after- hyperpolarisation current, which we called I AHP-UCL. Well, is it really there? After all, a model prediction is no use if you can’t test it.

17. Cue Allan and Kiho … Perforated patch, voltage-clamp traces, showing the evoked I AHP 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 I AHP in hippocampal pyramidal cells, but please don’t ask me about this.

18. What does the model predict? Block I AHP-SK, get longer bursts, spaced further apart. Block I AHP-UCL, get faster bursting, a bit messier.

19. Cue Allan and Kiho again... Block I AHP-SK, get longer bursts, spaced further apart. Block I AHP-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.

20. And so… There are two Ca 2+ -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.

21. 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.

22. Dendritic calcium responses So, no CICR in the dendrite.

23. How does this work? This is the big question

24. Our initial thoughts IP 3 receptors control when the burst stops (via CICR and activation of Ca 2+ -dependent K + channels). So, we said that IP 3 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 IP 3 receptors.

25. The most important parameter D=8000  m 2 /ms is the best fit.

26. 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.

27. Unfortunately... Allan was right... I hate it when that happens.

28. Lots of questions remain Our model requires a very particular kind of I AHP-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.

29. Reminder... excitable systems defines the slow manifold Because  << 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.

30. 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.

31. The fast oscillations Assume that c, c t 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.

32. 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.

33. 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.

34. The slow phase plane Looks exactly like a FitzHugh- Nagumo model. This jump in Ca i is caused by the bursting, which takes Ca i over the threshold and leads to a large increase in Ca i

35. 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.

36. One can continue...... to play this game for all the pharmacological perturbations, but there are no real surprises.

37. The End Thanks to: