The dynamic range of bursting in a network of respiratory pacemaker cells

1. The dynamic range of bursting in a network of respiratory pacemaker cells Alla Borisyuk Universityof Utah

2. Joint work with: Janet Best Jonathan Rubin David Terman Martin Wechselberger Mathematical Biosciences Institute (MBI), OSU

3. Previously… Biological data Existing model

4. Previously… Biological data Observations (Predictions) Numerical simulations Existing model

5. In this project Biological data Observations (Predictions) Numerical simulations Mathematical structure Existing model

6. In this project Biological data Observations (Predictions) New Predictions Numerical simulations Mathematical structure Existing model Advance available tools

7. Motivation: analyze model for neuronal activity in the Pre-Bötzinger complex Control of respiratory rhythm originates in this area

8. Motivation: analyze model for neuronal activity in the Pre-Bötzinger complex Individual neurons display variety of behaviors - - quiescent cells, spiking, bursting V

9. Motivation: analyze model for neuronal activity in the Pre-Bötzinger complex Population exhibits synchronous rhythms figure Question: How can a synchronous network bursting be supported by heterogeneous (e.g. spiking) cells?

10. Model for Each Cell Ca2+ Na+ CmV′ = - IL - IK - INa - INaP - Iton n′ = (n∞(V) – n)/n(V) h′ = (h∞(V) – h)/h(V) K+ Cl- IL = gL(V-VL) INa = gNam∞(V)3(1-n)(V-VNa) IK = gKn4(V-VK) INaP = gNaPm∞(V)3h(V-VNa) From: Butera et al. (1999) J. Neurophys. 81, 382-397

11. Model for Each Cell CmV′ = - IL - IK - INa - INaP - Iton n′ = (n∞(V) – n)/n(V) h′ = (h∞(V) – h)/h(V) IL = gL(V-VL) INa = gNam∞(V)3(1-n)(V-VNa) IK = gKn4(V-VK) INaP = gNaPm∞(V)3h(V-VNa) Iton(V) =gton(V-Vsyn) - Inputfrom other brain areas From: Butera et al. (1999) J. Neurophys. 81, 382-397

12. Single cell activity modes V quiescent gton = 0 bursting gton = .4 spiking gton = .6 time (ms)

13. Coupling the neurons CmV′ = - IL - IK - INa - INaP - Iton - Isyn s2 n′ = (n∞(V) – n)/n(V) h′ = (h∞(V) – h)/h(V) si′ = (1-si)H(Vi-)-si s1 Isyn = gsyn( si)(V-Vsyn) - Input from other network cells From: Butera et al. (1999) J. Neurophys. 81, 382-397

14. Coupling the neurons CmV′ = - IL - IK - INa - INaP - Iton - Isyn n′ = (n∞(V) – n)/n(V) h′ = (h∞(V) – h)/h(V) si′ = (1-si)H(Vi-)-si Isyn = gsyn( si)(V-Vsyn) - Input from other network cells gsyn =0  individual cells From: Butera et al. (1999) J. Neurophys. 81, 382-397

15. Full system CmV′ = - IL - IK - INa - INaP - Iton - Isyn n′ = (n∞(V) – n)/n(V) h′ = (h∞(V) – h)/h(V) si′ = (1-si)H(Vi-)-si Iton =gton(V-Vsyn) Isyn = gsyn( si)(V-Vsyn)

16. Observations: quiescence spiking gsyn (coupling strength) bursting gton (type of cell) From: Butera et al. 1999

17. Observations: • For a fixed gsyn transitions • from quiescence to • bursting to spiking gsyn (coupling strength) Burst duration gton (type of cell) From: Butera et al. 1999

18. Observations: • For a fixed gsyn transitions • from quiescence to • bursting to spiking • Network of spiking cells • can burst • (as in experiments) gsyn (coupling strength) single cell gton (type of cell) From: Butera et al. 1999

19. Observations: • For a fixed gsyn transitions • from quiescence to • bursting to spiking • Network of spiking cells • can burst • (as in experiments) gsyn (coupling strength) single cell gton (type of cell) From: Butera et al. 1999

20. Burst duration Observations: • For a fixed gsyn transitions • from quiescence to • bursting to spiking • Network of spiking cells • can burst • (as in experiments) • Sharp transition • in burst duration gsyn (coupling strength) gton (type of cell) From: Butera et al. 1999

21. Observations: • For a fixed gsyn transitions • from quiescence to • bursting to spiking • Network of spiking cells • can burst • (as in experiments) • Sharp transition • in burst duration gsyn (coupling strength) What are the mechanisms? gton (type of cell) From: Butera et al. 1999

22. Mathematical analysis • Self-coupled cell • - single cell • - synchronous network • Two cell network • - strong coupling • - weaker coupling

23. Mathematical analysis • Self-coupled cell • - single cell • - synchronous network • Two cell network • - strong coupling • - weaker coupling Questions • Transitions mechanism • quiescence  bursting  spiking • Why network is more bursty than a • single cell (shape of bursting border) • Sharp transition in burst duration

24. Network 1: self-connected cell CmV′ = - IL - IK - INa - INaP - Iton - Isyn n′ = (n∞(V) – n)/n(V) h′ = (h∞(V) – h)/h(V) s′ = (1-s)H(V-)-s Iton = gton(V-Vsyn) Isyn = gsyns(V-Vsyn)

25. Network 1: self-connected cell CmV′ = - IL - IK - INa - INaP - Iton - Isyn n′ = (n∞(V) – n)/n(V) h′ = (h∞(V) – h)/h(V) s′ = (1-s)H(V-)-s Iton = gton(V-Vsyn) Isyn = gsyns(V-Vsyn) Why is this an interesting case? • Includes individual neuron case (gsyn= 0) • Equivalent to a fully synchronized network • One slow variable (h) • /h(V)≪ 1/n(V) h is slower than V

26. Network 1: self-connected cell CmV′ = - IL - IK - INa - INaP - Iton - Isyn n′ = (n∞(V) – n)/n(V) h′ = (h∞(V) – h)/h(V) s′ = (1-s)H(V-)-s slow variable fast subsystem Iton = gton(V-Vsyn) Isyn = gsyns(V-Vsyn)

27. ′ V n s = F(V,n,s) h′ =  G (V,h) States of the fast subsystem with par. h V gsyn = 0 eriodics gton= 0.2 (Vmaxand Vmin) teady states

28. ′ V n s = F(V,n,s) h′ =  G (V,h) States of the fast subsystem with par. h V gsyn = 0 eriodics gton= 0.2 (Vmaxand Vmin) homoclinic teady states

29. ′ V n s = F(V,n,s) h′ =  G (V,h) Quiescence V gsyn = 0 eriodics gton= 0.2 (Vmaxand Vmin) h′ = 0 h′ > 0 h′ < 0 teady states

30. Transition to bursting gsyn gton gsyn (coupling strength) gton (type of cell)

31. gton Transition to bursting gsyn gton

32. Transition to bursting h′ = 0 gsyn gton

33. Bursting V gsyn gton h V t

34. Bursting V gsyn Square-wave bursting gton h V t

35. gsyn gton gsyn (coupling strength) gton (type of cell)

36. Transition to spiking V gsyn gton V t h Transition from bursting  spiking is when { h’=0 } crosses the homoclinic point Terman (1992) J. Nonlinear Sci.

37. gsyn gton gsyn (coupling strength) gton (type of cell)

38. gsyn (coupling strength) gton (type of cell)

39. Compare single cell to self-connected V gton gsyn > 0 gsyn = 0 h′ = 0 h Homoclinic point is higher for gsyn>0, i.e. transition to spiking ({ h’=0 } crosses the homoclinic point) will happen for larger gton

40. This explains wider range of bursting gsyn (coupling strength) gton (type of cell)

41. This explains wider range of bursting Or DOES IT??? gsyn (coupling strength) gton (type of cell)

42. Follow the transition curve in (gton,gsyn) space Where {h’=0} intersects the homoclinic point

43. Follow the transition curve in (gton,gsyn) space Underestimates bursting region Where {h’=0} intersects the homoclinic point

44. WHY? Because the synchronous solution is unstable

45. Network 2: two connected cells CmVi′ = - IL - IK - INa - INaP - Iton - Isyn ni′ = (n∞(Vi) – ni)/n(Vi) si′ = (1-si)H(Vi-)-si 2 slow variables: hi′ = (h∞(Vi) – hi)/h(Vi) Iton = gton(Vi-Vsyn) Isyn = gsynsj(Vi-Vsyn) i∈{1,2}, j=3-i

46. Simplification for larger gsyn: h1≈h2

47. h1≈h2 CmVi′ = - IL - IK - INa - INaP - Iton - Isyn ni′ = (n∞(Vi) – ni)/n(Vi) si′ = (1-si)H(Vi-)-si h′ = (h∞(Vi) – h)/h(Vi) Iton = gton(Vi-Vsyn) Isyn = gsynsj(Vi-Vsyn) i∈{1,2}, j=3-i

48. V1 h Bursting Synchronous h1≈h2 Anti-synchronous

49. Bursting NEW: Top-hat bursting h1≈h2

50. Features of top-hat bursting: • Square wave bursters, when coupled, • can generate top hat bursting h1≈h2