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. Biological data Existing model Previously…

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

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

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

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 I L = g L (V-V L ) I Na = g Na m ∞ (V) 3 (1-n)(V-V Na ) I K = g K n 4 (V-V K ) I NaP = g NaP m ∞ (V) 3 h(V-V Na ) n′ = (n ∞ (V) – n)/  n (V) h′ =  (h ∞ (V) – h)/  h (V) C m V′ = - I L - I K - I Na - I NaP - I ton From: Butera et al. (1999) J. Neurophys. 81, 382-397 Na + Ca 2+ K+K+ Cl -

11. Model for Each Cell I L = g L (V-V L ) I Na = g Na m ∞ (V) 3 (1-n)(V-V Na ) I K = g K n 4 (V-V K ) I NaP = g NaP m ∞ (V) 3 h(V-V Na ) I ton (V) = g ton (V-V syn ) - Input from other brain areas n′ = (n ∞ (V) – n)/  n (V) h′ =  (h ∞ (V) – h)/  h (V) C m V′ = - I L - I K - I Na - I NaP - I ton From: Butera et al. (1999) J. Neurophys. 81, 382-397

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

13. n′ = (n ∞ (V) – n)/  n (V) h′ =  (h ∞ (V) – h)/  h (V) s i ′ =  (1-s i )H(V i -  )-  s i C m V′ = - I L - I K - I Na - I NaP - I ton - I syn Coupling the neurons I syn = g syn (  s i )(V-V syn ) - Input from other network cells From: Butera et al. (1999) J. Neurophys. 81, 382-397 s1s1 s2s2

14. n′ = (n ∞ (V) – n)/  n (V) h′ =  (h ∞ (V) – h)/  h (V) s i ′ =  (1-s i )H(V i -  )-  s i C m V′ = - I L - I K - I Na - I NaP - I ton - I syn Coupling the neurons From: Butera et al. (1999) J. Neurophys. 81, 382-397 g syn =0  individual cells I syn = g syn (  s i )(V-V syn ) - Input from other network cells

15. n′ = (n ∞ (V) – n)/  n (V) h′ =  (h ∞ (V) – h)/  h (V) s i ′ =  (1-s i )H(V i -  )-  s i C m V′ = - I L - I K - I Na - I NaP - I ton - I syn Full system I ton = g ton (V-V syn ) I syn = g syn (  s i )(V-V syn )

16. Observations: g ton (type of cell) g syn (coupling strength) From: Butera et al. 1999 bursting spiking

17. Observations: g ton (type of cell) g syn (coupling strength) bursting spiking quiescence From: Butera et al. 1999

18. Observations: g ton (type of cell) g syn (coupling strength) For a fixed g syn transitions from quiescence to bursting to spiking Burst duration From: Butera et al. 1999

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

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

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

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

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

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

25. Network 1: self-connected cell n′ = (n ∞ (V) – n)/  n (V) h′ =  (h ∞ (V) – h)/  h (V) s′ =  (1-s)H(V-  )-  s C m V′ = - I L - I K - I Na - I NaP - I ton - I syn I ton = g ton (V-V syn ) I syn = g syn s(V-V syn )

26. Network 1: self-connected cell n′ = (n ∞ (V) – n)/  n (V) h′ =  (h ∞ (V) – h)/  h (V) s′ =  (1-s)H(V-  )-  s C m V′ = - I L - I K - I Na - I NaP - I ton - I syn I ton = g ton (V-V syn ) I syn = g syn s(V-V syn ) Why is this an interesting case? Includes individual neuron case (g syn = 0) Equivalent to a fully synchronized network One slow variable (h)  /  h (V) ≪ 1/  n (V) h is slower than V

27. Network 1: self-connected cell n′ = (n ∞ (V) – n)/  n (V) h′ =  (h ∞ (V) – h)/  h (V) s′ =  (1-s)H(V-  )-  s C m V′ = - I L - I K - I Na - I NaP - I ton - I syn I ton = g ton (V-V syn ) I syn = g syn s(V-V syn ) fast subsystem slow variable

28. g syn = 0 States of the fast subsystem with par. h g ton = 0.2 V teady states eriodics (V max and V min ) VnsVns ′ = F(V,n,s) h′ =  G (V,h)

29. g syn = 0 States of the fast subsystem with par. h g ton = 0.2 V teady states eriodics (V max and V min ) VnsVns ′ = F(V,n,s) h′ =  G (V,h) homoclinic

30. g syn = 0 Quiescence g ton = 0.2 V teady states eriodics h′ = 0 h′ < 0 h′ > 0 (V max and V min ) VnsVns ′ = F(V,n,s) h′ =  G (V,h)

31. g ton (type of cell) g syn (coupling strength) g syn Transition to bursting g ton

32. g syn g ton Transition to bursting g ton

33. g syn h′ = 0 g ton Transition to bursting

34. g syn h V Bursting g ton t V

35. g syn h V Bursting g ton t V Square-wave bursting

36. g ton (type of cell) g syn (coupling strength) g syn g ton

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

38. g ton (type of cell) g syn (coupling strength) g syn g ton

39. g ton (type of cell) g syn (coupling strength)

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

41. g ton (type of cell) g syn (coupling strength) This explains wider range of bursting

42. g ton (type of cell) g syn (coupling strength) This explains wider range of bursting Or DOES IT???

43. Follow the transition curve in (g ton,g syn ) space Where {h’=0} intersects the homoclinic point

44. Follow the transition curve in (g ton,g syn ) space Where {h’=0} intersects the homoclinic point Underestimates bursting region

45. WHY? Because the synchronous solution is unstable

46. Network 2: two connected cells n i ′ = (n ∞ (V i ) – n i )/  n (V i ) s i ′ =  (1-s i )H(V i -  )-  s i 2 slow variables: h i ′ =  (h ∞ (V i ) – h i )/  h (V i ) C m V i ′ = - I L - I K - I Na - I NaP - I ton - I syn I ton = g ton (V i -V syn ) I syn = g syn s j (V i -V syn ) i ∈ {1,2}, j=3-i

47. Simplification for larger g syn : h1 ≈h2

48. n i ′ = (n ∞ (V i ) – n i )/  n (V i ) s i ′ =  (1-s i )H(V i -  )-  s i h′ =  (h ∞ (V i ) – h)/  h (V i ) C m V i ′ = - I L - I K - I Na - I NaP - I ton - I syn I ton = g ton (V i -V syn ) I syn = g syn s j (V i -V syn ) i ∈ {1,2}, j=3-i h1 ≈h2

49. Synchronous Anti-synchronous h V1V1 Bursting h1 ≈h2

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

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

52. Features of top-hat bursting: h1 ≈h2 Frequency does not go to zero at the end of a burst

53. Features of top-hat bursting: h1 ≈h2 Different mechanism of (bursting  spiking)

54. Features of top-hat bursting: h1 ≈h2 Different mechanism of (bursting  spiking) - Reduce full system to equations for slow variables:

55. Features of top-hat bursting: h1 ≈h2 Different mechanism of (bursting  spiking) - Reduce full system to equations for slow variables: silent phase VinisiVinisi ′ = F (V i,n i,s i,h) h′ =  G (V i,h)  =  t

56. Features of top-hat bursting: h1 ≈h2 Different mechanism of (bursting  spiking) - Reduce full system to equations for slow variables: silent phase h′ = G (V i,h) 0 = F (V i,n i,s i,h)

57. Features of top-hat bursting: h1 ≈h2 Different mechanism of (bursting  spiking) - Reduce full system to equations for slow variables: active phase hLhL hRhR For h L < h < h R let: (V i (t,h),n i (t,h),s i (t,h)) - periodic orbit T(h) - period When   0 h′ = (1/T(h))∫ (h ∞ (V i (t,h))-h)/  h (V i (t,h)) dt ≡ a(h) 0 T(h) Bursting: a(h) < 0 for h L < h < h R

58. Features of top-hat bursting: h1 ≈h2 Different mechanism of (bursting  spiking) - Reduce full system to equations for slow variables: active phase hLhL hRhR For h L < h < h R let: (V i (t,h),n i (t,h),s i (t,h)) - periodic orbit T(h) - period When   0 h′ = (1/T(h))∫ (h ∞ (V i (t,h))-h)/  h (V i (t,h)) dt ≡ a(h) 0 T(h) Spiking: a(h R ) 0

59. Features of top-hat bursting: h1 ≈h2 Different mechanism of (bursting  spiking) - Reduce full system to equations for slow variables: active phase hLhL hRhR For h L < h < h R let: (V i (t,h),n i (t,h),s i (t,h)) - periodic orbit T(h) - period When   0 h′ = (1/T(h))∫ (h ∞ (V i (t,h))-h)/  h (V i (t,h)) dt ≡ a(h) 0 T(h) Transition: a(h L ) = 0

60. Follow the transition curve a(h L )=0 Predicts transition correctly for high g syn h1 ≈h2

61. Small and moderate g syn : h1  h2 Define region R in (h 1, h 2 ) space such that fast subsystem supports oscillations Reduce full system: for R in (h 1, h 2 ) Transition (bursting  spiking) can be understood by analyzing the phase planes of this system h 1 ′ = (1/T(h 1,h 2 ))∫ G(V 1,p (t,h 1,h 2 ),h 1 ) dt ≡ a 1 (h 1,h 2 ) 0 T(h1,h2) h 2 ′ = (1/T(h 1,h 2 ))∫ G(V 2,p (t,h 1,h 2 ),h 2 ) dt ≡ a 2 (h 1,h 2 ) 0 T(h1,h2)

62. Small and moderate g syn : h1  h2 g syn g ton R symmetric bursting asymmetric bursting asymmetric spiking symmetric spiking

63. This analysis explains: Transitions for small and moderate g syn

64. This analysis explains: Transitions for small and moderate g syn Sharp change in burst duration Predicts different types of bursting and spiking

65. Transitions diagram

66. Transitions for small and moderate g syn Sharp change in burst duration Predicts different types of bursting and spiking This analysis explains:

67. Conclusions New in networks of bursting cells: Coupled square-wave bursters can generate top-hat bursting Activity modes of coupled bursters can be characterized by considering phase space of averaged slow-variable equations New predictions for experiments: Isolated cell has infrequent spikes at the end of a burst, but a cell in the network does not In a pair of cells there can be two different types of bursting and two different types of spiking. Transitions can be made by changing g ton

68. To estimate g syn experimentally: large small

69. J. Best, J. Rubin, D. Terman, M. Wechselberger Supported by NSF (agreement No. 0112050) through Mathematical Biosciences Institute (MBI), OSU Acknowledgments

71. - Motivation (Pre-Botz) - Butera et al. model and some results (dynamic range? Freq jump?) -What is the mathematical structure that underlies it + uncoupled cells: square wave bursting, transition to spiking + coupled cells: what is the correct reduced model? If we had a self-coupled cell, transition to spiking would happen when h’=0 crosses homoclinic point. But is this a valid reduced model? Compute: anti-phase + Numerically: anti-phase solution is a top-hat burster. Transition to spiking is when the average at the saddle-node of periodics is zero – incorrect. + Next: h1 ~=h2 - It is good to know mathematical structure, also good for bio (predictions)

72. g ton (type of cell) g syn (coupling strength)

73. g syn g ton g syn Range of bistability increases

74. g syn g ton Can transition to spiking

75. g ton (type of cell) g syn (coupling strength) This explains vertical bursting to spiking transition

76. 2 experimental figures Diff figure from butera et al for burst duration Slide for H functions Correct figure with h’s Different figure from Janet Insert schematic of h1-h2 plane