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

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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, 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,

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, 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, 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 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 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 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 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 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 ) through Mathematical Biosciences Institute (MBI), OSU Acknowledgments

70

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


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