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

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

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Biological data Existing model Previously…

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

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

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

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

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

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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?

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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 -

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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,

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

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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

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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

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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 )

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

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

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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

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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

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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

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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

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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

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

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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

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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 )

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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

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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

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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)

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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

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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)

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

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g syn g ton Transition to bursting g ton

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g syn h′ = 0 g ton Transition to bursting

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g syn h V Bursting g ton t V

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

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

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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.

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

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

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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

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

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

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

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

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WHY? Because the synchronous solution is unstable

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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

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Simplification for larger g syn : h1 ≈h2

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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

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Synchronous Anti-synchronous h V1V1 Bursting h1 ≈h2

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Bursting h1 ≈h2 NEW: Top-hat bursting

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

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

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

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

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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

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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)

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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

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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

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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

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

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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)

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

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This analysis explains: Transitions for small and moderate g syn

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

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Transitions diagram

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

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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

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To estimate g syn experimentally: large small

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

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- 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)

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

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g syn g ton g syn Range of bistability increases

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g syn g ton Can transition to spiking

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

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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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