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

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

Biological data Existing model Previously…

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

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

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

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

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

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?

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 -

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,

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

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

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

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 )

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

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

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

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

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

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

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

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

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

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 )

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

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

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)

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

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)

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

g syn g ton Transition to bursting g ton

g syn h′ = 0 g ton Transition to bursting

g syn h V Bursting g ton t V

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

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

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.

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

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

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

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

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

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

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

WHY? Because the synchronous solution is unstable

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

Simplification for larger g syn : h1 ≈h2

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

Synchronous Anti-synchronous h V1V1 Bursting h1 ≈h2

Bursting h1 ≈h2 NEW: Top-hat bursting

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

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

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

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

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

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)

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

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

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

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

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)

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

This analysis explains: Transitions for small and moderate g syn

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

Transitions diagram

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

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

To estimate g syn experimentally: large small

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

- 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)

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

g syn g ton g syn Range of bistability increases

g syn g ton Can transition to spiking

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

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