using maps to predict activation order in multiphase rhythmsusing maps to predict activation order...

Using maps to predict activation order in multiphaserhythms

Jonathan E. RubinDept. of Mathematics, University of Pittsburgh

2012 SIAM Conference on the Life Sciences

collaborator: David Terman, Ohio Statefunding: National Science Foundation

Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 1 / 16

Motivation

respiratory rhythm generation circuit in mammalian brainstem

tune parameters to attain multi-phase rhythms


Given a network of coupled cells such that each can beactive or silent,

if a parameter set is inadequate, how can we adjust parametersto get desired rhythms?

if a parameter set gives the desired rhythm, how robust is it?

what rhythms are possible from a particular parameter tuning?

challenges: heterogeneous network, complicated rhythms, manyparameters

idea: use separation of timescales to answer these questions withmaps


Model equations - three cells coupled with inhibitionCv ′1 = FNaP(v1, h)− gI (b21i∞(v2) + b31i∞(v3))(v1 − VI )− gEd1(v1 − VE )

Cv ′2 = Fad(v2,m2)− gI (b12i∞(v1) + b32i∞(v3))(v2 − VI )− gEd2(v2 − VE )

Cv ′3 = Fad(v3,m3)− gI (b13i∞(v1) + b23i∞(v2))(v3 − VI )− gEd3(v3 − VE )

h′ = �(h∞(v1)− h)/τh(v1)

m′2 = �(m∞(v2)−m2)/τ2(v2)

m′3 = �(m∞(v3)−m3)/τ3(v3)

with

FNaP(v , h) = −(gNaPmp∞(v)h(v − VNa) + gKdrn4∞(v)(v − VK ) + gL(v − VL))

Fad(v ,m) = −(gadm(v − VK ) + gL(v − VL))

x∞(v) = (1 + exp[(v − θx)/σx ])−1, x ∈ {h,m,mp, n, i} ∼ H(v)

τj(v) = τa,j + τb,j/{1 + exp[(v − θτj )/στj ]}, j ∈ {h, 2, 3}.

NOTE: ∼ 30 parameters; θi importantJonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 4 / 16

Fast-slow decomposition

one cell jumps up at a time0 500 1000 1500 2000 2500

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−40

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0 500 1000 1500 2000 2500−60

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0 500 1000 1500 2000 2500

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release transitions occur when active cell reaches v = θi , withh = h∗,m2 = m

∗2, or m3 = m

∗3: THE RACE IS ON!

(RACE TIME)


mov_border_faster_h264.mp4Media File (video/mp4)

Fast-slow decomposition

one cell jumps up at a time0 500 1000 1500 2000 2500

−60

−40

−20

0 500 1000 1500 2000 2500−60

−40

−20

0 500 1000 1500 2000 2500

−60

−40

−20

!"#$$%"'($

)*$

)+$

),$

release transitions occur when active cell reaches v = θi , withh = h∗,m2 = m

∗2, or m3 = m

∗3: THE RACE IS ON!

!65 !60 !55 !50 !45 !40 !35 !30 !25 !200

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The race

Example:

while cell 1 is up, cells 2 and 3 live on v2- and v3-nullclines

when cell 1 jumps down, this converts m2,m3 into initial conditionsfor v2, v3:

v2(t = 0) =gadm2VK + gLVL + gIb12VIgadm2 + gL + gIb12 + gEd2

,

v3(t = 0) =gadm3VK + gLVL + gIb13VIgadm3 + gL + gIb13 + gEd3

the race: from these ICs, solve v2(t21) = θi to find t21(m2),v3(t31) = θi to find t31(m3)

compare t21(m2) vs. t31(m3) to determine whether cell 2 or cell 3wins race!


Predicting jumping sequences: six 2-d maps

t21(m2) vs. t31(m3) determines race outcome when cell 1 jumps down(similarly for other races)thus, can define curve C23 such that cell 2 (cell 3) wins if (m2,m3)above (below) C23:

C23 = {(m2,m3) : t21(m2) = t31(m3)}

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

m2

m3

C23


Predicting jumping sequences: six 2-d maps (cont.)

if (m2,m3) above C23 s.t. cell 2 wins, then define map

Π12 : (m2,m3) 7→ (h,m3)

from positions of cells 2,3 when cell 1 jumps down to positions ofcells 1,3 when cell 2 jumps down

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

h

m3

C13

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

m2

m3

C23Π

12

Π13

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

h

m2

C12

similarly, define Π13,Π21,Π23,Π31,Π32maps can be derived explicitly

images of boundaries of regions determine possible jump orders


Numerical example: 1,3,2,3,1,3,2,1,3,1

1 up 3 up0 0.05 0.1 0.15 0.2 0.25

0

0.05

0.1

0.15

0.2

0.25

0.3

m2

m3

15

8

10

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

h

m2

2

4

6

9

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

h

m3

3

7

2 up


Numerical example: 1→← 3

→← 2

1 up 3 up0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

m2

m3

1

3

(A)0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

h

m2

2

4

(B)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

h

m3

(C)

2 up


From six maps to onecan reduce to a single map Π from subset of (m2,m3) plane to itself

idea is to map from one jump down of cell 1 to the next

step 1: pick cell to jump next - sets domain in (m2,m3)

step 2: let N2,N3 denote number of jumps of cells 2,3 before 1 jumpsagain; e.g., if cell 3 follows cell 1, would have one of the following:

(a) N3 = N2 : Π(m2,m3) = Π21 ◦ Π32 ◦ (Π23 ◦ Π32)N3−1 ◦ Π13(m2,m3)

(b) N3 = N2 + 1 : Π(m2,m3) = Π31 ◦ (Π23 ◦ Π32)N3−1 ◦ Π13(m2,m3)

if time for cell 1 to jump up is independent of h(0), then can deriveΠ(m2,m3) explicitly (C12, C13 flat)connection with earlier calculation divides (m2,m3) into regions ofdifferent (N2,N3), with explicitly computed boundary curves


Four examples

blue = cell 1, green = cell 2, red = cell 3

132

1323

13123132

132313213


Four examplesred, blue, green = cell 1, 2, 3 (resp.) jumps down


Conclusions

general idea

• use race results to partition phase space based onnext-to-jump• images of boundaries reveal possible activation orders• influence of parameters on boundaries is key• heterogeneity is OK

specific case

• assumed planar dynamics with fast-slowdecomposition• assumed transitions by release• derived explicit formulas for boundary curves and six2-d maps• if jump-up time for one cell independent of slowvariable, compress to single map

reference: J. Rubin and D. Terman, J. Math. Neurosci., 2012, 2:4


Open directions

• transitions by escape and release• practicalities for larger networks/more slow variables• dynamics with noise: curves become blurred/transitionsbecome probabilistic

• smoothing the Heavisides• stability/contraction• chaos

reference: J. Rubin and D. Terman, J. Math. Neurosci., 2012, 2:4


using maps to predict activation order in multiphase rhythmsusing maps to predict activation order...

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