Using maps to predict activation order in multiphase rhythmsUsing maps to predict activation order...
Transcript of Using maps to predict activation order in multiphase rhythmsUsing maps to predict activation order...
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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
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Motivation
respiratory rhythm generation circuit in mammalian brainstem
tune parameters to attain multi-phase rhythms
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 2 / 16
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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
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 3 / 16
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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
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Fast-slow decomposition
one cell jumps up at a time0 500 1000 1500 2000 2500
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0 500 1000 1500 2000 2500−60
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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)
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 5 / 16
mov_border_faster_h264.mp4Media File (video/mp4)
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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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Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 6 / 16
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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!
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 7 / 16
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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
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m2
m3
C23
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 8 / 16
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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
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h
m3
C13
0 0.1 0.2 0.30
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m2
m3
C23Π
12
Π13
0 0.2 0.4 0.6 0.8 10
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h
m2
C12
similarly, define Π13,Π21,Π23,Π31,Π32maps can be derived explicitly
images of boundaries of regions determine possible jump orders
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 9 / 16
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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
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m2
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m2
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h
m3
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2 up
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 10 / 16
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Numerical example: 1→← 3
→← 2
1 up 3 up0 0.05 0.1 0.15 0.2 0.25 0.30
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m2
m3
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(A)0 0.2 0.4 0.6 0.8 10
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0 0.2 0.4 0.6 0.8 10
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2 up
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 11 / 16
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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
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 12 / 16
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Four examples
blue = cell 1, green = cell 2, red = cell 3
132
1323
13123132
132313213
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Four examplesred, blue, green = cell 1, 2, 3 (resp.) jumps down
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 14 / 16
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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
Jonathan E. Rubin (Pitt) Using maps to predict activation order August 10, 2012 15 / 16
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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
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