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Learning Cycle‐Linear Hybrid Automata for Excitable Cells
Sayan MitraJoint work with
Radu Grosu, Pei Ye, Emilia Entcheva, I V Ramakrishnan, and Scott Smolka
HSCC 2007Pisa, Italy
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Outline
• Excitable cells• Hybrid model for excitable cells• Conclusions and future directions
Excitable Cells
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Excitable Cells
• An excitable cell generates electrical pulses or action potentials in response to electrical stimulation• Examples: neurons, cardiac cells, smooth muscle cells
• Local regeneration allows electric signal propagation without damping
• Building block for electrical signaling in brain, heart, and muscles
Neurons of a squirrelUniversity College London
Artificial cardiac tissueUniversity of Washington
Excitable Cells
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Interaction of Excitable Cells• Action Potential (AP) depends on stimulus, membrane voltage of neighboring cells, state of cell itself
• Normal: synchronous pulses, spiral waves
• Abnormal: incoherent pulses,wave breakup• Leads to cardiac arrhythmia, epilepsy
time
volta
ge
failed initiation
Threshold
Resting potential
Stim
ulus
Schematic Action Potential
Excitable Cells
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Macro Models of Action Potentials
• Cellular automata• Oscillators and uniform coupling between cells [Kuramoto`84]
• Small‐world network of coupled oscillators [Watts & Strogatz`98]
NiNK
t
N
jiji
i
...1
)sin(1
=
−+=∂∂ ∑
=
θθωθ
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Na+ K+
Micro Models for Action Potentials• Membrane potential for squid giant axon [Hodgkin‐
Huxley`52] • Luo‐Rudy model (1991) for cardiac cells of guinea pig• Neo‐Natal Rat (NNR) model for cardiac cells of rat
Na K L
Inside
Outside
C
3 4.
( ) ( ) ( )N a N a K K L L stC V g m h V V g n V V g V V I= − + − + − +
1 0 1
80
0 1 0 011
0 125
.
( . . )( )
( ) .
n
V
n
V
V
V e
Ve
α
β
−
−
−=
−
=
2 5 0 1
18
2 5 0 11
4
. .
( . . )( )
( )
m
m
V
V
Ve
e
V
V
α
β
−
−
−=
−
=
20
3 0 1
0 07
11.
( ) .
( )
V
V
h
h V
e
e
Vα
β
−
−
=
=+
.( )m m mm mα β α= − + +
.( )h h hh hα β α= − + +
.( )n n nn nα β α= − + +
• Large state‐space• Nonlinear differential equations • Multiple spatial and temporal scales
V
Ist
INa
gNa gK gL CIL ICIK
VNa VLVK
Excitable Cells
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MacroAnalyzable but unrealistic
MicroRealistic, but not
analyzable. Simulation is slow.
Excitable Cells
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Linear Hybrid Approximations for Action Potentials
• Suppose, AP can be partitioned into modes so that in mode M, v can be approximated by:
xi = bMixiv = Sixi, M e {S,U,E,P,F,R}
• bMi’s can be found by Prony’s methodwhich fits sum of exponentials to data
• Mode switches• at the beginning and end of stimulus• when v crosses threshold voltages VM
• But, stimulus can appear at any M• State of cell at the time of arrival of
stimulus influences behavior of cell for the next AP
• bMi’s history dependent
time
volta
ge (v
)
Stimulated(S)
Upstroke(U)
EarlyRepol(E)
Plateau(P)
FinalRepol(F)
Resting(R)
U
E P
F
RS
v ¥
V U
v § VE
v §VF
v §
V R
Hybrid Automata Model for Excitable Cells
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History Dependence of APs• Frequency of stimulation determines
voltage (v0) at the time of appearance of stimulus, which influences shape of next AP
• Lower frequency: longer resting time and v0 closer to resting voltage results in longer AP
• Higher frequency: shorter AP
time
volta
ge
v0
Hybrid Automata Model for Excitable Cells
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History Dependence of APs• Frequency of stimulation determines
voltage (v0) at the time of appearance of stimulus, which influences shape of next AP
• Even higher frequency: conjoined AP, bifurcation
time
volta
ge
v0
stimulation frequency
APD
50 100 150 200
25
20
15
10
Hybrid Automata Model for Excitable Cells
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History Dependence and Restitution
• Frequency of stimulation determines voltage (v0) at the time of appearance of stimulus, which influences shape of next AP
• Restitution curve: APD vs. DI• Slope > 1 indicates breakup of
spiral waves under high frequency stimulation
• Local to global behavior
time
volta
ge
10% of peak
AP Duration (APD)
Diastolic Interval (DI)
v0
Hybrid Automata Model for Excitable Cells
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Cycle Linear Hybrid Automata (CLHA)
• Uncountable family of modes • = μ : Epoch and Regime
• ( , <) is a total order• Linear dynamics in each mode
• Unique g e that is visited infinitely many times
• There exists a snapshot function : Xö , such that for any switch (x1, e1, r1) ö (x2, e2, r2) (i) r2 = g and e2 = (x1), or(ii) r2 ∫g, e2 = e1 and r2 <r1
• = {S,U,E,P,F,R}determined by v0
Mxi = bMi(v0) xiv = Sixi
M e {S,U,E,P,F,R}
v 0:=v
v 0:=v
v 0 := v
v0 := v
Hybrid Automata Model for Excitable Cells
g
U
E P
F
RS
v § VE(v0) v §VF (v0 )
v §
V R(v 0)
v ¥
V T(v 0)
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Identifying CLHA Parameters for a single AP
• Curve segments are Convex, concave or both
• Consequences:• Solutions: might require at least two exponentials• Coefficients ai andbi: positive/negative orreal/complex
• Exponential fitting: Modified Prony’s method [Osborne and Smyth `95]
For each mode, we seek a solution for LTI:
Observable solution is a sum of
...
exponentials
:
1 1
1
1
, (0) ,( , ..., ), [ ]
i
Tn n
n
n b tii
ii
x bx x ab diag b b a a a
v
v a
x
e
=
=
= =
= =
=
= ∑
∑
&
Hybrid Automata Model for Excitable Cells
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Parameters as Functions of History Variable
• Parameters:• Threshold voltages VS, VE, …• Coefficients of differential equations bS1, bS2, bE1, …
• Coefficients in reset maps• From each stimulation frequency in
the training set, we get a corresponding value for bS1, bS2, …, VS, VE, …
• Apply Prony’s method (a second time) to obtain bS1 as a function of v0 :• bM1(v0) = cM1 exp (v0 dM1) + c’M1 exp (v0 d’M1),
for each M e {S,U,E,P,F,R} • VT(v0) = cT exp (v0 dT) + c’T exp (v0 d’T)
Hybrid Automata Model for Excitable Cells
v0
VTVE
VP
VL
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Contributions and Simulation Results
• CLHA as a model for almost periodic systems
• Iterative process to obtain excitable cell model with desired accuracy
• Simulation efficiency (> 8 times faster)[True, Entcheva, et al.]
• Biological interpretation of state variables x1, x2;restitution curve
• Spiral wave generation and breakup
Spiral waves
Breakup
Conclusions: Results
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Future Directions• CLHA for stimulation with different
shapes• CLHAs coupled through‐‐‐pulses or
diffusion‐‐‐for analyzing synchronization conditions
• Specification of spatiotemporal voltage patterns
• Distributed control through targeted stimulation
Conclusions: Future Directions