cycle linear hybrid for excitable cells
TRANSCRIPT
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
Outline
• Excitable cells• Hybrid model for excitable cells• Conclusions and future directions
Excitable Cells
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
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
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
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
MacroAnalyzable but unrealistic
MicroRealistic, but not
analyzable. Simulation is slow.
Excitable Cells
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
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
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
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
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)
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
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
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
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