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Learning Cycle-Linear Hybrid Automata for Excitable Cells
Radu GrosuSUNY at Stony Brook
Joint work with
Sayan Mitra and Pei Ye
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Motivation
• Hybrid automata: an increasingly popular formalism for approximating systems with nonlinear dynamics
– modes: encode various regimes of the continuous dynamics
– transitions: express the switching logic between the regimes
• Excitable cells: neuronal, cardiac and muscular cells
– Biologic transistors whose nonlinear dynamics is used to
– Amplify/propagate an electrical signal (action potential AP)
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Motivation
• Excitable cells (EC) are intrinsically hybrid in nature:– Transmembrane ion fluxes and AP vary continuously, yet
– Transition from resting to excited states is all-or-nothing
• ECs modeled with nonlinear differential equations:
– Invaluable asset to reveal local interactions
– Very complex: tens of state vars and hundreds of parameters
– Hardly amenable to formal analysis and control
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• Learn linear HA modeling EC behavior (AP):– Measurements readily available in large amounts
• Analyze HA to reveal properties of ECs:– Setting up new experiments for ECs may take months
• Synthesize controllers for ECs from HA:– Higher abstraction of HA simplifies the task
• Validate in-vitro the EC controllers:– Cells grown on chips provided with sensors and actuators
Project Goals
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Impact
• 1 million deaths annually:– caused by cardiovascular disease in US alone, or
– more than 40% of all deaths.
• 25% of these are victims of ventricular fibrillation:
– many small/out-of-phase contractions caused by spiral waves
• Epilepsy is a brain disease with similar cause: – Induction and breakup of electrical spiral waves.
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Ventricular Tachycardia / Fibrillation
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Mathematical Models
• Hodgkin-Huxley (HH) model (Nobel price):– Membrane potential for squid giant axon
– Developed in 1952. Framework for the following models
• Luo-Rudy (LRd) model:– Model for cardiac cells of guinea pig
– Developed in 1991. Much more complicated.
• Neo-Natal Rat (NNR) model:– Being developed at Stony Brook by Emilia Entcheva
– In-vitro validation framework. Very complicated, too.
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Active Membrane
Conductances vary w.r.t. time and membrane potential
Na+ K+
Na K L
Inside
Outside
C
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Action Potential
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Currents in an Active Membrane
V
Inside
OutsideIst
INa
gNa gK gL C
IL ICIK
VNa VLVK
st Na K L CI I I I I
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Currents in an Active Membrane
V
Inside
OutsideIst
INa
gNa gK gL C
IL ICIK
VNa VLVK
st Na K L CI I I I I
( - )Na Na NaI g V V
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Currents in an Active Membrane
V
Inside
OutsideIst
INa
gNa gK gL C
IL ICIK
VNa VLVK
st Na K L CI I I I I
( - )Na Na NaI g V V
( - )K K KI g V V
( - )L L LI g V V
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Currents in an Active Membrane
V
Inside
OutsideIst
INa
gNa gK gL C
IL ICIK
VNa VLVK
st Na K L CI I I I I
( - )Na Na NaI g V V
( - )K K KI g V V
( - )L L LI g V V
.
CI CV
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Currents in an Active Membrane
.
( ) ( ) ( )Na Na K K L L stV g V V g V V g V V I
V
Inside
OutsideIst
INa
gNa gK gL C
IL ICIK
VNa VLVK
st Na K L CI I I I I
( - )Na Na NaI g V V
( - )K K KI g V V
( - )L L LI g V V
.
CI CV
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Kinetics of a Gate Subunit
( )n V
( )n V
1 nn
1.
( ) n nn n n
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Kinetics of a Gate Subunit
.
( ) n n nn n ( )n V
( )n V
1 nn
1.
( ) n nn n n
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Kinetics of a Gate Subunit
801 0 1
0 1 0 010 125
1.
( . . )( ) ( ) .
V
n nV
VV V e
e
.
( ) n n nn n ( )n V
( )n V
1 nn
1.
( ) n nn n n
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The Full Hodgkin-Huxley Model
3 4.
( ) ( ) ( )Na Na K K L L stCV g m h V V g n V V g V V I
1 0 1
80
0 1 0 01
1
0 125
.
( . . )( )
( ) .
n
V
n
V
V
V e
Ve
2 5 0 1
18
2 5 0 1
1
4
. .
( . . )( )
( )
m
m
V
V
Ve
e
V
V
20
3 0 1
0 07
1
1.
( ) .
( )
V
V
h
h V
e
e
V
.
( )m m mm m .
( )h h hh h .
( )n n nn n
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Frequency Response
APD90: AP > 10% APm DI90: AP < 10% APm BCL: APD + DI
vn
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Frequency Response
APD90: AP > 10% APm DI90: AP < 10% APm BCL: APD + DI
S1S2 Protocol: (i) obtain stable S1; (ii) deliver S2 with shorter DI
vn
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Frequency Response
APD90: AP > 10% APm DI90: AP < 10% APm BCL: APD + DI
S1S2 Protocol: (i) obtain stable S1; (ii) deliver S2 with shorter DI
Restitution curve: plot APD90/DI90 relation for different BCLs
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• Training set: for simplicity 25 APs generated from the LRd
– BCL1 + DI2: from 160ms to 400 ms in 10ms intervals
• Stimulus: step with amplitude -80A/cm2, duration 0.6ms
• Error margin: within 2mV of the Luo-Rudi model
• Test set: 25 APs from 165ms to 405 ms in 10ms intervals
Learning Luo-Rudi
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Stimulated
Roadmap: One AP
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Stimulated
off Us v V ons
Uv V
Ev V
Pv V
Rv V
Fv V
Roadmap: Linear HA for One AP
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Stimulated
off Us v V ons
Uv V
Ev V
Pv V
Rv V
Fv V1 1 1
2 2 2
1 2
x b x
x b x
v x x
1 1
2 2
Pv V /
x a
x a
Roadmap: Linear HA for One AP
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Stimulated
off Us v V
Uv V
Ev V
Pv V
Rv V
Fv V1 1 1
2 2 2
1 2
x b x
x b x
v x x
1 1
2 2
Pv V /
x a
x a
Roadmap: Cycle-Linear HA for All APs
/ nons v v
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Stimulated
( )off U ns v V v / nons v v
( )U nv V v
( )E nv V v
Pv V
( )R nv V v
Fv V 1 1 1
2 2 2
1 2
( )
( )n
n
x b x
x b x
v x x
v
v
1 1
2 2
( )
( )
/
( )
n
n
n
P vv V
x a
x a
v
v
Roadmap: Cycle-Linear HA for All APs
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Finding Segmentation Pts
Null Pts: discrete 1st Order deriv.
Infl. Pts: discrete 2nd Order deriv.
Seg. Pts: Null Pts and Infl. Pts
Segments: between Seg. Pts
Problem: too many Infl. Pts
Problem: too many segments?
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Finding Segmentation Pts
Solution: use a low-pass filter-Moving average and spline LPF: not satisfactory
-Designed our own: remove pts within trains of inflection points
Solution: ignore two inflection points
Null Pts: discrete 1st Order deriv.
Infl. Pts: discrete 2nd Order deriv.
Seg. Pts: Null Pts and Infl. Pts
Segments: Between Seg. Pts
Problem: too many Infl. Pts
Problem: too many segments?
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Finding Segmentation Pts
Problem: some inflection points disappear in certain regimes
Solution: ignore (based on range) additional inflection points
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Finding Segmentation Pts
Problem: removing points does not preserve desired accuracy
Solution: align and move up/down inflection points
- Confirmed by higher resolution samples
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Finding Linear HA Coefficients
For each mode, we seek a solution fo
r LTI:
...
1 1
1
, (0) ,
( , ..., ), [ ]Tn n
n
ii
x bx x a
b diag b b a a a
v x
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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 a
b diag b b a a a
v
v a
x
e
Finding Linear HA Coefficients
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Exponential Fitting
• Exponential fitting: Typical strategy– Fix bi: do linear regression on ai
– Fix ai: nonlin. regr. in bi ~> linear regr. in bi via Taylor exp.
• Geometric requirements: curve segments are– Convex, concave or both – Upwards or downwards
• Consequences:– Solutions: might require at least two exponentials
– Coefficients ai and bi: positive/negative or real/complex
• Modified Prony’s method: only one that worked well
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Stimulated
off Us v V ons
Uv V
Ev V
Pv V
Rv V
Fv V1 1 1
2 2 2
1 2
x b x
x b x
v x x
1 1
2 2
Pv V /
x a
x a
Linear HA for One AP
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Finding CLHA Coefficients
Solution: apply mProny once again on each of the 25 points
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Stimulated
( )off U ns v V v / nons v v
( )U nv V v
( )E nv V v
Pv V
( )R nv V v
Fv V 1 1 1
2 2 2
1 2
( )
( )n
n
x b x
x b x
v x x
v
v
1 1
2 2
( )
( )
/
( )
n
n
n
P vv V
x a
x a
v
v
Cycle-Linear HA for All APs
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Stimulated
( )off U ns v V v / nons v v
( )U nv V v
( )E nv V v
Pv V
( )R nv V v
Fv V 1 1 1
2 2 2
1 2
( )
( )n
n
x b x
x b x
v x x
v
v
1 1
2 2
( )
( )
/
( )
n
n
n
P vv V
x a
x a
v
v
Cycle-Linear HA for All APs
11 12
21 22
1 11 12
2 21 22
( )
( )
n n
n n
v vn
v vn
b v e e
b v e e
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Frequency Response on Test Set
AP on test set: still within the accepted error margin
Restitution on test set: much better than we had before
Frequency response: the best we know for approximate models
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Biological Meaning of x1 and x2
1 1 2 2
1 1 2 1
2 1 2 1
1
2
1 2
2
1
0
.
( )
( )
( )
V x x
x V x
V b x b x
V b x b V x
a a
b V b x
V
b
Vb1 –b2b2 C
I2I1
.
V
b1
x1
x2
Two gates: with constant conductances distributed as above
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Outlook: Modeling Entire Range
• Modes 1&2: require 3 state variables (Na, K, Ca)
• Shape changes dramatically: modes are sidestepped
• Input: consider different shapes and intensities
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Outlook: Analysis and Control
• Safety properties:– How to specify: what kind of temporal/spatial properties?– How to verify: what kind of reachability analysis?
• Liveness properties:– Stability analysis: switching speed and stability/bifurcation
• Controllability:– Design centralized (distributed) controllers: from CLHA– Control task: diffuse spirals and ventricular fibrillation
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CLHA as a TIOA
variables m (1) is a set of including oode f type
(2 states
) is the set of
(3) is a nonempty set of start state
(4)
s
a tuple where:Structured Hybrid Automa on t :
( )
( , , , , , )
X
Q val X
Q
X Q A D P
is a set of
(5) is a set of
(6) is an indexed famil
actions
discrete transitions
, state modely of s
a collection
F of dState ifferemodel for X: ntial and ali
gebraicnvequations
i
A
D Q A Q
P F i
x val(X), solution trajectory of F starts from x.there that
olving X such tha
t
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X discrete variabl (1) includes a of called . Thee type epoch variab le
with and is an
state space , mode set snapshot map mode set SHA with Cycle-linear hybrid automaton (CLAH
s.t.
:A :
)
:X
S X
mode type unique visited
infinitely in any execution infinite mode switc
has ; there is a that is
many times with
(2) For each ( ,
hes.
linear state mo) d, i es a For l. , pFp
each action
the guard (reset map ) can be expressed as a
(r
,
linear predicate
function , coefficients functions of epochesp. ) on with that are
.
( mode swit (3) If is a, , ')
a a
a A
E R
X
x a x
1 2
with and
for some , , , '.epoc
ch .mode ( , ) '.mode ( , ),
if then othh ;
'.epo
erwise
ech pochepoch. The first MS type is call transition.
ed an1 2 2 2
1 1 2 2
: ( )
.
p p p x S x
x x
x p x p
CLHA as a TIOA