Learning Cycle-Linear Hybrid Automata for Excitable Cells Radu Grosu SUNY at Stony Brook

Learning Cycle-Linear Hybrid Automata for Excitable Cells

Radu GrosuSUNY at Stony Brook

Joint work with

Sayan Mitra and Pei Ye

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)

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

• 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

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.

Ventricular Tachycardia / Fibrillation

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.

Active Membrane

Conductances vary w.r.t. time and membrane potential

Na+ K+

Na K L

Inside

Outside

C

Action Potential

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


V

Inside

OutsideIst

INa

gNa gK gL C

IL ICIK

VNa VLVK


( - )Na Na NaI g V V


V

Inside

OutsideIst

INa

gNa gK gL C

IL ICIK

VNa VLVK



( - )K K KI g V V

( - )L L LI g V V


V

Inside

OutsideIst

INa

gNa gK gL C

IL ICIK

VNa VLVK



( - )K K KI g V V

( - )L L LI g V V

.

CI CV


.

( ) ( ) ( )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



( - )K K KI g V V

( - )L L LI g V V

.

CI CV

Kinetics of a Gate Subunit

( )n V

( )n V

1 nn

1.

( ) n nn n n


.

( ) n n nn n ( )n V

( )n V

1 nn

1.

( ) n nn n n


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

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

Frequency Response

APD90: AP > 10% APm DI90: AP < 10% APm BCL: APD + DI

vn

Frequency Response


S1S2 Protocol: (i) obtain stable S1; (ii) deliver S2 with shorter DI

vn

Frequency Response


S1S2 Protocol: (i) obtain stable S1; (ii) deliver S2 with shorter DI

Restitution curve: plot APD90/DI90 relation for different BCLs

• 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

Stimulated

Roadmap: One AP

Stimulated

off Us v V ons

Uv V

Ev V

Pv V

Rv V

Fv V

Roadmap: Linear HA for One AP

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

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

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

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?


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?


Problem: some inflection points disappear in certain regimes

Solution: ignore (based on range) additional inflection points


Problem: removing points does not preserve desired accuracy

Solution: align and move up/down inflection points

- Confirmed by higher resolution samples

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

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

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

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

Finding CLHA Coefficients

Solution: apply mProny once again on each of the 25 points

Stimulated


( )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

Stimulated


( )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

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

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

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

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

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

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

Learning Cycle-Linear Hybrid Automata for Excitable Cells Radu Grosu SUNY at Stony Brook

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