Learning Cycle-Linear Hybrid Automata for Excitable Cells Radu Grosu SUNY 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 Framework for the following models Luo-Rudy (LRd) model: –Model for cardiac cells of guinea pig –Developed in 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+K+ NaKL Inside Outside C

Action Potential

Currents in an Active Membrane V Inside Outside I st I Na g Na gKgK gLgL C ILIL ICIC IKIK V Na VLVL VKVK

Currents in an Active Membrane V Inside Outside I st I Na g Na gKgK gLgL C ILIL ICIC IKIK V Na VLVL VKVK

Currents in an Active Membrane V Inside Outside I st I Na g Na gKgK gLgL C ILIL ICIC IKIK V Na VLVL VKVK

Currents in an Active Membrane V Inside Outside I st I Na g Na gKgK gLgL C ILIL ICIC IKIK V Na VLVL VKVK

Currents in an Active Membrane V Inside Outside I st I Na g Na gKgK gLgL C ILIL ICIC IKIK V Na VLVL VKVK

Kinetics of a Gate Subunit

The Full Hodgkin-Huxley Model

Frequency Response APD90: AP > 10% AP m DI90: AP < 10% AP m BCL: APD + DI vnvn

Frequency Response APD90: AP > 10% AP m DI90: AP < 10% AP m BCL: APD + DI S1S2 Protocol: (i) obtain stable S1; (ii) deliver S2 with shorter DI vnvn

Frequency Response APD90: AP > 10% AP m DI90: AP < 10% AP m BCL: APD + DI 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 –BCL 1 + DI 2 : from 160ms to 400 ms in 10ms intervals Stimulus: step with amplitude -80  A/cm 2, 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 Roadmap: Linear HA for One AP

Stimulated Roadmap: Linear HA for One AP

Stimulated Roadmap: Cycle-Linear HA for All APs

Stimulated Roadmap: Cycle-Linear HA for All APs

Finding Segmentation Pts Null Pts: discrete 1 st Order deriv. Infl. Pts: discrete 2 nd Order deriv. Seg. Pts: Null Pts and Infl. Pts Segments: between Seg. Pts Problem: too many Infl. Pts Problem: too many segments?

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 1 st Order deriv. Infl. Pts: discrete 2 nd Order deriv. Seg. Pts: Null Pts and Infl. Pts Segments: Between Seg. Pts Problem: too many Infl. Pts Problem: too many segments?

Finding Segmentation Pts Problem: some inflection points disappear in certain regimes Solution: ignore (based on range) additional inflection points

Finding Segmentation Pts 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

Exponential Fitting Exponential fitting: Typical strategy –Fix b i : do linear regression on a i –Fix a i : nonlin. regr. in b i ~> linear regr. in b i via Taylor exp. Geometric requirements: curve segments are –Convex, concave or both –Upwards or downwards Consequences: –Solutions: might require at least two exponentials –Coefficients a i and b i : positive/negative or real/complex Modified Prony’s method: only one that worked well

Stimulated Linear HA for One AP

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

Stimulated Cycle-Linear HA for All APs

Stimulated Cycle-Linear HA for All APs

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 x 1 and x 2 V b 1 –b 2 b2b2 C I2I2 I1I1 b1b1 x1x1 x2x2 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