Bisimulation -Based Abstraction of Sodium-Channel Dynamics in Cardiac-Cell Models

1. Bisimulation-Based Abstraction of Sodium-Channel Dynamics in Cardiac-Cell Models Abhishek Murthy & Md. Ariful Islam Computer Science, Stony Brook University Joint work with: Ezio Bartocci, Flavio Fenton, Scott Smolkaand Radu Grosu Workshop on Systems Biology and Formal Methods (SBFM 2012)

2. Outline 1. Motivation • Computational modeling and analysis • Towers of abstraction • Cardiac cell modeling 2. Approach • Sodium channel abstraction • Methodology • Parameter Estimation from Finite Traces (PEFT) • Rate-Function Identification (RFI) 3. Results • Hodgkin-Huxley (HH)-type abstraction • Substitutivity via bisimulation 4. Ongoing Work and Summary

3. Motivation Minimal Model (ABSTRACT) Qualitative/ Quantitative Insights (Abstract parameter and state-space) Formal Analysis – Exhaustive exploration of state space Model Checking (MC), Abstract Interpretation (AI), Parameter Estimation. Computational Model Linear Hybrid Automata (LHA), Kripke structure, etc. Variables: 4 Parameters: 27 Hybridization, over-approximation, abstraction Tusscher-Noble-Panfilov-03 ... Mathematical Modeling Variables: 17 Parameters: 44 Mathematical Model (Possibly Non-linear) Biological Phenomena (Cardiac excitation: cell & tissue-level behavior) Iyer Model (DETAILED) Intermediate Models Variables: 67 Parameters: 94 • Physiological Insights • Root-cause detection • Personalized treatment • Pharmacology Tower of Abstraction for Cardiac Models Systematic Refinement Abstraction

4. Towers of Abstraction (Approximate bisimulation) Abstract model () Regions of interest (unsafe, invariants, etc.) series of abstractions State space of A Intermediate model 2 Mappings resulting from approx. bisimulation relation 2nd abstraction Intermediate model 1 1st abstraction Physiologically detailed (mechanistic) model () State space of M

5. Cardiac Electrophysiology Action Potential (AP): Myocyte’s response in time to supra-threshold stimulus, measured as membrane potential V • Macro (tissue) – level simulation • Isotropic diffusion of charge from excitable cells to neighbors

6. The Iyer Model Cell membrane (selective ion permeability) JSR NSR Buffer Buffer Subspace • Rate of change in membrane potential () = • - (sum of physiological currents due to ion flows across membrane) • Physiologically detailed - 67 variables • Difficult to simulate and formally analyze -Dominant current in Upstroke phase

7. The Minimal Model Abstract currents fast inward (fi) slow outward (so) Slow inward (si) Scaled membrane potential , , Dominant current in upstroke phase (corresponds to Iyer’s ) 1 0 • Amenable to formal analysis, post hybridization • Abstract variables – no physiological interpretation

8. Hodgkin-Huxley (HH) Formalismfor Sodium Channels Conductance Counts # of m-type particles inside & h-type particles outside extracellular space Na+ ions Lipid bi-layer of cell membrane C O Voltage-gated Na channel Activating (m) gate C O Inactivating (h) gate intracellular space

9. Sodium Channel Abstraction Stable invariant manifold of 8-state model HH-type abstraction Independent m-type and h-type gates Iyer’s 13-state model for Sodium Channel

10. Methodology Iyer’s 13-state model - Parameter Estimation from Finite Traces (PEFT) Set of finite representative behaviors produced at constant (under regularity assumptions) Rate-Function Identification (RFI) 2-state HH-type abstraction -

11. Parameter Estimation from Finite Traces (PEFT) Parameter Estimation from Finite Traces (PEFT) Set of finite representative behaviors produced at constant Solved using MATLAB’s FMINCON (steady state)

12. Parameter Estimation from Finite Traces (PEFT) Time step Time step

13. Rate-Function Identification (RFI) Rate-Function Identification (RFI)

14. Rate-Function Identification (RFI) PEFT PEFT RFI RFI V (mV) V (mV)

15. Rate-Function Identification (RFI) PEFT PEFT RFI RFI V (mV) V (mV)

16. Results Action Potential (AP)

17. Results V(mV) = Maximum L2 error at voltage

18. Substitutivity via Bisimulation- Labeled Transition Systems (LTS) – states – Initial states – inputs – transition relation – Outputs – Output map m h Time Voltage Time m h Initial states – as per literature – chosen by PEFT Time

19. Substitutivity via Bisimulation- Labeled Transition Systems (LTS) – states, – Initial states, – inputs – transition relation – Outputs – Output map m h Time At constant voltage Time (t) m h Initial states – as per literature – chosen by PEFT Time

20. LTS for Sodium channel • At compartmental matrix, law of mass action-based system for set of monomolecular reactions • For all , eigenvalues of non-positive (stable equilibrium)

21. Substitutivity via Bisimulation- Approximate Bisimulation • & approx. bisimilar with precision • if there exists approx. bisimulation relation such that for all : • where – distance metric • For all , there exists such that • For all , there exists such that

22. Substitutivity via Bisimulation Claim: For any , PEFT generates , such that • Initial conditions for chosen by PEFT • Steady state at guaranteed for all • ( - compartmental, system of monomolecular reactions) • returned by FMINCON • For all , band-limited, time discretization possible (steady state)

23. Ongoing Work Voltage Time • Establish regularity assumptions such that

24. Summary • Towers of abstraction – translate analysis results into physiological insights • Sodium channel – m-type and h-type gates • Modeled as being independent (HH-type, 8-state) or dependent (Iyer, 13-state) • 1st abstraction – enforce conditional independence between m-type and h-type • Proof-of-concept of establishing towers of abstraction • PEFT and RFI – optimization-based techniques to identify abstraction • Approximate bisimulation – notion of approximate system equivalence • Prove abstraction and original model approximately bisimilar • Approx. bisimulation ensures Substitutivity