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

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 Smolka and Radu Grosu Spring 2012 CMACS Virtual PI Meeting. Outline.

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

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  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 Spring 2012 CMACS Virtual PI Meeting

  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 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 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

  7. The Minimal Model Abstract currents fast inward (fi) slow outward (so) Slow inward (si) Scaled membrane potential 1 0 • Amenable to formal analysis, post hybridization • Abstract variables – no physiological interpretation

  8. Hodgkin-Huxley (HH) Formalismfor Sodium Channels 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 Parameter Estimation from Finite Traces (PEFT) Rate-Function Identification (RFI)

  11. Parameter Estimation from Finite Traces (PEFT) Parameter Estimation from Finite Traces (PEFT) Solved using MATLAB’s FMINCON

  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)

  18. Substitutivity via Bisimulation- Labeled Transition Systems (LTS) m h Time Voltage Time m h Time

  19. Substitutivity via Bisimulation- Labeled Transition Systems (LTS) – states, – Initial states, – inputs – transition relation – Outputs – Output map m h Time Time (t) m h Time

  20. Substitutivity via Bisimulation- Approximate Bisimulation

  21. Substitutivity via Bisimulation

  22. Ongoing Work Voltage Time

  23. 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

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