Bisimulation-Based Abstraction of Sodium-Channel Dynamics in Cardiac-CellModelsAbhishek Murthy & Md. Ariful IslamComputer Science, Stony Brook University

Joint work with: Ezio Bartocci, Flavio Fenton, Scott Smolka and Radu Grosu

Spring 2012 CMACS Virtual PI Meeting

Outline1. MotivationComputational modeling and analysisTowers of abstractionCardiac cell modeling2. ApproachSodium channel abstractionMethodologyParameter Estimation from Finite Traces (PEFT)Rate-Function Identification (RFI)3. ResultsHodgkin-Huxley (HH)-type abstractionSubstitutivity via bisimulation4. Ongoing Work and Summary*

MotivationMathematical ModelingMathematical Model (Possibly Non-linear)Hybridization, over-approximation, abstractionFormal Analysis Exhaustive exploration of state space

Model Checking (MC), Abstract Interpretation (AI), Parameter Estimation.Biological Phenomena (Cardiac excitation: cell & tissue-level behavior)Qualitative/ Quantitative Insights(Abstract parameter and state-space)Computational ModelLinear Hybrid Automata (LHA), Kripke structure, etc. *

*Towers of AbstractionIntermediate model1Intermediate model21st abstraction2nd abstractionseries of abstractions

*Cardiac ElectrophysiologyMacro (tissue) level simulation

Isotropic diffusion of charge from excitable cells to neighbors

*Cell membrane(selective ion permeability)The Iyer Model

*The Minimal ModelScaled membrane potentialAbstract currents fast inward (fi)slow outward (so)Slow inward (si)Amenable to formal analysis, post hybridization

Abstract variables no physiological interpretation

*Hodgkin-Huxley (HH) Formalismfor Sodium Channels

*Sodium Channel Abstraction

*MethodologyParameter Estimation from Finite Traces(PEFT)Rate-Function Identification(RFI)

*Parameter Estimation from Finite Traces (PEFT)Parameter Estimation from Finite Traces(PEFT)

*Parameter Estimation from Finite Traces (PEFT)Time stepTime step

*Rate-Function Identification (RFI)Rate-Function Identification(RFI)

*Rate-Function Identification (RFI)V (mV)V (mV)

*Rate-Function Identification (RFI)V (mV)V (mV)

*Results

*ResultsV(mV)

*Substitutivity via Bisimulation- Labeled Transition Systems (LTS)

*Substitutivity via Bisimulation- Labeled Transition Systems (LTS)mhTimeTime(t)

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*Substitutivity via Bisimulation- Approximate Bisimulation

*Substitutivity via Bisimulation

*Ongoing Work

*SummaryTowers 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

Weve assembled a world class team to combine and advance two mature, and powerful methods*