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Bisimulation-Based Abstraction of Sodium-Channel Dynamics in Cardiac-CellModels

Abhishek Murthy & Md. Ariful IslamComputer Science, Stony Brook University

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

Workshop on Systems Biology and Formal Methods (SBFM 2012)Weve assembled a world class team to combine and advance two mature, and powerful methods1

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

Motivation

Mathematical 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. Physiological InsightsRoot-cause detectionPersonalized treatmentPharmacologyIyer Model(DETAILED)Variables: 67Parameters: 94Minimal Model(ABSTRACT)Variables: 4Parameters: 27Tusscher-Noble-Panfilov-03Variables: 17Parameters: 44AbstractionSystematic Refinement Intermediate ModelsTower of Abstraction for Cardiac Models4

Towers of AbstractionIntermediate model1Intermediate model2State space of MState space of ARegions of interest(unsafe, invariants, etc.)Mappings resulting from approx. bisimulation relation1st abstraction2nd abstractionseries of abstractions5

Cardiac Electrophysiology

Action Potential (AP): Myocytes response in time to supra-threshold stimulus, measured as membrane potential V

Macro (tissue) level simulation

Isotropic diffusion of charge from excitable cells to neighbors6

BufferSubspaceJSRNSRBufferCell membrane(selective ion permeability)The Iyer Model7

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 interpretation108

Hodgkin-Huxley (HH) Formalismfor Sodium Channels

extracellular spaceintracellular spaceNa+ ionsLipid bi-layer of cell membraneActivating (m) gateInactivating (h) gateVoltage-gated Na channelCOCO9

Sodium Channel AbstractionStable invariant manifold of 8-state model

HH-type abstraction

Independent m-type and h-type gates

Iyers 13-state model for Sodium Channel

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MethodologyParameter Estimation from Finite Traces(PEFT)Rate-Function Identification(RFI)11

Parameter Estimation from Finite Traces (PEFT)Parameter Estimation from Finite Traces(PEFT)Solved using MATLABs FMINCON12

Parameter Estimation from Finite Traces (PEFT)

Time stepTime step13

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

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Rate-Function Identification (RFI)

PEFT RFI PEFT RFI V (mV)V (mV)

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Rate-Function Identification (RFI)

PEFT RFI PEFT RFI V (mV)V (mV)

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Results

Action Potential (AP)17

Results

V(mV)18

Substitutivity via Bisimulation- Labeled Transition Systems (LTS)TimeVoltageTimemhmhTime

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Substitutivity via Bisimulation- Labeled Transition Systems (LTS)TimemhmhTimeTime(t)20

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Substitutivity via Bisimulation- Approximate Bisimulation22

Substitutivity via Bisimulation23

Ongoing WorkTimeVoltage

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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