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