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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
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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*
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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. *
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*Towers of AbstractionIntermediate model1Intermediate model21st abstraction2nd abstractionseries of abstractions
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*Cardiac ElectrophysiologyMacro (tissue) level simulation
Isotropic diffusion of charge from excitable cells to neighbors
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*Cell membrane(selective ion permeability)The Iyer Model
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*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
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*Hodgkin-Huxley (HH) Formalismfor Sodium Channels
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*Sodium Channel Abstraction
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*MethodologyParameter Estimation from Finite Traces(PEFT)Rate-Function Identification(RFI)
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*Parameter Estimation from Finite Traces (PEFT)Parameter Estimation from Finite Traces(PEFT)
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*Parameter Estimation from Finite Traces (PEFT)Time stepTime step
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*Rate-Function Identification (RFI)Rate-Function Identification(RFI)
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*Rate-Function Identification (RFI)V (mV)V (mV)
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*Rate-Function Identification (RFI)V (mV)V (mV)
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*Results
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*ResultsV(mV)
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*Substitutivity via Bisimulation- Labeled Transition Systems (LTS)
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*Substitutivity via Bisimulation- Labeled Transition Systems (LTS)mhTimeTime(t)
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*Substitutivity via Bisimulation- Approximate Bisimulation
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*Substitutivity via Bisimulation
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*Ongoing Work
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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
Weve assembled a world class team to combine and advance two mature, and powerful methods*