Efficient Modeling of Excitable Cells Using Hybrid Automata Radu Grosu SUNY at Stony Brook

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Efficient Modeling of Excitable Cells Using
Hybrid AutomataRadu GrosuSUNY at Stony Brook

• Joint work with Pei Ye, Emilia Entcheva and Scott
  A. Smolka

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Talk Outline

1. Biological Background
2. Motivation
3. Computational Background
4. Hybrid Automata
5. HA Models of Excitable Cells
6. Simulation Results
7. Conclusions Future Work

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Main Goal

• Computational Efficiency
• Making large-scale simulation practical
• Formal Analysis (in the future)
• Reachability
• Safety
• Liveness

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Background

• Excitable cells
• Neurons
• Cardiac myocytes
• Skeletal muscle cells
• Different concentrations of ions inside and
  outside of cells form
• Trans-membrane potential
• Ion currents cross the cell membrane through
  channels

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Squid Giant Axon (Animation from Marine
Bilogical Laboratory, MA)
1. Squid at rest. 2. Mantle opens. Water enters
the mantle cavity. 3. A signal from the brain is
sent to the stellate ganglion which is
connected to nerve cells (axons) distributed
through the mantle. 4. Nerve impulses travel the
length of these axons. 5. The muscles contract
synchronously, rapidly closing the mantle. 6.
Water is forced out through the siphon, producing
a jet action.
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Cardiac Myocytes (WorldWide Anaesthetist Univ.
of British Columbia)
Gap Junctions
Cardiac Myocytes
Action Potential Propagation
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2D Simulations of Atrial Fibrillation (Kneller et
al., McGill)
Single Spiral Wave
Fast Spiral Wave
Spiral Wave Breakup
Atrial Fibrillation
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Motivation(Hofstra University, NY)

• 1 million deaths annually caused by
  cardiovascular disease in US alone, or more than
  40 of all deaths.
• Almost 25 of these are victims of ventricular
  fibrillation (VF).
• During VF, normal electrical activity of heart is
  masked by higher frequency activation waves,
  leading to small and out-of-phase localized
  contractions.

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Mathematical Models

• Hodgkin-Huxley (HH) model
• Membrane potential for squid giant axon
• Developed in 1952
• Framework for the following models
• Luo-Rudy (LRd) model
• Model for cardiac cells of guinea pig
• Developed in 1991
• Neo-Natal Rat (NNR) model
• Being developed in Stony Brook University by
  Emilia Entcheva et al.

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Who?
Alan Lloyd Hodgkin 1914 1998
Andrew Fielding Huxley 1917
Nobel Preis for Physiology or Medicine in 1963
"for their discoveries concerning the ionic
mechanisms involved in excitation and inhibition
in the peripheral and central portions of the
nerve cell membrane"
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Active Membrane(BiologyMad.com)
- Membrane acts like a capacitor - Discharge
creates an AP - Channels control the potential
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Active Membrane
Na
K
In an Active Membrane, some Conductances vary
with respect to time and the membrane potential
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Action Potential(HyperPhysics, Georgia State
University)
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Action Potential Propagation (BiologyMad.com)
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Action Potential Propagation (BiologyMad.com)
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Action Potential Propagation (BiologyMad.com)
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Currents in an Active Membrane
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The Potasium Channel (Pictures from B. Babadi,
Univ. of Teheran)

• Channel is open iff all 4 subunits are open.

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Kinetics of Potasium Subunits (Pictures from B.
Babadi, Univ. of Teheran)
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The Sodium Channel (Pictures from B. Babadi,
Univ. of Teheran)

• Has three similar fast subunits and a single slow
  subunit.

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The Full Hodgkin-Huxley Model
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Hodgkin-Huxley Model in Action (Applet of A.
Fodor, Stanford)
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Hybrid Automata (HA)(Alur, Henzinger, Sifakis
and others)

• Combine both
• Continuous behavior (Differential Equations)
• Discrete transitions
• Advantages
• Simplicity
• Rich descriptive ability

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Hybrid Automata (HA)

• HA consists of Variables Control graph having
  modes, switches Predicates init, inv, flow for
  each mode Jump conditions and Events for each
  switch.

Simple Thermostat example
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General HA Template
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Assumptions for the Flows

• Each mode corresponds to an open/closed
  configuration of the gates.
• Gate dependence on V is factored into the modes.
• Sodium and potasium gates (conductances) are
  mutually independent of each other.
• Gate (conductance) behavior within a mode is
  given by a linear differential equation
• A step function approximation is too crude.

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Assumptions for the Flows
Solution Assume the inward (INa) and outward
(IK IL) currents are linear!
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Is this justified? (Applet of A. Fodor, Stanford)
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Assumptions for the Flows
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HA for HH Model
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Simulation of HH Model
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Restitution Property (Frequency Response)
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New Features for HA Models

• Capture dependence on the Ca2 ion
• Add new flow variable vz
• Capture restitution nonlinearity
• Add new state variable vn remembering voltage
  value when stimulus occurs.
• Adjust AP slope with cycle constant f
• Adjust AP height duration with constants g, h

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HA for NNR Model
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Simulation for LRd Model
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Simulation for NNR Model
3 APs on a 22 cell array
Single cell, single AP
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Large-scale Spatial Simulation for NNR Model

• Re-entry on a 400400 cell array

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Performance Comparison
Run on a Pentium 4 CPU 3.00GHz, 1G Memory machine
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Conclusion

• Cell excitation used to be modeled by ODE systems
• Hodgkin-Huxley
• Luo-Rudy
• Neo-Natal Rat
• Hybrid automata approach combines
• Differential equations
• Discrete mode switches
• Simulation by using Hybrid automata
• Accurate
• Efficient
• Easily extended to other complex biological
  systems

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Future Work

• Use optimization techniques to automatically
  derive HA model parameters.
• Develop simpler spatial model to further improve
  efficiency (FDM vs. FEM).
• Formal analysis ventricular fibrillation as a
  reachability property.
• Long-term work improved pacemaker/defibrillator
  technology, communicate with prosthesis robots.

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Transmission of a nerve impulse
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Ions and Channels of Excitable Cells
Cell
Na
Na
Na
channel
Na
K
Na
Cell
Na
Ca2
Na
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The Giant Axon of Squid
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Action Potential (AP)

• Caused by ion fluxes - inward (Na, Ca2) and
  outward (K)
• 5 stages
• Resting
• Upstroke
• Early Repolarization
• Plateau
• Final Repolarization

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Restitution Property

• Excitable cells respond differently to stimuli
  with different frequency.
• Each cycle is characterized by
• Action Potential Duration (APD)
• Diastolic Interval (DI)
• Longer DI, longer APD

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Hodgkin-Huxley Model

• C Cell capacitance
• V Trans-membrane voltage
• gna, gk, gL Maximum channel conductance
• Ena, Ek, EL Reversal potential
• m, n, h Ion channel gate variables
• Ist Stimulation current

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Two Ways of Abstraction

• Rational method derive the flow functions from
  the differential equations in the original model
• Empirical method use curve-fitting techniques to
  get the flow functions with the form chosen (here
  we use the form ).

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General HA Template

• 4 control modes
• Resting and Final repolarization (FR)
• Stimulated
• Upstroke
• Early repolarization (ER) and Plateau
• Threshold voltage monitoring mode switches
• Vo, VT and VR
• Event VS represents the presence of stimulus

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HA for LRd Model
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New Features of HA for LRd and NNR Model

• Add vz to capture dependence on the Ca2 ion
• Use vn to remember the current voltage when the
  next stimulus occurs.
• 
  determines the time cell
• stays in mode ER and plateau
• Thus, APD will change with DI
• For NNR model, define
  and
• thus the
  threshold voltages are also influenced by DI.