Efficient Modeling of Excitable Cells Using Hybrid Automata
Radu GrosuSUNY at Stony Brook
Joint work with Pei Ye, Emilia Entcheva and Scott A. Smolka
Background
• Excitable cells– Neuron– Cardiac Cells
• Different concentrations of ions inside and outside of cells form:– Trans-membrane potential– Ion currents through channels across the cell
membrane
channel
Ions and Channels of Excitable Cells
Na+
Na+
Na+
Na+
Na+
Na+
Na+
K+
Ca2+
K+
K+
K+
K+
Ca2+
Ca2+
Ca2+
Cell
Cell
Action Potential (AP)
• Caused by positive ions moving in and then out of the cell membrane.
• 5 stages– Resting– Upstroke– Early Repolarization– Plateau– Final Repolarization
Restitution Property
• Excitable cells respond to different frequency stimuli.
• Each cycle is composed of:
– Action Potential Duration (APD)
– Diastolic Interval (DI)
• Longer DI, longer APD
Restitution Property
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.
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
Circuit for Hodgkin-Huxley Model
V
ELEna
C
EK
gLgKgna
Hybrid Automata (HA)
• Variables• Control Graph
– Modes– Switches
• Init, Inv and flow• Jumps and Actions• Events
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 ).
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
HA for HH Model
Simulation of HH Model
New Features of HA for LRd and NNR Model
• Adding vz to enrich modeling ability
• Using vn to remember the current voltage when the next stimulus is coming.
– Define , , 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.
HA for LRd Model
HA for NNR Model
Simulation for LRd Model
Simulation for NNR Model
Single cell, single AP 3 APs on a 2*2 cell array
Large-scale Spatial Simulation for NNR Model
Re-entry on a 400*400 cell array
Performance Comparison
Future Work
• Using Optimization techniques to derive the parameters for HA model automatically.
• Develop simpler spatial model to further improve efficiency.
Thank you
04/05/2005
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