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