Efficient Modeling of Excitable Cells Using Hybrid Automata Radu Grosu SUNY 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 + K+K+ Ca 2+ K+K+ K+K+ K+K+ K+K+ 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 g na, g k, g L : Maximum channel conductance E na, E k, E L : Reversal potential m, n, h: Ion channel gate variables I st : Stimulation current

Circuit for Hodgkin-Huxley Model V ELEL E na C EKEK gLgL gKgK g na

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 –V o, V T and V R Event V S represents the presence of stimulus

HA for HH Model

Simulation of HH Model

New Features of HA for LRd and NNR Model Adding v z to enrich modeling ability Using v n 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