ECE602 BME I Ordinary Differential Equations in Biomedical Engineering (Cont’d)

BME Model Examples Hodgkin – Huxley Model An empirical model of an action potential in a squid giant axon

BME Model Examples Hodgkin – Huxley Model Separation of charges across a membrane Negative ions along the inside of the cell membrane Positive ions along the outside of the cell membrane Elsewhere the negative and positive ions are approximately evenly distributed

BME Model Examples Hodgkin – Huxley Model Channels allow ions to pass through the membrane Selective Either passive or active

BME Model Examples Hodgkin – Huxley Model Passive Channels Always open Ion specific Responsible for resting potential (Nernst Equation)

BME Model Examples Hodgkin – Huxley Model Active Channels Open or close Ion specific Responsible for action potential

BME Model Examples Hodgkin – Huxley Model Action Potential (V) V Reaches threshold->Na + channel open->inward flow of Na + ->further depolarization->increases Na + conductance->more Na + current->driving V -> concludes with the closure of the Na + channel A similar, but slower change in K + conductance drives V back to the resting potential.

BME Model Examples Hodgkin – Huxley Model Electrical circuit model of the cell membrane V: the internal minus the external potential

BME Model Examples Hodgkin – Huxley Model The principle ionic currents: The sodium current (I Na ) The potassium current (I K ) The leakage current (I L ) g x : the membrane conductance for X V X : the membrane resting potential for X

BME Model Examples Hodgkin – Huxley Model The current flow through a population of channels: s: the proportion of open channels in a population c: the I-V curve of a single channel

BME Model Examples Hodgkin – Huxley Model (1-s) Closed s Open  (V)  (V) (Steady state value) (The time constant)

BME Model Examples Hodgkin – Huxley Model The sodium conductance

BME Model Examples Hodgkin – Huxley Model The potassium conductance

BME Model Examples Hodgkin – Huxley Model The leakage conductance

BME Model Examples Hodgkin – Huxley Model The complete model:

BME Model Examples Hodgkin – Huxley Model

BME Model Examples Hodgkin – Huxley Model

BME Model Examples Hodgkin – Huxley Model Reference: J. Keener and J. Sneyd, “Mathematical Physiology”, Springer: 1998.