# U of U BIOEN 6000 - Simulation of Cardiac Action Potentials Background Information (9 pages)

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## Simulation of Cardiac Action Potentials Background Information

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## Simulation of Cardiac Action Potentials Background Information

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9
School:
University of Utah
Course:
Bioen 6000 - System Physiology I- Cardiovascular, Respiratory and Renal Systems
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Simulation of Cardiac Action Potentials Background Information Rob MacLeod and Quan Ni February 7 2011 1 Introduction The goal of assignments related to this document is to experiment with a numerical simulation of the cardiac action potential The form of this simulation is just as described in class using the Hodgkin Huxley formalism and differential equations to reproduce the currents responsible for the action potential To simulate the cardiac action potential it is necessary to expand the number of channels from the simple squid giant axon case and also to alter the dynamic of these currents compared with the original work of Hodgkin and Huxley This background section and the Matlab code that you will use is primarily the work of Quan Ni a former Bioengineering graduate student and now engineer at Guidant Corporation 2 Derivation In this section we describe the derivation and additional material you need to know in order to simulate cardiac ventricular membranes This assumes you have already studied the text and or read the notes from the lecture on this material the notes from which are also available on the class website The basic equation is the familiar one describing all the currents that travel across the cell membrane of any excitable cell dV I Iion Cm 1 dt where Iion is the sum of all the ionic currents and Cm dV dt is the current that arises from the membrane capacitance There are six major and many minor ionic currents present in cardiac ventricular cell membranes IN a a fast sodium current Isi a slow inward largely calcium current IK a timedependent potassium current IK1 a second time independent potassium current IKp a plateau potassium current and Ib a time independent background current Thus we can write Iion from Equation 1 as Iion IN a Isi IK IK1 IKp Ib 2 We can assume that the instantaneous voltage current relation is linear i e the ionic current for any ion species x Ix t is related to the voltage across the membrane Vm by Ohm s law Ix t gx t V Vm Veqx 1 3 Simulation of Cardiac Action Potentials Bioengineering 6000 6460 where gx is the conductance of the particular ionic channel and Veqx is its Nernst or equilibrium potential The behavior of a cell membrane for most ion channels is not totally ohmic but instead shows some degree of rectification The use of Ohm s law is best justified by past simulation results that have successfully reconstructed membrane potential 1 however one has to be skeptical about the linear assumption under certain conditions such as large and fast changes in calcium concentration 2 The conductance of an ionic channel is determined by the maximal conductance g and the fraction of channels that are open The fraction of channels open is given by some combination of that Hodgkin and Huxley proposed as hypothetical activation variables For the sodium channel they assumed two gating variables m representing activation and h representing inactivation both raised to some integral power such that the resulting simulation matches measured values for membrane voltage and current gN a t V g m t V i h t V j 4 where i and j are positive integers They further assumed m and h to obey first order kinetics of the form dy y y t V 5 dt y V where y represents any gating variable y is the steady state value of y and y is its time constant To adapt this formalism to a particular care one has to determine the rate constants y and y as well as the powers i and j experimentally 3 Numerical Simulations The individual components of ionic currents in the cardiac membrane simulations described here are formulated in terms of Hodgkin Huxley type equations 1 The formulae for gating variables follow those of Beeler and Reuter 3 Ebihara and Johnson 4 and Luo and Rudy models 5 6 7 all of which are based on cardiac ventricular cells The primary goal of the simulation is to approximate the electrical behavior of a single piece of membrane that consists of a membrane capacity with six ionic currents To represent the ionic currents we use a coupled system of eight first order ordinary differential equations At each step in time an algorithm establishes a set of values for the variables involved and then we integrate them based on initial conditions and simulation parameters To determine the membrane potential we sum the individual ionic currents together with any externally applied current to arrive at the charging current for the membrane capacitance which then determines the derivative of the membrane potential Finally the Equation 1 contains all the necessary values and an integration steps yields Vm t It is also possible although outside the scope of this assignment to fix the membrane potential at a desired value and simulate a voltage clamp experiment This then permits the study of the characteristics of gate variable by using a simple exponential expression without the need for integration The integration algorithm used to solve the differential equations for gate variables is based on a hybrid method 8 Briefly for a sufficiently small change of membrane potential over the corresponding time interval t the gate variables remain essentially unchanged and an approximate solution for Equation 5 becomes a simple exponential of the form y t t y y y t e t y 2 6 Simulation of Cardiac Action Potentials Bioengineering 6000 6460 Steady state gate variables 1 0 8 minf 0 6 hinf 0 4 jinf 0 2 80 60 40 20 0 20 Membrane Potential mV Time constants 40 60 80 taum ms 0 15 100 80 60 taum 0 1 tauh 40 tauj 0 05 20 0 100 80 60 40 20 0 20 40 Membrane Potential mV 60 80 tauh tauj ms 0 100 0 100 Figure 1 Steady state values and time constants for the activation m inactivation h and slow inactivation j parameters of the fast inward sodium current IN a where y y y y 7 y 1 y y 8 and are the rate constants y and y depend on transmembrane potential V and are determined by fitting data from voltage clamp experiments 3 1 3 1 1 Ionic Currents The Fast Inward Sodium Current The sodium current in this particular model is described by the equation IN a g N a m3 h j Vm EN a 9 where g N a is the maximum conductance of the sodium channel m and h are activation and inactivation parameters respectively j is a slow inactivation gate for modeling the slow recovery EN a is the Nernst equilibrium potential for sodium 54 8 mV See Fig 1 for steady state values and time constants for these parameters 3 1 2 The Slow Inward Current The slow inward current Isi plays a dominant part in the creation of the

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