IntroductionDerivationNumerical SimulationsIonic CurrentsThe Fast Inward Sodium CurrentThe Slow Inward CurrentThe Time-dependent Potassium CurrentTime-independent Potassium Current, IK1Plateau Potassium CurrentBackground CurrentAction PotentialSimulation of Cardiac Action PotentialsBackground InformationRob MacLeod and Quan NiFebruary 7, 20111 IntroductionThe goal of assignments related to this document is to experiment with a numerical simulation ofthe cardiac action potential. The form of this simulation is just as described in class, using theHodgkin-Huxley formalism and differential equations to reproduce the currents responsible for theaction potential. To simulate the cardiac action potential, it is necessary to expand the numberof channels from the simple squid giant axon case, and also to alter the dynamic of these currentscompared with the original work of Hodgkin and Huxley.This background section and the Matlab code that you will use is primarily the work of QuanNi, a former Bioengineering graduate student and now engineer at Guidant Corporation.2 DerivationIn this section we describe the derivation and additional material you need to know in order tosimulate cardiac ventricular membranes. This assumes you have already studied the text and/orread the notes from the lecture on this material, the notes from which are also available on theclass website.The basic equation is the familiar one describing all the currents that travel across the cellmembrane of any excitable cell:I = Iion+ CmdVdt(1)where Iionis the sum of all the ionic currents and CmdVdtis the current that arises from the membranecapacitance.There are six major (and many minor) ionic currents present in cardiac ventricular cell mem-branes: INa, a fast sodium current; Isi, a slow inward (largely) calcium current; IK, a time-dependent potassium current; IK1, a second time-independent potassium current; IKp, a plateaupotassium current; and Ib, a time-independent background current. Thus we can write IionfromEquation 1 asIion= INa+ Isi+ IK+ IK1+ IKp+ Ib(2)We can assume that the instantaneous voltage-current relation is linear, i.e., the ionic currentfor any ion species x, Ix(t), is related to the voltage across the membrane, Vm, by Ohm’s lawIx(t) = gx(t, V )(Vm− Vxeq) (3)1Simulation of Cardiac Action Potentials Bioengineering 6000/6460where gxis the conductance of the particular ionic channel and Vxeqis its Nernst or equilibriumpotential. The behavior of a cell membrane for most ion channels is not totally ohmic but insteadshows some degree of rectification. The use of Ohm’s law is best justified by past simulation resultsthat have successfully reconstructed membrane potential [1], however, one has to be skepticalabout the linear assumption under certain conditions, such as large and fast changes in calciumconcentration [2].The conductance of an ionic channel is determined by the maximal conductance, g, and thefraction of channels that are open. The fraction of channels open is given by some combination ofthat 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 formembrane voltage and current.gNa(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 ofthe formdydt=y∞− y(t, V )τy(V ), (5)where y represents any gating variable, y∞is the steady-state value of y, and τyis its time constant.To adapt this formalism to a particular care, one has to determine the rate constants y∞and τy,as well as the powers, i and j, experimentally.3 Numerical SimulationsThe individual components of ionic currents in the cardiac membrane simulations described hereare formulated in terms of Hodgkin-Huxley type equations[1]. The formulae for gating variablesfollow 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 pieceof membrane that consists of a membrane capacity with six ionic currents. To represent the ioniccurrents, we use a coupled system of eight first order, ordinary differential equations. At each step intime, an algorithm establishes a set of values for the variables involved and then we integrate thembased 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 atthe charging current for the membrane capacitance, which then determines the derivative of themembrane potential. Finally, the Equation 1 contains all the necessary values and an integrationsteps yields Vm(t).It is also possible (although outside the scope of this assignment) to fix the membrane potentialat a desired value and simulate a voltage-clamp experiment. This then permits the study ofthe characteristics of gate variable by using a simple exponential expression without the need forintegration.The integration algorithm used to solve the differential equations for gate variables is based ona hybrid method[8]. Briefly, for a sufficiently small change of membrane potential over the corre-sponding time interval, ∆t, the gate variables remain essentially unchanged, and an approximatesolution for Equation 5 becomes a simple exponential of the formy(t + ∆t) = y∞− (y∞− y(t))e−∆t/τy, (6)2Simulation of Cardiac Action Potentials Bioengineering 6000/6460minfhinfjinf020406080taumtauhtauj−100 −80 −60 −40 −20 0 20 40 60 80 10000.050.10.15tauh,tauj (ms)Membrane Potential (mV)taum (ms)Time constants−100 −80 −60 −40 −20 0 20 40 60 80 10000.20.40.60.81Steady−state gate variablesMembrane Potential (mV)Figure 1: Steady state values and time constants for the activation (m), inactivation (h) and slowinactivation (j) parameters of the fast inward sodium current INa.wherey∞=αyαy+ βy(7)andτy=1αy+ βy(8)are the rate constants. αyand βydepend on transmembrane potential V and are determined byfitting data from voltage clamp experiments.3.1 Ionic Currents3.1.1 The Fast Inward Sodium CurrentThe sodium current in this particular model is described by the equationINa= gNa· m3· h · j · (Vm− ENa), (9)where gNais the