Cellular electrophysiology: modeling and simulation

Chapter 2 Cellular electrophysiology: modeling and simulation By Nico Kuijpers. 2.1 Introduction All cells have a difference in voltage across the cell membrane. This is the so-called membrane potential. Some cells, such as neurons and muscle cells, use the membrane potential as a signal. In that case, a cell may be stimulated by application of a small current during a short time period. If the current is sufficiently strong, the membrane potential goes through a large excursion, which is called an action potential, before returning to its resting value. Action potentials are used as signaling mechanism in the brain and to initiate muscular contraction. Cells that are able to generate an action potential are excitable. Examples of excitable cells are neurons, cardiac cells, and smooth and skeletal muscle cells. By establishing electrical connections between the cells, action potentials can propagate through a network of connected cells. Electrical connections between neurons are usually established via synapses, while cardiac muscle cells are electrically connected through so-called gap junctions [5]. Over the past 100 years, physiologists have studied the generation and propagation of action potentials. The first quantitative mathematical model of a propagating action potential was developed by Alan Hodgkin and Andrew Huxley [11]. In 1952, Hodgkin and Huxley published a series of articles describing their experiments and model of the squid giant action potential [10, 7, 6, 8, 9] for which they won the Nobel Prize in Physiology and Medicine in 1963 (shared with J.C. Eccles). Since the Hodgkin-Huxley model is a beautiful example of a physiological model, we discuss the model in these lecture notes. 2.2 Cell membrane models The cell membrane can be modeled as a capacitor in parallel with a number of ionic currents (Figure 2.1). The voltage difference across the membrane is the membrane potential Vmem and is defined 13 14 8E020 Inleiding Meten Extracellular medium Cm gK gNa EK ENa gL Iapp EL Intracellular medium Figure 2.1: Basic components of the Hodgkin−Huxley model. The cell membrane is represented as a capacitance Cm . Voltage-gated Na+ and K+ channels are represented by nonlinear conductances gNa and gK . Leak ion channels are represented by a linear conductance gL . The electrochemical gradients driving the flow of ions are represented by batteries ENa , EK , and EL . The applied stimulus current is represented by a current source Iapp . by Vmem = Vint − Vext , (2.1) where Vint is the intracellular potential and Vext the extracellular potential. The main ionic currents are the sodium (Na+ ) and potassium (K+ ) currents, denoted by INa and IK , respectively. Although there are other currents such as chloride (Cl− ) and calcium (Ca2+ ) currents, in the Hodgkin-Huxley model they are lumped together into one leakage current denoted by IL . From Figure 2.1, we derive the following differential equation for Vmem Cm dVmem + INa + IK + IL = Iapp , dt (2.2) where Cm is the membrane capacitance expressed in µF per cm2 membrane surface and Iapp is the applied current to generate an action potential. All ionic currents and the applied current are expressed in µA per cm2 membrane surface. 2.3 Ionic membrane currents When the cell is at rest, the membrane potential Vmem typically has a value between −80 and −90 mV. Consider ionic membrane current Iion describing the current flow of ion species ion over the membrane. In case the charge of ion is positive (e.g., Na+ , K+ , or Ca2+ ), Iion < 0 means positive charge is flowing into the cell and the membrane depolarizes, i.e., the membrane potential Vmem becomes less negative. In case Iion > 0, positive charge moves out of the cell and the membrane is repolarizing, i.e., Vmem returns to its resting value. The current size of Iion flowing into or out of the cell is related to the intracellular and extracellular concentrations of ion and, since ion is charged, on the membrane potential. The net force acting on Cellular electrophysiology: modeling and simulation 15 the ion thus depends on the electrical and chemical gradients and is referred to as the electrochemical gradient or driving force [2]. The driving force is defined as (Vmem − Eion ), where Eion is the equilibrium potential, or Nernst potential, of ion. Eion is given by the Nernst equation   RT [ion]e , (2.3) ln Eion = zion F [ion]i where R is the universal gas constant (8.3143 J·K−1 ·mol−1 ), T is the absolute temperature (310 K), zion is the valence of ion (1 for Na+ and K+ , and 2 for Ca2+ ), F is Faraday’s constant (96.4867 C/mmol), and [ion]e and [ion]i are the extracellular and intracellular concentrations of ion. The direction of Iion across the membrane is determined by the sign of (Vmem − Eion ). The current size depends on the driving force as well as the conductance gion of the membrane to ion, i.e., Iion = gion (Vmem − Eion ), (2.4) which is equivalent to Ohm’s law. gion depends on the number and states of the ion channels. Let γ denote the conductance of a single channel, N the number of channels, and p the probability of a channel being in the open state, then gion = γN p. (2.5) The product of γ and N determines the maximum conductance Gion = γN and equation (2.4) is usually written as Iion = Gion p (Vmem − Eion ). (2.6) The probability p of a channel being in the open state corresponds to the fraction of channels in the open state in the cell. It is assumed that the channel is controlled by a gate that can be either open or closed. Let α p denote the opening rate constant and β p the closing rate constant. Since p is the fraction of channels in the open state, the rate of opening is equal to α p (1 − p) and the rate of closing is equal to β p p. The dynamics of p are determined by the difference between the rates of opening and closing, i.e., dp = α p (1 − p) − β p p. dt (2.7) At steady-state, the rates of opening and closing are equal, i.e., α p (1 − p) = β p p, from which follows αp . p = p∞ = αp + βp (2.8) (2.9) Let p(t) denote the value of p at time t and p0 the value of p at time t0 , then the solution of differential equation (2.7) is   t − t0 , (2.10) p(t) = p∞ − (p∞ − p0 ) exp − τp 16 8E020 Inleiding Meten 1 9 τ m 0.9 m∞ 0.8 h∞ 0.7 n∞ 8 τ 7 τn h 6 τ [ms] m,h,n [−] 0.6 0.5 5 4 0.4 3 0.3 2 0.2 1 0.1 0 −20 0 20 40 60 Potential [mV] 80 100 120 0 −20 0 20 40 60 Potential [mV] 80 100 120 Figure 2.2: Steady-state values (left) and time constants (right) of gating variables m, h, and n as function of potential V. where time constant τ p is defined by τp = 1 . αp + βp (2.11) Opening rate constant α p and closing rate constant β p depend on Vmem and are usually fitted to experimental data using a Boltzmann-type equation [2]. All ionic membrane currents of the HodgkinHuxley model as well as models describing cardiac electrophysiology are described in this way using one or more gating variables. Examples of these models are the Di Francesco-Noble model of the Purkinje fiber [4], the Beeler-Reuter model of the mammalian ventricular action potential [1], the Priebe-Beuckelmann model [12] and the model by Ten Tusscher et al. [13, 14] describing the human ventricular action potential, and the Courtemanche-Ramirez-Nattel model [3] describing the human atrial action potential. There are many more models available that are based on the principles introduced by Hodgkin and Huxley. Extracellular and intracellular concentrations of Na+ , K+ , Cl− , and Ca2+ are influenced by the ionic membrane currents. In most modern models, the extracellular concentrations are assumed to be constant, while the intracellular concentrations are kept up-to-date (e.g. [3, 14]). In the Hodgkin-Huxley model, also the intracellular ionic concentrations are assumed to be constant, since the amount of ions crossing the membrane is relatively small [11]. Under the assumption that ionic concentrations are constant and not affected by the action potential, also the Nernst potentials remain constant. 2.4 The Hodgkin-Huxley equations In 1952, Hodgkin and Huxley published their model describing the action potential of the squid giant axon (”giant” because of the size of the axon, not the size of the squid) [9]. In the Hodgkin-Huxley model it is assumed that the potential V represents the deviation from the resting membrane potential Vrest . Thus, for the membrane potential Vmem , it holds Vmem = Vrest + V. (2.12) Cellular electrophysiology: modeling and simulation 17 The Hodgkin-Huxley model is defined by the following system of differential equations Cm dV dt dm dt dh dt dn dt = −GNa m3 h (V − VNa ) − GK n4 (V − VK ) − GL (V − VL ) + Iapp , (2.13) = αm (1 − m) − βm m, (2.14) = αh (1 − h) − βh h, (2.15) = αn (1 − n) − βn n. (2.16) Cm = 1.0 µF/cm2 is the membrane capacitance and GNa , GK , and GL are conductances expressed in mS (milli Siemens) per cm2 membrane surface (unit Siemens is defined by S = Ω−1 ). VNa , VK , and VL are adjusted equilibrium potentials for INa , IK , and IL , respectively, and m, h, and n are gating variables. Sodium current INa The sodium conductance is described by two gating variables m and h. Since m is close to zero at rest and increases at the beginning of the action potential, m is called the sodium activation gating variable. On the other hand, h is close to one during rest and decreases during the action potential. Therefore, h is called the sodium inactivation gating variable. The best fit to experimental data was found by describing sodium activation to the third power, i.e., INa = GNa m3 h (V − VNa ), (2.17) where GNa is the maximum INa conductance (GNa = 120 mS/cm2 ) and VNa is the adjusted equilibrium potential for Na+ (VNa = 115 mV). Opening rate constants αm and αh and closing rate constants βm and βh (unit (ms)−1 ) depend on V and are defined by αm = 0.1 βm αh βh 25 − V   (2.18) −1 exp 25−V 10  −V  = 4 exp 18  −V  = 0.07 exp 20 1   = exp 30−V +1 10 In Figure 2.2, the steady-state values m∞ = 1 and τh = αm +β are shown as function of V. m (2.19) (2.20) (2.21) αm αm +βm and h∞ = αh αh +βh and the time constants τm = 1 αm +βm Potassium current IK The potassium conductance is described by a single gating variable n which is called the potassium activation gating variable. The best fit to experimental data was found by describing potassium acti- 18 8E020 Inleiding Meten vation to the fourth power, i.e., IK = GK n4 (V − VK ), (2.22) where GK is the maximum IK conductance (GK = 36 mS/cm2 ) and VK is the adjusted equilibrium potential for K+ (VK = −12 mV). Opening rate constant αn and closing rate constant βn (unit (ms)−1 ) depend on V and are defined by αn = 0.01 exp 10 − V   βn = 0.125 exp 10−V 10 (2.23) −1  −V  (2.24) 80 In Figure 2.2, the steady-state value n∞ = of V. αn αn +βn and the time constant τn = 1 αn +βn are shown as function Leak current IL In contrast with the sodium and the potassium current, the leak current is not described by a gating variable. The conductance of the leak current is assumed to be constant, i.e., IL = GL (V − VL ), (2.25) where GL is the IK conductance (GL = 0.3 mS/cm2 ) and VL is the adjusted equilibrium potential (VL = 10.6 mV). 2.5 Action potential simulation By reformulating equation (2.7) using definitions (2.9) and (2.11), the dynamics of gating variable p are described by dp p∞ − p , = dt τp (2.26) where p∞ represents the steady-state value and τ p the time constant. Let p(k) denote the solution of p at time k∆t, then p(k+1) can be computed by   ∆t (k+1) (k) p = p∞ − (p∞ − p ) exp − . (2.27) τp Using this integration scheme, the action potential of the squid giant axon can be simulated by integrating the Hodgkin-Huxley equations over time. In Figure 2.3, action potentials are shown for various (constant) values of stimulus current Iapp . In addition, traces of the ionic membrane currents INa , IK , and IL are shown as well as traces of the gating variables m, h, and j during one action potential (Iapp = 10 µA/cm2 ). 19 Cellular electrophysiology: modeling and simulation Train of action potentials I app V [mV] 100 Action potential = 10 µA/cm2 100 2 Iapp = 10 µA/cm 50 80 0 0 20 40 60 80 50 0 0 20 40 60 80 I V [mV] 100 app 60 Potential [mV] 100 V [mV] 100 Iapp = 20 µA/cm2 100 40 20 = 50 µA/cm2 0 50 0 0 20 40 60 80 −20 100 0 5 Time [ms] 10 15 Time [ms] Ionic membrane currents Gating variables 800 1 INa m h n 0.9 IK 600 IL 0.8 400 0.7 0.6 m,h,j [−] 2 I [µA/cm ] 200 0 0.5 0.4 −200 0.3 −400 0.2 −600 −800 0.1 0 5 10 Time [ms] 15 0 0 5 10 15 Time [ms] Figure 2.3: Action potentials generated with Hodgkin-Huxley equations. Top-left: Train of action potentials for Iapp = 10, 20, and 50 µA/cm2 . Top-right: Action potential for Iapp = 10 µA/cm2 . Bottomleft: Corresponding traces of ionic membrane currents INa , IK , and IL . Bottom-right: Corresponding traces of gating variables m, h, and n. 20 8E020 Inleiding Meten Bibliography [1] G. W. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibers, J Physiol 268: 177–210 (1977) [2] M. R. Boyett, A. Clough, J. Dekanski, and A. V. Holden, Modelling cardiac excitation and excitability, in A. V. Panfilov and A. V. Holden, editors, Computational Biology of the Heart, pp. 1–47, John Wiley & Sons Ltd, Chichester, UK (1997) [3] M. Courtemanche, R. J. Ramirez, and S. Nattel, Ionic mechanisms underlying human atrial action potential properties: insights from a mathematical model, Am J Physiol Heart Circ Physiol 275: H301–H321 (1998) [4] D. DiFrancesco and D. Noble, A model of cardiac electrical activity incorporating ionic pumps and concentration changes, Phil Trans R Soc Lond B Biol Sci 307: 353–398 (1985) [5] A. Guyton and J. Hall, Textbook of Medical Physiology, W.B. Saunders Company, Philadelphia (1996) [6] A. L. Hodgkin and A. F. Huxley, The components of membrane conductance in the giant axion of Loligo, J Physiol 116: 473–495 (1952) [7] A. L. Hodgkin and A. F. Huxley, Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo, J Physiol 116: 449–472 (1952) [8] A. L. Hodgkin and A. F. Huxley, The dual effect of membrane potential on sodium conductance in the giant axion of Loligo, J Physiol 116: 497–506 (1952) [9] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J Physiol 117: 500–544 (1952) [10] A. L. Hodgkin, A. F. Huxley, and B. Katz, Measurements of current-voltage relations in the membrane of the giant axion of Loligo, J Physiol 116: 424–448 (1952) [11] J. P. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, New York (1998) [12] L. Priebe and D. J. Beuckelmann, Simulation study of cellular electric properties in heart failure, Circ Res 82: 1206–1223 (1998) [13] K. H. W. J. ten Tusscher, D. Noble, P. J. Noble, and A. V. Panfilov, A model for human ventricular tissue, Am J Physiol Heart Circ Physiol 286: H1573–H1589 (2004) [14] K. H. W. J. ten Tusscher and A. V. Panfilov, Alternans and spiral breakup in a human ventricular tissue model, Am J Physiol Heart Circ Physiol 291: H1088–H1100 (2006) 21