Mathematical Modeling of Action Potential with Transmission Equations and Hodgkin-Huxley Model BENG 221 Problem Solving ReportIntroduction Action potential, a process during which the electrical membrane potential rapidly rises and falls with a distinctive pattern, is almost a universal process in all organisms. Action potential exists in neurons, muscle cells, and other types of endocrine cells. In neurons, propagation of action potential enables the communication between neurons, leading to cognitive functions. In other types of cells, action potential triggers cascades of intracellular processes. For instance, in muscle cells, the propagation of action potential triggers the release of calcium and further results in muscle contraction. Action potential is generated by voltage-gated ion channels that respond to membrane potentials. When the incoming membrane potential is above a certain threshold, these ion channels open. The electrochemical gradient then drives sodium and potassium ions across membrane. Rushing in of the sodium ions is responsible for the depolarization phase (increase in membrane potential), while outward movement of potassium ions, which lags behind the movement of sodium ions, results in repolarization and hyperpolarization phases. The sodium/potassium ion transporters then actively transport these ions against their gradients to restore the original electrochemical gradients. Propagation of action potential is such an essential process in all organisms as they are responsible for cellular processes in multiple organs. The impairment of conduction of action potential can lead to diseases such as multiple sclerosis or even sudden death. Here we try to use transmission equations and Hodgkin-Huxley model to mathematically model the propagation of action potentials as a function of time and distance. As the model depends on a wide array of parameters, such as the input voltage, membrane capacitance, resistance, etc., with such a model established, one can simply modify the parameters, such as these affected by a certain disease, and examine the effects on propagation of action potential. Also, knowing the spatial and temporal distribution of action potential of a specific can let us reversely model the parameters and shed light on the physiological origins of certain diseases. Set-up An unmyelinated axon with radius a would be modeled. Current is allowed to leak back and forth across a cylindrical membrane, every point in the membrane, to the interstitial fluid through capacitive and ion-transport mechanisms. Modeling of membrane as a capacitor is reasonable because it is thin so that the accumulation of charged particles on one side will pull the oppositely charged particles to the other side of the membrane. Interstitial fluid is treated as a shunt, so it does not have resistance. The differential equations that will be derived are also Figure 1 Excitatory postsynaptic potential and an action potential.adapted to muscle fibers as muscle fibers are not myelinated. In our model, a voltage v is implemented at x=0 as an initial -15 mV sawtooth impulse returning to V=0 after 3 ms (more detailed initial and boundary conditions would be described in the next section). A sawtooth impulse is chosen to mimic the shape of an excitatory postsynaptic potential (EPSP), depicted in figure 1. In addition, as current passes through the axon, it generates self-inductance. In summary, the axon model can be described as the circuit diagram, which is also the circuit for telegrapher’s equations, in figure 2. The differential equations that are derived from the circuit are described as following: (1) (2) In which v is potential difference across membrane, i is membrane current per unit length, I is membrane current density, ia is axon current per unit length, r is resistance per unit length of axon material, R is specific resistance of axon, L is axon specific self-inductance, and Ca is axon self-capacitance per unit area per unit length. Solution Figure 3 is the circuit diagram of Hodgkin-Huxley model. The lipid bilayer is represented as a capacitance ( ). Voltage-gated and leak ion channels are represented by nonlinear (gn) and linear (gL) conductances, respectively. The electrochemical gradients driving the flow of ions are represented by batteries (E), and ion pumps and exchangers are represented by current sources (Ip). Figure 3 Circuit diagram for Hodgkin-Huxley model (Reference: http://en.wikipedia.org/wiki/File:Hodgkin-Huxley.jpg) Figure 2 Circuit diagram for axon model and telegrapher’s equations.The time derivative of the potential across the membrane is proportional to the sum of the currents in the circuit. This is represented as follows, The Hodgkin-Huxley expression for I can be separated into four parallel components, the capacitive current ( ), ion currents of potassium and sodium ( and ), and a smaller current ( ) made up of chloride and other ions. The parameters used in the Hodgkin-Huxley equation are as follows, the specific resistances corresponding to the component ion currents can be denoted by , , and . From here on the will be written as . , , and are quantities of empirical convenience. They can also be thought of as the probabilities of given ion in a specific location. (3) WhereAlso, let the Vk, VNa, and Vt denote the equilibrium potential of the corresponding ions and CM the membrane capacitance per unit area. The full Hodgkin-Huxley excitation equation is (4) Now we apply to equation (1), , Or . While using the left term of equation (2), we can get Since and , The transmission equation can be written as (5) Replacing the current by the value we got in equation (4), the following equation for membrane voltage can be obtained, which combines both the processes of transmission down the axoplasm and excitation across the membrane.The can be replaced by without significant loss since the ( ) is pretty small compared to . Let