1.4 The Equivalent Circuit of a Neuron PDF

1.4 The equivalent circuit of a neuron Dr. Veronica Campanucci Objectives for Section 1.4: To understand basic concepts of electricity To understand why/how a neuronal membrane can be represented as an electrical circuit. Readings: From Boron: Electrophysiology of the cell membrane...

1.4 The equivalent circuit of a neuron Dr. Veronica Campanucci Objectives for Section 1.4: To understand basic concepts of electricity To understand why/how a neuronal membrane can be represented as an electrical circuit. Readings: From Boron: Electrophysiology of the cell membrane Chapter 6 From Kandel: Membrane potential and the passive electrical properties of the neuron Chapter 9 Basics of Electricity Ions are mismatched in the number of protons and electrons, and are therefore electrically charged. Cations are positively charged and anions are negatively charged. Different charges attract, while similar charges repel each other. A potential difference (E) between two points is measured in volts (V) and refers to the potential to do work (moving charge from one place to the other). Current (I) is the net flow of charge from one point to the other and is measured in amperes (A). Resistance (R) is measured in ohms (Ω) and refers to the resistance to the movement of current. The functional properties of the neuron can be represented as an electrical equivalent circuit The usefulness of the Goldman equation is limited, it cannot determine how rapidly the membrane potential changes in response to a change in permeability, or the magnitude of the individual Na+, K+, and Cl− currents. This information can be obtained using the equivalent circuit, which represents all of the important electrical properties of the neuron by a circuit consisting of conductors or resistors, batteries, and capacitors: Plasma membrane = capacitor Channels = resistors (or conductors) Na/K pump = generator Capacitor Capacitance: the ability of a system to store an electric charge. The nonconducting phospholipid bilayer separates the cytoplasm and extracellular fluid, both of which are highly conductive. Figure 6-9 (Boron) The presence of a thin layer of opposing charges on the inside and outside surfaces of the cell membrane, acting as a capacitor, gives rise to the electrical potential difference across the membrane. The electrical potential difference or voltage across a capacitor is: V = Q/C “A charge separation of 1 C across a capacitor of 1 F produces a potential difference of 1 V”. Q, is the net charge (Coulombs; C) C, is the capacitance (Farads; F) Resistance/Conductance The membrane is a leaky capacitor because it has ion channels that can conduct charge. Thus, channels give membrane with conductance (opposite to resistance) Ohm’s law States that the electric current is directly proportional to the voltage and is inversely proportional to the resistance. I, current; V, voltage; V=IxR R, resistance This may also be expressed in terms of conductance (G), which is the inverse of resistance. This is sometimes more convenient when referring to ion channels since their presence adds a specific conductance to the membrane in which it is located. V=IxR V=I/G I=VxG Figure 9-6 Taken from Kandel, ER, Schwartz, JH, & Jessell, TM. Principles of Neural Science, 6th ed. Resistance/Conductance For a particular channel… Ohm’s law iK = Vm x  i, unitary current; Vm, membrane voltage, r, unitary resistance, , unitary conductance Conductance in Siemens (S) Resistance in Ohms () Figure 9-6 Taken from Kandel, ER. et al. Principles of Neural Science, 6th ed. Expression of ionic current Consider a K+ conductance: - There is a K+ concentration gradient, there is also a chemical force driving K+ across the membrane, represented in the equivalent circuit by a battery. - The electromotive force of this battery is given by EK, the Nernst potential for K+ For a neuron that has a membrane voltage and a K+ concentration gradient, the net K+ current is: iK = K × (Vm − EK) electrochemical driving force Figure 9-5 Taken from Kandel, ER. et al. Principles of Neural Science, 6th ed. Calculation of the Nernst potential for K+ : 58 mV [X]o EK = log10 z [X]i 4 mM EK = 58 mV x log10 = -91 mV 155 mM Note that without a concentration gradient, [X]o = [X]i, Ek = 0 mV iK = K × V m iK = K × (Vm − EK) Each type of channel can be represented in the equivalent circuit : unitary or single channel conductance gNa: cell Na conductance Figure 9-9 & 9-10 Taken from Kandel, ER. et al. Principles of Neural Science, 6th ed. The equivalent circuit of a neuron The so-called 'equivalent circuit' (EC) was first proposed by Hodgkin and Huxley (1952). According to this model, an RC circuit (a simple electrical circuit containing a resistor and a capacitor in a parallel configuration) can be used to simulate the main passive electrical properties of neuronal membranes. The equivalent circuit can be made more realistic by incorporating the active ion fluxes driven by the Na-K pump. The pump is electrogenic and keeps the ionic batteries charged. It can be added to the equivalent circuit in the form of a current generator. The equivalent circuit of a neuron Figure 9-11 The equivalent circuit can be used to calculate resting potential, Vm; single or total currents, single or total conductance and battery potential Er. Taken from Kandel, ER. et al. Principles of Neural Science, 6th ed.  Summary A simple electrical circuit containing a resistor and a capacitor in a parallel configuration can be used to simulate the main passive electrical properties of neuronal membranes. - Plasma membrane = capacitor - Channels = resistors (or conductors) - Na/K pump = generator A capacitor is a device that stores electrical energy by accumulating electric charges on two closely spaced surfaces insulated from each other. The membrane is a leaky capacitor because it has ion channels that conduct charge across it. Thus, channels give membrane with conductance (opposite to resistance). The electromotive force of a channel in the equivalent circuit is the battery given by the Nernst potential for the ion. Ionic current is defined by Ohm’s law: iK = gK × (Vm − EK) The equivalent circuit can be used to calculate resting potential, Vm; single or total currents, single or total conductance and battery potential EX. Self-testing questions 1) Why an RC system can be used to simulate the electrical properties of a neuronal membrane. 2) Explain the difference between resistance and conductance 3) Why is the battery represented as the Nernst equilibrium potential? 4) Explain what Ohm’s law states. 5) Explain two situations in which a current for a particular ion will be 0. 6) If ion channels for Ca2+ are blocked by a pharmaceutical agent, which part of the linear equation for the Ca2+ current will be affected? 7) Why does the linear relationship between I and V in slide #7 cross the graph at 0,0?