Koester - Iohn Koester APPENDIX Current Flow in Neurons...

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Unformatted text preview: Iohn Koester APPENDIX Current Flow in Neurons Definition of Electrical Parameters Potential Difference {V or E] Current (I) Conductance (g) Capacitance (Cl Rules for Circuit Analysis Conductance Current Capacitance Potential Difference Current Flow in Circuits with Capacitance Circuit with Capacitor Circuit with Resistor and Capacitor in Series Circuit with Resistor and Capacitor in Parallel his section reviews the basic principles of electrical circuit theory. Familiarity with this material is iin~ portant for understanding the equivalent circuit model of the neuron developed in Chapters 5 through 9. The sec- tion is divided into three parts: 1. The definition of basic electrical parameters. 2. A set of rules for elementary circuit analysis. 3. A description of current flow in circuits with capaci- tance. Definition of Electrical Parameters Potential Difference (V or 13) Electrical charges exert an electrostatic force on other charges: Like charges repel, opposite charges attract. As the distance between two charges increases, the force that is exerted decreases. Work is done when two charges that initially are separated are brought together: Negative work is done if their polarities are opposite, and positive work if they are the same. The greater the values of the charges and the greater their initial separation, the greater the work that is done (work = If fir) (it, where f is electrostatic force and r is the initial distance between the two charges). Potential difference is a measure of this work. The poten- tial difference between two points is the work that must be done to move a unit of positive charge {1 coulombi, from one point to the other, i.e., it is the potential energy of the charge. One volt [V] is the energy required to move 1 coulomb a distance of 1 meter against a force of 1 newton. 1034 Appendices Current (I) A potential difference exists within a system whenever positive and negative charges are separated. Charge sepa- ration may be generated by a chemical reaction (as in a battery} or by diffusion between two electrolyte solutions with different ion concentrations across a permeability~ selective barrier, such as a cell membrane. If a region of charge separation exists within a conducting medium, then charges move between the areas of potential differ- ence: positive charges are attracted to the region with a more negative potential, and negative charges go to the regions of positive potential. The resulting movement of charges is current flow, which is defined as the net move- ment of positive charge per unit time. In metallic conduc- tors current is carried by electrons, which move in the opposite direction of current flow. In nerve and muscle cells, current is carried by positive and negative ions in solution. One ampere {A} of current represents the move- ment of I coulomb (of charge} per second. Conductance (g) Any object through which electrical charges can flow is called a conductor. The unit of electrical conductance is the siemen [S]. According to Ohm’s law, the current that flows through a conductor is directly proportional to the potential difference imposed across it.I I=V><g Current {A} = Potential difference {V} X Conductance {8}. As charge carriers move through a conductor, some of their potential energy is lost,- it is converted into thermal energy due to the frictional interactions of the charge car- riers with the conducting medium. Each type of material has an intrinsic property called conductivity [0}, which is determined by its molecular structure. Metallic conductors have very high conductiv- ities,- they conduct electricity extremely well. Aqueous solutions with high ionized salt concentrations have somewhat lower values of o, and lipids have very low conductivities—they are poor conductors of electricity and are therefore good insulators. The conductance of an ‘bbiect is proportional to o times its cross-sectional area, divided by its length: _ Area 3 _ {0] X Length The length dimension is defined as the direction along —-—..a—._..._______~___m INote the analogy of this formula for current flow to the other formulas for describing flow,- e.g., bulk flow of a liquid due to a hydrostatic pressure; flow of a solme in response to a concentration gradient,- flow of heat in response to a temperature gradient, etc. In each case flow is proportional to the product of a driving force times a conductance factor. which one measures conductance [between a and b}: Area }"-‘- Length —"} For example, the conductance measured across a piece of cell membrane is less if its length {thickness} is increased, e.g., by niyelination. The conductance of a large area of membrane is greater than that of a small area of mem- branc. Electrical resistance [R] is the reciprocal of conduc- tance, and is a measure of the resistance provided by an object to current flow. Resistance is measured in oluns to]: 1 ohm e {1 siemenl”. Capacitance (C) A capacitor consists of two conducting plates separated by an insulating layer. The fundamental property of a capac- itor is its ability to store charges of opposite sign: positive charge on one plate, negative on the other. A capacitor made up of two parallel plates with its two conducting surfaces separated by an insulator (an air gap} is shown in Figure A—IA, part I. There is a net excess of positive charges on plate 1:, and an equal number of excess negative charges on plate y, resulting in a potential differ- ence between the two plates. One can measure this po- tential difference by determining how much work is required to move a positive test charge from the surface of y to that of X. Initially, when the test charge is at y, it is attracted by the negative charges on ,y, and repelled less strongly by the more distant positive charges on x. The result of these electrostatic interactions is a force I that opposes the movement of the test charge from y to X. As the test charge is moved _to the left across the gap, the attraction by the negative charges on y diminishes, but the repulsion by the positive charges on it increases, with the result that the net electrostatic force exerted on the test charge is constant everywhere between x and 3/ (Figure A—lA, part 2}. Work lW} is force times the distance {D} over which the force is exerted: W:fxD. Therefore, it is simple to calculate the work done in mov- ing the test charge from one side of the capacitor to the other. It is the shaded area under the curve in Figure A—IA, part 2. This work is equal to the difference in elec- trical potential energy, or potential difference, between X and y. Capacitance is measured in farads }F}. The greater the density of charges on the capacitor plates, the greater the force acting on the test charge, and the greater the result- ing potential difference across the capacitor [Figure A—lB}. Thus, for a given'capacitor, there is a linear relationship Appendix A. Current Flow in Neurons 1035 Force I vv V. v' '01'0‘0'6'3 '3 0:0:0°¢ 0’9 9’. ‘s ' ‘o‘o‘q 33? 0000900,; 66060. .150 0,0 c o o 0“. ‘ o’o’o‘o“o°o\’o .’.‘o§‘. , 0.0. 00000. c ‘ ’ 0 00 co fioo‘o‘o‘o‘ assessoahfisro a .O.‘AA.;I‘€A Q. 0‘ C O A“ a 0 D Distance 0 '0 Distance I' 1 2 Force ++++++++++ ++++++++++ ++++++++ .++++++ ++++ Distance '0 ‘0' 3‘0“. '0 ’0'... Waf‘wv Exists” o o ‘ “:40 ‘ofifigo‘o’o‘o‘o‘o‘o‘o‘o’o‘? :.o.g;.:.:::.;:;:;.::,¢a.;¢:, X3». o.o:«:e'-:o:o:o:o§:~:o. N 0999009. c coo 'o‘o‘ ‘c‘o’o‘o """ ’0‘ t u“ c 050‘. ‘9'.“ ‘ .9 ‘ave.‘3‘.fifiéA‘fififik{'2430.0.fl's302t1036102‘302020. D O 4. O Distance FIGURE AH] The factors that affect the potential difference between two plates of a capacitor. . A. As a test charge is moved between two charged plates [1], it must overcome a force (2). The work done against this force is the potential difference between the two plates. B. Increasing the charge density [1} increases the potential dif- ference (2). C. Increasing the area of the plates (ll increases the number 0! charges required to produce a given potential difference {2]. D. Increasing the distance between the two plates (1] increases the potential difference between them (2‘1. the area of the plates increases capacitance, because a greater amount of charge must be deposited on leach1 side g i to produce the same charge density, which is w tat eter— ' Q {coulombsl — C [farms] X V [mks] [Aflll mines the force {acting on the test charge {Figure A—ulA and C]. Increasing the distance D between the plates docs : not change the force acting on the test charge, but it does increase the work that must be done to move it from one side of the capacitor to the other {Figures A—lA and D}. between the amount of charge [Q] stored on its plates and the potential difference across it: where the capacitance, C, is a constant. The capacitance oi a parallel-plate capacitor is deter- { mined by two features of its geometry: the area [Al of the ‘ two plates, and the distance {D} between them. Increasing Appendices Therefore, for a given charge separation between the two plates, the potential difference between them is propor tional to the distance. Put another way, the greater the distance the smaller the amount of charge that must be deposited on the plates to produce a given potential‘dif- ference, and therefore the smaller the capacitance [Equa- tion A—l j. These geometrical determinants of capacitance can be summarized by the equation: A C 0‘ As shown in Equation Ami, the separation of positive and negative charges on the two plates of a capacitor results in a potential difference between them. The converse of this statement is also true: The potential difference across a capacitor is determined by the excess of positive and neg- ative charges on its plates. in order for the potential across a capacitor to change, the amount of electrical charges stored on the two conducting plates must change first. Rules for Circuit Analysis A few basic relatioriships that are used for circuit analysis are listed below. Familiarity with these rules will help in understanding the electric circuit examples that follow. Conductance This is the symbol for a conductor: O_"\/\/\/\/-—O A variable conductor is represented this way: W A pathway with infinite conductance (zero resistance) is called a short circuit, and is represented by a line: 0—-——O Conductances in parallel add: “ A o 53 105 ml Conductances in series add reciproeally: A ‘ - B 0-.—’\ANV‘—-O—WW—o 53 105 i+r=a SM} 5 10 10 gAB = 3.3s. Resistances in series add, while resistances in parallel add reciprocally. Current An arrow denotes the direction of current flow (net move- ment of positive charge). Ohni’s law is r=vg:,—‘,’. When current flows through a conductor, the end that the current enters is positive with respect to the end that it leaves: () Current generatOr o c The algebraic sum of all currents entering or leaving a junction is zero [we arbitrarily define current approaching a junction as positive, and current leaving a junction as negative). in this circuit for junction X, IA=+5A 1,,=—5A h+h=0 WWI— l ( In this circuit for junction y Ia=+3A Ib=-2A 1c=~1A In+Ib+Ic=tl Current follows the path of greateSt conductance [least resistance}. For conductance pathways in parallel, the cur- rent through each path is proportional to its conductance value divided by the total conductance of the parallel com- bination: IT=10A 53:38 gb=25 gc=ss 3:: 1:1 --—--—-—= a Tga'l'gb'l'ge 3A Sb I =1 ——-—=2A b Tga+gh+gc 8r: I=I—-—$5A. c Tga+3b+gc Capacitance This is the symbol for a capacitor: as» Appendix A. Current Flow in Neurons 1037 The potential difference across a capacitor is propor- tional to the charge stored on its plates: Potential Difference This is the symbol for a battery, or electromotive force. It is often abbreviated by the symbol E. The positive pole is always represented by the longer bar. Batteries in series add algebraically, but attention must be paid to their polarities. If their polarities are the same, their absolute values add: 5V 10V VA]; : V. If their polarities are oppositer they subtract: 5 V 10 V VA]; = _5 V. [The convention used here for potential difference is that Van 3 (VA T Val‘l _ A battery drives a current around the circuit from Its positive to its negative terminal: For purposes of calculating the total resistance of a cir— cuit the internal resistance of a battery is set at zero. ,—-\. 1038 Appendices The potential differences across parallel branches of a circuit are equal: vi. : vs. As one goes around a closed loop in a circuit, the alge- braic sum of all the potential differences is zero: 2V+3V+5V—IOV=0. FIGURE A—Z Time course of charging a capacitor. A. Circuit before the switch (S) is closed. B. Immediater after the switch is closed. Current Flow in Circuits with Capacitance Circuits that have capacitive elements are much more complex than' those that have only batteries and conduc- tors. This complexity arises because current flow varies with time in capacitive circuits. The time dependence of the changes in current and voltage in capacitive circuits is illustrated qualitatively in the following three examples. Circuit with Capacitor Current does not actually flow across the insulating gap in a capacitor; rather it results in a build-up of positive and negative charges on the capacitor plates. However, we can measure a current flowing into and out of the terminals of a capacitor. Consider the circuit shown in Figure A—2A. When switch 8 is closed {Figure A—ZBJ, a net positive charge is moved by the batter}l E onto plate a, and an equal amount of net positive charge is withdrawn from plate b. The result is current flowing counterclockwise in the cir— cuit. Since the charges that carry this current flow into or out of the terminals of a capacitor, building up an excess of plus and minus charges on its plates, it is called a capac- itive current {16]. Because there is no resistance in this circuit, the battery E can generate a ver)r large amplitude of current, which will charge the capacitance to a value Q = E X C in an infinitesimally short period of time [Figure A—ED]. Circuit with Resistor and Capacitor in Series Now consider what happens if a resistor is added in series with the capacitor in the circuit shown' in Figure A—SA. The maximum current that can be generated when switch C. After the capacitor has become fully charged. 1). Time course of changes in I, and VE in response to closing of the switch. g: rpm FIGURE A-3 Time course of charging a capacitor in series with a resistor, from a constant voltage source (13). 7 A. Circuit before the switch {5) is closed. B. Shortly after the switch is closed. S is closed (Figure A—3B] is now limited by Ohm's law (I = V/R]. Therefore, the capacitor charges more slowly. When ’the potential across the capacitor has finally reached the value Vc = Q/C = E [Figure Ami-3C), there is no longer a difference in potential around the loop,- i.e., the battery voltage (E) is equal and opposite to the voltage across the capacitor, V,. The two thus cancel out, and there is no source of potential difference left to drive a current around the loop. Immediately after the switch is closed the po- tential difference is greatest, so current flow is at a maxi- mum. As the capacitor begins to charge, however, the net potential difference (V, + E] available to drive a current becomes smaller, so that current flow is reduced. The re- sult is that an exponential change in voltage and in cur— rent flow occurs across the resistor and the capacitor. Appendix A. Current Flow in Neurons rmx=£ Va l I i A B C Time —p C. After the capacitor has settled at its new potential. D. Time course of current flow, of the increase in charge depos- ited on the capacitor, and of the increased potential differences across the resistor and the capacitor. Note that in this circuit resistive current must equal ca» pacitative current at all times {see Rules for Circuit Anal- ysis, above]. Circuit with Resistor and Capacitor in Parallel Consider now what happens if we place a parallel resistor and capacitor combination in series with a constant cur- rent generator that generates a current l-r (Figure A4). When switch S is closed [Figure A—4BJ, current starts to flow around the loop. Initially, in the first instant of time after the current flow begins, all of the current flows into the capacitor, i.e., IT = 1,. However, as charge builds up on the plates of the capacitor, a potential difference Vc is gen- 1040 Appendices A 0 Current 0 utput of I -I |generator current m" T generator H S from: h I1 1: B s In I lrrnaic ; Ll In G A a VITHX a ITR Vn JR = IT FIGURE A—4 Time course of charging a capacitor in parallel with a resistor, from a constant current source. A. Circuit before the switch (S) is closed. B. Shortly after'thc switch is closed. erated across it. Since the resistor and capacitor are in parallel, the potential across them must be equal,- thus, part of the total current begins to flow through the resis- tor, such that [RR = VR = Vt. As less and less current flows into the capacitor, its rate of charging will become slower, I Close 3 I l Time—av C. After the charge deposited on the capacitor has reached its final value. D. Time course of changes in 1,, Vs, IR, and VR in response to closing of the switch. this accounts for the exponential shape of the curve of voltage versus time. Eventually, a plateau is reached at which the voltage no longer changes. When this occurs, all of the current flows through the resistor, and ‘VE = V“ = JTR. ...
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This note was uploaded on 04/21/2008 for the course BIO 101 taught by Professor Pott-santone during the Spring '08 term at Northeastern.

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Koester - Iohn Koester APPENDIX Current Flow in Neurons...

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