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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 crosssectional 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 parallelplate 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 buildup 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 A3 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 Ami3C), 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 lr (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 Pottsantone during the Spring '08 term at Northeastern.
 Spring '08
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