Description
The class is called math and the brain. Do a final report on the application of cable equation in signaling. Use the provided pdf file(MODELING SIGNAL) as your main reference file, write more mathematical analysis, use more equations and icons, and explain that you understand the experimental method. Then use 2-3 individual references. word count around 1500
Unformatted Attachment Preview
Modelling Signal Transmission in Passive Dendritic
Fibre Using Discretized Cable Equation
1st Satyabrat Malla Bujar Baruah
2nd Deepshikha Nandi
3rd Soumik Roy
Department of Electronics and
Communication Engineering,
Tezpur University
Tezpur,Assam India
[email protected]
Department of Electronics and
Communication Engineering
Tezpur University
Tezpur, Assam, India
[email protected]
Department of Electronics and
Communication Engineering,
Tezpur University
Tezpur, Assam, India
[email protected]
Abstract—Signal transmission and propagation in neuronal
fibres is one of the complex and fascinating process that leads
to varied dendritic signal dynamics depending on the complexity
of the dendritic arbour. Such process of individual fibres can
be modelled with the cable equation and passive cable representation. In the proposed work, emphasis is given in modelling
of such individual dendritic fibre, since such dendritic arbour
might affect the neuronal dynamics by a varied range. Hence,
an attempt is made to model the signal propagation in passive
dendritic fibre, considering the length and diameter dependence
instead of length constants and time constants of the fibre.
Considering the electrical equivalent cable model, the differential
equation for passive fibre dynamics is modelled. Further, in order
to incorporate the length and diameter of the fibre into the
model, the modelled equation is discretized assuming the passive
fibre to be linear. The results of the modelled equation shows
that the propagation loss in a uniform cable minimizes with
increase in the diameter thus describing the cause of losses in thin
distal dendritic branches whereas the propagation losses increases
with increasing length describes the successful transmission as a
cumulative sum of signals in the primary dendritic branches
connected directly to the soma.
Index Terms—Passive membrane, Cable equation, Signal propagation, Passive fiber, Discretized model.
I. I NTRODUCTION
Cable equation and its solutions to model a passive nerve
fiber dynamics is one of the distinguished development. The
dendritic arbor is one of the important aspect that can be
modeled using the infamous cable equation and assumed
to be one of the major contributor to neural computation.
A small change in the dendritic arbor morphology and its
equivalent electrical circuit might change the overall response
to a wide range of input signals ranging from its propagation
delay to backpropagation in dendritic junctions. These effects
significantly affects the type of computation a cluster of
dendrites performed on incoming signals which might result
in operations such as coherent detection, signal convolution,
filtering and so on. Dendrites at distal end are thin and works
basically as input signal accumulator whereas dendrites with
comparatively larger diameters acts as filters. Thus these
complex dendritic arbor structures are the prime contributors
involved in such complex dynamics and rich computations
such as edge detection etc. in case of neurons in striat cortex.
Goldstein [1] and Rall [2] [3] modelled the change in action
potential shape and velocity of propagation due to change in
cable geometry used the cable equation. This model gives the
idea of 3/2 rule of Rall and Ralls cylinder. Blocking condition
for propagating action potential in terms of geometric ratios
has also been computed in a bifurcating junction. Later, Ralls
cylinder has been the basis for transient behaviour analysis in
Rinzel [4] neuronal dendritic tree. The 3/2 rule of Rall has
been extensively used in order to model dendritic bifurcation.
Ramon [5], Abbot [6], Guy Major [7], Monai [8] Sweilam
[9] are some of the literature that provide a good insight
into the role of the cable equation in modelling of passive
cable properties in neurons. Propagation of action potential
is also one of the important aspect of cable properties in
terms of local signal processing in dendrite arbour. Therefore a
discretized model for emulating propagation of action potential
in passive dendritic arbour has been formulated. Formulation
of the discretized model is based on the assumption that
the passive dendritic cable is linear in nature. In the first
section, basics of cable theory has been discussed which plays
a significant role in modelling of action potential propagation.
In the method section the derivation of the proposed model has
been discussed finally followed by the result and discussion
section where qualitative comparison with real neurons are
discussed.
II. T HE C ABLE E QUATION
Fig.1 is the equivalent representation for a differential length
of a nerve fibre. Considering point A and point B in Fig.1
voltage and current equation are
dV
= −rlon Ia
dx
dIa
= IT
dx
The transmembrane current is
dV
+ Iion
IT = c m
dt
where Iion due to a passive membrane is
Iion = gL (V − EL )
978-1-7281-0744-8/19/$31.00 ©2019 IEEE
Authorized licensed use limited to: Kelvin Smith Library @ CASE. Downloaded on December 15,2023 at 20:59:45 UTC from IEEE Xplore. Restrictions apply.
138
(1)
(2)
(3)
functions of fibre length(L) and diameter(D). RLon , GL and
Cm are considered linear in case of a passive fibre.
Rlon (L, D)
lon L
= 4rπD
2
GL (L, D)
= gL πDL
Cm (L, D)
= cm πDL
where, rlon , gL and m are characteristic parameters, L is the
length from the point of initiation to any point on the cable
where the voltage transient is to be measured and D is the
diameter of the fibre.
B. Tappered Passive Cable
Again comparing a tapered passive fibre with Fig.1, for a
differential length of tapered passive fibre, the voltage and
current equation can be written as
Fig. 1. A section of nerve fibre and its equivalent cable representation in
terms of membrane characteristic parameters.
Equating equation (2) in equation (3) and equation (1) in
equation (2) we get the cable equation for a passive fibre as
dV
1 d2 V
+ gL (V − EL ) = 0
+ cm
2
rlon dx
dt
(4)
A. Uniform Passive Cable
Considering uniform passive cable and comparing with
Fig.1, for a differential length of uniform passive fibre, the
voltage and current equation is as given in equation(1) and
the transmembrane current is given as in equation(3). Equating
for solution, the cable equation for a uniform passive fibre is
given as
1 d2 V
dV
+ gL (V − EL ) = 0
+ cm
rlon dx2
dt
(5)
The equation (5) is the transient response at a given point
provided the propagating current at the point of interest is
known. Assuming propagating current to be Ia at the point of
interest, the above equation describing the transients can be
written as
1 d2 V
dV
+ gL (V − EL ) = Ia
+ cm
rlon dx2
dt
(6)
Considering the propagating velocity Vprop to be very much
greater than the differential length dx, the instantaneous membrane potential can be considered equipotential and the above
equation can be summarized as
Cm
dV
+ GL (V − EL ) = Ia
dt
(7)
out
such that Ia = VinR−V
.
Lon
The equation (7) is the propagation equation where RLon ,
GL and Cm are the lumped parameters and are dependent
4Ia rl on
dv
=−
(8)
dx
πD
dIa
= IT πD
(9)
dx
Equating the above equations, cable equation for a tapered
fibre with a tapering of dD
dx can be written as
πD d2 v
dv
1 dD
+Cm +(V − EL ) GL = Ia 1 +
(10)
4rlon dx2
dt
D dx
Again assuming the differential length of the cable to be
equipotential for instantaneous time, the above equation is
reduced to
dv
1 dD
+ (V − EL ) GL = Ia 1 +
(11)
Cm
dt
D dx
out
as Ia is the
where Ia can be calculated as Ia = VinR−V
Lon
resultant axial current flow between the two point of interest.
In the above equation, if considered a uniform cable, dD
dx = 0
and the equation results is the cable equation for a passive
fibre. Equation (11) also describes a lot about the amount
of axial current flow into the circuit. When the change in
diameter of the fibre is positive, the flow of current increases
which suggests the decrease in axial resistance with increase
in diameter. On the other hand, when the change in diameter
of the fibre is negative, the equation (11) shows the flow of
axial current decreases which suggests that the resistance to
the flow of current in a fibre to gradually decrease as diameter
increases.
III. R ESULTS AND D ISCUSSION
The proposed work is an attempt to model and simulate
signal propagation in a passive dendritic fibre which gives
a comparative model for simulating signal propagation in
a uniform passive cable with a tapered passive cable. The
propagation phenomena is modelled using differential equation
for the uniform as well as tapered passive fibre. The equation
for the tapered fibre is similar to the uniform passive fibre with
incorporation of a gradual linear change in fibre diameter and
is described as ratio of change in diameter with differential
change in length of the fibre.
Authorized licensed use limited to: Kelvin Smith Library @ CASE. Downloaded on December 15,2023 at 20:59:45 UTC from IEEE Xplore. Restrictions apply.
139
length of 100μm and diameters of 2.5μm, 5μm and 7.5μm
respectively. It can be seen from Fig.4, Fig.5 and Fig.6 that
the propagation losses are very high in case of a fibre with
small diameter as compared to a fibre with larger diameter.
This also suggests that the impedance of a thinner cable is
very much higher than a thicker cable as reported in literature
[10] [11].
60
Action potential at x=0μm
Potential at x=50μm
Potential at x=65μm
Potential at x=80μm
Potential at x=100μm
potential(mV)
40
20
0
-20
-40
-60
-80
0
5
10
15
20
25
30
35
60
40
Time(mSec)
Action potential
Propagating potential in an uniform fiber of L=100μm and D=2.5μm
40
20
0
-20
-40
-60
-80
0
2
4
6
8
10
12
14
16
18
20
Time(mSec)
Fig. 4. Propagation response of a nerve fibre of length x = 100μm and
diameter D = 2.5μm of the cable.
60
Action potential
Propagating potential in an uniform fiber of L=100μm and D=5μm
40
potential(mV)
Shown in Fig.2 is the response of the modelled propagation
equation(7) where the input action potential is generated by
injecting 20 nA current step input, output is the potential at
different length of a passive nerve fibre. Membrane potential
measured at some distance x due to an input injected at a
distal dendritic branch shows resemblance with the modelled
passive response as shown in Fig.2. On the other hand, Fig.3
gives clear indication of more attenuation in small fibres as
compared to a fibre with larger diameter. Henzel [10] and
Stuart [11] synaptic EPSP attenuation in pyramidal arbour
found that the attenuation in primary dendrites is about 50%
whereas secondary dendrite signals are attenuated to about
93%. Similarly, Stuart [11] also described this attenuation due
to the passive characteristics of dendrites. Similarly, in the
proposed model, it can be seen that attenuation in uniform
passive fibres with small diameter is much abrupt as compared
to uniform passive fibres with larger diameters due to decrease
in axial resistance with increase in size. The proposed model
holds true for the mentioned literature. On the other hand,
similar findings are reported for signal propagation in dendrites
of mitral cells [12]. It can be seen in Fig.2 and Fig.3 that the
Potential(mV)
Fig. 2. Propagation response of a uniform nerve fibre at different length of
the cable.
20
0
-20
-40
-60
-80
0
2
4
6
8
10
12
14
16
18
20
Time(mSec)
Fig. 5. Propagation response of a nerve fibre of length x = 100μm and
diameter D = 5μm of the cable.
60
Action Potential
Potential at x=80μm when D=8μm
Potential at x=80μm when D=5μm
Potential at x=80μm when D=3μm
Potential at x=80μm when D=1μm
20
60
Action potential
Propagating potential in an uniform fiber of L=100μm and D=7.5μm
40
0
potential(mV)
potential(mV)
40
-20
-40
-60
20
0
-20
-40
-80
0
2
4
6
8
10
12
14
16
18
20
-60
Time(mSec)
-80
0
Fig. 3. Propagation response of a uniform nerve fiber of length x = 80μm
for different diameter of the cable.
major change in attenuation is dependent on the length of fibre
but the spread of the signal is due to change in propagating
fibre diameter. Even though attenuation is also seen in case of
changing fibre diameter, but change in shape of the signal is
basically due to the diameter of the fibre.
Shown in Fig.4, Fig.5 and Fig.6 are the propagation response of action potential in a passive fibre with constant
2
4
6
8
10
12
14
16
18
20
Time(mSec)
Fig. 6. Propagation response of a nerve fibre of length x = 100μm and
diameter D = 7.5μm of the cable.
Shown in Fig.7, are the transients of a passive tapered fibre
with different tapering ratios. It can be seen from the Fig.7 that
increase in tapering ratio along the direction of propagation
results in less severe attenuation as compared to its negative
counterpart.
Authorized licensed use limited to: Kelvin Smith Library @ CASE. Downloaded on December 15,2023 at 20:59:45 UTC from IEEE Xplore. Restrictions apply.
140
ACKNOWLEDGEMENT
60
Action potential
Propagating potential for tappered fiber D=5μm L=100μm tapper ratio=6
Propagating potential for tappered fiber D=5μm L=100μm tapper ratio=3
Propagating potential for tappered fiber D=5μm L=100μm tapper ratio=0
Propagating potential for tappered fiber D=5μm L=100μm tapper ratio=-2
Propagating potential for tappered fiber D=5μm L=100μm tapper ratio=-4
potential(mV)
40
20
This publication is an outcome of the R&D work undertaken
project under the Visvesvaraya PhD Scheme of Ministry of
Electronics & Information Technology, Government of India,
being implemented by Digital India Corporation.
0
-20
R EFERENCES
-40
-60
-80
0
2
4
6
8
10
12
14
16
18
20
time(mSec)
Fig. 7. Propagation response of a tapered nerve fibre of length x = 100μm
and diameter D = 5μm with different tapering ratios.
Comparing propagation of action potential in uniform fibre
Fig.4, Fig.5 and Fig.6 with its tapered or flared counterpart
Fig.7, it can be seen that attenuation in a flared dendrite is
minimized as compared to its uniform or tapered counterparts.
Shown in Fig.2 and Fig.3 are the transient responses of a
dendritic section modelled using the proposed sets of equation
and the responses are similar to those found in the literature
[10] [11] [14]. On the other hand, Fig.2 and Fig.3 are the
dynamics of a dendritic cylinder when an action potential
enters a dendritic section propagates along the cylinder. The
modelled propagating voltage at different location of dendrite
satisfies the responses reported in literature [10] [11] [14]. On
the other hand, the response of the active membrane is also
consistent with the literature. The proposed model shows easy
implementation of active and passive channels to construct
more realistic neuron model. All modelling and simulation of
the proposed model is done in XPP differential equation solver.
The parameters for simulation of the model is taken from the
literature [13].
IV. C ONCLUSION
[1] Goldstein, S. S., & Rall, W. (1974). Changes of action potential shape
and velocity for changing core conductor geometry. Biophysical journal,
14(10), 731-757.
[2] Rall, W. (1962). Theory of physiological properties of dendrites. Annals
of the New York Academy of Sciences, 96(4), 1071-1092.
[3] Rall, W. (2011). Core conductor theory and cable properties of neurons.
Comprehensive physiology, 39-97.
[4] Rinzel, J. (1975). Voltage transients in neuronal dendritic trees. In
Membranes, Ions, and Impulses (pp. 71-83). Springer, Boston, MA.
[5] Ramon, F., Joyner, R. W., & Moore, J. W. (1975). Propagation of
action potentials in inhomogeneous axon regions. In Membranes, Ions,
and Impulses (pp. 85-100). Springer, Boston, MA.
[6] Cao, B. J., & Abbott, L. F. (1993). A new computational method for
cable theory problems. Biophysical journal, 64(2), 303-313.
[7] Major, G., Evans, J. D., & Jack, J. J. B. (1993). Solutions for transients
in arbitrarily branching cables. Biophysical Journal, 65(1), 450-468.
[8] Monai, H., Omori, T., Okada, M., Inoue, M., Miyakawa, H., & Aonishi,
T. (2010). An analytic solution of the cable equation predicts frequency
preference of a passive shunt-end cylindrical cable in response to extracellular oscillating electric fields. Biophysical journal, 98(4), 524-533.
[9] Sweilam, N. H., Khader, M. M., & Adel, M. (2014). Numerical simulation
of fractional Cable equation of spiny neuronal dendrites. Journal of
advanced research, 5(2), 253-259.
[10] Henze, D. A., Cameron, W. E., & Barrionuevo, G. (1996). Dendritic
morphology and its effects on the amplitude and risetime of synaptic
signals in hippocampal CA3 pyramidal cells. Journal of Comparative
Neurology, 369(3), 331-334.
[11] Stuart, G., & Spruston, N. (1998). Determinants of voltage attenuation in
neocortical pyramidal neuron dendrites. Journal of Neuroscience, 18(10),
3501-3510.
[12] Djurisic, M., Antic, S., Chen, W. R., & Zecevic, D. (2004). Voltage
imaging from dendrites of mitral cells: EPSP attenuation and spike trigger
zones. Journal of Neuroscience, 24(30), 6703-6714.
[13] Gerstner, W., & Kistler, W. M. (2002). Spiking neuron models: Single
neurons, populations, plasticity. Cambridge university press.
[14] Golding, N. L., Mickus, T. J., Katz, Y., Kath, W. L., & Spruston,
N. (2005). Factors mediating powerful voltage attenuation along CA1
pyramidal neuron dendrites. The Journal of physiology, 568(1), 69-82.
Signal transmission and propagation is one of the important
aspect that contributes vastly to the aspect of neuronal signal processing. As signal attenuation and propagation delay
might contribute to the function formation in neuronal assemblies. Apart from signal attenuation and delay, successful
transmission of information with minimal loss for accurate
interpretation of transmitted signal is desired. Therefore a
detailed understanding of the transmission process and its
dependence on the shape and length of the fibre is necessary.
Results from the model shows very efficient transmission with
less attenuation in case of a flared fibre as compared to its
tapered and uniform counterpart which may be one of the
probable scenario as signal propagates from dendrites to the
cell body. Whereas, as the signal propagates in a tapered
fibre, attenuation of the signals increases which shows higher
attenuation in case of signals propagating from the cell body
toward the axon or dendrites, which might be the case for
limiting and controlling amount of current transmitted from
one cell to another.
Authorized licensed use limited to: Kelvin Smith Library @ CASE. Downloaded on December 15,2023 at 20:59:45 UTC from IEEE Xplore. Restrictions apply.
141
C H A P T E R
17
Cable Properties and Information
Processing in Dendrites
Michael Beierlein
Neurons form elaborate extensions termed dendrites that receive and integrate a myriad of inhibitory
and excitatory synaptic inputs. The overall size and
branching pattern can differ dramatically between different neuronal types (Fig. 17.1) suggesting that dendritic trees are critical for the computational tasks
carried out by a given neuron. One of the main challenges in modern neuroscience is to understand how
the properties of dendritic trees shape informationprocessing functions in individual neurons. An important step toward this goal is to reveal how different
parts of a neuron interact.
Neurons carry out five basic functions (Fig. 17.2):
1. Generate intrinsic activity (at any given site in the
neuron through voltage-dependent membrane
properties and internal second-messenger
mechanisms).
2. Receive synaptic inputs (mostly in dendrites, to
some extent in cell bodies, and in some cases in
axon hillocks, axon initial segments and axon
terminals).
3. Integrate signals by combining synaptic responses
with intrinsic membrane activity (in dendrites, cell
bodies, axon hillocks and axon initial segments).
4. Encode output patterns in graded potentials or
action potentials (at any given site in the neuron).
5. Distribute synaptic outputs (from axon terminals
and, in some cases, from dendrites).
The electrotonic properties of neurons that underlie
the spread of electrical current are often referred to as
cable properties. Electrotonic theory was first applied
mathematically to the nervous system in the late 19th
century for spread of electric current through nerve
fibers. By the 1930s and 1940s, it was applied to simple
invertebrate axons—the first steps toward the development of the HodgkinHuxley equations (Chapters 12
and 14) for the action potential in the axon.
Mathematically, it is impractical to apply cable theory to branching dendrites, but in the 1960s Wilfrid
Rall showed how this problem could be solved by the
development of computational compartmental models
(Rall, 1964; Rall, 1967; Rall, 1977; Rall and Shepherd,
1968). These models have provided the basis for a theory of dendritic function (Rall et al., 1995).
SPREAD OF STEADY-STATE SIGNALS
Cable Theory Depends on Simplifying
Assumptions
This chapter will first outline a quantitative description of the passive electrical properties of dendritic
trees, which influence synaptic integration. We will
then consider the properties of active dendrites and
their role in more complex types of information
processing.
From Molecules to Networks.
DOI: http://dx.doi.org/10.1016/B978-0-12-397179-1.00017-8
BASIC TOOLS: CABLE THEORY AND
COMPARTMENTAL MODELS
The successful application of cable theory to neurons requires that it be based as closely as possible on
the structural and functional properties of dendritic
processes, despite their obvious complexity. The key to
describing the spread of electrical current through dendrites is a set of carefully chosen simplifying assumptions, which allow the construction of an equivalent
circuit of the electrical properties of such a segment.
These are summarized in Box 17.1 (Rall et al., 1995).
509
© 2014 Elsevier Inc. All rights reserved.
510
17. CABLE PROPERTIES AND INFORMATION PROCESSING IN DENDRITES
FIGURE 17.1 Cell-type specific dendritic arborizations. (A) Alpha motor neuron in cat spinal cord. (B) Interneuron in locust mesothoracic ganglion. (C) Granule cell in mouse olfactory bulb. (D) Spiny projection neuron in rat striatum. (E) Layer 5 pyramidal cell in rat neocortex. (F) Ganglion cell in cat retina. (G) Amacrine cell in salamander retina. (H) Neuron in human nucleus of Burdach. (I) Purkinje neuron in
human cerebellum. (J) Relay neuron in rat ventrobasal thalamus. (K) Purkinje neuron in mormyrid fish. Adapted from Mel (1994).
Electrotonic Spread Depends on the
Characteristic Length Constant
We begin by using the assumptions in Box 17.1 to represent a segment of a process by electrical resistances: an
internal resistance ri connected to the ri of the neighboring
segments and through the membrane resistance rm to
ground (see Fig. 17.3B). We will first consider the spread
of electrotonic potential under steady-state conditions
(Fig. 17.3C). In standard cable theory, this is described by
V5
r m d2 V
U
ri dx2
ð17:1Þ
This equation states that if there is a steady-state
current input at point x 5 0, the electrotonic potential
(V) spreading along the cable is proportional to the
second derivative of the potential (d2V) with respect to
distance and the ratio of the membrane resistance (rm)
to the internal resistance (ri) over that distance. The
steady-state solution of this equation for a cable of infinite extension for positive values of x gives
V 5 V0 e2x=λ ;
ð17:2Þ
where λ is defined as the square root of rm/ri (in centimeters) and V0 is the value of V at x 5 0.
Inspection of this equation shows that when x 5 λ,
the ratio of V to V0 is e21 5 1/e 5 0.37. Thus, λ is a critical parameter defining the length over which the electrotonic potential spreading along an infinite cable
decays (is attenuated) to a value of 0.37 of the value at
the site of the input. It is referred to as the characteristic
length constant of the cable. The higher the value of the
specific membrane resistance (Rm), the higher the value
of rm for that segment, the larger the value for λ, and
the greater the spread of electrotonic potential through
that segment (Fig. 17.4). Specific membrane resistance
(Rm) is thus an important variable in determining how
efficient activity can spread through a dendritic tree.
Most of the electrotonic current crossing the membrane may be carried by K1 “leak” channels, which
are largely responsible for holding the cell at its resting
potential. However, the dendrites of a cell are targeted
by numerous synapses, whose activation will lead to
the opening of a variety of additional ligand and
voltage-gated channels. Thus, the effective Rm can
vary from values of less than 1,000 Ωcm2 to more than
100,000 Ωcm2 at different times and in different
subregions of a dendritic tree. Note that λ varies with
the square root of Rm, so a 100-fold difference in Rm
translates into only a 10-fold difference in λ.
II. PHYSIOLOGY OF ION CHANNELS, EXCITABLE MEMBRANES AND SYNAPTIC TRANSMISSION
511
SPREAD OF TRANSIENT SIGNALS
between rm and Rm, and ri and Ri, discussed in the
preceding section,
rffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi
rm
Rm d
ð17:3Þ
5
U
λ5
ri
Ri 4
Neuronal processes vary widely in diameter. In the
mammalian nervous system, the thinnest processes are
the distal branches of dendrites and the necks of some
dendritic spines; these processes may have diameters
of only 0.1 μm or less. In contrast, the largest dendritic
trunks of mammalian neurons may have diameters as
large as 20 to 25 μm. This means that the range of diameters is approximately three orders of magnitude
(1,000-fold). Note, again, that λ varies with the square
root of d; thus, for a 10-fold change in diameter, the
change in λ is only about 3-fold.
SPREAD OF TRANSIENT SIGNALS
Electrotonic Spread of Transient Signals
Depends on Membrane Capacitance
FIGURE 17.2 Neurons have four main regions and five main
functions. Electrotonic potential spread is fundamental for coordinating the regions and their functions. See text for details.
Conversely, the higher the value of the specific
internal resistance (Ri), the higher the value of ri for
that segment, the smaller the value of λ, and the less
the spread of electrotonic potential through that segment. The value of Ri in mammalian neurons is estimated to be at 200 Ωcm. Unlike Rm, biological changes
in Ri are assumed to be small and are therefore not
critical in determining changes in the spread of current
through dendritic trees. However, the presence of
intracellular organelles in the cytoplasm may alter the
effective Ri, particularly for very thin processes, such
as distal dendritic branches.
Until now, we have considered only the spread of
steady-state inputs. However, most neural signals
change rapidly. In mammalian neurons, fast synaptic
potentials last from 1 to 30 milliseconds. The spread of
such transient signals through the dendritic tree not
only depends on all of the factors discussed for steadystate signals, but also on the membrane capacitance
(cm), which is due to the lipid moiety of the plasma
membrane. The value for the specific membrane capacitance (Cm) is estimated at B1 μF/cm2.
The simplest case demonstrating the effect of membrane capacitance on transient signals is that of a single segment. In the equivalent electrical circuit for a
neural process, the membrane capacitance is placed in
parallel with ohmic components of the membrane
conductance and the driving potentials for ion flows
through
those
conductances
(Fig.
17.3B).
Again neglecting the resting membrane potential, we
take as an example the injection of a current step into a
segment; in this case, the time course of the current
spread to ground is described by the sum of the
capacitive and resistive current (plus the input
current, Ipulse):
dVm
Vm
1
5 Ipulse
dt
R
ð17:4Þ
dVm
1 Vm 5 Ipulse UR;
dt
ð17:5Þ
C
Electrotonic Spread Depends on the Diameter of
a Process
The length constant (λ) depends not only on the
internal and membrane resistance, but also on the
diameter of a process. Thus, from the relations
Rearranging,
RC
where RC 5 τ (τ is the time constant of the membrane).
II. PHYSIOLOGY OF ION CHANNELS, EXCITABLE MEMBRANES AND SYNAPTIC TRANSMISSION
512
17. CABLE PROPERTIES AND INFORMATION PROCESSING IN DENDRITES
BOX 17.1
B A S I C A S S U M P T I O N S U N D E R LY I N G C A B L E T H E O R Y
1. Segments are cylinders. A segment is assumed to
be a cylinder with constant radius.
This is the simplest assumption; however,
compartmental simulations can readily incorporate
different geometrical shapes with differing radii if
needed (Fig. 17.3B).
2. The electrotonic potential is due to a change in the
membrane potential. At any instant of time, the
“resting” membrane potential (Er) at any point on
the neuron can be changed by several means:
injection of current into the cell, extracellular
currents that cross the membrane, and changes in
membrane conductance (caused by a driving force
different from that responsible for the membrane
potential). Electric current then begins to spread
between that point and the rest of the neuron, in
accord with
V 5 Vm 2 Er
where V is the electrotonic potential and Vm is the
changed membrane potential.
Since membrane potential is rarely at rest under
physiological conditions, “resting” potential refers to
the membrane potential at any given instant of time
other than during an action potential or rapid
synaptic potential.
3. Electrotonic current is ohmic. Passive electrotonic
current flow is usually assumed to be ohmic, i.e., in
accord with the simple linear equation
E 5 IR;
where E is the potential, I is the current, and R is the
resistance. This relation is largely inferred from
macroscopic measurements of the conductance of
solutions having the composition of the intracellular
medium, but is rarely measured directly for a given
nerve process. Also largely untested is the likelihood
that at the smallest dimensions (0.1 μm diameter or
less), the processes and their internal organelles may
acquire submicroscopic electrochemical properties
that deviate significantly from macroscopic fluid
conductance values; compartmental models permit
the incorporation of estimates of these properties.
4. In the steady state, membrane capacitance is
ignored. The simplest case of electrotonic spread
occurs from the point on the membrane of a
steady-state change (e.g., due to injected current, a
change in synaptic conductance, or a change in
voltage-gated conductance) so that time-varying
properties (transient charging or discharging of the
membrane) due to the membrane capacitance can be
ignored (Fig. 17.3C).
5. The resting membrane potential can usually be
ignored. In the simplest case, we consider the
spread of electrotonic potential (V) relative to a
uniform resting potential (Er) so that the value of the
resting potential can be ignored. Where the resting
membrane potential may vary spatially, V must be
defined for each segment as
V 5 E m 2 Vr
6. Electrotonic current divides between internal and
membrane resistance. In the steady state, at any
point on a process, current divides into two local
resistance paths: further within the process through
an internal (axial) resistance (ri) or across the
membrane through a membrane resistance (rm) (see
Fig. 17.3C).
7. Axial resistance is inversely proportional to
diameter. Within the volume of the process, current
is assumed to be distributed equally (in other
words, the resistance across the process, in the Y
and Z axes, is essentially zero). Because resistances
in parallel sum reciprocally to decrease the overall
resistance, axial current (I) is inversely proportional
to the cross-sectional area (I ~ A1 ~ πr12 ); thus, a
thicker process has a lower overall axial resistance
than a thinner process. Because the axial resistance
(ri) is assumed to be uniform throughout the
process, the total cross-sectional axial resistance of a
segment is represented by a single resistance,
ri 5 Ri =A;
where ri is the internal resistance per unit length of
cylinder (in ohms per centimeter of axial length), Ri
is the specific internal resistance (in Ωcm), and A
(5πr2) is the cross-sectional area.
The internal structure of a process may contain
membranous or filamentous organelles that can
raise the effective internal resistance. In voltageclamp experiments, the space clamp eliminates
current through ri, so that the only current
remaining is through rm, thereby permitting
isolation and analysis of different ionic membrane
conductances, as in the original experiments of
Hodgkin and Huxley (Fig. 17.3D).
8. Membrane resistance is inversely proportional to
membrane surface area. For a unit length of
II. PHYSIOLOGY OF ION CHANNELS, EXCITABLE MEMBRANES AND SYNAPTIC TRANSMISSION
SPREAD OF TRANSIENT SIGNALS
BOX 17.1
cylinder, the membrane current (im) and the
membrane resistance (rm) are assumed to be
uniform over the entire surface. Thus, by the same
rule of the reciprocal summing of parallel
r
