Passive Properties of Axons: The neuron
described as a wire.
• Passive processing or cable properties.
• Power point taken (in part) from Fundamental
Neuroscience, 3rd Ed. Squires, Berg, Bloom, du
Lac, Ghosh & Spitzer, Academic Press
(Elsevier), 2008
Areas of Neural Processing
1) Generate intrinsic activity
• At any given sight in the neuron intrinsic
activity can be generated through voltagedependant membrane properties and internal
second messenger mechanisms.
2) Receive synaptic input
• Occurs mostly in dendrites, to some extent in
cell bodies, and in some cases in axon hillocks,
initial segments, and axons
3) Integrate signals by combining synapticresponses with intrinsic membrane activity
• Occurs in dendrites, cell bodies, axon hillocks
and initial axon segments.
4) Encodes output patterns in graded potentials
or action potentials
• Happens at any given site in the neuron.
5) Distribute synaptic outputs
• Occurs at axon terminals and, in some cases,
from cell bodies and dendrites.
Toward a theory of neuronal information
processing
• The basic tool: Understanding cable theory
and compartmental models
Spread Of Stead-State Signals
• Are main interest is in the rapid spread of
electric current.
• This spread of current is called electrotonic
properties
• There are 11 assumptions about this spread of
potentials that are not always explicitly
acknowledged.
• The neuron can be compartmetnalized.
A given segment or compartment of a neuron
(1) Segments are cylinders
• A segments is assumed to be a cylinder with
constant radius
The electrotonic potential is due to a
change in the membrane potential
• AT any instant in time the “resting” membrane
potential (Er) at any point on the neuron can be
changed by several means:
•
(a) injection of current into the neuron
•
(b) extracellular currents that cross the
•
membrane
•
(c) changes in membrane conductance
•
(different from that responsible for the
•
membrane potential)
2 (cont.)
• Neurobiologist do not believe the membrane
is ever “at rest”. “At rest” practically speaking
means any potential not including action
potentials and rapid synaptic potentials.
3) Electrotonic current is ohmic
• Passive eletrotonic current flow is usually
assumed to be ohmic; follows ohms law;
•
E = IR
• Where E is the potential (in volts), I is the
current (in amps) and R is resistance (in
ohms).
3 (cont.)
• The membrane following ohms law is inferred
from macroscopic measurements of
conductance of solutions having compositions
of intracellular mediums. Rarely is this tested
for a given neuron.
3 (cont)
• Largely untested is the likelihood that at the
smallest dimensions (0.1μm diameter or less)
the processes and their internal organelles
may acquire submicroscopic electrochemical
properties that deviate significantly from
macroscopic fluid conductance values.
• Modeling the neuron into compartments
allows for estimates of these properties in
conductance terms.
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 injection
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 capacitive current can be ignored.
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) such that the
resting potential can be ignored.
• Where the resting membrane potential may
vary spatially, V must be defined for each
segment as
•
V = Em - Vr
6) Electronic current divides between
internal and membrane resistances.
• 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).
7) Axial current is inversely
proportional to diameter
• Basically, this say as the diameter increases
the resistance decreases.
• 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 = Ri/A
(7 cont)
• Where ri is the internal resistance per unit
length of ri cylinder (in ohms per centimeter
of axial length), Ri is the specific internal
resistance (in ohms centimeter, or ohms cm),
and A (= πr2 ) is the cross-sectional area.
(7 cont.)
• In voltage clamp experiments, the space
clamp eliminates current through ri so that the
only current remaining is through rm there by
permitting isolation and analysis of different
ionic membrane conductances. As in the
original H & H studies.
A given segment or compartment of a neuron
8) Membrane current is inversely proportional
to membrane surface area.
For a unit length of cylinder, the membrane current
(im) and the membrane resistance (rm) are assumed
to be uniform over the entire surface. Using the
summing rule for parallel resistances, the membrane
current is inversely proportional to the membrane
area of the segment so that a thicker process has a
lower overall membrane resistance. Thus
• a thicker process has a lower overall membrane
resistance. Thus,
•
rm = Rm/c
8 cont.
• where rm is the membrane resistance for unit length
of the cylinder (in ohms cm of axial length), Rm is the
specific membrane resistance (in ohms cm) and c (=
2πr) is the circumference of the cylinder.
• Membrane resistance is considered a point; that is
no axial flow within a segment only between
segments
8 cont.
• Membrane current passes through ion channels in
the membrane. The density and types of channels of
channels vary in different processes and may vary
locally in different segments and branches. These
differences are incorporated into compartmental
representations of the processes.
9) The external medium along the process
is assumed to have zero resistivity
• In contrast to the internal axial resistance (ri) which is
relatively high because of the small dimensions of
most nerve processes, the external medium has a
relatively low resistivity for current because of its
relatively large volume. For this reason the resistivity
of the paths either along a process or to ground
generally is regarded as negligible and the potential
outside the membrane is assumed to be everywhere
equivalent to ground.
10) Driving forces on membrane
conductances are assumed to be constant
• Only in small restricted compartments, either
extracellular or intracellular, would this
assumption not hold true.
11) Cables have different boundary
conditions
• The relatively short length of spines and dendrites
impose boundary conditions on solutions when
apply cable theory to neurons.
• The boundary conditions impose significantly effect
on e lectronic spread.
Characteristic length, or space constant (λ),
of a fiber (axon)
• The question is;
what is the
spread of the
current down a
axon from the
site of the
input?
• Standard cable theory uses the relationship
of:
•
V = (rm/ri) (d2V/dx2)
• Given the stead state with input at point x = 0,
the electrotonic potential (V) spreads 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)
•
Space Constant (cont)
• over that distance.
• The steady-state solution of the equation for
a cable of infinite length and positive values of
x is
•
V = Voe-x/λ
• Where lambda is defined as the square root of
rm/ri (in centimeters) and Vo is the value at x=
0
• The solution is relatively easy. Note when x =
λ, -x/λ = -1, the ratio of V to Vo is e-1 = 1/e =
0.37.
• Lambda is a crititcal parameter defining the
length over which the electrotonic potential
will spread along a cable of infinite length.
Lambda is the characteristic length or space
constant of the spread of the potential.
Space constant and real axons
• Because of the open passive channels of axons or
other parts of the neuron and the constant activity of
the neuron the electrotonic current can be carried
through many open channels. The effective Rm can
vary from less than 1000Ω cm2 to more than
1000,000Ω cm2 in different neurons or parts of
neurons.
• The space constant λ depends not only on the
internal and membrane resistance, but also on the
diameter of the process, or
Resistance changes, change lambda
• λ = (rm/ri)1/2 = ((Rm/Ri) (d/4))1/2
• (see assumption 7).
Three different space constants
that very because of specific
internal resistance changes.
Diameter changes that effect lambda
•
•
•
•
Increasing the
diameter,
significantly
changes λ
Summary of space constant λ
• The key to understanding the space constant λ, is
understanding that time is allowed to vary. The only
variable held constant is the amplitude of the
current. If the current’s amplitude is considered 1.0
at the place of injection, λ will be the length between
where the current started and the spot where the
current has fallen to 0.37.
Distinguish between “spread of a current’ vis-avi propagation of the action potential.
•
•
•
•
Spread of current is passive.
Propagation means active regeneration.
An action potential can be called a ‘transient”
(temporary) signal
Time constant of “transient” signals depends on
membrane capacitance
• In the compartmental model of the axon, the
electrical equivalent circuit places the membrane
capacitance in parallel with ohmic components
(conductance of Na and K) and the driving potentials
for ion flows through those components. Neglecting
the resting membrane potential, inject a current into
the soma of a neuron: The time course of the current
spread to ground is described by the sum of
capacitative and resistive currents, plus the input
current, Ipulse
Time constants mathematics
• That is:
•
C(dVm/dt) +Vm/R = Ipulse
• Rearranging for R gives:
•
RC(dVm/dt) = Vm = (Ipulse) (R)
• Where RC = τ (τ is the time constant of the
membrane).
• The solution of this equation though simple
mathematically is not necessary for this class.
The significance of tau
• The significance of tau is shown in the next slide; it is
the time required for the voltage change across the
membrane to reach 1/e = 0.37 of its final value. This
time constant of the membrane defines the transient
voltage response of a segment of the membrane to a
current step in terms of the electrotonic properties
of the segment. It is analogous to the way that the
length constant defines the spread of voltage over
distance.
Time constant (cont.)
• Transient signals are oscillating signal that is
“on”/”off” signals. The RC in the equation
means it has resistive and capacitive
components which turn on then off.
Change in voltage to 1/e
Applying tau to a two compartment model
• Assume current is injected into compartment A.
• Positive charges in comp. A attempt to flow outward
across the membrane partially opposing the negative
charge on the inside of the lipid membrane
• This depolarizes the membrane capacitance (Cm) at
that site.
• At the same time the charge begins to flow as
current across the membrane through the resistance
channels of the ionic membrane (Rm) at that site.
Compartmental analysis of tau (cont.)
• The proportion of charge divided between Cm and Rm
determines the rate of charge of the membrane, that
is the membrane constant τ(tau).
• Charge also starts to flow through internal resistance
Ri in to compartment B where the current again
divides between capacitance and resistance.
• The charging and discharging of compartment A
changes τ, being faster, because of the of the
impedance (resistance and capacitance) load in
compartment B is smaller than that of compartment
A. Thus the time constant is not the same as the
Time constant (cont.)
• beginning when only compartment A was
considered.
• Impedance measures are used only in in currents and
voltages that oscillate (will have a Hz when
describing the system).
•