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LECTURE 2: NEURONS and SYNAPSES
1)
Neuronal Electrical Behavior
The membrane
The membrane separates the inside of the cell from the outside. The membrane contains many conducting
channels which each have a given permeability to certain ions. In addition to these conductive channels, the
lipid layer of the membrane separates internal and external conducting solutions by an extremely thin
insulating layer. Such a narrow gap between two conductors forms a significant electrical capacitor.
The membrane potential can be measured by inserting a microelectrode into the inside of a cell’s
membrane and comparing the voltage at the outside to that at the inside of the neuron. This voltage,
typically several tens of mV’s, is called the membrane potential. The inside of the cell is negative with
respect to the outside.
The difference in voltage between the inside and the outside of the cell is due to a difference in ion
concentrations between the inside and the outside of the cell. This difference can exist because the
membrane is not completely permeable to these ions.
Outside
Inside
Na+
Na+
K+
K+
Capacitance is a measure of how much charge needs to be transferred from one conductor (one side of the
membrane) to another to set up a given potential difference. Capacity (C) is defined by charge/voltage, its
unit is a the farad (F). A 1 F capacitor will be charged to 1 Volt when +1.0 C of charge is one side and –1.0
C on the other side.
Electrically, the membrane is often represented as a resistance (representing the conductivity to ions) and a
capacity (representing the capacity to separate charges) in parallel.
Inside
Capacity (C)
Resistance (R) or Conductance (G)
Em
Outside
In general, capacitance slows down the voltage response to any given current by a characteristic time t that
depends on the product of RC.
Example: RC lowpass filter
R
C
Vin
Vout
Resistance
Capacitance
Voltage
Vin
Vout
O.63 V in
τ=RC
V out = Vin (1-e- t/RC) when Vin > 0
V out = Vin e-t/RC
when Vin = 0
time
Example: Current injection into a simplified membrane
Inside
Vm
Outside
Resistance
Capacitance
Current Injection
Membrane potential
From: Book of Genesis, James E. Bower and David Beeman (eds), Springer Verlag.(page 88)
Nernst Potential
Two forces act on the ions at both sides of the membrane: 1) an electrical force due to the difference in
electrical potential between the inside and the outside of the cell attracts positive ions to the more negative
inside of the cell and 2) a force resulting from the concentration difference between the ions on the
outside and on the inside tends to equilibrate the ion concentrations (for example K+ ions would move from
the inside of the cell to the outside if they could freely pass the membrane).
Outside
Inside
[Ion]+
Electrical attraction
Low
High
[Ion]+
Concentration difference
When a neuron is “at rest”, it has found an equilibrium point defined by the permeability of the various ions
and their concentration differences. This equilibrium is often called the resting membrane potential.
However, this equilibrium point is not stable. This means that if a perturbation occurs (for example a
current injection, a change in voltage or a change in permeability), the concentrations differences and the
voltage across the membrane will change.
Imagine that suddenly the permeability for a given ion, for example K+, is infinite. Then K+ ions will want
to leave the cell’s inside because of the concentration difference, but they will also want to rush into the
cell because of the voltage difference. All other parameters being constant, these two tendencies of K+ will
equilibrate when the cell reaches a certain voltage called the “Nernst potential” (or reversal potential or also
zero current potential). For each type of ion, the Nernst potential can be calculated. It’s value tells you
towards which voltage the membrane will potential will move if suddenly the permeability of the
membrane for a given ion is higher.
If the Nernst potential for a given ion (for example K+) is more negative then the resting membrane
potential, then the cell will hyperpolarize (become more negative) when the permeability to that ion
increases. If the Nernst potential is less negative then the resting potential, then the cell will depolarize
(become more positive).
Example:
The membrane potential of a frog satorious muscle fiber is held at various levels by two microelectrode
voltage clamp. The motor nerve is stimulated by an electric schock (indicated by the arrow in the current
record). About 1 ms later, the nerve action potential reaches the nerve terminal, releasing transmitter
vesicles and opening postsynaptic endplate channels transiently. The endplate current reverses sign near
0mV. From Bertil Hille, Ionic Channels of Excitable Membranes, Sinauer Associates page 146.
Electrically, the permeability can be modeled as a conductance, or inverse resistance. The conductance
measures how permeable the membrane is to a given ion. The Nernst potential can be modeled as a
“driving force”, or a battery. Remember that the Nernst potential is the voltage at which no more ions of a
given kind would flow across the membrane; this means that of the membrane potential is equal to the
Nernst potential, then even for an “infinite conductance” (no resistance), no more ions (current) will flow.
Thus the term “driving force”, which is given by the difference between the Nernst potential and the
membrane potential.
Inside
Capacity (C)
Resistance (R) or Conductance (G)
Em = -70 mV
Battery ( Ek )
Ek
Ik = g K (Em - Ek ) current through the conductance
Outside
Why is this important? Because the current that will flow when a conductance change occurs, depends,
among other things, on the difference between the Nernst potential of the conductance and the membrane
potential. Note that the current itself will change the membrane potential!
All together:
Outside
Variable conductance
gNA
gK
Rm
ENA
EK
Em
Cm
Inside
Most often, you will see a complicated looking differential equation describing the dynamical behavior of
the circuit shown above:
d
C V
dt
m
m
=
(E − V
R
m
m
)
+ (E − V m) g + (E
K
K
Na
− V Na) g
Na
m
This equation describes how the membrane potential of the modeled neuron (or part of a neuron) changes
when conducatnces change. Usually, computer algorithms are used to solve such equations.
2) Modeling synapses
The variable conductances can change as a function of a number of parameters, for example voltage or the
presence of a neurotransmitter. The representation of voltage dependent conductances often necessitates a
number of inter-dependent differential equations.
[A differential equation expresses how one variable changes as a function of the changes of one or more
other variables. An example is the expression of speed as a change of space occurring during a change of
time. Simple differential equations like those for the RC low pass filter can be solved analytically, but often
numerical solutions have to be calculated on a computer. ]
Some conductance changes occur at synapses when an action potential on a presynaptic terminal leads to
the release of neurotransmitter, changing the conductance of a particular ion channel, allowing ions to
move across the membrane more freely. The change in conductance leads to a current entering or leaving
the cell (depending on the Nernst potential of the ion under consideration), which in turn leads to a change
in voltage. Remember that the change in voltage is delayed with respect to the change in current because of
the membrane capacitance!
In the case of a fast synapse, the release of neurotransmitter from the presynaptic terminal directly results in
the opening of chemically gated ion channels at the postsynaptic membrane. The input produced is
primarily expressed as a local conductance change. Specific ions can flow through these channels and
produce what is called the postsynaptic potential (PSP).
From: From: Book of Genesis, James E. Bower and David Beeman (eds), Springer Verlag.(page 94)
[Note that although both the synaptic input and the electrode which injects current result in the flow of ions
across the membrane, the two types of inputs differ in a significant way. While the electrode across the
membrane does not change the properties of the membrane, the synaptic input changes the characteristics
of the postsynpatic cell: it opens new channels there. ]
If we were to consider that the release of neurotransmitter at the presynaptic terminal produced a synaptic
conductance change which is a rectangular pulse, the observed phenomena are close to those described
above for a current injection. The conductance change is however better described by a smooth function
rather than a rectangular pulse. A fairly good approximation may be obtained b y an analytical function
called the alpha function, which is a special case of a double exponential function.
g
syn
( t) =
g
t
max
(1− t / tp )
t e
p
is called the alpha function. This function increases rapidly to a maximum of
gmax at t = tp. Following its peak, g syn (t) decreases more slowly to zero. A slow synapses will be modeled
by a large t p , a strong synapse by a large g max.
The dual exponential equation is a more general form often used for the description of synapses:
g
syn
=
Ag
(e
τ −τ
max
1
τ1−
−t /
e
τ 2) for
−t /
τ >τ
1
1
; where A is a normalization factor chosen so that g syn reaches a
2
maximum of g max.
3) Space
Thus far, we have considered local processes as isolated temporal events. However, a neuron extends in
space, and the electrical signals produced by synaptic interactions or current injection have to travel along
the dendrites to the soma and to the axon. How the electrical signals change across a dendrite considered
passive (no voltage dependent conductances) can be described by means of a calculation first developed by
engineers: cable theory. Briefly, the flow of the electrical signal along the dendrite has behaviors similar to
that across the membrane described above, and can be described by means of a differential equation. To
this purpose, the dendrite is decomposed into a large number of small cylinders which are all attached to
each other via resistors. A current injection then results in currents flowing across the membrane as well as
currents flowing along the membrane. Both decay in time, and the current flowing along the membrane
also decays with distance.
From: From: From: Book of Genesis, James E. Bower and David Beeman (eds), Springer Verlag.(page 70)
Each compartment is modeled by an equation similar to those described above, but in addition, terms are
added that take into account the currents flowing in and out of the compartment via neighboring
compartments:
Outside
Variable conductance
gN A
gK
Rm
EN A
EK
Em
Cm
Ra
Ra’
Vm
Vm’’
Vm ’
Inside
The equation describing the dynamical behavior of this circuit becomes:
C
dV m
m
dt
=
(E − V
R
m
m
m
)
+ ( E −V m ) g + ( E
K
K
Na
− V Na ) g
Na
+
(V '−V
R'
m
a
m
)
+
(V
m
' '−V m)
R
a
Time and space are neglected in many neural network models! Synaptic integration, spatial integration in
the dendrites, refractory periods are only a few of the many complicated things that neurons do that
modelers often neglect!
When dendrites are modeled in detail, the location and temporal interactions of synaptic events can play an
important role in the resulting neural activity.
Example (from Anderson, page 43-45)
:
The electrical activation seen at the soma depends on the location of the conducatnces on the dendrite. The
signal decays both in space and in time.
The summation of several synaptic events as seen at the some depends on the order in which these events
occur:
3 The Single Neuron
3.1 The neuron as an electrical device
• Potential difference across membrane due to differences in ion concentrations between the inside
and outside of the cell
•
•
•
•
•
[SH88 Fig. 4.3 pg 67]
Membrane has a different level of permeability to different ions
Changes in permeability lead to the flow of ions across the membrane with a resultant change in
the voltage (membrane potential)
Action potential is a sharp deviation from rest caused by a rapid change in permeability
Main ions are sodium (Na +), potassium (K+), calcium (Ca 2+) and chloride (Cl−)
Nernst equation describes equilibrium (reversal) potential for a single ionic species:
[X+]o
RT
EX =
ln
zF
•
•
[X+]i
Different ions have different reversal potentials e.g. ENa = 55mV, EK = −75mV
Individual ionic currents
o permeability ≡ electrical conductance
IX = (Vm−EX)GX
•
•
Different ion channels for different ions
Membrane potential due to the combined permeability of different ionic species is given by the
Goldman-Hodgkin-Katz equation:
RT
Vm =
zF
•
Ko +[p Na/p K ]Na o +[p Cl /p K ]Cli
ln
Ki +[p Na/p K ]Na i +[p Cl /p K ]Clo
Total membrane current is given by the sum of individual currents:
Im = INa+IK +ICl
•
•
Rest potential of cell due to combined current through different ion channels summing to zero
o typically around -65mV
A small patch of membrane can be described by an equivalent electrical circuit consisting of:
o capacitance
o batteries in series with conductances representing the different ion channels
o biochemical pumps
[SH88 Fig. 4.6 pg 71]
3.2 Electrical signalling
• Electrical signals are changes in the membrane potential at specific points in the neuron
• Such changes are due to the opening and closing of ion channels
o the inputs from other neurons create synaptic potentials
o the output from a neuron is called the action potential
•
[CS92 Fig. 2.19 pg 44]
The action potential:
o Signal from one neuron to another consisting of a sharp voltage wave 100mV in
amplitude and 1-2msec in duration
[BR92 Fig. 1.20 pg 25]
Generated by a rapid opening and then inactivation and closing of voltage-dependent
sodium channels which generate a depolarizing current, followed by the slower opening
and closing of potassium channels which generate a repolorizing current
o Initiated in axon hillock if membrane potential reaches a threshold value above rest due
to excitatory input to the neuron
§ high density of sodium channels in axon hillock
§ threshold of around 20mV above rest is determined by voltage dependence of
sodium channels
o Travels along the axon to the synaptic terminals at velocities of from 1 (unmyelinated
axon, continuous conduction) to 120m/sec (myelinated axon, saltatory conduction)
o Complete output signal from a neuron generally consists of many action potentials, either
in regular bursts or a steady frequency of single action potentials
Excitatory inputs:
o Excitatory response to a presynaptic action potential is an excitatory postsynaptic
potential (EPSP)
§ generated by influx of sodium and/or calcium through ligand-gated ion channels
at the synapse
§ reversal potentials of Na and Ca are well above rest, so an EPSP is a
depolarizing response
o Excitatory inputs increase the probability of a neuron firing an action potential
o Excitatory synapses (connections) are on the dendrites
o EPSPs propagate passively (like electricity down a wire) along the dendrites to the soma
o Separate EPSPs sum together (possibly non-linearly) on the way to the soma and at the
soma
o If the EPSPs raise the membrane potential at the axon hillock (next to the soma) to above
threshold an action potential is fired
o May take 20 to 100 coincident EPSPs to fire an action potential
Inhibitory inputs:
o Inhibitory response to a presynaptic action potential is an inhibitory postsynaptic
potential (IPSP)
§ generated by influx of potassium and/or chloride ions through ligand-gated ion
channels at the synapse
§ reversal potentials of K and Cl are near or below rest, so an IPSP is mostly a
conductance change, with little change in the membrane potential, but small
hyperpolarizing responses may be seen
o Inhibitory synapses (connections) may be on the dendrites, the soma or the axon
o Inhibitory inputs decrease the probability of a neuron firing an action potential
o IPSPs may increase, decrease or have no effect on the membrane potential, but will
decrease the amplitude of a coincident EPSP
o
•
•
•
The synapse:
o Presynaptic action potential causes calcium entry in the presynaptic terminal
o Calcium (in some way) causes vesicles filled with neurotransmitter to bind to the
presynaptic membrane and release their transmitter into the synaptic cleft
o Transmitter diffuses across the cleft and binds to receptors on the postsynaptic membrane
o This causes, either directly (ionotropic) or indirectly (metabotropic), ion channels to open
o The consequent flow of ions results in a voltage response (EPSP or IPSP) in the
postsynaptic membrane
[SH88 Fig. 6.14 pg 122 and BR92 pg 29]
3.3 Places to Start
• Brodal (1992) The Central Nervous System, Chapt. 1: Cellular elements of nervous tissue and
their functions
• Shepherd (1994) Neurobiology, Chapters 3-7
• Levitan and Kaczmarek (1991) The Neuron: Cell and Molecular Biology, Oxford University Press