Notes on electrical activity of neurons for Math 4600
Alla Borisyuk
February 2011
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Neurons
Today’s biological theme is neurons. Neurons are special cells in the body that are
electrically active and are the core units in the nervous system.
A typical neuron possesses a cell body (often called the soma), dendrites, and an
axon (see pictures online). Dendrites arise from the cell body, often extending for
hundreds of micrometres and branching multiple times. An axon is a special cellular
filament that arises from the cell body and travels for a distance, as far as 1 meter in
humans or even more in other species. There are axons, for example that run from
your spine to your toes. As a general rule, dendrites receive inputs to the neuron and
the axon is used to transmit the output. There are, however, many exceptions to this
rule.
Neurons are electrically active. They produce large electrical signals called “action potentials” or “spikes” or “nerve impulses” that can travel down the axon and
are reliably transmitted to other neurons. Action potentials are considered to be
stereotypical and are the main communication units in the nervous system. When a
neuron produces a spike it is said to “fire a spike”.
When neurons are active they can produce a variety of spike patterns (firing
patterns) - they can spike tonically (periodically), burst, fire randomly. Different
firing patterns may communicate different signals to the rest of the system.
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Electrical activity of neurons at rest.
There are many charged particles (ions) both inside and outside of the neurons. To
retain capacity for generating electrical signals, neurons maitain difference in ions
concentration between the inside and the outside of the cell. Main ionic species
that participate in neuronal activity are K+ , Na+ , Ca2+ and Cl− . When a neuron
is not sending a signal, it is ”at rest.” When a neuron is at rest, there are more
K+ ions and negatively charged proteins inside and more Na+ , Ca2+ and Cl− ions
on the outside. Overall, at rest the inside of the neuron is negative relative to the
outside. This means that there is a negative difference in the voltage between the
inside and outside of the neuron. That voltage difference is simply called “membrane
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potential” or sometimes simply “voltage”. We will use the notation V . Resting value
of the membrane potential is usually near -65 mV. The charge difference between the
inside and the outside also means that the cell membrane works as a capacitor (with
.
capacitance C) producing the capacitive current C dV
dt
Ions can move into and out of the cell through ionic channels - special protein
complexes, spanning the cellular membrane that allow through only particular kinds
of ions. There are several different types of channels for each of the common ionic
species, and some channel types pass more than one type of ion. For example, people
often refer to the “leakage channel” which may be a non-specific mix of potassium
and chloride channels. This type of channel is important as it is always functional
(open) and thus sets the base for the resting potential of the neuron.
Movement of ions through the channels creates electrical currents. To keep the
system in balance, the currents have to satisfy the conservation of charge rule or the
Kirchhoff’s first rule (sometimes called current balance equation):
C
dV
+ Iion = 0,
dt
(1)
where Iion is the sum of all the ionic currents.
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Ohm’s law for ionic currents
Let’s consider a cell with just one ionic current. If we think of ionic channels as
resistors, the simplest (and fairly accurate) model for the ionic currents is the Ohm’s
law:
v
Iion = ,
R
stating that the current is equal to voltage over the resistance. In neuronal models it
is customary to use the inverse of the resistance – the conductance gion = 1/R – as
parameter, giving
Iion = gion v.
The value of the ionic conductance depends on the type of the channel, the density
of the channels in a given cell, and also on the conformation of the channel proteins
(whether the channel is “open” or “closed”, see below). The voltage v in this equation
is the membrane voltage relative to the “reversal potential” for each ionic species:
Iion = gion (V − Vion ).
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(2)
The reversal potential is the voltage gradient built across the membrane by the difference in ionic concentrations inside and outside. If the membrane voltage exactly
compensates such that V = Vion for particular type of ions, then the corresponding ionic current is zero. Reversal potentials for potassium, sodium and calcium are
approximately VK = −80 mV, VN a = 40 mV, VCa = 20 mV.
Generalizing eqiation (2) to multiple ionic species, current balance equation (1)
becomes
X
dV
=−
gi (V − Vi ),
(3)
C
dt
i
with summation over all distinct ionic currents with corresponding conductances gi
and reversal potentials Vi .
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Finding resting voltage
Once again, let us consider a cell with just one ionic current, leakage, which is “passive” (its conductance is constant):
C
dV
= −gL (V − VL ).
dt
(4)
Exercise. Analyze equation 4. Find steady states, their stability and long term
behavior of solution both graphically and analytically (this equation can be solved
explicitly).
The results of the exercise should show that the the solution of the equation (4)
will go to VL , regardless of the initial conditions, i.e. the ionic current wants to pull
V towards its reversal potential. The resting potential for this cell would be equal to
VL .
With multiple constant-conductance ionic currents
C
X
dV
=−
gi (V − Vi ),
dt
i
(5)
the steady state value for V would be
P
gi VI
,
i gI
V = Pi
∗
i.e. the weighted average of the reversal potentials.
Exercise. Analyze equation (5) to confirm steady state (6) and its stability.
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(6)
P
gV
The resting potential for this cell is set at V ∗ = Pi igI I , at a “compromise” value
i
between the currents’ reversal potentials. In particular, if one of the currents is
dominant (e.g. g1 gi , i > 1), then the voltage will tend to a value that is close
to the corresponding reversal potential V ∗ ≈ V1 .
Notice that the conclusions of this sections in the case of time-varying conductances are still valid “instantaneously” (i.e. for a short amount of time).
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