Neurophysics
Adrian Negrean
doctoral student
CNCR, VU University Amsterdam
Office address:
Department of Integrative Neurophysiology (INP)
Centre for Neurogenomics and Cognitive Research (CNCR)
Neuroscience Campus Amsterdam (NCA)
VU University Amsterdam
De Boelelaan 1087
1081HV Amsterdam
adrian.negrean@cncr.vu.nl
Contents
1.
Aim of this class
2.
A first order approximation of neuronal biophysics
2.1
2.2
2.3
2.4
2.5
2.6
3.
And then a second order approximation
3.1
3.2
3.3
4.
Introduction
Electro-chemical properties of neurons
Ion channels and the Action Potential
The Hodgkin-Huxley model
The Cable equation
Compartmental models
Neuronal networks and diversity
Neuronal network oscillations
Learning models
Neurotechniques
1.
Aim of this class
The study of brains and cognitive processes has been traditionally the preoccupation of biologists
and psychologists and only recently, in the last 60 years these issues started to be looked at from a
physical perspective. There has been a lot of effort in popularizing physics and mathematics in the
biology community (such as in the field of computational biology) and many good textbooks have
been written with the aim to present various physical and mathematical concepts (dynamical
systems, graph theory) to biologists in a friendlier manner. However amusing it may seem, there is
also a need to present biology to physicists in a friendlier manner as well, and my task for the next
classes will be therefore to convince you that there’s a lot of physics going on in the brain.
2.
A first order approximation of neuronal biophysics
2.1 Introduction
Neurons share many features with other cells, such as having a cellular membrane composed of
phospholipids that separates the cytosol containing various cellular elements from the environment.
A striking difference between neurons and other cells is their characteristic morphology (Fig. 1),
with multiple branches (2, 3 and 4) that extend from the cell body (1), connecting neurons together
via synapses (5) through which neurons communicate with each other.
In a simple picture, the dendrites of a neuron on which axons of other neurons form synapses,
represent the ‘input’ to the neuron that the cell body ‘integrates’ and then ‘outputs’ the result via the
axon to other neurons. The problem with this simple picture that will have to suffice for the moment
is that the shaping of synaptic inputs is already happening at the level of dendrites and that the
whole neuron takes part in ‘processing’ its inputs. This ‘little’ detail is often ignored in the field of
artificial intelligence that uses overly simplistic models of single neurons to justify their use in large
scale neuronal networks simulations.
(4)
(3)
(5)
(1)
(2)
Figure 1: A histological staining of a single so called pyramidal neuron from a mouse
observed under a light microscope (photo credits: Cristiaan de Kock, CNCR, VU,
Amsterdam). (1) Cell body, (2) Axon, (3) Apical dendrite, (4) Basal dendrite, (5)
Synapses.
To a large extent communication between neurons occurs through the initiation of an Action
Potential (AP) in the axon initial segment which is an electro-chemical wave of excitation that
propagates throughout the axonal tree much like the ignition of a dynamite fuse. When this wave of
excitation reaches a synapse, it causes the release of neurotransmitters that in turn activate electrochemical processes in the post-synaptic neuron. Depending on the type of neurotransmitter, the
postsynaptic neuron may be excited or inhibited to produce an AP.
Check out this link for some nice animations and explanations (report if broken):
http://www.youtube.com/watch?v=DF04XPBj5uc&feature=related
There’s a great diversity of neurons (Fig. 2) that can be readily observed from differences in
morphology, anatomical location and function which can often make their classification difficult.
Moreover, at the level of a single neuron, there is a great diversity of cellular components that will
determine the input-output transformations of a neuron, i.e. the way it reacts to synaptic inputs. A
full understanding of how brains work must be able to explain how these details fit together and at
the same time by accounting for all the details, it is often easy to miss the big picture.
Figure 2: Diversity of neurons that can be readily distinguished based on their
morphology.
2.2 Electro-chemical properties of neurons
The neuronal membrane like other cellular membranes is composed of a bilayer of phospholipids
that insulates the cytosol from the external environment (Fig. 3). The insulation provided by the
phospholipid membrane allows the cell’s cytosol to have a different ionic composition than the
external medium (i.e. different concentrations of Na+, K+, Ca2+, Cl-). As it will be explained shortly,
this ionic imbalance will lead to an electric potential difference across the membrane, which in turn
will create an electric field across the membrane. Considering these properties, the cellular
membrane will act as the electrical equivalent of a capacitor. The capacitance of a simple cell
having a soma with a surface area A can be calculated from the well known formula of parallelplate capacitors:
C
0 A
d
(1)
Where  is the dielectric constant of the phospholipid membrane,  0 , the dielectric constant of
vacuum and d the thickness of the membrane (which is about 2.3 nm). A handier way of calculating
the capacitance of a simple cell is to remember that cellular membranes have a specific capacitance
of about 1.0 μF/cm2 which can be multiplied with the surface area of the cell.
Figure 3: The neuronal membrane is composed of a bilayer of
phospholipids that insulates the contents of the cytosol from the
environment.
The insulation of cellular membranes is not perfect and additionally the membrane contains many
pore-forming proteins through which ions can flow that are collectively referred to as ion-channels.
Ion-channels can actively control the flow of ions depending on factors like the membrane potential
in the case of voltage-gated ion-channels (Fig. 4), external ligands such as neurotransmitters that
bind to ligand-gated ion-channels, intracellular factors in the case of metabotropic ion-channels or
simply, their conductance remains constant, referred to as passive ion-channels or leak-channels.
Figure 4: Structure of a voltage-gated potassium selective ion-channel
from the Kv1.2 gene. A: Side-view with the extracellular part above and
the intracellular part below. B: Top-view from the extracellular side. The
channel is a tetramer, each subunit shown in a different colour with TM
representing the integral membrane component of the complex, T1 a
cytoplasmic domain, a β subunit and transmembrane helices S1-S6 [Long
et. al. 2005].
The membrane capacitance together with leak-channels (detailed in the next section) gives rise to
the passive electrical properties of the cell (Fig. 5). Note that the electrical circuit shown in (Fig. 5)
includes a voltage source Vrest which takes into account that there is a potential difference between
the two sides of the membrane.
A
B
Figure 5: A: Electrical circuit equivalent of a simple cell with leakchannels. B: Basic electrical circuit describing a small membrane patch.
As it has been mentioned before, the membrane potential is due to a difference in the ionic
composition of the cytosol and the external solution. When a salt, such as NaCl is dissolved in
water, it dissociates into its ionic constituents Na+ and Cl- and the solution is overall electrically
neutral. Let’s make a thought experiment that will clarify the relation between ions and
electrochemical potentials: suppose that you have a jar filled with water, which is separated in two
compartments by a membrane permeable only to Cl- and you add NaCl only to one of the
compartments (Fig. 6). Initially all the Na+ and Cl- ions will be in one compartment and each
compartment will contain the same number of ions, making the two compartments electrically
neutral (Fig. 6A). As time passes, Cl- ions will diffuse across the membrane (increasing the entropy
of the system) and an electric potential will develop between the two compartments. The increase in
potential difference creates in turn an electric field across the membrane that will oppose the
increase in ionic imbalance. Eventually the system reaches a dynamic equilibrium, where diffusion
of Cl- is balanced by the electric potential difference.
A
B
E=0
Na+Cl
 -
Na+Cl
 -
Na+Cl
 -
water
E=ECl
Na+Cl
 -
Na+
Na+
Cl-
Cl-

Cl- permeable
membrane
Figure 6: Ions and electrochemical potentials. A: When salt is added initially only to one
compartment, both compartments are electrically neutral and there is no potential difference
between them. B: As time passes, Cl- ions diffuse across the membrane permeable only to Cland a potential difference between the two compartments will be established. In the end the
system reaches equilibrium where diffusion across the membrane is balanced by the established
electric field across the membrane.
The value of the equilibrium potential Eion of a system similar to (Fig. 6), such as a simple cell,
having a membrane permeable only to a particular ion can be calculated using Nernst’s equation:
Eion 
 [ion]out 
RT

 ln 
zF
 [ion]in 
(2)
where z is the valence of the ion, R is the gas constant (8.315 J K-1mol-1), T is the
temperature (K), F is Faraday’s constant (96.485 C mol-1) and [ion]o and [ion]i are the
concentrations of the ion inside and outside of the cell respectively.
The cytosol of a realistic cell and the extracellular medium contain several ion species such as K+,
Na+, Cl-, Ca2+ (Fig. 7) and when their membranes are simultaneously permeable to several ions, the
steady state equilibrium potential (or the cell’s membrane potential) is described by the GoldmanHodgkin-Katz (GHK) equation:
  
  
 
 
  
   
 pK K  o  p Na Na  o  pCl Cl  i
RT
Vm 
 ln 



F
 pK K i  pNa Na i  pCl Cl o
(3)
where in addition to the terms described in Nernst’s equation, the relative contribution of
each ion is determined also by the relative permeability of the membrane. For example, in the squid
giant axon, where the propagation of neuronal impulses termed Action Potentials (AP’s) has been
first described, at resting membrane potential (no AP) the membrane relative permeability ratios are
pK:pNa:pCl=1.00:0.04:0.45. Thus, the cell has two means of controlling its membrane potential: 1) by
changing the ionic composition of its cytosol, 2) by changing the membrane permeability to
different ions using ion-channels that were introduced previously.
Figure 7: Intracellular and extracellular concentrations of different ions
(millimoles) given in parentheses for a typical mammalian neuron and their
Nernst equilibrium potentials (mV). The permeability of the membrane to
different ions is regulated by various ion-channels inserted in the membrane
and the ionic composition of the cytosol is actively maintained through ionic
pumps, in this case, a Na+-K+ exchange pump.
To understand better the relation between membrane permeability to different ions, equilibrium
potentials and movement of ions, I’ll give an example situation based on a cell having typical ionic
concentrations and equilibrium potentials shown in (Fig. 7). Suppose that based on the GHK
equation, the calculated membrane potential is -64 mV. This membrane potential is more positive
than the equilibrium potential for K+ of -102 mV, which has the consequence that an increase in the
membrane permeability for K+ will cause an outward flow of K+. Only if the membrane potential
would be forced to be -102 mV, a change in the permeability for K+ would cause no flux of K+
across the membrane. What would happen with K+ if the cell is hyperpolarized to -120 mV?
2.3 Ion-channels
To make an analogy between electrical circuits and neurons, ion-channels are to a neuron what
transistors are for microprocessors in computers. Ion-channel function forms the basis of
understanding how neurons respond and adapt to synaptic inputs and any theory aiming to describe
how brains work, must take this aspect into account.
As briefly introduced before, ion-channels are membrane-bound proteins that have three important
properties: 1) they conduct ions, 2) they are selective for specific ions (Fig. 8) and 3) they open and
close in response to specific electrical, mechanical or chemical signals.
Figure 8: Ions can cross the plasma membrane only through specialized poreforming proteins called ion-channels. Ions in solution are surrounded by
several water molecules forming a hydration shell that prevents the ion to cross
the lipid membrane by creating a large potential energy barrier. A binding site
inside the pore selects a certain ion based on the size and charge of the ion,
hydration shell size and hydration energy.
Early studies of ion-channel properties revealed that the flow of ions can occur in discrete steps
depending whether the channel is in an open or closed state. In these experiments (early 1960s), a
thin lipid bilayer created over a small hole in a non-conducting barrier between two salt solutions
contained a low concentration of a 15-amino acid peptide, gramicidin A, that had the property of
forming pores when two-such peptides associated in the membrane (Fig. 9C). Creating a potential
difference between the two salt (e.g. NaCl) solutions, the occasional formation of a pore could be
observed as a discrete increase in current (Fig. 9A) that lasted a certain amount of time until the
two subunits of the pore dissociated again (note that there is a dynamical balance between the
single and associated form of the peptide, depending on the reaction rates). Varying the potential
difference across the membrane, the peak current flowing through the channel varied linearly (Fig.
9A,B) and current-voltage relationship could be described by Ohm’s law i=V/R, demonstrating that
such an ion-channel behaves effectively like a resistor in an electronic circuit. Now it becomes
understandable that a resistor has been included in the electrical circuit equivalent of a simple cell
to model leak-ion-channels (Fig. 5B).
Figure 9: C: Gramicidin A peptide has been
added to a phospholipid bilayer membrane to
form trans-membrane channels that allow passage
of ions. A: The formation of functional
Gramicidin A channels can be seen as random
step-increases in current when a potential
difference is applied to the membrane. B: The
size of the current steps is related to the applied
potential through Ohm’s law.
Gramicidin A is a cation-selective channel (check Wikipedia if you want to know what it’s good
for). Looking at the single-channel currents recorded at various potentials in (Fig. 9A), what is the
reversal potential for this current? Generally when describing single channel currents, the reversal
potential (see eq. (2) and eq. (3)) has to be also taken into account since there may be an ionic
imbalance between the two sides of the membrane:
I L  g L E  Erev 
(4)
where g L is the single-channel conductance (measured in Siemens, being the inverse of the
resistance), E the holding potential and Erev the reversal potential of the current.
While leak ion-channels having an Ohmic behaviour are the simplest to describe, the great majority
of ion-channels require more elaborate biophysical models to describe them. In the following I will
focus on the biophysical modelling of two voltage-gated ion-channels that are essential in the
production of action potentials that neurons use to communicate with each other. First I will give an
overview of what happens during an action potential and then we’ll have a closer look at what the
ion-channels are doing. Don’t worry if it’s not immediately clear, I’ll explain it step by step.
After the activation of several excitatory synapses of a neuron, a depolarizing current travels
through the dendritic tree and depolarizes the soma which had a resting membrane potential around
-64 mV (Fig. 10). When the somatic membrane potential reaches about -45 mV, Na+ permeable
channels open, Na+ ions enter the cell and further depolarize the soma such that within a
millisecond the membrane potential reaches a value around +20 mV. With a slight delay, K+
permeable ion channels open and the outflow of K+ ions repolarises the membrane potential to a
value close to -64 mV ending the action potential.
Let’s start with the biophysics of K+ channels involved in action potential generation. During the
opening and closing of ion-channels, several subunits within the membrane-spanning protein must
change their conformation (shape) at the same time for the channel to open or close. In the simplest
case then, assuming that the subunits change their conformation independently of each other, the
probability for the channel to open is:
PK  n k
(5)
where k is the number of subunits (in this case k=4) and n is the probability that a single subunit
switches to an open conformation. The subunit probability n is related to the membrane potential
through a kinetic scheme where the closed to open transition occurs at a voltage-dependent rate
 n (V ) and the reverse transition open to closed occurs at a voltage-dependent rate  n (V ) .
 n (V )
C
O
 n (V )
dn
  n (V )  (1  n)   n (V )  n
dt
(6)
In practice, the rate coefficients  n (V ) and  n (V ) are obtained from fitting experimental data,
however thermodynamical arguments can be also used to a good approximation to describe the rate
coefficients. Voltage-gated ion-channels are able to respond to a change in the membrane potential
by having one or more charged domains (charged amino acids) that couple with the membrane
electric field. Thus, a conformational change involves the movement of a certain fraction B of
charge q within the membrane electric field. The potential energy difference (barrier) qBV that
separates the two conformational states is then proportional to the membrane potential. This means
that the probability of thermal fluctuations to have enough energy to overcome this potential barrier
and change the conformational state of the subunit, will be proportional to the Boltzmann factor
exp( qB V / k BT ) . Using such a thermodynamical argument,  n is expected to have the form:
 n (V )  A exp( qBV / kBT )
(7)
You may have already come across a similar equation in chemistry where it is known as the
Arrhenius equation describing reaction rates. A similar equation can be written for  n (V ) .
A
B
(1)
(2)
(3)
(4)
time
(5)
Figure 10: The production of action potentials in neurons. A: (1) Incoming action potentials release an
excitatory neurotransmitter (for example, glutamate) at the synapse that open glutamate-gated ion-channels
from the post-synaptic neuron shown in green. The opening of glutamate-gated ion-channels allows Na+ ions to
enter the post-synaptic neuron, which depolarize the dendritic tree and flow towards the soma (2). As time
passes, the accumulation of Na+ ions depolarizes the soma (3) until the membrane potential reaches about -45
mV. At this point, a large number of Na + gated ion-channels present in the axon hillock (4) open, further
depolarizing the cell and initiating an action potential that propagates down the axon (5) toward other neurons.
The action potential is terminated through the opening of K + ion-channels that bring back the neuron’s
membrane potential to the resting value of about -64 mV. B: The membrane potential of the neuron is shown to
change during an action potential (red). This change in membrane potential is due to the change in the number
of open Na+ and K+ over time (magenta), where initially Na + channels open, followed by K+ channels.
Now that the forms of the rate coefficients are given, it is useful to rewrite eq. (6) in a more
meaningful way by dividing it with  n (V )   n (V ) :
 n (V )
dn
 n (V )  n
dt
(8)
 n (V ) 
1
 n (V )   n (V )
(9)
n (V ) 
 n (V )
 n (V )   n (V )
(10)
In this way it is easier to notice that for a fixed voltage V, n approaches the limiting value n (V )
exponentially with a time constant  n (V ) which is easy to measure and fit to experimental data.
The second type of ion-channels involved in the generation of action potentials we’re going to
discuss are the Na+ permeable channels. They are “more special” than the previously discussed K+
channels in that besides an activation gate they also have a blocking gate which makes these
channels to be transiently opened upon depolarization with a probability:
PNa  m k h
(11)
where m is the probability that any of the k activation gates open and h is the probability that a
blocking gate closes the channel. The description of gating variables m and h has the same form as
for the K+ channel gating variable n. In the following section I will summarize all the results we
have so far which make up the Hodgkin-Huxley model for the generation of action potentials.
2.4 The Hodgkin-Huxley model
The Hodgkin-Huxley model is a set of nonlinear differential equations that describe the generation
of action potentials in (simple) neurons. The model (Fig. 11) consists of a somatic compartment
containing leak channels, Na+ and K+ permeable voltage-gated ion-channels.
Figure 11: Electrical circuit equivalent of a neuron consisting of a soma and
three ion-channel types required to generate action potentials. The model
describes the evolution of the membrane potential V depending on the injected
current Iinj.
Using Kirchhoff’s laws, the following equation can be written from (Fig. 11):
I Na
IK
IL


 
 

4
3

CV  I inj  g K n (V  EK )  g Na m h(V  E Na )  g L (V  EL )
(12)
together with the evolution of the gating variables as described in the previous section:
n   n (V )(1  n)   n (V )n
(13)
   m (V )(1  m)   m (V )m
m
(14)
h  h (V )(1  h)  h (V )h
(15)
where
 n (V )  0.01
10  V
 10  V 
exp 
 1
 10 
 V 

 80 
 n (V )  0.125 exp 
 m (V )  0.1
25  V
 25  V 
exp 
 1
 10 
 V 

 18 
 m (V )  4 exp 
 V 

 20 
1
30
 V
exp 
 10
(17)
(18)
(19)
 h (V )  0.07 exp 
 h (V ) 
(16)

 1

(20)
(21)
with typical values for:
- the maximal ion-channels conductances:
g K  36 mS/cm2,
g Na  120 mS/cm2,
g L  0.3 mS/cm2
- specific membrane capacitance:
C= 1 F/cm2
- reversal potentials (the membrane potential at which there is no net current flow through a certain
ion-channel):
Ek= -77 mV, ENa= 50 mV and EL=-54 mV
To solve the set of equations (12-21) numerical integration is required which can be done using e.g.
Matlab or specialized neurocomputational software, e.g. Neuron. To understand better how the
model described the membrane potential evolution, the Hodgkin-Huxley equations have been
solved in the case when two current pulses are applied (injected) to the neuron (Fig. 12).
Figure 12: Action potential in the Hodgkin-Huxley model. The first injected current pulse is too small for an action
potential to occur while the second pulse is large enough to open Na+ channels and cause an action potential. For
convenience the membrane potential has been shifted in the model by 65 mV such that V m= 0 mV at rest instead of
Vm= -65 mV.
While the Hodgkin-Huxley model captures the essence of neuronal action potential generation, in
practice, neurons show a much larger diversity in the pattern of action potentials which is due to
the specific morphology of the dendritic tree and to other types of ion-channels that were omitted
from the model (Fig. 13).
Figure 13: Neurons in the mammalian brain show widely varying electrophysiological properties that can
be readily seen in the pattern of generated action potentials in response to a depolarizing current injection.
A: Intracellular injection of a depolarizing current pulse in a cortical pyramidal cell results in a train of
action potentials that slow down in frequency. This pattern of activity is known as “regular firing”. B:
Some cortical cells generate bursts of three or more action potentials, even when depolarized only for a
short period of time. C: Cerebellar Purkinje cells generate high-frequency trains of action potentials. D:
Thalamic relay cells may generate action potentials either as bursts or E: as tonic trains of action
potentials due to the presence of Ca2+ permeable ion-channels. F: Medial habenular cells generate action
potentials at a steady and slow rate in a “pacemaker” fashion.
2.5 The Cable equation
In the previous section, the mechanism of action potential generation has been described using the
Hodgkin-Huxley model, which assumes that the neuron has a somatic compartment containing Na+and K+-permeable voltage gated ion-channels as well as passive leak channels. Obviously this
simplified, yet very useful picture neglects the passive and active contribution of the dendritic tree
that hosts a multitude of voltage-gated ion-channel. Examining again eq. (12) in the situation when
a dendritic tree is added to the somatic compartment, several differences appear when injecting a
current pulse in the soma. First, a part of the somatic injected current would “flow” in the dendritic
tree which would slow down the charging of the somatic compartment. Second, the addition of the
dendritic tree would change the passive properties of the cell by increasing its overall capacitance
(due to more membrane) and by lowering the overall resistance (due to the additional leak channels
in the dendritic membrane). Finally, the current injected in the soma and flowing in the dendritic
tree can depolarize the dendrites and open other voltage-gated ion channels that could further
depolarize the dendrites and lead to a locally generated action-potential. Other situations where the
contribution of the dendritic tree has to be taken into account include the back-propagation of action
potentials into the dendritic tree (Fig. 14A) after their generation in the axon-hillock (where the
density of Na+-channels is the highest) and the propagation of excitatory synaptic potentials within
the dendritic tree (Fig. 14B) that eventually reach the soma, depolarize it and produce action
potentials.