Untangling Complex Systems, page 56
+ + + + + + + + + + + +
FIGURE 9.21 Schematic structure of a neuron. The grey arrows indicate the flow of information from the
dendrites up to the synapsis through the axon.
14 There are neurons whose axon can be 1.5 m long! For example, those connecting our toes with the base of the spine.
15 Note that in pure water D would be ≈ 100 μm2/s, but the crowded environment of a cell reduces the velocity of diffusion significantly, especially for a macromolecule. Read Box 9.4 of this chapter.
276
Untangling Complex Systems
integrated into the axon hillock. Neurons, like the other cells in our body, are electrically polar-
ized: they have an electric potential gradient between the exterior and the interior of the cellular
membrane. This gradient of electric potential is made possible by the presence of proteins within
the membrane; they work as ionic pumps and channels.
The information that is integrated into the axon hillock can trigger an electrochemical wave
that crosses the entire axon quickly without weakening and reaches the bottom part of the neuron
where there are the synapses. At the synapses, the electrochemical information is transduced back
in chemical signals, and neurotransmitters are released to the dendrites of other neurons.
Every excitable portion of the neural membrane has two critical values of electric potential: one is
the resting potential and the other is the threshold potential. The resting potential represents the value
that the membrane maintains as long as it is not perturbed; usually, in the axon hillock, it is −70 mV.
The inputs, which a neuron receives, induce either a hyperpolarization or depolarization of the mem-
brane. In other words, the inputs either decrease or increase its electric potential. When a sufficiently
strong excitatory input depolarizes the membrane, and its potential overcomes the threshold value
(that is ≈ −55 mV in the axon hillock), the neuron triggers an action potential (Figure 9.22). The electric potential of the axon hillock soars quickly, in roughly 1 ms or less, up to reach a positive value, as
large as +50 mV. Then, it drops to values that are more negative than the resting potential. Finally, it
recovers its initial state, and the neuron is ready to discharge another action potential.
The discharge of an action potential involves voltage-gated ion channels and pumps that are
transmembrane proteins shown schematically in Figure 9.23. We start from the resting state (a)
with the membrane that is hyperpolarized. If the neuron receives an excitatory input, the membrane
potential raises because the sodium channels open (see step b in Figure 9.23). The increase of the membrane potential induced by the income of the first sodium ions exerts a positive feedback action.
In fact, it causes a further opening of the Na+ channels and a stronger depolarization. Overall, this
autocatalytic action determines a sharp jump of the potential membrane (see part c of Figure 9.23).
Then, within a fraction of one-thousandth of a second, the sodium channels switch to an inactivated
state due to the highly positive membrane potential (see part d of Figure 9.23). In the inactivated state, the sodium channels are closed and also refractory to reopening. At the same time, potassium
channels open. They allow K+ ions to leave the neuron. Such transfer repolarizes the membrane.
Finally, the ion pumps restore the initial transmembrane gradients of K+ and Na+ ions by exploiting
energy supplied by the cell. After all, the portion of the axon where all the described events have
occurred is ready to transmit another action potential (see part e of Figure 9.23). Meanwhile, the transmitted action potential propagates only forwards. The refractory states of the sodium channels prevent the depolarization from spreading backward along the axon. The action potentials
propagate vectorially towards the synapsis, as genuinely electrochemical waves.
+50
V)
al (m
tenti
0
po
−55
−
Membrane
70
0
1
2
3
4
Time (ms)
FIGURE 9.22 Schematic profile of an action potential.
The Emergence of Order in Space
277
K+ Channel
Na+ Channel
Ion pump
Na+
V) +50
Na+
K+
Out
Na+ Na+ Na+
Na+
ial (m
Na+
tentpo 0
K+
−55
−70
In
K+
K+
K+
Na+
Membrane
(a)
0
1
2
3
4
Resting state
Time (ms)
V) +50
Na+
K+
Out
Na+ Na+ Na+
Na+
ial (m
Na+
tentpo 0
K+
−55
−70
In
K+
K+
Membrane
K+
Na+
Na+
(b)
0
1
2
3
4
Depolarization
Time (ms)
V) +50
Na+
K+
Out
Na+
ial (m
Na+
tentpo 0
K+
−55
−70
In
K+
K+
Membrane
K+
Na+
Na+
(c)
Na+
Further
0
1
2
3
4
Time (ms)
Na+
Na+
depolarization
K+
Na+
V)
K+
+50
K+
Out
Na+
Na+
ial (m
tent
Na+
po
0
−55
K+
−70
In
K+
Membrane
Na+
(d)
Na+
0
1
2
3
4
Na+ Na+
Repolarization
Time (ms)
Na+
V) +50
Na+
K+
Na+
Out
Na+ Na+
Na+
ial (m
Na+
tentpo 0
K+
−55
−70
In
K+
K+
K+
Membrane
Na+
(e)
0
1
2
3
4
Restoring the resting potential
Time (ms)
FIGURE 9.23 Schematic description of five relevant steps in the discharge of an action potential: resting
state (a); depolarization (b); further depolarization (c); repolarization (d), and restoring the resting potential (e).
When the action potential reaches the synapsis, voltage-gated Ca + 2 channels open and calcium
ions enter the axon terminals. Ca + 2 entry causes neurotransmitter-containing synaptic vesicles to
move to the plasma membrane, fuse with it and release their content according to the process of
exocytosis. The neurotransmitter diffuses and binds to ligand-gated ion channels of the dendrites of
other neurons, exerting either an inhibitory or an excitatory action.
278
Untangling Complex Systems
Our understanding of how voltage-gated ion channels give rise to propagating action potentials is
primarily due to the physiologists Alan Hodgkin and Andrew Huxley. They were the first to record
the potential difference across the membrane of a squid giant axon by inserting a fine capillary
electrode in it. It was the year 1939. The Second World War began only a few weeks after their first
experiments were published in Nature (Hodgkin and Huxley 1939). Therefore, they were forced
to quit their research. After the war, Hodgkin and Huxley could restart their fruitful collaboration
that culminated with the publication of their mathematical model of the action potential in 1952
(Hodgkin and Huxley 1952). The electrical behavior of the axon membrane may be described by the
electrical circuit represented in Figure 9.24. Current flows through the membrane either by charging the membrane capacity ( I
(
))
C = CM dE dt
or by movement of ions through resistances that are in
parallel with the capacity. The ionic current consists of three contributions: sodium ( I ), potassium
Na
( I ) and leakage ( I ) current, the latter being due to other ions, such as chloride anions. Each term of K
L
the ionic current is given by Ohm’s law:
1
I
(
)
Na =
E − ENa
[9.48]
RNa
1
I
(
)
K =
E − EK
[9.49]
RK
1
I
(
)
L =
E − EL
[9.50]
RL
In the equations [9.48 through 9.50], E is membrane potential, whereas E , E , and E are the equi-Na
K
L
librium potentials for sodium, potassium, and leakage ions, respectively. The terms (1 /R ), (1 /R )
Na
K
and (1 /R ) are the sodium, potassium and leakage ions conductances.
L
The sodium and potassium conductances are time- and membrane potential-dependent. In
fact, depolarization causes a temporary rise in sodium conductance, whereas repolarization
determines an increase in potassium conductance but also a decrease in sodium conductance.
If we refer all the potentials to the absolute value of the membrane resting potential Er, and
we fix V = E − Er, VNa = ENa − Er, VK = EK − Er, VL = EL − Er, the equation that gives the total membrane current I is:
Extracellular
I
INa
IK
IL
RNa
RK
RL
CM
E
−
+
+
−
+
−
ENa
EK
EL
Intracellular
FIGURE 9.24 Electrical circuit proposed by Hodgkin and Huxley as a model to explain the formation of
action potentials in axons.
The Emergence of Order in Space
279
dV
1
1
1
I = C
(
)
(
)
(
)
M
+
V − VNa +
V − VK +
V − VL
[9.51]
dt
RNa
RK
RL
After defining the best non-linear analytic functions for (1 /R ) and (1 /R ) by fitting experimental
Na
K
data of potassium and sodium conductances, Hodgkin and Huxley succeeded in reproducing the
shape of an action potential. For the description of how an action potential propagates through an
axon, we must consider the concatenation of circuits like that of Figure 9.24. If we indicate with r the spatial coordinate that represents the principal axis of the cylindrical axon, perpendicular to the
membrane, the current flowing along r is
1
∂ V
2
I =
[9.52]
2
2π ( R
)
out + Rin
r
∂
where R and R are the external and internal resistances of the axon.16 If we consider an axon out
in
surrounded by a large amount of conducting fluid, R is negligible compared with the resistance
out
of the axoplasm (that is the cytoplasm within the axon of a neuron), i.e., R . If we indicate with a
in
the radius of the axoplasm and with ρ its resistivity, the final equation describing the propagation
in
of an action potential is
a ∂ V
2
V
1
1
1
=
∂
C
(
)
(
)
(
)
M
+
V − VNa +
V − VK +
V − VL
[9.53]
2
2
ρ in r
∂
t
∂
RNa
RK
RL
Equation [9.53] can be rearranged in the following form:
∂ V
2
=
a
∂ V + f V()
[9.54]
∂ t
2 C
2
ρ in M ∂ r
9.7.2 The fisher-kolmogorov eQuaTion
Equation [9.54] is formally equivalent to the equation that describes a traveling wave. If you remem-
ber equations [9.42 and 9.43], it is evident that the traveling waves of a chemical concentration x that
diffuses and reacts, are expressed by
16 It is important to notice that, so far, we have learned and modeled how electrochemical signals propagate within an axon of small diameter. In large diameter axons, the propagation of electrochemical signals is assisted by the myelin sheath.
The myelin sheath is made of cells wrapping the axon with concentric layers of their cell membranes. Such layers are
made of lipids. Therefore, they are not conductive. In a myelinated neuron, the portions of the axon not covered are called nodes of Ranvier (see Figure in this note). The layer of insulation generated by the myelin coat avoids current leakage from the axon by blocking the movement of ions through the axon membrane. As a result, action potentials propagate
faster in a myelinated axon rather than an unmyelinated one.
Nodes of ranvier
Myelinated axon
Unmyelinated axon
280
Untangling Complex Systems
∂ x
2
= ∂ x
D
+ f ( x)
[9.55]
∂ t
∂ r 2
where D is the diffusion coefficient, and f ( x) represents the kinetic law. The simplest case of a kinetic law that generates propagating chemical waves involves an autocatalytic reaction of the type:
X S
ka
+ →
2 X
[9.56]
The net reaction is S → X. S is the reacting substrate and X is the autocatalytic species. If we have S distributed homogeneously in our system, the addition of a small amount of X at one point can
trigger the autocatalytic reaction, like a burning fire. Then, the autocatalysis coupled with diffusion
makes the rest, and chemical waves spread throughout the system. If the initial concentration of S
is s 0 and that of X is negligible, the mass balance allows us to write s 0 = s + x. Therefore, equation
[9.55] becomes
∂ x
2
= ∂ x
D
+ kax
[9.57]
2
( s 0 − x)
∂ t
∂ r
where k is the kinetic constant of the autocatalytic reaction. We may rewrite equation [9.57] in
a
terms of the dimensionless variable ξ = x/ s 0 and obtain
∂ξ
ξ
2
= ∂
D
+ k ξ
a
1
( − ξ )
[9.58]
∂ t
∂ r 2
Equation [9.58] was proposed by the British statistician and mathematical biologist Ronald Fisher
(1937) for describing the spatial spread of an advantageous mutant gene through a population of
interbreeding organisms in a linear habitat, such as a shoreline. He wrote a partial differential equa-
tion like [9.58], where ξ represents the frequency of the advantageous mutant gene X at the expense
of the allelomorph S occupying the same locus;17 k represents the rate of mutation and D is the coef-a
ficient of diffusion of the advantageous gene. Fisher showed that waves of stationary shape advance
through the population with speeds
v ≥ 2 kaD
[9.59]
The same result was achieved by the Russian mathematician Andrey Kolmogorov and his co-
workers, exactly in the same year (Kolmogorov et al. 1937). This is the reason why equation [9.58]
is now known as Fisher-Kolmogorov equation. 18 It is surprising that the result [9.59] was anticipated
17 In genetics, allelomorphs (or shortly, alleles) are alternative forms of a gene that can occupy the same locus on a particular chromosome and control the same character.
18 A detailed mathematical treatment of the Fisher-Kolmogorov equation that does not have an analytical solution can be found in chapter 13 of the book “Mathematical Biology I” by Murray (2002), and in the feature article by Scott and Showalter (1992) published in the Journal of Physical Chemistry. In the latter, the authors deal not only with a quadratic but also a cubic autocatalytic process. A representation of propagation of chemical waves according to the Fisher-Kolmogorov equation is shown in the Figure of this note.
FIGURE 9.21 Schematic structure of a neuron. The grey arrows indicate the flow of information from the
dendrites up to the synapsis through the axon.
14 There are neurons whose axon can be 1.5 m long! For example, those connecting our toes with the base of the spine.
15 Note that in pure water D would be ≈ 100 μm2/s, but the crowded environment of a cell reduces the velocity of diffusion significantly, especially for a macromolecule. Read Box 9.4 of this chapter.
276
Untangling Complex Systems
integrated into the axon hillock. Neurons, like the other cells in our body, are electrically polar-
ized: they have an electric potential gradient between the exterior and the interior of the cellular
membrane. This gradient of electric potential is made possible by the presence of proteins within
the membrane; they work as ionic pumps and channels.
The information that is integrated into the axon hillock can trigger an electrochemical wave
that crosses the entire axon quickly without weakening and reaches the bottom part of the neuron
where there are the synapses. At the synapses, the electrochemical information is transduced back
in chemical signals, and neurotransmitters are released to the dendrites of other neurons.
Every excitable portion of the neural membrane has two critical values of electric potential: one is
the resting potential and the other is the threshold potential. The resting potential represents the value
that the membrane maintains as long as it is not perturbed; usually, in the axon hillock, it is −70 mV.
The inputs, which a neuron receives, induce either a hyperpolarization or depolarization of the mem-
brane. In other words, the inputs either decrease or increase its electric potential. When a sufficiently
strong excitatory input depolarizes the membrane, and its potential overcomes the threshold value
(that is ≈ −55 mV in the axon hillock), the neuron triggers an action potential (Figure 9.22). The electric potential of the axon hillock soars quickly, in roughly 1 ms or less, up to reach a positive value, as
large as +50 mV. Then, it drops to values that are more negative than the resting potential. Finally, it
recovers its initial state, and the neuron is ready to discharge another action potential.
The discharge of an action potential involves voltage-gated ion channels and pumps that are
transmembrane proteins shown schematically in Figure 9.23. We start from the resting state (a)
with the membrane that is hyperpolarized. If the neuron receives an excitatory input, the membrane
potential raises because the sodium channels open (see step b in Figure 9.23). The increase of the membrane potential induced by the income of the first sodium ions exerts a positive feedback action.
In fact, it causes a further opening of the Na+ channels and a stronger depolarization. Overall, this
autocatalytic action determines a sharp jump of the potential membrane (see part c of Figure 9.23).
Then, within a fraction of one-thousandth of a second, the sodium channels switch to an inactivated
state due to the highly positive membrane potential (see part d of Figure 9.23). In the inactivated state, the sodium channels are closed and also refractory to reopening. At the same time, potassium
channels open. They allow K+ ions to leave the neuron. Such transfer repolarizes the membrane.
Finally, the ion pumps restore the initial transmembrane gradients of K+ and Na+ ions by exploiting
energy supplied by the cell. After all, the portion of the axon where all the described events have
occurred is ready to transmit another action potential (see part e of Figure 9.23). Meanwhile, the transmitted action potential propagates only forwards. The refractory states of the sodium channels prevent the depolarization from spreading backward along the axon. The action potentials
propagate vectorially towards the synapsis, as genuinely electrochemical waves.
+50
V)
al (m
tenti
0
po
−55
−
Membrane
70
0
1
2
3
4
Time (ms)
FIGURE 9.22 Schematic profile of an action potential.
The Emergence of Order in Space
277
K+ Channel
Na+ Channel
Ion pump
Na+
V) +50
Na+
K+
Out
Na+ Na+ Na+
Na+
ial (m
Na+
tentpo 0
K+
−55
−70
In
K+
K+
K+
Na+
Membrane
(a)
0
1
2
3
4
Resting state
Time (ms)
V) +50
Na+
K+
Out
Na+ Na+ Na+
Na+
ial (m
Na+
tentpo 0
K+
−55
−70
In
K+
K+
Membrane
K+
Na+
Na+
(b)
0
1
2
3
4
Depolarization
Time (ms)
V) +50
Na+
K+
Out
Na+
ial (m
Na+
tentpo 0
K+
−55
−70
In
K+
K+
Membrane
K+
Na+
Na+
(c)
Na+
Further
0
1
2
3
4
Time (ms)
Na+
Na+
depolarization
K+
Na+
V)
K+
+50
K+
Out
Na+
Na+
ial (m
tent
Na+
po
0
−55
K+
−70
In
K+
Membrane
Na+
(d)
Na+
0
1
2
3
4
Na+ Na+
Repolarization
Time (ms)
Na+
V) +50
Na+
K+
Na+
Out
Na+ Na+
Na+
ial (m
Na+
tentpo 0
K+
−55
−70
In
K+
K+
K+
Membrane
Na+
(e)
0
1
2
3
4
Restoring the resting potential
Time (ms)
FIGURE 9.23 Schematic description of five relevant steps in the discharge of an action potential: resting
state (a); depolarization (b); further depolarization (c); repolarization (d), and restoring the resting potential (e).
When the action potential reaches the synapsis, voltage-gated Ca + 2 channels open and calcium
ions enter the axon terminals. Ca + 2 entry causes neurotransmitter-containing synaptic vesicles to
move to the plasma membrane, fuse with it and release their content according to the process of
exocytosis. The neurotransmitter diffuses and binds to ligand-gated ion channels of the dendrites of
other neurons, exerting either an inhibitory or an excitatory action.
278
Untangling Complex Systems
Our understanding of how voltage-gated ion channels give rise to propagating action potentials is
primarily due to the physiologists Alan Hodgkin and Andrew Huxley. They were the first to record
the potential difference across the membrane of a squid giant axon by inserting a fine capillary
electrode in it. It was the year 1939. The Second World War began only a few weeks after their first
experiments were published in Nature (Hodgkin and Huxley 1939). Therefore, they were forced
to quit their research. After the war, Hodgkin and Huxley could restart their fruitful collaboration
that culminated with the publication of their mathematical model of the action potential in 1952
(Hodgkin and Huxley 1952). The electrical behavior of the axon membrane may be described by the
electrical circuit represented in Figure 9.24. Current flows through the membrane either by charging the membrane capacity ( I
(
))
C = CM dE dt
or by movement of ions through resistances that are in
parallel with the capacity. The ionic current consists of three contributions: sodium ( I ), potassium
Na
( I ) and leakage ( I ) current, the latter being due to other ions, such as chloride anions. Each term of K
L
the ionic current is given by Ohm’s law:
1
I
(
)
Na =
E − ENa
[9.48]
RNa
1
I
(
)
K =
E − EK
[9.49]
RK
1
I
(
)
L =
E − EL
[9.50]
RL
In the equations [9.48 through 9.50], E is membrane potential, whereas E , E , and E are the equi-Na
K
L
librium potentials for sodium, potassium, and leakage ions, respectively. The terms (1 /R ), (1 /R )
Na
K
and (1 /R ) are the sodium, potassium and leakage ions conductances.
L
The sodium and potassium conductances are time- and membrane potential-dependent. In
fact, depolarization causes a temporary rise in sodium conductance, whereas repolarization
determines an increase in potassium conductance but also a decrease in sodium conductance.
If we refer all the potentials to the absolute value of the membrane resting potential Er, and
we fix V = E − Er, VNa = ENa − Er, VK = EK − Er, VL = EL − Er, the equation that gives the total membrane current I is:
Extracellular
I
INa
IK
IL
RNa
RK
RL
CM
E
−
+
+
−
+
−
ENa
EK
EL
Intracellular
FIGURE 9.24 Electrical circuit proposed by Hodgkin and Huxley as a model to explain the formation of
action potentials in axons.
The Emergence of Order in Space
279
dV
1
1
1
I = C
(
)
(
)
(
)
M
+
V − VNa +
V − VK +
V − VL
[9.51]
dt
RNa
RK
RL
After defining the best non-linear analytic functions for (1 /R ) and (1 /R ) by fitting experimental
Na
K
data of potassium and sodium conductances, Hodgkin and Huxley succeeded in reproducing the
shape of an action potential. For the description of how an action potential propagates through an
axon, we must consider the concatenation of circuits like that of Figure 9.24. If we indicate with r the spatial coordinate that represents the principal axis of the cylindrical axon, perpendicular to the
membrane, the current flowing along r is
1
∂ V
2
I =
[9.52]
2
2π ( R
)
out + Rin
r
∂
where R and R are the external and internal resistances of the axon.16 If we consider an axon out
in
surrounded by a large amount of conducting fluid, R is negligible compared with the resistance
out
of the axoplasm (that is the cytoplasm within the axon of a neuron), i.e., R . If we indicate with a
in
the radius of the axoplasm and with ρ its resistivity, the final equation describing the propagation
in
of an action potential is
a ∂ V
2
V
1
1
1
=
∂
C
(
)
(
)
(
)
M
+
V − VNa +
V − VK +
V − VL
[9.53]
2
2
ρ in r
∂
t
∂
RNa
RK
RL
Equation [9.53] can be rearranged in the following form:
∂ V
2
=
a
∂ V + f V()
[9.54]
∂ t
2 C
2
ρ in M ∂ r
9.7.2 The fisher-kolmogorov eQuaTion
Equation [9.54] is formally equivalent to the equation that describes a traveling wave. If you remem-
ber equations [9.42 and 9.43], it is evident that the traveling waves of a chemical concentration x that
diffuses and reacts, are expressed by
16 It is important to notice that, so far, we have learned and modeled how electrochemical signals propagate within an axon of small diameter. In large diameter axons, the propagation of electrochemical signals is assisted by the myelin sheath.
The myelin sheath is made of cells wrapping the axon with concentric layers of their cell membranes. Such layers are
made of lipids. Therefore, they are not conductive. In a myelinated neuron, the portions of the axon not covered are called nodes of Ranvier (see Figure in this note). The layer of insulation generated by the myelin coat avoids current leakage from the axon by blocking the movement of ions through the axon membrane. As a result, action potentials propagate
faster in a myelinated axon rather than an unmyelinated one.
Nodes of ranvier
Myelinated axon
Unmyelinated axon
280
Untangling Complex Systems
∂ x
2
= ∂ x
D
+ f ( x)
[9.55]
∂ t
∂ r 2
where D is the diffusion coefficient, and f ( x) represents the kinetic law. The simplest case of a kinetic law that generates propagating chemical waves involves an autocatalytic reaction of the type:
X S
ka
+ →
2 X
[9.56]
The net reaction is S → X. S is the reacting substrate and X is the autocatalytic species. If we have S distributed homogeneously in our system, the addition of a small amount of X at one point can
trigger the autocatalytic reaction, like a burning fire. Then, the autocatalysis coupled with diffusion
makes the rest, and chemical waves spread throughout the system. If the initial concentration of S
is s 0 and that of X is negligible, the mass balance allows us to write s 0 = s + x. Therefore, equation
[9.55] becomes
∂ x
2
= ∂ x
D
+ kax
[9.57]
2
( s 0 − x)
∂ t
∂ r
where k is the kinetic constant of the autocatalytic reaction. We may rewrite equation [9.57] in
a
terms of the dimensionless variable ξ = x/ s 0 and obtain
∂ξ
ξ
2
= ∂
D
+ k ξ
a
1
( − ξ )
[9.58]
∂ t
∂ r 2
Equation [9.58] was proposed by the British statistician and mathematical biologist Ronald Fisher
(1937) for describing the spatial spread of an advantageous mutant gene through a population of
interbreeding organisms in a linear habitat, such as a shoreline. He wrote a partial differential equa-
tion like [9.58], where ξ represents the frequency of the advantageous mutant gene X at the expense
of the allelomorph S occupying the same locus;17 k represents the rate of mutation and D is the coef-a
ficient of diffusion of the advantageous gene. Fisher showed that waves of stationary shape advance
through the population with speeds
v ≥ 2 kaD
[9.59]
The same result was achieved by the Russian mathematician Andrey Kolmogorov and his co-
workers, exactly in the same year (Kolmogorov et al. 1937). This is the reason why equation [9.58]
is now known as Fisher-Kolmogorov equation. 18 It is surprising that the result [9.59] was anticipated
17 In genetics, allelomorphs (or shortly, alleles) are alternative forms of a gene that can occupy the same locus on a particular chromosome and control the same character.
18 A detailed mathematical treatment of the Fisher-Kolmogorov equation that does not have an analytical solution can be found in chapter 13 of the book “Mathematical Biology I” by Murray (2002), and in the feature article by Scott and Showalter (1992) published in the Journal of Physical Chemistry. In the latter, the authors deal not only with a quadratic but also a cubic autocatalytic process. A representation of propagation of chemical waves according to the Fisher-Kolmogorov equation is shown in the Figure of this note.
