A model of a myelinated nerve axon thres
Journal of
J. Math. Biology (1985) 23:119-135
Mathematical
61ology
9 Springer-Verlag 1985
A model of a myelinated nerve axon: threshold behaviour
and propagation
P. Grindrod* and B. D. Sleeman
Department of Mathematical Sciences, University of Dundee, Dundee DD1 4HN, U.K.
Abstract. A model of a myelinated nerve axon is developed on the basis of
F i t z H u g h - N a g u m o dynamics under the assumption that the nodes of Ranvier
are of small but finite width. It is shown that a periodic excited state may not
exist if the width of the nodes is too small and the leakage across the myelin
sheath is too great. The propagation of a super threshold pulse is prevented
in the absence of nodes. Global stability of the resting equilibrium state is
investigated as well as the propagation of "wave front", type solutions.
Key words: Myelinated nerve axons - - Nodes o f R a n v i e r - - FitzHugh- Nagumo
dynamics - - Threshold b e h a v i o u r - - Global stability - - Wave fronts
O. Introduction
A myelinated nerve axon consists of a long cylindrical membrane which is
surrounded by a sheath of lipoprotein, called myelin, formed from a condensation
of Schwann cell membranes. This fatty layered tissue insulates the axon from
the external ionic fluid. At approximately millimetre intervals there are small
gaps called nodes of Ranvier. These nodes expose the extracellular fluid to the
excitable axon membrane. At the nodes the axon membrane is selectively permeable to the charged ions within the axoplasm and outer ionic fluid. Hence
ionic currents may pass through the membrane in the same way as for the
somewhat simpler unmyelinated axon (see Fig. 1). Between the nodes the myelin
sheath prevents ionic transport, except possibly for some leakage from the
myelin
axopbsm
~\ V/////////A V/////////~ V//////////~
Fig. 1. A myelinated nerve axon
\
membrane
node
* Currept address: Mathematical Institute, University of Oxford, Oxford, U.K.
120
P. Grindrod and B. D. Sleeman
axoplasm, through the membrane, to small pockets of plasma held within the
folds of lipoprotein. Stimulation at one node generates longitudinal ionic currents
which excite neighbouring nodes. Because of the insulating nature of the myelihated segments impulses have the character of jumping from node to node. This
form of conduction is referred to as saltatory conduction.
In that which follows we allow the nodes to have a finite longitudinal length
and derive a model for the behaviour of the transmembrane potential along the
axon.
The approach adopted in this paper differs from that of FitzHugh [4] and
Bell and Cosner [2] who assume the nodes to be point sources of excitation.
Here we are able to allow the node width to be as small as we please and use it
as a parameter to control the degree of excitation possible at each node. In order
to derive our model we consider equivalent electric circuits for the myelinated
and unmyelinated segments separately and apply r - c cable theory.
Figure 2 shows the equivalent circuit for a myelinated segment. Here ~ ( x )
and ~o(X) denote the longitudinal axoplasmic and external fluid potentials
respectively; r~ and ro respectively denote the longitudinal resistivities of the
axoplasm and external fluid and cm denotes the capacitance per unit length of
the myelinated membrane. I(x) denotes the effective membrane current density
while i~(x) and io(x) denote the longitudinal axoplasmic and external currents
respectively. R denotes the resistivity of the myelinated membrane.
On applying Ohm's law and Kirchhoff's laws we obtain the following equation
governing the transmembrane potential, viz. v = ~ - ~o where
1
Gray, (ro+ r~) vxx v/ R.
=
-
(0.1)
Figure 3 shows the equivalent electric circuit for a node segment. Here J(x)
denotes the ionic current density present in the membrane at a distance x along
the axon. It is this current which provides the excitation. We remark that energy
is used up in driving such a current.
In this circuit CN denotes the capacitance per unit length of the membrane
(CN > CM).
(~i (X}
ri
4~i(x+Z~x}
AX
i~'• 'wvvwx,
I
m •
io(x)
4'o(x)
1
'wwv'
ro&x
4~o(x*Ax)
.4
Fig. 2. A myelinated segment of nerve axon: the equivalent electric circuit
Model of a rnyelinated nerve axon
ii(•
121
riA•
ImAX
Ci~h
x
"~JAx
F~
~ A X ~
j
Fig. 3. An unmyelinated (node) segment of nerve axon: the equivalent electric circuit
Again by applying Ohm's law and Kirchhoff's laws we obtain
CNV, =
1
to+ ri
Vxx-- J.
(0.2)
In the model (0.2) we must specify the form of the ionic membrane current
density J. Usually J is modelled by Hodgkin-Huxley dynamics but this requires
the introduction of three auxiliary variables which represent the rise and fall of
the membrane permeability to sodium and potassium ions. Alternatively we may
use the simpler FitzHugh-Nagumo dynamics which have been successfully
employed in the study of unmyelinated nerve axons. Thus we set
J= -f(v)+z,
Zt =
(0.3)
O-V -- ~/Z,
where o-, y are positive constants and f(v) is a non-linear function of v whose
qualitative behaviour is shown in Fig. 4. For example we could take f(v)=
v(1-v.)(v-a), 0 < a < 8 9 The dependent variable z represents the level of
potassium permeability of the membrane and serves as a recovery variable. Thus
enabling the action potential to return to equilibrium after excitation.
In Eqs. (0.3) the constants or and 7 are usually small and the recovery equation
for z is "slow" in comparison with the potential Eq. (0.2). Henceforth we shall
not include the recovery process in the present model, but will instead restrict
f(v)
0
Fig. 4. The qualitative form o f f
122
P. G r i n d r o d a n d B. D. S l e e m a n
our analysis to the "excitory-phase" for the action potentials at the nodes. That
is, we will use (0.2) together with
J = -f(v).
(0.4)
The same assumption is made in [2].
For the unmyelinated axon, the speed and shape of the leading front of
propagating pulses may be approximated by a monotone travelling front obtained
from the FitzHugh-Nagumo equations without recovery (see [4]), so we may
expect a similar approximation for myelinated axons to be attainable from the
present model, (see Sect. 4). The "excitory-phase" should be strong enough to
shift the transmembrane potential from rest up to some excited steady state, from
where the recovery process would eventually return it to rest.
To complete the model we match equations (0.1), (0.2) across the myelin-node
junctions by applying the following continuity conditions at a typical junction x =
X0 9
(a)
v(xo, t)l+_=-lim{v(Xo+e, t)-V(Xo-e, t)}=0,
~.--."0
(b)
Vx(Xo,t)l+=-lim{vx(xo+e, t)-vx(xo-e, t)}=0.
-
(0.5)
e..~O
Condition (0.5a) demands that the transmembrane potential v be continuous
across junctions while (0.5b) follows from the local continuity of the longitudinal
currents ii(x) and io(x) at x = Xo.
Once the positions and widths of the nodes have been specified we may use
(0.1), (0.2), (0.4) and (0.5) to model an infinitely long myelinated axon. The
simplest case to consider, and the one adopted in this paper is that in which the
nodes have uniform width and are located at periodic intervals along the axon.
Since the recovery process has been neglected a wave of excitation along the
axon corresponds to a shift in the potential v away from the resting state (v = 0
in this case) to some positive excited state. We note that if recovery is incorporated
in the model then this would return the potential to the resting equilibrium after
some time.
We also expect to observe a threshold phenomenon. That is, if v(x, 0) is small,
then in either the "sup norm" or the L2-norm, the solution evolves to the zero
resting equilibrium.
In Sect. 1 we investigate threshold behaviour and show that a periodic excited
steady state may not exist if the width of the nodes is too small and the leakage
across the myelin sheath is too great. In Sect. 2 we show how propagation of a
superthreshold solution may be prevented in the absence of nodes. In Sect. 3 we
derive an explicit condition on the system parameters which ensures that the
resting equilibrium is globally stable. In all these cases we are in effect indicating
how the nerve axon may degenerate. Finally, in Sect. 4 we discuss the propagation
of "wave-front" type solutions. Such solutions travel with a constant speed along
the axon, while the wave front profile oscillates with time.
Central to the theory developed in Sects. 1, 2 and 3 is the following comparison
principle stated here for convenience.
Model of a myelinated nerve axon
123
Comparison theorem. Let g: ~ -~ [0, 1] be a continuous function satisfying g(O) =
g(1) = 0. Suppose u( x, t) and v( x, t) are continuously differentiable functions from
R • R + to [0, 1] satisfying the differential inequalities
CNut - u~x - g(u) >i CNv, - Vx~ - g(v),
for t>~O, x~(O, 0) mod(1),
CMut - u~ + u / R >! Csavt - Vxx + v / R
for t >>-O,x c (0, 1) mod(1), where 0 ~ (0, 1), CN, CM and R are positive constants.
Then u(x, O) >1v(x, O) for all x ~ R implies
u(x,t)>~v(x,t)
f o r a l l x 6 R , t>~O.
The p r o o f of the above theorem follows in the same way as used by Aronson
and Weinberger in [1], except that the continuity of ut, vt and ux, vx must be used
to show that v(x, t) - u ( x , t) cannot have a local minimum value of zero at x = 0
or 0 mod(1).
1. Superthreshold steady-state solutions
Consider an infinitely long uniform myelinated axon and assume that the independent variable x has been scaled so that the nodes have width 0O.
CMu, = Ux~ - u / R ,
x ~ (0, 1) mod(1),
t~>O,
U(X, t)[ + = U~(X, t)]_+ = O,
x = O, 0 mod(1),
(1.1)
t~>O.
where the initial data u(x, O) satisfies
0 0 is assumed to be large a n d f i s of the form described in the introduction.
The comparison theorem applied to (1.1) shows that [0, 1] is an invariant set
for solutions u. That is 0~< u(x, 0)~< 1 for all x c ~ implies O 1 ,
q E (0, exp(-Xa/~/-R));
q
then there exists a steady state solution to (2.3), (2.4) such that q(O) = 1, q(oo) = 0
and q'(x) 0
then the resting equilibrium u-~O of Eqs. (1.1) is asymptotically stable in
Ilu(., t)ll >-O for all z, if and only if c = x / - ~ ( 1 - 2 a ) / x / 2 .
In (4.1) we set u(x, t ) = ( a ( z ) + p ( z , t), where z = x + c t ,
it follows that
P, = Pz~ - cpz + 5 ( f ( p + r - f ( c k ) ) + eft(z - c t ) f ( p + r )
together with the conditions limlzl_,oop(z, t ) = 0 for all t.
Now set p = eQ, then
Qt = Qz~ - cQz + 6f'(q~ )Q + [3(z - ct)f( ga) + O ( e ) .
To order e, Q satisfies the linear perturbed equation
Q, + A Q = ft(z - ct )f( ck).
(4.4)
where A = { - d 2 / d z : + c ( d / d z ) S f ' ( c k ) is considered as a linear closed, densely
defined operator in the Banach space BCo = {w(z)] w(z) is continuous on R, and
limlzl_,oo w ( z ) = 0}, endowed with the uniform topology IIwll B~o = supper I W(z)l,
Model of a myelinated nerve axon
133
It can be shown that the spectrum of A contains an isolated simple eigenvalue
at the origin, with eigenfunction r
and that the remaining eigenvalues and
essential spectrum are bounded to the right of the imaginary axis (see [5], p. 131,
for example).
Lemma 4.1. I f fl0 •(X) dx = O, then (4.4) has a solution Qo(z, t) such that
(a) Qo(z, t) ~ BCo for all t ~ R.
(b) Qo(z, t)= Q ( z , t+ 1 ) f o r a l l t ~ R .
Proof. The proof of the lemma rests on a general result concerning periodic
solutions for periodically forced parabolic equations, which will be published
elsewhere [7]. The point is that (4.4) has a solution Qo satisfying (a) and (b)
above if and only if
f l/c
0=
dO
P[fl(z - ct)f(r
dt
where P is the projection of BCo onto the linear subspace spanned by r
fl/c f l ( z - c t ) dt = O, and the lemma
Since P is linear this is certainly true when Jo
follows.
Immediately we have
Theorem 4.2 I f a ( x ) = ~ + eft ( x ), where 6 > 0 and ~~ fl ( x ) dx = O, then for all e ~ ~,
(4.1) has a solution of the form
u(x, t) = if(z, t) = r
where r
t)+ O(e2),
(4.5)
is the front solution of (4.3), and Qo is given by Lemma 4.1.
In particular when e is small then the function (4.5) approaches a travelling
front solution with an oscillatory profile in time.
Remarks. 1. Although the above result applies only when e > 0 is assumed small
numerical evidence suggests that such solutions exist when a ( x ) is an arbitrary
1-periodic non-negative function (and in particular the step function introduced
in (4.2)).
We conjecture that if a ( x ) is any non-negative one-periodic function with
average 6 =~1o a ( x ) d x , then (4.1) admits a travelling front type solution with
speed c = v/-6(1-2a)/,r
and a time-oscillating profile of period 1/c.
Figures 10 and 11 depict such solutions for a = 0 . 1 and a ( x ) = 3 2 for x~
[0, 0.25)mod(1) and zero otherwise. In these figures we have changed to the
variables (z, t) so that the solutions appear to be standing oscillatory fronts.
2. These results show that the speed of propagation c is proportional to the
square root of the nodal length. However, since our model has been nondimensionalised the actual speed of propagation in the myelinated fibre is "fast"
due to the fact that the diameter of the axon is small compared to that of an
unmyelinated axon.
d
"0
"0
o
'o
"0
{3
o
i_
=.
o
- -
o
o
o
o
o
i
~J
o
o'-
c~
o--
a-
c~
J. Math. Biology (1985) 23:119-135
Mathematical
61ology
9 Springer-Verlag 1985
A model of a myelinated nerve axon: threshold behaviour
and propagation
P. Grindrod* and B. D. Sleeman
Department of Mathematical Sciences, University of Dundee, Dundee DD1 4HN, U.K.
Abstract. A model of a myelinated nerve axon is developed on the basis of
F i t z H u g h - N a g u m o dynamics under the assumption that the nodes of Ranvier
are of small but finite width. It is shown that a periodic excited state may not
exist if the width of the nodes is too small and the leakage across the myelin
sheath is too great. The propagation of a super threshold pulse is prevented
in the absence of nodes. Global stability of the resting equilibrium state is
investigated as well as the propagation of "wave front", type solutions.
Key words: Myelinated nerve axons - - Nodes o f R a n v i e r - - FitzHugh- Nagumo
dynamics - - Threshold b e h a v i o u r - - Global stability - - Wave fronts
O. Introduction
A myelinated nerve axon consists of a long cylindrical membrane which is
surrounded by a sheath of lipoprotein, called myelin, formed from a condensation
of Schwann cell membranes. This fatty layered tissue insulates the axon from
the external ionic fluid. At approximately millimetre intervals there are small
gaps called nodes of Ranvier. These nodes expose the extracellular fluid to the
excitable axon membrane. At the nodes the axon membrane is selectively permeable to the charged ions within the axoplasm and outer ionic fluid. Hence
ionic currents may pass through the membrane in the same way as for the
somewhat simpler unmyelinated axon (see Fig. 1). Between the nodes the myelin
sheath prevents ionic transport, except possibly for some leakage from the
myelin
axopbsm
~\ V/////////A V/////////~ V//////////~
Fig. 1. A myelinated nerve axon
\
membrane
node
* Currept address: Mathematical Institute, University of Oxford, Oxford, U.K.
120
P. Grindrod and B. D. Sleeman
axoplasm, through the membrane, to small pockets of plasma held within the
folds of lipoprotein. Stimulation at one node generates longitudinal ionic currents
which excite neighbouring nodes. Because of the insulating nature of the myelihated segments impulses have the character of jumping from node to node. This
form of conduction is referred to as saltatory conduction.
In that which follows we allow the nodes to have a finite longitudinal length
and derive a model for the behaviour of the transmembrane potential along the
axon.
The approach adopted in this paper differs from that of FitzHugh [4] and
Bell and Cosner [2] who assume the nodes to be point sources of excitation.
Here we are able to allow the node width to be as small as we please and use it
as a parameter to control the degree of excitation possible at each node. In order
to derive our model we consider equivalent electric circuits for the myelinated
and unmyelinated segments separately and apply r - c cable theory.
Figure 2 shows the equivalent circuit for a myelinated segment. Here ~ ( x )
and ~o(X) denote the longitudinal axoplasmic and external fluid potentials
respectively; r~ and ro respectively denote the longitudinal resistivities of the
axoplasm and external fluid and cm denotes the capacitance per unit length of
the myelinated membrane. I(x) denotes the effective membrane current density
while i~(x) and io(x) denote the longitudinal axoplasmic and external currents
respectively. R denotes the resistivity of the myelinated membrane.
On applying Ohm's law and Kirchhoff's laws we obtain the following equation
governing the transmembrane potential, viz. v = ~ - ~o where
1
Gray, (ro+ r~) vxx v/ R.
=
-
(0.1)
Figure 3 shows the equivalent electric circuit for a node segment. Here J(x)
denotes the ionic current density present in the membrane at a distance x along
the axon. It is this current which provides the excitation. We remark that energy
is used up in driving such a current.
In this circuit CN denotes the capacitance per unit length of the membrane
(CN > CM).
(~i (X}
ri
4~i(x+Z~x}
AX
i~'• 'wvvwx,
I
m •
io(x)
4'o(x)
1
'wwv'
ro&x
4~o(x*Ax)
.4
Fig. 2. A myelinated segment of nerve axon: the equivalent electric circuit
Model of a rnyelinated nerve axon
ii(•
121
riA•
ImAX
Ci~h
x
"~JAx
F~
~ A X ~
j
Fig. 3. An unmyelinated (node) segment of nerve axon: the equivalent electric circuit
Again by applying Ohm's law and Kirchhoff's laws we obtain
CNV, =
1
to+ ri
Vxx-- J.
(0.2)
In the model (0.2) we must specify the form of the ionic membrane current
density J. Usually J is modelled by Hodgkin-Huxley dynamics but this requires
the introduction of three auxiliary variables which represent the rise and fall of
the membrane permeability to sodium and potassium ions. Alternatively we may
use the simpler FitzHugh-Nagumo dynamics which have been successfully
employed in the study of unmyelinated nerve axons. Thus we set
J= -f(v)+z,
Zt =
(0.3)
O-V -- ~/Z,
where o-, y are positive constants and f(v) is a non-linear function of v whose
qualitative behaviour is shown in Fig. 4. For example we could take f(v)=
v(1-v.)(v-a), 0 < a < 8 9 The dependent variable z represents the level of
potassium permeability of the membrane and serves as a recovery variable. Thus
enabling the action potential to return to equilibrium after excitation.
In Eqs. (0.3) the constants or and 7 are usually small and the recovery equation
for z is "slow" in comparison with the potential Eq. (0.2). Henceforth we shall
not include the recovery process in the present model, but will instead restrict
f(v)
0
Fig. 4. The qualitative form o f f
122
P. G r i n d r o d a n d B. D. S l e e m a n
our analysis to the "excitory-phase" for the action potentials at the nodes. That
is, we will use (0.2) together with
J = -f(v).
(0.4)
The same assumption is made in [2].
For the unmyelinated axon, the speed and shape of the leading front of
propagating pulses may be approximated by a monotone travelling front obtained
from the FitzHugh-Nagumo equations without recovery (see [4]), so we may
expect a similar approximation for myelinated axons to be attainable from the
present model, (see Sect. 4). The "excitory-phase" should be strong enough to
shift the transmembrane potential from rest up to some excited steady state, from
where the recovery process would eventually return it to rest.
To complete the model we match equations (0.1), (0.2) across the myelin-node
junctions by applying the following continuity conditions at a typical junction x =
X0 9
(a)
v(xo, t)l+_=-lim{v(Xo+e, t)-V(Xo-e, t)}=0,
~.--."0
(b)
Vx(Xo,t)l+=-lim{vx(xo+e, t)-vx(xo-e, t)}=0.
-
(0.5)
e..~O
Condition (0.5a) demands that the transmembrane potential v be continuous
across junctions while (0.5b) follows from the local continuity of the longitudinal
currents ii(x) and io(x) at x = Xo.
Once the positions and widths of the nodes have been specified we may use
(0.1), (0.2), (0.4) and (0.5) to model an infinitely long myelinated axon. The
simplest case to consider, and the one adopted in this paper is that in which the
nodes have uniform width and are located at periodic intervals along the axon.
Since the recovery process has been neglected a wave of excitation along the
axon corresponds to a shift in the potential v away from the resting state (v = 0
in this case) to some positive excited state. We note that if recovery is incorporated
in the model then this would return the potential to the resting equilibrium after
some time.
We also expect to observe a threshold phenomenon. That is, if v(x, 0) is small,
then in either the "sup norm" or the L2-norm, the solution evolves to the zero
resting equilibrium.
In Sect. 1 we investigate threshold behaviour and show that a periodic excited
steady state may not exist if the width of the nodes is too small and the leakage
across the myelin sheath is too great. In Sect. 2 we show how propagation of a
superthreshold solution may be prevented in the absence of nodes. In Sect. 3 we
derive an explicit condition on the system parameters which ensures that the
resting equilibrium is globally stable. In all these cases we are in effect indicating
how the nerve axon may degenerate. Finally, in Sect. 4 we discuss the propagation
of "wave-front" type solutions. Such solutions travel with a constant speed along
the axon, while the wave front profile oscillates with time.
Central to the theory developed in Sects. 1, 2 and 3 is the following comparison
principle stated here for convenience.
Model of a myelinated nerve axon
123
Comparison theorem. Let g: ~ -~ [0, 1] be a continuous function satisfying g(O) =
g(1) = 0. Suppose u( x, t) and v( x, t) are continuously differentiable functions from
R • R + to [0, 1] satisfying the differential inequalities
CNut - u~x - g(u) >i CNv, - Vx~ - g(v),
for t>~O, x~(O, 0) mod(1),
CMut - u~ + u / R >! Csavt - Vxx + v / R
for t >>-O,x c (0, 1) mod(1), where 0 ~ (0, 1), CN, CM and R are positive constants.
Then u(x, O) >1v(x, O) for all x ~ R implies
u(x,t)>~v(x,t)
f o r a l l x 6 R , t>~O.
The p r o o f of the above theorem follows in the same way as used by Aronson
and Weinberger in [1], except that the continuity of ut, vt and ux, vx must be used
to show that v(x, t) - u ( x , t) cannot have a local minimum value of zero at x = 0
or 0 mod(1).
1. Superthreshold steady-state solutions
Consider an infinitely long uniform myelinated axon and assume that the independent variable x has been scaled so that the nodes have width 0O.
CMu, = Ux~ - u / R ,
x ~ (0, 1) mod(1),
t~>O,
U(X, t)[ + = U~(X, t)]_+ = O,
x = O, 0 mod(1),
(1.1)
t~>O.
where the initial data u(x, O) satisfies
0 0 is assumed to be large a n d f i s of the form described in the introduction.
The comparison theorem applied to (1.1) shows that [0, 1] is an invariant set
for solutions u. That is 0~< u(x, 0)~< 1 for all x c ~ implies O 1 ,
q E (0, exp(-Xa/~/-R));
q
then there exists a steady state solution to (2.3), (2.4) such that q(O) = 1, q(oo) = 0
and q'(x) 0
then the resting equilibrium u-~O of Eqs. (1.1) is asymptotically stable in
Ilu(., t)ll >-O for all z, if and only if c = x / - ~ ( 1 - 2 a ) / x / 2 .
In (4.1) we set u(x, t ) = ( a ( z ) + p ( z , t), where z = x + c t ,
it follows that
P, = Pz~ - cpz + 5 ( f ( p + r - f ( c k ) ) + eft(z - c t ) f ( p + r )
together with the conditions limlzl_,oop(z, t ) = 0 for all t.
Now set p = eQ, then
Qt = Qz~ - cQz + 6f'(q~ )Q + [3(z - ct)f( ga) + O ( e ) .
To order e, Q satisfies the linear perturbed equation
Q, + A Q = ft(z - ct )f( ck).
(4.4)
where A = { - d 2 / d z : + c ( d / d z ) S f ' ( c k ) is considered as a linear closed, densely
defined operator in the Banach space BCo = {w(z)] w(z) is continuous on R, and
limlzl_,oo w ( z ) = 0}, endowed with the uniform topology IIwll B~o = supper I W(z)l,
Model of a myelinated nerve axon
133
It can be shown that the spectrum of A contains an isolated simple eigenvalue
at the origin, with eigenfunction r
and that the remaining eigenvalues and
essential spectrum are bounded to the right of the imaginary axis (see [5], p. 131,
for example).
Lemma 4.1. I f fl0 •(X) dx = O, then (4.4) has a solution Qo(z, t) such that
(a) Qo(z, t) ~ BCo for all t ~ R.
(b) Qo(z, t)= Q ( z , t+ 1 ) f o r a l l t ~ R .
Proof. The proof of the lemma rests on a general result concerning periodic
solutions for periodically forced parabolic equations, which will be published
elsewhere [7]. The point is that (4.4) has a solution Qo satisfying (a) and (b)
above if and only if
f l/c
0=
dO
P[fl(z - ct)f(r
dt
where P is the projection of BCo onto the linear subspace spanned by r
fl/c f l ( z - c t ) dt = O, and the lemma
Since P is linear this is certainly true when Jo
follows.
Immediately we have
Theorem 4.2 I f a ( x ) = ~ + eft ( x ), where 6 > 0 and ~~ fl ( x ) dx = O, then for all e ~ ~,
(4.1) has a solution of the form
u(x, t) = if(z, t) = r
where r
t)+ O(e2),
(4.5)
is the front solution of (4.3), and Qo is given by Lemma 4.1.
In particular when e is small then the function (4.5) approaches a travelling
front solution with an oscillatory profile in time.
Remarks. 1. Although the above result applies only when e > 0 is assumed small
numerical evidence suggests that such solutions exist when a ( x ) is an arbitrary
1-periodic non-negative function (and in particular the step function introduced
in (4.2)).
We conjecture that if a ( x ) is any non-negative one-periodic function with
average 6 =~1o a ( x ) d x , then (4.1) admits a travelling front type solution with
speed c = v/-6(1-2a)/,r
and a time-oscillating profile of period 1/c.
Figures 10 and 11 depict such solutions for a = 0 . 1 and a ( x ) = 3 2 for x~
[0, 0.25)mod(1) and zero otherwise. In these figures we have changed to the
variables (z, t) so that the solutions appear to be standing oscillatory fronts.
2. These results show that the speed of propagation c is proportional to the
square root of the nodal length. However, since our model has been nondimensionalised the actual speed of propagation in the myelinated fibre is "fast"
due to the fact that the diameter of the axon is small compared to that of an
unmyelinated axon.
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