=Paper=
{{Paper
|id=Vol-1638/Paper80
|storemode=property
|title=Three scenarios for changing of stability in the dynamic model of nerve conduction
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper80.pdf
|volume=Vol-1638
|authors=Elena A. Shchepakina
}}
==Three scenarios for changing of stability in the dynamic model of nerve conduction==
https://ceur-ws.org/Vol-1638/Paper80.pdf
Mathematical Modelling
THREE SCENARIOS FOR CHANGING OF
STABILITY IN THE DYNAMIC MODEL OF NERVE
CONDUCTION
E.A. Shchepakina
Samara National Research University, Samara, Russia
Abstract. The paper deals with the specific cases of the changing of stability of
slow integral manifold of singularly perturbed systems of ODE via the dynamic
model of nerve conduction. It is shown that the proper choice of the additional
parameters of the system allows us to construct the slow integral manifold with
multiple change of its stability.
Keywords: singular perturbations, canards, delaying of the loss of stability,
critical phenomena, Hindmarsh-Rose model.
Citation: Shchepakina EA. Three scenarios for changing of stability in the dy-
namic model of nerve conduction. CEUR Workshop Proceedings 2016; 1638:
664-673. DOI: 10.18287/1613-0073-2016-1638-664-673
Introduction
The usual assumption of the singular perturbation theory [1-3] is based on the fact,
that the main functional determinant of the fast subsystem is different from zero.
However, in many applications this condition is violated that implies the critical phe-
nomena. A violation of this condition can lead to a delaying effect of loss of stability
[4]. Phenomenon of the delay of the loss of stability is based on the fact that the actual
escape of the phase point from the position of equilibrium, which lost its stability,
does not occur immediately. Two scenarios for delaying of the loss of stability are
well-known [5].
The first case corresponds to the transition of one real eigenvalue of the linearized fast
subsystem through zero when the slow variables are changed. This scenario for delay-
ing of the loss of stability in the singularly perturbed systems is associated with the
canards or duck-trajectories [6-16]. In the second case a pair of complex conjugate
eigenvalues passes from the left complex half-plane to the right one [17-19].
In this paper we also consider a scenario of change of stability when the real parts as
well as the imaginary parts of a pair of complex conjugate eigenvalues become zero
followed by the appearance of multiple zero root and then by the birth of a pair of real
eigenvalues of opposite signs.
All three scenarios of change of stability are discussed via the Hindmarsh-Rose model
of nerve conduction [20].
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Mathematical Modelling Shchepakina EA. Three scenarios…
Delay phenomenon of loss of stability in multirate systems
Consider the system of ordinary differential equations with a small positive parame-
ter:
x X ( x, y, ), y Y ( x, y , ), (1)
where x and y are vectors in Euclidean spaces R m and R n , respectively, ε is a small
positive parameter, the vector-functions X and Y are sufficiently smooth and their
values are O(1) as 𝜀 → 0. The slow and fast subsystems are represented by the first
and the second equation of (1), respectively. Such systems are called the singularly
perturbed systems [1-3].
A smooth surface S in R m R n R is called an integral manifold of the system (1) if
any integral curve of the system that has at least one point in common with S lies
entirely on S. Only the integral manifolds of system (1) with dimension m (the dimen-
sion of the slow variable x) that can be represented as graphs of functions
y h ( x, ),
are discussed here.
We will assume that h( x, ) is a sufficiently smooth function of . Such integral
manifolds are called the manifolds of slow motions.
The surface described by the degenerate equation
Y ( x, y, 0) 0
(2)
is called a slow surface (or a slow curve when the dimension of this surface is equal to
one). The slow surface can be considered as a zero-order ( 0) approximation of
the slow integral manifold, i.e., h( x, 0) ( x), where ( x ) is an isolated root of the
degenerate equation (2).
The subset of the slow surface is stable (or attractive) if the spectrum of the Jacobian
matrix
Y
J x, ( x), 0 (3)
y
is located in the left half-plane. If there is at least one eigenvalue of the Jacobian ma-
trix (3) with a positive real part then the subset of the slow surface is unstable (or
repulsive). The subset of the slow surface given by
Y
det x, ( x), 0 0
y
determines a breakdown surface (or jump points in the scalar case) [1, 2].
In an "–neighborhood of a stable (unstable) subset of the slow surface there exists a
stable (unstable) slow integral manifold.
A slow integral manifold can change its stability in some specific cases. The mecha-
nism of two cases of change of stability is described below.
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The case of a zero root. Canards
Consider the autonomous singularly perturbed system
x X ( x, y, , ), y Y ( x, y, , ), (4)
where is small positive parameter, is an additional parameter, for which an
equilibrium of the fast equation becomes unstable with transition of one real eigen-
value of (3) through zero when the slow variables are changed.
For simplicity we consider the case when the variables x and y are scalar. The stable
and unstable subsets of the slow curve are separated by the jump point(s).
The presence of the additional scalar parameter 𝛼 provides the possibility of gluing
the stable and unstable slow invariant manifolds at a jump point to form a single tra-
jectory, the canard.
A canard is a trajectory of a singularly perturbed system of differential equations if it
follows at first a stable invariant manifold, and then an unstable one. In both cases the
length of the trajectory is more than infinitesimally small.
The term “canard” (or duck–trajectory) had been originally given by French mathe-
maticians to the intermediate periodic trajectories of the van der Pol equation between
the small and the large orbits due to their special shapes [6]. However, in our work a
canard is a one-dimensional slow invariant manifold of variable stability [1, 2].
As the simplest system with a canard we propose
x 1, y xy .
It is clear that the trajectory y 0 is a canard. The left part (x < 0) is stable and the
right part (x > 0) is unstable. These two parts are divided by a jump point, which sep-
arates stable and unstable parts of the slow curve, at x 0 .
The case of a pair of purely imaginary roots
Consider the slow-fast system
x X ( x, y, ), y Y ( x, y, ), (5)
for which a singular point of the equation of fast motions becomes unstable with tran-
sition of a pairs of eigenvalues through the imaginary axis when the slow variables
are changed [4, 17-19]. It should be noted that the system (5) can be obtained from (1)
by time scaling transformation.
For analytical systems the positive semi-trajectories in a certain region of phase space
tend to the curves of the degenerate system having a comparable length of motions
near the stable and unstable parts of the slow surface. This describes the trajectories
similar to canards, which are described above.
Slow surface of the system (5) is divided into stable and unstable regions. The first
one consists of a stable equilibrium of the fast subsystem and the other one consists of
the unstable steady states, their common border is called as the breakdown surface.
There is an open set of points on the stable part of the slow surface starting from
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which the phase curves of the slow system intersect the breakdown surface and, under
changing of x along the slow curve, the pair of eigenvalues of a singular point of the
equation of fast motions passes through the imaginary axis with a non-zero rate. The
phase point of the system (5) starting close to the stable part of the slow surface fast
(during the time of O(|ln ε|) as ε → 0) tends to the ε-neighborhood of the slow surface
and then moves along the slow trajectory. If the system (4) is analytic, then the further
movement has an interesting and unusual phenomenon - the delay. It consists in the
fact that the phase point continues to move along the unstable part of the slow surface
in the ε-neighborhood of the slow path, more time on the order O(ε−1) after crossing
by the slow path of the stability border. And this slow path of the trajectory along the
unstable part of the slow surface has a distance of order O(1). Only then can happen
jump, i.e., for a time of order O(|ln ε|) (slow variables are changed by a small amount
of the order O(ε|ln ε|) as ε → 0), avoiding the slow surface at a distance of the order of
O(1).
This situation observed is similar to the situation with canard. The canard also moves
at first along the stable part of the slow surface, and then moves along the unstable
part. But canards are found in two-dimensional system with an additional parameter
and they are quite rare: they exist for an exponentially small range of parameter val-
ues. And canards are the result of the transition of one real eigenvalue from the left
half-plane to the right one.
The change of stability on the Hindmarsh-Rose
The Hindmarsh-Rose model [20] describes the basic properties of individual neurons,
the generation of spikes and a constant level potential. In this model, Kirchhoff's law
is written for each ionic currents flowing through the cell membrane.
The dimensionless form of the Hindmarsh-Rose model is
x y ax3 bx 2 z I ,
2
y c dx y , (6)
z s( x ) z,
where х is a transmembrane neuron potential, y and z are the characteristics of ionic
currents dynamic, I is ambient current, other parameters reflect the physical features
of the neurons. The typical values of the positive parameter are small ( 1) [21].
It should be noted that the system (6) has the type of (4) with scalar slow variable and
two-dimensional fast variable.
Slow curve of system (7) is described by:
y ax 3 bx 2 z I 0,
(7)
c dx y 0.
2
The plot of the slow curve is presented on Fig. 1.
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Fig. 1. The slow curve (7) of the system (6); a=1, b=3, c=1, d=5, I=2.7
From (7) we get
ax3 (d b) x2 z I c 0,
which allow us to investigate a projection of the slow curve on the XOZ-plane, see
Fig.2.
Fig. 2. The projection of the slow curve on the XOZ-plane. The parameters’ values are the
same as for Fig. 1
The Jacobian matrix (3) for the system (6)
3ax 2 2bx 1
J
2dx 1
has a characteristic equation
2 (1 3ax2 2bx) 3ax2 2(d b) x 0. (8)
The necessary condition of stability (which is also the sufficient in this case) for the
polynomial (8) is
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Mathematical Modelling Shchepakina EA. Three scenarios…
1 3ax 2 2bx 0,
(9)
3ax 2d 2b x 0.
From (9) we can determine the abscissas of the points of an expected change of stabil-
ity of the slow curve:
2(b d ) b b 2 3a
x1 , x2 0, x3,4 .
3a 3a
These points divide the slow curve into several parts, see Fig. 3.
Fig. 3. The projection of the slow curve on the XOZ-plane and the points of changing of
stability
We check the sign of real part of eigenvalues of the matrix J for all parts to find out
whether the region is stable or unstable. One of the two real eigenvalues of the Jaco-
bian matrix of the fast system changes its sign at the points A1 and A2 , and at these
points the slow curve changes its stability. Thus, A1 and A2 are the jump points. The
proper choice of an additional parameter of the system allows us to glue the stable and
the unstable integral manifolds of the system that exist in the ε-neighborhood of the
stable and unstable regions of the slow curve. As a result of this gluing we obtain a
canard.
However, there are other points of the stability’s change (see B1 and B2 on Fig. 3),
which are not jump points, because the trajectory does not immediately escape the
slow integral manifold as soon as it reaches these points, compared to the previous
case. One can find them by equating the real parts of the complex conjugate eigenval-
ues of J
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Mathematical Modelling Shchepakina EA. Three scenarios…
(1 3ax 2 2bx) (1 3ax 2 2bx) 2 4(3ax 2 2dx 2bx)
(10)
2
to zero. The phenomenon of the delay of loss of stability occurs as the trajectory goes
through these points.
There are points C1 (between points A2 and B1 ) and C 2 (to the right from the point
B2 ) on the slow curve, at which the pair of real eigenvalues of the Jacobian matrix J
becomes the pair of the complex conjugate eigenvalues, see Fig. 3.
From (10) it is possible to find the relations between parameters values
3a
d
, (11)
2 b b 2 3a
under which the points A2 , B1 and C1 coincide. In that case the real and the imagi-
nary parts of the pair of complex conjugate eigenvalues become equal to zero simul-
taneously, and a multiple zero root arises with the following emergence of the pair of
real eigenvalues with the opposite sign.
Under condition (11) and the proper choice of the values of the parameters it is possi-
ble to construct the slow integral manifold with multiple change of stability: we need
one parameter, say s, to glue the stable and unstable slow integral manifolds at the
point A1 using the techniques of canards (see Figs. 4 and 5), and we need two more
parameters, say a and I, for gluing integral manifolds at the point A2 with help the
method described in [22, 23].
Fig. 4. The projections of the canard without head (blue line) and the slow curve (red line)
of the system (6) under the condition (11) and a=1, b=3, c=1, α=-1.2, I= 2.7, ε= 0.01,
s=3.0810445478558141214
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Fig. 5. The projections of the canard with head (blue line) and the slow curve (red line) of
the system (6) under the condition (11) and a=1, b=3, c=1, α=-1.2, I= 2.7, ε= 0.01,
s=3.0810445478558141213
As result of these gluing procedures we get the slow integral manifold with multiple
change of stability (see Fig. 6) that looks like a canard cascade [24]. The difference
between these two objects consists in that for a canard cascade we apply the canard
technique only.
Fig. 6. The projections of the integral manifold with multiple change of stability (blue line)
and the slow curve (red line) of the system (6)
Conclusion
In this paper we investigated the different scenarios of the changing of stability of
slow integral manifold via the Hindmarsh-Rose model. We have shown that in this
singularly perturbed system the equilibrium of the fast subsystem loses its stability
with the passage of one or a pair of complex conjugate eigenvalues through the imag-
inary axis when the slow variables are changed.
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Moreover, it was shown that there is the scenario of change of stability when the real
parts as well as the imaginary parts of a pair of complex conjugate eigenvalues be-
come zero followed by the appearance of multiple zero root and then by the birth of a
pair of real eigenvalues of opposite signs. The crucial result of present investigation is
that it is possible to construct the slow integral manifold with multiple change of its
stability for some values of the additional parameters of the singularly perturbed sys-
tem.
Acknowledgment
This work is supported in part by the Russian Foundation for Basic Research (grant
14-01-97018-p) and the Ministry of Education and Science of the Russian Federation
under the Competitiveness Enhancement Program of Samara University (2013–2020).
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