CHAOS 19, 023124 共2009兲
On the dynamics of chaotic spiking-bursting transition
in the Hindmarsh–Rose neuron
G. Innocenti and R. Genesio
Dipartimento di Sistemi e Informatica, Università di Firenze, via S. Marta 3, 50139 Firenze, Italy
and Centro per lo Studio di Dinamiche Complesse (CSDC), Università di Firenze, Italy
共Received 20 March 2009; accepted 1 June 2009; published online 26 June 2009兲
The paper considers the neuron model of Hindmarsh–Rose and studies in detail the system dynamics which controls the transition between the spiking and bursting regimes. In particular, such a
passage occurs in a chaotic region and different explanations have been given in the literature to
represent the process, generally based on a slow-fast decomposition of the neuron model. This
paper proposes a novel view of the chaotic spiking-bursting transition exploiting the whole system
dynamics and putting in evidence the essential role played in the phenomenon by the manifolds of
the equilibrium point. An analytical approximation is developed for the related crucial elements and
a subsequent numerical analysis signifies the properness of the suggested conjecture. © 2009
American Institute of Physics. 关DOI: 10.1063/1.3156650兴
Many biological neuron models have been developed in
the last decades for an accurate description and prediction of biological phenomena. From the basic contribution of Hodgkin and Huxley, the simplified model of
Hindmarsh and Rose has turned out to be quite accurate
in capturing the features of electrical data from experiments. This model is a three-dimensional oscillator, particularly aimed to study the spiking-bursting behavior of
the neuron membrane potential. Indeed, these operating
modes and their relationships are key questions in understanding the different mechanisms which regulate the
neural coding of information in many biological processes. The Hindmarsh–Rose model exhibits the spikingbursting passage in a chaotic region and such transition
has been explained in the past by applying a slow-fast
decomposition of the related equations. This paper formulates a novel conjecture to detect the phenomenon.
From the complete model, by considering the equilibrium
point and its manifolds, a kind of trajectory “separatrix”
is analytically derived in the state space for the occurrence of the transition. The method is developed for a
general class of polynomial systems, while a numerical
analysis, which appears to fully confirm the conjecture, is
performed for the specific case in study.
I. INTRODUCTION
The study of the neural activity and its response mechanism to external stimuli has produced in the literature a variety of neuronal models. In this respect it is to underline the
importance of the early work of Hodgkin and Huxley,1
whose model surely represents a milestone. However, since
such a model turns out to be quite complex, several authors
have attempted to provide simpler approximations.2–6 These
models are designed to guarantee a qualitative description of
some neuronal phenomena in order to derive useful insights
about them. Therefore, it is natural that the first ones were
low-dimensional models, namely, second order systems such
as the FitzHugh–Nagumo and Morris–Lecar neurons.
1054-1500/2009/19共2兲/023124/8/$25.00
In this work we refer to the Hindmarsh–Rose neuronal
model. Its first version is dated in 1982 共Ref. 7兲 and again it
is a second order form. Despite the genesis of several simplified models which corresponds to an approximation or
reduction of the Hodgkin–Huxley equations, the Hindmarsh–
Rose system is the result of biological experiments performed on the pond snails.8 Remarkably, this model has
some important similarities with other neurons such as the
S-shaped nullclines and the bistability property 共see, e.g.,
Refs. 8–10 for detailed analyses兲. However, the second order
model is not able to reproduce some interesting phenomena
such as terminating by itself a triggered state of firing.
Hence, to this aim the authors added a third dynamical component, whose role is to tune the above subsystem over the
mono- and bistability regions in order to activate or terminate the neuronal response.9 Since such an additional term is
designed to act on a longer time scale, the model results to be
naturally divided into a fast and a slow subsystem. Notably,
the recent paper Ref. 11 shows that the Hindmarsh–Rose
neuron can capture both qualitative and quantitative aspects
of experimental data.
Due to the particular structure of the model, several phenomena can be accurately described just by studying the fast
subsystem, eventually taking care of some interactions with
the slow component. However, this approach is not able to
explain the complete variety of the behaviors exhibited by
the model. To this aim the previous work Ref. 10 made an
effort to provide a comprehensive description of the neuron
phenomena in terms of its fully coupled equations, describing the generation of the different regimes in terms of bifurcation theory.12
In this paper we present the chaotic spiking-bursting
transition as a specific expression of the whole third order
dynamics. Section II introduces the mathematical model and
its main properties. In Sec. III the most important features
concerning the geometry of the bursting and spiking attractors are analyzed in details. Section IV introduces a dynami-
19, 023124-1
© 2009 American Institute of Physics
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023124-2
Chaos 19, 023124 共2009兲
G. Innocenti and R. Genesio
I = 3.29
3.5
3.5
3.4
3.4
3.3
3.3
3.2
3.2
z
z
I = 3.30
3.1
2
3.1
3
2
3
0
2.9
0
-2
0
-4
(a)
0
2.9
-6
y
-8
-2
x
-2
-4
(b)
-6
y
-2
-8
x
FIG. 1. 共Color online兲 共a兲 A three-dimensional view of the chaotic spiking attractor for I = 3.30 and its position with respect to the equilibrium point. 共b兲 The
chaotic bursting attractor and the equilibrium point for I = 3.29, i.e., after the spiking-to-bursting chaotic transition described in Sec. II. The comparison
between cases 共a兲 and 共b兲 shows the sharp modification of the attractor size along the z direction. Besides, the equilibrium point position has a negligible
variation.
cal explanation of the transition. Finally, Sec. V is devoted to
compute a suitable analytical approximation of the weakly
unstable manifold of the equilibrium and the following numerical analysis quantifies its essential influence on the phenomenon. The conclusions of Sec. VI end the paper.
II. THE MODEL ANALYSIS
The third order Hindmarsh–Rose model equations are
ẋ = y − z + I + bx2 − ax3 ,
共1兲
ż = rsx − rz − rsx0 ,
where x is the membrane potential, I is the external dc current, and y and z are the recovery and the adaptation current,
respectively. The constant parameters a, b, c, and d define
the underlying original second order system and the value x0
as well. In particular, they are set to the common values
a = 1,
b = 3,
c = 1,
d = 5,
x0 = − 1.6.
3.4
3.35
共2兲
The variable z is the additional component and the coefficients r and s have the critical role of tuning the neuron
response. For a proper choice of r and s the model is able to
display both regular and chaotic bursting and firing as the
current I varies.9 It is worth noticing that r is usually set to a
very small value, so dividing the model into a fast subsystem
共i.e., the second order model described by the first two equations兲 and a slow subsystem 共the third equation兲.
The Hindmarsh–Rose neuron exhibits a chaotic spikingbursting transition over a wide range of its parameters r and
s. In particular, this phenomenon has already been studied in
the literature for r 苸 共0.001, 0.006兲 and s = 4. Besides, the
model has been extensively studied by varying I in Refs. 13
and 14, while two parameter analyses have been recently
reported in Refs. 15 and 16, denoting the present interest for
the subject.
3.3
3.25
3.2
Min(z)
ẏ = − y + c − dx2,
Hereafter, we will assume r = 0.0021 and s = 4 in accordance to Ref. 10. For such values a chaos-chaos transition
happens when I varies in the very narrow interval 共3.29,
3.30兲, generally assumed as the passage between bursting
and spiking 共chaotic兲 regimes. This transition boils down
into a sharp modification of the chaotic attractor size, as
illustrated in Fig. 1. Since the main variation in the attractor
size is observed along the z component, this can be properly
illustrated by reporting the minimum of z versus the current
I as in Fig. 2. To this regard, Ref. 13 considers that such a
geometric modification may be related to an interior crisis
and due to the continuity with respect to the external current
I, this is called a “continuous internal crisis.” In Ref. 14 an
extensive and quantitative analysis of the transition is provided according to the fast and slow subsystem division approaches. The phenomenon is strongly related to the pres-
3.15
3.1
3.05
3
2.95
3.28
3.285
3.29
3.295
I
3.3
3.305
3.31
FIG. 2. 共Color online兲 The variation in the attractor minimum value along
the z direction as the current I varies in a narrow range around the transition
interval. The related attractor modification appears to be continuous in
nature.
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Chaos 19, 023124 共2009兲
A transition in the Hindmarsh–Rose model
ence in the fast subsystem of a homoclinic bifurcation and
what happens in terms of the whole system dynamics is not
investigated. A similar observation has already been expressed in Ref. 17. Moreover, this author introduces a different perspective as well. He suggests that the stable and unstable manifolds of the 共unique兲 equilibrium point of the
system may play an important role in determining the dynamics. Besides, the shape of the chaotic attractor may depend on the envelopment provided by such manifolds. According to this perspective, on one side of the transition
interval the attractor is forced to stay in a certain region of
the state space, while on the other side it is allowed to reach
a further region, so modifying its shape.
3.45
3.4
3.35
3.3
3.25
z
023124-3
3.2
3.15
3.1
3.05
3
2.95
III. THE TRANSITION BEHAVIOR
In this section we provide a detailed description of the
neuron model 共1兲 by illustrating the different shapes of the
state space trajectories on the two sides of the chaotic
spiking-bursting transition. Then, we will focus on the critical region where they differ to propose a dynamical explanation of the process.
Let us consider the system behavior for the decreasing
values of the external current I in the range 共3.30, 3.29兲.
Figure 1共a兲 illustrates the chaotic spiking attractor at I
= 3.30. The trajectory turns along a spiral, where each coil
nearly lies on a plane parallel to the x-y subspace and moves
having z as dominant direction. When the trajectory reaches
the top, a reinjection happens and the system state is rapidly
brought back to the bottom. During this motion the z dynamics is leading. Therefore, the chaotic spiking attractor results
in a spiral component ruled by the fast subsystem dynamics
and a reinjection path where the slow subsystem prevails.
In Fig. 1共b兲 the state space representation of the chaotic
bursting attractor is depicted for I = 3.29. As in the previous
case a spiral shape is recognized. The coils are almost parallel to the x-y plane and their motion nearly follows the z
coordinate. The reinjection process can still be highlighted
once the trajectory reaches the top of the spiral: while in the
spiking case the paths are always similar, in the bursting case
a variety of different routes are observed. In particular, they
differ in the length, the dominant direction, and the z heights
to which the bring back the trajectory.
Summing up, when the state reaches the top of the spiral,
it may undergo a reinjection process ranging between two
main dynamics. In the first case a shorter path analogous to
the spiking one is covered. In the second scenario the state
takes a longer path. At the beginning, it assumes a different
direction, except for a later turn, which gets the trajectory
into the spiral at a lower z height. In Fig. 3 a close look at the
chaotic bursting attractor reveals that the critical region is at
the spiral’s top, where the reinjection process is about to
start. In particular, the way that the trajectories follow to
approach such region affects how they are brought back to
the bottom. A comparison between the paths highlighted in
the figure reveals the different contributions to the attractor’s
shape as well.
These observations and in particular the dominance of
the z dynamics along the reinjection process underline that
1
0
-1
-2
-3
-4
y
-5
-6
-7
-8
-9
FIG. 3. 共Color兲 Two-dimensional projection of the bursting chaotic attractor
onto the y-z plane at I = 3.29. Two different routes are highlighted to illustrate the possible behaviors at the spiral top. Red curve: the trajectory follows a short reinjection path reaching a z height of about 3.34. Green curve:
after an initial decrease in the y component the trajectory gets the bottom of
the spiral at about z ⬇ 3.
the fast/slow subsystem technique is not suitable to explain
the nature of the transition.
For the following developments we briefly recall the nature of the unique equilibrium point of Eqs. 共1兲 and 共2兲 during the transition 共see Refs. 9 and 10 for further details兲.
During the process such fixed point almost remains constant
as well as its stability properties. Also the eigenvalues 共two
unstable and one stable兲 and the related eigenvectors undergo
negligible variations and they can be considered as invariants
along this passage. Although the equilibrium lies distant
from the attractor, the direction of its eigenvectors indicates
that it may play a fundamental role. Indeed, roughly speaking, on the side where the reinjection happens the spiral attractor is “squeezed” against the completely unstable eigenspace of the equilibrium 共i.e., the plane tangent to its
completely unstable manifold兲. Moreover, the trajectory approaches the starting region of the reinjection by a direction
parallel to the stable eigenspace and the weakly unstable
eigenspace turns out to pass close to that region too.
In Ref. 17 the author suggests that the shape of the attractor is due to the passage of the trajectory in two different
regions of the state space. Furthermore, he chooses the plane
formed by the stable and the weakly unstable eigenspaces of
the equilibrium point to describe the related separatrix. This
surface is just the linear approximation of the stable/weakly
unstable manifold and it plays a passive role in the explanation of the phenomenon. Indeed, according to the author’s
idea the dynamics may be completely attributed to only the
fast subsystem. The Poincaré map built in Ref. 17 and derived for r = 0.0021 in Fig. 4共a兲 provides just a simplified tool
to represent the candidate separatrix, which turns out to detect the transition fairly well. However, this happens because
such a map is able to catch the attractor’s size variation along
z, as shown in Fig. 4共b兲. Indeed, moving from the spiking
regime the transition is described in terms of the map’s intersection with the stable/weakly unstable line. However,
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023124-4
Chaos 19, 023124 共2009兲
G. Innocenti and R. Genesio
I = 3.295
I = 3.295
3.5
3.45
3.35
3.5
3.3
3.4
3.25
3.3
z
z
3.4
3.2
3.2
3.1
3.15
3.1
(a)
0
2.9
3.05
3
-5.5
2
3
0
-5
-4.5
-4
y
-3.5
-3
-2.5
(b)
-2
-4
-6
y
-8
-2
x
FIG. 4. 共Color online兲 共a兲 The Poincaré map obtained by the procedure described in Ref. 17 when the system is close to the transition. The related plane is
parallel to the y-z one at a distance of 10−5 from the equilibrium point. The straight line in the map is the intersection with the plane provided by the stable
and weakly unstable eigenvectors. 共b兲 The three-dimensional representation of the attractor and the Poincaré map illustrates their respective position in the
state space.
since the trajectories are not allowed to pass through a dynamical manifold, the situation depicted in Fig. 4共a兲 is not
related to any real crossing phenomenon. Nevertheless, observing the corresponding positions in Fig. 4共b兲, it is straightforward to derive that such an event corresponds to a change
in the attractor’s shape and that it occurs when the spiral’s
bottom reaches a sufficiently low z. Since the transition is
detected by the position of the attractor’s lowest part, such a
Poincaré map appears not suitable to explain the behavior in
the critical region at the top of the spiral, where really the
spiking and bursting regimes are set up.
On the base of these observations, in Sec. IV we will
introduce a novel conjecture for such a passage.
curves on ⌽. Then, assume that the system trajectory approaches ⌽ approximately according to the direction of S.
The three different situations illustrated in Fig. 5 are considered.
共a兲
共b兲
IV. THE CONJECTURE
From the analysis of Sec. III, the phenomenon can be
treated according to the following conjecture. The reinjection
process concerning the chaotic spiking-bursting transition
occurs near the completely unstable manifold of the equilibrium point. In particular, the critical region at the top of the
spiral is close to the weakly unstable component. Hence, in
the first stage the reinjection is driven by such dynamics,
while in the second stage the strongly unstable dynamics
prevails. In the spiking case the trajectory approaches the
completely unstable surface on the same side of the weakly
unstable curve, while in the bursting case it may fall on both
sides. Then, in the latter scenario the trajectory may diverge
according to both directions of the strongly unstable component, resulting in two different reinjection paths.
In the following we address the considered transition
more precisely by means of the scheme illustrated in Fig. 5.
Let E be the equilibrium point and S, W, and U be its stable,
weakly unstable, and strongly unstable manifolds, respectively. We denote by U1 and U2 the two branches of the
curve U from E and by ⌽ the surface corresponding to the
completely unstable manifold. Observe that U and W are
共c兲
As the system trajectory approaches ⌽ the influence of
the unstable dynamics of the equilibrium point becomes dominant. When the approaching region is sufficiently distant from W, the strongly unstable component prevails and the motion diverges along the branch
U 1.
The system trajectory tends to ⌽ very near to W.
Hence, at the beginning the weakly unstable dynamics
prevails and the motion diverges according to that
curve. Then, the strongly unstable component necessarily dominates and the trajectory finally follows its dynamics, namely, along U1.
The system state approaches ⌽ close to the weakly
unstable manifold, here on the opposite side of W with
respect to the case b. The trajectory follows initially the
curve W but, as the strongly unstable dynamics becomes leading, it diverges along the opposite branch of
U, namely, U2.
Therefore, according to this scenario, the chaotic spiking
is characterized at the spiral’s top by trajectories as that of
case 共b兲. Conversely, in the chaotic bursting the motion may
exhibit in that critical region both the behaviors 共b兲 and 共c兲.
In particular, case 共b兲 is related to the shortest reinjection
paths, while 共c兲 is responsible for the longest ones.
Section V will be devoted to a rigorous analytical approach of these system dynamics in order to validate the
above conjecture. Since the basic elements of such a phenomenon have been singled out in the scheme of Fig. 5, we
will develop our approach for a more general class of systems, then deriving specific results for the Hindmarsh–Rose
model.
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023124-5
Chaos 19, 023124 共2009兲
A transition in the Hindmarsh–Rose model
a
b
c
stable manifold S
strongly unstable manifold
(branch U1)
stable eigenvector Vs
completely unstable
manifold Φ
strongly unstable manifold
(branch U2)
equilibrium point E
strongly unstable
eigenvector Vu
weakly unstable
eigenvector Vw
weakly unstable manifold W
FIG. 5. 共Color兲 The scheme addressed in Sec. IV to explain the dynamical mechanism which separates the spike trajectories from the burst ones. All the basic
elements playing an active role in defining the phenomenon have been represented. Notice that the eigenvectors of the equilibrium point are tangent to the
related manifolds.
V. THE MANIFOLD GEOMETRY AND THE TRANSITION
At first, we derive an approximation of the weakly unstable manifold W to verify its presence in the critical region
of the reinjection process. To this aim, we develop hereafter
an analytical procedure designed for a more general class of
dynamical systems, just presenting aside specific results for
the Hindmarsh–Rose model.
Let us consider the class
+⬁
␰˙ = ⌳␰ + 兺 ␥ng关n兴共␰兲,
共3兲
n=2
where ␰ 苸 RN, ␥n 苸 RN, g关n兴共␰兲 : RN → R is a homogeneous
polynomial of the nth order in the components of ␰, and ⌳
苸 RN⫻N is a diagonal matrix such that all its eigenvalues are
negative but two. For the sake of simplicity we assume N
= 3, so that
冤冥 冤
␰s
␰ = ␰u ,
␰w
␭s
0
0
⌳ = 0 ␭u 0
0 0 ␭w
冥
with ␭s ⬍ 0 ⬍ ␭w ⬍ ␭u.
Let us prove that the Hindmarsh–Rose system admits
model 共3兲. To this aim first observe that it can be compactly
written as
冤冥 冤
x
␨⬟ y ,
z
0
1
−1
冥
A⬟ 0 −1 0 ,
rs 0 − r
f共␨兲 ⬟
冤
I + bx2 − ax3
− dx2
− rsx0
冥
.
Moreover, let the equilibrium at a certain value of the external current I be denoted as
冤 冥
1共I兲
E共I兲 ⬟ 2共I兲 .
3共I兲
共4兲
For the sake of simplicity, we will not indicate further this
dependence on I. Then, let ␭s, ␭u, and ␭w, respectively, be the
stable, the strongly unstable, and the weakly unstable eigenvalues of E at the considered value. In the transition interval
they have the real values ␭u ⬇ 0.1925, ␭w ⬇ 0.004 366, and
␭s ⬇ −0.645. Moreover, let Vs , Vu , Vw 苸 R3 be the three column eigenvectors related to ␭s, ␭u, and ␭w, respectively. Finally, define the modal matrix
冤
v11 v12 v13
T ⬟ 关Vs,Vu,Vw兴 = v21 v22 v23
v31 v32 v33
冥
共5兲
and let us denote its inverse by
冤
冥
␯11 ␯12 ␯13
T = ␯21 ␯22 ␯23 .
␯31 ␯32 ␯33
−1
共6兲
Then, we perform the linear affine change in coordinates
␨˙ = A␨ + f共␨兲
just by defining
␰ = T−1共␨ − E兲,
obtaining
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023124-6
Chaos 19, 023124 共2009兲
G. Innocenti and R. Genesio
␰˙ = T−1AT␰ + T−1AE + T−1 f共T␰ + E兲.
共7兲
After tedious though straightforward computations Eq. 共7兲
boils down to the equation system
␰˙ s = ␭s␰s + 共␯11共b − 3a1兲 − d␯12兲共v11␰s + v12␰u + v13␰w兲2
␰˙ u = ␭u␰u + 共␯21共b − 3a1兲 − d␯22兲共v11␰s + v12␰u + v13␰w兲2
− a␯21共v11␰s + v12␰u + v13␰w兲3 ,
共8兲
␰˙ w = ␭w␰w + 共␯31共b − 3a1兲 − d␯32兲共v11␰s + v12␰u + v13␰w兲2
− a␯31共v11␰s + v12␰u + v13␰w兲3
in the new state variables ␰s, ␰u, and ␰w. Observe that Eq. 共8兲
admits the representation 共3兲 just by defining the functions
g关2兴共␰s, ␰u, ␰w兲 ⬟ 共v11␰s + v12␰u + v13␰w兲2 ,
g 共␰s, ␰u, ␰w兲 ⬟ 共v11␰s + v12␰u + v13␰w兲 ,
3
g关n兴共␰s, ␰u, ␰w兲 ⬟ 0,
共9兲
冤冥 冤冥 冤冥
冤冥 冤冥
␥11
␯11
␯12
␥2 ⬟ ␥12 = 共b − 3a1兲 ␯21 − d ␯22 ,
␥13
␯31
␯32
共10兲
W = 兵共␰s, ␰u, ␰w兲 苸 R3:␰s = hs共␰w兲, ␰u = hu共␰w兲其,
⳵ hs
⳵ hu
共0兲 =
共0兲 = 0,
⳵␰w
⳵␰w
n=2
depend on the nature of the functions g关n兴 and on the parameters 兵␣n其 and 兵␤n其 as well.
only
depends
on
Remark:
Each
kn
共␣n−1 , ␤n−1 , . . . , ␣2 , ␤2兲.
For the Hindmarsh–Rose system Eq. 共14兲 reduces to
␥2g关2兴共hs共␰w兲,hu共␰w兲, ␰w兲 + ␥3g关3兴共hs共␰w兲,hu共␰w兲, ␰w兲
= 兺 kn␰wn ,
共15兲
n=2
⬁
␰˙ s共␰w兲 = 兺 共␣n␭s + k1n兲␰wn ,
共11兲
共12兲
␰˙ u共␰w兲 = 兺 共␤n␭u + k2n兲␰wn ,
共13兲
n=2
Deriving a suitable approximation of W turns into the probM
M
lem of computing the sets 兵␣n其n=2
and 兵␤n其n=2
for a sufficiently high number M − 1 of elements, with a related error
decreasing as M increases. To this aim, consider the nonlinear part of system 共3兲 and evaluate it into the points of the
共16兲
n=2
⬁
␰˙ w共␰w兲 = 兺 k3n␰wn ,
n=1
where k31 ⬟ ␭w.
Since the manifold W is such that
⳵ hs ˙
␰˙ s =
␰ w,
⳵␰w
⳵ hu ˙
␰˙ u =
␰w ,
⳵␰w
共17兲
by computing Eq. 共17兲 with Eq. 共13兲 and using the third
equation in Eq. 共16兲, we also obtain that
冉
兺冉
⬁
n
n=2
m=2
⬁
n
冊
冊
␰˙ s共␰w兲 = 兺 ␰wn 兺 共m␣mk3共n−m+1兲兲 ,
⬁
hu共␰w兲 = 兺 ␤n␰wn .
n⬎1
⬁
and they are the goal of our research. Let us represent hs and
hu by power series developments
h s共 ␰ w兲 = 兺
冤冥
k1n
kn = k2n ,
k3n
n=2
where the functions hs and hu must satisfy the tangency conditions
␣n␰wn ,
共14兲
n=2
␥n ⬟ 0, ∀ n ⬎ 3.
To develop a suitable approximation of the weakly unstable manifold, let us recall that W is a curve passing
through E with direction parallel to the eigenvectors of ␭w.
Now, system 共3兲 has the origin as equilibrium point and the
weakly unstable eigenvectors are directed along the axis ␰w.
Therefore, W can locally be parametrized by the coordinate
␰w as follows:
⬁
共hs共␰w兲,hu共␰w兲, ␰w兲 = 兺 kn␰wn ,
where g关2兴, g关3兴 and ␥2, ␥3 are defined as in Eqs. 共9兲 and 共10兲,
respectively. The expressions of the corresponding coeffi8
, i = 1 , 2 , 3, are listed in Table I.
cients 兵kin其n=2
Substituting the power development 共13兲 in the system
equation 共3兲 and exploiting expression 共14兲 for the nonlinear
term, we obtain
and the vectors
hs共0兲 = hu共0兲 = 0,
兺 ␥ ng
n=2
关n兴
⬁
∀ n ⬎ 3,
␥21
␯11
␥3 ⬟ ␥22 = − a ␯21 ,
␥23
␯31
⬁
+⬁
where the coefficients
− a␯11共v11␰s + v12␰u + v13␰w兲3 ,
关3兴
manifold 共11兲. Exploiting the representation 共13兲 for hs and
hu, we obtain a ␰w power series,
␰˙ u共␰w兲 =
n=2
共18兲
␰wn 兺 共m␤mk3共n−m+1兲兲 .
m=2
Then, we equate Eq. 共18兲 to the first two equations of Eq.
共16兲 and we balance the ␰w powers. Exploiting the above
remark, the coefficients 共␣n , ␤n兲, which define W in Eqs.
共11兲–共13兲, are finally obtained as
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023124-7
Chaos 19, 023124 共2009兲
A transition in the Hindmarsh–Rose model
TABLE I. The table reports the first coefficients implicitly defined by Eq. 共15兲.
i = 1 , 2 , 3:
ki2 = v213␥i2
ki3 = 2v13共␣2v11 + ␤2v12兲␥i2 + v313␥i3
ki4 = ␥i2共2v13共␣3v11 + ␤3v12兲 + 共␣2v11 + ␤2v12兲2兲 + 3v213共␣2v11 + ␤2v12兲␥i3
ki5 = ␥i2共2v13共␣4v11 + ␤4v12兲 + 2共␣2v11 + ␤2v12兲共␣3v11 + ␤3v12兲兲 + ␥i3共3v213共␣3v11 + ␤3v12兲 + 3v13共␣2v11 + ␤2v12兲2兲
ki6 = ␥i2共2v13共␣5v11 + ␤5v12兲 + 2共␣2v11 + ␤2v12兲共␣4v11 + ␤4v12兲 + 共␣3v11 + ␤3v12兲2兲
+␥i3共3v213共␣4v11 + ␤4v12兲 + 3v13共␣2v11 + ␤2v12兲共␣3v11 + ␤3v12兲 + 共␣2v11 + ␤2v12兲3兲
ki7 = ␥i2共2v13共␣6v11 + ␤6v12兲 + 2共␣2v11 + ␤2v12兲共␣5v11 + ␤5v12兲 + 2共␣3v11 + ␤3v12兲共␣4v11 + ␤4v12兲兲
+␥i3共3v213共␣5v11 + ␤5v12兲 + 3v13共␣2v11 + ␤2v12兲共␣4v11 + ␤4v12兲 + 3v13共␣3v11 + ␤3v12兲2 + 3共␣2v11 + ␤2v12兲2共␣3v11 + ␤3v12兲兲
ki8 = ␥i2共2v13共␣7v11 + ␤7v12兲 + 2共␣2v11 + ␤2v12兲共␣6v11 + ␤6v12兲 + 2共␣3v11 + ␤3v12兲共␣5v11 + ␤5v12兲 + 共␣4v11 + ␤4v12兲2兲
+␥i3共3v213共␣6v11 + ␤6v12兲 + 3v13共␣2v11 + ␤2v12兲共␣5v11 + ␤5v12兲 + 3v13共␣3v11 + ␤3v12兲共␣4v11 + ␤4v12兲 + 3共␣2v11 + ␤2v12兲2共␣4v11 + ␤4v12兲兲
共+3共␣2v11 + ␤2v12兲共␣3v11 + ␤3v12兲2兲
1
␣n =
␭s − n␭w
␤n =
1
␭u − n␭w
冉兺
冉兺
冊
冊
n−1
共m␣mk3共n−m+1兲兲 − k1n ,
m=2
n−1
共19兲
共m␤mk3共n−m+1兲兲 − k2n .
m=2
Figure 6 illustrates the approximation of W for the
Hindmarsh–Rose system obtained by using the coefficients
␣n and ␤n listed in Table I,
Ŵ =
再
8
8
冎
共␰s, ␰u, ␰w兲 苸 R3:␰s = 兺 ␣n␰wn, ␰u = 兺 ␤n␰wn .
n=2
n=2
d ⬟ min 储␩ − ␰储2 .
共20兲
In particular, Fig. 6共a兲 provides a comparison between Eq.
共20兲 and the spiking chaotic attractor at I = 3.30. The figure
shows that Ŵ is extremely close to the trajectories passing in
the critical region. Moreover, it highlights that the reinjection
paths are always placed on the same side with respect to Ŵ.
(a)
Figure 6共b兲 depicts the weakly unstable manifold approximation in comparison with the bursting chaotic attractor at I
= 3.29. The manifold Ŵ appears fully embedded in the trajectories of the critical region, but now they may pass on
both its sides. In particular, on one side the reinjection paths
follow a trail analogous to the spiking regime ones, while on
the other, they follow a different and longer route that
reaches lower z values.
In conclusion, let us define d as the minimum distance
between Ŵ and the system attractor ⌫, i.e.,
共21兲
␩苸Ŵ
␰苸⌫
The analysis of d provides useful insights about the influence
of the weakly unstable manifold of the equilibrium point
onto the neuron regime. To this aim, the diagram of d as the
external current I varies is reported in Fig. 7. Notably, the
distance almost vanishes in the chaotic bursting interval.
(b)
FIG. 6. 共Color兲 a兲 A three-dimensional view of the weakly unstable manifold approximation 共red line兲 compared to the chaotic spiking attractor at I = 3.30. 共b兲
A detail of the critical region at the spiral’s top for I = 3.29. The curve Ŵ 共red line兲 is fully embedded in such a region as conjectured.
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023124-8
10
Chaos 19, 023124 共2009兲
G. Innocenti and R. Genesio
-4
x 10
8
6
d
4
2
0
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
I
FIG. 7. 共Color online兲 The diagram of the distance d between the weakly
unstable manifold W estimated by using Eq. 共20兲 and the system attractor as
a function of the current I.
system approach appears not suitable to comprehend the dynamical mechanism of the transition. Therefore, a detailed
study of the trajectory evolution in the critical passage has
been carried out, leading to formulate a conjecture about the
influence of the equilibrium point onto the system regimes.
Then, classical mathematical tools of nonlinear dynamics
have been exploited to approximate the weakly unstable
manifold of the fixed point as the significant element which
governs the transition. Such an approach has been developed
for a more general class of polynomial systems, deriving
specific results for the Hindmarsh–Rose model. A numerical
analysis has been performed to provide quantitative results
as well. The accordance between such results and the conjectured behavior is remarkable and suggests the correctness
of the dynamical explanation for the chaotic spiking-bursting
transition. Moreover, the analysis points out that the considered phenomenon could be detected in other dynamical models, where sudden but continuous changes in the attractor
shape occur.
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1
2
Moreover, it has a minimum in the chaotic spiking-bursting
transition window 共3.29, 3.30兲, denoting the influence of the
weakly unstable manifold on the system dynamics, in accordance with the conjecture of Sec. IV. By decreasing I from
higher values the model exhibits the chaotic spiking regime
with trajectories passing on the same side of Ŵ and shows a
continuous distance reduction. Then, in the neighborhood of
the transition d vanishes and the orbits tend to overcome Ŵ,
passing by both its sides during the chaotic bursting regime.
When this behavior disappears, d is again growing and several local minima are experienced for lower values of I corresponding to other changes in the system dynamics, whose
study is beyond the scope of the paper.
VI. CONCLUSIONS
The paper has introduced a novel explanation about the
chaotic spiking-bursting transition in the Hindmarsh–Rose
neuron model. A qualitative analysis of the system behavior
in the state space has highlighted that the fast/slow sub-
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