an alternative bifurcation analysis of the rose–hindmarsh model

Chaos, Solitons and Fractals 23 (2005) 1643–1649

www.elsevier.com/locate/chaos

An alternative bifurcation analysis of theRose–Hindmarsh model

Svetoslav Nikolov *

Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria

Accepted 22 June 2004

Abstract

The paper presents an alternative study of the bifurcation behavior of the Rose–Hindmarsh model using Lyapunov–

Andronov�s theory. This is done on the basis of the obtained analytical formula expressing the first Lyapunov�s value(this is not Lyapunov exponent) at the boundary of stability. From the obtained results the following new conclusions

are made: Transition to chaos and the occurrence of chaotic oscillations in the Rose–Hindmarsh system take place

under hard stability loss.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

Many cell types exhibit more complex behavior, characterized by brief bursts of oscillatory activity interspersed with

quiescent periods during which the membrane potential changes only slowly. This behavior is called bursting [1,2].

Although bursting has been studied extensively for many years, an alternative approach is to construct a polynomial

model that retains the important qualitative features but is simpler to analyze and understand. This is accomplished in

the Rose–Hindmarsh model (RHM) for neuron cell activity [3], which has the form

0960-0

doi:10.

* Te

E-m

_x ¼ a1 þ y � zþ a2z � a3x3;

_y ¼ a4 � y � a5x2;

_z ¼ �a6a7a8 þ a6a8x� a6z;

ð1Þ

where x, y and z are dimensionless membrane potential, a recovery variable and the adaptation current, respectively.

Here, a1 to a8 are dimensionless constants. From mathematical point of view it is an example to constant forcing [4]

(and references there).

Bifurcation theory describes qualitative changes in phase portraits that occur as parameters are varied in the defi-

nition of a dynamical system [5]. The problem of investigating the evolution and bifurcation behavior of system (1)

(which is with proved suddenly qualitative changes in our chaotic behavior) has been addressed by the present authors

and others in [6–16]. According to [6,7,11], the system (1) can display a rich diversity of periodic and chaotic solutions

dependent upon the specific values of one or more bifurcation (control) parameters. Also, this system an example how

779/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

1016/j.chaos.2004.06.080

l.: +35 92 979 6428; fax: +35 92 870 7498.

ail address: [email protected]

mailto:[email protected]

1644 S. Nikolov / Chaos, Solitons and Fractals 23 (2005) 1643–1649

small variations in a map can result in sudden drastic changes in an attractor. These changes are called crises and can

include the sudden appearance or disappearance of the attractor or a discontinuous change in its size or shape [9,11,17].

It is easy to see that the equilibrium (steady state) points of the system are

�y ¼ a4 � a5�x2; �z ¼ a8ð�x� a7Þ;

�x3 þ a5 � a2a3

�x2 þ a8a3

�x� a1 þ a7a8a3

¼ 0:ð2Þ

Thus Eq. (2) presents ‘‘fixed points’’ in the phase space of the system (1). The divergence of the flow (1) is

D3 ¼o _xox

þ o _yoy

þ o_zoz

¼ �1� a6 þ xð2a2 � 3a3xÞ: ð3Þ

The system (1) is dissipative, when D3 < 0.

In the present paper, we continue the investigation (qualitatively and numerically) of the dynamics and bifurcation

behavior of RHM. According to [18], we calculate the so-called first Lyapunov value (this is not Lyapunov exponent––

see for a detailed discussion [19] or appendix in [20]) at the boundary of stability regions R = r = 0 of the system (1).

Following [18,19], we have: (i) the sign of Lyapunov�s value determines the character (stable or unstable) of equilibriumstate at R = 0; (ii) at the boundary of stability r = 0, two cases occur––(a) of the first Lyapunov�s value is different fromzero, then the equilibrium state is unstable double point, the system has irreversible behavior and the boundary r = 0 is

dangerous, (b) if the first Lyapunov�s value is zero, then the equilibrium state is stable; (iii) the character of equilibrium

state, at R = r = 0, qualitatively determines the reconstruction of phase space (including stability or instability of limit

cycle) at transition from R < 0 (r < 0) to R > 0 (r > 0) [18,19].

The plan of the paper is as follows: In Section 2 we make an analytical study of the first Lyapunov value for RHM.

In Section 3 we present the numerical results for different chose of parameters of the system (1). Finally, in Section 4, we

discuss and unify results from the previous sections.

2. Analytical study of the first Lyapunov value for Rose–Hindmarsh model

In this section, we investigate the system (1), which presents an autonomous 3D dynamical model. Here, we note

that all constants from a1 to a8 of this model are real and can be negative, zero or positive.

In order to determine the character of fixed points (Eq. (2)) we make the following substitutions into (1)

x ¼ �xþ x1; y ¼ �y þ x2; z ¼ �zþ x3: ð4Þ

Hence, after some transformations the system (1) has the form

_x1 ¼ c1x1 þ x2 � x3 þ c2x21 þ a3x31;

_x2 ¼ �c3x1 � x2 � a5x21;

_x3 ¼ a6a8x1 � a6x3;

ð5Þ

where

c1 ¼ 2a2�x� 3a3�x2; c2 ¼ a2 � 3a3�x; c3 ¼ 2a5�x: ð6Þ

According to [18], the Routh–Hurwitz condition for stability of (2) can be written in the form

p ¼ 1þ a6 � c1 > 0; ð7Þ

q ¼ c3 þ a6ð1þ a8Þ � ð1þ a6Þc1 > 0; ð8Þ

r ¼ a6ða8 þ c3 � c1Þ > 0; ð9Þ

R ¼ pq� r > 0: ð10Þ

Here the notations p, q, r and R are taken from [18]. When conditions (9) or (10) are not valid, the steady states (2)

become unstable. In order to define the type of stability loss of steady states (2) it is necessary to calculate the so-called

S. Nikolov / Chaos, Solitons and Fractals 23 (2005) 1643–1649 1645

first Lyapunov value [18,19,21]. In case of three first-order differential equations, this value can be determined analyt-

ically by the formula in [18]

L1ðk0Þ ¼p4q

2 Að2Þ33 A

ð3Þ33 � Að2Þ

22 Að3Þ22

� �þ 2Að2Þ

23 Að2Þ22 þ Að2Þ

33

� �� Að3Þ


33

� �þ 3

ffiffiffiq

pAð2Þ222 þ Að3Þ

333 þ Að2Þ233 þ Að2Þ

223

� �h i

þ p4p

ffiffiffiq

p ðp2 þ 4qÞ p2 2Að1Þ22 3Að2Þ

12 þ Að3Þ13

� �þ 2Að1Þ

33 Að2Þ12 þ 3Að3Þ

13

� �þ 4Að1Þ


12

� �h in

þ4p ffiffiffiq

pAð1Þ22 � Að1Þ

33

� �Að2Þ13 þ Að3Þ

12

� �þ 2Að1Þ

23 Að3Þ13 � Að2Þ

12

� �h iþ 16q Að1Þ

22 þ Að1Þ33

� �Að2Þ12 þ Að3Þ

13

� �o; ð11Þ

where k0 is defined as a value of a1 or a6. for which the relation R = 0 takes place. The coefficients Anij and An

ijk

(i, j,k,n = 1,2,3) are defined by corresponding formulas presented in [18]. After accomplishing some transformations

and algebraic operations for the first Lyapunov value L1(k0) (for the system (5)) we obtain

L1ðk0Þ ¼ S0½S1 þ S2ðS3 þ S4 þ S5Þ�; ð12Þ

where

S0 ¼p

2D20

ffiffiffiq

p ;

S1 ¼1ffiffiffiq

p fm2m3ða413 � a412Þ þ ða212 þ a213Þ½a12a13ðm22 � m2

3Þ � 1:5a3D0

ffiffiffiq

pm4�g;

S2 ¼a11m1

pðp2 þ 4qÞ ;

S3 ¼ p2½a212ð3a12m2 þ a13m3Þ þ a213ða12m2 þ 3a13m3Þ þ 2a12a13ða13m2 þ a12m3Þ�;S4 ¼ 2p

ffiffiffiq

p ½ða212 � a213Þða13m2 þ a12m3Þ þ 2a12a13ða13m3 � a12m2Þ�;S5 ¼ 8qða212 þ a213Þða12m2 þ a13m3Þ:

ð13Þ

Here we note that

a11 ¼ a6 � c1; a21 ¼ c3; a31 ¼ ðc1 þ pÞðp � 1Þ þ c3; a12 ¼ c1 � c3 þ a6ðc1 � a8Þ; a22 ¼ �a6c3;

a32 ¼ a6a8; a13 ¼ �ð1þ a6Þffiffiffiq

p; a33 ¼ �a6a8

ffiffiffiq

p; a23 ¼ c3

ffiffiffiq

p ð14Þ

and

D0 ¼ det

a11 a12 a13a21 a22 a23a31 a32 a33

��

��;

m1 ¼ a011c2 � a0

12a5; m2 ¼ a021c2 � a0

22a5;

m3 ¼ a031c2 � a0

32a5; m4 ¼ a021a12 þ a0

31a13:

ð15Þ

Consequently, (15) follows from

a011 ¼ a22a33 � a23a32; a0

12 ¼ a13a32 � a12a33; a021 ¼ a23a31 � a21a33;

a022 ¼ a11a33 � a13a31; a0

31 ¼ a21a32 � a22a31; a032 ¼ a12a31 � a11a32:

ð16Þ

The first Lyapunov value (in (12)) can be negative or positive. If L1(k0) is negative, then in case of transition throughthe boundary R = 0 from positive values to negative ones, a stable limit cycle (self-oscillations) emerges. Inversely, in

case of a transition from negative values to positive ones the stable limit cycle disappears, i.e. the self-oscillations cease

[18,20]. In theory of dynamic systems, the type of bifurcation behavior near the boundary R = 0 is often called soft loss

of stability, i.e. when the bifurcation parameter k0 changes, the system has reversible behavior. If L1(k0) is positive, thenin case of transition through the boundary R = 0 from positive values to negative ones, (i) an unstable and one stable

cycle emerge––hard self-excitation, or (ii) an unstable limit cycle emerges. Inversely, in case of transition from negative

values to positive ones the unstable limit cycle disappears. This type of bifurcation behavior near the boundary R = 0 is

often called hard loss of stability, i.e. the system has irreversible behavior and the boundary R = 0 is dangerous. In the

next section we demonstrate numerically these types of behavior of the model (5) which is topologically equivalent to

(1). Also, we obtain that in some intervals (different from these in [6,9,11]) of variation of the parameters a1 and a6, the

system (1) has interesting bifurcation chaotic behavior.


3. Bifurcation analysis––numerical study

In the previous section we introduced the analytical tools that will be used in our bifurcation analysis. Basically, all

we need for our purposes is the expression of the first Lyapunov value calculated on the boundary of stability R = 0 in

Eq. (12). We obtain our main results by examining plots of scale a12 [1,5.3345], a62 [0.0005,0.1563] versusL1(k0 = a1,a6), in which the system (1) changes its behavior. We fix the parameters a2 = 3, a3 = 1, a4 = 1, a5 = 5,

a7 = �1.6, a8 = 4.In Fig. 1, L1(k0) is shown for different values of the bifurcation parameters a1 and a6. It can be seen that, L1(k0)

passes through regions for which it is negative or positive. A more detailed investigation of L1(k0) in these regions willbe shown in the following figures.

The dependence of L1(k0) on parameters a1 and a6 is shown in Fig. 2(a) and (b). Initially, for a12 [4.0153,5.3345],a62 [0.0396,0.1563] (see Fig. 2(a)) L1 is negative. Therefore, following the terms introduced in [18,19,22], we have fora12 [4.0153,5.3345], a62 [0.0396,0.1563] that a soft stability loss takes place.

In Fig. 2(b) the parameter L1 is negative (in the figure ‘‘ ’’) at the end of the interval (a12 [1,4.0153],a62 [0.0005,0.1563]) and it is seen that L1 changes its sign from ‘‘�’’ into ‘‘+’’ for a1 = 2.8978 and a6 = 0.1078. The sign

‘‘+’’ is kept till the beginning of the interval (see ‘‘+’’ in the figure), i.e. here we have hard stability loss and the boundary

R = 0 is dangerous.

Unfortunately, the previous two Figs. 1 and 2 give only a most general idea about the bifurcation behavior of RHM.

The information supplied cannot fully illustrate the approach to chaos of this system. Hence, in Fig. 3 we show the

change of boundary of stability R = 0 from dangerous (L1 > 0) to secure (L1 < 0) one. To be exact, L1 becomes positive

after a6 < 0.1078, i.e. for values of a6 smaller then 0.1078, the boundary of stability R = 0 is dangerous. What could one

Fig. 1. The graph of the first Lyapunov value L1 versus the bifurcation parameters, a12 [1,5.3345], a62 [0.0005,0.1563] using Eq. (12).

Fig. 2. Illustrating dependence of L1 on the parameters a1, and a6, where: (a) a12 [4.0153,5.3345], a62 [0.0396,0.1563];(b) a12 [1,4.0153], a62 [0.0005,0.1563].

Fig. 3. A two parameter bifurcation diagram of equilibrium points (2). (a1 versus a6). R = 0––boundary of stability; L1––first

Lyapunov value. The regions are described in the text.


observe in the figure? When we cross the boundary of stability R = 0 (from R > 0 to R < 0) in dangerous zone (dashed

line) then the system (1) (after changing of the bifurcation parameters a1 and a6) can be in chaotic or regular regime (see

Fig. 5). It is interesting to note that here the limit cycles are with period one, two, three or more (see Fig. 4(a)–(c)). If we

cross the boundary of stability R = 0 in secure zone (continue line), then the system (1) has only the regular solutions

with period one (see Fig. 4(d)). Comparing our result with data submitted by other authors [1,2,11,23], who study the

bifurcation behavior of RHM, we conclude that those data are in accordance with the results obtained in our study.

Hence, we may conclude that the transformation into chaos is related to a hard loss of stability.

The bifurcation diagram shown in Fig. 5 need additional comments. Now, the values of zn are plotted against a1regarded as a continuously varying control parameter. Here we note that the parameter a6 = 0.007 is fixed and we vary

Fig. 4. Phase portrait of the system (1) when: (a) a1 = 2.75, a6 = 0.01; (b) a1 = 2.75, a6 = 0.015; (c) a1 = 2.75, a6 = 0.02; (d) a1 = 3.5,

a6 = 0.12. For other details, see text.

Fig. 5. (a) Bifurcation diagram zn versus a1 generated by computer solution of the Rose–Hindmarsh system (1) at a12 [2.1,4], a2 = 3,a3 = 1, a4 = 1, a5 = 5, a6 = 0.007, a7 = �1.6, a8 = 4, x0 = y0 = 0.1, z0 = �0.1 and (b) strange attractor of the system (1) at a1 = 3.27,

a6 = 0.007.


a12 [2.1,4]. For these values of the parameters a1 and a6 the system (1) passes (from R > 0 to R < 0) through the dan-

gerous zone of the boundary R = 0 i.e. L1(k0) > 0. For a12 [2.1,2.6] and a12 [3.5,4] the system has regular behavior with

period one. It is interesting that for a1 � 2.3 the system has so-called ‘‘hard self-excitation’’. This is to large extent in

agreement with Lyapunov–Andronov�s theory [18,19,22], i.e. that this behavior can be obtained only if L1(k0) > 0. As a1increased further a62 [2.6,3.25] the right and inverse bifurcations take place. After that for a6 � 3.26, the system (1)

suddenly passes in chaotic regime. Note that this behavior in [11,23] is obtained, when the bifurcation parameter is

a6. Therefore, the system (1) has similar behavior (suddenly passes in chaotic regime) when bifurcation parameter is

a1. A confirmation of our conclusions is the strange attractor shown in Fig. 5(b) (for a1 = 3.27 and a6 = 0.007) and

obtained for this case maximal Lyapunov exponent kmax. For the numerical calculation of kmax we use the TISEANsoftware package [24]. The obtained maximal Lyapunov exponent (per unit time) is: +0.0179 ± 0.00011. Here we note

that initial conditions were x0 = y0 = 0.1, z0 = �0.1 in all simulations.

4. Discussion and conclusions

The paper presents an alternative study of the bifurcation behavior of RHM using Lyapunov–Andronov�s theory.This is done on the basis of the obtained analytical formula expressing the first Lyapunov�s value (Eq. (12)). The sub-stantial difference between the analysis performed and that proposed by other authors [1,8,11,23] consists of the follow-

ing: The basic result is not found after obtaining a set of numerical solutions of the RHM for different values of the

bifurcation parameters a1 and a6, but this is done by using analytical formula derived directly from Lyapunov–Andro-

nov theory. The new conclusions following from the results presented in Sections 2 and 3 are:

1. Transition to chaos and the occurrence of chaotic oscillations in the Rose–Hindmarsh system (1) take place under

hard stability loss (L1(k0) > 0).2. When the bifurcation parameter a6 takes values larger than a6 = 0.1061 (for fixed a2 = 3, a3 = 1, a4 = 1, a5 = 5,

a7 = �1.6, a8 = 4 and a12 [2.8728,5.3345]) one may assume that RHM passes only into regular regime with period

one––soft stability loss (L1(k0) > 0).3. The RHM has similar behavior (suddenly passes in chaotic regime) when bifurcation parameter is a1 or a6.

Acknowledgment

This work was supported by the National Science Fund of the Ministry of Education and Science (Bulgaria), project

MM 1302/2003.


References

[1] Keener J, Sneyd J. Mathematical physiology. Springer-Verlag; 1998.

[2] Fall C, Marland E, Wagner J, Tyson J. Computational cell biology. NY: Springer-Verlag; 2002.

[3] Hindmarsh JL, Rose RM. A model of neuronal bursting using three coupled first order differential equations. Proc R Soc London

Ser B 1984;221:87–102.

[4] Panchev S, Spassova T. Reaction of two simple nonlinear dynamical systems to constant forcing. Atmosfera 1999;12:161–71.

[5] Guckenheimer J, Worfolk P. Dynamical systems: some computational problems. In: Schlomiuk D, editor. Bifurcations and

periodic orbits of vector fields. Kluwer Academic Publishers; 1992. p. 241–79.

[6] Holden A, Fan Y. From simple to simple bursting oscillatory behavior via chaos in the Rose–Hindmarsh model for neuronal

activity. Chaos, Solitons & Fractals 1992;2(3):221–36.

[7] Holden A, Fan Y. From simple to complex oscillatory behavior via intermittent chaos in the Rose–Hindmarsh model for neuronal

activity. Chaos, Solitons & Fractals 1992;2(4):349–69.

[8] Fan Y, Holden A. Bifurcations burstings, chaos, and crises in the Rose–Hindmarsh model for neuronal activity. Chaos, Solitons &

Fractals 1993;3(4):439–49.

[9] Fan Y, Chay T. Crisis transitions in excitable cell models. Chaos, Solitons & Fractals 1993;3(6):603–15.

[10] Holden A. Nonlinear phenomena in excitable physiological systems. Chaos, Solitons & Fractals 1995;5(3–4):317–8.

[11] Fan Y, Chay T. Crisis and topological entropy. Phys Rev E 1995;51(2):1012–9.

[12] Fan Y, Holden A. An asymmetrical phase locking structure for a non-excitable cell model. Chaos, Solitons & Fractals

1998;9(10):1637–50.

[13] Sabbagh H. Control of chaotic solutions of the Hindmarsh–Rose equations. Chaos, Solitons & Fractals 2000;11(8):1213–8.

[14] Longtin A. Autonomous stochastic in bursting neurons. Phys Rev E 1997;55(1):868–76.

[15] Raghavachari S, Glazier J. Waves in diffusively coupled bursting cells. Phys Rev Lett 1999;82(14):2991–4.

[16] de Souza S, Caldas I, Viana R, Balthazar J. Sudden changes in chaotic attractors and transient basins in a model for rattling in

gear boxes. Chaos, Solitons & Fractals 2004;21(3):763–72.

[17] Alligood K, Sauer T, Yorke J. Chaos, an introduction to dynamical system. Springer-Verlag; 1996.

[18] Bautin N. Behavior of dynamical systems near the boundary of stability. Moscow: Nauka; 1984.

[19] Andronov A, Vitt A, Khaikin S. Theory of oscillations. Reading, MA: Addison-Wesley; 1966.

[20] Nikolov S, Petrov V. New results about route to chaos in Rossler system. Int J Bifurcat Chaos 2004;14(1):293–308.

[21] Neimark Yu, Landa P. Stochastic and chaotic oscillations. Kluwer Academic Publishers; 1992.

[22] Arnold V. Theory of crises. Moscow University; 1983.

[23] Panchev S. Theory of chaos. Bulgarian Academic Press; 2001.

[24] Hegger R, Kantz H, Schreiber T. Practical implementation of nonlinear time series methods: the TISEAN package. Chaos

1999;9:413–35.

an alternative bifurcation analysis of the rose–hindmarsh model

Documents