On the dynamics of chaotic spiking-bursting transitionin the Hindmarsh–Rose neuron

G. Innocenti and R. GenesioDipartimento di Sistemi e Informatica, Università di Firenze, via S. Marta 3, 50139 Firenze, Italyand Centro per lo Studio di Dinamiche Complesse (CSDC), Università di Firenze, Italy

�Received 20 March 2009; accepted 1 June 2009; published online 26 June 2009�

The paper considers the neuron model of Hindmarsh–Rose and studies in detail the system dynam-ics which controls the transition between the spiking and bursting regimes. In particular, such apassage occurs in a chaotic region and different explanations have been given in the literature torepresent the process, generally based on a slow-fast decomposition of the neuron model. Thispaper proposes a novel view of the chaotic spiking-bursting transition exploiting the whole systemdynamics and putting in evidence the essential role played in the phenomenon by the manifolds ofthe equilibrium point. An analytical approximation is developed for the related crucial elements anda subsequent numerical analysis signifies the properness of the suggested conjecture. © 2009American Institute of Physics. �DOI: 10.1063/1.3156650�

Many biological neuron models have been developed inthe last decades for an accurate description and predic-tion of biological phenomena. From the basic contribu-tion of Hodgkin and Huxley, the simplified model ofHindmarsh and Rose has turned out to be quite accuratein capturing the features of electrical data from experi-ments. This model is a three-dimensional oscillator, par-ticularly aimed to study the spiking-bursting behavior ofthe neuron membrane potential. Indeed, these operatingmodes and their relationships are key questions in under-standing the different mechanisms which regulate theneural coding of information in many biological pro-cesses. The Hindmarsh–Rose model exhibits the spiking-bursting passage in a chaotic region and such transitionhas been explained in the past by applying a slow-fastdecomposition of the related equations. This paper for-mulates a novel conjecture to detect the phenomenon.From the complete model, by considering the equilibriumpoint and its manifolds, a kind of trajectory “separatrix”is analytically derived in the state space for the occur-rence of the transition. The method is developed for ageneral class of polynomial systems, while a numericalanalysis, which appears to fully confirm the conjecture, isperformed for the specific case in study.

I. INTRODUCTION

The study of the neural activity and its response mecha-nism to external stimuli has produced in the literature a va-riety of neuronal models. In this respect it is to underline theimportance of the early work of Hodgkin and Huxley,1

whose model surely represents a milestone. However, sincesuch a model turns out to be quite complex, several authorshave attempted to provide simpler approximations.2–6 Thesemodels are designed to guarantee a qualitative description ofsome neuronal phenomena in order to derive useful insightsabout them. Therefore, it is natural that the first ones werelow-dimensional models, namely, second order systems suchas the FitzHugh–Nagumo and Morris–Lecar neurons.

In this work we refer to the Hindmarsh–Rose neuronalmodel. Its first version is dated in 1982 �Ref. 7� and again itis a second order form. Despite the genesis of several sim-plified models which corresponds to an approximation orreduction of the Hodgkin–Huxley equations, the Hindmarsh–Rose system is the result of biological experiments per-formed on the pond snails.8 Remarkably, this model hassome important similarities with other neurons such as theS-shaped nullclines and the bistability property �see, e.g.,Refs. 8–10 for detailed analyses�. However, the second ordermodel is not able to reproduce some interesting phenomenasuch as terminating by itself a triggered state of firing.Hence, to this aim the authors added a third dynamical com-ponent, whose role is to tune the above subsystem over themono- and bistability regions in order to activate or termi-nate the neuronal response.9 Since such an additional term isdesigned to act on a longer time scale, the model results to benaturally divided into a fast and a slow subsystem. Notably,the recent paper Ref. 11 shows that the Hindmarsh–Roseneuron can capture both qualitative and quantitative aspectsof experimental data.

Due to the particular structure of the model, several phe-nomena can be accurately described just by studying the fastsubsystem, eventually taking care of some interactions withthe slow component. However, this approach is not able toexplain the complete variety of the behaviors exhibited bythe model. To this aim the previous work Ref. 10 made aneffort to provide a comprehensive description of the neuronphenomena in terms of its fully coupled equations, describ-ing the generation of the different regimes in terms of bifur-cation theory.12

In this paper we present the chaotic spiking-burstingtransition as a specific expression of the whole third orderdynamics. Section II introduces the mathematical model andits main properties. In Sec. III the most important featuresconcerning the geometry of the bursting and spiking attrac-tors are analyzed in details. Section IV introduces a dynami-

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cal explanation of the transition. Finally, Sec. V is devoted tocompute a suitable analytical approximation of the weaklyunstable manifold of the equilibrium and the following nu-merical analysis quantifies its essential influence on the phe-nomenon. The conclusions of Sec. VI end the paper.

II. THE MODEL ANALYSIS

The third order Hindmarsh–Rose model equations are

x = y − z + I + bx2 − ax3,

�1�y = − y + c − dx2, z = rsx − rz − rsx0,

where x is the membrane potential, I is the external dc cur-rent, and y and z are the recovery and the adaptation current,respectively. The constant parameters a, b, c, and d definethe 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. �2�

The variable z is the additional component and the coeffi-cients r and s have the critical role of tuning the neuronresponse. For a proper choice of r and s the model is able todisplay both regular and chaotic bursting and firing as thecurrent I varies.9 It is worth noticing that r is usually set to avery small value, so dividing the model into a fast subsystem�i.e., the second order model described by the first two equa-tions� and a slow subsystem �the third equation�.

The Hindmarsh–Rose neuron exhibits a chaotic spiking-bursting transition over a wide range of its parameters r ands. In particular, this phenomenon has already been studied inthe literature for r� �0.001,0.006� and s=4. Besides, themodel has been extensively studied by varying I in Refs. 13and 14, while two parameter analyses have been recentlyreported in Refs. 15 and 16, denoting the present interest forthe subject.

Hereafter, we will assume r=0.0021 and s=4 in accor-dance to Ref. 10. For such values a chaos-chaos transitionhappens when I varies in the very narrow interval �3.29,3.30�, generally assumed as the passage between burstingand spiking �chaotic� regimes. This transition boils downinto a sharp modification of the chaotic attractor size, asillustrated in Fig. 1. Since the main variation in the attractorsize is observed along the z component, this can be properlyillustrated by reporting the minimum of z versus the currentI as in Fig. 2. To this regard, Ref. 13 considers that such ageometric modification may be related to an interior crisisand due to the continuity with respect to the external currentI, this is called a “continuous internal crisis.” In Ref. 14 anextensive and quantitative analysis of the transition is pro-vided according to the fast and slow subsystem division ap-proaches. The phenomenon is strongly related to the pres-

-2

0

2

-8-6-4-20

2.9

3

3.1

3.2

3.3

3.4

3.5

z

yx

I = 3.30

-2

0

2

-8-6-4-20

2.9

3

3.1

3.2

3.3

3.4

3.5

z

yx

I = 3.29

(a) (b)

FIG. 1. �Color online� �a� A three-dimensional view of the chaotic spiking attractor for I=3.30 and its position with respect to the equilibrium point. �b� Thechaotic bursting attractor and the equilibrium point for I=3.29, i.e., after the spiking-to-bursting chaotic transition described in Sec. II. The comparisonbetween cases �a� and �b� shows the sharp modification of the attractor size along the z direction. Besides, the equilibrium point position has a negligiblevariation.

3.305 3.31

2.95

3

3.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.28 3.285 3.29 3.295 3.3

Min(z)

I

FIG. 2. �Color online� The variation in the attractor minimum value alongthe z direction as the current I varies in a narrow range around the transitioninterval. The related attractor modification appears to be continuous innature.

023124-2 G. Innocenti and R. Genesio Chaos 19, 023124 �2009�

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ence in the fast subsystem of a homoclinic bifurcation andwhat happens in terms of the whole system dynamics is notinvestigated. A similar observation has already been ex-pressed in Ref. 17. Moreover, this author introduces a differ-ent perspective as well. He suggests that the stable and un-stable manifolds of the �unique� equilibrium point of thesystem may play an important role in determining the dy-namics. Besides, the shape of the chaotic attractor may de-pend on the envelopment provided by such manifolds. Ac-cording to this perspective, on one side of the transitioninterval the attractor is forced to stay in a certain region ofthe state space, while on the other side it is allowed to reacha further region, so modifying its shape.

III. THE TRANSITION BEHAVIOR

In this section we provide a detailed description of theneuron model �1� by illustrating the different shapes of thestate space trajectories on the two sides of the chaoticspiking-bursting transition. Then, we will focus on the criti-cal region where they differ to propose a dynamical expla-nation of the process.

Let us consider the system behavior for the decreasingvalues of the external current I in the range �3.30, 3.29�.Figure 1�a� illustrates the chaotic spiking attractor at I=3.30. The trajectory turns along a spiral, where each coilnearly lies on a plane parallel to the x-y subspace and moveshaving z as dominant direction. When the trajectory reachesthe top, a reinjection happens and the system state is rapidlybrought back to the bottom. During this motion the z dynam-ics is leading. Therefore, the chaotic spiking attractor resultsin a spiral component ruled by the fast subsystem dynamicsand a reinjection path where the slow subsystem prevails.

In Fig. 1�b� the state space representation of the chaoticbursting attractor is depicted for I=3.29. As in the previouscase a spiral shape is recognized. The coils are almost paral-lel to the x-y plane and their motion nearly follows the zcoordinate. The reinjection process can still be highlightedonce the trajectory reaches the top of the spiral: while in thespiking case the paths are always similar, in the bursting casea variety of different routes are observed. In particular, theydiffer in the length, the dominant direction, and the z heightsto 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 twomain dynamics. In the first case a shorter path analogous tothe spiking one is covered. In the second scenario the statetakes a longer path. At the beginning, it assumes a differentdirection, except for a later turn, which gets the trajectoryinto the spiral at a lower z height. In Fig. 3 a close look at thechaotic bursting attractor reveals that the critical region is atthe spiral’s top, where the reinjection process is about tostart. In particular, the way that the trajectories follow toapproach such region affects how they are brought back tothe bottom. A comparison between the paths highlighted inthe figure reveals the different contributions to the attractor’sshape as well.

These observations and in particular the dominance ofthe z dynamics along the reinjection process underline that

the fast/slow subsystem technique is not suitable to explainthe nature of the transition.

For the following developments we briefly recall the na-ture of the unique equilibrium point of Eqs. �1� and �2� dur-ing the transition �see Refs. 9 and 10 for further details�.During the process such fixed point almost remains constantas well as its stability properties. Also the eigenvalues �twounstable and one stable� and the related eigenvectors undergonegligible variations and they can be considered as invariantsalong this passage. Although the equilibrium lies distantfrom the attractor, the direction of its eigenvectors indicatesthat it may play a fundamental role. Indeed, roughly speak-ing, on the side where the reinjection happens the spiral at-tractor is “squeezed” against the completely unstable eigens-pace of the equilibrium �i.e., the plane tangent to itscompletely unstable manifold�. Moreover, the trajectory ap-proaches the starting region of the reinjection by a directionparallel to the stable eigenspace and the weakly unstableeigenspace turns out to pass close to that region too.

In Ref. 17 the author suggests that the shape of the at-tractor is due to the passage of the trajectory in two differentregions of the state space. Furthermore, he chooses the planeformed by the stable and the weakly unstable eigenspaces ofthe equilibrium point to describe the related separatrix. Thissurface is just the linear approximation of the stable/weaklyunstable manifold and it plays a passive role in the explana-tion of the phenomenon. Indeed, according to the author’sidea the dynamics may be completely attributed to only thefast subsystem. The Poincaré map built in Ref. 17 and de-rived for r=0.0021 in Fig. 4�a� provides just a simplified toolto represent the candidate separatrix, which turns out to de-tect the transition fairly well. However, this happens becausesuch a map is able to catch the attractor’s size variation alongz, as shown in Fig. 4�b�. Indeed, moving from the spikingregime the transition is described in terms of the map’s in-tersection with the stable/weakly unstable line. However,

2.95

3

3.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

z

-9-8-7-6-5-4-3-2-101y

FIG. 3. �Color� Two-dimensional projection of the bursting chaotic attractoronto the y-z plane at I=3.29. Two different routes are highlighted to illus-trate the possible behaviors at the spiral top. Red curve: the trajectory fol-lows 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 ofthe spiral at about z�3.

023124-3 A transition in the Hindmarsh–Rose model Chaos 19, 023124 �2009�

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since the trajectories are not allowed to pass through a dy-namical manifold, the situation depicted in Fig. 4�a� is notrelated to any real crossing phenomenon. Nevertheless, ob-serving the corresponding positions in Fig. 4�b�, it is straight-forward to derive that such an event corresponds to a changein the attractor’s shape and that it occurs when the spiral’sbottom reaches a sufficiently low z. Since the transition isdetected by the position of the attractor’s lowest part, such aPoincaré map appears not suitable to explain the behavior inthe critical region at the top of the spiral, where really thespiking and bursting regimes are set up.

On the base of these observations, in Sec. IV we willintroduce a novel conjecture for such a passage.

IV. THE CONJECTURE

From the analysis of Sec. III, the phenomenon can betreated according to the following conjecture. The reinjectionprocess concerning the chaotic spiking-bursting transitionoccurs near the completely unstable manifold of the equilib-rium point. In particular, the critical region at the top of thespiral is close to the weakly unstable component. Hence, inthe first stage the reinjection is driven by such dynamics,while in the second stage the strongly unstable dynamicsprevails. In the spiking case the trajectory approaches thecompletely unstable surface on the same side of the weaklyunstable curve, while in the bursting case it may fall on bothsides. Then, in the latter scenario the trajectory may divergeaccording to both directions of the strongly unstable compo-nent, resulting in two different reinjection paths.

In the following we address the considered transitionmore 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, respec-tively. We denote by U1 and U2 the two branches of thecurve U from E and by � the surface corresponding to thecompletely unstable manifold. Observe that U and W are

curves on �. Then, assume that the system trajectory ap-proaches � approximately according to the direction of S.The three different situations illustrated in Fig. 5 are consid-ered.

�a� As the system trajectory approaches � the influence ofthe unstable dynamics of the equilibrium point be-comes dominant. When the approaching region is suf-ficiently distant from W, the strongly unstable compo-nent prevails and the motion diverges along the branchU1.

�b� The system trajectory tends to � very near to W.Hence, at the beginning the weakly unstable dynamicsprevails and the motion diverges according to thatcurve. Then, the strongly unstable component necessar-ily dominates and the trajectory finally follows its dy-namics, namely, along U1.

�c� The system state approaches � close to the weaklyunstable manifold, here on the opposite side of W withrespect to the case b. The trajectory follows initially thecurve W but, as the strongly unstable dynamics be-comes leading, it diverges along the opposite branch ofU, namely, U2.

Therefore, according to this scenario, the chaotic spikingis characterized at the spiral’s top by trajectories as that ofcase �b�. Conversely, in the chaotic bursting the motion mayexhibit in that critical region both the behaviors �b� and �c�.In particular, case �b� is related to the shortest reinjectionpaths, while �c� is responsible for the longest ones.

Section V will be devoted to a rigorous analytical ap-proach of these system dynamics in order to validate theabove conjecture. Since the basic elements of such a phe-nomenon have been singled out in the scheme of Fig. 5, wewill develop our approach for a more general class of sys-tems, then deriving specific results for the Hindmarsh–Rosemodel.

-5.5 -5 -4.5 -4 -3.5 -3 -2.53

3.05

3.1

3.15

3.2

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3.5z

y

I = 3.295

x-2

0

2

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2.9

3

3.1

3.2

3.3

3.4

3.5

I = 3.295

y

z

(a) (b)

FIG. 4. �Color online� �a� The Poincaré map obtained by the procedure described in Ref. 17 when the system is close to the transition. The related plane isparallel 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 stableand weakly unstable eigenvectors. �b� The three-dimensional representation of the attractor and the Poincaré map illustrates their respective position in thestate space.

023124-4 G. Innocenti and R. Genesio Chaos 19, 023124 �2009�

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V. THE MANIFOLD GEOMETRY AND THE TRANSITION

At first, we derive an approximation of the weakly un-stable manifold W to verify its presence in the critical regionof the reinjection process. To this aim, we develop hereafteran analytical procedure designed for a more general class ofdynamical systems, just presenting aside specific results forthe Hindmarsh–Rose model.

Let us consider the class

� = �� + �n=2

+�

�ng�n���� , �3�

where ��RN, �n�RN, g�n���� :RN→R is a homogeneouspolynomial of the nth order in the components of �, and ��RN�N is a diagonal matrix such that all its eigenvalues arenegative but two. For the sake of simplicity we assume N=3, so that

� = � �s

�u

�w�, � = ��s 0 0

0 �u 0

0 0 �w�

with �s�0��w��u.Let us prove that the Hindmarsh–Rose system admits

model �3�. To this aim first observe that it can be compactlywritten as

= A + f��

just by defining

� �x

y

z�, A � � 0 1 − 1

0 − 1 0

rs 0 − r�, f�� � �I + bx2 − ax3

− dx2

− rsx0� .

Moreover, let the equilibrium at a certain value of the exter-nal current I be denoted as

E�I� � ��1�I��2�I��3�I�

� . �4�

For the sake of simplicity, we will not indicate further thisdependence on I. Then, let �s, �u, and �w, respectively, be thestable, the strongly unstable, and the weakly unstable eigen-values of E at the considered value. In the transition intervalthey have the real values �u�0.1925, �w�0.004 366, and�s�−0.645. Moreover, let Vs ,Vu ,Vw�R3 be the three col-umn eigenvectors related to �s, �u, and �w, respectively. Fi-nally, define the modal matrix

T � �Vs,Vu,Vw� = �v11 v12 v13

v21 v22 v23

v31 v32 v33� �5�

and let us denote its inverse by

T−1 = �11 12 13

21 22 23

31 32 33� . �6�

Then, we perform the linear affine change in coordinates

� = T−1� − E� ,

obtaining

completely unstablemanifold Φ

weakly unstable manifold W

equilibrium point E

strongly unstable manifold(branch U2)

stable manifold S

strongly unstable manifold(branch U1)

a b c

stable eigenvector Vs

strongly unstableeigenvector Vu

weakly unstableeigenvector Vw

FIG. 5. �Color� The scheme addressed in Sec. IV to explain the dynamical mechanism which separates the spike trajectories from the burst ones. All the basicelements playing an active role in defining the phenomenon have been represented. Notice that the eigenvectors of the equilibrium point are tangent to therelated manifolds.

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� = T−1AT� + T−1AE + T−1f�T� + E� . �7�

After tedious though straightforward computations Eq. �7�boils down to the equation system

�s = �s�s + �11�b − 3a�1� − d12��v11�s + v12�u + v13�w�2

− a11�v11�s + v12�u + v13�w�3,

�u = �u�u + �21�b − 3a�1� − d22��v11�s + v12�u + v13�w�2

− a21�v11�s + v12�u + v13�w�3, �8�

�w = �w�w + �31�b − 3a�1� − d32��v11�s + v12�u + v13�w�2

− a31�v11�s + v12�u + v13�w�3

in the new state variables �s, �u, and �w. Observe that Eq. �8�admits the representation �3� just by defining the functions

g�2���s,�u,�w� � �v11�s + v12�u + v13�w�2,

g�3���s,�u,�w� � �v11�s + v12�u + v13�w�3, �9�

g�n���s,�u,�w� � 0, ∀ n � 3,

and the vectors

�2 � ��11

�12

�13� = �b − 3a�1��11

21

31� − d�12

22

32� ,

�10�

�3 � ��21

�22

�23� = − a�11

21

31�, �n � 0, ∀ n � 3.

To develop a suitable approximation of the weakly un-stable manifold, let us recall that W is a curve passingthrough E with direction parallel to the eigenvectors of �w.Now, system �3� has the origin as equilibrium point and theweakly unstable eigenvectors are directed along the axis �w.Therefore, W can locally be parametrized by the coordinate�w as follows:

W = ��s,�u,�w� � R3:�s = hs��w�,�u = hu��w� , �11�

where the functions hs and hu must satisfy the tangency con-ditions

hs�0� = hu�0� = 0,�hs

��w�0� =

�hu

��w�0� = 0, �12�

and they are the goal of our research. Let us represent hs andhu by power series developments

hs��w� = �n=2

�

�n�wn , hu��w� = �

n=2

�

n�wn . �13�

Deriving a suitable approximation of W turns into the prob-lem of computing the sets �nn=2

M and nn=2M for a suffi-

ciently high number M −1 of elements, with a related errordecreasing as M increases. To this aim, consider the nonlin-ear part of system �3� and evaluate it into the points of the

manifold �11�. Exploiting the representation �13� for hs andhu, we obtain a �w power series,

�n=2

+�

�ng�n��hs��w�,hu��w�,�w� = �n=2

�

kn�wn , �14�

where the coefficients

kn = �k1n

k2n

k3n�, n � 1

depend on the nature of the functions g�n� and on the param-eters �n and n as well.

Remark: Each kn only depends on��n−1 , n−1 , . . . ,�2 , 2�.

For the Hindmarsh–Rose system Eq. �14� reduces to

�2g�2��hs��w�,hu��w�,�w� + �3g�3��hs��w�,hu��w�,�w�

= �n=2

�

kn�wn , �15�

where g�2�, g�3� and �2, �3 are defined as in Eqs. �9� and �10�,respectively. The expressions of the corresponding coeffi-cients kinn=2

8 , i=1,2 ,3, are listed in Table I.Substituting the power development �13� in the system

equation �3� and exploiting expression �14� for the nonlinearterm, we obtain

�s��w� = �n=2

�

��n�s + k1n��wn ,

�u��w� = �n=2

�

� n�u + k2n��wn , �16�

�w��w� = �n=1

�

k3n�wn ,

where k31��w.Since the manifold W is such that

�s =�hs

��w�w, �u =

�hu

��w�w, �17�

by computing Eq. �17� with Eq. �13� and using the thirdequation in Eq. �16�, we also obtain that

�s��w� = �n=2

� ��wn �

m=2

n

�m�mk3�n−m+1��� ,

�18�

�u��w� = �n=2

� ��wn �

m=2

n

�m mk3�n−m+1��� .

Then, we equate Eq. �18� to the first two equations of Eq.�16� and we balance the �w powers. Exploiting the aboveremark, the coefficients ��n , n�, which define W in Eqs.�11�–�13�, are finally obtained as

023124-6 G. Innocenti and R. Genesio Chaos 19, 023124 �2009�

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�n =1

�s − n�w��

m=2

n−1

�m�mk3�n−m+1�� − k1n� ,

�19�

n =1

�u − n�w��

m=2

n−1

�m mk3�n−m+1�� − k2n� .

Figure 6 illustrates the approximation of W for theHindmarsh–Rose system obtained by using the coefficients�n and n listed in Table I,

W = ��s,�u,�w� � R3:�s = �n=2

8

�n�wn , �u = �

n=2

8

n�wn� . �20�

In particular, Fig. 6�a� provides a comparison between Eq.�20� and the spiking chaotic attractor at I=3.30. The figure

shows that W is extremely close to the trajectories passing inthe critical region. Moreover, it highlights that the reinjection

paths are always placed on the same side with respect to W.

Figure 6�b� depicts the weakly unstable manifold approxima-tion in comparison with the bursting chaotic attractor at I

=3.29. The manifold W appears fully embedded in the tra-jectories of the critical region, but now they may pass onboth its sides. In particular, on one side the reinjection pathsfollow a trail analogous to the spiking regime ones, while onthe other, they follow a different and longer route thatreaches lower z values.

In conclusion, let us define d as the minimum distance

between W and the system attractor �, i.e.,

d � min��W

���

�� − ��2. �21�

The analysis of d provides useful insights about the influenceof the weakly unstable manifold of the equilibrium pointonto the neuron regime. To this aim, the diagram of d as theexternal current I varies is reported in Fig. 7. Notably, thedistance almost vanishes in the chaotic bursting interval.

TABLE I. The table reports the first coefficients implicitly defined by Eq. �15�.

i=1,2 ,3:

ki2=v132 �i2

ki3=2v13��2v11+ 2v12��i2+v133 �i3

ki4=�i2�2v13��3v11+ 3v12�+ ��2v11+ 2v12�2�+3v132 ��2v11+ 2v12��i3

ki5=�i2�2v13��4v11+ 4v12�+2��2v11+ 2v12���3v11+ 3v12��+�i3�3v132 ��3v11+ 3v12�+3v13��2v11+ 2v12�2�

ki6=�i2�2v13��5v11+ 5v12�+2��2v11+ 2v12���4v11+ 4v12�+ ��3v11+ 3v12�2�+�i3�3v13

2 ��4v11+ 4v12�+3v13��2v11+ 2v12���3v11+ 3v12�+ ��2v11+ 2v12�3�

ki7=�i2�2v13��6v11+ 6v12�+2��2v11+ 2v12���5v11+ 5v12�+2��3v11+ 3v12���4v11+ 4v12��+�i3�3v13

2 ��5v11+ 5v12�+3v13��2v11+ 2v12���4v11+ 4v12�+3v13��3v11+ 3v12�2+3��2v11+ 2v12�2��3v11+ 3v12��

ki8=�i2�2v13��7v11+ 7v12�+2��2v11+ 2v12���6v11+ 6v12�+2��3v11+ 3v12���5v11+ 5v12�+ ��4v11+ 4v12�2�+�i3�3v13

2 ��6v11+ 6v12�+3v13��2v11+ 2v12���5v11+ 5v12�+3v13��3v11+ 3v12���4v11+ 4v12�+3��2v11+ 2v12�2��4v11+ 4v12���+3��2v11+ 2v12���3v11+ 3v12�2�

(a) (b)

FIG. 6. �Color� a� A three-dimensional view of the weakly unstable manifold approximation �red line� compared to the chaotic spiking attractor at I=3.30. �b�A detail of the critical region at the spiral’s top for I=3.29. The curve W �red line� is fully embedded in such a region as conjectured.

023124-7 A transition in the Hindmarsh–Rose model Chaos 19, 023124 �2009�

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Moreover, it has a minimum in the chaotic spiking-burstingtransition window �3.29, 3.30�, denoting the influence of theweakly unstable manifold on the system dynamics, in accor-dance with the conjecture of Sec. IV. By decreasing I fromhigher values the model exhibits the chaotic spiking regime

with trajectories passing on the same side of W and shows acontinuous distance reduction. Then, in the neighborhood of

the transition d vanishes and the orbits tend to overcome W,passing by both its sides during the chaotic bursting regime.When this behavior disappears, d is again growing and sev-eral local minima are experienced for lower values of I cor-responding to other changes in the system dynamics, whosestudy is beyond the scope of the paper.

VI. CONCLUSIONS

The paper has introduced a novel explanation about thechaotic spiking-bursting transition in the Hindmarsh–Roseneuron model. A qualitative analysis of the system behaviorin the state space has highlighted that the fast/slow sub-

system approach appears not suitable to comprehend the dy-namical mechanism of the transition. Therefore, a detailedstudy of the trajectory evolution in the critical passage hasbeen carried out, leading to formulate a conjecture about theinfluence of the equilibrium point onto the system regimes.Then, classical mathematical tools of nonlinear dynamicshave been exploited to approximate the weakly unstablemanifold of the fixed point as the significant element whichgoverns the transition. Such an approach has been developedfor a more general class of polynomial systems, derivingspecific results for the Hindmarsh–Rose model. A numericalanalysis has been performed to provide quantitative resultsas well. The accordance between such results and the con-jectured behavior is remarkable and suggests the correctnessof the dynamical explanation for the chaotic spiking-burstingtransition. Moreover, the analysis points out that the consid-ered phenomenon could be detected in other dynamical mod-els, where sudden but continuous changes in the attractorshape occur.

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2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7

0

2

4

6

8

10x 10

-4

d

I

FIG. 7. �Color online� The diagram of the distance d between the weaklyunstable manifold W estimated by using Eq. �20� and the system attractor asa function of the current I.

023124-8 G. Innocenti and R. Genesio Chaos 19, 023124 �2009�

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