Reciprocal inhibitory coupling: Measure and control of chaos on a biophysically motivated model of...

Commun Nonlinear Sci Numer Simulat 14 (2009) 2734–2746

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Commun Nonlinear Sci Numer Simulat

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Reciprocal inhibitory coupling: Measure and control of chaoson a biophysically motivated model of bursting

Jorge Duarte a,*, Cristina Januário a, Nuno Martins b

a ISEL-High Institute of Engineering of Lisbon, Department of Chemistry, Mathematics Unit, Rua Conselheiro Emídio Navarro, 1949-014 Lisboa, Portugalb Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Department of Mathematics, Instituto Superior Técnico,Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 September 2008Received in revised form 29 October 2008Accepted 30 October 2008Available online 8 November 2008

PACS:87.18.�h05.45.Ac87.18.Vf

Keywords:Reciprocal inhibitory couplingSymbolic dynamicsTopological entropyChaos

1007-5704/$ - see front matter � 2008 Elsevier B.Vdoi:10.1016/j.cnsns.2008.10.020

* Corresponding author.E-mail addresses: [email protected] (J. Duar

Bursting activity is an interesting feature of the temporal organization in many cell firing pat-terns. This complex behavior is characterized by clusters of spikes (action potentials) inter-spersed with phases of quiescence. As shown in experimental recordings, concerning theelectrical activity of real neurons, the analysis of bursting models reveals not only patternedperiodic activity but also irregular behavior [Holden AV, Winlow W, Haydon PG. The induc-tion of periodic and chaotic activity in a molluscan neurone. Biol Cybern 1982;43:169–73;Chay TR, Rinzel J. Bursting, beating and chaos in an excitable membrane model. Biophys J1985;47:357–66]. The interpretation of experimental results, particularly the study of theinfluence of coupling on chaotic bursting oscillations, is of great interest from physiologicaland physical perspectives. The inability to predict the behavior of dynamical systems in pres-ence of chaos suggests the application of chaos control methods, when we are more interestedin obtaining regular behavior. In the present article, we focus our attention on a specific classof biophysically motivated maps, proposed in the literature to describe the chaotic activity ofspiking–bursting cells [Cazelles B, Courbage M, Rabinovich M. Anti-phase regularization ofcoupled chaotic maps modelling bursting neurons. Europhys Lett 2001;56:504–9]. More pre-cisely, we study a map that reproduces the behavior of a single cell and a map used to examinethe role of reciprocal inhibitory coupling, specially on two symmetrically coupled burstingneurons. Firstly, using results of symbolic dynamics, we characterize the topological entropyassociated to the maps, which allows us to quantify and to distinguish different chaoticregimes. In particular, we exhibit numerical results about the effect of the coupling strengthon the variation of the topological entropy. Finally, we show that complicated behavior aris-ing from the chaotic coupled maps can be controlled, without changing of its original proper-ties, and turned into a desired attracting time periodic motion (a regular cycle). The control isillustrated by an application of a feedback control technique developed by Romeiras et al.[Romeiras FJ, Grebogi C, Ott E, Dayawansa WP. Controlling chaotic dynamical systems.Physica D 1992;58:165–92]. This work provides an illustration of how our understandingof chaotic bursting models can be enhanced by the theory of dynamical systems.

� 2008 Elsevier B.V. All rights reserved.

1. Motivation and preliminaries

Bursting oscillations are ubiquitous and extremely important in physical and biological systems (see [5] and the very re-cent studies carried out in [6–8]). The biological significance and dynamical complexity of bursting cells have stimulated

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te), [email protected] (C. Januário), [email protected] (N. Martins).

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J. Duarte et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 2734–2746 2735

challenging mathematical investigations, including the construction of detailed biophysical models, which represent thehigh dimensional dynamics of nonlinear processes responsible for variations in the ionic currents across the membrane.The description of such complex bursting activity is usually based on either realistic ionic-based models or phenomenolog-ical models. The ionic-based models are designed to replicate the physiological mechanisms of the membrane, with func-tions and parameters derived from experimental data. Some of these models are given by a system of many nonlineardifferential equations. Due to the high dimensionality of the phase space, it is difficult to understand the collective behaviorof such dynamical systems[9–11]. As an alternative approach, phenomenological models are constructed to retain the mostimportant qualitative behavior, with minimal complexity of the equations [12]. A special type of phenomenological modelsconsists of low-dimensional maps. In fact, there have been proposed only few maps capable of generating significant featuresof bursting dynamics (for instance, see [3,13–15]).

The electrical activity of a population of neurons is the result of the intrinsic properties of individual cells as well as of thenature of coupling. The cooperative dynamics of coupled chaotic bursting neurons, for increasing values of the couplingstrength, is somewhat unexpected in the sense that it can be more organized than the activity of the individual cells.

In order to isolate essential aspects of the chaotic bursting dynamics, Cazelles, Courbage and Rabinovich introduced a newclass of maps (see [3]) which describes the chaotic activity of bursting neurons observed in neurophysiological experiments.

In recent years, there has been a considerable research effort into the analysis of chaotic systems. For instance, control,targeting, synchronization and forecasting of chaotic motion have proved well established results in the fields of, physics,applied mathematics and engineering. In particular, since the publication of the seminal paper of Ott et al. [16], there hasbeen a great deal of progress in the development of techniques for the control of chaotic phenomena, with applications,for example, to physiology, biochemistry, cardiology, communications, physics laboratories and turbulence. Indeed, thedynamical control of cardiac [17] and neuronal tissues [18] has been one of the most practical application of chaos controltechniques to biological systems. By exploring the natural dynamics of the systems, these techniques were used to stabilizeor destabilize heart beats and neuronal firing with minimum perturbation. The central question addressed in the theory ofchaos control is: given a chaotic system, how can we obtain improved performance and achieve a desired attracting timeperiodic motion by making small controlling temporal perturbations in an accessible system parameter? [4].

In the context of the electrical activity of neuron cell bodies, irregular bursting and spiking solutions of some represen-tative biophysical models can be converted efficiently to periodic spike trains (regular beating and periodic bursting), usingpractical methods of the new and exciting field of chaos control. Particularly, in order to control the chaotic motion, we donot need to change the fundamental characteristics of the system, we only have to impose upon the dynamics some small (ortiny) perturbations. The application of small external adjustment to the model leaves the main features unchanged and it isable to eliminate chaotic bursts.

The aim of the present article is to provide a contribution for the detailed analysis of the Cazelles–Courbage–Rabinovich(CCR) maps introduced in [3] through a comprehensive study in terms of symbolic dynamics theory of the single-cell modeland the analysis of important features of the system representing the dynamics of bursting neural assemblies connected to-gether via reciprocal inhibitory coupling.

We notice that we will focus our attention on the study of the two-cell system, used to analyze the effect of inhibitorycoupling on two symmetrically coupled bursting cells. A compelling reason to examine two-cell systems methodicallyand in detail is due to the fact pointed out in [19,20]: simulations of larger clusters of cells have shown that the results ob-tained for two-cell systems can be carried over to large populations (many cell systems).

Indeed, we can gain some fundamental qualitative insights about the principles and mechanisms underlying chaoticbursting behavior by studying low-dimensional maps, that incorporate representative dynamical properties of thephenomenon.

A quantifier for the complex orbit structure – an attribute used to define chaos – is the topological entropy. This measureof the amount of chaos in a dynamical system is the most important numerical invariant related to the orbit growth and itsvariation with particular parameters gives us a finer distinction between states of complexity. The topological entropy can beefficiently used to examine the role of coupling and to perform chaos control strategies.

In this work we are going to exhibit an application of the pole placement technique, initially proposed by Romeiras et al.[4] as an extension of the Ott–Grebogi–Yorke (OGY) method carried out in [16]. As far as our study is concerned, this methodis applied to the discrete time system of symmetrically coupled cells and uses a linear approximation, obtained from theinduced two-dimensional dynamics, in the neighborhood of the desired periodic orbit.

2. Symbolic dynamics. Topological entropy of chaotic bursting maps

For the sake of clarity, the next paragraph starts with an overview of the uncoupled map replicating chaotic burstingactivity (for more details, the reader is referred to the paper [3]).

2.1. The single-cell model of a burster

The behavior of a single cell can be described by the following one-dimensional map xnþ1 ¼ f ðxnÞ where f ðxnÞ is a piece-wise linear map (see Fig. 1) given by

0 0.2 0.4 0.6 0.8 1xn

0.2

0.4

0.6

0.8

1

xn 1

Fig. 1. Piecewise linear map with a fast-slow regime and a cobweb orbit. The parameter values are: c0 ¼ 0, c1 ¼ 0:4, c2 ¼ 0:6, c3 ¼ 0:7, c4 ¼ 1:0, a1 ¼ 0,a2 ¼ 1� 0:4b2, a3 ¼ �0:6b3 and a4 ¼ 0:7� 0:7b4. In this figure b1 ¼ 1:05, b2 ¼ �1:25, b3 ¼ 1:5 and b4 ¼ �1:0.

2736 J. Duarte et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 2734–2746

f ðxnÞ ¼

b1xn þ a1 if c0 6 xn < c1;




8>>><>>>:

ð1Þ

with ci 2 R (i ¼ 1;2;3;4). The coefficients of the ith branch of xnþ1 ¼ f ðxnÞ are the real numbers ai and bi. A typical regime oftemporal behavior of the map is shown in Fig. 2.

At this point, using symbolic dynamics, in particular some results concerning to admissible Markov partitions associatedto piecewise monotone maps, we characterize the topological entropy of the fast and slow regimes of the map xnþ1 ¼ f ðxnÞ.For more details see [21–23].

The map f is discontinuous piecewise monotonic on the interval I ¼ ½c0; c4�. This interval I is subdivided into four subin-tervals: IL ¼ ½c0; c1½; IM ¼�c1; c2½; IN ¼�c2; c3½; IR ¼�c3; c4� in such a way that the restriction of f to each interval IL or IN is strictlyincreasing and in the other intervals IM or IR, f is strictly decreasing. Each such maximal intervals on which the function f ismonotone is called a lap of f, and the number ‘ ¼ ‘ðf Þ of distinct laps is called the lap number of f.

Denoting by c1, c2 and c3 the three turning points of f, we consider the orbits

Oðc�1 Þ ¼ fx�j : x�j ¼ f jðc�1 Þ; j 2 Ng;Oðc�2 Þ ¼ fy�j : y�j ¼ f jðc�2 Þ; j 2 Ng;Oðc�3 Þ ¼ fz�j : z�j ¼ f jðc�3 Þ; j 2 Ng;

where Oðc�i Þ is the left orbit of the turning point ci and Oðcþi Þ is the right orbit of the turning point ci (i ¼ 1;2;3).With the aim of studying the topological properties of these orbits we associate to each orbit Oðc�i Þ a sequence of symbols

S ¼ S1S2; . . . ; Sj; . . . ; where

100 200 300 400 500n

0.2

0.6

1xn

Fig. 2. Wave forms of temporal behavior of individual cells. The parameter values are exactly the same as in Fig.1.


Sj ¼ C0 if f jðc�i Þ ¼ 0;

Sj ¼ L if c0 < f jðc�i Þ < c1;

Sj ¼ C1 if f jðc�i Þ ¼ c1;

Sj ¼ M if c1 < f jðc�i Þ < c2;


Sj ¼ N if c2 < f jðc�i Þ < c3;


Sj ¼ R if c3 < f jðc�i Þ < c4; and

Sj ¼ C4 if f jðc�i Þ ¼ 1:

The dynamics of the interval is characterized by the symbolic sequences associated to the orbits of c�i ði ¼ 1;2;3Þ. We denoteby nMRðSÞ the number of symbols of S corresponding to subintervals where the map f is monotone decreasing. Furthermore,we define the parity of this sequence, qðSÞ ¼ ð�1ÞnMRðSÞ, according to whether nMRðSÞ is even or odd. Here, we use an orderrelation defined in R ¼ fC0; L;C1;M;C2;N;C3;R;C4g that depends on parity. Therefore, for two of such sequences, P and Qin R, let m be such that Pm–Q m and Pj ¼ Qj for j < m. If the parity of the block P1; . . . ; Pm�1 ¼ Q1; . . . ;Q m�1 is even (that is,qðP1; . . . ; Pm�1Þ ¼ þ1), we say that P < Q if Pm < Q m in the order

C0 < L < C1 < M < C2 < N < C3 < L < C4:

If the parity of the same block is odd (that is, qðP1; . . . ; Pm�1Þ ¼ �1), we say that P < Q if Pm > Qm:

If no such index m exists, then P ¼ Q . When Oðc�i Þ is a k-periodic orbit we obtain a sequence of symbols that can be char-acterized by a block of length k, SðkÞ ¼ S1; . . . ; Sk�1Ci with i ¼ 0;1;2;3;4.

Now, we consider the topological entropy. As we pointed out before, this important numerical invariant is related to theorbit growth and allows us to quantify the complexity of the phenomenon. It represents the exponential growth rate for thenumber of orbit segments distinguishable with arbitrarily fine but finite precision. In a sense, the topological entropy de-scribes in a suggestive way the total exponential complexity of the orbit structure with a single number.

A definition of chaos, in the context of one-dimensional dynamical systems, states that a system is called chaotic if itstopological entropy is positive. Therefore, the topological entropy can be computed to express whether a map has chaoticbehavior.

To each value of the parameters, the dynamics is characterized using the kneading data. This kneading data determines aMarkov partition of the interval, considering the orbits

Oðc�1 Þ ¼ fx�i gi¼1;2;...;p�1;

Oðc�2 Þ ¼ fy�i gi¼1;2;...;p�2;

Oðc�3 Þ ¼ fz�i gi¼1;2;...;p�3;

and ordering the elements x�i , y�i and z�i of these orbits. With this procedure we obtain the partition

fIk ¼ ½wk;wkþ1�gk¼1;2;...;p�1þp�2 þp�3;

of the interval I ¼ ½yþ1 ; xþ1 �, since the point yþ1 is the leftmost point possible and the point xþ1 is the rightmost point possible forall the iterations. The transitions between the subintervals are represented by a transition matrix Mðf Þ. The topological en-tropy of f, denoted by hðf Þ, can be given by

hðf Þ ¼ ln½kmaxðMðf ÞÞ�;

where kmaxðMðf ÞÞ is the spectral radius of the transition matrix Mðf Þ. This quantity, kmaxðMðf ÞÞ, is equal to the growth num-ber sðf Þ (the growth rate of the number of intervals on which f k is monotone) given by sðf Þ ¼ limk!1

ffiffiffiffiffiffiffiffiffiffi‘ðf kÞk

p, (see [23–25]).

In order to illustrate the previous considerations, we discuss the following example.

Example 1. Let us consider a map f with b1 ¼ b3 ¼ 1:37497, b2 ¼ �2:0 and b4 ¼ �1. The orbits of the points c�1 , c�2 and c�3define the following symbolic sequences

ðc�1 ; cþ1 ; c�2 ; cþ2 ; c�3 ; cþ3 Þ ! ðMC13 ; ðC4C1Þ1;C12 ;C10 ; ðL

4MRMNL7MRMRMC3Þ1;C13 Þ:

Putting the orbital points in order we obtain the Markov partition:

C0 < z�9 < z�10 < z�11 < z�12 < z�1 < z�13 < z�2 < z�14;

< z�3 < z�15 < z�4 < C1 < z�16 < z�18 < z�5 < z�20 ¼ x�1 ;

< z�7 < C2 < z�8 < C3 ¼ x�2 ¼ z�21 < z�6 < z�19 < z�17 < C4:

The corresponding transition matrix is obtained considering the images of these points. The characteristic polynomial is gi-ven by

1.1 1.2 1.3 1.4 1.5d

0.1

0.2

0.3

0.4

0.5

ha

b

Fig. 3. Variation of the topological entropy associated to different regimes: (a) fast regime where b2 ¼ �d with 1:01 < d < 1:5, b1 ¼ 1:05, b3 ¼ 1:5 andb4 ¼ �1; and (b) slow regime where b1 ¼ b3 ¼ d with 1:01 < d < 1:5, b2 ¼ �1:25 and b4 ¼ �1. Other parameter values are exactly the same as in Fig. 1.

Fig. 4.parame


pðkÞ ¼ detðMðf Þ � kIÞ ¼ ð�1þ kÞð1þ kÞð1� k4 � k5 � k6 � k7 � k8 � k9 � k10 � k11 � k12 � k13 � 2k14 � k21 þ k22Þ:

The growth number sðf Þ – the spectral radius of matrix Mðf Þ – is 1:40502; . . . ;. Therefore, the value of the topological entropycan be given by hðf Þ ¼ ln sðf Þ ¼ 0:340057; . . ..

Several situations of the variation of the topological entropy with the slopes that characterize the fast and slow regimes ofmap f are depicted in Fig. 3.

In all situations, the topological entropy hðf Þ is increasing with the value of d. This study reveals that the fast regime ischaracterized by higher values of the topological entropy when the slope b2 ¼ d is larger.

We are now in position to study the influence of coupling on chaotic maps modelling bursting neurons.

2.2. The two-cell model of coupled bursters

As pointed out by Cazelles et al. in [3], the reciprocal inhibitory coupling, e.g., each neuron inhibits the other throughchemical synapse, has interesting properties which are the result of the competition of two inhibitory coupled neurons.The model suggested by these authors for such coupling (between two neurons x and y) is written as a map, T, in the form:

ðxnþ1; ynþ1Þ ¼ Tðxn; ynÞ;

where

0 0.2 0.4 0.6 0.8 1xn

0

0.2

0.4

0.6

0.8

1

yn

Iterates of the coupled system (2) in the two-dimensional space for a ¼ 0:018 and g ¼ 0:01. In our study of coupling we consider the remainingter values as in Fig. 1.


xnþ1 ¼ f ðxnÞ þ aðx2n þ y2

nÞ � gxnyn ðaÞ;ynþ1 ¼ f ðynÞ þ aðx2

n þ y2nÞ � gxnyn ðbÞ;

ð2Þ

and f is the piecewise linear map introduced in the previous section. The value of a ð0 6 a < 1Þ controls the stability of themap under the action of the synaptic coupling and g is the coupling strength. Due to reasons previously explained, concern-ing the importance of studying a pair of coupled cells (see [19,20]), we have explored the behavior of two coupled identicalmaps with g 2 ½0;0:08�. In fact, the results for two-cell systems of identical bursters are significant in the study of coupling.For further informations concerning coupled systems consisting of two identical bursting cells, the reader is referred to thearticles [26,27].

Using a numerical approach, different structures emerge when the iterates ðxn; ynÞ are visualized in the two-dimensionalspace (see one example in Fig. 4). With the purpose of understanding important features of the dynamics, we construct themap of Fig. 5. This iterated map consists of pairs ðxn; xnþ1Þ, obtained from the successive first coordinates of the points definedby the system (2). We emphasize that this map is piecewise linear and due to its one-dimensionality, it plays an extremelyimportant role in our study. More specifically, these features of the map allow us to apply the procedure previously ex-plained for the computation of the topological entropy. It is also relevant to notice that the piecewise linear character is ob-tained for g 2 ½0;0:08�. This fact justifies the motivation of choosing these specific values of the coupling strength.

In Fig. 6 we show the burst pattern of xn under the action of coupling. The wave form for yn is similar. The active and silentphases seem to be sensitive to the application of the coupling strength. Notice that the clusters of spikes, in the wave form oftemporal behavior of xn, maintain the same shape compared with Fig. 2.

We are now in position to study the effect of reciprocal inhibitory coupling on two identical bursting cells using the pro-cedure established previously for the computation of the topological entropy. In Fig. 7 we show the variation of the topolog-ical entropy with the slopes associated to the fast and slow regimes.

0 0.2 0.4 0.6 0.8 1xn

0

0.2

0.4

0.6

0.8

1

xn 1

Fig. 5. Representation of the iterates of the map 2(a) in the coupled system for a ¼ 0:018 and g ¼ 0:01.

100 200 300 400n

0.2

0.6

1xn

Fig. 6. Wave form of temporal behavior of xn regarding the two-cell model (2), with a ¼ 0:018 and g ¼ 0:01.

1.1 1.2 1.3 1.4 1.5d

0.1

0.2

0.3

0.4

0.5

h

a

b

Fig. 7. Variation of the topological entropy for the coupled map associated to different regimes: (a) fast regime where b2 ¼ d with1:01 < d < 1:5; b1 ¼ 1:05; b3 ¼ 1:5 and b4 ¼ �1; and (b) slow regime where b1 ¼ b3 ¼ d with 1:01 < d < 1:5, b2 ¼ �1:25 and b4 ¼ �1. The values ofcoupling parameters are a ¼ 0:018 and g ¼ 0:01.


In a similar way to the case of uncoupled maps, the topological entropy is increasing with the value of the slope d. Nev-ertheless, there is an inversion on the features pointed out by the different regimes. With the introduction of the couplingstrength the slow regime is now characterized by higher values of the topological entropy when the slope d is larger.

Let us consider two identical maps, namely, the same as the map shown in Fig. 1. The effect of coupling strength on thispair of bursting maps is depicted in Fig. 8. Our numerical simulations reveal that, when we plot the topological entropy as afunction of the coupling strength, the coupled cell is associated with lower values of the topological entropy, which representa different dynamical behavior from the single cell. The introduction of the strength of reciprocal inhibitory coupling reducessignificantly the topological entropy. The coupling strength plays a crucial role in modifying the topological entropy (see[28]). Indeed, it is known from the analysis of the biological data, that synaptically coupled chaotic bursting neurons canbe more organized than individual cells.

The topological entropy allows us to identify different levels of the map complexity.

3. Chaos control of the two-cell model

In this section we apply the pole placement method, due to Romeiras et al. [4], to the discrete two-cell map (2)

Fig. 8.a ¼ 0:0

ðxnþ1; ynþ1Þ ¼ Tðxn; ynÞ;

with

xnþ1 ¼ f ðxnÞ þ aðx2n þ y2

nÞ � gxnyn;

ynþ1 ¼ f ðynÞ þ aðx2n þ y2

nÞ � gxnyn;

in order to stabilize a unstable periodic orbit embedded in the chaotic attractor of the coupled map. In their work, theseauthors emphasize the fact that a chaotic attractor typically has embedded densely within it an infinite number of unstableperiodic orbits. With small controlling perturbations to the system, the aim of this process is not to create new orbits with

0.01 0.03 0.05 0.07g

0.05

0.1

0.15

0.2

0.25

0.3

h

Topological entropy as a function of the coupling strength when two maps with the same topological entropy are coupled. The parameter values are18 and 0 < g < 0:08.


very different properties from the already existing ones, but to exploit the existing unstable periodic orbits in the absence ofcontrol. By applying a small adequate chosen perturbation to the dynamical system, the original chaotic trajectory can beconverted into the desired stable periodic orbit. Therefore, through the intervention of the control technique, it is possibleto regularize the timing and the wave form of bursts.

Since the control is usually designed for parameter values where the system exhibits chaotic motion we fix, for illustrativepurposes, the parameter values a ¼ 0:006 and g ¼ 0:03; with the remaining values as in Fig. 1. The parameter g is consideredthe control parameter which is available to vary in some small interval j g � g0 j hd; di0, around the nominal value g0 ¼ 0:03,for which the map has a chaotic attractor (see Fig. 9). In the next lines, we illustrate a numerical exploration with the purposeto stabilize a unstable period-two orbit. Concerning the role and importance of periodic orbits on neuronal dynamics theauthor is referred to [29].

3.1. The application of the pole placement method

The pole placement technique (see [4,30]), which is a feedback control method, extends the OGY method, allowing for amore general choice of the so called feedback matrix. In our attractor (Fig. 9) the unstable period-two orbit to be stabilized isdefined by two fixed points located approximately at

Fig. 9

ðxð1Þ; yð1ÞÞ ¼ ð0:407938; 0:987211Þ;ðxð2Þ; yð2ÞÞ ¼ ð0:987203; 0:407969Þ:

These two points verify

Tðxð1Þ; yð1ÞÞ ¼ ðxð2Þ; yð2ÞÞ;Tðxð2Þ; yð2ÞÞ ¼ ðxð1Þ; yð1ÞÞ:

The control strategy consists in finding two stabilizing local feedback control laws, which are linear maps, obtained by usingleast squares fitting on the sampled data in a small neighborhood of each one of the fixed points ðxð1Þ; yð1ÞÞ and ðxð2Þ; yð2ÞÞ. Forclarity reasons, we explain the construction of a control linear law associated to one of the fixed points, for instance ðxð1Þ; yð1ÞÞ.

A stabilizing local feedback control map in a small neighborhood of the fixed point ðxð1Þ; yð1ÞÞ is given by

xtþ1 � xð1Þ

ytþ1 � yð1Þ

" #¼ A1

xt � xð1Þ

yt � yð1Þ

" #þ B1ðq� q0Þ; ð3Þ

where

A1 ¼0:0815727 0:244921�0:370971 0:413833

� �; B1 ¼

0:1328730:730003

� �; ð4Þ

0 0.2 0.4 0.6 0.8 1xn

0

0.2

0.4

0.6

0.8

1

yn

. Iterates of the coupled system (2) in the two-dimensional space for a ¼ 0:006 and g ¼ 0:03. The remaining parameter values are as in Fig. 1.


and ðq� q0Þ corresponds to a parameter available for small perturbations applied to the control law (3). The ergodic natureof the chaotic dynamics ensures that the state trajectory eventually enters into the neighborhood of the fixed point. Onceinside, we apply the stabilizing feedback control law in order to steer the trajectory towards the desired orbit.

Now, we verify whether the system is controllable. A system is called controllable if a matrix K1�n can be found such that½A� BK� has any desired eigenvalues. This is possible if rankðCÞ ¼ n, where n is the dimension of the state space and C is theðn� nÞ matrix

C ¼ ½B...AB..

.A2B ..

.. . . ..

.An�1B�:

In our case it follows that

C1 ¼ ½B1...A1B1� ¼

0:132873 ...

0:189632...

0:730003 ...

0:252808

26664

37775; ð5Þ

which has rank 2, and so the system is controllable. This matrix C1 is the controllability matrix.Assume, in a small neighborhood around the fixed point ðxð1Þ; yð1ÞÞ, that

q� q0 ¼ �Kxt � xð1Þ

yt � yð1Þ

" #; ð6Þ

where K ¼ ½ k1 k2 � is a constant vector to be determined. The linearized map becomes

xtþ1 � xð1Þ

ytþ1 � yð1Þ

" #¼ ½A1 � B1K� xt � xð1Þ

yt � yð1Þ

" #; ð7Þ

with ½A1 � B1K� given by

0:081572� 0:132873k1 0:244921� 0:132873k2

�0:370971� 0:730003k1 0:413833� 0:730003k2

� �;

which shows that the fixed point is then stable as long as the ð2� 2Þ�matrix ½A1 � B1K� is asymptotically stable, that is, all itseigenvalues have modulus less than unity.

The determination of K, such that the eigenvalues of the matrix ½A1 � B1K� have specified values is called, in the theory ofcontrol systems, pole placement technique. The eigenvalues k1 and k2 of the matrix ½A1 � B1K� are called the regulator poles,and the problem of placing these poles at the desired location, by choosing K with A1 and B1 given, is the pole placementproblem.

In our particular case, the characteristic polynomial, associated to the matrix ½A1 � B1K�; is given by

pðkÞ ¼ k2 þ ð�0:495406þ 0:132873k1 þ 0:730003k2Þkþ ð0:124616þ 0:123806k1 � 0:10884k2Þ:

Since the eigenvalues verify the equations

k1k2 ¼ 0:124616þ 0:123806k1 � 0:10884k2; andk1 þ k2 ¼ �ð�0:495406þ 0:132873k1 þ 0:730003k2Þ;

the lines of marginal stability can be determined by solving the equations

-7.5 -5 -2.5 0 2.5 5 7.5 10k1

-3

-2

-1

0

1

2

3

k2 1

Fig. 10. The bounded region X1; related to fixed point ðxð1Þ; yð1ÞÞ; that corresponds to stable regulator poles.

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

n

xn

Fig. 11. Time series data for variable xn . The control is activated in a neighborhood of the fixed point ðxð1Þ; yð1ÞÞ, after the 295th iterate.


k1 ¼ �1 and k1k2 ¼ 1:

These conditions guarantee that the eigenvalues k1 and k2 have modulus less than unity for k1 and k2 within a certain region.This region is define by the three lines of marginal stability:

k2 ¼ �8:04282þ 1:13750k1; k2 ¼ �1:01295� 0:41322k1; and k2 ¼ 1:93126� 0:01081k1:

We obtain stable eigenvalues considering k1 and k2 within the triangular region depicted in Fig. 10. Selecting, for example,k1 ¼ 3:0 and k2 ¼ 0:0 well inside the triangular region, X1 and applying the control linear law (3) we obtain the desired timeperiodic orbit (see Figs. 11 and 12).

At this stage it should be pointed out that, depending on the values of k1 and k2 in the basin of attraction X1, the controlledorbit will converge towards the fixed point but takes different periods of time in order to fully accomplish the convergenceprocess. The chaotic trajectory will also converge to the desired fixed point if, in contrast, we consider fixed values of k1 andk2 and randomly choose some initial conditions inside the neighborhood of ðxð1Þ; yð1ÞÞ.

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

n

yn

Fig. 12. Time series data for variable yn . The control is activated in a neighborhood of the fixed point ðxð1Þ; yð1ÞÞ; after the 295th iterate.

-4 -2 0 2 4 6k1

-4

-2

0

2

4

6

k2

2

Fig. 13. The bounded region X2; related to fixed point ðxð2Þ; yð2ÞÞ; that corresponds to stable regulator poles.


Theoretically, after switching on the control, the orbit continues to perform chaotic behavior for some time, unchangedfrom the uncontrolled case, because it is no close enough to the fixed point. After some steps this is eliminated and the orbitis rapidly brought to the fixed point.

With the same procedure used to establish the first control map, we obtain the following stabilizing local feedback lawassociated to the fixed point ðxð2Þ; yð2ÞÞ

xtþ1 � xð2Þ

ytþ1 � yð2Þ

" #¼ ½A2 � B2K� xt � xð2Þ

yt � yð2Þ

" #; ð8Þ

with ½A2 � B2K� given by

0:0578313� 0:87604k1 0:132535� 0:876041k2

�0:186186� 0:562917k1 0:0707288� 0:562917k2

� �;

considering k1 and k2 within the triangular region X2 depicted in Fig. 13.

0 50 100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0

n

xn

Fig. 14. Time series data for variable xn . The period-two orbit is obtained after the 245th iterate.

0 50 100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0

n

yn

Fig. 15. Time series data for variable yn . The period-two orbit is obtained after the 245th iterate.

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

n

xn

Fig. 16. Time series data for variable xn without and with control. The control is switched off after the 350th iterate.

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

n

yn

Fig. 17. Time series data for variable yn without and with control. The control is switched off after the 350th iterate.


Gathering the previous information, we are able to convert efficiently the irregular dynamics arising from the chaotic cou-pled map to the desired period-two orbit, which is a regular spike train, by using alternately, for each iterate, Eqs. (7) and (8)(see Figs. 14 and 15). Turning off the control at any stage will result in chaotic behavior which can be controlled onto anotherperiodic motion (see Figs. 16 and 17). Especially, important is the fact that the control can be carried out without any needfor a change to the system configuration.

The pole placement control strategy works very well for the chaotic coupled model. In fact, our numerical simulationsrevealed a fast convergence for different initial conditions in the neighborhood of the two fixed points and different valuesof k1 and k2.

4. Final considerations

In this note we have studied a class of one-dimensional phenomenological maps, which are constructed to examine thebehavior of the single cell and the effect of reciprocal inhibitory coupling on bursting neurons. This model addresses inter-esting and important mathematical and physical questions. In particular, there are no comprehensive studies about a collec-tion of questions pertaining to chaotic behavior in terms of symbolic dynamics and measurements of complexity (seequestions left open in [3]) and in terms of chaos control theory. This work has constituted an opportunity to make a contri-bution for the careful study of the Cazelles–Courbage–Rabinovich maps in terms of symbolic dynamics theory of the singlecell model and of important features of the system representing the dynamics of bursting neural assemblies connected to-gether via reciprocal inhibitory coupling.

In the field of life sciences, where quantitatively predictive theories are rare, the use of powerful tools for the analysis ofdynamic models, such as the symbolic dynamics theory, stands out to be extremely effective for the computation of animportant numerical invariant related to the exponential orbit growth – the topological entropy.

As established in [3], the map replicates correctly the main features of the cooperative behavior of coupled bursters ob-served in biological experiments. A rigorous characterization of the complexity of the coupled system consisting of two iden-tical bursting maps became possible by the study of the variation of the topological entropy. Our numerical simulationsrevealed that the topological entropy is sensitive to the application of the coupling strength g. Indeed, coupled cells exhibitsignificant lower values of the topological entropy. An important issue in the study of coupled cells is to understand how it ispossible that the potentially very complex behavior which might transpire when chaotic neurons are coupled, can lead in adynamical way to rather simpler, often well organized motion. In fact, the model exhibits positive topological entropy, whichmeans that in certain conditions the bursting behavior of neurons has a chaotic nature.

With the evidence of the existence of chaos, it is interesting to think about its role in neurophysiology, in particular inneural systems. The investigations pointed out in previous works (for instance, [31]) suggest that nature uses complexdynamics of neural assemblies in promoting principles of adaptability and reliability as well as in providing rapid responseto changing external stimuli for information processing and response. These arguments state the physiological relevance ofthe quantification of chaos in neural models.

Although chaos is unpredictable over long time periods, its deterministic nature often can be exploited by control tech-niques to obtain desired results. Chaos control techniques, which are ‘‘ model-independent” because they do not requireknowledge of a system’s underlying equations, have been applied successfully to a wide range of physical systems. Such suc-cess has fostered interest in applying model-independent control techniques to stabilize the fluctuations of excitable phys-iological systems, which are often well-understood qualitatively, but for which quantitative relationships between systemcomponents are usually incomplete. Motivated by the chaotic structure of the model, we have applied the pole placementcontrol method (which is a model-independent feedback control) in order to obtain predictable behavior – the stabilizedperiod-two spike train. We showed, numerically, that the complicated motion that emerges from the dynamics of the cou-pled model can be controlled by small parameter perturbations in a control linear law. The perturbation performed by the


control procedure has no residual effects, i.e., the state point behaves as if it arrived on the stable orbit naturally. We empha-size that, with the application of the chaos control technique, the model exhibits fast convergence for different initial con-ditions and different values of the control parameters. The chaotic dynamics could be converted, by using just a smallfeedback control, to motion on a desired periodic orbit.

Acknowledgement

The authors would like to thank Professor B. Cazelles for his enlightenments and valuable information. The first authoracknowledges the Centro de Investigação em Matemática e Aplicações (CIMA-UE) for financial support. The third authorwas partially supported by the Fundação para a Ciência e a Tecnologia (FCT) through the Program POCI 2010/FEDER.

References

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