The influence of coupling on chaotic maps modelling bursting cells

Chaos, Solitons and Fractals 28 (2006) 1314–1326

www.elsevier.com/locate/chaos

The influence of coupling on chaotic maps modellingbursting cells

Jorge Duarte a,1, Luıs Silva b,2, J. Sousa Ramos c,*,3

a Departamento de Eng. Quımica, Seccao de Matematica, Instituto Superior de Engenharia de Lisboa,

Rua Conselheiro Emıdio Navarro 1, 1949-014 Lisboa, Portugalb Departamento de Matematica, Universidade de Evora, Rua Romao Ramalho, 59, 7000-671 Evora, Portugalc Departamento de Matematica, Instituto Superior Tecnico, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal

Accepted 8 August 2005

Abstract

Bursting behavior is ubiquitous in physical and biological systems, specially in neural cells where it plays an impor-tant role in information processing. This activity refers to a complex oscillation characterized by a slow alternationbetween spiking behavior and quiescence. In this paper, the interesting phenomena which transpire when two cellsare coupled together, is studied in terms of symbolic dynamics. More specifically, we characterize the topologicalentropy of a map used to examine the role of coupling on identical bursters. The strength of coupling leads to the intro-duction of a second topological invariant that allows us to distinguish isentropic dynamics. We illustrate the significanteffect of the strength parameter on the topological invariants with several numerical results.� 2005 Elsevier Ltd. All rights reserved.

1. Motivation and preliminaries

Bursting oscillations have received a lot of attention in recent years, in particular in the context of physiology. Thiscomplex behavior is seen to be the primary mode of behavior of a wide variety of excitable cells.

The chaotic activity of bursting cells has provided challenging mathematical investigations on several levels, includ-ing the development of detailed biophysical models that describe the high dimensional dynamics of nonlinear eventsresponsible for variations in the ionic currents across the membrane. The characterization of such activity is usuallybased on either realistic ionic-based models or phenomenological models. The ionic-based models proposed for a singlecell are designed to replicate the physiological mechanisms of the membrane, with the parameters and functions derivedfrom experimental data. Some of these models consist of a system of many nonlinear differential equations. The highdimensionality of the phase space is a significant obstacle in understanding the collective behavior of such dynamical

0960-0779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2005.08.188

* Corresponding author.E-mail addresses: [email protected] (J. Duarte), [email protected] (L. Silva), [email protected] (J. Sousa Ramos).

1 Partially supported by Instituto Superior de Engenharia de Lisboa.2 Partially supported by Universidade de Evora and FCT/POCTI/FEDER.3 Partially supported by FCT/POCTI/FEDER.

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J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326 1315

systems [9]. The phenomenological models are constructed to retain the important qualitative features with minimalcomplexity of the equations [8]. A special type of phenomenological models is based on low-dimensional maps. Therehave been proposed only few explicit maps capable of generating essential aspects of bursting dynamics (for instance,see [19,20,3] and [10]). The models are designed with the aim of gaining a deeper understanding of the mathematicalstructure underlying the oscillations. In this work, the focus will be on providing a study of the role of coupling onbursting cells. Neurons and endocrine cells rarely act alone, but rather as members of a population connected togethervia gap-junctional or synaptic coupling. The electrical activity observed in the population is the result of the intrinsicproperties of individual cells as well as of the nature of coupling.

The cooperative behavior of coupled cells is somewhat unexpected and can be much more organized than the activ-ity of the individual neurons. The isolated neural cells often exhibit chaotic motions, as observed in the characteristics ofintracellular voltage measurements.

Our goal is to provide a contribution for the detailed analysis of a family of maps, introduced in [19] and studied insubsequent papers (for instance, see [4] and [6]), which produces chaotic bursting patterns similar to those observed inneurons and endocrine cells. More precisely, using techniques of symbolic dynamics [7], we compute the topologicalentropy and a second invariant, denoted by r, in order to elucidate the effect of mean field coupling on identical burst-ers. The topological invariant r allows us to distinguish different systems with equal topological entropy. The two topo-logical invariants are quantitative measures of different states of complexity that arise through the coupling. Attentionwill be focussed on two-cell systems. As pointed out in [5] and [2], numerical simulations have demonstrated that theanalytical results for two-cell systems carry over to many-cell systems. We study a system of two identical cells and showthe influence of the coupling strength on the variation of the topological invariants.

In order to facilitate the study and make this note self-contained, we describe briefly some aspects of the discrete-time model replicating chaotic bursting (for further informations see [4] and [19]).

A group of irregularly bursting cells with different individual properties can be modeled using two-dimensional maps(for each cell i = 1,2, . . . ,N) of the form

Fig. 1

xnþ1;i ¼ ai1þx2n;i

þ yn;i þ �N

PNj¼1

xn;j;

ynþ1;i ¼ yn;i � rixn;i � bi;

8><>: ð1Þ

where xn,i and yn,i are, respectively, the fast and slow dynamical variables of the ith cell, the parameter � is the strengthof global coupling, and N is the total number of cells. The x-variable replicates the dynamics of the membrane potentialand the y is the recovery variable. The slow evolution of yn,i is a result of the small values of the positive parameters biand ri, which are on the order of 0.001. In other words, the time course of yn,i is much slower than that of xn,i. Thevalues of the parameter ai are selected on the interval [1.5,8.0]. We note that the considered mechanism of burstingis similar to the oscillations in the well-known Hindmarsh–Rose model of biological neuron, where the role of param-eter a is played by a hyperpolarization current I [8]. The cells are coupled to each other through the mean field.

Depending on the value of parameter a, each single cell (that is, when � = 0 in (1)) demonstrates two qualitativelydifferent regimes of behavior, namely continuous oscillations (spiking) and bursts (square-wave bursting). The model (1)contains a mix of slow and fast dynamics to describe the bursting and spiking behavior of observations in neural sys-tems. A typical regime of temporal behavior of the fast variable x for the full two-dimensional map (1) for a single cell(� = 0) is shown in Fig. 1.

As pointed out in [19], since yn,i changes slowly, the time evolution of xn,i can be considered independently of mapyn+1,i = yn,i � ri xn,i � bi, assuming that yn,i is a control parameter c = yn,i. Thus, important insights about the fastdynamics of each coupled cell can be obtained from the analysis of the three-parameters family of maps

. Wave forms of temporal behavior of individual cells, regarding the full system (1) with a = 4.1, r = b = 0.001, and � = 0.

Fig. 2. Wave forms of temporal behavior of coupled identical cells, regarding the full system (1) with a = 4.1, r = b = 0.001, and� = 0.2.

1316 J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326

xnþ1;i ¼ F a;c;�ðxn;iÞ ¼ai

1þ x2n;iþ ci þ

�

N

XNj¼1

xn;j. ð2Þ

This approach was pioneered by Rinzel [16] in a study of continuous bursting models, and it is extensively used onthe analysis of single and coupled systems.

Coupling between cells influences the fast dynamics of each cell by adding the value �N

PNj¼1xn;j to the parameter ci. As

mentioned above, we will concentrate on the study of the behavior of two identical bursting cells (that is, N = 2, a1 = a2,and c1 = c2), when they are coupled via the mean field. The results for two-cell systems of identical bursters are signif-icant in the study of coupling. For further informations concerning coupled systems consisting of two identical burstingcells, the reader is referred to the papers [17] and [18].

The solution behavior of the full system (1) for two identical cells, namely those of Fig. 1 with � = 0.2, is shown inFig. 2.

The wave form for xn,2 is similar to the one shown for xn,1 and bursts (clusters of spikes) are synchronized. The activeand silent phases are considerably longer when the cells are coupled. Of less significance, but still noticeable, is theobservation that the amplitude of the burst oscillation has increased.

As pointed out in [19] and [4], yn,1 � yn,2 and we are justified in studying the fast subsystem

xnþ1;1 ¼ a1þx2

n;1þ cþ �

2ðxn;1 þ xn;2Þ;

xnþ1;2 ¼ a1þx2n;2

þ cþ �2ðxn;1 þ xn;2Þ;

8<: ð3Þ

where a = a1 = a2. When both cells start with initial conditions that satisfy jx0,1j = jx0,2j, the evolution of x in each cellcan be described by the family of maps

xnþ1 ¼ Ga;c;�ðxnÞ ¼a

1þ x2nþ cþ �xn. ð4Þ

We are now in a position to study the effect of coupling on two identical bursting cells using techniques of symbolicdynamics theory.

2. Topological invariants of coupled identical bursters. Isentropic dynamics

Let us consider the interesting region of the parameter space

X� ¼ fða; cÞ 2 R2 : �4:4 < c < 0:0 and 1:5 < a < 8:0g.

When � = 0 in (4) we obtain the function

Ga;c;0ðxnÞ ¼a

1þ x2nþ c; ð5Þ

which has the shape of an unimodal map (continuous map on the interval with two monotonic subintervals and oneturning point (relative maximum)). There have been used techniques of symbolic dynamics to study this map [6]. How-ever, with the introduction of the coupling strength �, there is a subregion of X�, denoted by X�

� , where Ga,c,� has theshape of a bimodal map (continuous map on the interval with three monotonic subintervals and two turning pointsc1 and c2 (c1 the relative maximum and c2 the relative minimum)). In our study, we consider


X�� ¼ fðc; aÞ 2 X� : Ga;c;�ðc1Þ > c2 and Ga;c;�ðGa;c;�ðc1ÞÞ < Ga;c;�ðc1Þ and Ga;c;�ðc2Þ < c1 and Ga;c;�ðGa;c;�ðc2ÞÞ> Ga;c;�ðc2Þg.

The regions X� and X�� are presented in Fig. 3 and a typical map of the family Ga,c,�, with ðc; aÞ 2 X�

� , is depicted inFig. 4. The values of the coupling strength are selected on the interval [0.0,0.45].

At this point, using some results concerning to Markov partitions associated to bimodal maps we characterize thetopological entropy of Ga,c,�(xn), and we show situations of the variation of this numerical invariant with the parametersa and c for different values of the coupling strength.

A bimodal map f on the interval I = [c0,c3] is piecewise monotone and I is subdivided into three subintervals:

L ¼ ½c0; c1½; M ¼�c1; c2½; R ¼�c2; c3�

in such a way that the restriction of f to each interval L or R is strictly increasing and in the other interval M is strictlydecreasing. Each such maximal intervals on which the function f is monotone is called a lap of f, and the number ‘ = ‘(f)of distinct laps is called the lap number of f.

Denoting by c1 and c2 the two turning points (relative extrema) of f, we obtain the orbits

Oðc1Þ ¼ fxi : xi ¼ f iðc1Þ; i 2 Ng and Oðc2Þ ¼ fyi : yi ¼ f iðc2Þ; i 2 Ng.

With the aim of studying the topological properties of these orbits we associate to each orbit O(ci) a sequence of sym-bols S = S1S2 . . .Sj . . . where Sj = L if fj(ci) < c1, Sj = A if fj(ci) = c1, Sj = M if c1 < fj(ci) < c2, Sj = B if fj(ci) = c2 andSj = R if fj(ci) > c2. The points c1 and c2 play an important role. The dynamics of the interval is characterized by thesymbolic sequences associated to the orbits of points c1 and c2. We denote by nM(S) the frequency of the symbol Min S and we define the M-parity of this sequence, qðSÞ ¼ ð�1ÞnM ðSÞ, according to whether nM(S) is even or odd. Thus,in the first case we have q(S) = +1 and in the second q(S) = �1. In our study we use an order relation defined inR ¼ fL;A;M ;B;RgN that depends onM-parity. Thus, for two of such sequences, P and Q in R, let i be such that Pi 5 Qi

Fig. 3. Representation of the regions X� and X�� , with � = 0.2.

Fig. 4. Map Ga,c,�(xn) for a = 7.981, c = �3.206 and � = 0.2. The turning points are c1 = 0.0125337. . . and c2 = 4.14618. . ..


and Pj = Qj for j < i. If the M-parity of the block P1 . . .Pi�1 = Q1 . . .Qi�1 is even (that is, q(P1 . . .Pi�1) = +1), we saythat P < Q if Pi < Qi in the order L < A < M < B < R. If the M-parity of the same block is odd (that is,q(P1 . . .Pi�1) = �1), we say that P < Q if Pi < Qi in the order R < B < M < A < L. If no such index i exists, thenP = Q. If a finite symbolic sequence S has n symbols, it is usual to write jSj = n. When O(ci) is a k-periodic orbit weobtain a sequence of symbols that can be characterized by a block of length k,S(k) = S1 . . .Sk�1Ci, with i = 1,2. In whatfollows, we restrict our study to the case where the two critical points are periodic (respectively, eventually periodic),O(c1) is p-periodic and O(c2) is q-periodic (respectively, fp(c1) = c2 or fq(c2) = c1). Note that O(c1) is realizable if theblock P = P1 . . .Pp�1A is maximal, that is, ri(P) 6 P, where 1 6 i 6 p and r(PiPi+1Pi+2 . . .) = Pi+1Pi+2 . . . is the usualshift operator. On the other hand, O(c2) is realizable if the block Q = Q1 . . .Qq�1B is minimal, that is, rj(Q)P Q,where 1 6 j 6 q. Finally, note that the pair of sequences that are realizable satisfies the following conditionsri(P) P Q, 1 6 i 6 p and rj(Q) 6 P, 1 6 j 6 q. The set of such pair of sequences is denoted by R(A,B). Wedesignate by kneading data the pairs (P(p),Q(q)) 2 R(A,B), where P

(p) = P1 . . .Pp�1A, Q(q) = Q1 . . .Qq�1B, the bistable se-

quence P1 . . .Pp�1BQ1 . . .Qq�1A, and the eventually periodic sequence P1 . . .Pp�1BQ1 . . .Qq�1B or Q1 . . .Qq�1AP1 . . .Pp�1A.

Now we consider the topological entropy. This numerical invariant measures the quantitative amount of chaos. Apossible definition of chaos in the context of one-dimensional dynamical systems state that a system is called chaotic ifits topological entropy is positive. Thus, the topological entropy can be computed to express whether a map has chaoticbehavior.

Let Ga,c,� be the 3-parameters family of maps such that (c,a) 2 X�. To each values of the parameters, the dynamics ischaracterized using the kneading data. This kneading data determines a Markov partition of the interval, consideringthe orbits O(c1) = {xi}i=1,2, . . ., p and O(c2) = {yi}i=1,2, . . ., q, and ordering the elements xi, yi of these orbits. With this pro-cedure we obtain the partition {Ik = [zk,zk+1]}k=1,2, . . ., p+q of the interval I = [y1,x1]. The transitions between the subin-tervals are represented by a matrix MðGa;c;�Þ. The topological entropy of Ga,c,�, denoted by htop(Ga,c,�), can be given by

htopðGa;c;�Þ ¼ ln kmaxðMðGa;c;�ÞÞ ¼ ln sðGa;c;�Þ;

where kmaxðMðGa;c;�ÞÞ is the spectral radius of the transition matrix MðGa;c;�Þ and s(Ga,c,�) is the growth rate

sðGa;c;�Þ ¼ limk!1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi‘ðGk

a;c;�Þkq

of the number of intervals on which Gka;c;� (kth-iterate of Ga,c,�) is monotone. We have sðGa;c;�Þ ¼ kmaxðMðGa;c;�ÞÞ (see

[11,13,14]).To illustrate the previous considerations, we discuss the following example.

Example 1. Let us consider the map of Fig. 4. The orbits of the turning points define the pair of sequences(RLLLLLLA,LLLLLLLA). Putting the points of the orbits in order we obtain:

y1 < x2 ¼ y2 < x3 ¼ y3 < x4 ¼ y4 < x5 ¼ y5 < x6 ¼ y6 < x7 ¼ y7 < c1 ¼ x8 ¼ y8 < c2 < x1.


The corresponding transition matrix is

MðGa;c;�Þ ¼

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 1

1 1 1 1 1 1 1 1 1

1 0 0 0 0 0 0 0 0

266666666666666664

377777777777777775

;

which has the characteristic polynomial

pðkÞ ¼ detðMðGa;c;�Þ � kIÞ ¼ kð2� kÞð1þ kÞð1þ k2Þð1þ k4Þ.

The growth number s(Ga,c,�) (the spectral radius of matrix MðGa;c;�Þ) is 2. Therefore, the value of the topologicalentropy can be given by

htopðGa;c;�Þ ¼ ln sðGa;c;�Þ ¼ 0:693147 . . . .

To see the long term behavior for different values of the parameters a and c, we plot, in Figs. 5 and 6 bifurcationdiagrams for � = 0.2.

These bifurcation diagrams suggest the existence of an inversion in the usual chaos ordering (for instance, note theinverted period-doubling bifurcations). Several situations of the variation of the topological entropy with each of the

Fig. 5. Bifurcation diagram for xn as a function of a, with a 2 [2.0,8.0], c = �1.85, and � = 0.2.

Fig. 6. Bifurcation diagram for xn as a function of c, with a = 3.8, c 2 [�3.6,�1.0], and � = 0.2.


parameters a and c for different values of � are depicted in Figs. 7 and 8. In all situations, the topological entropyhtop(Ga,c,�) has an absolute maximum value.

With the last numerical results it becomes apparent that coupling strength plays a crucial role in modifying thetopological entropy of the family of maps (4). This study reveals that the maximum value of the entropy decrease whencoupling strength is larger.

Let us consider two single cells with the same topological entropy,

Fig. 7� = 0.2

Fig. 8� = 0.2

htopðGa� ;c� ;�Þ ¼ ln sðGa� ;c� ;�Þ ¼ ln1þ

ffiffiffi5

p

2

!¼ 0:481212 . . . ;

with a* and c* fixed values of the parameters. The effect of the coupling strength on this pair of bursting cells is shownin Figs. 9 and 10. The coupling can convert chaotic cells to non-chaotic cells.

. Variation of the topological entropy for a 2 [2.6,6.5], c = �1.85, and different values of the parameter �: (a) � = 0.25, (b), (c) � = 0.15, (d) � = 0.

. Variation of the topological entropy for a = 3.8, c 2 [�3.4,�1.5], and different values of the parameter �: (a) � = 0.25, (b), (c) � = 0.15, (d) � = 0.

Fig. 9. Bifurcation diagram for xn as a function of �, with a = 3.55 and c = �1.85.

Fig. 10. Topological entropy as a function of the coupling strength when two cells with the same topological entropy are coupled. Thetopological entropy of the single cells is given by htopðGa� ;c� ;�Þ ¼ lnð1þ

ffiffi5

p

2Þ ¼ 0:481212 . . ., with a* = 3.55 and c* = �1.85.


Indeed, it can be readily verified by numerical simulations that the topological entropy is not very robust to a changein the coupling strength. In this regard, Figs. 11–13 show pertinent features of some isentropic curves (the levels of topo-logical entropy) in region X� for small periods n (n 6 5), which can arise through the coupling. The topological entropyremains constant over each curve. We remind that when we have the symbolic sequence RL1 of the turning point (forthe unimodal map), the dynamics of the iterates is a full shift of two symbols and the topological entropy is one (in thesubset A [ CRL1 of X�, see Fig. 11).

We note the role of coupling in enlarging the non-chaotic region of the parameter space.Situations of isentropic dynamics, in the study of a dynamical system, can raise interesting questions. To illustrate

this idea, we are going to study a topological entropy level set. More specifically, we will consider a subset of the param-eter space X�, denoted by K2, for which the corresponding maps of the family Ga,c,� have growth number 2, i.e., K2 is thetopological entropy level set for htop(Ga,c,�) = ln2.

Now consider a bimodal map Ga,c,� with kneading data (P,Q) such that B � P and Q � A. Then, as pointed out in[12], the following statements are equivalent:

(i) c; aÞ 2 K2,(ii) G2

a;c;�ðc1Þ ¼ G2a;c;�ðc2Þ,

(iii) rðP Þ ¼ rðQÞ.

Fig. 11. Curves in the parameter space corresponding to periodic orbits of the turning point (periods n 6 5) for � = 0. The labels arethe respective periods.

Fig. 12. Curves in the parameter space corresponding to periodic orbits of the turning point (periods n 6 5) for � = 0.25. The periodsfollow the ordering showed in Fig. 11.


Thus, the maps of the family Ga,c,� satisfying the relation G2a;c;�ðc1Þ ¼ G2

a;c;�ðc2Þ have topological entropy ln2. The curveK2 is shown in Fig. 14.

At this point of our study, we emphasize that for (c,a) 2 K2 the maps Ga,c,� have chaotic behavior and the topolog-ical entropy has exactly the same value. One question appears naturally: how can we distinguish these isentropic maps?It is our purpose to address a contribution to the answer to this question.

Fig. 13. Curves in the parameter space corresponding to periodic orbits of the turning point (periods n 6 5) for � = 0.4. The periodsfollow the ordering showed in Fig. 11.

Fig. 14. The isentropic level set K2 (dark) in parameter space.


The topological entropy by itself is no longer sufficient to classify the maps introduced. We need to consider a secondtopological invariant in order to distinguish the maps with the same entropy.

The study of topological classification for bimodal maps f leads to the introduction of two topological invariants:one of them is the well known growth number sðf Þ ¼ ehtopðf Þ and the other numerical quantity, denoted by r, is asso-ciated to the relative positions of the turning points of the map. The topological invariant r is introduced using thehypothesis s(f) > 1 and the Milnor–Thurston map k that topologically semi-conjugate f to a piecewise linear mapFe,s having slope ±s(f) everywhere (see [1,13,15]). There exists one and only one map

F e;s : ½0; 1� ! ½0; 1� so that F e;sðkðxÞÞ ¼ kðf ðxÞÞ

for every x 2 I = [0,1]. The map Fe,s is piecewise linear with slope ±s everywhere and is defined by

F e;sðyÞ ¼sy if 0 6 y < kðc1Þ�sy þ e if kðc1Þ 6 y < kðc2Þsy þ 1� s if y P kðc2Þ;

8><>:


where k(c1) = e/(2s), k(c2) = (e + s � 1)/(2s) and e = r + (s + 1)/2, that is, r = e � (s + 1)/2. Then, to each bimodal mapf, characterized by a kneading data (P,Q), we can associate two topological invariants. One of them is the growth num-ber s(f), as we saw, and the other is the invariant r(f).

The definition of e can be seen in [1] and [15]. However, in the study of the topological entropy level set forh(Ga,c,�) = ln2 there are practical formulas to compute e given a kneading sequence (see [12]). More precisely, givena map f satisfying s(f) = 2, with finite kneading data (S,T) such that jSj = n + 1, let

fðS; T Þ ¼ qðSÞ 2n�1 þXki

ð�1Þi2n�ki

!;

with 1 6 ki 6 n the integers such that Ski ¼ M (when there is no ki such that Ski ¼ M , f(S,T) = 2n�1q(S)). We define also

nðS; T Þ ¼Xni

qðrniðSÞÞ2n�ni ;

with 1 6 ni 6 n the integers such that Sni ¼ R. According to [12], the value of e(f) is given by

eðf Þ ¼ 4nðS; T Þ4fðS; T Þ � 1

if S ¼ S1 . . . SnA

and

eðf Þ ¼ 4nðS; T Þ þ 1

4fðS; T Þ � 1if S ¼ S1 . . . SnB.

Now regarding the previous considerations, the maps of the family Ga,c,� can be topologically classified by the pair oftopological invariants (s, r). We discuss the following example which illustrate well the nature of our work.

Example 2. Let us consider the kneading data (RLLLLLLA,LLLLLLLA) associated to the map of Fig. 4. We showedpreviously that s = 2 and we have jSj = n + 1 = 8. The kneading data determines f(S,T) = 26 and n(S,T) = 26.Therefore,

e ¼ 4nðS; T Þ4fðS; T Þ � 1

¼ 4� 26

4� 26 � 1

and

r ¼ e� 3

2¼ �0:4960784314 . . .

In this case, Ga,c,� is characterized by

s ¼ 2 and r ¼ �0:4960784314 . . .

We present in Figs. 15 and 16 some numerical results of the variation of the topological invariant r with each of theparameters a and c, for (c,a) 2 K2.

With the invariant r it is possible to distinguish the isentropic maps.

Fig. 15. Variation of the topological invariant r with a, for (c,a) 2 K2.

Fig. 16. Variation of the topological invariant r with c, for (c,a) 2 K2.


3. Final considerations

In this paper we have provided a contribution for the detailed analysis of a family of maps, which is used to examinethe influence of mean field coupling on bursting cells.

A rigorous characterization of the complexity of a coupled system consisting of two identical bursting cells becamepossible using techniques of symbolic dynamics. We studied the topological entropy and we introduced the parameterspace ordering of the dynamics that arose through the coupling. Our numerical simulations revealed that couplingstrength plays a significant effect on the variation of the topological entropy. The larger the coupling strength, the smal-ler the region of the parameter space corresponding to positive topological entropy (which means chaotic behavior).With the coupling strength, we introduced a second topological invariant as a tool to distinguish isentropic maps(applied to the subset K2 of the parameter space).

In the context of coupled bursting models, what is the meaning of the topological invariant r and what does it rep-resent? This is an interesting question for which we do not have any answer yet, but hope to address in forthcomingresearch.

A central issue in the analysis of coupled cells is to understand how it is possible that the potentially very complexbehavior which might transpire when chaotic neurons are coupled, can lead in a dynamical way to rather simpler, oftenwell organized motion.

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The influence of coupling on chaotic maps modelling bursting cells

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