Chaos, Solitons and Fractals 28 (2006) 1314–1326
www.elsevier.com/locate/chaos
The influence of coupling on chaotic maps modelling
bursting cells
Jorge Duarte
a,1
, Luı́s Silva
b,2
, J. Sousa Ramos
c,*,3
a
Departamento de Eng. Quı́mica, Secção de Matemática, Instituto Superior de Engenharia de Lisboa,
Rua Conselheiro Emı́dio Navarro 1, 1949-014 Lisboa, Portugal
b
Departamento de Matemática, Universidade de Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
c
Departamento de Matemática, Instituto Superior Técnico, 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 important role in information processing. This activity refers to a complex oscillation characterized by a slow alternation
between spiking behavior and quiescence. In this paper, the interesting phenomena which transpire when two cells
are coupled together, is studied in terms of symbolic dynamics. More specifically, we characterize the topological
entropy of a map used to examine the role of coupling on identical bursters. The strength of coupling leads to the introduction of a second topological invariant that allows us to distinguish isentropic dynamics. We illustrate the significant
effect of the strength parameter on the topological invariants with several numerical results.
Ó 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. This
complex 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, including the development of detailed biophysical models that describe the high dimensional dynamics of nonlinear events
responsible for variations in the ionic currents across the membrane. The characterization of such activity is usually
based on either realistic ionic-based models or phenomenological models. The ionic-based models proposed for a single
cell are designed to replicate the physiological mechanisms of the membrane, with the parameters and functions derived
from experimental data. Some of these models consist of a system of many nonlinear differential equations. The high
dimensionality of the phase space is a significant obstacle in understanding the collective behavior of such dynamical
*
1
2
3
Corresponding author.
E-mail addresses: [email protected] (J. Duarte), [email protected] (L. Silva), [email protected] (J. Sousa Ramos).
Partially supported by Instituto Superior de Engenharia de Lisboa.
Partially supported by Universidade de Évora and FCT/POCTI/FEDER.
Partially supported by FCT/POCTI/FEDER.
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.chaos.2005.08.188
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 minimal
complexity of the equations [8]. A special type of phenomenological models is based on low-dimensional maps. There
have 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 mathematical
structure underlying the oscillations. In this work, the focus will be on providing a study of the role of coupling on
bursting cells. Neurons and endocrine cells rarely act alone, but rather as members of a population connected together
via gap-junctional or synaptic coupling. The electrical activity observed in the population is the result of the intrinsic
properties 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 activity of the individual neurons. The isolated neural cells often exhibit chaotic motions, as observed in the characteristics of
intracellular voltage measurements.
Our goal is to provide a contribution for the detailed analysis of a family of maps, introduced in [19] and studied in
subsequent papers (for instance, see [4] and [6]), which produces chaotic bursting patterns similar to those observed in
neurons and endocrine cells. More precisely, using techniques of symbolic dynamics [7], we compute the topological
entropy and a second invariant, denoted by r, in order to elucidate the effect of mean field coupling on identical bursters. The topological invariant r allows us to distinguish different systems with equal topological entropy. The two topological invariants are quantitative measures of different states of complexity that arise through the coupling. Attention
will be focussed on two-cell systems. As pointed out in [5] and [2], numerical simulations have demonstrated that the
analytical results for two-cell systems carry over to many-cell systems. We study a system of two identical cells and show
the 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 discretetime 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
8
N
>
< xnþ1;i ¼ ai þ y þ P xn;j ;
n;i
N
1þx2n;i
ð1Þ
j¼1
>
:y
nþ1;i ¼ y n;i ri xn;i bi ;
where xn,i and yn,i are, respectively, the fast and slow dynamical variables of the ith cell, the parameter is the strength
of global coupling, and N is the total number of cells. The x-variable replicates the dynamics of the membrane potential
and the y is the recovery variable. The slow evolution of yn,i is a result of the small values of the positive parameters bi
and 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. The
values of the parameter ai are selected on the interval [1.5, 8.0]. We note that the considered mechanism of bursting
is similar to the oscillations in the well-known Hindmarsh–Rose model of biological neuron, where the role of parameter 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 qualitatively
different 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 systems. 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 map
yn+1,i = yn,i ri xn,i bi, assuming that yn,i is a control parameter c = yn,i. Thus, important insights about the fast
dynamics of each coupled cell can be obtained from the analysis of the three-parameters family of maps
Fig. 1. Wave forms of temporal behavior of individual cells, regarding the full system (1) with a = 4.1, r = b = 0.001, and = 0.
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J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326
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.
xnþ1;i ¼ F a;c; ðxn;i Þ ¼
N
ai
X
þ ci þ
xn;j .
2
N j¼1
1 þ xn;i
ð2Þ
This approach was pioneered by Rinzel [16] in a study of continuous bursting models, and it is extensively used on
the analysis of single and coupled systems.
P
Coupling between cells influences the fast dynamics of each cell by adding the value N Nj¼1 xn;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 significant in the study of coupling. For further informations concerning coupled systems consisting of two identical bursting
cells, 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 in
Fig. 2.
The wave form for xn,2 is similar to the one shown for xn,1 and bursts (clusters of spikes) are synchronized. The active
and silent phases are considerably longer when the cells are coupled. Of less significance, but still noticeable, is the
observation 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
8
< xnþ1;1 ¼ 1þxa 2 þ c þ 2 ðxn;1 þ xn;2 Þ;
n;1
ð3Þ
: xnþ1;2 ¼ 1þxa 2 þ c þ 2 ðxn;1 þ xn;2 Þ;
n;2
where a = a1 = a2. When both cells start with initial conditions that satisfy jx0,1j = jx0,2j, the evolution of x in each cell
can be described by the family of maps
a
þ c þ xn .
ð4Þ
xnþ1 ¼ Ga;c; ðxn Þ ¼
1 þ x2n
We are now in a position to study the effect of coupling on two identical bursting cells using techniques of symbolic
dynamics 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
a
Ga;c;0 ðxn Þ ¼
þ c;
1 þ x2n
ð5Þ
which has the shape of an unimodal map (continuous map on the interval with two monotonic subintervals and one
turning point (relative maximum)). There have been used techniques of symbolic dynamics to study this map [6]. However, with the introduction of the coupling strength , there is a subregion of X, denoted by X , where Ga,c, has the
shape of a bimodal map (continuous map on the interval with three monotonic subintervals and two turning points
c1 and c2 (c1 the relative maximum and c2 the relative minimum)). In our study, we consider
J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326
1317
X ¼ fðc; aÞ 2 X : Ga;c; ðc1 Þ > c2 and Ga;c; ðGa;c; ðc1 ÞÞ < Ga;c; ðc1 Þ and Ga;c; ðc2 Þ < c1 and Ga;c; ðGa;c; ðc2 ÞÞ
> Ga;c; ðc2 Þg.
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 in
Fig. 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 the
topological entropy of Ga,c,(xn), and we show situations of the variation of this numerical invariant with the parameters
a 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 ; c1 ½;
M ¼c1 ; c2 ½;
R ¼c2 ; 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 strictly
decreasing. 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
Oðc1 Þ ¼ fxi : xi ¼ f i ðc1 Þ; i 2 Ng and
Oðc2 Þ ¼ fy i : y i ¼ f i ðc2 Þ; i 2 Ng.
With the aim of studying the topological properties of these orbits we associate to each orbit O(ci) a sequence of symbols 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 and
Sj = R if fj(ci) > c2. The points c1 and c2 play an important role. The dynamics of the interval is characterized by the
symbolic sequences associated to the orbits of points c1 and c2. We denote by nM(S) the frequency of the symbol M
in S and we define the M-parity of this sequence, qðSÞ ¼ ð1ÞnM 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 in
R ¼ fL; A; M; B; RgN that depends on M-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.
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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 . . . Pi1 = Q1 . . . Qi1 is even (that is, q(P1 . . . Pi1) = +1), we say
that 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 . . . Pi1) = 1), we say that P < Q if Pi < Qi in the order R < B < M < A < L. If no such index i exists, then
P = 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 we
obtain a sequence of symbols that can be characterized by a block of length k,S(k) = S1 . . . Sk1Ci, with i = 1, 2. In what
follows, 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 the
block P = P1 . . . Pp1A is maximal, that is, ri(P) 6 P, where 1 6 i 6 p and r(PiPi+1Pi+2 . . .) = Pi+1Pi+2 . . . is the usual
shift operator. On the other hand, O(c2) is realizable if the block Q = Q1 . . . Qq1B 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 conditions
ri(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). We
designate by kneading data the pairs (P(p),Q(q)) 2 R(A,B), where P(p) = P1 . . . Pp1A, Q(q) = Q1 . . . Qq1B, the bistable sequence P1 . . . Pp1BQ1 . . . Qq1A, and the eventually periodic sequence P1 . . . Pp1BQ1 . . . Qq1B or Q1 . . . Qq1AP1 . . .
Pp1A.
Now we consider the topological entropy. This numerical invariant measures the quantitative amount of chaos. A
possible definition of chaos in the context of one-dimensional dynamical systems state that a system is called chaotic if
its topological entropy is positive. Thus, the topological entropy can be computed to express whether a map has chaotic
behavior.
Let Ga,c, be the 3-parameters family of maps such that (c, a) 2 X. To each values of the parameters, the dynamics is
characterized using the kneading data. This kneading data determines a Markov partition of the interval, considering
the 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 procedure we obtain the partition {Ik = [zk,zk+1]}k=1,2, . . ., p+q of the interval I = [y1,x1]. The transitions between the subintervals are represented by a matrix MðGa;c; Þ. The topological entropy of Ga,c,, denoted by htop(Ga,c,), can be given by
htop ðGa;c; Þ ¼ ln kmax ðMðGa;c; ÞÞ ¼ ln sðGa;c; Þ;
where kmax ðMðGa;c; ÞÞ is the spectral radius of the transition matrix MðGa;c; Þ and s(Ga,c,) is the growth rate
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sðGa;c; Þ ¼ lim k ‘ðGka;c; Þ
k!1
of the number of intervals on which Gka;c; (kth-iterate of Ga,c,) is monotone. We have sðGa;c; Þ ¼ kmax ðMðGa;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:
y 1 < x2 ¼ y 2 < x3 ¼ y 3 < x4 ¼ y 4 < x5 ¼ y 5 < x6 ¼ y 6 < x7 ¼ y 7 < c1 ¼ x8 ¼ y 8 < c2 < x1 .
J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326
The corresponding
2
0
60
6
6
60
6
6
60
6
MðGa;c; Þ ¼ 6
60
6
60
6
60
6
6
41
1
transition matrix is
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
1
0
1
0
0
1
1
0
1319
3
0
07
7
7
07
7
7
07
7
07
7;
7
07
7
17
7
7
15
0
which has the characteristic polynomial
pðkÞ ¼ detðMðGa;c; Þ kIÞ ¼ kð2 kÞð1 þ kÞð1 þ k2 Þð1 þ k4 Þ.
The growth number s(Ga,c,) (the spectral radius of matrix MðGa;c; Þ) is 2. Therefore, the value of the topological
entropy can be given by
htop ðGa;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 bifurcation
diagrams for = 0.2.
These bifurcation diagrams suggest the existence of an inversion in the usual chaos ordering (for instance, note the
inverted 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.
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J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326
parameters a and c for different values of are depicted in Figs. 7 and 8. In all situations, the topological entropy
htop(Ga,c,) has an absolute maximum value.
With the last numerical results it becomes apparent that coupling strength plays a crucial role in modifying the
topological entropy of the family of maps (4). This study reveals that the maximum value of the entropy decrease when
coupling strength is larger.
Let us consider two single cells with the same topological entropy,
pffiffiffi!
1þ 5
htop ðGa ;c ; Þ ¼ ln sðGa ;c ; Þ ¼ ln
¼ 0:481212 . . . ;
2
with a* and c* fixed values of the parameters. The effect of the coupling strength on this pair of bursting cells is shown
in Figs. 9 and 10. The coupling can convert chaotic cells to non-chaotic cells.
Fig. 7. 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)
= 0.2, (c) = 0.15, (d) = 0.
Fig. 8. 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)
= 0.2, (c) = 0.15, (d) = 0.
J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326
1321
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
pffiffi two cells with the same topological entropy are coupled. The
topological entropy of the single cells is given by htop ðGa ;c ; Þ ¼ lnð1þ2 5Þ 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 change
in the coupling strength. In this regard, Figs. 11–13 show pertinent features of some isentropic curves (the levels of topological entropy) in region X for small periods n (n 6 5), which can arise through the coupling. The topological entropy
remains constant over each curve. We remind that when we have the symbolic sequence RL1 of the turning point (for
the unimodal map), the dynamics of the iterates is a full shift of two symbols and the topological entropy is one (in the
subset A [ C RL1 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 parameter space X, denoted by K2, for which the corresponding maps of the family Ga,c, have growth number 2, i.e., K2 is the
topological entropy level set for htop(Ga,c,) = ln 2.
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) G2a;c; ðc1 Þ ¼ G2a;c; ðc2 Þ,
(iii) rðP Þ ¼ rðQÞ.
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J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326
Fig. 11. Curves in the parameter space corresponding to periodic orbits of the turning point (periods n 6 5) for = 0. The labels are
the 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 periods
follow the ordering showed in Fig. 11.
Thus, the maps of the family Ga,c, satisfying the relation G2a;c; ðc1 Þ ¼ G2a;c; ðc2 Þ have topological entropy ln 2. The curve
K2 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 topological 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.
J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326
1323
Fig. 13. Curves in the parameter space corresponding to periodic orbits of the turning point (periods n 6 5) for = 0.4. The periods
follow 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 second
topological 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 ehtop ðf Þ and the other numerical quantity, denoted by r, is associated to the relative positions of the turning points of the map. The topological invariant r is introduced using the
hypothesis s(f) > 1 and the Milnor–Thurston map k that topologically semi-conjugate f to a piecewise linear map
Fe,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;s ðkðxÞÞ 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
8
if 0 6 y < kðc1 Þ
>
< sy
if kðc1 Þ 6 y < kðc2 Þ
F e;s ðyÞ ¼ sy þ e
>
:
sy þ 1 s if y P kðc2 Þ;
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J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326
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 map
f, characterized by a kneading data (P, Q), we can associate two topological invariants. One of them is the growth number 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 for
h(Ga,c,) = ln 2 there are practical formulas to compute e given a kneading sequence (see [12]). More precisely, given
a map f satisfying s(f) = 2, with finite kneading data (S, T) such that jSj = n + 1, let
!
X
i nk i
n1
fðS; T Þ ¼ qðSÞ 2 þ
;
ð1Þ 2
ki
with 1 6 ki 6 n the integers such that S k i ¼ M (when there is no ki such that S ki ¼ M, f(S, T) = 2n1q(S)). We define also
X
qðrni ðSÞÞ2nni ;
nðS; T Þ ¼
ni
with 1 6 ni 6 n the integers such that S ni ¼ R. According to [12], the value of e(f) is given by
eðf Þ ¼
4nðS; T Þ
4fðS; T Þ 1
if S ¼ S 1 . . . S n A
eðf Þ ¼
4nðS; T Þ þ 1
4fðS; T Þ 1
if S ¼ S 1 . . . S n B.
and
Now regarding the previous considerations, the maps of the family Ga,c, can be topologically classified by the pair of
topological 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 showed
previously that s = 2 and we have jSj = n + 1 = 8. The kneading data determines f(S, T) = 26 and n(S, T) = 26.
Therefore,
e¼
4nðS; T Þ
4 26
¼
4fðS; T Þ 1 4 26 1
and
r ¼e
3
¼ 0:4960784314 . . .
2
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 the
parameters 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.
J. Duarte et al. / Chaos, Solitons and Fractals 28 (2006) 1314–1326
1325
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 examine
the influence of mean field coupling on bursting cells.
A rigorous characterization of the complexity of a coupled system consisting of two identical bursting cells became
possible using techniques of symbolic dynamics. We studied the topological entropy and we introduced the parameter
space ordering of the dynamics that arose through the coupling. Our numerical simulations revealed that coupling
strength plays a significant effect on the variation of the topological entropy. The larger the coupling strength, the smaller 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 represent? This is an interesting question for which we do not have any answer yet, but hope to address in forthcoming
research.
A central issue in the analysis of coupled cells is to understand how it is possible that the potentially very complex
behavior which might transpire when chaotic neurons are coupled, can lead in a dynamical way to rather simpler, often
well organized motion.
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