Chaotic itinerary

• Ichiro Tsuda, Hokkaido University, Japan

Dr. Ichiro Truda accepted the invitation on 19 July 2007 (self-imposed deadline: 19 January 2008).

According to the development of measurement techniques in neuroscience, the dynamic activity of neurons or neural assemblies called chaotic itinerancy has become clearly observed, for example, as dynamically switching cortical states, as the nonstationary transitions between synchronization and desynchronization of oscillatory field potentials within single-band frequencies and also in distinct bands-frequencies, as chaotic population dynamics, as dynamic change of coherent activity in neuron assemblies, and as chaotic interspike intervals, i.e., chaotic fluctuations of membrane potentials. These have been observed to be not merely random, but transitory dynamics accompanying with nonstationary, repetitive, itinerant, and chaotic transitions. These transitions can be described by chaotic itinerancy. First, we briefly touch upon the finding of chaotic itinerancy. Second, we describe chaotic itinerancy as a new dynamical state in high-dimensional dynamical systems. Third, we briefly introduce similar transitory behaviors in the brain. Finally, we propose five scenarios for the appearance of chaotic itinerancy, which provides typical transitory dynamics.

A short note on the finding of chaotic itinerancy

At least up to the late 1980s, complex and dynamic behaviors in high-dimensional dynamical systems had attracted attention in various fields such as nonlinear physics, hydrodynamics, condensed matter physics, physical chemistry, and even neuroscience.

Ikeda studied optical turbulence and found complex phenomena in his model of a delayed-feedback optical system, one of which showed transitory dynamics among several optical modes [1]. A similar phenomenon was observed in a delayed-feedback system by Anderson [2] and was further studied by Davis with a model system [3]. Anderson called this transitory phenomenon a “day dream” because of its chaotic appearance of embedded optical modes. Independently, Tsuda found a similar transitory phenomenon in a skew-product type of a nonequilibrium neural network model of associative memory [4]. Before this work, a mathematical model of associative memory succeeded in describing a single association. On the other hand, in the case of successive association, a rule for order of association was given [5,6,7]. However, any emergent properties could not be treated by these theories .

Tsuda’s model describes a dynamically successive association of memories, whose rule was self-organized to emerge. This kind of successive association of memories can be described by dynamic transitions between dynamical memory states. The dynamic process and its transition rule were studied, and found a circle map with criticality as a transition rule. Criticality appeared twofold: a circle map was in a critical stage between a chaotic map and a stable one; and fixed points representing memories were described by indifferent fixed points, which were similar to those introduced by Milnor as a typical example of attractor extended to include even an invariant set with neutral stability [8]. After the work of successive association of memories, Körner and his colleagues [9] observed a similar transitory behavior in a neural network model of parallel-in-sequence processing of information, which was proposed as a dynamic model of internal attention of vision [10,11]. Aihara and colleagues made a model for dynamic associative memories [12] in terms of the network of chaotic neurons invented by Aihara [13], and observed a similar chaotic transition between memories. A similar, but not transient, behavior was also observed in a neural network model for association of memories, where each memory was represented by a limit cycle attractor [14]. In Nara-Davis’s model, when decreasing the number of synaptic connections, a transition occurs from attractor dynamics subserving a single association of memory to chaotic dynamics subserving a dynamic and successive association of memories. Horn and Opur further proposed a similar model, where they found successive associations of memories were noise-induced [15].

Kaneko invented coupled map lattices (CML) and globally coupled maps (GCM) to describe turbulence-like complex, dynamic, and bifurcation phenomena appearing in various fields. He found transitory behaviors in intermediate regions between a fully chaotic region and an ordered region of two-dimensional parameter space consisting of the coupling strength among elementary individual maps and a degree of nonlinearity commonly inherent in each map [16]. In GCM, such a transition occurs when multiple attractors are weakly destabilized, where underlying states in the transitions were considered to be attractors in a Milnor’s sense.

Kaneko, Ikeda, Davis, and Tsuda discussed the universality of this kind of transitory behavior. Common features were as follows: the successive transitions among quasi-attractors [17,18] are chaotic; the stability of the states changes via transitions; there are distribution functions for the transitions such as transition probabilities and distributions of residence-time of orbits in a neighborhood of quasi-attractors; the changes of the Lyapunov dimensions, etc. They called the phenomena characterized by these factors a “chaotic itinerancy” [1,16,19], where the quasi-attractors were named “attractor ruins”. Shortly after, Tito Arecchi studied the chaotic transitions in optical turbulence in terms of this concept [20].

The necessity of chaotic itinerancy as a new dynamical concept

In dynamical systems such as described by differential equations, that is, vector fields, dimensionality is crucial to produce a variety of solutions. Here we discuss dissipative dynamical systems, but do not need to change any essential points on the dimensionality in the case of conservative dynamical systems. The dynamical systems with less than three dimensions possess three types of attractors: fixed points, limit cycles, and tori. The dynamical systems with three dimensions can produce strange attractors, namely chaotic attractors. The presence of chaos is essentially new in three-dimensional dynamical systems. Are there an essentially new dynamics in higher-dimensional dynamical systems?

Rössler [21] proposed the idea of hyperchaos as a newly appearing attractor in dynamical systems with more than three dimensions. Hyper-chaos is defined by the presence of a multiple number of positive Lyapunov exponents, namely the presence of a multiple number of independently expansive directions. Thus, hyper-chaos is considered as describing a general class of higher-dimensional chaos. Then, the question arises: what is a specific behavior among higher-dimensional chaotic behaviors? In other words, what is a classification of hyperchaos, or could the specificity exist in dynamical systems with higher degrees of freedom than three? Chaotic itinerancy can provide a specific transitory dynamics among “quasi-attractors” [17,18].

Let us consider the dynamical representation of complex phenomena in far-from-equilibrium systems. One can describe various dynamical states in far-from-equilibrium systems in terms of the concept of attractors in dynamical systems. The steady state is described by a fixed point, the periodic state by a limit cycle, the quasi-periodic state by a torus, and the irregular state by a strange attractor. In order to describe the transitory phenomena, one needs another dynamical concept.

A saddle-connection provides a description of transition between unstable states, which are represented by saddles. Generically, in low-dimensional systems, a saddle-connection is structurally unstable, so that it cannot be an appropriate model for the transitions. However, if the system has some kind of symmetry, the saddle-connection becomes structurally stable [22]. In such a case, successive transitions between states represented by saddles can occur, but are not always chaotic. What is a generic structure allowing for chaotically successive transitions between dynamical states, whose states must be what we should call “quasi-attractors”, but not low-dimensional saddles? See section 4 for considerations of possible mechanisms of chaotic itinerancy.

Transitory dynamics cannot be explained by geometric attractors because the transition should be associated with the instability of such a state itself. On the other hand, the concept of chaotic itinerancy expresses the chaotic transitions between “quasi-attractors”. Quasi-attractors are also called “attractor ruins” since these are the states appearing soon after the destabilization of geometric attractors. The orbits behave as if attractors still exist, in the sense that a positive measure of orbits is attracted to attractor ruins after a certain length of time. However, such an attracted area is not asymptotically stable.

Furthermore, this idea of attractor ruin is similar to the Milnor attractor concept [8], However, if a Milnor attractor exits, the orbits converge to a Milnor attractor unless it has a riddled basin, and then cannot leave from a neighborhood of attractor without additional perturbations.

Chaotic itinerancy has been numerically observed in many systems [23]. Typical systems include GCM [16], CML [14,24], networks of neuron maps [12], coupled differential equations [26,27,28], delay-differential equations [1], and skew product transformations [30,4].

Characteristics of chaotic itinerancy have also been clarified (see, for example, reference [29]). The distribution of the residence time in attractor ruins follows a power law [30] or an exponential law [24], depending on the models. The chaotic transition usually occurs in high-dimensional phase space, but, for the case where chaotic orbits are confined in a “narrow tube”-like structure in phase space, the transition can be described by low-dimensional chaos [31,4]. On the Lyapunov spectrum, the following three specific characteristics have been known. (1) Many of the Lyapunov exponents accumulate in a neighborhood of zero [16,30]. (2) The zero exponents beside the direction of orbit (in the case of flow) show large fluctuations and never converge [32]. (3) Even the largest exponent fluctuates, and shows extremely slow convergence [24].

Transitory phenomena in the brain

The complex transitions between distinct states of brain activity have been observed to be not merely random, but transitory dynamics with some features such as nonstationary, repetitive, and chaotic transitions. Typical phenomena observed in laboratories are chaotic transitions between quasi-attractors in rat and rabbit olfactory system, particularly in olfactory bulb [33,34,35,36], irregular transitions between synchronization and desynchronization of subthreshold dynamics in cat visual cortex [37], irregular reentry of synchronization of phase differences in human electro-encephalogram (EEG) [38], and the task-related propagation of wave packets consisting of \(\gamma\)-waves with around 30-90 Hz oscillations and \(\beta\)-waves with around 10–30Hz oscillations in rat olfactory and widely connected areas [39,40]. These chaotic transitions appear, accompanying with external inputs.

In particular, Freeman and colleagues have observed chaotic transitions during active behaviors of animals. Rats learn odor, which is represented by a limit cycle attractor in the olfactory bulb, synchronized with inhalations. After the learning of several odor inputs, the activity of olfactory bulb becomes a chaotic wandering among learned states if an input is new, but the activity converges to one of the learned states, that is, the activity is represented by a limit cycle attractor if an input is an already learned one [33,34]. This shows a typical attended dynamics of the animal. Kay proved the existence of state transitions in the field potentials in the olfactory bulb, hippocampus, and entorhinal cortex of rats. The transitions occur during successive periods of anticipation of odor inputs, perception of odor, judgment for action, and actual action [39,40]. The transitions may reflect the representation of the animal’s experience, i.e., episode.

On the other hand, experimental evidence on the transition without any external inputs, that is, as spontaneous cortical activity or as ongoing activity has also accumulated. This has been measured via quantities reflecting field potentials such as LFP, calcium imaging, EEG and ECoG. The brain changes its activity in the absence of stimuli such that the spontaneously activated pattern or ongoing activity is similar to what would appear if the stimulus were actually presented [41,42,43,38].

In particular, Kenet et al. showed that ongoing activity contains a set of dynamically switching cortical states in V1 [41], where these dynamic states were suggested to reflect expectations about the sensory inputs. This finding may indicate that the brain is always in an active idling state, with possible responsive patterns being evocated to enable quick responses to any stimuli. Spontaneous cortical activity has also been suggested to appear in accordance with wandering mental process because of the activation of default networks [43].

Other kinds of spontaneous activity have also been observed. One kind comes from the study by Freeman and Zhai, who observed spontaneous activity of animal and human brains [44], and conducted a data analysis in terms of a random number moderated by refractory periods. They found that the spontaneous activity could be characterized by black noise, whose power spectrum density follows \(1/f^x\), where \(x \ge 2\). The appearance of black noise activity means that extremely rare events predominate. Another kind has been observed in the culture of the hippocampal CA3 [45]. The transition occurred, in the presence of much carbachol, which is an agonist to muscarinic acetylcholine receptors, among five kinds of states: random firing states, up–down states, steady firing states, \(\theta\) rhythm activity, and partially synchronized states. The transition was not regular but chaotic such as shown in the presence of noisy tent map. In place of carbachol, the input of atropin, which is an antagonist to muscarinic acetylcholine receptors, prohibited the transition and had a strong tendency to force the CA3 network to a single state among the above five kinds of states, depending on the initial conditions. The finding of this spontaneous transition in CA3 is important because hippocampal CA3 can be considered as playing a role in the internal reconstruction of episodes.

Rabinovich and colleagues have studied another dynamical system which may account for a certain kind of cortical transition observed in olfactory system of insects [46,47]. For representation of the transition, they have proposed a heteroclinic cycle of saddles. In such a case, the transition is not always chaotic. This mechanism for the transition is based on a generic property of saddle connections. Usually, saddle connections are not structurally stable, because an unstable manifold of a saddle can coincide with a stable manifold of the other saddle with measure-zero in phase space. However, it may appear to be structurally stable under the presence of symmetry [22]. This holds under the condition that the sum of the dimensions of the unstable manifolds of one saddle and the stable manifolds of the other exceeds the dimension of phase space. In each invariant subspace of symmetric dynamical systems, one can confirm this condition. Using this theory, the transition by means of saddle connections may occur in some areas of the brain. However, the presence of such symmetry in the brain must be investigated in more detail.

One may still discuss the relation of the appearance of chaotic itinerancy to heteroclinic cycles [51]. It may be interesting to note the memory capacity of networks of competing neuron groups. Rabinovich et al. estimated it at approximately \(e(N-1)!\), where \(N\) is the number of neurons and \(e\) is a Napier’s constant, \(2.71828\cdots\), calculating the possible numbers of heteroclinic cycles [46]. On the other hand, to calculate the critical dimensionality of the appearance of chaotic itinerancy, Kaneko [49] estimated two factors that are supposed to determine the dimensionality for the chaotic transition. Let \(N'\) be the system’s dimension. Let us assume that the number of states in each dimension is two, taking into account the presence of two stable states separated by a saddle. The number of admissible orbits cyclically connecting the subspaces, using, say heteroclinic cycles, increases in proportion to \((N'-1)!\), whereas the number of states increases in proportion to \(2^{N'}\). If the former number exceeds the latter, then not all orbits can necessarily be assigned to each of the states, thus causing the transitions. In this situation, we expect itinerant motions between states. This critical number is six for chaotic itinerancy [49,23]. We identify \(N'\) with \(N\). In such a case, one may conclude that the transition via heteroclinic cycles appears when the memory capacity is less than the number of states, whereas chaotic itinerancy appears in the opposite condition.

Five scenarios on the genesis of chaotic itinerancy

An attractor ruin cannot be expressed as a geometric attractor, because a dynamical mechanism must allow transitions between attractor ruins. Is there a mathematical concept that can represent an attractor ruin? Possible scenarios of the appearance of chaotic itinerancy have been discussed in relation to information processing in the brain [50,51]. Here, we summarize this issue from a mathematical viewpoint(See also Tsuda [51]). We expect the question of whether or not other possibilities of scenarios exist to be investigated.

Scenario 1

A three-tuple (chaotic invariant set, unstable Milnor attractors [8], riddled basins) yields chaotic transitions [52] between attractor ruins.

Any transition from a Milnor attractor is impossible without external perturbations because it is an invariant set. External perturbations can be provided by interactions with other systems, or by external noise. Here we treat chaotic itinerancy in the GCM as a typical case produced by scenario 1. Then it is clarified that an unstable Milnor attractor associated with a riddled basin [53] brings about chaotic itinerancy.

A GCM is defined as follows. For a one-dimensional map \(g^{(i)}: R \rightarrow R\) for \(1 \le i \le N\), \(G: R^N \rightarrow R^N, x_{n+1} = G(x_n)\) is determined by the relation: \(x_{n+1}^{(i)} = (1 - \epsilon )g^{(i)}(x_n^{(i)}) + \frac{\epsilon}{(N - 1)} \sum_{j \neq i}g^{(j)}(x_n^{(j)}),\quad \quad (1 \le i \le N), \) where \(n\) is a discrete time, \(i, j\) are indices of the map, and \(N\) is the number of individual elementary maps.

Kaneko first investigated the case of each \(g^{(i)}\) being an identical logistic map producing chaos, and numerically found chaotic itinerancy [54].

A GCM is invariant under the substitution \(s\) of individual elementary maps. In other words, a group action \(s\) commutes with a dynamical rule \(G\), i.e., \(Gs = sG\). In this sense, a GCM is a symmetric system. This type of symmetric system has been widely studied by Ashwin and others [55,56,57]. In a symmetric dynamical system (M, h), an invariant set under a group action is also invariant for the development of the dynamical system [52].

A synchronization state of all elementary individual maps is realized by a one-dimensional dynamics, which is invariant under any substitution of elementary individual maps. Hence all-synchronized state is invariant under dynamical development, which is here denoted by \(H_1\). If an elementary individual map is provided to produce chaos, as Kaneko proposed, \(H_1\) is a chaotic invariant set. In the GCM, there are many other invariant sets representing partially synchronized states. For example, two different synchronized states can appear. These two synchronized states construct a two-dimensional invariant subspace \(H_2\). In a neighborhood of \(H_2\), stagnant motion can occur. Now, assume that a partially synchronized state is stable and represented by an attractor. On the other hand, if the sign of the normal Lyapunov exponent of \(H_1\) changes from positive to negative values via a blowout bifurcation [58], and if this bifurcation is local, then the basin of attraction of chaos representing an all-synchronized state becomes riddled. Therefore, the chaotic invariant set becomes an unstable Milnor attractor. Because we assume that a normal Lyapunov exponent to \(H_2\) remains negative because of the locality of the blowout bifurcation, a similar situation happens in a neighborhood of \(H_2\). If the partially synchronized state is chaotic, its basin of attraction may also become riddled [59].

Thus, in the GCM, transitions between partially synchronized states becomes chaotic, influenced by a chaotic invariant set representing the all-synchronized state. In particular, partially synchronized states look like attractor ruins. Thus, one realization of attractor ruin is a chaotic saddle [60].

Related behaviors but much simpler cases have been described as on–off intermittency [61,62], and as in–out intermittency [63]. It was pointed out by Ott [58] that the riddled basin accompanies on–off intermittency, and by Ashwin [63] that , for the case of in–out intermittency, the basin of attraction of a chaotic invariant set can become riddled, but that it is an open set for the basin of attraction of a periodic orbit or a fixed point. These remarks are crucial for the appearance of chaotic transitions.

Scenario 2

The interacting fixed point type of unstable Milnor attractors can yield chaotic transitions between tori or local chaotic attractors.

For the scenario 1, chaotic transitions between partially synchronized states are caused by chaos as an all-synchronized state. Then it is valuable to investigate whether such a chaotic transition can occur when an all-synchronized state is represented by a fixed-point type of unstable Milnor attractor. The appearance of a riddled basin in a neighborhood of such a fixed point cannot be expected. A motivation of the study is whether or not a coupled map system under this condition can produce chaotic itinerancy. What has been observed are chaotic transitions between tori and also between chaos, which are produced by the interactions of fixed-point type of unstable Milnor attractors [24]. The basin yielded was a mixture of riddled basins and open sets. Chaotic transitions caused by crisis-induced chaos were associated with a set of riddled basins and open sets of basins. Stagnant motion can occur in a neighborhood of tori.

Scenario 3

A heterodimensional cycle may produce chaotic itinerancy.

Let us consider a heterodimensional cycle [67,68]. A diffeomorphism \(h\) has a heterodimensional cycle associated with two saddles \(S_1\) and \(S_2\) if the saddles have different indices. Here, different indices mean different dimensions of those unstable manifolds of those saddles, \(n_1^u \neq n_2^u\), where \(n_1^u\) and \(u_2^u\) are dimensions of unstable manifolds of saddles \(S_1\) and \(S_2\), respectively. A coindex 1 cycle is defined by a heterodimensional cycle with \(n_1^u = n_2^u \pm 1\). A heterodimensional cycle is not typically robust for the same reason as in a heteroclinic cycle. However, Bonatti and Diaz proved the following theorem.

Theorem 1 (Bonatti and Diaz)

Let \(h\) be a \(C^1\) -diffeomorphism having a coindex 1 cycle associated with a pair of saddles. Then there are diffeomorphisms arbitrarily \(C^1\) -close to \(h\) that have robust heterodimensional coindex 1 cycles.

By the following theorem, in a neighborhood of a diffeomorphism with a co-index 1 cycle, chaotic behaviors are expected.

Theorem 2 (Bonatti and Diaz)

Let \(h\) be a diffeomorphism with a coindex 1 cycle that has real central eigenvalues. Then there are diffeomorphisms arbitrarily \(C^1\) -close to \(h\) that have strong homoclinic intersections associated with saddle-node or with flips.

In this case, we may expect transitory behaviors such as chaotic itinerancy because of the presence of heteroclinic intersections and the possibility of the appearance of stagnant motion in a neighborhood of heteroclinic tangency. One weak point of this scenario for chaotic itinerancy is that it is not clear that there would be stagnant motion under the appearance of heteroclinic tangency, although the presence of tangency indicates slow motion in a normal direction. Therefore, further studies are necessary to confirm this assertion. We further expect that dynamics in other coindex cases are investigated in relation to chaotic itinerancy.

Scenarios 4

A normally hyperbolic invariant manifold (NHIM) can yield chaotic itinerancy.

A NHIM is an extended concept of saddle in high-dimensional phase space such that the normal Lyapunov exponents to an invariant manifold are greater than the tangential ones in such a manifold [64]. Thus, the transitions between high-dimensional saddles can easily occur in a normal direction to an invariant manifold. If a chaotic motion occurs in a neighborhood of the saddle via heteroclinic intersections of the saddles, the transition can be chaotic. Because, one can expect stagnant motion in a neighborhood of a NHIM, and the motion in a NHIM can be chaotic, a NHIM can provide a mechanism for chaotic itinerancy. Actually, Komatsuzaki and Toda insisted a NHIM as a mechanism of chaotic itinerancy [65] in the context of the study of molecular dynamics of chemical reactions. Also in this case, a problem of how stangant motion occurs in a neighborhood of NHIM remains. Further studies are necessary to resolve this problem.

Scenario 5

Unstable Milnor attractors associated with fractal basin boundaries may yield noise-induced chaotic itinerancy.

The appearance of a negative Lyapunov exponent in the direction normal to the chaotic invariant set was essential for the chaotic transition in symmetric dynamical systems, but it has been pointed out that the presence of positive normal Lyapunov exponents brings about a curious transition phenomenon [66]. The case is known that a separation of multiple attractors is caused by fractal basin boundaries [69]. In this context, Feudel and colleagues found a chaotic itinerancy-like phenomenon in a double-rotor system with weak noise [69], where many periodic orbits coexist, together with higher periodic orbits possessing very tiny basins. The addition of noise may bring about the disappearance of such basins, leaving only low periodic orbits. Similar behavior was found in the KIII model by Kozma and Freeman [70]. Because of fractal basin boundaries, long chaotic transients appear before the system falls into a periodic orbit. Furthermore, orbits are trapped for some time in the vicinity of periodic attractors, but are eventually kicked out by noise, following which the orbits become chaotic again because of the fractality of the basin boundary. Thus chaotic transitions between periodic attractors occur, possessing a distribution of residence time in a neighborhood of periodic orbits, which implies the appearance of stagnant motion. This may be called noise-induced chaotic itinerancy. This type of chaotic transition can occur also with the fixed-point type of unstable Milnor attractors, as observed in nonequilibrium neural networks [30,4].

Acknowledgment

This work was partially supported by HFSP(RGP0039/2010). This work was also partially supported by a Grant-in-Aid for Scientific Research on Innovative Areas “The study on the neural dynamics for understanding communication in terms of complex hetero systems (No. 4103) ( 21120002 )” of The Ministry of Education, Culture, Sports, Science, and Technology, Japan.

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