Rhythmic control of oscillatory sequential dynamics in heteroclinic motifs

Highlights

• We study networks with heteroclinic oscillatory dynamics under periodic stimulation.

• Rhythmic stimulation controls coherence and coordination of sequential switching.

• Spectrum control, dynamical filtering and encoding enhancement are observed.

• We suggest analyses of sequential activity in rhythmic input experimental protocols.

• This can help interpreting data from transcranial stimulation and evoked potentials.

Abstract

Cognitive/behavioral brain functions are implemented through temporary correlated sequential activity of many brain elements that form universal anatomical and functional motifs, i.e., characteristic functional interactions among brain nodes, at different levels of the neural hierarchy. Such motif dynamics is determined by both the interconnections among nodes and their intrinsic oscillations. This paper focuses on heteroclinic motifs, i.e., those built in networks of oscillatory nodes that interact through asymmetric inhibitory coupling in a winnerless competitive way. We introduce a basic rate-phase motif model – based on a generalization of the well-known ecological Lotka–Volterra model – for the analysis and prediction of control processes that emerge in interacting heteroclinic motifs under periodic stimulation. This approach describes both intensity and phase in each node. We study how a rhythmic signal, which can be linked to internal or external sources, can functionally change the heteroclinic network and produce a rich gallery of motifs in the form of coordinated sequential activations. In computer simulations of the model in a “master-slave” approximation, we report phenomena such as dynamical filtering, encoding enhancement and transition to chaos. Our results are relevant in the context of several experimental protocols related to the role of brain rhythms and/or the use of external rhythmic stimulation, in particular in the context of transcranial control and evoked potentials, to assess cognitive functions and their associated pathologies.

Introduction

The human brain is composed of a vast number of elements that form hierarchical networks that cooperate and compete in wide spatial and temporal scales [1], [2], [3], [4], [5]. The dynamical features of brain networks, from microcircuits to global systems, provide the performance of diverse important cognitive functions, which involves robust transient sequential activity among different coordinated brain regions [6], [7], [8], [9], [10]. Key mechanisms of robust brain sequential dynamics are: (i) winnerless competition (WLC) [11], (ii) entrainment of oscillatory activity at different levels of the hierarchy [12], [13], and (iii) binding of different cognitive/perception modalities [10], [14]. Modern nonlinear dynamical theory can suggest several tools for the analysis of multiscale transient brain activities (e.g., see [9], [15], [16]). However, keeping in mind that theoretical and computational complexity exponentially increases with the number of degrees of freedom, investigators require constructive network analysis tools and reduced approaches based on key dynamical principles [17], [18], [19], [20], [21]. Approximations in this context can profit from the consideration of: (i) the topological reduction of the structural organization of complex networks through the representation of the corresponding graphs as sets of connected functional networks; and (ii) the low-dimensionality of oscillatory dynamics, which in many cases arises from asymmetric inhibitory interactions [9], [22], [23].

The term “network” is used in this paper in a general sense to denote a set of interconnected neural groups in the brain (a physical space), while the term “heteroclinic network” denotes a set of invariant objects/structures produced in the phase space by the dynamics of the physical network constituents. In this phase space, nodes in the network are metastable states connected through separatrices. We call motifs the recurrent patterns of interactions in such modular heteroclinic hierarchical networks. The modern theory of dynamical systems considers both robust (i.e., structurally stable) and non-robust heteroclinic networks. Robust heteroclinic structures are mathematical objects that often appear as attractors in the phase space of dissipative dynamical systems with invariant subspaces or manifolds. Investigating the link between physical brain networks and the presence of robust heteroclinic structures in the phase space has been a hot topic in the last years both in dynamical system theory and neuroscience [11], [24], [25], [26], [27], [28]. Here, we focus on transient sequential dynamics phenomena in global network oscillatory activity controlled by an endogenous rhythm or by an external rhythmic stimulus. Rhythmic stimulation is widely used both in cognitive studies as well as in clinical or rehabilitation procedures (e.g., see [29], [30], [31]). Recent modeling efforts have pointed out the importance of taking into account ongoing rhythms in brain stimulation protocols [32]. However, the effect of rhythmic stimulation on sequential brain dynamics based on oscillatory motifs has not been addressed yet. To motivate experimental research in this direction, the basic model proposed in this paper allows studying the effect of periodic input on the switching dynamics of metastable network oscillatory activity. Such activity is the result of temporal entrainment of different oscillatory nodes. The heteroclinic dynamics of the metastable state sequential switching (see Fig. 1) builds up the phases of the node oscillations resulting in coordinated motif dynamics.

Brain rhythms are considered to be involved in higher cognitive activities ranging from functional coordination and integration to attentional control [1], [33], [34], [35], [36]. The corresponding information processing phenomena are thought to be implemented mainly through neural synchronization mechanisms (e.g., see [37], [38], [39]), not necessarily involving synaptic transmission [40], and include transient coordination processes between hemispheres [41]. Microelectrode recordings and modern brain imaging, in particular fMRI, have unveiled the transient functional reorganization of large scale brain networks during the performance of different cognitive tasks (e.g., see [42], [43]). The dynamical mechanisms underlying the fast functional restructuring of flexible cognitive networks are still unclear. Departing from observed metastable informational patterns in the nervous system [21], [44], [45], our basic dynamical model helps to understand different dynamical phenomena that can be related to behavioral/cognitive programs in the framework of the same structural heteroclinic network. In particular, our results suggest that by changing the frequency of an input field that models a brain rhythm band or an external rhythmic input, it is possible to alter the level of coherence between different oscillating competitive patterns in the heteroclinic motif network and dynamically control their sequential interactions while preserving the coordination.

We assume that the human cognitive control system employs multiple strategies that include brain rhythms as a mechanism for the robustness and functional changing of circuits in specialized networks like those that build up working memory, information binding and attention processes. Brain functional networks require having a dependence on the phase, as this is probably the only way to be sensitive to entrainment and perform frequency control. However, it is not clear that neural sequential activity can keep its stability when modulated by extrinsic rhythms, e.g., by ephaptic interactions [40], [46]. Inhibition plays a key role in shaping cognitive network architectures [23] and can be responsible for stable dynamics and sequential behavior, as a key factor for suppressing noise, but at the same time can prevent the interaction with excitation that depends on extrinsic rhythms. We also proceed from the hypothesis that brain networks display WLC, a competitive process through inhibitory interactive nodes that are only transient winners during a finite time [10], [47], [48]. Thus, the network generates sequential dynamics through node switching activity [49], [50]. From this view, top-down approaches, although far from a biophysical detailed description, provide the required ingredients to model cognitive dynamics, being also suitable for the interpretation of activation sequences in regions of interest in EEG and fMRI (see also [16], [51], [52]).

External rhythmic stimulation has been suggested as a control paradigm to manipulate the brain's intrinsic oscillatory properties of networks driven via a variety of input-driven mechanisms [31]. Noninvasive protocols such as steady state visually evoked potentials (SSVEPs) [53], [54] and transcranial alternating current stimulation are examples where an external rhythm drives brain dynamics [29], [30], [31], [55]. Rhythmic light stimulation evoking SSVEPs is used in cognitive studies dealing with visual attention [56], [57]. Transcranial periodic stimulation allows modulating brain oscillations and, in turn, influence cognitive processes to assess their causal link. In particular, transcranial rhythmic protocols have been used to modulate basic motor and sensory processes as well as higher cognitive processes like memory, ambiguous perception, and decision making [58], [59], [60].

Here, we propose a complementary view on brain rhythms from the perspective of their interaction with neural sequential dynamics involved in a wide variety of information processing tasks [50], [61], [62], [63]. Thus, the network that we consider has a general function, the generation of input rhythm-specific coordinated sequential activity, which can be adapted to many specific neural systems. In the next sections, we show that a multifunctional network with sequential activations entrained by external rhythms, described by a simple frequency and amplitude, can evolve through distinct dynamical states. These states can be related to different brain functions and characterized by the broadness of their frequency spectrum, their level of regularity and the specific features of the sequential activations generated in a heteroclinic network.