Dynamical bridge between brain and mind.

TICS-1453; No. of Pages 9

Opinion

Dynamical bridge between brain and mind Mikhail I. Rabinovich1, Alan N. Simmons2,3, and Pablo Varona4 1

BioCircuits Institute, University of California San Diego, 9500 Gilman Drive 0328, La Jolla, CA 92093-0328, USA Department of Psychiatry, University of California San Diego, 9500 Gilman Drive 0603, La Jolla, CA 92093-0603, USA 3 Center of Excellence in Stress and Mental Health, VA San Diego Healthcare, 3350 La Jolla Village Drive, San Diego, CA 92161, USA 4 Grupo de Neurocomputacioń Biolo´gica, Departamento de Ingenierıá Informa´tica, Escuela Politećnica Superior, Universidad Autońoma de Madrid, 28049 Madrid, Spain 2

The bridge between brain structures as computational devices and the content of mental processes hinges on the solution of several problems: (i) inference of the cognitive brain networks from neurophysiological and imaging data; (ii) inference of cognitive mind networks – interactions between mental processes such as attention and working memory – based on cognitive and behavioral experiments; and (iii) the discovery of general dynamical principles for cognition based on dynamical models. In this opinion article, we focus on the third problem and discuss how it provides the bridge between the solutions to the first two problems. We consider the possibility of creating low-dimensional dynamical models from multidimensional spatiotemporal data and its application to robust sequential cognitive processes in the context of finite processing capacity of the mind. Nonlinear dynamics in cognition Experimental neuroscience and cognitive science are currently based on the premise that neural mechanisms underlying human perception, emotion, and cognition are well approximated by activity measurements of specific neuronal groups or brain centers. However, recent brain imaging and neurophysiological data indicate that cognition is neither a property of a single brain center nor of the entire brain [1–3]. Modern experiments have shown that cognitive functions arise from integrated processes in distributed circuits of interconnected brain areas [4,5], that is, the cooperative activity of many elements that form temporal associations for specific cognitive tasks. Brain dynamics generates spatiotemporal patterns with a high level of coherency; we can refer to these as cognitive modes. These modes interact with each other during cognitive processing. The number of interacting modes that represent the activity of large-scale functional brain networks at a given time is usually not very high. Thus, the number of corresponding variables to model these cooperative cognitive modes and describe the performance of task-dependent cognitive functions is much smaller than Corresponding author: Rabinovich, M.I. ([email protected]). Keywords: cognitive dynamical principles; transient brain dynamics; robust cognitive processing; sequential stability and winnerless competition; functional cognitive networks; stable heteroclinic channel. 1364-6613/ Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.tics.2015.06.005

the number of basic network elements. This means that the dynamics of these variables can be investigated in the framework of low-dimensional models, whose logic is illustrated in Figure 1. As suggested by existing work, breaking out low-dimensional network dynamics in conjunction with a flexible dynamical model that includes environmental and intrinsic variables is needed to effectively predict behavior. Diverse and complex dynamics can emerge from excitatory and inhibitory connections between these cognitive modes (see Glossary). The level of mode excitation is usually stabilized by inhibition. In general, excitation is responsible for bringing information to active modes, and inhibition is responsible for their competitive interaction [6]. Because of such interaction, the thinking brain demonstrates very rich temporal activity. The robust performance of task-dependent cognitive functions can be viewed as a dynamical process that happens through a sequence of transient states. Such states are important elements of cognitive processes and are associated with the temporal clusterization of brain centers that execute a specific cognitive task. In cognitive science, these temporal clusters can be named as dynamical modes and the corresponding transient states are named as metastable states [7,8]. Sequential transient states have two main features: they are resistant to noise and, at the same time, they are input-specific and convey information about what caused them. Thus, such dynamical processes are stable and reproducible, that is, robust. The intuitive understanding that human cognition is a transient dynamical process was articulated more than a century ago in 1890 by William James: ‘Thought is in constant change . . . no state once gone can recur and be identical with what it was before’ [9]. In other words, we move continuously from one relatively stable thought to another. Following James, many scientists have emphasized the crucial role of itinerant brain activity in human cognition [10]. Recently, the traditional perspective of temporal patterns of thought that are based on the characterization of reproducible rhythmic activity is giving way to one that tries to understand observable neural phenomena as robust transient mind dynamics [11–15] (Figure 2). fMRI data and sequential mental processes fMRI brain imaging data are collected in 3D pixels (voxels) over a time dimension. The time series for these voxels are Trends in Cognitive Sciences xx (2015) 1–9

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associated with either an external stimulus (using a correlation approach) [16–18] or an internal stimulus [19,20]. These approaches attempt to determine meaningful data-derived signals. Until recently, much of the analysis on these matrixes has treated the voxel in isolation and time has only had limited utility. Recent developments have attempted to infer more information from spatial [21,22] and temporal [23–25] patterns. The role of time during information processing in the brain is fundamental for all aspects of mental activity – perception, motor programming, cognition, and emotion [26]. In fact, brain dynamics can be considered as taskdependent sequential activations of metastable states, that is, states where system variables reach and temporarily hold stationary values [27–31]. Learning and generating ordered sequences of metastable states can be considered a core component of cognition. This view is supported by novel results in brain imaging, multielectrode recordings, and modeling experiments [15,22,32,33] (Figure 3). The sequential low-frequency competition between different modes in default-mode brain networks was initially identified based on fMRI analyses [32] (Figure 2A). The application of the concepts related to robust transient dynamics to the analyses of perception, working memory, learning, behavior, speech production, and other types of mind activities has demonstrated its effectiveness and broad scope [8]. Two key events have advanced the application of dynamical systems approach to neuroscience: (i) experiments suggesting that macroscopic phenomena in the brain are sequential and represent transient interactions of mental modes (i.e., patterns of activity) [8,25,34,35]; and (ii) research in genetics, ecology, brain, and other sciences have led to the discovery of reproducible and robust transients that are at the same time sensitive to informational signals [15,36,37]. The performance of most cognitive and behavioral tasks needs the sequential participation of several specialized brain networks (e.g., Figure 2B). The switching from one network activity to the next is controlled by intrinsic goals, or external information [38,39]. A well-known example of such switching is the competition between the defaultmode network – which is thought to support internally oriented processing – and external attention or salience networks that mediate attention to exogenous stimuli [40]. These networks show anticorrelated activity across a range of experimental paradigms. Based on fMRI experiments, control networks such as the frontoparietal brain network (FPN) would involve variable connectivity across networks and across tasks [38], acting as a ‘flexible hub mechanism’. The sequential interaction between the FPN and specialized brain networks is a basis for the performance of complex task-dependent cognitive processes. The next step for the interpretation of these dynamics is to infer the meaning behind the sequential switching among networks. It is through meaningful shifts between these metastabilities that brain networks can organize to represent a multitude of cognitive functions. To understand and predict the temporal characteristics of sequential cognitive processes, such as temporal overlapping of different functional networks activity, coordination, and stability against noise, it is necessary to build a general model that also incorporates the description of

Attractor: attractors are the regions of the phase space of a dynamical system (see below) towards which trajectories tend to evolve as time passes. As long as parameters are unchanged, if the system passes close enough to the attractor, that is, in the basin of attraction, it will never leave that region. Two examples of attractors are: (i) a stable fixed point and (ii) a stable periodic orbit (limit cycle) that, in particular, represents oscillatory activity of neurons in tonic spiking regime. Binding: the process of combining informational items from different sources into one unified block or chunk. Brain hubs: brain networks that transiently shift their functional connectivity patterns to implement control across a variety of cognitive tasks. Chunking: the reduction of hierarchical complexity through the subgrouping of similar proximal pieces of information into singular units to allow further processing. Cognitive mode: temporary stable activity pattern of correlated elements in a cognitive network. Because of the high level of intrinsic coherency, the dynamics of complex cognitive modes can be described with a small number of variables in a model. This number depends on the hierarchical structure of the cognitive process. Cognitive network: task-dependent distributed brain network that participates in the performance of a specific cognitive function. Dissipative dynamical system: if a system is closed, it does not exchange flows of energy, mass, information, etc. with the environment, and the intrinsic volume of the flow is preserved in the phase space. A system with internal friction, inhibition, or radiation is called a dissipative system. In such open systems, the volume of the flow contracts in the phase space. When time goes to infinity, the activity of this system can be represented by attractor dynamics. Dynamical models of cognition are exclusively dissipative systems. Dynamical system (model): a mathematical description of how a point in a representative space (e.g., phase space) depends on time. The evolution of this system in time corresponds to a unique trajectory that is determined by initial conditions. Fixed point in phase space: represents an equilibrium state of the modeled system. Such equilibrium can be stable, for example, a gymnast hanging headup from a gymnastics horizontal bar, unstable, for example, a gymnast headdown griping over the bar, and metastable, for example, the ball in the saddle landscape of Figure 4A. Generalized Lotka–Volterra (GLV) model: a mathematical framework for a dissipative dynamical system that can describe species’ competition in ecology, chemical reactions, and economic and neural processes. Hidden Markov model: a model that describes the temporal evolution of a system with a finite set of states with random variables. Transitions among these states are governed by a set of probabilities. In these types of models, the state is only partially observable (hidden) and, in particular, can be used to characterize the sequential activity of fMRI spatiotemporal patterns. Metastability: in a metastable state, dynamical system variables reach and temporarily hold stationary values. It is characterized by slowing down the system motion in the vicinity of the stationary state. On the time series, this phenomenon is represented by a plateau or pause. The image of a metastable state in the phase space is a saddle point and its neighborhood (Figure 4A–C). Phase space of a dynamical system: a space in which all possible states of the system are represented. Each possible state of the system corresponds to one point in the phase space and close points in the phase space represent close system states. The system evolving over time forms a phase space trajectory. As a whole, the phase portrait represents all behaviors that the system can demonstrate. Robust transients: trajectories in a phase space of a dynamical model that are disposed in the vicinity of each other when initial conditions are varied. These trajectories are robust against noise. Examples of such transients are the trajectories inside the stable heteroclinic channel. Saddle point: a stationary fixed point characterized by the coexistence of two types of trajectories in its neighborhood – one set of trajectories is going in, and the other set corresponds to trajectories going out. Those trajectories that intersect with fixed points are named stable and unstable separatrices (Figure 4A–C). Stable heteroclinic channel (SHC): a transient attractor formed by a sequence of saddle states and their vicinity. If the compressing of the phase volume around the SHC is stronger than the stretching of the volume along the SHC, the trajectories that are attracted by the SHC cannot leave it. SHC denotes the image of robust transient behavior in a dynamical system. Winnerless competition (WLC): a general dynamical phenomenon that denotes sequential switching of prevalence among participants. For example, if in a head-to-head competition, boxer A beats boxer B, boxer B beats boxer C, and finally boxer C beats boxer A, all participants are ‘winners’ for a finite time, but there is no overall winner such as in ‘winner takes all’.

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Figure 1. Brain imaging to a low-dimensional model of cognition and behavior. Characterizing and predicting cognitive processes as measured with modern imaging techniques require the inclusion of temporal sequencing in the description of brain networks and their conceptualization as nonlinear dynamical systems. The bidirectional interaction among experimental and theoretical approaches will lead to a better understanding of the dynamics of cognitive processes. Top feedback arrow represents the influence of cognitive processes, such as learning and attention focusing, on the functional brain networks architecture. The dynamical model is represented in the general form of coupled kinetic equations for the activities Ai(t) of the cognitive modes. In this representation, function F characterizes the competition of the modes that can depend on internal and external conditions represented by the parameters in parentheses.

chunking dynamics, that is, a means to reduce hierarchical complexity by subgrouping informational items to allow further processing. The model has to be hierarchical and take into account the sequential interaction of different cognitive functions on subsequent sequential cognitive episodes. It is important to note that anatomical connectivity creates limitations, which act to constrain and stabilize network configurations towards a limited number of robust states. Reconfigurable coupling in a network can lead to significant modifications in functional output [41] (Figure 2B). Numerous functional networks, including hubs, have been identified using different approaches [2,42]. One vital observation of these networks is that they are hierarchically constructed. Specific nodes have been identified as critical points in the connective network [42–44]. These nodes are often shared across multiple networks and represent an energy economy compromise [45]. There is a growing recognition that these adaptive networks change over time and that some aspects of these connections may be transient [46]. These can also be seen as interconnected active dynamical states with hierarchical connective properties with other nodes that carry out more of a sensory or reactive role [47]. These hierarchical states appear to be adaptive and temporally linked through a cascade of activation [48]. In fact, it has been suggested that to effectively adapt to the environment a temporally dynamic set of clusters is required [49]. Thus, this hierarchical network may have transients – metastable states – that change over time [50] in response to the perceived and predicted needs of the system [47].

An interesting approach is based on a state space model for cognitive functions that yield not only spatial maps of activity but also its temporal structure along with spatially varying estimates of the hemodynamic response [51]. Similarly, a low-dimensional feature space for representing the data has been proposed based on cognitive activity across modalities [52]. Using fMRI data collected during mental arithmetic, researchers have demonstrated the ability of this neurophysiologically inspired model to represent spatiotemporal information about specific mental processes. Including time in the cognitive coding space is crucially important for most cognitive functions, for example, for language comprehension and decoding, which requires the dynamic interaction between multiple brain regions [53]. Recent research using hidden Markov models has also shown that rapid state fluctuations (100–200 ms) can be found in magnetoencephalography and that these states have recognizable sequential patterns [33] (Figure 2C,D). Cognitive inhibition and winnerless competition: transient dynamics Inhibition plays a key role in the goal-directed control of thought and behavior, memory, decision-making, speech production, and other cognitive tasks [54–56]. Around a decade ago, a dynamical principle, based on inhibition, was suggested to be a leading mechanism of many sequential cognitive activities: the winnerless competition (WLC) principle [27,37,57]. WLC is a general dynamical phenomenon that leads to sequential activity among participants. These participants 3


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Figure 2. Transient functional dynamics in the brain. (A) Time series of anticorrelated switching in different functional networks during resting state (reprinted from [32], copyright 2005 National Academy of Sciences). Arrows indicate intraparietal sulcus (IPS), posterior cingulate/precuneus (PCC), and medial prefrontal cortex (MPF). (B) Illustration of the stimulus-dependent reorganization of the functional connectivity by the frontoparietal brain network (FPN) among visual, auditory, and motor systems across two different tasks. Global variable connectivity is depicted by the shifting connectivity pattern (red lines connecting FPN to other brain networks; see [38] for details). The importance of sequential switching between network arrangements is signified by blue lines between the two networks. Adapted from [38], with permission from Macmillan Publishers Ltd, copyright 2013. (C) Using a hidden Markov model for detection of state change in magnetoencephalography scanning during resting state revealed that four to 14 states could summarize brain behavior very well (adapted from [33], copyright Baker et al.). An eight-state model is shown in (C). The spatial extent of these brain states in the physical space show notable similarity to the ‘traditional’ resting state networks shown using fMRI. (D) The current state was predictive of a subsequent state, suggesting chains of states during resting state cognition. The increased probability of sequential visual networks (8-to-7 and 2-to-6) suggests the completion of a motivated visual search behavior (adapted from [33], copyright Baker et al.).

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Figure 3. Winnerless competition (WLC) dynamics observed in several nervous systems. In the context of sensory systems, the response to a stimulus is provided by the receptors to the insect antennal lobe. It is the intrinsic transient dynamics of the complex antennal lobe system that maps such input to a sequential representation as seen in these single-trial responses of 110 antennal lobe neurons to one odor shown in (A) (gray bar, 1 s). Panel (B) shows the projections of neuron trajectories, representing the succession of states visited by this neural network in response to one odor. Red lines, individual trials; black line, average of ten trials. Abbreviations: B, baseline state; FP, fixed point, reached after 1.5 s. (A,B) Adapted from [27], reprinted, with permission, from the American Association for the Advancement of Science. Panel (C) shows the taste-specific robust sequential pattern observed in neurons of the gustatory cortex of the rat (adapted from [28], copyright 2007 National Academy of Sciences). These panels show the sequential WLC activity among ten cortex neurons in response to four taste stimuli. A model of joint temporal activity (the ticks denoting the action potentials) reveals that the network behavior is best represented by four discrete states in a WLC setting. The dashed horizontal line denotes the threshold (P = 0.8) above which the network is considered to be occupying the corresponding state (i.e., the state becomes the winner). The second row lists the outcome of the four replicates of the previous experiment on the same network and confirms the reproducibility of the sequential activity: the order of the observed states is the same in each trial.

can be groups of neurons, information items, or different decisions. In the brain, it is organized by inhibitory processes [57]. For example, in the context of insect olfactory perception, the response to a stimulus is provided by the receptors to the antennal lobe. It is the intrinsic transient dynamics of perception systems that maps such input to a sequential representation (Figure 3). WLC plays a key role for the formation of robust, that is, stable and reproducible, cognitive sequences. Similarly, complex cognitive processes such as binding and decision-making can be viewed as a result of transient sequential competition between metastable states (i.e., saddles; Figure 4). Dynamical principles and basic model To create a mathematical model bridging the mind and the brain, we need to specify the requirements of the system. Nonlinear dynamical modeling of human mental activity (cognition, emotion, perception, and their interaction) can be based on the following general principles or concepts [36,58]: (i) dynamical equations have to incorporate variables – activity in brain modes – that represent the evolution of brain elements in their temporal coherency and the model has to have solutions that correspond to metastable patterns in the brain; (ii) sequential switching between metastable states must have robust dynamics. Such transient dynamics can be based on a nonlinear process of mutual interaction of many informational items or spatiotemporal modes (i.e., WLC); (iii) dynamics are controlled through inhibition of cognitive processes [27,37,57,59], which can be represented by nonsymmetrical inhibitory connectivity balanced by excitation; and (iv) dynamics are sensitive to prior neural (i.e., memory) and ongoing environment information. This approach determines the structure of a basic dynamical model of mental processes.

In the case when it is possible to separate the dependence of the mode variables from their temporal and spatial coordinates, the basic model can be written in the simplest form as generalized Lotka–Volterra (GLV) equations: # " N X d Ai ¼ Ai mi ðSÞ (1) ri j A j þ hðtÞ dt j1 Here, Ai (t) characterizes the temporal evolution of the cooperative dynamics of the i-th mode activity, N is the number of modes, mi(S) is the parameter that represents both intrinsic and external excitation, rij describes a mutual inhibition between the cognitive modes, h(t) is noise perturbing the system, and S is the informational input. Such a set of equations can also incorporate the evolution of cognitive resources, for example, attention, working memory, language, and their hierarchical organization [58,60,61]. The interaction of different modalities, such as emotion and cognition, which is important for the understanding of normal and pathological mental dynamics, can be described by the same type of equations [62,63]. Note that WLC dynamics can also be implemented with a wide variety of models (e.g., see [64–67]). How high-dimensional networks with complex connectivity give rise to functionally meaningful dynamics is a key question for understanding cognition. Low-dimensional functional cognitive dynamics, in fact, arises as a result of the timescale separation of the transients and emergent slave variables. As a result, the stable low-dimensional space in which the functional dynamics occurs is regarded as a stable subspace onto which the entire systems dynamics can be collapsed. It is important to note that lowdimensional functional dynamics can also be obtained from 5


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Figure 4. Convenient landscape metaphors to describe transient dynamics with metastable states are based on a mathematical object called saddle. Panel (A) shows a saddle with two stable and two unstable separatrices. A separatrix is a surface or curve that refers to the boundary separating two modes of behavior. A set of saddles can be sequentially connected by unstable separatrices (B) to form a stable heteroclinic channel (C). Panel (D) illustrates the low-dimensional heteroclinic dynamics of a large neuronal model network – 200 excitatory/inhibitory neuronal clusters in this example [64]. Panel (E) illustrates the transient dynamics of attention in the framework of the basic model described in Equation 1; in this case, one cognitive modality out of three requires full attention [58]. Panel (F) represents attention sharing (sequential switching of attention among three different modalities) in the same model with different intrinsic/extrinsic inputs [58]. Heteroclinic dynamics may serve an appropriate mathematical framework for robust transient processes that can be treated as an itinerary pass through metastable states. A heteroclinic channel is robust provided that the compression of the phase volume in the vicinity of the metastable states is stronger than the stretching, and trajectories that come to this area become prisoners, and thus unable to leave it [15,36].

firing rate neuron models by placing biologically realistic constraints (i.e., inhibition) on the coupling [66]. Multimodality dynamics Transient dynamics of multimodal cognitive processes can be illustrated on an example of the sequential working memory capacity problem. Everybody has the experience that it is impossible to memorize large multistep driving directions, that is, to perform a sequence such as this: make a left turn after the second light, then go straight for three lights, and make a right turn after the first light, then make a right turn again, etc. Usually after four or five steps the driver is lost. Why? To answer this question it is critically important to remember that correct recall of a loaded sequence is a transient dynamical process, and recalling a sequence of information items without order mistakes requires the stability of this process. Analyses of the sequential stability of the chain of items in the framework of Equation 1 have shown that with an increasing number of items the probability to correctly recall the order of the sequence drops exponentially to zero after some plateau [68,69]. The exact finite capacity of working memory is equal to the length of the plateau. Of course, the number of the correctly recalled items can depend on many factors – personal features, specificity of the performed cognitive function, and level of the complexity of items and their associations with habitual items in long-term memory. In particular, if the driver is given additional information about turning points together with 6

the directions, such as colors and architecture of the buildings on the side of intersecting streets, the length of the correctly recalled chain can be substantially longer. Such information makes the creation of ‘chunkable’ information easier. To determine how contextualized information can improve working memory capacity, it is necessary to analyze the stability of the chunk sequence. This is a multimodality problem – one has to consider the sequential dynamics of the chunks, the interaction between elementary items that form a chunk, that is, binding and dynamics of buffers that determine the interaction between elementary items and chunk sequences. Thus, to model chunking/binding dynamics, one can use several levels of hierarchical organization based on Equation 1 [60,70]. It can be hypothesized that the finite capacity of sequential memory is determined by the stability of the sequential item competition. This concept is supported by results that demonstrate a systematic relationship between working memory capacity and the level and method of chunking [68]. The quality of recall of the chunking sequences depends on the recall conditions. In particular, a too-fast recall may disrupt the sequence order [71]. In general, a multimodality dynamical model for cognitive processes has to take into account the following interactions: (i) a competitive interaction between modes within one modality that guarantees a sequence; (ii) the interaction through mutual excitation among modalities; and (iii) the competition for attentional resources that are partially separated for different modalities. Such


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Figure 5. Multimodality interactions. Panel (A) shows an example of three modality binding networks in the context of the discussed model (filled black circles represent inhibitory connections, unfilled circles represent excitatory connections). Panel (B) is the corresponding phase portrait of the binding dynamics. The unstable separatrices that connect different metastable states are 2D in this case [80]. Q and G represent the metastable states and S are the corresponding separatrices. Panel (C) illustrates an example of a three-level chunking hierarchical network architecture. It can be, for example, text creation: with the first level representing the organization of sentences; second level, paragraph creation; and upper level, chapter organization. Filled black circles represent inhibitory connections; triangles represent excitatory connections responsible for the choosing of informational items. Spheres represent the informational items or units (metastable stables). Different colors indicate different chunks. All connections inside the elementary items are inhibitory. Panel (D) is the corresponding phase portrait of the chunking activity in the phase space of auxiliary variables [60,70]. Blue trajectories represent the dynamics inside the chunk. Green trajectories represent the chunk sequential switching.

competition cannot be too strong so that it remains possible to achieve a synchronized collaboration in time of all components [72]. In their combined phase space, these interactive modalities form a specific network of heteroclinic channels. Uncertainty, prediction, and hierarchical thinking One of the core functions of the brain is the prediction of future states [73]. We have to consider how predictive patterns of the mind result in predictable structured patterns in the brain (Figure 1). Efficacious models of cognition should be able to accurately represent the predictive capacities and tendencies of the brain. The successful dynamical description of sequential cognitive activity provides an opportunity to make these probabilistic predictions. The corresponding dynamical model has to be hierarchical so as to link the processes that form elementary informational items (binding) to create the chunks –blocks of elementary items (Figure 5). The structure of the heteroclinic networks in these cases is determined by specific additional unstable separatrices of metastable states forming cognitive networks which, for example, can be modeled as a heteroclinic cylinder or a heteroclinic hierarchical pyramid. The thoughts

generated over time are related to the chaotic changing of cognitive topics in the framework of a given semantic space, and we can conceptualize these as ‘chunks of thoughts’. These chunks implement the semantics and thus have a slower timescale. In some cognitive activities, chunk dynamics have to be irregular to describe seemingly random thought switching (i.e., as that seen in resting state [33,74]). The structure of the semantic network keeps the information about past cognitive experience that provides a cognitive basis for prediction. This is a lower level hierarchical fast network, responsible for the probabilistic prediction of specific events in the framework of each chunk. As the level of excitation in the cognitive network increases, there are many possible ways for the cognitive chain to continue from the moment corresponding to the beginning of the chunk to future events in which new unstable separatrices emerge. To find the probability of realization for one of such ways, it is necessary to estimate the speed of the information flow along the new separatrix – possible steps to the future – and make a probabilistic prediction step by step [60]. Thus, one can use this dynamical approach and extrapolate into the future based on the current state and rate of change of a dynamical system [75]. 7


Opinion Concluding remarks Dynamical models can provide a much-needed bridge between our concepts of mind and measurable brain functioning. However, the creation of such models requires a close bidirectional interaction of the dynamical model and the experimental observations (Figure 1). The experimental basis for cognitive modeling needs state-of-the-art tools for clustering neural population activity based on their temporal dynamics. In general, cognitive activity is transient. To describe such activity, a low-dimensional dynamical model has to be based on several key principles such that its variables represent the evolution of brain elements in their temporal coherency, and its solutions correspond to metastable patterns in the brain. The sequential activity between these metastable states has to be robust, which can be controlled by inhibition through WLC. The resulting dynamics are sensitive to ongoing environment information. The application of these basic principles in designing dynamical models will help improve our understanding of brain activity. A similar idea has also been used in [76] where the authors proposed a model of behavioral sequence generation. Of course, the intensive ‘meeting of the minds’ – between nonlinear dynamics and cognitive experimentalists – can influence the development of new methodological tools in both fields. For example, novel real-time closed-loop technologies in electroencephalography (EEG) and fMRI are already contributing in this direction [77–79]. This will hopefully allow future work to better integrate the brainbased models presented here with specific cognition or behavioral output in an iterative model. Many recent studies have adopted dimensionality reduction to analyze brain population dynamics and to find transient dynamical features that are not apparent at the level of individual elements. Nonlinear dynamical theory also has strong reasons to develop mathematical bases for the robustness and controllability of task-dependent transient motions. The merging of these sciences and technologies may provide new ways to understand and predict cognitive dynamics and provide a bridge between the mind and the brain. Acknowledgments The authors acknowledge anonymous reviewers and Rebecca Schwarzlose for very useful comments to improve this manuscript. M.I.R. acknowledges support from ONR (Office of Naval Research) grant N00014310205. The salary of A.N.S. was supported by grant 1I01CX000715 from the Department of Veterans Affairs Office of Research and Development. P.V. was supported by Spanish MINECO (Ministry of Economy and Competitiveness) TIN2012-30883.

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