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Mental Modes: Dynamical Variables and Phase Space: Numerous attempts have been made to quantify cognition, i.e., problem solving by information processing, and emotion, i.e., spontaneous motivation and subsequent implementation of a behavior. Being directly related to the processing of auxiliary information, cognition has attracted relatively more attention compared to emotion in these efforts, particularly in the form of decision-making tasks [27], [28]. Although several tests aiming to assess emotions exist (see, for example, [29]), these have often been confounded by concomitant cognitive processes, such as appraisal [30], [31], decision making [32], or memory [33], [34]. Which variables we needed to describe the evolution of the emotional and cognitive modes while capturing their functional complexity? To answer this question, we look at an example of complex systems in non-living nature, such as turbulent flows [35]. A macroscopic description of turbulence can be made using equations for coarse-grain liquid particles; the micro details of the molecular dynamics are irrelevant. Of course, these micro details are important because they determine the parameters of the macroscopic model. However, the basic coarse-grain equations are much simpler and transparent. Although the situation regarding mental dynamics is much more complex, we can still apply the turbulent flow analogy. Using this approach, a neural mass model has been suggested for the simulation of cortical activity [36]–[38]. Our approach – based on mental mode interaction - is also coarse-grain. The dynamical variables, i.e., the amplitude of the different mental modes describing emotion, cognition, and mental resources consumed by them, form a joint state space (or phase space). We assume that a specific cognitive activity (e.g., appraisal or sequential navigation) can be described by the interaction of a finite number (N) of cognitive modes and that such interaction is both reproducible and distinguishable over time. Thus, a spatio-temporal movie of such cognitive activity can be captured, for example, by a series of functional Magnetic Resonance Imaging (fMRI) snapshots taken at consecutive times while the subject engages in a specific cognitive task. There are several efficient ways to extract the modes from the experimental data, e.g., by principal or independent components analyses of temporal brain activity [39]–[41]. Thus, the cognitive activity at time t can be represented as , where Ui(k) is a function that characterizes the averaged relative activities of k participants of a distributed neuronal set that forms the i-th cognitive mode, and Ai(t) is the level of activity of this mode at time t. Some of the cognitive modes, can be responsible for the interaction with emotion, for example, arousal and generation of any given coping strategy (see also [42]). The number N of these modes depends on the level of details that we wish to describe. Emotional activity can be represented in the same way - , where Bj(t) are dynamical variables and Vj(l) is a function that characterizes the structure of the j-th emotional mode. The ensemble of emotional modes includes both positive and negative emotions in our model. Resources are represented in a similar manner.
A Canonical Model of the Mental Dynamics: In the last few years, the nonlinear dynamical theory has formulated the concept of stable transients that are robust against noise, yet sensitive to the external signal [12], [24]. The mathematical object that corresponds to such stable transients is a sequence of the metastable states that are connected by special trajectories named separatrices (see Fig. 1). Under proper conditions (as outlined in the Appendix), all trajectories in the neighborhood of the metastable states that form the chain remain in their vicinity, ensuring robustness and reproducibility in a wide range of the control parameters. Because such sequence is possibly the only dynamical object that satisfies the dynamical principles that were formulated above, we assume that from the dynamical point of view, mental activity is also a sequence of the metastable states. The following is the formulation of the desired features of the model: the model must be dissipative with an unstable trivial state (origin) in the phase space and the corresponding linear increments must be stabilized by the nonlinear terms organized by self- and mutual-inhibition (mode competition); the phase space of the system must include the metastable states that represent the activity of an individual mode when other modes are passive; and finally these metastable states must be connected by separatrices to build a sequence. Well-known rate models in neuroscience satisfy these conditions in some regions of the control parameter space [43], [44]. Thus, the canonical model describing the mode dynamics employs the nonlinear rate equations:(1)where Ai≥0, i = 1,…,N, represents the cognitive modes, Bi, i = 1,…,M, represents the emotional modes and Ri, i = 1,…,K, represents the resources consumed by these mental processes. Fi, Φi, and Qi are functions of Ai, Bi, and Ri, respectively. The collections of N cognitive modes, M emotional modes, and K resource items are encapsulated in A, B, and R, respectively. When initiated properly, this set of equations ensures that all the variables remain non-negative. The vector S represents the external or/and internal inputs to the system and τA, τB, and θ are the time constants. First, let us apply the model (1) to just one form of the mental activity when cognition-emotion interaction is negligible. Let us imagine a situation where cognition changes over time while emotion remains more or less constant over time. Keeping in mind that the competition between the different modes of cognitive activity can be described in the simplest form of functions on the right side of equation (1), i.e., F(A,S) being linear, we can present the first set of equations (1) in a standard form of a generalized Lotka-Volterra (GLV) model [45]:(2)Here μi(S) is the increment that represents both intrinsic and external excitation, ρij is the competition matrix between the cognitive modes, η(t) is a multiplicative noise perturbing the system, S is the input that captures the sources of internal or external effects on the increment. A similar model can describe the competition between the emotional modes when cognition does not influence the emotion. The model (2) has many remarkable features, which we will use to build and understand the canonical model; depending on the control parameters, it can describe a vast array of mental behaviors. In particular, when connections are nearly symmetric, i.e., ρij≈ρji, two or more stable states can co-exist, yielding multi-stable dynamics where the initial condition determines the final state. When the connections are strongly non-symmetric, a stable sequence of the metastable states can emerge [12] (see Fig. 1). The non-symmetric inhibitory interaction between the modes helps to solve an apparent paradox related to the notion that sensitivity and reliability in a network can coexist: the joint action of the external input and a stimulus-dependent connectivity matrix defines the stimulus-specific sequence. Dynamical chaos can also be observed in this case [46]. Furthermore, a specific kind of the dynamical chaos, where the order of the switching is deterministic, but the lifetime of the metastable states is random, is possible [47]. Similar “timing chaos with serial order” has been observed in vivo in the gustatory cortex [23]. For the model (2), the area in the control parameter space with a structural stability of the transients has been formulated in [12] (see the Appendix in File S1). Describing the interaction between the cognitive modes, emotional modes, and the resources consumed by these mental processes, we are particularly interested in a structurally stable transient mental activity, which can effectively describe the reproducible activation patterns during normal mental states and identify specific instabilities that correspond to mental disorders. Based on the GLV model (2), we introduce the system (1) as follows:(3)(4)(5)(6) The proposed model (3)–(6) reflects a mutual inhibition and excitation within and among these three sets of modes (see Table 1). These modes depend on the inputs through parameter S (that may represent, for example, stress, cognitive load, physical state of the body). The variables RiA and RiB characterize the KA and KB resource items that are allocated to cognition and emotion, respectively. The vectors RA and RB are the collections of these items that gate the increments of the cognitive and emotional modes in competition. The characteristic times θ of the different resources may vary. The coefficients φA and φB determine the level of competition between cognition and emotion for these resources. Each process is open to the multiplicative noise denoted by η and d terms in the equations.
Table data removed from full text. Table identifier and caption: 10.1371/journal.pone.0012547.t001 Model parameters and their values used the simulations. The values of the increments σi and ζi depend on the stimuli and/or the intensity of the emotional and cognitive modes, respectively. The only design constraint that we can impose on the increments σi and ζi is that they must stay positive. Three types of interactions are described by the model (3)–(6): (i) a competitive interaction within each set of modes; (ii) the interaction through excitation (increments); and (iii) the competition for resources. For the latter, which occurs via variables RA and RB, one only needs a proper selection of the parameters φA and φB. Despite the computational simplicity in their selection, they appear to be highly individual- and task-specific. The time constants are the decisive parameters of the model and should be determined ad hoc experimentally. The values of the control parameters of the model, which ensure stability of the transients (for normal behavior), can be obtained from the inequalities that describe the ratio between compressing and stretching of the phase volume in the vicinity of the metastable states [12]. The effective number of the parameters can be much smaller than that listed in the model. The brain imaging data available today does not reveal the detailed structure of the modes and values of parameters, most importantly, the connectivity matrix to specify the model for different mental functions and disorders; therefore, a complete theoretical description and prediction is not possible today. In spite of this, the model has a large dynamical repertoire and has just enough number of parameters to demonstrate possible behaviors and transitions among them, i.e., bifurcations. This capability, together with demonstrated success in representing some key phenomena observed in the real brain, is a valuable qualitative prediction by itself and can be useful for understanding the origin of observed mental phenomena such as depression, working memory, and decision making in a changing environment [48], [49]. The dynamical objects in the phase space of the model representing mental processes are influenced by the intrinsic brain dynamics and by the external stimuli. For example, during sequential decision-making, sequential working memory or navigation, the image of the cognitive dynamics is a stable transient, while other common cognitive activities, such as those pertaining to music [50] or linguistic functions [51] can be represented by the recurrent dynamics. Emotion can also demonstrate a whole range of the dynamical behaviors: transient regimes similar to cognitive ones, recurrent regular or irregular recurrence dynamics corresponding to mood changes; and long lasting equilibria associated with clinical cases of deep depression or constant excitement.
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