Dynamical Principles of Emotion-Cognition Interaction: Mathematical Images of Mental Disorders

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1 Dynamical Principles of Emotion-Cognition Interaction: Mathematical Images of Mental Disorers Mikhail I. Rabinovich 1 *, Mehmet K. Muezzinoglu 1, Irina Strigo 1,2, Alexaner Bystritsky 3 1 BioCircuits Institute, University of California San Diego, La Jolla, California, Unite States of America, 2 Psychiatry Department, University of California San Diego, La Jolla, California, Unite States of America, 3 Department of Psychiatry an Biobehavioral Science, University of California Los Angeles, Los Angeles, California, Unite States of America Abstract The key contribution of this work is to introuce a mathematical framework to unerstan self-organize ynamics in the brain that can explain certain aspects of itinerant behavior. Specifically, we introuce a moel base upon the coupling of generalize Lotka-Volterra systems. This coupling is base upon competition for common resources. The system can be regare as a normal or canonical form for any istribute system that shows self-organize ynamics that entail winnerless competition. Crucially, we will show that some of the funamental instabilities that arise in these couple systems are remarkably similar to enogenous activity seen in the brain (using EEG an fmri). Furthermore, by changing a small subset of the system s parameters we can prouce bifurcations an metastable sequential ynamics changing, which bear a remarkable similarity to pathological brain states seen in psychiatry. In what follows, we will consier the coupling of two macroscopic moes of brain activity, which, in a purely escriptive fashion, we will label as cognitive an emotional moes. Our aim is to examine the ynamical structures that emerge when coupling these two moes an relate them tentatively to brain activity in normal an non-normal states. Citation: Rabinovich MI, Muezzinoglu MK, Strigo I, Bystritsky A (2010) Dynamical Principles of Emotion-Cognition Interaction: Mathematical Images of Mental Disorers. PLoS ONE 5(9): e oi: /journal.pone Eitor: Vlaimir Brezina, Mount Sinai School of Meicine, Unite States of America Receive May 21, 2010; Accepte August 11, 2010; Publishe September 21, 2010 Copyright: ß 2010 Rabinovich et al. This is an open-access article istribute uner the terms of the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original author an source are creite. Funing: M.I.R. an M.K.M. acknowlege the support from ONR grant N M.K.M. acknowleges the support from the Jet Propulsion Laboratory ( ). I.S. is supporte by the National Institutes of Health (NIH) grant NH A.B. is partially supporte by Saban Family Founation. The funers ha no role in stuy esign, ata collection an analysis, ecision to publish, or preparation of the manuscript. Competing Interests: The authors have eclare that no competing interests exist. * mrabinovich@ucs.eu Introuction The view that the brain is an active system that entails the acquisition an maintenance of information for responing to environmental events has a long history [1 4]. On a coarse grain level of escription, mental brain activity can be represente by a ynamical moel as the activity of a complex nonequilibrium system [5,6]. In spite of the fact that the brain is a noisy place, i.e., iniviual responses of single neurons to stimuli are highly variable, the cooperative activity of a large number of neurons is robust against noise an reproucible [6,7]. The principles of mental activity an, in particular those regaring the cognitionemotion interaction that we are going to iscuss in this paper, are base on experimental observations supporting the following statement: the human brain is intrinsically organize into active, interactive functional networks an its effective coarse grain activity can be escribe by a ynamical moel. Traitional efforts in moeling ynamical phenomena in the brain are preominantly base on the premise that ynamical systems ten to converge to stable fixe points or ynamical states (limit cycles or strange attractors) where the ensity of all flows (matter, energy, or information) are balance an o not change (see, for example, [8]). Active neuronal networks in some specific conitions - (with symmetric reciprocal interactions) give rise to a convergent mental activity involving multiple attractors [9,10]. There may be some cognitive activities, such as associative memory [11], which fits the attractor-oriente escription. However, mental computing with attractors generally limits the use of complex ynamical networks. Once the attractor (or its vicinity) is reache, the ynamical nature of the brain becomes irrelevant; therefore, when attractors mark the terminal states of mental process, this behavior coul be formulate equally effectively by an algebraic cause-response mapping. Furthermore, this scheme overlooks the qualities of the (transient) path from the initial conition to the attractor, an important phase where the brain coul exploit its remarkable repertoire of behaviors. Thus, confining ynamical moels of the brain with global an symmetric coupling is not only unrealistic, but also rules out a continuum of opportunities for moeling an unerstaning mental activity. In this paper, we propose an alternative paraigm, i.e., stable transient ynamics, which can be implemente in the realistic case of non-symmetric connections between interacting mental agents. One of the most intriguing components of mental transients is the metastable state, which provies the structural stability of transient behavior [12]. Metastability is a system-level phenomenon, which is becoming increasingly popular in neuroscience for the eluciation of human information processing an pattern recognition. Metastability imposes semi-transient signals in the brain, which persist briefly an iffer from the usual equilibrium state [13]. The metastable activity of the cortex can also be inferre from behavior [14]. Metastability is a principle that escribes the brain s ability to make sense out of seemingly ranom environmental cues [15,16]. The existence of mental PLoS ONE 1 September 2010 Volume 5 Issue 9 e12547

2 metastable states, supporte by interactions observe among iverse brain centers or neuron groups [17 21], is the result of selforganization in very complex neuronal systems. The temporal orer of metastable states is etermine by the functional connectivity of the unerlying networks an their causality structure [22]. The mathematical image of a metastable state is a sale set in the working (state) space of the brain; the transition between these sales occurs via unstable separatrices connecting them (see Fig. 1). There is a substantial experimental support [13,23,24] (also outline below) that metastability an transient ynamics are key phenomena in brain ynamics. Neuroimaging an multi-electroe recoring experiments reporte in the last ecaes have shown that various brain functions an psychiatric isorers are represente by ifferent spatio-temporal brain activity patterns. Such patterns are a sequence of metastable states. Each metastable state is the result of a coorinate interaction between emotional, cognitive an perceptional moes; that is, a set of neuronal groups in ifferent parts of the brain, which are self-organize for the execution of a specific mental function. The moes interact with each other accoring to the following general principles: (1) they compete for limite mental resources (attention, memory), (2) they emonstrate a stable an reproucible ynamics that is sensitive to the informational signals; an (3) their ynamics is transient an evolves via metastable states. Base on these principles, we introuce a ynamical moel of interaction of moes that we label as emotional an cognitive. The moel has a form of couple generalize Lotka-Volterra kinetic equations an has emonstrate a spectrum of qualitatively ifferent activity patterns an bifurcations epening on the value of a moerate number of control parameters. We wish to examine the ynamical structures that emerge when coupling cognitive an emotional moes, an relate them tentatively to brain activity in normal an pathological states (see, for example, [25,26]). The paper is organize as follows: first, we introuce a canonical moel for emotion- cognition moe interaction base on the above-mentione main principles. Then we analyze its ynamical features an the corresponing ynamical objects in phase space which correlate with specific types of mental activities in healthy an isorere brains. We will illustrate the moel abilities on examples: the spontaneous resting-state ynamics characterize by low-frequency pulsations, an the ynamics of anxiety isorers, such as panic attack an obsessive-compulsive isorer (OCD). We also analyze a cognitive performance arousal interaction focusing on a hysteresis phenomenon. Materials an Methos Mental Moes: Dynamical Variables an Phase Space Numerous attempts have been mae to quantify cognition, i.e., problem solving by information processing, an emotion, i.e., spontaneous motivation an subsequent implementation of a behavior. Being irectly relate to the processing of auxiliary information, cognition has attracte relatively more attention compare to emotion in these efforts, particularly in the form of ecision-making tasks [27,28]. Although several tests aiming to assess emotions exist (see, for example, [29]), these have often been confoune by concomitant cognitive processes, such as appraisal [30,31], ecision making [32], or memory [33,34]. Which variables we neee to escribe the evolution of the emotional an cognitive moes 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 escription of turbulence can be mae using equations for coarse-grain liqui particles; the micro etails of the molecular ynamics are irrelevant. Of course, these micro etails are important because they etermine the parameters of the macroscopic moel. However, the basic coarse-grain equations are much simpler an transparent. Although the situation regaring mental ynamics is much more complex, we can still apply the turbulent flow analogy. Using this approach, a neural mass moel has been suggeste for the simulation of cortical activity [36 38]. Our approach base on mental moe interaction - is also coarse-grain. The ynamical variables, i.e., the amplitue of the ifferent mental moes escribing emotion, cognition, an mental resources consume 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 escribe by the interaction of a finite number (N) of cognitive moes an that such interaction is both reproucible an istinguishable over time. Thus, a spatiotemporal movie of such cognitive activity can be capture, 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 moes from the experimental ata, e.g., by principal or inepenent components analyses of temporal brain activity [39 41]. Thus, the cognitive activity at time t can be represente as PN i~0 A i ðþu t i ðkþ, where U i (k) is a function that characterizes the average relative activities of k participants of a istribute Figure 1. Heteroclinic chain. (Left panel) The simplest sequence of two metastable states (i.e., sales), where the unstable separatrix of the first sale is owne by the stable separatrix of the secon. (Right panel) The sequence can be (structurally) stable, i.e., attractive for any point in its vicinity marke by the ark stripe. oi: /journal.pone g001 PLoS ONE 2 September 2010 Volume 5 Issue 9 e12547

3 neuronal set that forms the i-th cognitive moe, an A i (t) is the level of activity of this moe at time t. Some of the cognitive moes, can be responsible for the interaction with emotion, for example, arousal an generation of any given coping strategy (see also [42]). The number N of these moes epens on the level of etails that we wish to escribe. Emotional activity can be represente in the same way - P M j~0 B j ðþv t j ðþ, l where B j (t) are ynamical variables an V j (l) is a function that characterizes the structure of the j-th emotional moe. The ensemble of emotional moes inclues both positive an negative emotions in our moel. Resources are represente in a similar manner. A Canonical Moel of the Mental Dynamics In the last few years, the nonlinear ynamical theory has formulate the concept of stable transients that are robust against noise, yet sensitive to the external signal [12,24]. The mathematical object that correspons to such stable transients is a sequence of the metastable states that are connecte by special trajectories name separatrices (see Fig. 1). Uner proper conitions (as outline in the Appenix), all trajectories in the neighborhoo of the metastable states that form the chain remain in their vicinity, ensuring robustness an reproucibility in a wie range of the control parameters. Because such sequence is possibly the only ynamical object that satisfies the ynamical principles that were formulate above, we assume that from the ynamical point of view, mental activity is also a sequence of the metastable states. The following is the formulation of the esire features of the moel: the moel must be issipative with an unstable trivial state (origin) in the phase space an the corresponing linear increments must be stabilize by the nonlinear terms organize by self- an mutual-inhibition (moe competition); the phase space of the system must inclue the metastable states that represent the activity of an iniviual moe when other moes are passive; an finally these metastable states must be connecte by separatrices to buil a sequence. Well-known rate moels in neuroscience satisfy these conitions in some regions of the control parameter space [43,44]. Thus, the canonical moel escribing the moe ynamics employs the nonlinear rate equations: t Ai t A iðþ~a t i ðþ t : F i ða,b,r,sþ t Bi t B iðþ~b t i ðþ t : W i ða,b,r,sþ h i t R iðþ~r t i ðþ t : Q i ða,b,r,sþ where A i $0, i = 1,,N, represents the cognitive moes, B i, i = 1,,M, represents the emotional moes an R i, i = 1,,K, represents the resources consume by these mental processes. F i, W i, an Q i are functions of A i, B i, an R i, respectively. The collections of N cognitive moes, M emotional moes, an K resource items are encapsulate in A, B, an R, respectively. When initiate properly, this set of equations ensures that all the variables remain non-negative. The vector S represents the external or/an internal inputs to the system an t A, t B, an h are the time constants. First, let us apply the moel (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 min that the competition between the ifferent moes of cognitive activity can be escribe in the simplest form of functions on the ð1þ right sie of equation (1), i.e., F(A,S) being linear, we can present the first set of equations (1) in a stanar form of a generalize Lotka-Volterra (GLV) moel [45]: " # t A t A i~a i m i ðsþ{ XN r ij A j zgðþ t Here m i (S) is the increment that represents both intrinsic an external excitation, r ij is the competition matrix between the cognitive moes, g(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 moel can escribe the competition between the emotional moes when cognition oes not influence the emotion. The moel (2) has many remarkable features, which we will use to buil an unerstan the canonical moel; epening on the control parameters, it can escribe a vast array of mental behaviors. In particular, when connections are nearly symmetric, i.e., r ij <r ji, two or more stable states can co-exist, yieling multistable ynamics where the initial conition etermines 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 moes helps to solve an apparent paraox relate to the notion that sensitivity an reliability in a network can coexist: the joint action of the external input an a stimulus-epenent connectivity matrix efines the stimulus-specific sequence. Dynamical chaos can also be observe in this case [46]. Furthermore, a specific kin of the ynamical chaos, where the orer of the switching is eterministic, but the lifetime of the metastable states is ranom, is possible [47]. Similar timing chaos with serial orer has been observe in vivo in the gustatory cortex [23]. For the moel (2), the area in the control parameter space with a structural stability of the transients has been formulate in [12] (see the Appenix in File S1). Describing the interaction between the cognitive moes, emotional moes, an the resources consume by these mental processes, we are particularly intereste in a structurally stable transient mental activity, which can effectively escribe the reproucible activation patterns uring normal mental states an ientify specific instabilities that correspon to mental isorers. Base on the GLV moel (2), we introuce the system (1) as follows: t Ai t A iðþ~a t i ðþ t : s i ðs,b,r A i~1,...,n t Bi t B iðþ~b t i ðþ t : f i ðs,b,r B i~1,...,m Þ{ XN j~1 " # j~1 Þ{ XM j~1 r ij A j ðþ t " # j ij B j ðþ t za i ðþg t A ðþ, t zb i ðþg t B ðþ, t " # h i A t Ri A ðþ~ri t AðÞ: t XN A j ðþ{ t XK A R m A {w X K B A R m B z AðÞ t, j~1 m~1 m~1 ð5þ i~1,...,k A ð2þ ð3þ ð4þ PLoS ONE 3 September 2010 Volume 5 Issue 9 e12547

4 " # h i B t Ri B ðþ~ri t BðÞ: t XM B j ðþ{ t XK B R m B {w X K A B R m A z BðÞ t, j~1 m~1 m~1 ð6þ i~1,...,k B The propose moel (3) (6) reflects a mutual inhibition an excitation within an among these three sets of moes (see Table 1). These moes epen on the inputs through parameter S (that may represent, for example, stress, cognitive loa, physical state of the boy). The variables R i A an R i B characterize the K A an K B resource items that are allocate to cognition an emotion, respectively. The vectors R A an R B are the collections of these items that gate the increments of the cognitive an emotional moes in competition. The characteristic times h of the ifferent resources may vary. The coefficients w A an w B etermine the level of competition between cognition an emotion for these resources. Each process is open to the multiplicative noise enote by g an terms in the equations. The values of the increments s i an f i epen on the stimuli an/or the intensity of the emotional an cognitive moes, respectively. The only esign constraint that we can impose on the increments s i an f i is that they must stay positive. Three types of interactions are escribe by the moel (3) (6): (i) a competitive interaction within each set of moes; (ii) the interaction through excitation (increments); an (iii) the competition for resources. For the latter, which occurs via variables R A an R B, one only nees a proper selection of the parameters w A an w B. Despite the computational simplicity in their selection, they appear to be highly iniviual- an task-specific. The time constants are the ecisive parameters of the moel an shoul be etermine a hoc experimentally. The values of the control parameters of the moel, which ensure stability of the transients (for normal behavior), can be obtaine from the inequalities that escribe the ratio between compressing an 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 liste in the moel. The brain imaging ata available toay oes not reveal the etaile structure of the moes an values of parameters, most importantly, the connectivity matrix to specify the moel for ifferent mental functions an isorers; therefore, a complete theoretical escription an preiction is not possible toay. In spite of this, the moel has a large ynamical repertoire an has just enough number of parameters to emonstrate possible behaviors an transitions among them, i.e., bifurcations. This capability, together with emonstrate success in representing some key phenomena observe in the real brain, is a valuable qualitative preiction by itself an can be useful for unerstaning the origin of observe mental phenomena such as epression, working memory, an ecision making in a changing environment [48,49]. The ynamical objects in the phase space of the moel representing mental processes are influence by the intrinsic brain ynamics an by the external stimuli. For example, uring sequential ecision-making, sequential working memory or navigation, the image of the cognitive ynamics is a stable transient, while other common cognitive activities, such as those pertaining to music [50] or linguistic functions [51] can be represente by the recurrent ynamics. Emotion can also emonstrate a whole range of the ynamical behaviors: transient regimes similar to cognitive ones, recurrent regular or irregular recurrence ynamics corresponing to moo changes; an long lasting equilibria associate with clinical cases of eep epression or constant excitement. Results Moulation Instability: possible ynamical origin of lowfrequency resting-state oscillations Let us first analyze spontaneous mental ynamics in a stationary environment where the brain is not engage in a particular cognitive function, i.e., resting-state brain ynamics. Such resting state is relate to the ynamics of the Default-Moe Network (DMN), which is a set of specific brain regions whose activity is preominant uring the resting state [52]. We have chosen the control parameters of the moel (3) (6) in the area of the control parameter space where the basic ynamics of both cognitive an emotional moes emonstrate simple rhythmic activity (oscillations with a characteristic time scale of 2 3 sec). Such inepenent emotional an cognitive activity has been observe uring weak competition. When competition becomes larger than the critical value, this simple rhythmic activity becomes unstable ue to moulation instability, thereby a stable limit cycle appears on the phase plane of the mean activity: At ~ ðþ~ 1 X N A i an N i~1 ~B ðþ~ t 1 X M B i. This moulation instability leas to stable Low M i~1 Frequency Oscillations (LFO), as shown in Fig. 2 (see also [53]). Table 1. Moel parameters an their values use the simulations. Parameter Role Range of values in simulations t Ai Time constants for cognitive moes 1e-2 to1e-1 t Bi Time constants for cognitive moes 1e-2 to 1e-1 h i A, hi B Time constants for resource ynamics 1 r, j Competition matrices inucing metastable state sequence Selecte accoring to the inequalities in Appenix, assuming all increment values equal unity s i Increments to the cognition moes locating the metastable states Depenent variable within 0 an 1: proportional to exogenous input S, inversely proportional to either a specific emotion moe, or the total emotional activity, i.e., S B i f i Increments to the emotion moes Depenent variable within 0 an 1: either a constant of 1 or inversely proportional to S A i g A,g B, A, B Noise components Uniform ranom terms within 0 an u, where u is set between 1e-6 an 1e-3 w A,w B Regulate the resource moes competition Constants set to 1, except in hysteresis simulation, where w A = 0.33 an w B = 1.0 oi: /journal.pone t001 PLoS ONE 4 September 2010 Volume 5 Issue 9 e12547

5 Figure 2. Anti-phase Low-Frequency Oscillation of mental activity. In the resting state, it is a result of two groups of efault moes competition moulational instability (S is constant, N = M = 5 in the moel (3) (6)). The black envelope on the mile plot is the total cognitive activity as preicte by the moel. Its competition with the emotion moes (top row) results in a pulsation as observe in many EEG an fmri stuies. The bottom figure, reconstructe base on the ata presente in [53], shows one such observation in the brain s resting state. oi: /journal.pone g002 The average-in-time time series inicates that the observe robust moulation process is close to the quasi-perioic LFO. An increasing number of EEG an resting state fmri stuies in both humans an animals inicate that spontaneous low frequency fluctuations in cerebral activity at Hz represent a funamental component of brain functioning. In particular, resting state fmri measures show stable properties of LFO (see Fig. 2), the nature of which are only beginning to be uncovere an have prouce a lot of ebates. Observe LFO in fmri, for instance, coul simply be ue to the ban-pass filtering effect of the hemoynamic response function. In general, the LFO fluctuations observe with fmri are not the same as the unerlying neuronal fluctuations because they have been passe through a hemoynamic response function. However, some ata [54 57] supports the hypothesis that LFO are correlate with the network s activity, i.e. moes cooperative ynamics (e.g. ue to their moulation or synchronization). The moulation instability that we have observe in the computer experiments iscloses a plausible ynamical origin of low frequency mental moe ynamics in the resting states, possibly relate to the cortical-subcortical crosstalk [58]. Importantly, iscovering changes in the resting state ynamics in various psychiatric isorers may provie a new tool for the iagnosis of psychopathology or for ientifying iniviual variations in physiological arousal [59 62]. Psychopathology an Emotional Instability Let us consier anxiety isorers that have been associate with abnormal activity in istinct brain networks or moes incluing hippocampus anterior cingualte, insula, basal ganglia, an some others [63,64]. Although there is large symptomatic overlap between ifferent anxiety isorers, each isorer can be characterize by a specific quality of anxiety ynamics, i.e., specific emotional instability, an, thus, can be represente by its own ynamical object in the phase space of the canonical moel with appropriate value of the control parameters. Below we will explain a possible ynamical escription for panic attack an Obsessive-Compulsive Disorer (OCD). From the ynamical theory point of view, normal emotion- an cognitionrelate activities occur between pathological but stable states (like eep epression, coma, etc.) an the ege of chaos which is also a pathological. Psychopathology, like panic attack an OCD, is associate with irregular ynamical behavior an is often more consistent with chaotic ynamics (see, for example [65 68]). Mathematical images of such ynamics are transient chaos or strange attractors [69]. Panic attacks occur in many ifferent types of anxiety isorers. They have a suen onset an typically peak within 10 minutes. There are several interesting views an moels of panic attack in the literature [70,71]. In particular, Callahan an Sashi [71] have PLoS ONE 5 September 2010 Volume 5 Issue 9 e12547

6 suggeste an emotion moel base on the catastrophe theory which represents some qualitative characteristics of pathologic affective response. While such perspectives are useful for the integration of biophysical formulations of the interaction between the biology an social experience [72], our view is that any escription of emotion as a stan-alone phenomenon, isolate from other players of mentality, woul be limite in utility. Base on equations (3) (6), we moele the interaction of moes that can be relate to panic attack with moes of sequential cognitive activity (- e.g., sequential ecision-making). When the interaction cognition with emotion is negligibly small, the ynamical object representing such cognitive activity in the phase space is a robust perioic or quasi-perioic sequence of the metastable states. The moel reprouces this activity when the (s,r) pair satisfies certain stability conitions (see Appenix) to construct a stable chain of metastable states in the cognitive subspace of the joint phase space. Suppose now that the interaction function s(b) is nonmonotonic an large for some B components, an the resource competition is mil (i.e., w A an w B are small). As can be seen in Fig. 3, the ynamical object corresponing to the temporal interaction between emotion an cognition uring a panic attack is transient chaos, i.e., both the cognitive activity an the emotion become unpreictable for a finite time (see the time series in Fig. 3). Such chaotic mental activity can be quantitatively characterize by the maximal transient Kolmogorov-Sinai entropy, which is approximately 0.34 (uring the panic attack perio) in our example. The OCD is a type of an anxiety isorer that traps people in the enless cycles of repetitive feelings, unwante thoughts an unwante repetitive acts which the sufferer realizes are unesirable but is unable to resist compulsive rituals [73,74]. The compulsive rituals characteristic of OCD are performe in an attempt to prevent the obsessive thoughts or make them go away. Although the ritualistic behavior may make the anxiety go away temporarily, the person must perform the ritualistic behavior again when the obsessive thoughts return. People with OCD may be aware that their obsessions an compulsions are senseless or unrealistic, but they cannot stop themselves. An attractor-base escription of the OCD has been attempte recently in [75]. To moel OCD, we introuce a sequence of sales that is, metastable states, in the cognitive subspace. Each sale in this Figure 3. A simulation of a panic isorer on the emotional or the cognitive moe evolution. The panic attack arises ue to an external isturbance (represente as an instantaneous kick marke by a re ticker in the inset above) at time t = 1000s. The attack is characterize by an irregularity in both the cognition an the emotion ynamics. The magnitues (thus, the switching orer) of each group of moes are noneterministic uring this perio. In this example, the system returns to its regular pace after some perio (i.e., t.10,000 s). The top row shows an early phase of the irregular activity ue to the attack both in time an in a phase portrait. The situation after turning to normal is shown on the bottom row. oi: /journal.pone g003 PLoS ONE 6 September 2010 Volume 5 Issue 9 e12547

7 sequence has a two-imensional unstable manifol, by construction, as escribe in the Appenix. One of these imensions forms the separatrix leaing to the next cognitive metastable state along a cognitive sequence, whereas the secon unstable separatrix targets the emotional sale that represents the entry to the ritual, which is moele as a ifferent stable chain of the emotional metastable states. The ritual terminates at a sale that has many unstable separatrices, each yieling to a cognitive moe (see Fig. 4). As a result, the OCD ynamics is represente by a (N+M) imensional transient, which qualitatively istinguishes itself from the normal behavior an from other isorers characterize by a specific instability that leas to uncertainty. The result of moeling preicts that in OCD the interaction of the sequential cognitive activity (e.g., sequential ecision making), with emotion is characterize by intermittent ynamical instability. The corresponing ynamical images an phase portraits are represente in Fig. 4. The OCD moel inclues enough parameters to escribe the competition between the cognitive process an the ritual in great etail. It is possible to make the moel non-circular by arranging the unstable separatrices of the terminal ritual moe. This ajustment, however, requires clinical ata that reveal the return to the cognitive function from the ritual (see also [76]). As one can observe from Figs. 3 an 4, both a panic attack an the OCD have noneterministic components. However, the mechanisms of the chaotic behavior in these cases are ifferent: the ynamical object that represents the emotion-cognition interaction uring a panic attack is transient chaos, whereas a very specific object, which we name intermittent transient, escribes the emotion-cognition interaction in OCD. Along such an intermittent transient, a chain of metastable cognitive moes is interrupte by the ritualistic behavior characteristic of OCD with an unpreictable returning. Cognitive performance arousal interaction. Hysteresis It is well known that emotions can suenly switch cognition from one regime to another. Let us use the moel (3) (6) for the analysis of such a phenomenon, i.e., the epenence of cognitive performance on the level of arousal. The popular point of view is Figure 4. Dynamical representation of Obsessive Compulsive Disorer (OCD). (Top Left) The propose ynamical image of the OCD. Here, while the cognitive task evolves on stable transients, the ynamics shift towars a ominant intermittent transient sequence (i.e., the ritual) whose initial moe lies on the unstable manifols of the cognitive sales. During this ritual, the cognition halts an upon its completion the iniviual returns to the cognitive process, not necessarily through the last cognitive moe visite. (Bottom) A simulation of OCD by the propose moel. Here the iniviual performs a normal cognitive task represente by the five moes colore yellow-to-re. At certain perios, the iniviual performs a ritual as illustrate by four ark-colore moes in a prescribe orere. The system can enter this ritual sequence from any cognitive moe, an, upon completion of the ritual, returns back to the cognitive process via an arbitrary moe. (Top right) A phase portrait of this ynamical behavior. oi: /journal.pone g004 PLoS ONE 7 September 2010 Volume 5 Issue 9 e12547

8 that Yerkes-Doson law ictates an iniviual s reaction to a stressor: the performance improves with physiological or mental arousal, but only up to a point; when levels of arousal become too high, performance ecreases. The process is often illustrate graphically as an inverte U-shape curve. Despite its plausibility, the Yerkes-Doson law is ifficult to test empirically (see, for example, [77]). Usually the inverte-u behavior is realistic enough when the cognitive anxiety is low, i.e., when the human is not worrie. Since we are intereste in the emotion-cognition interaction, we focus on comparing the moel s preiction with previously reporte experiments in the case when physiological arousal interacts with cognitive anxiety, i.e. the human is worrie to influence his/her performance. To the best of our knowlege, currently there is no ynamical moel capable of escribing an emotion-cognitive hysteresis (see Fig. 5b). Hysteresis phenomenon is well known among sociologists an sport psychologists [78]. To explain it we nee to estimate regions or attraction of two equilibria A9 an B9. Two levels of arousal, where one region (omain) of attraction becomes much larger than the other one an vice versa, etermine the hysteresis loop. Our analysis is base on the fact that ynamics of the average levels of emotional an cognitive activities largely coincies with the ynamics of available mental resources (represente by variables R A an R B in the moel, respectively). The phase portraits of the system in the phase plane (R A, R B ) are shown in Fig. 5a. One can observe that the area of attraction of state A9 ominates uner low arousal levels. As arousal increases, the areas of attraction A9 an B9 become comparable, thus manifesting the bistability - further increase in arousal level results in a monostable conition A9 = 0. Since the system remembers the initial conitions, the ecreasing level of arousal will lea to equilibrium (calm state of min) only when the area of attraction A9 greatly excees area of attraction B9. Discussion In conclusion, we have seen that the suggeste coarse-grain moel base on the formulate principles is potentially very powerful. This moel operates on the activities of correlate neuronal groups istribute in the brain, abstractly terme mental moes. In principle, it is possible to escribe the anatomical structure of many cognitive an emotion moes, base on imaging ata that are available toay. However, from the practical point of view, at neuronal level it still may not be helpful to make a irect connection of our phenomenological moel with neurobiology. This is because it is currently impossible to etermine the functional connections between ifferent moes. Thus, at just glance, our approach (base on the observation of behavioral transformations) seems like the only reasonable one presently. A key object in the moel ynamics is a stable transient realize through a reproucible serial orer of metastable states. As a generic ynamical phenomenon, which is rare in simple systems yet common in complex ones, the sequential switching among metasable states can provie concise an constructive formulations in a variety of mental problems [79]. The prototype ynamical moels are wiely accepte in neuroscience [80]. The key physiological mechanism unerlying the winnerless competition (WLC) an sequential switching in the brain is nonsymmetric inhibition, which is known to exist in neural systems from microto macroscopic levels [81 85]. WLC is observe in biological systems, in particular, in two in vivo experiments, e.g., gustatory an olfactory sensory systems [23,86,87]. Figure 5. Hysteresis relation between cognitive performance an arousal. (a) The cognitive performance an arousal can have a hysteresis relation. The mechanism responsible from this relation is the inirect (resources) competition among the R A an R B variables as introuce in the moel. A sequence of phase portraits that correspons to ecreasing stressor (arousal) intensity is shown on the left panel. The attractors are inicate by A9 an B9. Panel (b) illustrates the hysteresis relation observe in the propose moel. The moel can successfully reprouce this isorer when (s,r) an (f,j) pairs satisfy SHC conitions; s(b) an f (A,S) are non-smooth an large for some A, B; f(a,s) increases with S; an the resource competition is strong with Q A?Q B. oi: /journal.pone g005 PLoS ONE 8 September 2010 Volume 5 Issue 9 e12547

9 Accoring to the stanar efinition, a ynamical system is a moel of a real system that unergoes a time evolution from the initial moment to infinity. The cortex, strictly speaking, is not a ynamical system uner this efinition. However, many mental functions emonstrate ynamical features an can be escribe by appropriate ynamical moels over finite intervals of time. Furthermore, the brain is somehow capable of coorinating the results of interrupte ynamical activities as a sequence of sequences (see also [88]). One can hypothesize that, in such cases, mental moes coorinate their activities in accorance with the universal principles that we have alreay iscusse in this paper. To hanle the uncertainty that is typical for the interrupte sequences one nees aitional principles - presumably the minimum preictive error, the minimum information principle [89], or the free energy principle [90]. References 1. James W (1890) The principles of psychology Henry Holt & Company. 2. 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