On the Dynamic Interplay between Positive and Negative Affects

On the Dynamic In terpla y b et w een P ositiv e and Negativ e Aﬀects Jonathan T oub oul 1,2, ∗ , Alb erto Romagnoni 3,1,4, , Rob ert Sc h wartz 5, Abstract Emotional disorders and psyc hological ﬂourishing are the result of complex interactions b etw een positive and negativ e aﬀects that depend on external ev en ts and the sub ject’s internal representations. Based on psychological data, we mathematically model the dynamical balance b et w een p ositiv e and negative aﬀects as a function of the resp onse to external p ositiv e and negativ e even ts. This mo deling allows the in vestigation of the relative impact of tw o leading forms of therapy on aﬀect balance. The model uses a delay diﬀerential equation to analytically study the complete bifurcation diagram of the system. W e compare the results of the mo del to psychological data on a single, recurrently depressed patient that was administered the t w o t yp es of therapies considered (viz., coping-fo cused vs. aﬀect-fo cused). The mo del leads to the prediction that stabilization at a normal state may rely on ev aluating one’s emotional state through an historical ongoing emotional state rather than in a narrow presen t windo w. The simple mathematical model prop osed here oﬀers a theoretically grounded quantitativ e framew ork for inv estigating the temp oral pro cess of change and parameters of resilience to relapse. 1. In tro duction Human functioning is regulated b y a dialectical tension b etw een b oth p ositiv e and negative states. T ra- ditional psyc hology focused primarily on the negative dimension [1], whereas p ositiv e psyc hology has recently shifted the emphasis to include p ositiv e exp erience [2]. Rather than developing these tw o dimensions along indep enden t lines, psychology needs theories that systematically integrate b oth concurrently . Describing the pro cess of c hange during psychotherap y thus requires the dev elopmen t of theoretically based models that cap- ture the dynamical in teraction of positive and negative dimensions along with mathematical to ols to analyze the ev olution of these states. First eﬀorts in this directions w ere undertook with the Balanced State of Mind mo del (BSOM, see [3, 4]), an integrativ e model of p ositiv e P and negative N cognition and aﬀect. This static mo del has demonstrated that distinct ratios diﬀerentiate psychopathological, normal and optimal states. Dra wing on Lefebvre’s [5] mathematically based theory of consciousness, the BSOM mo del uses a ratio, the Emotional Balance E B = P / ( P + N ), to deﬁne emotional and cognitiv e balance. Numerous studies hav e shown that clients progress from lo w pre-treatment balances to normal or optimal balances dep ending on the success of the therap y [6, 7, 8]. The dynamical evolution of these v ariables are extremely imp ortan t, but still largely ignored. Recent works ha ve nev ertheless rep orted naturally o ccurring rhythms in daily and w eekly mo od [9, 10, 11] in resp onse to stresses: the mo del predicts a temp orary increase in amplitude of the oscillations as the p erson ﬂuctuates b et w een more intense negative emotions dealing with loss and a more positive orientation of restoring normal adjustmen t [9, 12, 13], and as the loss is gradually integrated, the tra jectory of emotional expression is “damped” ∗ Corresponding Author Email addr esses: jonathan.touboul@college-de-france.fr (Jonathan T oub oul), alberto.romagnoni@college-de-france.fr (Alberto Romagnoni), robsch77@gmail.com (Rob ert Sch w artz) 1 Mathematical Neuroscience T eam Coll` ege de F rance, Centre for Interdisciplinary Research in Biology , UMR CNRS 7241/IN- SERM 1050, Lab ex MemoLife, PSL Research Univ ersity , 11 place Marcelin Berthelot, 75231 Paris. 2 Mycenae T eam, Inria P aris-Ro cquencourt. 3 Group for Neural Theory , LNC INSERM Unit ´ e 960, D´ epartemen t d’ ´ Etudes Cognitiv es, ´ Ecole Normale Sup ´ erieure, Paris, F rance 4 Quantum Research Group, School of Physics and Chemistry , Universit y of KwaZulu-Natal, Durban, 4001, South Africa. 5 Universit y of Pittsburgh School of Medicine Pr eprint submitted to Elsevier Octob er 18, 2021 to return to a steady state. Moreo ver, it was sho wn that during eﬀective psychotherap y , esp ecially early in treatment, aﬀect tra jectories were c haracterized by extreme and unev en ﬂuctuations rather than smo oth oscillations. Using a dynamic systems mo del of change to in vestigate cognitive therapy of depression, Hay es and Strauss [14] found that greater “destabilization” of depressive patterns and increased aﬀect in tensity early in treatmen t predicted sup erior treatmen t outcome. Although relative amplitudes of oscillations and their damping may adequately describ e aﬀectiv e resp onses to normally o ccurring stressors, iden tifying additional phases of v ariability and stability may b e important to b etter understand psyc hopathology and the pro cess of c hange. In 2005, F redrikson and Losada [15] prop osed a mathematical mo del for the emergence of these ratios. The mo del w as based on a classical, chaotic dynamical system, the Lorentz’ equation, and op ened an interesting and inﬂamed debate in the comm unity . This mo del w as decisively criticized and retracted b y F redrickson b ecause it w as sho wn to be an inappropriate extrap olation from physics to psyc hology and failed to meet an y of the criteria required to apply a mathematical mo del to data [16]. But despite these criticisms, the attempt w as fundamen tal in promoting mathematical mo deling to explain the w ell established phenomenon of quan titativ ely precise ratios in psychology . Although this particular mo del of aﬀect balance was ﬂa wed, mathematic al mo deling of the laws of human psyc hology remain essen tial to adv ance the theoretical understanding of emotional dynamics and mo od disorders. This is precisely the topic of the presen t manuscript. W e introduce a mathematical mo del of the evolution in time of positive and negative aﬀect levels and ho w they ev olv e according to external ev ents. F rom the mo del, we can ev aluate a quantit y analogous to the Emotional Balance that clinicians can ev aluate on patients. W e compared the dynamics of the mo del to the ev olution of the balance betw een positive and negative aﬀect during psyc hotherapy to ascertain whether the emotional tra jectory con tained distinct patterns that characterize diﬀeren t phases of treatment. F ollo wing the growing trend that focuses on in tra-individual structures and dynamics or “p ersonalit y architecture”, we adopted an idiographic, single-case quasi-exp erimental design to p erform a detailed analysis of patient’s c hange tra jectory [17, 18]. The exp erimen tal data we consider in the presen t manuscript sho w the ev olution in time of the emotional balance of a recurrently depressed individual who w as sequentially administered three increasingly intensiv e forms of therap y . The typical ev olution of the emotional balance was examined in relationship to the type of treatmen t, stage of therap y and critical even ts o ccurring in the patient’s life. Our mathematical mo del accurately repro duces the qualitative features observed clinically. The developmen t of a mathematical theory and its clinical conﬁrmation is a complex task. A complete ap- plication of a theory to psyc hological data would require extensiv e experiments that are not y et a v ailable. As an initial step in this direction, the curren t pap er fo cuses primarily on in tro ducing the mo deling and mathematical analysis of p ositive and negativ e states with illustrative, empirical data that b oth supp orted and guided our in vestigations. These results illuminate the dynamical process associated with phase transitions and optimal outcome during therap y , and this dynamical analysis reﬁnes the BSOM conclusions: reaching health y lev els of emotional balance ma y not b e a stable steady state, and deeper therapies building up on self-image ma y lead to small amplitude lo w frequency oscillations of the EB that sho w increased stability . In other words, ﬂexibility of EB and emergence of oscillations leads to a more stable outcomes than rigid steady states, or as Confucius put it,“The green reed whic h b ends in the wind is stronger than the might y oak which breaks in a storm”. 2. Material and metho ds 2.1. Balanc e d States-of-Mind Mo del The Balanced States-of-Mind Mo del (BSOM; [4], dra wing up on a sp eciﬁcally psychological theory of con- sciousness or self-a wareness dev elop ed by mathematician-psychologist Vladimir Lefebvre [5], is a natural frame- w ork that related ratios of p ositiv e and negative aﬀects to psychopathology and optimal functioning. Theoret- ically deriv ed predictions of levels of p ositive aﬀects in distinct situations were compared to empirical scores deriv ed from cognitive and aﬀectiv e measuremen t instrumen ts. Lefeb vre et al. [19] used this theory to mo del and empirically replicate existing exp erimen tal results [20]. Applying Bo olean computations to the c haracters, 2 they generated ratios representing the likelihoo d that individuals will ev aluate themselves p ositively under ﬁve distinct mo od states: p ositive ev aluations of self in deep-p ositiv e mo od E B = 0 . 875; p ositive ev aluations of self in p ositiv e mo o d E B = 0 . 813; positive ev aluations of self in neutral mo o d E B = 0 . 719; positive ev aluations of self in negativ e mo o d E B = 0 . 625; p ositiv e ev aluations of self in deep-negative mo o d E B = 0 . 500. These ratios constitute the basic parameters of the BSOM Mo del used to represent functionally distinct states of mind (SOM) that account for conditions ranging from psychopathology to optimal functioning: Sup er Optimal, E B = 0 . 88; Optimal E B = 0 . 81; Normal E B = 0 . 72; Subnormal or Coping E B = 0 . 62; and P athological E B = 0 . 50 or b elo w. See [4, 8] for further details of the model and empirical results. With the exception of the extreme p ositiv e SOMs that are diﬃcult to obtain with instruments sensitiv e to extreme states, the mo del’s quan titative parameters hav e receiv ed considerable empirical supp ort (see [3, 16]). More recen tly , Sc hw artz and collab orators [8] trac ked depressed men during cognitive and pharmacotherap y and found that at p ost-treatment a group of a priori deﬁned “a v erage” responders ac hiev ed an emotional balance E B = 0 . 70, close to predicted normal ratio E B = 0 . 72. The predeﬁned group of “optimal” resp onders evinced an emotional balance exactly at the theoretically predicted optimal ratio E B = 0 . 81. Similarly , [21] rep orted that p eople who rated their happiness 80% were more successful on measures of income, education and p olitical inv olv emen t than those that rated themselv es either low er or higher. These ratios provide a verage, cross sectional lev els of p ositiv e and negativ e aﬀects that characterize e motional states at a giv en p oin t in time (e.g. b efore and after psychotherap y). These states presumably build up as a resp onse to external p ositiv e and negativ e ev en ts o ccurring in the individuals liv es. How ev er, the existing models do not account for the presumably nonlinear manner in whic h these states of mind progress o ver time. One imp ortan t con tribution of our work is to propose a mathematical model of ho w these states of mind build up from exp erience, the individual’s nonlinear resp onses to p ositive and negative even ts and how these dep end on the emotional balance lev els. 2.2. Mathematic al Mo del of Emotional Balanc e Dynamics This section is devoted to the introduction of our nov el mathematical mo del characterizing the evolution in time of the psyc hological state of an individual. The mo del is based on the level of p ositiv e and negativ e aﬀects, t wo quantities directly observed b y the clinicians during therapy , and on which the BSOM mo del is grounded. These v ariables evolv e dep ending on external even ts that may b e p ositiv e or negative, and that o ccur randomly in time at a rate denoted λ p and λ n , and on the individual’s response depending on their emotional balance level. Our mo del is based on the simple psychological observ ation schematically depicted in Fig. 1: dep ending on the emotional balance of a patien t, positive and negative even ts distinctively aﬀect its state of mind. Sp eciﬁcally , depressed patients are strongly aﬀected by negative even ts that can impact their emotional state for longer p eriods of time than that of non-depressed persons; in con trast, p ositiv e ev ents barely aﬀect their mo od, and are only eﬀective for a brief perio d of time [22]. The opp osite arises for non-depressed p eople, who are able to deal with negativ e even ts and b etter sustain pleasant even ts. This v ariable integration of p ositiv e and negative ev ents is central in the understanding of psychological resilience. P time N h i gh E B low EB Figure 1: Mo del of Emotional Balance dynamics. P ositive (up) and Negativ e (down) life events aﬀect the sub ject’s positive or negative aﬀects with an amplitude and a duration depending on their state of mind (high or low EB, see text). 3 F rom the mathematical viewp oin t, w e mo del the ev olution of self-assessed intensit y of p ositive P ( t ) and negativ e aﬀects N ( t ), in relationship with the emotional balance ratio: E B ( t ) = P ( t ) P ( t ) + N ( t ) . The time ev olution of the v ariables P and N is characterized by tw o main features: 1. It is driv en by random p ositive and negativ e even ts o ccurring in the patient’s life. In the absence of determinism, we consider such ev en ts o ccurring as t wo indep endent Poisson processes Π P and Π N with the same in tensity λ 0 . 2. The wa y these even ts are integrated in the patient emotional state. If a p ositiv e or negativ e even t o ccurs at time t , t wo main quantities will describ e its eﬀect on the v ariables: • the amplitude of the mo diﬁcation of the v ariables P and N subsequent to this even t. These am- plitude dep end on the emotional balance at time t . W e denote these amplitudes b y q P ( E B ( t )) and q N ( E B ( t )), resp ectively corresp onding to the eﬀect of a p ositive (negativ e) even t on P ( t ) ( N ( t )) for a patien t in a current emotional balance E B ( t ). • the timescale c haracterizing the impact, in time, of this even t. These also dep end on the emotional balance lev el of the patient at time t , and are denoted τ P ( E B ( t )) and τ N ( E B ( t )). T ypically , depressed patients are more aﬀected by negative even ts, and for longer times, than non-depressed patien ts, and are less aﬀected by positive ev en ts. In other words, b oth q P and τ P are increasing functions of the emotional balance, and b oth q N and τ N decreasing functions taking v alues in a b ounded interv al. W e will c ho ose for simplicit y q P and q N as smo oth error functions (see Fig. 2A), and constant τ P and τ N . The dynamics of the the p ositive and negative aﬀect levels therefore satisﬁes the sto c hastic diﬀeren tial equation with jumps: ( dP ( t ) = − 1 τ P ( E B ( t )) P ( t ) + q P ( E B ( t )) d Π P ( t ) dN ( t ) = − 1 τ N ( E B ( t )) N ( t ) + q N ( E B ( t )) d Π N ( t ) . This mo del makes the implicit assumption that the individuals ev aluate instan taneously their state of mind. A more realistic mo del would b e that there exists an internal represen tation of the emotional balance, the internal b alanc e I B ( t ), emerging from an internal representation of p ositivity I P and negativity I N , and which gov erns the w a y individuals see themselves and feel external ev ents. In addition to biasing the integration of external ev ents, the internal balance creates self-induced p ositivity when the emotional balance is ab o ve the internal balance (corresp onding to the feeling of b eing b etter than w e thought) or self-induced negativity of b eing b elo w our exp ectations. This leads to the mo del:      dP ( t ) = − 1 τ P ( I B ( t )) P ( t ) + g ( P ( t ) − I P ( t )) + q P ( I B ( t )) d Π P ( t ) dN ( t ) = − 1 τ N ( I B ( t )) N ( t ) + g ( N ( t ) − I N ( t )) + q N ( I B ( t )) d Π N ( t ) E B ( t ) = P ( t ) P ( t )+ N ( t ) . (1) T o complete the mo del, one needs to mo del the evolution of the internal represen tations as a function of the actual p ositiv e and negative aﬀects. Essentially , the in ternal p ositiv e and negativ e aﬀect levels follow the p ositive and negative aﬀect lev els, but with a dela y dep ending on the time needed to incorp orate these mo diﬁcations in our self-image. F or simplicity , we will simply consider I P ( t ) = P ( t − t d ) and I N ( t ) = N ( t − t d ) where t d is the t ypical time needed to take into account emotional c hanges in our representations. In a ﬂuid limit appro ximation, one obtains that the system has an av eraged b ehavior given by the simple system of nonlinear ordinary diﬀeren tial equations:      dP dt = − 1 τ P ( E B ( t − t d )) P ( t ) + g 0 ( P ( t ) − P ( t − t d )) + λ 0 q P ( E B ( t − t d )) dN dt = − 1 τ N ( E B ( t − t d )) N ( t ) + g 0 ( N ( t ) − N ( t − t d )) + λ 0 q N ( E B ( t − t d )) E B ( t ) = P ( t ) P ( t )+ N ( t ) . (2) 4 T o ﬁx ideas, w e choose for q P the three parameters sigmoid function q P ( x ) = αx 2 1 + β x 2 + c (3) where c is the v alue at zero, α is a scale parameter and β con trols the slop e of the sigmoid at the inﬂection p oin t, i.e. the sharpness of the changes b etw een the wa y depressed or non-depressed patients integrate p ositiv e and negative even ts, see inset in Fig. 2B. A typical example that we will inv estigate numerically throughout the pap er is shown in Fig. 2A. In order to unco v er the role of diﬀeren t parameters in the dynamics and the eﬀect of therap y , we will use new psyc hological data describ ed b elo w. 2.3. Ther ap eutic al data and analysis W e analyze the dynamics of emotional balance data from a recurrently depressed patien t. The ﬁrst inv esti- gations of this individual app eared in [4] and used visual examination of the aﬀect balance tra jectory to identify diﬀeren t therapy phases. In the present work, we add up t w o datasets, and moreo v er developed and employ ed qualitativ e metho ds v alidated by mathematical and statistical to ols. 2.3.1. Particip ant Our study focuses on an individual patient (JR) who w as treated on three separate o ccasions ov er a ten- y ear p erio d during which he t wice relapsed. JR presen ted as a brigh t, o ver-ideational, 41-year-old Caucasian male, married with tw o b oys, and the son of a reno wned scientist. He w as mo derately depressed and anxious, exhibited time urgency , and strained interpersonal relationships b ecause he was constantly comp eting to pro ve his sup eriorit y . His mother was chronically depressed and he lived under the shado w of his renowned father. Because intellectual eﬃciency was central to his self-esteem, he felt distressed b y rumination that inhibited pro ductivit y and generated fear of failure. • First p eriod of treatment: Coping fo cuse d ther apy (Ther apy 1) . The ﬁrst instance of cognitiv e-dynamic therap y administered to JR, describ ed in [4], lasted ﬁve months and focused on developing coping strategies with minimal emphasis on psyc ho dynamic exploration. JR learned anxiety managemen t techniques to deal with stress, cognitive strategies to reduce worry , and communication skills to enhance interpersonal functioning. Later stages of therapy addressed the theme of JR b eing driven b y comp etitiv e needs to surpass his un usually successful father. When his symptoms abated, JR prematurely terminated the therap y noting that he w as accustomed to a “den tal c hec kup” model that in volv ed brief treatment with follo w-up if distress recurred. • Second p eriod of treatmen t: Mixed therapy . The second instance of therap y b egan 3 years after the initial treatmen t when JR exp erienced a recurrence of mo od disorder and work inhibition. This in tegrative ther- ap y that lasted for 2 y ears expanded on the coping strategies in troduced previously and shifted the balance to wards the psychodynamic spectrum. Dynamic issues fo cused on JR’s extreme need to demonstrate his adequacy that was driven by his father complex. The treatmen t uncov ered anger at abusive p eers and at his depressed mother b ecause of her “grey mo o ds” and inability to protect him. Deeper dynamic issues surfaced with dream themes of depriv ation, rage and mortalit y fears. Although JR was still struggling with these conﬂicts and exhibited dep endency and interpersonal problems, he was no longer depressed or anxious. He somewhat abruptly terminated this instance of therapy , p erhaps in ﬂigh t from uncov ering deep er lay ers of unmet dep endency needs and accompanying rage. • Third p erio d of treatment: Dynamic fo cuse d t her apy (Ther apy 2) . The second instance of therapy b egan ﬁv e years later with JR experiencing the deep est lev el of anxiety and depression when he saw that his grandiose exp ectations w ouldn’t b e realized. He recognized that he needed to fundamentally “re-in ven t himself professionally and p ersonally”.Although his previous treatments provided reasonably enduring symptomatic relief, they ended prematurely and did not fully address his underlying p ersonalit y structures that predisp osed him to moo d disturbances. Thus, w e agreed to use a more psychodynamic fo cus from 5 the start, to p enetrate to a deep er lev el and to work through these issues to a mutually agreed conclusion. This therap y b egan with a prolonged p erio d of emotionally charged sessions of intense grieving ab out his mother’s death and not b eing as successful as he though t she exp ected. He b ecame aw are of his narcissistic p ersonalit y structure, compulsiv e achiev emen t striving and interpersonal conﬂicts. JR work ed through emotionally charged dreams with classical psychodynamic themes of oral depriv ation, Oedipal con tent (killing father and ﬂirting with mother) and raw images of dehumanization. The end stage of therap y was diﬀerent in that when JR b ecame asymptomatic, he contin ued consolidating treatment gains b y working through p ersonalit y issues and dream material. T ermination was not hastened, o ccurring at a time considered rip e by b oth patient and therapist. F or the last therapy a log was pro duced from the clinical notes to identify session conten t and critical even ts of the patient’s life. Three main cognitive-aﬀectiv e domains consisten tly relate to well b eing, namely emotion [23], self-image [24] and optimism [25]. W e reasoned that an individual with high lev els of these balances will b e functioning well and therefore monitored their evolution using diﬀerent inv en tories. Most existing measures of cognition and aﬀect were not designed to assess balances. Therefore, they p ose sev eral problems when one w ants to compute ratios. First, they don’t include an equal n umber of p ositive and negative items. Second, they use a Likert-scale anc hored at 1 rather than 0, whic h creates a nonlinear, artiﬁcial ceiling and ﬂo or in the ratio, since the p ositive and negative scores are forced to a non-null minimum (i.e. 1 x the num b er of items in the in ven tory , see [26]). And third, they contain only mo derately p ositive and negative moo ds, such as “c heerful” or “scared” (e.g., P ANAS [27]). This preven ts the ev aluation of p otentially dysfunctional extreme states suc h as excess positivity . T o ov ercome these dra wbacks, new measures w ere designed, as reported in detail in [4]. These measures include an expanded v ersion the Emotional Balance Inv entory (EBI-E), constructed by adding to each sub-scale extreme aﬀects such as passionate or infuriated, in order to capture intense states. The resulting EBI-E is a 36-item in ven tory consisting of 18 p ositiv e and 18 negativ e mo o d terms of diﬀeren t in tensity , categorized in to 3 p ositive (Happ y , Vital, F riendly) and 3 negative (F earful, Sad, Angry) sub-scales. JR indicated on a 5-p oint Likert scale ho w frequently he felt each emotion during the past week (0 = not all to 4 = almost alwa ys). Clinical symptoms were assessed pre-and p ost treatmen t and p erio dically as needed using the Bec k Depression Inv en tory (BDI: [28]) and the Beck Anxiet y Inv en tory (BAI: [29]). 2.4. Automatic Se gmentation of Ther apy Phases The emotional balance tra jectory presents three phases, describ ed by the following concepts: emotional balance trend, lo cal v ariabilit y and oscillation. Patien ts were instructed to complete the balance inv en tories w eekly and freely c hose the day of the monitoring b etw een sessions, which made the data unevenly sampled in time. Missed sessions and p erio dic extended v acations further increased the irregularity in the data collection. The in terv al b etw een tw o consecutive measurements for JR ranged from 3 to 27 days, with a mean of 8.6 days and a mode of 7 da ys, thus approximating w eekly ratings. In order to ev aluate the dimensions of in terest (trend, v ariabilit y and oscillation), we used the following data analysis to ols: • The emotional balance trend w as ev aluated using a sliding mean. Sp eciﬁcally , at time t, this quan tity is equal to the lo cal mean of the emotional balance in a time windo w cen tered on t, with a range of sev en w eeks, including the three previous and the three subsequent sessions. The choice of this window size w as a compromise b etw een the need to ha ve suﬃcient p oin ts in the signal to compute a meaningful v alue and few enough p oin ts to track with suﬃcient sensitivit y the trend of the signal. Note that the results obtained do not dep end signiﬁcantly on the choice of this time window. • The ﬂuctuation level is estimated from the signal b y computing a sliding standard deviation in the same fashion as used for calculating the sliding mean. This sliding standard deviation is computed on a cen- tered signal obtained by subtracting the aforementioned sliding mean from the original signal in order to accurately distinguish v ariations linked with the trend from random v ariabilit y around the trend. • The automatic segmentation metho d distinguishing a high v ariabilit y phase and a low v ariabilit y (i.e., relativ e stability) phase is based on optimizing the p-v alue across the p ossible segmentations using the standard Brown-F orsythe test for equality (or homogeneit y) of v ariance. If the resulting p-v alue of the 6 Bro wn-F orsythe test is less than some critical v alue (e.g., 0.05), the obtained diﬀerence in sample v ariances is unlikely to ha v e o ccurred based on random sampling; thus, the hypothesis of equal v ariances is rejected, and it is concluded that there is a diﬀerence betw een the v ariances in the population [30]. The follo wing automatic segmentation algorithm was used: Considering a sample of N v alues ( x i , i = 1. . . N), we aim at detecting whether this sample is comp osed of tw o distinct subgroups characterized by diﬀeren t v ariances: Phase 1 corresponding to i = 1 . . . t and Phase 2 corresponding to i = t + 1 . . . N . W e w ant to ﬁnd this time t, and chec k if the segmentation obtained presents a signiﬁcan t diﬀerence of v ariance. F or each time t considered, we p erform the Brown-F orsythe test, and com pute the p-v alue. The segmentation time t is c hosen as the v alue that minimizes this p-v alue. The p-v alue corresp onding to this optimal segmentation is then compared to our threshold (0.05) to assess statistical signiﬁcance of the segmentation. • The presence of oscillations w as assessed by computing a sliding lo cal Lom b-Scargle transform on a neigh b oring window around the time p oint of in terest [31, 32]. Because of the time scale in volv ed in the oscillations (around 7 w eeks) and the necessity to hav e around three cycles in a time window to assess the presence of oscillations, we chose a window of 20 weeks. The Lom b-Scargle transform derives from the classical F ourier transform that is widely used for signal analysis applications. It extends the F ourier transform to unevenly spaced data, and aims at revealing the frequencies that are present in a signal. It is asso ciated with statistical signiﬁcance tests of the detected oscillation (see details in App endix B). The signiﬁcance level ev aluated assumes that each frequency constitutes an indep enden t test and corrects the signiﬁcance lev el by the n umber of tests p erformed in order to control for Type I error. The fact that the Lomb-Scargle handles unev enly spaced data mak es it the metho d most appropriate for our clinically deriv ed data, and our ability to automatically compute signiﬁcance levels provides an ob jective criterion for ascertaining the presence of oscillations in the signal. The sliding transform allows identifying the onset of signiﬁcant oscillations in the signal in the time-window considered, and highlights a p eriod where the signal presents signiﬁcant oscillations in each time-windo w. In order to fully assess the presence of signiﬁcan t oscillations throughout the identiﬁed segmen t of the signal, w e compute a standard (non-sliding) Lom b-Scargle transform on the whole phase and derive from this test the actual statistical signiﬁcance of these oscillations. Note that visual insp ection of a noisy signal in the raw data can sometimes pro duce the spurious impression that the signal is oscillating. The analysis of the amplitude of Lomb-Scargle transforms and the application of the presen t tests diﬀerentiates ob jectiv ely the tw o phenomena (see App endix B). 3. Theory In this section we inv estigate the dynamics and bifurcations of the system, in order to characterize the p ossible states and transitions b etw een these. The ﬁrst step is to characterize the p ossible stationary solutions. At the equilibria of system (2), the emotional balance necessarily satisﬁes the implicit equation: E B = q P ( E B ) τ P ( E B ) q P ( E B ) τ P ( E B ) + q N ( E B ) τ N ( E B ) . Heuristically , equilibrium emotional balance therefore do not dep end on the in tensity of positive and negativ e ev ents occurring (as long as these are equal, otherwise only depends on their ratio), but only on the w a y emotions are integrated. The righ thand side of the equation is a strictly increasing function of E B . Dep ending on its slop e, it can ha v e either one or three ﬁxed points. W e recall our c hoice q P ( x ) = αx 2 1+ β x 2 + c as depicted in Fig. 2A. The related equilibria and their stabilit y for t d = 0 for the system of Eq 2 are depicted in Fig. 2B. In order to precisely understand the dynamics of the system as a function of the diﬀerent parameters, we no w slightl y simplify the mo del to reduce it to a system which can b e analytically solved. T o this end, we mak e the simplifying assumption that the total amount of aﬀect, P + N , is almost constan t in time b oth in sim ulations of the full system tend and in psychological data. This assumption together with ﬁxing constan t 7 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 Em ot iona l B a lanc e β 1 2 3 4 5 0 0.4 0.8 0.2 0.6 1 0 0.5 1 3 2 1 0 Em ot iona l B a lanc e q Am pl it udes of P and N m odi f ic at ions q P N A B β =1 β =3 β =5 β =1 β =3 β =5 Figure 2: (A): Solid lines represent the sigmoid functions q P ( E B ) (blue) and q N ( E B ) (red) used for simulations and analytical results. Dashed lines represent the corresponding functions q P ( E + 0 . 2) and q N ( E + 0 . 2). (B): Bifurcation diagram as function of the parameter β . Mo diﬁcations in the shap e of the map as β is v aried are shown in inset: the map b ecomes shap er (up) and is globally scaled (b ottom). Red solid lines corresp ond to stable ﬁxed p oint, while the black dashed line indicates an unstable ﬁxed point. This unstable point is the separatrix b et ween depressed and non-depressed states (blue arrows). The grey shaded zone correspond to the range of parameter β inv estigated within simulations shown in Fig. 8 and discussed in Sec. 4. F or b oth panels, parameters are ﬁxed as: α = 10 , c = 0 . 2 , λ = 4 , τ P = τ N = 10. In panel (A) β = 2 . 7, corresp onding to the vertical black solid line in panel B. timescales τ P ( x ) = τ N ( x ) ≡ τ allow to derive a simpliﬁed version of the mo del by reducing it to one diﬀerential equation. Starting from the original equation for P and denoting T the constant v alue of P + N , we hav e: dP dt = − P τ + λ 0 q P  P ( t − t d ) T  + g 0 T ( P ( t ) − P ( t d )) . (4) Deﬁning p = P T , λ = λ 0 ατ T , g = g 0 τ T and t 0 = t d τ , c = 0 and after rescaling the time, we obtain dp dt = ( g − 1) p + λ p 2 d 1 + β p 2 d − g p d , (5) where p = p ( t ) and p d = p ( t − t 0 ), with 0 ≤ p, p d ≤ 1. In this simpliﬁed mo del, we can characterize analytically the b eha vior of the system and its bifurcations as a function of the diﬀeren t parameters. 3.1. Ste ady states and stability Steady states are indep enden t on the v alue of g and depend only from the m utual relation betw een λ and β . Simple algebra allows showing the following: Prop osition 3.1 (Fixed p oints). Possible ﬁxe d p oints ar e given by: p 0 = 0 p ± = λ ± p λ 2 − 4 β 2 β . (6) T aking into ac c ount the c onstr aint that p + ≤ 1 , we have: (i) for λ < 2 , β > λ − 1 and for λ ≥ 2 , β > λ 2 4 : p 0 is the unique e quilibrium (ii) for λ < 2 , β ≤ λ − 1 and for λ ≥ 2 , β < λ − 1 : the system has two e quilibria p 0 , p − 8 (iii) for λ ≥ 2 , λ − 1 ≤ β ≤ λ 2 4 : the system has thr e e e quilibria p 0 , p − and p + Mor e over, in the absenc e of delays, we have that p 0 and p + ar e always stable for β 6 = λ 2 4 , wher e as p − is always unstable. However, for β = λ 2 4 , p + = p − = 2 /λ is left-unstable sadd le p oint. This prop osition is summarized in Fig. 3. The proof is elementary and details left to the reader. It only amoun ts to solving a p olynomial equation to ﬁnd the expressions of the ﬁxed p oin ts, and inv estigating the sign of the diﬀeren tial of the vector ﬁeld at the ﬁxed p oin ts. W e note that the along the b oundary of the parameter region (iii) given by λ = 2 and β = λ 2 4 , we hav e p − = p + (hence, p ossibly indicating saddle-node bifurcation) whereas β = λ − 1 corresponds to p + = 1. W e will sho w that this is indeed the case, and moreov er will show that the delays ma y induce destabilization of the stable ﬁxed p oin t in fa vor of an oscillating solution. 0 24 68  g 0.5 0.7 0.9 1.1 Figure 3: Parameter space. (A): Parameter space regions for equilibrium states. The solid line corresp ond to p + = 1, the dashed line to p 0 = p − and the dotted line to p − = p + . (B): Parameter space regions for Hopf bifurcation, for a ﬁxed v alue λ = 4. The solid line corresp ond to β = λ − 1, the dashed line to the combination of g = 1 − √ λ 2 − 4 β 2 λ (for g < 1) and β = λ 2 4 (for g ≥ 1). 3.2. Delay-induc e d Hopf instability W e now characterize the stability of the equilibrium p + in the presence of delays. W e therefore concentrate on parameters within region (iii). W e show the following: Theorem 3.2. F or any p ar ameters satisfying the ine qualities: λ > 2 λ − 1 < β < λ 2 4 g > 1 − p λ 2 − 4 β 2 λ (7) ther e exists a unique value t d of the delay for which the system under go es a generic sup er critic al Hopf bifur c ation. Pro of In order to demonstrate this result, w e need to ﬁrst identify the p oin ts at whic h the linearized system has a single pair of purely imaginary eigenv alues, and then reduce to the normal form of the Hopf bifurcation at this p oint in order to characterize the type of Hopf by computing the ﬁrst Lyapuno v co eﬃcien t and showing that it is negative. In particular, we ha v e here one of the seldom cases in which w e can derive a closed-form and relatively compact expression for this co eﬃcient. The calculations b eing tedious yet not very standard due to the presence of dela ys, we rep ort the reduction to normal form to the App endix C. 9 Line ar Stability Analysis The linear stabilit y analysis amounts to ﬁnding the c haracteristic equation obtained b y linearizing the system at the ﬁxed p oin t p + . The linearized equation is: dx dt = ( g − 1) x (0) − " ( g − 1) + p λ 2 − 4 β λ # x ( − t 0 ) (8) where we used the notation x ( θ ) = p ( t + θ ) − p + . The disp ersion relationship is obtained when lo oking for solutions of the form X e ζ t : ζ = ( g − 1) − " ( g − 1) + p λ 2 − 4 β λ # e − ζ t 0 (9) The possible Hopf bifurcations corresp ond to purely imaginary v alues of ζ = ± i ω (ﬁxing for example ω > 0). Eq. 9 can then b e solved equating real and imaginary part. Indeed, one obtains: ω =   p λ 2 − 4 β  p λ 2 − 4 β + 2 λ ( g − 1)  λ 2   1 / 2 (10) t 0 = tan − 1  ω g − 1  + 2 k π ω (11) with k = 0 , 1 , 2 , . . . . These equations hence provide v alues of the parameters ( λ, β , g , t 0 ) related to p ossible Hopf bifurcations, see Fig. 4. In particular, it is easy to see that the conditions the parameters hav e to satisfy to allow for Hopf bifurcations at p + < 1 (then for ω to b e real and non-v anishing) are given by the announced set of relationships (7). 0 5 5 10 t 0 β g 3 3.5 2 1.5 1 4 β 3 3.4 3.8 0 10 20 0 8 4 t 0 λ = 4 A B C λ = 4 t 0 0 1.5 3 g λ = 4 Figure 4: Time delay at Hopf bifurcation for a ﬁxed v alue of the parameter λ = 4. (A): Time dela y (for k = 0) as a function of the parameters β and g . (B) Sections of (A) as function of β , along g = 0 . 8 (solid line), g = 1 (dashed line) and g = 2 (dotted line). (C) Sections of (A) as function of g , along β = 0 . 99 λ 2 / 4 = 2 . 23 (solid line) and β = λ − 1 = 3 (dotted line). The reduction to normal form along the curve obtained is made explicit in App endix C. W e obtain there a form ula that is exploited to numerically show negativity of the ﬁrst Lyapuno v co eﬃcient.  The formulae obtained for the curve of Hopf bifurcations and the Lyapuno v co eﬃcien ts app ear relatively complex at this lev el of generality . W e make it slightly more explicit in the sp eciﬁc case g = 1. 10 The constrain ts of Eq. 7 are satisﬁed as long as λ > 2 , 0 < γ = p λ 2 − 4 β < λ − 2 . (12) In this case the diﬀeren tial equation dep ends only on the delay ed v ariable dp dt =  − 1 + 4 λp d 4 + ( λ 2 − γ 2 ) p 2 d  p d , (13) the p ositiv e stable ﬁxed p oin t is no w given by p + = 2 λ − γ , (14) and the disp ersion relation reads ζ = − γ λ e − ζ t 0 . (15) In this case, the corresp onding frequencies (substituting ζ = ± i ω , ω > 0), and delays associated to the Hopf bifurcation are: ω = γ λ (16) t 0 = (4 k + 1) π 2 ω = (4 k + 1) π λ 2 γ , k = 0 , 1 , 2 , . . . . (17) The equations for the Ly apunov co eﬃcien t largely simplify and an easy formula can b e written: α g =1 Ly ap = ( λ − γ ) 3  2(7 π (4 k + 1) − 8) γ 3 − 30 π (4 k + 1) γ 2 λ + 3(4 − 11 π (4 k + 1)) γ λ 2 + (4 − 11 π (4 k + 1)) λ 3  80 (4 + π 2 ) γ λ 3 (18) In particular, for k = 0, it easy to see that α g =1 Ly ap = a 0 ( λ − γ ) 3 ( γ − a 1 λ )( γ 2 − 2 a 2 γ λ + ( a 2 2 + a 2 3 ) λ 2 ) (19) with a 0 = (7 π − 8) 40(4+ π 2 ) γ λ 3 , a 1 ' 4 . 2, a 2 ' − 0 . 42 and a 3 ' 0 . 29. In particular, imp osing the constrain ts of Eq. 12, it is straightforw ard to see that ( λ − γ ) > 0, ( γ − a 1 λ ) < 0 and therefore α g =1 Ly ap < 0. Similar results are obtained for the general k 6 = 0 case. 3.3. Co dimension two BT bifur c ation In theorem 3.2, we ha ve iden tiﬁed an op en region of the parameter space in which the system undergo es a generic Hopf bifurcation. Along the b oundaries of this domain, the Hopf bifurcation disapp ears through tw o diﬀeren t scenarios: (1) β = λ 2 4 : the system undergo es a sup ercritical Bogdanov-T ak ens bifurcation ( p − = p + ), as shown b elo w in theorem 3.3 (2) g = 1 − √ λ 2 − 4 β 2 λ : This case do not corresp ond to a bifurcation, since the corresp onding solution for the dela y t 0 is either negativ e ( k = 0) or inﬁnite ( k 6 = 0). Theorem 3.3. F or g > 1 , the system under go es a sup er critic al Bo gdanov-T akens bifur c ation at β = λ 2 / 4 and ( g − 1) t 0 = 1 . Pro of Around this p oin t, it is easy to see that the linearized equation reads: ˙ u = g ( u t − u t − t 0 ) s which is very close from the linearized system inv estigated, in a more general setting, by F aria and Magalhaes in [33, section 6]. The same calculations can b e dev elop ed with minor ob vious mo diﬁcations to get to the conclusion.  11 4. Results W e in terpret the analysis of the model and confront its predictions to our psychological data. This points to wards the idea that more resilience to relapse is reac hed when delays are increased, a prediction which w e test thoroughly with extensiv e simulations of the mo del. 4.1. Psycholo gic al Interpr etations of the Mathematic al The ory Let us ﬁrst interpret the diﬀerent equilibria in psychological terms. W e recall that our mo del revealed the presence of t wo stable and attractiv e emotional balance states, one lying around normal levels and one in the depressed range (Fig. 2), separated b y an unstable equilibrium delineating initial v alues of the emotional balance evolving to normal or to depressed states. W e note that the precise v alue of the equilibrium do es not sensitiv ely depend on the individual’s capacit y to react positively or negativ ely to even ts (parameter β ), but the existence of these equilibria do es dep end on individual’s emotional pro cessing. In detail, we hav e lo ok ed at the role of the parameter β on the existence of these ﬁxed p oints (see their eﬀect on emotional pro cessing in the inset of Fig 2). This parameter for small v alues of β , the in tegration of positive and negativ e ev ents do es not sharply dep end on the emotional balance, and patients ev olv e to depressed states, while when the dependence is sensitiv e enough, patients alw a ys end up at normal states. Interestingly , as β is v aried, the equilibrium do es not progress gradually from depressed to normal: the system shows a h ysteresis shap e and only relev an t equilibria of normal and depressed p ossibly exist. Within the region in which both depressed and non-depressed states are stable outcomes, a third virtual ﬁxed p oint, unstable, exists and separates the attraction basins of the tw o states (blue arrows in Fig. 2(B)): the patien t progresses tow ards a depressed or normal state depending on whether its emotional balance is b elo w or ab o ve this critical v alue, and as β increases, the attraction basin of the non-depressed state increases and the patien t is more likely to b e non-depressed. This h ysteresis diagram also allo ws understanding ho w depression may o ccur. Indeed, assuming that a patien t, initially at a normal state, suﬀers a series of negativ e ev ents, his emotional balance will decrease and ma y exceed the critical v alue and enter the attraction basin of the stable depressed state. It is then very hard to escap e this equilibrium without any external interv en tion. This is where the therapy takes place. In the follo wing sections w e sp eciﬁcally describ e the therapies (and their impact on JR’s emotional balance), dev elop a mathematical mo del and inv estigate how they can stabilize patients to normal levels. 4.2. The thr e e phases of depr ession r e c overy Before we pro ceed in the description of therap eutical conten ts and their mo deling, we in tro duce the concept of therap eutical phases that will be very useful in interpreting the experimental data. Indeed, we will see that the dynamics of the emotional balance during therap y crosses three very diﬀerent phases: • Phase 1: V ariability : is deﬁned b y a relativ ely large v alue of the standard deviation of the cen tered emotional balance. The actual v alue of the standard deviation v aries across therapies, the emphasis b eing on relatively higher v alues compared to adjacent phases of the rest of the signal, in particular to the subsequen t phase. • Phase 2: Stability : is deﬁned by a relativ ely low degree of v ariabilit y of the emotional balance. Similar to ab o v e, the emphasis is on v alues relatively low er than adjacent phases of the signal. • Phase 3: Oscil lations is deﬁned by the presence of p erio dic oscillations of the emotional balance. W e describ e in detail the metho dology used in order to automatically detect and v alidate statistically the presence of these phases in the exp erimen tal data in section 2.4. 4.3. Coping fo cuse d ther apy The ﬁrst instance of treatmen t w as of type 1: coping focused. W e start b y describing the ev olution of the EB of JR during the course of this treatment b efore mo deling the mo diﬁcations it induces in the mathematical mo del. 12 4.3.1. Exp erimental data The dynamics of JR’s emotional balance during the coping-fo cused treatmen t presents tw o consecutiv e sequences of Phase 1 (V ariability) and Phase 2 (Stability) (see Fig. 5). The ﬁrst Phase 1 (noted 1a) lasted ﬁve w eeks, sho ws a rapid increase in emotional balance from a Negative ratio ( E B = 0 . 34) to a Successful Coping ratio ( E B = 0 . 65), and despite high v ariabilit y achiev es an o verall mean ratio for the phase of E B = 0 . 49, close to the Conﬂicted set-p oin t of E B = 0 . 50, asso ciated with mild psychopathology . This rapid initial increase of the emotional balance indicates the likelihoo d of successful cognitive therap y , as established in [34]. The patien t then stabilized at the Successful Coping ratio of E B = 0 . 62 and sustained this ratio during four weeks of Phase 2 (noted 2a). Then another Phase 1 (noted 1b) started afresh with the patien t dropping into a Conﬂicted ratio ( E B = 0 . 48), but rapidly reb ounding to a P ositive ratio of E B = 0 . 72, asso ciated with normal (but not optimal) functioning. Finally , he stabilized at this level in another o ccurrence of Phase 2 (noted 2b). F rom a mathematical p oint of view, the emotional balance reac hed a stationary state corresp onding to E B = 0 . 72 ± 0 . 05. W e tested if the segmentation obtained by this tra jectory analysis is statistically signiﬁcant. T o this end, we used the classical F-test of equality of v ariance and obtained F = 13 . 017, which corresp onds to a p-v alue p < 0 . 01, thus v alidating that the ﬂuctuations in the ﬁrst and second phase are signiﬁcantly diﬀerent. The therap eutic conten t and the life even ts corresp onding to this dynamics can b e summarized as: • Phase 1a. The initial increase in emotional balance w as attributed in our earlier study [4] to the pro cess of “remoralization” that often characterizes the ﬁrst stage of treatment [35]. The transition from Phase 1a to 2a was not triggered by any identiﬁable critical even ts, but with a change in the patient’s attitude. Sp eciﬁcally , the patient started to constructively analyze the source of his problems rather than ruminate ab out them. • Phase 2a. During this phase the patient stabilized his aﬀect by applying anxiety management techniques, comm unication skills and cognitiv e strategies to reduce w orry . No dreams were rep orted. Moun ting job pressures and in terp ersonal conﬂicts at w ork triggered the transition from Phase 2a to 1b. • Phase 1b. JR entered Phase 1b with his mo o d reb ounding rapidly to the normal balance level of E B = 0 . 72 where it stabilized, as opp osed to Phase 1a that reached only a subnormal level. The transition from Phase 1b to 2b did not app ear to b e triggered b y external life even ts per se, but by gro wing self-conﬁdence resulting from cognitive restructuring of his attitudes tow ards his critical father and teasing from p eers. In what pro ved correct, JR expressed concern that these improv emen ts might be transitory b ecause he attributed them to situational factors (i.e., w ork success), rather than to internal changes. • Phase 2b. JR consolidated his new coping strategies and impro ved mo od, resulting in a stable emotional balance at a normal (but not optimal) lev el ( E B = 0 . 72). At this p oin t, he somewhat abruptly decided to terminate therapy . F ollo w up assessmen ts conducted at three, four and ﬁve months sho wed a sus- tained, normal emotional balance, but with an increasing standard deviation, probably asso ciated with destabilization of the steady state reac hed at the end of treatment and a p ossible return to Phase 1. 4.3.2. Mathematic al mo del Depression induces an im balance in the p erception of p ositiv e and negative even ts that the coping-fo cused therap y aims at coun terbalancing. Due to the depressed state, the eﬀect of p ositiv e (resp ectively negative) even ts are to o small (resp. to o large) and gov erned b y the map q P (resp. q N ) mo deling the amplitude of the positive and negative aﬀects. By helping the patien t “sa v or” positive even ts and stop fo cusing on negative even ts, the therap y mo diﬁes the v alue of q P and q N bac k to those corresp onding to the normal range. In our approac h, this can be modeled by considering that therap y leads to in tegrate positive and negativ e ev en ts in a wa y more similar to non-depressed situations. Mathematically , this can b e tak en in to account b y considering that a p ositive even t is sav ored with increased in tensity leading to a larger amplitude of p ositiv e aﬀects increase q P ( E B + a ) with a > 0 the net eﬀect of the therapy . Analogous eﬀect on the intensit y of negative even ts in tegration is considered and we mathematically replace in equations (2) q N ( E B ) by q N ( E B + a ) during therapy . Numerical simulations sho w that even small shifts a are suﬃcient to induce the desired changes in the dynamics and bring a depressed 13 ! Figure 5: Emotional balance tra jectory for JR’s ﬁrst p eriod of treatmen t (Coping F ocused therapy). (A) Emotional Balance data present tw o iterations of Phase 1 and 2 sequences. Red = raw emotional balance data; Black = sliding mean; Blue = standard deviation (multiplied by t wo for legibility). (B) Sliding Lomb-Scargle (LS) transform shows no signiﬁcant oscillatory activity (the transform is at a 0.50 signiﬁcance level for b oth phases) and (C) Non-sliding LS transform on the whole phase 1a (purple) and 2b (blue). 14 patien t to normal states. F or instance, Fig. 7 shows simulation results with a = 0 . 2 applied during a weeks to a month consistently results in a stabilization of the patien t’s emotional state. Of course, stopping therap y at this p oin t (by resetting a to 0) do es not destabilize the patient’s emotional state that remains in the normal range, since the non-depressed state is a stable attractor. Imp ortan t p erturbations suc h as an accumulation of negativ e even ts can nevertheless destabilize this attractor and bring the patient bac k to depressed levels, as we sho w in section 4.5. 4.4. Dynamic aﬀe ct-fo cuse d ther apy Three y ears after the ﬁrst therapy , JR relapsed and was administered a mixed coping focused/ aﬀect fo cused therap y . W e describe the therap eutical con ten t and evolution of the emotional balance in App endix App endix A. The time course essen tially concatenates the t ypical features seen in therapy 1 and 2. W e concentrate here on a third instance of treatmen t that w as essen tially psychodynamic, after the patient relapsed 5 years after the mixed therapy , and returned to the clinician with an emotional balance at the lo west level. W e again start by discussing the time-course of the emotional balance during this therapy b efore discussing its mo deling. 4.4.1. Exp erimental data The dynamic fo cused therapy resulted in a tra jectory that diﬀered from the other treatmen t. The initial stage lasted longer than the previous treatments (5 months) and presen ted a highly v ariable but globally increasing emotional balance that gradually reac hed the normal ratio of E B = 0 . 72 (see Fig. 6). The emotional balance stabilized in Phase 2 (note the sharp decrease in v ariability) and smoothly climbed to a ratio of E B = 0 . 79, close to the optimal set p oint of E B = 0 . 81. Phase 3 follo wed with mo derate v ariabilit y , intermediate b etw een Phase 1 and 2. The emotional balance began oscillating smoothly b etw een the normal ( E B = 0 . 72) and optimal ( E B = 0 . 814) ratios with a p erio d cycle of seven weeks and a high statistical signiﬁcance ( p < 0 . 01). This pattern was sustained for seven months during treatmen t and later conﬁrmed at the six-month and one-year follo w-up assessments (see Fig. 6). The Bro wn-F orsythe test for equalit y of v ariance on the segmentation obtained ﬁnds a Phase 1 with a standard deviation of 0.0142 (sample size 22) and a Phase 2 with a standard deviation of 0.020 (sample size: 11). The statistical test conﬁrms that the diﬀerence in v ariance b etw een the t wo phases is highly signiﬁcan t (F-test F = 6 . 36, p-v alue p = 0 . 017). The therap eutic conten t and the life even ts corresp onding to this dynamics can b e summarized as: • Phase 1. The client engaged in more emotional expression than in earlier treatmen ts, with in tense sobbing ab out his mother’s death and not succeeding at the level he thought she exp ected of him. A proliferation of emotionally charged dreams o ccurred with themes of maternal depriv ation, conﬂict with father and a wareness of narcissistic strivings to succeed. As can be seen in Fig. 6, this phase was c haracterized b y extreme v ariabilit y in emotional balance, with the sliding mean showing a gradual, steady increase. Sev eral weeks prior to the transition from Phase 1 to 2, the client work ed through dreams, sometimes t wice weekly , that progressed from female ﬁgures who were inconsistently present and asso ciated with bad fo o d to recov ery themes of eating steak (goo d fo o d) to get in to shap e. JR made progress in shifting from external and uncontrollable sources of self-esteem to b ecoming more self-v alidating, yielding a more stable sense of self. Sev eral weeks prior to the phase transition, he exp erienced stressful even ts, including am biv alence ab out his father’s re-marriage (whic h he did not attend) that led to a precipitous drop in emotional balance to E B = 0 . 38. A dramatic surge in optimism to an optimal lev el ( E B = 0 . 81) triggered a reco very in mo o d that then stabilized at the normal ratio ( E B = 0 . 72). • Phase 2. JR engaged in increased positive activities and a shift from dependence on v alidation from others to self-v alidation. Mourning losses contin ued, but w ere diminished in intensit y and JR fo cused less on mother and more on working through his dep endence on his wife. He developed more insigh t into his narcissistic preo ccupation with self-esteem management that diminished his sensitivity to others. Dreams rev ealed early concerns about loneliness as a child and throughout college, as w ell as peer rejection. The transition from Phase 2 to 3 (Oscillation) was mark ed by a startling exp erience: JR announced in the session prior to the transition that he had a transformative spiritual experience of increased God a wareness 15 ! Figure 6: Emotional balance tra jectory for JR’s third p erio d of treatmen t (Dynamic F ocused therapy). (A) Depicts a single sequence of Phase 1-Phase 2-Phase 3. (B) Lomb Scargle transform identiﬁes a prolonged Phase 3 of oscillations with a six-week perio d sustained until the end of therapy . (C) Non-sliding LS transform on the whole Phase 1 (purple), Phase 2 (blue) and Phase 3 (black) that presen ts a statistically signiﬁcance level ( p < 0 . 01). follo wing the inspiring story of a colleague who faced his death with tranquility and p ositive attitude. He dream t that his mother was alive, and that he felt greater acce ptance of his parents b eing in the pro cess of dying. Immediately after this, his ov erall emotional balance b egan consistently oscillating b et w een the normal ( E B = 0 . 72) and optimal ( E B = 0 . 81) ratios (see Fig. 6) • Phase 3. With his emotional balance oscillating, JR contin ued to work on resolving residual issues of narcissism, depriv ation and childhoo d anger. He was less stressed by w ork and learned to maintain some joy even while engaged in the more thankless asp ects of his job. Imp ortantly , JR reported less en vy , increased h umility , social graciousness, acceptance of self and others, and spiritual transcendence. Although he contin ued to recall and pro cess similar dream themes during this phase, his mo o d remained p ositiv e and his in terp ersonal functioning was v astly impro v ed, with the exception of some residual tensions in his marriage. The Happiness sub-scale p eaked at its highest level ever and he felt less constricted and more creativ e in his approac h to w ork. With his Beck Depression and Anxiety In v entory scores at zero, we w orked tow ards a planned termination with the recommendation that he monitor marital issues. F ollow- up at six months and one year sho wed a sustained pattern of oscillation b et w een the normal and optimal ratios with all measures remaining at similar levels, indicating a resilien t treatment outcome thus far. 16 T h1 T h1+ T h2 T h1+ s tr e ss T h1+ T h2+ s tr e ss T h1+ s tr e ss T h1+ T h2+ s tr e ss 1 2 3 4 5 6 8 9 10 7 0 1 2 3 4 5 6 8 9 10 7 0 y e ar s 0 0.2 0.3 0.4 0.1 0.5 0.7 0.8 0.9 0.6 1 0 0.2 0.3 0.4 0.1 0.5 0.7 0.8 0.9 0.6 1 Em ot iona l B a lanc e A B y e ar s Figure 7: Numerical sim ulations modeling the dynamics of the emotional balance during 10 years. After an initial perio d of 6 mon th, with lo w EB initial conditions, we administer a 6 months p erio d of Therapy 1 (grey shaded zone). In panel (A) four diﬀerent cases are discussed. Blue and red (mostly hidden by the magenta) lines correspond to the diﬀerent treatmen t, without or with Therapy 2 ( t d = 0 or t d = 21). Cy an and magenta correspond to the same cases, when a stress p eriod (green shaded zone, number of negativ e equal to 3 times the usual one, ﬁxed by λ 0 ) of 3 weeks is taken into account after 5 years. The parameter of the mo del of Eq. 2, are ﬁxed as: α = 10 , β = 2 . 7 , c = 0 . 2 , λ 0 = 4 , τ P = τ N = 10 , g 0 = 13. In panel (B) we show a diﬀeren t dynamics when parameters are changed and an equivalen t stress perio d is taken into account: α = 1 . 5 , β = 2 . 5 , c = 0 . 04 , λ 0 = 20 , τ P = τ N = 7 , g 0 = 11. Blue and red solid line resp ectively correspond to delays t d = 0 and t d = 21 days. 4.4.2. Mathematic al mo del The aﬀect-fo cused therapy encourages the patient to re-exp erience and reconsider the origins of his troubles so he can put current ev en ts in p ersp ectiv e. In other words, the therapy will lead the patien t to consider his emotional balance in a broader p erspective. F orm the mathematical mo del p oint of view, this is equiv alen t to a non-v anishing co eﬃcien t t d in Eq. 2. The net eﬀect on the dynamics is the app earance of oscillations around a stable ﬁxed p oin t, for t d ≥ t c d , where t c d is a critical v alue determined by all the other parameters in the model. F rom a mathematical p oint of view, this corresp onds to a Hopf bifurcation, as discussed in details in Sec.3 and in App endix C. This can b e seen in the t w o examples shown in Fig. 7. After the p erio d of coping fo cused therapy (grey zone) that allo wed a progression from the lo w er ﬁxed point to the upper one, when the dela y is turned on, the emotional balance v alue oscillates around that v alue. Notice how c hanging the parameters (cf. panel Fig. 7A and Fig. 7B), oscillations can b e more or less eviden t and distinguishable from the ﬂuctuations due to the random p ositiv e and negativ e aﬀects. 4.5. Stability The ab o v e mo dels predict and reproduce with accuracy the time course of the emotional balance of a patien t undergoing therapy . W e ha ve noted that more than 10 years after the end of the last aﬀect-fo cused treatment, the patient did not relapse into depression, which ma y indicate that therapy 2 leads to a more stable state. This is a coun ter-in tuitiv e prediction: indeed, the emotional balance oscillations visible in the exp erimen tal data tend to indicate a destablization of the normal state, and also leads the individual to low er emotional balances. Moreo ver, if the adv an tage brought b y the ﬁrst coping-fo cused therapy is clear, it is muc h less so in the second treatmen t. In fact, as sho wn for example b y the blue and red lines in Fig. 7A, or by the left part (from 0 to 5 y ears) of Fig. 7B, the tw o p ossibilities (only therap y 1 or b oth therapies) can hav e v ery similar tra jectories and the ﬁrst therap y alone stabilizes the non-depressed state. In order to test whether therap y 2 can ha v e an impact on the stabilit y on the non-depressed state, we incorp orated to the mo del a stress p erio d in which the frequency of negative ev ents is increased. Two instance are pro vided in Fig. 7 (cy an and magenta lines for panel A and blue and red for panel B). In our sim ulations w e consider a p erio d of 3 w eeks in which the negative even ts hav e a rate of o ccurrence which is three times as 17 0 1 2 3 4 5 6 7 8 9 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 1 2 3 4 5 6 7 8 9 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 1 2 3 4 5 6 7 8 9 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 1 2 3 4 5 6 7 8 9 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 1 2 3 4 5 6 7 8 9 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 β=2.5 β=2.3 β=2.7 β=2.9 β=3.1 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 T h1 T h1+ T h2 1 2 3 4 5 Fr ac tion not depr e ss ed at the end 0 0.5 1 1 2 3 4 5 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 Fact or of ave r age nega ti ve e vent s dur i ng stress per i od Figure 8: Statistical numerical analysis of the eﬀect of Therapy 2 on the resistance to a 2 weeks stress p eriod. F raction of ﬁnal states near the normal stable ﬁxed p oint, for diﬀeren t v alues of the parameter β , as a function of j , the ratio betw een the a verage num ber of negative even ts during the stress perio d and that of the normal one ( λ 0 ). The studied scenario is the same as in Fig. 7A, with all other parameters ﬁxed as: α = 10 , β = 2 . 7 , c = 0 . 2 , λ 0 = 4 , τ P = τ N = 10 , g 0 = 13. m uch as the p ositive even ts frequency . Such a stress p erio d decreases signiﬁcantly the emotional balance in the depression range. But the response to this stress perio d is v ery diﬀeren t when therapy 2 has o ccurred. The dela ys, and particularly the presence of oscillations seem in fact to stabilize the normal state b y pro ducing an increased resistance to drifting a w ay from an originally stable state. And the greater the delay t d , the longer and stronger p erio d of exceptional negative ev ents can b e o vertak en. In the same wa y , the sp eciﬁc shap e of the sigmoid function Eq. 3 can determine the limits tow ard whic h also this therap y b ecomes ineﬃcient. In order to hav e an idea of this eﬀect, w e hav e sim ulated several tra jectories from random depressed initial conditions, b y letting v arying the parameter β of Eq. 3 and a parameter j = λ 0 n ; S tr ess λ 0 , where λ 0 n ; S tress ( λ 0 ) is the rate of negativ e pro cesses during the stressful (normal) p eriod. The results are shown in Fig. 8. F or each v alue of the parameter β and j we ha ve considered a set of 50 numerical sim ulations of random p oisson processes and plotted the num ber of ﬁnal states around the p ositiv e state of mind v alue of E B . The stabilizing eﬀect of the second therap y is therefore manifest, esp ecially for small v alues of β and high v alues of j , i.e. the extreme situations (in terms of patient and in terms of negative ev ents) in which depression is more lik ely , or in other words when the basin of attraction of the lo w er stable p oin t is more imp ortant than that of the upp er one. 5. Discussion This article introduced a theoretical framework to mathematically mo del and empirically in vestigate the complex in terpla y of positive and negativ e aﬀects in normal and depressed situations. As an illustrativ e example of the mo del’s p otential, we pro vided a quasi-exp erimen tal study of a recurrently depressed patient undergoing m ultiple treatments demonstrating that the most eﬀective therapy progressed from an initial p eriod of extreme mo od ﬂuctuation to a smo oth, enduring oscillation in emotional balance. The mo del is based on a few basic observ ations of reactivity to p ositive and negative life even ts and on ho w these v ary with the emotional state of the individual. The present ﬁndings build on a research tradition demonstrating the eﬀects of p ositiv e and negative even ts on depressed individuals [36, 37] and more recen tly delineating the diﬀerential temp oral course of sustaining p ositiv e and negative aﬀect in normal and depressed p ersons [22]. Clinical research on p ositive and negative 18 information pro cessing is progressing from a static approac h to more dynamic analyses of the evolution of these states ov er time. Our work oﬀers an imp ortan t scientiﬁc step in dev eloping a psyc hologically grounded theory of self aw areness that can mathematically mo del the temp oral progression of a p erson’s positive and negative aﬀect during the course of life ev ents such as psychotherap y and related changes [38, 21]. A t a critical juncture when mathematical mo deling of psychological dynamics has come under attack [39], our theoretical mo del oﬀers a viable alternative mo del with mathematical to ols for understanding the dynamics of emotion. The SOM Mo del dra ws upon psychological models and data (as opp osed to extrapolating from ﬂuid dynamics) supp orted in a v ariet y of areas [40], and oﬀers theoretical predictions of empirically substantiated ratios that distinguish normal and dysfunctional states [41, 42, 8]. Our mathematical approach opens the w ay to a further understanding of the emotions dynamics. F or in- stance, the model provides a theoretical prediction of the existence of normal and depressed ratios. These corresp ond to the t wo stable equilibria of the system, that may co-exist, meaning that individuals can be in normal or depressed state, from which it can b e hard to escape in the absence of an external stim ulation (life ev ents or therap y). The mo del also motiv ated us to go beyond static levels and consider the time ev olution of the emotional balance in the c ourse of therapy . This was not previously addressed in the context of psyc hother- ap y , whic h has fo cused more on cross sectional data to demonstrate, for example, that psychotherap y clients progressed from a sp eciﬁc negativ e balance prior to treatment to a normal or optimal balance at p ost-treatmen t. None has delineated the temp oral progression of these ratios. The preliminary clinical results show that emo- tional balance tra jectories during treatment reveal three distinct phases: v ariabilit y , stability and oscillation, phases that correlated in meaningful wa ys with the type of therapy , stage of treatment, critical external even ts and inner states of the client. W e found a more intensiv e and eﬀective treatment yielded end-state oscillations in emotional balance and reasoned that these oscillations ma y b e the source of resilience to depression after stressful ev ents. In order to in vestigate this hypothesis, we used extensive numerical simulations of the mo del and quantiﬁed that in regimes during which the emotional balance sho ws low magnitude oscillations, the non-depressed state is more stable and resists perio ds of intense stress. In the model, these oscillations are related to the fact that the individual ev aluates his current state not only at presen t time, but within a longer window of ongoing history . These considerations suggest that Phase 2, characterized b y non-sustained p eriods of relatively high stabilit y , can b e designated as a stage of consolidation when prior gains of the therapy are b eing in tegrated, rather than an ultimate treatment outcome. Overall, the stabilit y phase was not maintained for very long and w as terminated by the o ccurrence of p ositiv e or negativ e in ternal states or life even ts. When learning a new skill, a novice may initially prefer to maintain a more ﬁxed set of circumstances un til acquiring the conﬁdence and ﬂexibilit y that allows engaging more v aried situations. In contrast to the stability phase, the oscillation phase, designated as “resilience”, is asso ciated with w ell b eing and ﬂourishing. This is supported by the patient’s psychological progress to an alleviation of mo od disorder, enhanced interpersonal sensitivit y , and heightened spiritual a wareness. Since the oscillations of the emotional balance are b ounded b y the normal and optimal set-p oints, no striking c hange w as observ ed in the patien t’s state during these smooth, mo derate shifts in moo d. This capabilit y to oscillate allo ws one to maintain a p ositiv e state while ﬂexibly reacting to b oth go o d and bad even ts. This adaptive prop ert y can b e likened to a reed that remains in tact despite sev ere wind. Unlike the thick er but rigid branches of a tree that might snap, the thin and seemingly fragile reed will b end and oscillate with the breeze. It is this ﬂexibilit y that enables it to smo othly survive life’s vicissitudes. T o the best of our knowledge, the curren t study is the ﬁrst to mathematically demonstrate oscillation in aﬀect as a p ost-therap y outcome. The results join the increasingly observ ed phenomenon of normally o ccurring damp ed oscillations in mo o d during natural stressors [9, 10], as w ell as in biological systems suc h as cardio- v ascular heart rate and brain rh ythms [43]. The use of dynamic mo dels to study the ev olution of cognitive and aﬀectiv e states is relatively new, so it is lik ely that diﬀerent tra jectories will deﬁne diverse disorders and pro cesses such as b erea vemen t, unnatural trauma, or c hange during psyc hotherap y . The data from the therapies presen ted here indicate that during treatmen t emotions do not evolv e as a transformation in only the magni- tude of the emotional balance, but rather that progress through distinct phases of v ariabilit y early in therap y , (non-oscillating) stabilit y during mid-therapy and oscillation only at the ﬁnal stage of intensiv e therapy . 19 Man y questions remain ab out what constitutes an optimal tra jectory for v arious life ev en ts. F or example, Stro ebe and Sch ut’s dual pro cess mo del [12] suggests that “optimal adjustment” in coping with stress is as- so ciated with a mo derate lev el of oscillation betw een loss (negativ e) and restoration (positive) resp onses. But Bonanno et al. [44] contend that c hronic and dysfunctional loss reactions are associated with “more extreme and unregulated forms of oscillation” and more enduring negative aﬀect (p.803). These approaches diﬀerentiate b et w een stability and oscillation, but do not distinguish true oscillation from random ﬂuctuation or what we ha ve termed “v ariabilit y” [45]. W e believe that some confusion may be caused b y the not distinguishing betw een the non-technical meaning of oscillation as “wa v ering b etw een conﬂicting courses of action” and its meaning in ph ysics as “an eﬀect expressible as a quantit y that rep eatedly and regularly ﬂuctuates ab ov e and b elo w some mean v alue” ([46], italics added). If we limit our scien tiﬁc use of the term “oscillation” to its meaning in physics, then Bonanno et al’s. [44] refers to ﬂuctuations. A more precise rendering based on this clariﬁcation suggests that a mo derate level of oscillation represen ts the optimal tra jectory and that more extreme and unregulated ﬂuctuations or v ariabilit y – not necessarily larger oscillations – are asso ciated with dysfunctional adaptation. In sum, the present study quantiﬁed the fact that a high emotional balance or p ositiv e ratio alone is not alw ays a suﬃcient indicator that treatment has achiev ed a sustainable, optimal result. Psyc hological resilience and resistance to relapse dep ends on the ability to sustain this high level of p ositiv e moo d and the presence of oscillations in the system provides the ﬂexibilit y needed to accomplish this. Presumably the p erson in an oscillating state has cultiv ated the necessary strategies to monitor and regulate his or her state so it remains optimally balanced. The mathematical to ols introduced here allow a more precise assessment of the levels and dynamics of aﬀect that can b e used in future studies that delineate optimal treatment outcomes and predict the lik eliho o d of relapse. The strength of this approac h that allo ws a detailed analysis of the complexit y of the c hange process also brings corresponding limitations. The question that arises, as in any idiographic study , is the univ ersality of this discov ery . Although the present ﬁndings statistically demonstrate an oscillatory phenomenon in the end-state functioning of a highly successful treatmen t, these observ ations need to b e further inv estigated on additional patients and with diﬀeren t therapists. Since the dynamic fo cused treatmen t temp orally follow ed t wo less in tensiv e therapies, the presence of oscillation in aﬀect and general ﬂourishing cannot b e conclusiv ely attributed to the greater depth and duration of the ﬁnal treatmen t, as opp osed to its position as the culmination of a long pro cess of self-developmen t. Although the current data is consistent with this conclusion, only group design studies can conﬁrm the connection b et w een type of therapy and the generation of end-state oscillation in aﬀect. Larger scale studies need to recruit more individuals and devise simpler, more accessible data recording systems. The time scale of the oscillations (on the order of 7 weeks) represen ts a practical limitation for large-scale exp erimen tation that complicates further analysis of this “macro” oscillatory phenomenon. This initial, in tensive study raises new questions with implications for p ositive and classical psyc hology ab out the dialectical eﬀects of p ositiv e and negativ e aﬀects considered separately: Are the oscillations driven by p ositive or negativ e aﬀects or are they an emerging prop ert y of b oth? Are phase transitions induced more by p ositive v ersus negative states? Do diﬀeren t conten t categories of p ositive v ersus negative even ts (e.g., b erea vemen t, job loss, relationship enhancement, spiritual uplifts, etc.) hav e diﬀerential impact on phase transitions in depressed and normal p ersons? W e encourage further intra-individual studies to delineate how v ariability , stability and oscillation in human systems ev olv e during diﬀerent stages of therapy with diﬀerent disorders and p ersonalit y t yp es. 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Lunel, H.-O. W alther, Delay equations: functional-, complex-, and nonlinear analysis, None. [51] S. A. Campbell, Calculating cen tre manifolds for dela y diﬀerential equations using maple, in: Delay Dif- feren tial Equations, Springer, 2009, pp. 1–24. 23 App endix A. JR mixed therap y Three years after the ﬁrst (coping focused) therapy , JR came bac k to consultation. His therapy started with a coping-fo cused phase follow ed by an aﬀect-fo cused phase, and this is why we refer to this therapy as mixed. When JR returned after a three-y ear hiatus, he b egan with an emotional balance E B = 0 . 64 placing him within the subnormal, but Successful Coping range. Although not clinically depressed, he was struggling with negativ e mo o d, w ork inhibition, worry and sleep disturbance. His tra jectory was mo derately v ariable (Phase 1a), progressing gradually to the normal range at which p oin t he en tered a prolonged perio d of reduced v ariabilit y and stabilization around the normal ratio of E B = 0 . 72 (Phase 2a). As Phase 2a progressed, the patien t en tered a phase that evinced the highest v ariabilit y of the treatment and some of the lo west emotional balances that ﬂuctuated b et ween the Conﬂicted and Successful Coping ranges, with o ccasional p eaks into the low Normal range. During this four-month phase he display ed no oscillation. The v ariabilit y then dropp ed to its low est lev el and the emotional balance stabilized in Phase 2b for a p erio d of one and a half mon ths. Finally , the patien t completed the iterativ e pro cess b y (presumably) entering Phase 3 for the ﬁrst time. During this phase of six and one half mon ths duration, the v ariabilit y increased to a mo derate level and the emotional balance b egan oscillating b etw een the Optimal ( E B = 0 . 81) and Super-Optimal ( E B = 0 . 88) levels with a sev en-w eek p eriod. The premature end of therapy in terrupted the tw o p erio ds of putative, visually detected oscillations. Th us, the Lomb-Scargle transform, though presenting a clear p eak compared to the rest of the signal, (see Figure Fig. A.9), did not achiev e signiﬁcance ( p = 0 . 17) (see Fig. A.9). The Brown-F orsythe test corresp onding to this segmentation shows a v ariance of 0.0249 for Phase 1 (merged data from 1a and 1b) and 0.0027 for Phase 2 (merged data from 2a and 2b) that yields a signiﬁcant diﬀerence (F-test: F = 5 . 79, p = 0 . 025), th us statistically v alidating the phase segmentation for this treatment. The analysis of JR data is exemplary of how visual insp ection can be used together with the statistical to ols. When analyzing the full dataset, our automatic segmen tation algorithms failed to segmen t the therapy in to a single Phase 1 (v ariabilit y) follo wed by Phase 2 (stabilit y), which is conﬁrmed by the visual insp ection of the data. Instead, the data are c haracterized b y a phase of high v ariability , follow ed by a second phase where the signal is not oscillating, but the v ariabilit y is slo wly increasing to a standard deviation b etw een typical v alues of Phase 1 and Phase 2. This sequence is then follow ed by another v ariabilit y phase, a second stability phase of short duration, and ﬁnally a brief p erio d of oscillation. W e are therefore able to visually detect the segmentation prop osed in Fig. A.9. This visually detected segmen tation was then tested and statistically v alidated, using the Brown-F orsythe indicator. The therap eutic conten t and the life even ts corresp onding to this dynamics can b e summarized as: • Phase 1a. Therapy focused initially on a review of cognitive-behavioral coping strategies. After three mon ths, the client explored in terp ersonal themes and engaged in emotional and dream expression (e.g., themes of fo o d, need for unconditional appro v al and disturbing images of a primitive nature), but no deep er psyc hodynamic or dream analysis w as done. The transition from Phase 1a to 2a is mark ed b y a stabilization in moo d and self-image noted in the clinical log. Themes were emerging of oral depriv ation and wishful thinking that others w ould “read his mind” so they could satisfy his needs. • Phase 2a. The client no longer rep orted dreams during most of this phase. Apparen tly picking up on the themes from earlier dreams of oral (maternal) depriv ation, the patient work ed on c hildhoo d loss because of his mother’s depression and her current illness, as well as marital conﬂicts. The emotional and optimism balances reached an optimal level, as he cop ed b etter with frustration and reduced his compulsivity . Otherwise free of dream activity , at mid-phase he rep orted dreams in four successive sessions with themes of tw o dying people presumed to b e his paren ts, childhoo d y earnings for atten tion from his self-centered mother, and fear of world destruction. After this ﬂurry of dream work ceased, JR shifted to a prolonged fo cus on here and now issues of ambiv alence in current p eer relationships. The phase transition 2a to 1b, unlik e the prior phase transition in to a p ositive and stable state, regressed to an extended p erio d of v ariabilit y and lo w emotional balance. It w as not preceded by a dream, but w as instead triggered by a v erbal lashing from a colleague and the clien t’s aw areness that his interpersonal insensitivity pro vok ed conﬂict. 24 • Phase 1b. During this second phase of high v ariabilit y , the client w orked on curren t in terp ersonal conﬂicts and mounting anxiety , depression and sleep problems caused b y his realization that his lifelong, grandiose am bitions were unlikely to b e realized. The transition from Phase 1b to 2b was triggered during the last w eek of Phase 1b when the clien t’s moo d reac hed the subnormal, but Successful Coping level of E B = 0 . 62 and he recalled a dream ab out his wife’s salary increase that made him feel diminished. Also, a critical ev ent o ccurred that the client exp erienced as transformational. His son had a signiﬁcant accident, but his surviv al and recov ery led the client to b ecome more attuned to the needs of his family (esp ecially his wife), to slo wing down his hectic pace, and softening his comp etitiv eness and interpersonal brusqueness. • Phase 2b. This stable phase of normal emotional balance ( E B = 0 . 72) ﬁnds the client engaging in less “name calling” when he mak es mistak es, enjoying family v acations more because he is less self-centered and more ﬂexible, and communicating b etter with his wife. Tw o weeks prior to the phase transition from 2b to 3, the client rep orted successiv e weeks of dreams. The ﬁrst is of tw o dogs dying that the patient related to fear ab out the death of his aging paren ts and to conﬂicts with his mother as a “suﬀo cating, amorphous and ill-deﬁned problem”. The second dream is of a woman falling through a dam and his not b eing able to rescue her, reﬂecting his mother’s precarious mo o ds and his inabilit y to sav e her. • Phase 3. Dream recall and pro cessing contin ued throughout this ﬁnal phase of oscillating aﬀect. The clinical log notes decreased self-fo cus, increased ability to live in the moment and b etter connection with his wife. Parado xically , these improv emen ts were accompanied by dreams of disconnection from mother, p erhaps reﬂecting work tow ards resolution. He w as able to express his emotions more directly regarding grief ab out his grandmother’s recen t death. F or the ﬁrst time his wife commented appreciatively about his progress. JR requested a May termination that app eared inﬂuenced more by the academic year than by his psychological state. Despite signiﬁcant gains, the client had a dream revealing oral rage ab out maternal depriv ation and paren tal inatten tion. Such recurrences of old themes when terminating therap y are not unusual, but the ov erall termination summary raised questions ab out the need to con tin ue w orking on self-conﬁdence, dependency issues and in terp ersonal st yle, as w ell as marital counseling. Interestingly , his ﬁnal emotional balance of E B = 0 . 74 w as near the normal (not optimal) ratio of E B = 0 . 72, but his Happiness sub-score was one-half lo wer than the other p ositiv e scales for Vitality and F riendliness. Th us, he achiev ed a normal aﬀect balance, but was still not optimally balanced and remained deﬁcien t in happiness. App endix B. The Lom b-Scargle T ransform Since the clinical protocol allo wed patients to freely monitor their emotional balance b etw een consultation sessions, the obtained assessments were not evenly spaced. In the case of irregularly sampled data, the classical F ourier transform (see [38, 47] fails to provide the frequency con tent of the signal. T o handle such cases, we used the Lomb-Scargle transform, a very eﬃcient metho d that was dev elop ed for the study of astrophysical data [31, 32, 48]. This metho d is based on the following principles we now mak e explicit. Consider that we observ e a contin uous phenomenon though a given scalar measurement h. The contin uous time phenomenon pro duces a contin uous time measure h(t), but we only ha ve access to a discrete set of N v alues of this function sampled at unev enly spaced times { t i , i = 1 . . . N } . F or this set of N m easuremen ts { h i := h ( t i ) , i = 1 . . . N } , of mean denoted b y ¯ h and of standard deviation denoted σ , the Lomb-Scargle transform p erforms a pro jection on sines and cosines ev aluated only at times t i where data are actually measured. In detail, the Lomb normalized p eriodogram is deﬁned by: P N ( ω ) = 1 σ 2       P N j =1 ( h j − ¯ h ) cos ( ω ( t j − t ))  2 P N j =1 cos 2 ( ω ( t j − τ ( ω ))) +  P N j =1 ( h j − ¯ h ) sin ( ω ( t j − t ))  2 P N j =1 sin 2 ( ω ( t j − τ ( ω )))      (B.1) 25 ! Figure A.9: Emotional balance tra jectory for JR’s second perio d of treatment (Mixed therapy). (A) Depicts two iterations of Phase 1-Phase 2 and a trend tow ard a Phase 3 sequence (see B and C). (B) Note that the short duration of the recorded oscillating phases prevents the Lomb-Scargle transform to reac h signiﬁcant levels. (C) : Non-sliding LS transform on the whole Phase 1b (purple) , Phase 2b (blue), and Phase 3 (black). 26 ! Figure B.10: Lomb-Scargle transforms. (A) On oscillating data, the Lomb-Scargle transform presents a p eak, at a frequency ω related to the period of oscillations, and whose amplitude is related to the statistical signiﬁcance of the observ ed oscillations (Data of JR, Phase 3, see Results). (B) Non-oscillating data presen t a shuﬄed Lomb-Scargle transform with small amplitude, corresponding to low lev els of signiﬁcance. (Note that the scale of the tw o images is diﬀerent, for the sake of legibilit y). where τ is deﬁned by the relation: τ ( ω ) := 1 2 ω arctan P N j =1 sin(2 ω t j ) P N j =1 sin(2 ω t j ) ! (B.2) The amplitude of the transform P N ( ω ) gives access to the oscillatory con ten t of the signal. A peak in the transform at frequency ω indicates that the signal presen ts oscillations at this frequency , and the bigger the amplitude of the p eak, the more signiﬁcant the oscillations. Oscillating signals presen t highly peaked transforms, whereas non-oscillating signals pro duce ﬂat, generally noisy p erio dograms (see Fig. B.10). P eaks are therefore related to oscillations and indicate the p otential frequencies in a signal. The statistical signiﬁcance of these oscillations can b e rigorously ev aluated under the assumption that the data are samples of a p erio dic signal p erturb ed by a Gaussian white noise. This estimator has a closed form, i.e. a formula pro vides levels of P N ( ω ) directly related to statistical signiﬁcance levels of the observed oscillation. More precisely , the probability of a p eak with amplitude z to b e a false alarm of oscillation detection can b e written P ( > z ) = 1 − (1 − e − z ) M where M is the n um b er of indep endent frequencies considered, usually chosen to b e equal to 2 N , i.e. twice the num ber of observ ations. T o evidence the app earance of oscillations in the course of treatment we p erformed a sliding Lom b-Scargle transform (instead of a Lomb-Scargle transform on the full time series). This means that for eac h time t , w e compute the Lomb-Scargle transform of the recorded data in a time interv al (window) around this time, pro viding for each time t the related p erio dogram of the window ed signal. W e will therefore represent this transform as a three dimensional graph. F or eac h time t and each frequency ω will corresp ond P N ( ω ) the v alue of the Lomb-Scargle transform given b y Eq. B.1 of the signal in the time window around t . The onset of oscillations at time t will pro duce a hill in the 3D surface of the transform that will persist for the whole oscillating phase, and that will b e lo cated around the oscillation frequency (see Fig. B.11 for an artiﬁcial example). App endix C. Reduction to normal form at the Hopf bifurcation W e now compute the normal form reduction of the system deﬁned in Eq. 5 in the neighborho od of the putativ e Hopf bifurcation p oint found by the linear stabilit y analysis. Reduction to normal form for dela y diﬀeren tial equations is not a simple task, since these are dynamical systems in inﬁnite dimensions. How ever, 27 ! Figure B.11: Sliding Lomb Scargle transform, 3D representation. Sliding Lomb-Scargle transform, on the function f deﬁned piecewise by: f ( t ) = sin( φt ) with φ = ω 1 = 2 π for t ≤ t 1 and φ = ω 2 = 4 π for t > t 1 . W e can clearly see the transition at time t = t 1 , from oscillations with frequency ω 1 to oscillations at frequency ω 2 . Note the imprecision in the observed frequencies and the decreased amplitude of the transform at the transition, linked with the presence of multiple frequencies in the signal. 28 the theory is now well developed, and relatively classical metho ds are av ailable to p erform these reductions [49, 50, 51]. In detail, and using the standard notations in the domain, we w ork in the Banach space C of contin uos functions from [ − t 0 , 0] to R endow ed with the uniform norm k x k = sup θ ∈ [ − t 0 , 0] | x t | , where the norm on the righ t side is the Euclidean norm on R . The dela yed diﬀerential equation is expressed as a functional diﬀerential equation on this space. W e denote by x t the p ortion of the solution x ( t ) = p ( t ) − p + , t > 0, restricted to the in terv al [ t − t 0 , t ], with the deﬁnition x t ( θ ) = x ( t + θ ) , − t 0 ≤ θ ≤ 0 . (C.1) W e rewrite the Eq. 5 as: d dt x t ( θ ) = ( d dθ x t ( θ ) − t 0 ≤ θ < 0 A 0 x t (0) + A 1 x t ( − t 0 ) + f ( x t (0) , x t ( − t 0 )) θ = 0 (C.2) where we treat separately the linear terms dep ending on x t (0) and x t ( − t 0 ), and the nonlinear part f ( x t (0) , x t ( − t 0 )): A 0 = g − 1 A 1 = − ( g − 1) − p λ 2 − 4 β λ f ( x t (0) , x t ( − t 0 )) = λ ( p + + x t ( − t 0 )) 2 1 + β ( p + + x t ( − t 0 )) 2 − (1 − p λ 2 − 4 β λ ) x t ( − t 0 ) . (C.3) F or our purp oses, we will need only few terms of the expansion of the non-linear part f ( x (0) , x ( t 0 )) near the origin, and in particular w e can write: f ( x t (0) , x t ( − t 0 )) = B 2 x t ( − t 0 ) 2 + B 3 x t ( − t 0 ) 3 + O ( x t ( − t 0 ) 4 ) B 2 = 4 β 2  8 β − 3 λ  p λ 2 − 4 β + λ  λ 2  p λ 2 − 4 β + λ  3 B 3 = 2 β  λ 2  p λ 2 − 4 β − λ  − 2 β  p λ 2 − 4 β − 2 λ  λ 3 . (C.4) The basis for the cen ter eigenspace N is given by: Φ = (cos ( θ ω ) , sin ( θ ω )) (C.5) The corresp onding center manifold is given by: M = { φ ∈ C , φ = Φ u + h ( u ) } , (C.6) where u = ( u 1 , u 2 ) t are the coordinates of N relative to the basis Φ, and h ( u ) ∈ S , the inﬁnite-dimensional stable eigenspace, for k u k suﬃcien tly small. The solution of the functional diﬀerential equation on the center manifold are then giv en by x ( t ) = x t (0), where x t ( θ ) is a solution of C.2 satisfying x t ( θ ) = Φ( θ ) u ( t ) + h ( θ , u ( t )) (C.7) The basis for the centre eigenspace of the transp ose system Ψ( ξ ) , ξ ∈ [ − t 0 , 0], is easily derived after imposing the normalisation condition h Ψ , Φ i = I , asso ciated to the bilinear form h ψ , φ i = ψ (0) φ (0) + Z 0 − t 0 ψ ( σ + t 0 ) A 1 φ ( σ ) dσ. (C.8) 29 0 -50 0 50 100 0 -2.5 -5 0 2.5 5 α L y a p t 0 -100 Figure C.12: Scatter plot for the ev aluation of the co eﬃcien t α Lyap as function of the delay t 0 at the bifurcation. One million of points are distributed in the parameter space for 2 . 01 ≤ λ ≤ 20 and g ≤ 10. A zo om of the region near the origin is sho wn in the inset. In particular, for the follo wing calculations, we only need Ψ(0), which reads: Ψ t (0) =   1 A 1 ( 2 t 2 0 ω 2 +cos(2 t 0 ω ) − 1 ) 4 ω ( t 0 ω cos( t 0 ω ) − sin( t 0 ω )) + 1 , − 4 t 0 ω 2 sin( t 0 ω ) 4 t 0 cos( t 0 ω ) ω 2 − 4 sin( t 0 ω ) ω + A 1 (2 t 2 0 ω 2 + cos(2 t 0 ω ) − 1)   . (C.9) In terms of the co ordinates u , it can b e prov en that the diﬀerential equation translates into ˙ u ( t ) = B u ( t ) + Ψ(0) f (Ψ( θ ) u ( t ) + h ( θ , u ( t ))) (C.10) where B =  0 ω − ω 0  . (C.11) F ollo wing the standard approach in centre manifold theory , we assume that h ( θ , u ) and f ma y b e expanded in p o w er series in u . In particular, h ( θ , u ) = h 2 ( θ , u ) + h 3 ( θ , u ) + . . . , and f = B 2 + B 3 + . . . . In the same w ay , each B i can be expanded in a T aylor series around Φ( θ ) u . In order to inv estigate the criticality of the bifurcation, only the third order in u is necessary . Therefore Eq. C.10 reads: ˙ u ( t ) = B u ( t ) + Ψ(0) [ B 2 (Ψ( θ ) u ) + h 2 ( θ , u ) B 0 2 (Ψ( θ ) u ) + B 3 (Ψ( θ ) u )] + O ( k u k 4 ) (C.12) W e can then express h 2 ( θ , u ) = h 11 ( θ ) u 2 1 + h 12 ( θ ) u 1 u 2 + h 22 ( θ ) u 2 2 . Using classical methods [49], h ij can b e found solving a system of linear ordinary diﬀerential equations in terms of arbitrary constants that are then ﬁxed b y setting suitable b oundary conditions. The explicit expressions ha ve b een computed with Mathematica R  . Collecting all these terms, and confron ting them with the standard form for the system of Eq. C.12: ˙ u 1 = ω u 2 + c 1 20 u 2 1 + c 1 11 u 1 u 2 + c 1 02 u 2 2 + c 1 30 u 3 1 + c 1 21 u 2 1 u 2 + c 1 12 u 1 u 2 2 + c 1 03 u 3 2 ˙ u 2 = − ω u 1 + c 2 20 u 2 1 + c 2 11 u 1 u 2 + c 2 02 u 2 2 + c 2 30 u 3 1 + c 2 21 u 2 1 u 2 + c 2 12 u 1 u 2 2 + c 2 03 u 3 2 , (C.13) w e can derive the ﬁrst Lyapuno v co eﬃcient, whose sign determines the criticality of the bifurcation: α Ly ap = 1 8  3 c 1 30 + c 1 12 + c 2 21 + 3 c 2 03  − 1 8 ω  c 1 11 ( c 1 20 + c 1 02 ) − c 2 11 ( c 2 20 + c 2 02 ) − 2 c 1 20 c 2 20 + 2 c 1 02 c 2 02  . (C.14) 30 In our case, it can b e written as: α Ly ap = ω 8 ( A 1 (2 t 2 0 ω 2 + cos(2 t 0 ω ) − 1) + 4 t 0 ω 2 cos( t 0 ω ) − 4 ω sin( t 0 ω )) 2 ×  4 t 0 ω  A 1 B 2 h 22 ( − t 0 )  8 t 2 0 ω 2 − 5  + 6 A 1 B 3 t 2 0 ω 2 − 3 A 1 B 3 − 4 B 2 2  + 2 sin(2 t 0 ω )  2 A 1 B 2 h 22 ( − t 0 )  1 − 2 t 2 0 ω 2  − 6 A 1 B 3 t 2 0 ω 2 + 3 A 1 B 3 + 4 B 2 2  +4 t 0 ω cos(2 t 0 ω )  2 A 1 B 2 h 22 ( − t 0 )  3 − 2 t 2 0 ω 2  + 3 A 1 B 3 + 4 B 2 2  − sin(4 t 0 ω )  2 A 1 B 2 h 22 ( − t 0 ) + 3 A 1 B 3 + 4 B 2 2  +4 B 2 h 11 ( − t 0 )(2 t 0 ω (cos(2 t 0 ω ) + 2) − 3 sin(2 t 0 ω ))  A 1  2 t 2 0 ω 2 + cos(2 t 0 ω ) − 1  + 4 t 0 ω 2 cos( t 0 ω ) − 4 ω sin( t 0 ω )  − 8 B 2 h 12 ( − t 0 ) sin( t 0 ω )(2 t 0 ω cos( t 0 ω ) − sin( t 0 ω ))  A 1  2 t 2 0 ω 2 + cos(2 t 0 ω ) − 1  + 4 t 0 ω 2 cos( t 0 ω ) − 4 ω sin( t 0 ω )  − 4 A 1 B 2 t 0 ω h 22 ( − t 0 ) cos(4 t 0 ω ) − 4 ω cos(3 t 0 ω )  2 h 22 ( − t 0 )  2 B 2 t 2 0 ω 2 + B 2  + 3 B 3  +4 ω cos( t 0 ω )  2 h 22 ( − t 0 )  6 B 2 t 2 0 ω 2 + B 2  + 3  4 B 3 t 2 0 ω 2 + B 3  − 4 t 0 ω 2 sin( t 0 ω )(22 B 2 h 22 ( − t 0 ) + 15 B 3 ) +4 t 0 ω 2 sin(3 t 0 ω )(2 B 2 h 22 ( − t 0 ) − 3 B 3 )  . (C.15) Once the explicit expressions for h in , A 1 , B 2 , B 3 , t 0 and ω are inserted, a v ery long formula is obtained in terms of the parameters λ, β and g of the mo del of Eq. 5, which is very hard to study analytically . Nonetheless, an extensive n umerical in vestigation allo w to argue that in the parameter region of Eq. 7 the ﬁrst Lyapuno v co eﬃcien t α Ly ap is alwa ys negativ e and then the Hopf bifurcation is alw a ys supercritical. This can b e seen in Fig. C.12, where we show a scatter plot in which we hav e ev aluated the co eﬃcient α Ly ap of Eq. C.15 for a million of p oints uniformly distributed in the ( λ, β , g )parameter space, for 2 . 01 ≤ λ ≤ 20 and for 1 . 01 ∗ ( λ − 1) < β < 0 . 99 ∗ ( λ 2 4 ) and 1 − √ λ 2 − 4 β 2 λ < g ≤ 10, as function of the delay t 0 at the bifurcation (for k = 0). The b oundaries for the co eﬃcien t β hav e b een chosen in order to av oid numerical errors and instabilities (see Sec. 3). 31

On the Dynamic Interplay between Positive and Negative Affects

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