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Jean-Louis Nandrinoa,b, Fabrice Leroya,b and Laurent Pezardb,c,d (a) UPRES “Temps, ´emotion et cognition”, Universit´e Lille 3 (c) Neurosciences Cognitives et Imagerie c´er´ebrale LENA-CNRS UPR 640 (d) Institut de Psychologie, Universit´e Paris 5 Address for correspondance: L. Pezard, LENA-CNRS UPR 640, 47 Bd de l’Hˆopital, 75651 Paris cedex 13. France.
The development of the mathematics of dynamical systems now offers arigourous framework to deal with complex phenomenon evolving with time.
The possible euristic value of applying dynamical concepts to the field ofpsychopathology is investigated here. Three levels of applications foundin the literature are reviewed: metaphoric, qualitative and quantitative.
Psychopathology seems indeed a field where the concepts of dynamics canoffer important tools, both theoretical and empirical. Nevetheless, specificproblems should be emphasized to obtain a more profound insight in normaland pathological mental phenomenon.
1.1 Explanation levels in psychopathology . . . . . . . . . . . . .
1.2 How dynamic are mental diseases? . . . . . . . . . . . . . . .
1.3 From dynamical diseases to psychopathology . . . . . . . . .
2.1 The ’Self’ as a dynamical system . . . . . . . . . . . . . . . .
2.2 Dynamical metaphors for the psychotherapeutic processes . .
2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Gradient systems . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . . .
Models of syndromes . . . . . . . . . . . . . . . . . . .
Time-course of affective disorders . . . . . . . . . . . . 10 3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.1 Data fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Time series analysis . . . . . . . . . . . . . . . . . . . . . . . 12 Brain dynamics . . . . . . . . . . . . . . . . . . . . . . 12 Symptoms dynamics and therapies . . . . . . . . . . . 13 Dynamics of cognitive processes . . . . . . . . . . . . . 14 Clinical Interviews . . . . . . . . . . . . . . . . . . . . 15 Family system . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 The science of the mind is usually fond of importing new concepts fromother disciplines. In the last thirty years, the development of the scientificinterest in the behavior of complex systems has led to the emergence ofnotions such as chaos, attractors, sensitivity to initial conditions, etc. andto related numerical methods. The goal of this article is to estimate, on thebasis of a literature review1, the possible heuristic value, for psychopathologyof the tools developed within the mathematical and physical framework ofdynamical systems theory.
Since mental diseases have been studied from biological to social level, psy-chopathology stands at the border between natural and human sciences.
From the point of view of natural sciences, mental troubles are to be re-duced to biological phenomena such as Korsakov syndrome or dementia inAlzheimer’s disease. For the human sciences, mental disease are thought tobe due to “mind” troubles or to be related to social factors such as rela-tionships with close relatives (i.e. family) or to more general factors such associal frustrations. Nevertheless, the search for a linear causality from onelevel to another (from biology to social or backwards) has obviously failed.
For example, no biological indicators are available yet to unambiguouslydecide for a specific mental trouble.
Such problems have led to emphasize the need for a multidisciplinary in- vestigation of the bio-psycho-social nature of mental troubles (Engel, 1980;Freedman, 1995). These approaches usually explain the whole disease as thesum of each individual factor: biological, social and psychological. Never-theless, a complex phenomenon, such as a mental disease, can hardly fit intoa linear model and a co-determination of levels seems more probable. It isthus necessary to find tools to deal with circular causality and interactionsbetween levels.
The hallmark of mental troubles is the compulsive repetition of actions,fantasies or patterns of discourse which can be considered as successive con-scious or unconscious acts. Mental diseases have an onset, evolve and canfinally disappear. Moreover, specific temporal patterns appear in mental dis-eases whatever the observation scale: from milliseconds (response to stimuli,biochemical modulation or neuronal electrical activity) through minutes or 1The literature was scanned using two data bases: “pubmed” (url:) and “PsychInfo” (url:). Key words were: chaos, nonlinear dynamics, catastrophe theory, psychopathology,psychiatry, depression, schizophrenia, personality disorders, mood disorders, addiction.
hours (clinical interview) to years (time course of recurrence) or generations.
During the acute period, changes in biological and behavioral rhythms areobserved and during the whole life, specific alternations between disease andremission are also observed (Keller et al., 1986). The number of recurrencesincreases as a function of previous episodes and the illness patterns becomemore rhythmic with cycle acceleration finally resulting in rapid cycling orultradian mood patterns (Kramlinger and Post, 1996; Huber et al., 2001a).
As an explanation for the occurrence and evolution of specific patholog- ical patterns, several models have underlined the importance of initial con-ditions. In the psychoanalytic tradition, or even in cognitive psychotherapy,the possible influence of interactions and learning in infancy are assumedas important vulnerability factors for the development of mental disorders.
Nevertheless, a longitudinal study, of more than one hundred subjects, frominfancy to early adulthood, showed that the onset of behavioral disorder washighly variable (from 2 to 16 years). In most of the cases appearing duringthe adolescence, data revealed neither any prodromal or pathogenic symp-toms nor excessive stress in earlier period (Thomas and Chess, 1984). Thestructural hypothesis of universal development stages and of early determin-ism of mental disorders is thus severely challenged. In fact, the evolutionof mental troubles are highly contextualized and related to supports or con-straints continuously acting on individuals.
From dynamical diseases to psychopathology The application of dynamical systems theory to the modeling of physiologi-cal systems led to the definition of “dynamical diseases” (Mackey and Glass,1977; May, 1978; Mackey and Milton, 1987). The hallmark of a dynamicaldisease is a sudden qualitative change in the temporal pattern of physiolog-ical variables (B´elair et al., 1995). From a dynamical point of view, suchchanges are related to modifications in the control parameters that leadto abnormal dynamics. This kind of dynamical changes have been clearlyobserved in neurological diseases (Milton et al., 1989).
In a review of 32 neurological and psychiatric diseases, two main char- acteristics have been considered as landmarks for a “dynamical disease”(Milton and Black, 1995): the recurrence of symptoms (10/32) and the os-cillations appearing in the functioning of nervous systems (22/32). Withinthese fields, epilepsy and affective disorders are the best candidates for theapplication of the “dynamical disease” concept.
Such a framework can be generalized so that psychopathology may fit into the framework of “dynamical disease”. As in the case of physiologicalfunctioning, it may be hypothesized that a mental structure is an emer-gent property associated to an underlying dynamics. Clinical signs andsymptoms, observed in psychiatry, would thus correspond to qualitativedynamical changes related to modifications in control parameters (Globus and Arpaia, 1994; Moran, 1991; Schmid, 1991). Within such a conception,mental disorders and changes in mental states (such as changes followingpsychotherapeutic activity) in emotional states or in developmental stagesmay be influenced by parameters acting at several levels from physiologi-cal to social one. The presence of a symptom would thus emphasize thestability of the system in a specific parameter domain and thus be seen asan attractor. The articulation between levels of observation would thus bedefined on the basis of changes in dynamical observables.
We describe the psychopathological literature, dealing with time evo- lution of psychopathological phenomema, using mathematical and physicalconcepts from dynamical systems theory. We will distinguish three levelsof application: firstly, the study of the dynamics of complex systems canoffer a set of metaphors for the description of mental phenomena, secondlyqualitative insights of the behavior of systems can be obtained with thestudy of various models (such as neural networks or catastrophe models)and then quantitative characteristics of dynamical behaviors can be inferedusing nonlinear modeling and time series analysis. At last, criticisms and in-terests are given in order to favor a rigorous development of the applicationof dynamical concepts to psychopathology.
On the basis of the similarities between general properties of nonlinear dy-namical systems and temporal phenomena observed in mental life, metaphor-ical associations between concepts have been undertaken. We distinguishdifferent attempts using dynamical system paradigm as a metaphor in psy-chopathology. It has been proposed to understand the Self as an emergentproperty issued from dynamics of multiple iterations of brain processes, per-ceptual and social experience. Moreover, psychotherapists have used termsfrom chaos theory as an analogy for phenomena emerging during the courseof psychotherapies (psychoanalytical and systemic).
Object-relation psychoanalysts (Mahler, 1968; Klein, 1948) have underlinedhow the extended system of personal relationship influence personality de-velopment throughout life. Intersubjectivity theory (Stolorow et al., 1994)examines how the interplay between the subjective worlds of the patientsand the analyst gather into a new system. These two points of view led toconceptualize the ’Self’ as adaptive and multi-stable state of consciousnessabout oneself and the ’world’. Thus, the ’Self’ is able to adopt successively aset of discrete states evolving on the basis of contextual influences from mi-croscopic level of physiology (Freeman, 1990) through macroscopic levels ofpsychology, social or cultural organization. This psychic structure could thus be conceptualized as an open, complex, dynamical system (Marks-Tarlow,1999). Healthy selves self-organize and evolve to the edge of chaos, wherethey are capable of flexible reorganization in response to unpredictable socialan environmental contingencies (Goldstein, 1997).
In these conditions, the ’Self’ finds its origin in the continuous interac- tions between biological roots and the history of the subject. ’Self’ is thuslinked to preconscious and preverbal roots. Nevertheless, language is nec-essary to make the ’Self’ conscious (Schwalbe, 1991). Consciousness, as arecursive process operating upon internal objects and external influences,does not precede acts but emerges out of it. An iterative loop of perception-action-reflection may lead to the emergence of a new level of complexity: aconsciousness of consciousness.
Dynamical metaphors for the psychotherapeutic processes The course of psychotherapy is not a linear progression towards a new health-ier mental state. Psychotherapy is a multidimensional process involving bi-ological factors, psychological and social experiences leading the subjectstowards a new state (B¨ utz, 1993). The course towards this change is an enchainment of stable and instable periods that could be described as a nonlinear dynamic phenomenon (Langs, 1986; B¨ Spruiell, 1993; Levinson, 1994). Analysts perceive patients as different alongthe psychotherapy course; this change can be conceptualized as a qualita-tive shift in patients’ state i.e. a bifurcation in the dynamical systems theory(Moran, 1991; Priel and Schreiber, 1994; Verhuslt, 1999).
Moreover, the process of interpretation during a psychotherapy can make the psychological system more sensitive to new perturbations (B¨ Verhuslt, 1999). The psychotherapist’s function, especially through his in-terventions, is to stabilize or destabilize patients’ mental processes and theirway of thinking or telling their narrative. The therapeutical situation canthus be viewed as a dynamical process where a common system is co-createdin the interaction between therapist and patient (Elka¨ım, 1990; Lonie, 1991).
The therapeutic frame (regular appointments and stable environment) is designed to allow the emergence of a sampling of the patient’s inner world.
This phenomenon has been interpreted through the concept of self-similarity.
At any level of examination: within the whole case history, during a singlesession or a single dream, one can observe the patient own ”signature”,a recognizable pattern of his/her mental life (Lonie, 1991; Moran, 1991).
Certain aspects of psychoanalytical situation such as unconscious fantasieshave also been viewed as a form of strange attractor (Moran, 1991; Quinodoz,1997; Galatzer-Levy, 1995) or the repetition of some themes in the courseof the therapy as a limit cycle (Lonie, 1991).
The sensitivity to initial conditions and the unpredictability of complex phenomena is an important analogy between nonlinear dynamical systems individual’s mental life and behaviour is powerfully affected and determinedby precocious experiences, repetitions are not strictly identical and somesmall elements could make the evolution unpredictable. The evocation of thehistory of the patient or the focalization on certain events or feelings can haveunpredictable effects. Therapy can thus be considered as an extended seriesof well-timed perturbations which serve gradually to disrupt the strangeattractors characteristic of the patient’s fantasy-behavioral coupling (Moran,1991).
Systemic therapy has used the concepts from the general systems theory for a long time. The models from nonlinear dynamical systems are thus akind of “natural” extension for this practice (Koopmans, 1998; Miller et al.,2001). The time evolution of a family system goes through ordered anddisordered phases (Brabender, 2000) where the symptom signs the inabilityof the group to overcome crisis. Family therapist can be considered as acatalytic factor for changes in the family functioning leading the emergenceof a new state (Ricci and Selvini-Palazzoli, 1984; Elka¨ım, 1990).
The properties of nonlinear dynamical systems are obviously appealing forthe description of complex mental phenomena. In fact, the metaphoricaluse of dynamical concepts might be a first movement to get away fromstrictly medical models based on a linear explanation of the onset and theevolution of mental disorders. In that sense, nonlinear dynamical analogiescan offer new tools to deal with complex situations encountered in the clinicalpractice.
Nevertheless, several caveats need to be avoided. The distance be- tween mathematical concepts and psychological (or psychoanalytical) theo-ries needs to be questioned precisely (Denman, 1994; Kincanon and Powel,1995). Does mathematics throw a light on psychology or does it darken it?What is exactly the nature of the explanation expected from such analogies(Gardner, 1994)? It is important to avoid errors due to superficial compre-hension of precise scientific concepts (Sokal and Bricmont, 1999).
Finally, such analogies can be used as a starting point for a scientific enquiry into mental phenomena and should be tested on qualitative modelingor empirical quantitative studies.
Qualitative models are related to the introduction of explicit constraints onthe definition of a specific dynamical system supposed to model the em-pirical system taken into account. Two types of dynamical systems havebeen taken as models in psychopathology: first, gradient systems related to “catastrophe theory” have been considered, then the development of neuralnetwork introduced another kind of modeling.
The state of a system at time t can be described by a set of variables ψ(t) ={ψi(t)} (ψi are thus called state variables) and that a set of parameters,denoted (1 ≤ α ≤ k), controls the qualitative properties of the system’stime evolution (are thus called control parameters). The dynamics of thesystem is said to be described by a dynamical system when2: with f = {fi}. The general study of systems represented by equation (1) is avery difficult problem. It can be made more tractable when two assumptionsare added (Gilmore, 1981): 1. If the functions fi are considered as independent of time, the dynam- ical system is now an autonomous dynamical system and powerfulstatements can be made about such systems which depend on a smallnumber of parameters (k ≤ 4).
2. It can be noticed that in equation (1) the functions fi look as the components of a force. With the assumption, inspired from mechanics,that all the functions fi can be derived as the negative gradient (withrespect to the ψi) of some potential function V (ψj, cα): ψ = −∇ψV ). This kind of system is much more tractable than the other systems described previously.
Dynamical systems theory deals with the solutions ψ1(t), ψ2(t), . . . , ψn(t) of equation (1) which define trajectories (i.e. time evolution) of thesystem. Of particular interest are the equilibria (dψi/dt = 0) of dynamicaland gradient systems. They define the states where the system can settlein, either, a stable or unstable manner.
2For a more general statement about the time evolution of a system and the hypothesis that lead to the somehow reduced dynamical system description, see Gilmore (1981, p. 3–5).
Elementary catastrophe theory is the study of how the equilibria ψej() of V (ψj, cα) change as the control parameters cα change for gradient sys-tems. In that sense, elementary catastrophe theory is a quasi-static theorysince it is only concerned by the equilibrium points of a dynamics and howthey change when the control parameters are varied (Thom, 1977a; Arnol’d,1992).
These models have been mainly used to model the emergence of discon- tinuous behaviors out of continuous parameter variations. The applicationof catastrophe theory to concrete phenomenon can be divided into the ’meta-physical’ way and the ’physical’ way (Thom, 1977b).
The metaphysical way considers the generality of elementary catastro- phe as justifying the use of archetype situations to describe phenomenonwhere the nature of the dynamical systems that produce them is un-known. This method lead to qualitative models that can be usedanalogically with real situations.
The dichotomy between anorexia and bulimia is an archetypic example(Zeeman, 1977). The starting point of the model was the observationthat an anorexic loses access to normal attitudes toward food and thatmany sufferers develop bulimic phase. During theses cycles attitudestoward food switch catastrophically from one extreme to the other,and they never take on normal intermediate values. These are thehallmarks of the cusp catastrophe which was used to model this be-havioral trouble. A more sophisticated model added the sleep/wakecycle to the preceding cusp model and thus develop a geometrical non-trivial double cusp model (Callahan, 1982).
Catastrophe theory has also been used in a set of other models inclinical psychology (Weiner, 1977; Galatzer-Levy, 1978; Scott, 1985).
Catastrophe model based on the attention focus has been proposed todeal with manic/depressive illness (Johnson, 1986). Emotional numb-ing associated with post-traumatic stress disorder (Glover, 1992) andother emotional responses (Lanza, 1999) have also been modeled usingcusp catastrophe such as the relationship between alcohol intoxicationand suicidal behavior (Hufford, 2001). In the case of schizophrenia,catastrophe models provided ways in which neurochemical and envi-ronmental influences could interact so that very small changes in eithervariable may produce the rapid changes in intensity of psychosis (Mac-Culloch and Waddington, 1979) The dopaminergic hypothesis has alsobeen investigated using this framework (King et al., 1981).
From a more general standpoint, the possible heuristic value of Thom’sdynamical theory to the Freudian metapsychology has been evaluated(Porte, 1994). On the basis of a careful parallel between both authors,it can be estimated that positivists caveats of Freud’s theory find a natural solution in modern dynamical theories.
The physical way applies when the dynamics is indeed described by a gradient system. It is the case for example in physical systems forphase transitions in thermodynamics or caustics in optics (Poston andStewart, 1978; Gilmore, 1981).
An exemplary modeling of alcohol consumption follows such a per-spective (an der Heiden et al., 1998). The model is based on themathematical expressions relating general phenomenon supposed todrive alcohol consumption (denoted A). The authors reported severalstage of a qualitative model which final expression is: where F (frustration) is considered as a constant force driving alcoholconsumption (such as life conditions, habits, social environment.),r is related to the disagreement of alcohol intake (illness, social val-ues.) and the last term with parameter σ is a nonlinear auto-catalyticmodel. The study of the equilibria of this model leads to describe thephenomenology of drinkers typology and a cusp catastrophe was foundin the description of the bifurcations. The discussion of the model showhow control parameters can be varied to change the drinking behaviorand thus may be of interest in the clinical practice. Moreover, Thisstudy demonstrates that the interaction of very few “mechanisms” re-sults in a large manifold of different kinds of behavior.
The first use of neural networks has been devoted to provide models of brainfunctioning. Two major class of models can be differentiated: parallel dis-tributed processing (PDP) models (McClelland et al., 1986a,b) and attractorneural network (ANN) models (Hopfield, 1982; Amit, 1989). We will onlyreview here some models using ANN to deal with psychiatric syndromes(other models can be found in Rialle and Stip (1994), Aakerlund and Hem-mingsen (1998) or Huberman (1987)). Neural networks have also been usedas models of symptoms dynamics.
Attractor neural network models are based on systems such as (Hopfield, where Si(t) is the state of “neuron”i at time t, wij is the “synaptic weight”between neurons i and j and θi is the threshold. Such system has compu-tational abilities since memories are stored as attractors of its dynamics; sothat, as an content-addressable memory: memories (or patterns) are retrieved according to similarity to the generalizations based on different memories are possible memories are distributed across all neurons, and are not localized An alterations of these functions, related to changes in the control parame-ters, may thus simulate cognitive impairments in some mental disorders.
Manic-depressive illness An interpretation of manic behavior has beenproposed on the basis of a classical Hopfield network (Hoffman, 1987). Theincrease of noise (related to the steepness of the slope of the transitionfunction) causes an increase of transitions between attractors. This behaviorof the network has thus been related to the transitions between thoughts inmanic patients.
Another model consider depression-like and manic-like behavior as at- tractors of a dynamical system (Globus and Arpaia, 1994). The same for-malism is thus used at a higher level where attractors represent the overallbehavior. It must be emphasized, that this model is clearly similar to acatastrophe model.
Schizophrenia On the basis of ANN, schizophrenia has been interpretedas the result of the overloading of the network memorization abilities (Hoff-man, 1987). In fact, overloading causes the creation of spurious attractorsfrom which the network cannot escape. Delirium has been associated withsuch a process. Troubles in cortical pruning, during development, lead toa decrease of cortical synaptic contacts and would thus decrease the mem-orization ability of the cortical network in schizophrenic patients (Hoffmanand Dobscha, 1989; Hoffman and McGlashan, 1994, 2001). The presenceof spurious attractors could be the analog of the three types of symptoms:strange outputs, independent submodules, and independence of modulesfunctioning in front of inputs. This model has been discussed in David(1994). A network based on spreading activation was also proposed to modelhow an initial paranoid state becomes crystallized into a fixed delusion inschizophrenia (Vinogradov et al., 1992).
The defect of generalization and/or of taking into account the context in schizophrenic patients has been related to dysregulation of dopamine trans-mission (Cohen and Servan-Schreiber, 1992). Such changes in the interac-tions between cortical and sub-cortical structures could reduce the size ofattractors in patients when compared to controls (Tassin, 1996). Neverthe-less, the observed increased variability in behavior among schizophrenics,could also been related to chaotic dynamics in the central dopaminergicneuronal system (King et al., 1984).
Episodes of affective disorders have been analogically compared to firing inneuronal networks (Huber et al., 2000b,a, 1999). A mathematical modelbased on a nonlinear dynamical system influenced by noise has been pro-posed: i wi(x − xi) + S + gw where τx is a relaxation time constant, aνi represents the activation states (ν = 1 or ν = 2), i ranges over four different states, wi are coupling con-stants and xi describes different activation levels. S represents the controlparameter (corresponding to an ongoing disease process), and gw representsa Gaussian white noise to take into account environmental or endogenousstochastic influences.
The dynamic behavior shows that, in the course of the illness, noise might amplify sub-clinical vulnerabilities into disease onset and could in-duce transitions to rapid-changing mood pattern. In this model, based oncooperative effects between deterministic and random dynamics, noise in-creases the spectrum of dynamic behaviors.
Furthers modifications of this model, based on a feedback mechanism for episode sensitization, permits to strongly support the importance of episodesensitization as fundamental mechanism for the disease’s progression in af-fective disorders (Huber et al., 2001a,b).
The introduction of specific kind of dynamical systems as models in psy-chopathology provide a global framework for the description of changes inpsychopathology. Based on the generality of the formalism it is thus possibleto describe various levels of observations within the same model. Neverthe-less, even if these models introduce more constraints than in analogical useof dynamical concepts, it is not always clear whereas they constitute realmodel or mere elaborated metaphors.
These models thus need development towards empirical empirical tests.
The introduction of quantitative methods may fill the gap between qualita-tive modeling and empirically observed dynamics.
Empirical studies quantifying the characteristics of observed dynamics areneeded to estimate the scientific and clinical value of dynamical paradigmin psychopathology.
The presence of a “catastrophe” can be infered either on the basis of obser-vation or from the study of a model. Empiricists would prefer that “catas-trophe” could be proved and measured on the basis of experimental data.
The theoretical analysis of the behavior of systems in the neighborhood of singularities allow one to define critical phenomena that should be ob-served for a catastrophe model to apply. These phenomena have been calledcatastrophe flags (Gilmore, 1981). The first ones (modality, inaccessibil-ity, sudden jump) have usually been taken as qualitative indices for the’metaphysical’ application of catastrophe theory. The other one (diver-gence, hysteresis, divergence of linear response, critical slowing down andmode softening, anomalous variance) are usually more difficult to observeor to describe. Such ’flags’ have been infered in development stages (van derMaas and Molenaar, 1992).
Three quantitative approaches to the problem of testing the fit of be- havioral data to catastrophe models have been developed. The first hastaken stochastic difference equations as a basis and uses the methods of mo-ment to estimate model parameters (Cobb and Watson, 1980). The seconduses polytope search curve-fitting procedure to obtain maximum likelihoodestimates of the model from the observed data (Oliva et al., 1987; Langeet al., 2001). The third approach is in the form of least-square regression(Guastello, 1982, 1987). This last method has been discussed in Alexanderet al. (1992) and Guastello (1992).
The analysis of a cusp catastrophe used to model adolescent alcohol use have shown that dispositions should be viewed as the normal parameterand situation pressure as the splitting parameter of the cusp (Clair, 1998).
Statistical analysis of empirical data using polynomial regression have shownthat the cusp model better fit the data than the alternative linear models(Clair, 1998). Such procedure have also been used in the test of anxietytheory in the context of sport performance (Hardy, 1996).
It is out of the scope of this review to develop a complete methodologicaloverview. For complete references, see Kantz and Schreiber (1997); Grass-berger et al. (1991); Abarbanel et al. (1993); Ott et al. (1994); Badii andPoliti (1997).
Time series analysis deals with the quantification of the ’complexity’ in the sequence of observed data. From the dynamical point of view, the firststep is the reconstruction of the trajectory of the system within its phasespace, then geometrical indices (such as dimensions) or dynamical indices(such as entropies) are computed. It has been shown that these indicesshould be statistically validated using surrogate data methods (for a review see: Schreiber and Schmitz, 2000). When data are discrete (or when thecontinuous dynamical system is ’properly’ discretized), the characterizationof the dynamics uses symbolic methods (Badii and Politi, 1997).
The central nervous system can be considered as a complex system whichcan be modeled within the dynamical system theory. For example, nonlineardynamics provides new methods for the investigation of EEG signals.
Depression Studies of brain dynamics in depression have mainly showna decrease of the first Lyapunov exponent for sleep stage IV in depressed pa-tients when compared either to controls (Roschke et al., 1995b) or schizophrenicpatients (Roschke et al., 1994). Unipolar depression is characterized by spe-cific brain dynamical patterns of low complexity which evolve during phar-macological treatments (Nandrino et al., 1994; Pezard et al., 1996). Never-theless, the recovery of a healthy brain dynamics is dependent upon clinicalhistory: in the case of patients with recurrent episodes, even after a clinicalimprovement similar to that of first episode patients, brain dynamics did notrecover the complexity level of control subjects. Changes in brain dynamicshave been correlated with clinical evaluation of depressive mood in threedepressed patients (Thomasson et al., 2000). These results were confirmedin the case of a 48-hour rapid cycling patient (Thomasson et al., 2002).
Schizophrenia Brain dynamics was studied in schizophrenic patients bothduring sleep and awake states. REM sleep in schizophrenic patients is char-acterized by a lower Lyapunov exponent (Roschke et al., 1995a). This alteredbrain dynamics could correspond to an impairment of the safety function ofdreams (Keshavan et al., 1990). In addition, it has been shown that EEG’sdimensionality was reduced during sleep stages and REM in schizophrenicpatients (Roschke and Aldenhoff, 1993).
During awake states, nonlinearity and correlation dimension computed with spatial embedding of EEG data are lower in schizophrenia (Lee et al.,2001b; Jeong et al., 1998). Moreover, Lyapunov exponents also decreasein schizophrenia (Kim et al., 2000). When time embedding is used, spa-tial heterogeneities are demonstrated by correlation dimension (Lee et al.,2001a).
Finally, using mutual cross prediction (Le Van Quyen et al., 1998), it has been shown that the driving system was shifted to the frontal channelafter 4-week trial with clozapine in schizophrenia (Kang et al., 2001).
Other physiological indices Time series of heart period and respiratoryrhythms obtained from normal controls and patients with panic disorder were analyzed (Yeragani et al., 2000, 2002). Results showed that approx-imate entropy and largest Lyapunov exponents were higher in patients innormal breathing condition (Yeragani et al., 2002).
Mood disorders The alternation between depressed and manic episodesin bipolar troubles constitutes an important illustration of symptoms dy-namics (e.g. Wehr and Goodwin, 1979; Wehr et al., 1982). In order toassess whether the time evolution of mood modifications in bipolar troubleare related to stochastic or deterministic dynamics, daily scores to analogicalmood scales have been recorded from one to two years and a half (Gottschalket al., 1995). Linear (autocorrelation function and power spectra) and non-linear (phase space embedding, correlation dimension, recurrence plots andsurrogate data testing) were performed on the data obtained from sevenrapid-cyclers and twenty-eight control subjects. Six out of the seven pa-tients depicted convergent estimates of the correlation dimension whereasnone of the controls did. Together with the complex power spectra this re-sult indicates that mood in patients with bipolar disorder is not really cycliccontrary to the current opinion. Nonetheless, self-rated mood in patientsis more organized than in control subjects and can be characterized as alow-dimensional chaotic process.
In a similar study (Woyshville et al., 1999), patients and control gener- ated time series data, using a visual analog scale to quantify their mood.
The results showed that patients display more variability but less complexity(measured by fractal dimension) in their time series than controls.
Schizophrenia Time-course of schizophrenic episodes can be investigatedas a non-linear phenomenon. Daily assessment of psychotic derealizationin fourteen schizophrenics have been studied during a period lasting be-tween 200 and 770 days. Phase space reconstruction, nonlinear forecastingmethods and surrogate data testing were applied to these time series. Timeevolution of psychotic symptoms were classified as non-linear dynamics (8patients out of 14), linear dynamics (4/14), and stochastic evolution (2/14).
These results show that schizophrenia can be considered as a nonlinear dy-namical disease, controlled by a low dimensional attractor (Tschacher et al.,1997). More descriptive methods might also be valuable to the interpre-tation of symptoms trajectories in schizophrenia (Tschacher and Kupper,2002; Kupper and Tschacher, 2002) Addiction Single-case studies have shown that daily alcohol consumptionassessed during a five-year period can be modeled using multi-scale nonlinearmethods (Warren et al., 2003; Warren and Hawkins, 2002).
Psycho-social crisis intervention In a sample of 40 in-patients of apsychosocial crisis intervention unit, time series data were obtained by self-rated evaluation on mood, tension and cognitive orientation (Tschacher andJacobshagen, 2002). In crisis intervention, outward cognitive orientationgenerally preceded improved mood so that cognitive orientation is responsi-ble of the experienced affective effects of crisis intervention.
Psychotherapy courses To test empirically the proposal that psychother-apy can be viewed as a self-organized dynamical system, 28 psychotherapycourses have been evaluated (Tschacher et al., 1998). The course of the ther-apies was characterized by a decrease of degree of freedom and an increaseof order. Moreover, these results were independent of the kind of therapyand increase of order was related to positive outcomes of therapy.
Time series generated, in a simple binary choice task, by schizophrenics weremore interdependent than that of controls, suggesting that their behavior isless complex (Paulus et al., 1996, 1999). Moreover, schizophrenic patientsexhibited significantly less consistency in their response selection and order-ing, characterized by a greater contribution of both highly perseverative andhighly unpredictable subsequences of responses within a test session (Pauluset al., 1996). Schizophrenic patients also are significantly less influenced byexternal stimuli than are normal comparison subjects (Paulus et al., 1999).
This dysregulation is stable over time and independent of psychosocial fac-tors and symptomatic fluctuations (Paulus et al., 2001).
In motor and perceptual tasks, schizophrenic patients exhibit a higher instability in their movement’s process (horizontal finger oscillations) anda higher reversal rate in the perception of an ambiguous figure (the Ru-bin vase) compared to matched controls. Moreover, motor and perceptualmeasures were unrelated. These results suggest that alterations observed inthe motor and perceptual dynamics in schizophrenia are be supported by acommon underlying mechanism (Keil et al., 1998).
Dynamical quantification of language in schizophrenia (Leroy et al., 2003) have shown that the probability transition between macro-clauses andmicro-clauses is lower in schizophrenic patients than in controls. This re-sult can be view as a deficit in the dynamical access to the context level inschizophrenia.
During clinical interview, one can focus either on the patient himself, or onthe patient-therapist interaction.
Brain dynamics In a pilot study (Rockstroh et al., 1997), time serieswere obtained from electroencephalographic records during clinical inter-views with 10 schizophrenic (6 paranoid, 4 disorganized) and 2 depressivepatients. The time sequence of thought disorders (unusual thought contents,sudden change in topic, thought stopping,. . . ) were also assessed.
The paranoid subgroup has been characterized by a lower complexity but more critical transitions in the EEG when compared to disorganizedand depressive patients. But, such results are hardly correlated with aparticular symptom, or to an underlying cognitive process. Furthermore,sudden phase transitions in brain activity were significantly enhanced priorto expressions of thought disorders that were detected by the interviewerand an observer in the conversation, compared with time periods during theinterview without such symptoms.
Cardiac dynamics Since cardiological markers are related to the emo-tional behavior, they might be of interest to assess the complexity of patient-therapist interaction (Redington and Reidbord, 1992; Reidbord and Reding-ton, 1992, 1993). Patient’s cardiac dynamics is less complex when talkingabout important topics than for more distant topics. In the case of the ther-apist, it has been shown that cardiac dynamics depict a higher complexitywhen the therapist feels something with the patient rather than about thepatient. Similar results were found in a study of 20 patients where variationin the complexity of heart’s dynamics was observed when topics changes(Pincus, 1991).
Patient-therapist interaction The communicative process between pa-tient and therapist needs to be studied (Langs and Badalamenti, 1994). Tocontribute to the construction of research methodology, patient-therapistinteractions were encoded by means a matrix, in which each column repre-sent a time series obtained by responses at questions about the sequence ofinteractions (Rapp et al., 1991). By this method, time series were obtainedand a complexity score was computed.
Psychotherapy is also viewed as a chaotic process, and tools of non-linear dynamics are used to quantify this qualitative hypothesis. A single case wasanalyzed, by means of a time series obtained from the patient-therapistinteractions (Schiepek et al., 1997). It has been shown that the time seriesis non-periodic, and the technique of surrogate data demonstrates that thisnon-periodicity is caused by a chaotic dynamics, and not by a stochasticprocess (or by noise). Fractal dimension and largest Lyapunov exponentsrevealed the presence of an attractor, which characterized the chaotic processof the therapy. Nevertheless, from a clinical point of view, the goal of atherapy is to lead the patient toward change rather than to stability, thusthe methods used to characterize stationary dynamical systems are hardly adapted. The same data were thus re-analyzed (Kowalik et al., 1997) anddemonstrate that, critical transitions appear during the therapeutic process,so that a non-stationary approach of the phenomena is necessary.
Family systems may be described by a 5-R’s model where the four com-ponents (rules, roles, relationships and realities) are determining the fifthR (response pattern). In order to test the basic assumption of this modela family discussion was video-taped and analyzed (Pincus, 2001) using theorbital decomposition procedure (Guastello et al., 1998). The author makethe hypothesis that the family response patterns during the discussion willshow evidence of both coherence and complexity.
The family conversation was transformed into a symbolic sequence. En- tropy measurements demonstrate the existence of a local coherence for stringlengths equal to 3 and provide evidence for low dimensional chaos withinthe global family discussion.
These studies demonstrate the importance of temporal evolution in psy-chopathology. Aside from methodological drawbacks, dynamical processeshave been characterized at several levels from physiological to linguistic one.
Moreover, several studies have shown correlation between dynamical pro-cesses at different levels: brain dynamics and mood assessment, cardiacdynamics and emotions induced during interviews.
We have explored three ways of using mathematical and numerical dynam-ical concepts in psychopathology. We can conclude that the metaphoricaldescription of mental troubles and changes are beginning to be modeled andtested empirically. More efforts are still needed to introduce an adaptedmethodology to the field of psychopathology. In fact, empirical tests de-cribed here are usually either data fitting to models or time series analysis(either of continous or discrete data). These two approaches are mainly“data-driven” i.e. they do not rely on a “theoretical” model to be testedin the data exploration (even when they are based on a model such as acatastrophe model). This interaction between models and data explorationis certainly a promising perspective of the application of dynamical systemsto psychopathology.
The application of dynamical methodology to the “human sciences” are, however, still in its infancy. Several problems are to be worked out: 1. The development of accurate quantitative tools on short time series are clearly needed since the numerical methods imported from physicsare highly data demanding.
2. The emphasis has been mainly given to deterministic modelling be- cause of the fascinating properties of deterministic chaos. Never-theless, stochastic or deterministic description are only a problem ofscale and choice (in physics, molecular dynamics are deterministic butstochastic and statistical description of a gas is usually prefered formacroscopic scale). Thus, the choice of a model should not be ob-scured by some ’fascination’.
3. Quantification has long being the ideal of science. However, carefully designed qualitative models might be more informative than the com-putation of (ill-founded) quantitative indices.
Psychopathology is an adapted field for dynamics since it deals with entitieswith clear time evolution. Nevertheless, it could be misleading to imaginethat dynamics can be directly imported in the field of psychopathology with-out considering its specificity. Different scales usually means that differenttools adapted are to each kind of measures.
The behavioral, biological and clinical data that are mostly used in the study of mental troubles are observed from one sample at a single timepoint. Those data are informative but lack sensitivity to the frequency ofbehavior and hence to its temporal organisation. Thus the measurementsof dynamical complexity are complementary to the first kind of empiricaldata. These studies are an useful tool for the comprehension of mental andbehavioral changes. They allow one the study of the interaction betweenseveral factors and thus avoid the reduction of mental trouble to the effectof one single factor.
Because several levels interact, it is important to focalize attention on break or changes of state. The ruptures or the dynamical changes are ob-servable at the different observation levels. Clinical data are concordantwith such a point of view since changes are simultaneously observed in neu-rophysiology, in the strategy of thinking, the kind of beliefs, the types ofbehavior or the transactional activities. The only common point susceptibleto be study is these ruptures in dynamics.
Moreover, open systems are by definition coupled with their environ- ment. Studying human being implies that researchers take into account thecontexts in which a behavior is developed. We must not have only knowl-edge about the system itself but also about the way it uses to interact withits environment. Contexts are necessary the broadest possible and implyphysiological parameters, ecological, familial, social and cultural elements.
A last point must be underlined: the role of observers. An observer placedin an environment has necessarily an effect on the observed system.
The generalisation of the “dynamical disease” concept to mental troubles 1. From the point of view of diagnostics, the possibility of defining dy- namical characteristics specific of a disease (such as a specific rhythmin a biological functionning) would offer a tool for the biological sideof psychopathology.
2. From the point of view of therapeutics, the isolation of factors that may influence the behavioral and/or mental changes would offer, tothe clinicians, several paths of action. In that case, changes would bepossible either on the basis of a changes in the control parameters or onthe basis of a perturbation depending upon the level of the intervention(biological, psychological or social). It is thus possible to imagine newtherapeutical ways based on valid models of the dynamics underlyingthe mental trouble.
3. From a theoretical point of view, the model of a “dynamical disease” underlying mental troubles seems more legitime than a linear “med-ical” point of view. Clinical signs or symptoms can be considered asdiscontinuous changes based on continuous changes in control param-eters. Thus dynamical systems theory seems particularly well adaptedto the study of mental troubles.
It is thus important to develop the methodology of dynamical systemstowards (rigorous) applications in the “human sciences” and then tointegrate these tools into more classical psychopathological studies.
It seems particularly important to emphasize the study of temporaldimension of psychopathological phenomena.
Such a dynamical point of view decrease the ontological gap that hasbeen hypothesized between normal and pathological mental activities:it favors an underlying continouous point of view even if the behavioralobservables are clearly discontinuous.
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A velha lei no novo testamento

UM ESTUDO SUPLEMENTAR A VELHA LEI NO NOVO TESTAMENTO OWEN D. OLBRICHT “Agora, com efeito, obteve Jesus ministério tan- digo: até que o céu e a terra passem, nem um i ou to mais excelente, quanto é ele também Mediador um til jamais passará da Lei, até que tudo se de superior aliança instituída com base em supe- riores promessas. Porque, se aquela primeira Essa passage

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