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Pavlidis E, Campillo F, Goldbeter A, Desroches M. Multiple-timescale dynamics, mixed mode oscillations and mixed affective states in a model of bipolar disorder. Cogn Neurodyn 2022. [DOI: 10.1007/s11571-022-09900-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022] Open
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2
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Nunes A, Singh S, Allman J, Becker S, Ortiz A, Trappenberg T, Alda M. A critical evaluation of dynamical systems models of bipolar disorder. Transl Psychiatry 2022; 12:416. [PMID: 36171199 PMCID: PMC9519533 DOI: 10.1038/s41398-022-02194-4] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 04/18/2022] [Revised: 09/18/2022] [Accepted: 09/20/2022] [Indexed: 12/02/2022] Open
Abstract
Bipolar disorder (BD) is a mood disorder involving recurring (hypo)manic and depressive episodes. The inherently temporal nature of BD has inspired its conceptualization using dynamical systems theory, which is a mathematical framework for understanding systems that evolve over time. In this paper, we provide a critical review of the dynamical systems models of BD. Owing to the heterogeneity of methodological and experimental designs in computational modeling, we designed a structured approach that parallels the appraisal of animal models by their face, predictive, and construct validity. This tool, the validity appraisal guide for computational models (VAG-CM), is not an absolute measure of validity, but rather a guide for a more objective appraisal of models in this review. We identified 26 studies published before November 18, 2021 that proposed generative dynamical systems models of time-varying signals in BD. Two raters independently applied the VAG-CM to the included studies, obtaining a mean Cohen's κ of 0.55 (95% CI [0.45, 0.64]) prior to establishing consensus ratings. Consensus VAG-CM ratings revealed three model/study clusters: data-driven models with face validity, theory-driven models with predictive validity, and theory-driven models lacking all forms of validity. We conclude that future modeling studies should employ a hybrid approach that first operationalizes BD features of interest using empirical data to achieve face validity, followed by explanations of those features using generative models with components that are homologous to physiological or psychological systems involved in BD, to achieve construct validity. Such models would be best developed alongside long-term prospective cohort studies involving a collection of multimodal time-series data. We also encourage future studies to extend, modify, and evaluate the VAG-CM approach for a wider breadth of computational modeling studies and psychiatric disorders.
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Affiliation(s)
- Abraham Nunes
- Department of Psychiatry, Dalhousie University, Halifax, NS, Canada.
- Faculty of Computer Science, Dalhousie University, Halifax, NS, Canada.
| | - Selena Singh
- Department of Psychology, Neuroscience & Behaviour, McMaster University, Hamilton, ON, Canada
| | - Jared Allman
- Department of Psychiatry, Dalhousie University, Halifax, NS, Canada
| | - Suzanna Becker
- Department of Psychology, Neuroscience & Behaviour, McMaster University, Hamilton, ON, Canada
| | - Abigail Ortiz
- Department of Psychiatry, University of Toronto, Toronto, ON, Canada
- Centre for Addiction & Mental Health, Toronto, ON, Canada
| | | | - Martin Alda
- Department of Psychiatry, Dalhousie University, Halifax, NS, Canada
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3
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Bottemanne H, Barberousse A, Fossati P. [Multidimensional and computational theory of mood]. Encephale 2022; 48:682-699. [PMID: 35987716 DOI: 10.1016/j.encep.2022.02.002] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2021] [Revised: 01/31/2022] [Accepted: 02/04/2022] [Indexed: 12/22/2022]
Abstract
What is mood? Despite its crucial place in psychiatric nosography and cognitive science, it is still difficult to delimit its conceptual ground. The distinction between emotion and mood is ambiguous: mood is often presented as an affective state that is more prolonged and less intense than emotion, or as an affective polarity distinguishing high and low mood swinging around a baseline. However, these definitions do not match the clinical reality of mood disorders such as unipolar depression and bipolar disorder, and do not allow us to understand the effect of mood on behaviour, perception and cognition. In this paper, we propose a multidimensional and computational theory of mood inspired by contemporary hypotheses in theoretical neuroscience and philosophy of emotion. After suggesting an operational distinction between emotion and mood, we show how a succession of emotions can cumulatively generate congruent mood over time, making mood an emerging state from emotion. We then present how mood determines mental and behavioral states when interacting with the environment, constituting a dispositional state of emotion, perception, belief, and action. Using this theoretical framework, we propose a computational representation of the emerging and dispositional dimensions of mood by formalizing mood as a layer of third-order Bayesian beliefs encoding the precision of emotion, and regulated by prediction errors associated with interoceptive predictions. Finally, we show how this theoretical framework sheds light on the processes involved in mood disorders, the emergence of mood congruent beliefs, or the mechanisms of antidepressant treatments in clinical psychiatry.
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Affiliation(s)
- Hugo Bottemanne
- Paris Brain Institute - Institut du Cerveau (ICM), Sorbonne University/CNRS/Inserm, Paris, France; Department of philosophy, Sciences Normes Démocratie research unit, Sorbonne university/CNRS, Paris, France; Department of psychiatry, DMU Neuroscience, Pitié-Salpêtrière hospital, Sorbonne university/Assistance publique-Hôpitaux de Paris (AP-HP), Paris, France.
| | - Anouk Barberousse
- Department of philosophy, Sciences Normes Démocratie research unit, Sorbonne university/CNRS, Paris, France
| | - Philippe Fossati
- Paris Brain Institute - Institut du Cerveau (ICM), Sorbonne University/CNRS/Inserm, Paris, France; Department of psychiatry, DMU Neuroscience, Pitié-Salpêtrière hospital, Sorbonne university/Assistance publique-Hôpitaux de Paris (AP-HP), Paris, France
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4
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Abstract
Mood is an integrative and diffuse affective state that is thought to exert a pervasive effect on cognition and behavior. At the same time, mood itself is thought to fluctuate slowly as a product of feedback from interactions with the environment. Here we present a new computational theory of the valence of mood-the Integrated Advantage model-that seeks to account for this bidirectional interaction. Adopting theoretical formalisms from reinforcement learning, we propose to conceptualize the valence of mood as a leaky integral of an agent's appraisals of the Advantage of its actions. This model generalizes and extends previous models of mood wherein affective valence was conceptualized as a moving average of reward prediction errors. We give a full theoretical derivation of the Integrated Advantage model and provide a functional explanation of how an integrated-Advantage variable could be deployed adaptively by a biological agent to accelerate learning in complex and/or stochastic environments. Specifically, drawing on stochastic optimization theory, we propose that an agent can utilize our hypothesized form of mood to approximate a momentum-based update to its behavioral policy, thereby facilitating rapid learning of optimal actions. We then show how this model of mood provides a principled and parsimonious explanation for a number of contextual effects on mood from the affective science literature, including expectation- and surprise-related effects, counterfactual effects from information about foregone alternatives, action-typicality effects, and action/inaction asymmetry. (PsycInfo Database Record (c) 2022 APA, all rights reserved).
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Affiliation(s)
| | | | - Yael Niv
- Princeton Neuroscience Institute and Department of Psychology
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5
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Gruichich TS, Gomez JCD, Zayas-Cabán G, McInnis MG, Cochran AL. A digital self-report survey of mood for bipolar disorder. Bipolar Disord 2021; 23:810-820. [PMID: 33587813 PMCID: PMC8364560 DOI: 10.1111/bdi.13058] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 07/31/2020] [Revised: 11/13/2020] [Accepted: 02/02/2021] [Indexed: 12/23/2022]
Abstract
OBJECTIVES Bipolar disorder (BP) is commonly researched in digital settings. As a result, standardized digital tools are needed to measure mood. We sought to validate a new survey that is brief, validated in digital form, and able to separately measure manic and depressive severity. METHODS We introduce a 6-item digital survey, called digiBP, for measuring mood in BP. It has three depressive items (depressed mood, fidgeting, fatigue), two manic items (increased energy, rapid speech), and one mixed item (irritability); and recovers two scores (m and d) to measure manic and depressive severity. In a secondary analysis of individuals with BP who monitored their symptoms over 6 weeks (n = 43), we perform a series of analyses to validate the digiBP survey internally, externally, and as a longitudinal measure. RESULTS We first verify a conceptual model for the survey in which items load onto two factors ("manic" and "depressive"). We then show weekly averages of m and d scores from digiBP can explain significant variation in weekly scores from the Young Mania Rating Scale (R2 = 0.47) and SIGH-D (R2 = 0.58). Lastly, we examine the utility of the survey as a longitudinal measure by predicting an individual's future m and d scores from their past m and d scores. CONCLUSIONS While further validation is warranted in larger, diverse populations, these validation analyses should encourage researchers to consider digiBP for their next digital study of BP.
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Bottemanne H, Chevance A, Joly L. [Psychiatry without mind?]. Encephale 2021; 47:605-612. [PMID: 34579938 DOI: 10.1016/j.encep.2021.05.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2021] [Revised: 05/08/2021] [Accepted: 05/21/2021] [Indexed: 10/20/2022]
Abstract
Philosophy of Mind is currently one of the most prolific fields of research in philosophy and has witnessed a progressive hybridization with cognitive science. It focuses on fundamental questions to neuroscience and psychiatry, such as the nature of mental states and cognitive processes, or the relationships between mental states and the world. Anticipating the accumulation of experimental data from neuroscience, it provides a framework for the generation of theories in cognitive science. Philosophy of mind has thus laid the foundations of the conceptual space within which cognitive sciences have spread: a large part of contemporary theories in cognitive science result from a hybridization of conceptions forged by philosophers of mind and data produced by neuroscientists. Yet contemporary psychiatry is still reluctant to feed on the philosophy of mind, other than through the fragments that emerge from neuroscience. In this paper, we describe the evolution of contemporary philosophy of mind, and we detail its contributions around three central themes for psychiatry: naturalization of mind, mental causality, and subjectivity of mental states. We show how philosophy of mind provide the conceptual framework to link different levels of explanation in psychiatry: from biological to functional, from neurophysiology to cognition, from matter to mind.
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Affiliation(s)
- H Bottemanne
- Paris Brain Institute-Institut du Cerveau (ICM), UMR 7225/UMRS 1127, Sorbonne University/CNRS/Inserm, Paris, France; Sorbonne University, Department of Philosophy, SND Research Unit, UMR 8011, CNRS, Paris, France; Sorbonne University, Department of Psychiatry, Pitié-Salpêtrière Hospital, DMU Neuroscience, Assistance Publique-Hôpitaux de Paris (AP-HP), Paris, France.
| | - A Chevance
- Centre of Research in Epidemiology and Statistics Sorbonne Paris Cité, Institute for Health and Medical Research, and French National Institute of Research for Agriculture, University of Paris, Paris, France
| | - L Joly
- Sorbonne University, Department of Psychiatry, Saint Antoine Hospital, DMU Neuroscience, Assistance Publique-Hôpitaux de Paris (AP-HP), Paris, France
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Psychiatric Illnesses as Disorders of Network Dynamics. BIOLOGICAL PSYCHIATRY: COGNITIVE NEUROSCIENCE AND NEUROIMAGING 2021; 6:865-876. [DOI: 10.1016/j.bpsc.2020.01.001] [Citation(s) in RCA: 14] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2019] [Accepted: 01/06/2020] [Indexed: 01/05/2023]
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8
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Anýž J, Bakštein E, Dally A, Kolenič M, Hlinka J, Hartmannová T, Urbanová K, Correll CU, Novák D, Španiel F. Validity of the Aktibipo Self-rating Questionnaire for the Digital Self-assessment of Mood and Relapse Detection in Patients With Bipolar Disorder: Instrument Validation Study. JMIR Ment Health 2021; 8:e26348. [PMID: 34383689 PMCID: PMC8386400 DOI: 10.2196/26348] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 12/09/2020] [Revised: 04/23/2021] [Accepted: 05/10/2021] [Indexed: 11/13/2022] Open
Abstract
BACKGROUND Self-reported mood is a valuable clinical data source regarding disease state and course in patients with mood disorders. However, validated, quick, and scalable digital self-report measures that can also detect relapse are still not available for clinical care. OBJECTIVE In this study, we aim to validate the newly developed ASERT (Aktibipo Self-rating) questionnaire-a 10-item, mobile app-based, self-report mood questionnaire consisting of 4 depression, 4 mania, and 2 nonspecific symptom items, each with 5 possible answers. The validation data set is a subset of the ongoing observational longitudinal AKTIBIPO400 study for the long-term monitoring of mood and activity (via actigraphy) in patients with bipolar disorder (BD). Patients with confirmed BD are included and monitored with weekly ASERT questionnaires and monthly clinical scales (Montgomery-Åsberg Depression Rating Scale [MADRS] and Young Mania Rating Scale [YMRS]). METHODS The content validity of the ASERT questionnaire was assessed using principal component analysis, and the Cronbach α was used to assess the internal consistency of each factor. The convergent validity of the depressive or manic items of the ASERT questionnaire with the MADRS and YMRS, respectively, was assessed using a linear mixed-effects model and linear correlation analyses. In addition, we investigated the capability of the ASERT questionnaire to distinguish relapse (YMRS≥15 and MADRS≥15) from a nonrelapse (interepisode) state (YMRS<15 and MADRS<15) using a logistic mixed-effects model. RESULTS A total of 99 patients with BD were included in this study (follow-up: mean 754 days, SD 266) and completed an average of 78.1% (SD 18.3%) of the requested ASERT assessments (completion time for the 10 ASERT questions: median 24.0 seconds) across all patients in this study. The ASERT depression items were highly associated with MADRS total scores (P<.001; bootstrap). Similarly, ASERT mania items were highly associated with YMRS total scores (P<.001; bootstrap). Furthermore, the logistic mixed-effects regression model for scale-based relapse detection showed high detection accuracy in a repeated holdout validation for both depression (accuracy=85%; sensitivity=69.9%; specificity=88.4%; area under the receiver operating characteristic curve=0.880) and mania (accuracy=87.5%; sensitivity=64.9%; specificity=89.9%; area under the receiver operating characteristic curve=0.844). CONCLUSIONS The ASERT questionnaire is a quick and acceptable mood monitoring tool that is administered via a smartphone app. The questionnaire has a good capability to detect the worsening of clinical symptoms in a long-term monitoring scenario.
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Affiliation(s)
- Jiří Anýž
- National Insitute of Mental Health, Klecany, Czech Republic
| | | | | | - Marián Kolenič
- National Insitute of Mental Health, Klecany, Czech Republic
| | | | - Tereza Hartmannová
- National Insitute of Mental Health, Klecany, Czech Republic.,Mindpax s.r.o, Prague, Czech Republic
| | - Kateřina Urbanová
- National Insitute of Mental Health, Klecany, Czech Republic.,Mindpax s.r.o, Prague, Czech Republic
| | - Christoph U Correll
- Department of Psychiatry, The Zucker Hillside Hospital, Glen Oaks, NY, United States.,Department of Psychiatry and Molecular Medicine, Donald and Barbara Zucker School of Medicine at Hofstra/Northwell, Hempstead, NY, United States.,Department of Child and Adolescent Psychiatry, Charité Universitätsmedizin Berlin, Berlin, Germany
| | - Daniel Novák
- Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic
| | - Filip Španiel
- National Insitute of Mental Health, Klecany, Czech Republic
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9
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Sulis W. The Continuum Between Temperament and Mental Illness as Dynamical Phases and Transitions. Front Psychiatry 2021; 11:614982. [PMID: 33536952 PMCID: PMC7848037 DOI: 10.3389/fpsyt.2020.614982] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 10/07/2020] [Accepted: 12/21/2020] [Indexed: 12/31/2022] Open
Abstract
The full range of biopsychosocial complexity is mind-boggling, spanning a vast range of spatiotemporal scales with complicated vertical, horizontal, and diagonal feedback interactions between contributing systems. It is unlikely that such complexity can be dealt with by a single model. One approach is to focus on a narrower range of phenomena which involve fewer systems but still cover the range of spatiotemporal scales. The suggestion is to focus on the relationship between temperament in healthy individuals and mental illness, which have been conjectured to lie along a continuum of neurobehavioral regulation involving neurochemical regulatory systems (e.g., monoamine and acetylcholine, opiate receptors, neuropeptides, oxytocin), and cortical regulatory systems (e.g., prefrontal, limbic). Temperament and mental illness are quintessentially dynamical phenomena, and need to be addressed in dynamical terms. A meteorological metaphor suggests similarities between temperament and chronic mental illness and climate, between individual behaviors and weather, and acute mental illness and frontal weather events. The transition from normative temperament to chronic mental illness is analogous to climate change. This leads to the conjecture that temperament and chronic mental illness describe distinct, high level, dynamical phases. This suggests approaching biopsychosocial complexity through the study of dynamical phases, their order and control parameters, and their phase transitions. Unlike transitions in physical systems, these biopsychosocial phase transitions involve information and semiotics. The application of complex adaptive dynamical systems theory has led to a host of markers including geometrical markers (periodicity, intermittency, recurrence, chaos) and analytical markers such as fluctuation spectroscopy, scaling, entropy, recurrence time. Clinically accessible biomarkers, in particular heart rate variability and activity markers have been suggested to distinguish these dynamical phases and to signal the presence of transitional states. A particular formal model of these dynamical phases will be presented based upon the process algebra, which has been used to model information flow in complex systems. In particular it describes the dual influences of energy and information on the dynamics of complex systems. The process algebra model is well-suited for dealing with the particular dynamical features of the continuum, which include transience, contextuality, and emergence. These dynamical phases will be described using the process algebra model and implications for clinical practice will be discussed.
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Affiliation(s)
- William Sulis
- Collective Intelligence Laboratory, Department of Psychiatry and Behavioural Neuroscience, McMaster University, Hamilton, ON, Canada
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10
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Sreenivas NK, Rao S. Analyzing the effects of memory biases and mood disorders on social performance. Sci Rep 2020; 10:20895. [PMID: 33262387 PMCID: PMC7708996 DOI: 10.1038/s41598-020-77715-6] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2020] [Accepted: 10/21/2020] [Indexed: 12/04/2022] Open
Abstract
Realistic models of decision-making and social interactions, considering the nature of memory and biases, continue to be an area of immense interest. Emotion and mood are a couple of key factors that play a major role in decisions, nature of social interactions, size of the social network, and the level of engagement. Most of the prior work in this direction focused on a single trait, behavior, or bias. However, this work builds an integrated model that considers multiple traits such as loneliness, the drive to interact, the memory, and mood biases in an agent. The agent system comprises of rational, manic, depressed, and bipolar agents. The system is modeled with an interconnected network, and the size of the personal network of each agent is based on its nature. We consider a game of iterated interactions where an agent cooperates based on its past experiences with the other agent. Through simulation, the effects of various biases and comparative performances of agent types is analyzed. Taking the performance of rational agents as the baseline, manic agents do much better, and depressed agents do much worse. The payoffs also exhibit an almost-linear relationship with the extent of mania. It is also observed that agents with stronger memory perform better than those with weaker memory. For rational agents, there is no significant difference between agents with a positive bias and those with a negative bias. Positive bias is linked with higher payoffs in manic and bipolar agents. In depressed agents, negative bias is linked with higher payoffs. In manic agents, an intermediate value of mood dependence offers the highest payoff. But the opposite is seen in depressed agents. In bipolar agents, agents with weak mood dependence perform the best.
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Affiliation(s)
| | - Shrisha Rao
- International Institute of Information Technology - Bangalore, Bangalore, India.
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11
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Ossola P, Garrett N, Sharot T, Marchesi C. Belief updating in bipolar disorder predicts time of recurrence. eLife 2020; 9:e58891. [PMID: 33168133 PMCID: PMC7655098 DOI: 10.7554/elife.58891] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2020] [Accepted: 10/23/2020] [Indexed: 11/13/2022] Open
Abstract
Bipolar disorder is a chronic relapsing condition in which mood episodes are interspersed with periods of wellbeing (euthymia). Shorter periods of euthymia are associated with poorer functioning, so it is crucial to identify predictors of relapse to facilitate treatment. Here, we test the hypothesis that specific valence-dependent learning patterns emerge prior to the clinical manifestation of a relapse, predicting its timing. The ability to update beliefs in response to positive and negative information was quantified in bipolar patients during euthymia, who were then monitored for 5 years. We found that reduced tendency to update beliefs in response to positive relative to negative information predicted earlier relapse. Less updating in response to positive information may generate pessimistic beliefs, which in turn can lead to more severe prodromal symptoms (e.g. sleep disturbance, irritability etc.). The results suggest that measuring valence-dependent belief updating could facilitate risk prediction in bipolar disorder.
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Affiliation(s)
- Paolo Ossola
- Psychiatry Unit, Department of Medicine and Surgery, Università di ParmaParmaItaly
| | - Neil Garrett
- Department of Experimental Psychology, University of OxfordOxfordUnited Kingdom
| | - Tali Sharot
- Affective Brain Lab, Department of Experimental Psychology, University College LondonLondonUnited Kingdom
| | - Carlo Marchesi
- Psychiatry Unit, Department of Medicine and Surgery, Università di ParmaParmaItaly
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12
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Victory A, Letkiewicz A, Cochran AL. Digital solutions for shaping mood and behavior among individuals with mood disorders. CURRENT OPINION IN SYSTEMS BIOLOGY 2020; 21:25-31. [PMID: 32905495 PMCID: PMC7473040 DOI: 10.1016/j.coisb.2020.07.008] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
Abstract
Mood disorders present on-going challenges to the medical field, with difficulties ranging from establishing effective treatments to understanding complexities of one's mood. One solution is the use of mobile apps and wearables for measuring physiological symptoms and real-time mood in order to shape mood and behavior. Current digital research is focused on increasing engagement in monitoring mood, uncovering mood dynamics, predicting mood, and providing digital microinterventions. This review discusses the importance and risks of user engagement, as well as barriers to improving it. Research on mood dynamics highlights the possibility to reveal data-driven computational phenotypes that could guide treatment. Mobile apps are being used to track voice patterns, GPS, and phone usage for predicting mood and treatment response. Future directions include utilizing mobile apps to deliver and evaluate microinterventions. To continue these advances, standardized reporting and study designs should be considered to improve digital solutions for mood disorders.
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Affiliation(s)
- Amanda Victory
- Department of Psychiatry, University of Michigan, Ann Arbor, MI, US
| | | | - Amy L Cochran
- Department of Population Health Sciences, Department of Math, University of Wisconsin, Madison, WI, US
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