Phenotypic clustering of bipolar disorder supports stratification by lithium responsiveness over diagnostic subtypes.

INTRODUCTION: The aim of this study was to determine whether the clinical profiles of bipolar disorder (BD) patients could be differentiated more clearly using the existing classification by diagnostic subtype or by lithium treatment responsiveness. METHODS: We included adult patients with BD-I or II (N = 477 across four sites) who were treated with lithium as their principal mood stabilizer for at least 1 year. Treatment responsiveness was defined using the dichotomized Alda score. We performed hierarchical clustering on phenotypes defined by 40 features, covering demographics, clinical course, family history, suicide behaviour, and comorbid conditions. We then measured the amount of information that inferred clusters carried about (A) BD subtype and (B) lithium responsiveness using adjusted mutual information (AMI) scores. Detailed phenotypic profiles across clusters were then evaluated with univariate comparisons. RESULTS: Two clusters were identified (n = 56 and n = 421), which captured significantly more information about lithium responsiveness (AMI range: 0.033 to 0.133) than BD subtype (AMI: 0.004 to 0.011). The smaller cluster had disproportionately more lithium responders (n = 47 [83.8%]) when compared to the larger cluster (103 [24.4%]; p = 0.006). CONCLUSIONS: Phenotypes derived from detailed clinical data may carry more information about lithium responsiveness than the current classification of diagnostic subtype. These findings support lithium responsiveness as a valid approach to stratification in clinical samples.

A computational model to characterize the time-course of response to rapid antidepressant therapies.

Our objective is to propose a method capable of disentangling the magnitude, the speed, and the duration or decay rate of the time course of response to rapid antidepressant therapies. To this end, we introduce a computational model of the time course of response to a single treatment with a rapid antidepressant. Numerical simulation is used to evaluate whether model parameters can be accurately estimated from observed data. Finally, we compare our computational modelling-based approach with linear mixed effects modelling in terms of their ability to detect changes in the magnitude and time-course of response to rapid antidepressant therapies in simulated randomized trials. Simulation experiments show that the parameters of our computational model can be accurately recovered using nonlinear least squares. Parameter estimation accuracy is stable over noise levels reaching as high as 25% of the true antidepressant effect magnitude. Comparison of our approach to mixed effects modelling using simulated randomized controlled trial data demonstrates an inability of linear mixed models to disentangle effect magnitude and time course, while our computational model accurately separates these response components. Our modelling approach may accurately identify the (A) magnitude, (B) speed, and (C) durability or decay rate of response to rapid antidepressant therapies. Future studies should fit this model to data from real clinical trials, and use resulting parameter estimates to uncover predictors and causes of different elements of the temporal course of antidepressant response.

A pilot study examining the impact of lithium treatment and responsiveness on mnemonic discrimination in bipolar disorder.

INTRODUCTION: Mnemonic discrimination (MD), the ability to discriminate new stimuli from similar memories, putatively involves dentate gyrus pattern separation. Since lithium may normalize dentate gyrus functioning in lithium-responsive bipolar disorder (BD), we hypothesized that lithium treatment would be associated with better MD in lithium-responsive BD patients. METHODS: BD patients (N = 69; NResponders = 16 [23 %]) performed the Continuous Visual Memory Test (CVMT), which requires discriminating between novel and previously seen images. Before testing, all patients had prophylactic lithium responsiveness assessed over ≥1 year of therapy (with the Alda Score), although only thirty-eight patients were actively prescribed lithium at time of testing (55 %; 12/16 responders, 26/53 nonresponders). We then used computational modelling to extract patient-specific MD indices. Linear models were used to test how (A) lithium treatment, (B) lithium responsiveness via the continuous Alda score, and (C) their interaction, affected MD. RESULTS: Superior MD performance was associated with lithium treatment exclusively in lithium-responsive patients (Lithium x AldaScore ß = 0.257 [SE 0.078], p = 0.002). Consistent with prior literature, increased age was associated with worse MD (ß = -0.03 [SE 0.01], p = 0.005). LIMITATIONS: Secondary pilot analysis of retrospectively collected data in a cross-sectional design limits generalizability. CONCLUSION: Our study is the first to examine MD performance in BD. Lithium is associated with better MD performance only in lithium responders, potentially due to lithium's effects on dentate gyrus granule cell excitability. Our results may influence the development of behavioural probes for dentate gyrus neuronal hyperexcitability in BD.

Granule cells perform frequency-dependent pattern separation in a computational model of the dentate gyrus.

Mnemonic discrimination (MD) may be dependent on oscillatory perforant path input frequencies to the hippocampus in a "U"-shaped fashion, where some studies show that slow and fast input frequencies support MD, while other studies show that intermediate frequencies disrupt MD. We hypothesize that pattern separation (PS) underlies frequency-dependent MD performance. We aim to study, in a computational model of the hippocampal dentate gyrus (DG), the network and cellular mechanisms governing this putative "U"-shaped PS relationship. We implemented a biophysical model of the DG that produces the hypothesized "U"-shaped input frequency-PS relationship, and its associated oscillatory electrophysiological signatures. We subsequently evaluated the network's PS ability using an adapted spatiotemporal task. We undertook systematic lesion studies to identify the network-level mechanisms driving the "U"-shaped input frequency-PS relationship. A minimal circuit of a single granule cell (GC) stimulated with oscillatory inputs was also used to study potential cellular-level mechanisms. Lesioning synapses onto GCs did not impact the "U"-shaped input frequency-PS relationship. Furthermore, GC inhibition limits PS performance for fast frequency inputs, while enhancing PS for slow frequency inputs. GC interspike interval was found to be input frequency dependent in a "U"-shaped fashion, paralleling frequency-dependent PS observed at the network level. Additionally, GCs showed an attenuated firing response for fast frequency inputs. We conclude that independent of network-level inhibition, GCs may intrinsically be capable of producing a "U"-shaped input frequency-PS relationship. GCs may preferentially decorrelate slow and fast inputs via spike timing reorganization and high frequency filtering.

Mechanisms of Sustained Increases in γ Power Post-Ketamine in a Computational Model of the Hippocampal CA3: Implications for Ketamine's Antidepressant Mechanism of Action.

Subanaesthetic doses of ketamine increase γ oscillation power in neural activity measured using electroencephalography (EEG), and this effect lasts several hours after ketamine administration. The mechanisms underlying this effect are unknown. Using a computational model of the hippocampal cornu ammonis 3 (CA3) network, which is known to reproduce ketamine's acute effects on γ power, we simulated the plasticity of glutamatergic synapses in pyramidal cells to test which of the following hypotheses would best explain this sustained γ power: the direct inhibition hypothesis, which proposes that increased γ power post-ketamine administration may be caused by the potentiation of recurrent collateral synapses, and the disinhibition hypothesis, which proposes that potentiation affects synapses from both recurrent and external inputs. Our results suggest that the strengthening of external connections to pyramidal cells is able to account for the sustained γ power increase observed post-ketamine by increasing the overall activity of and synchrony between pyramidal cells. The strengthening of recurrent pyramidal weights, however, would cause an additional phase shifted voltage increase that ultimately reduces γ power due to partial cancellation. Our results therefore favor the disinhibition hypothesis for explaining sustained γ oscillations after ketamine administration.

Differential characteristics of bipolar I and II disorders: a retrospective, cross-sectional evaluation of clinical features, illness course, and response to treatment.

BACKGROUND: The distinction between bipolar I and bipolar II disorder and its treatment implications have been a matter of ongoing debate. The aim of this study was to examine differences between patients with bipolar I and II disorders with particular emphasis on the early phases of the disorders. METHODS: 808 subjects diagnosed with bipolar I (N = 587) or bipolar II disorder (N = 221) according to DSM-IV criteria were recruited between April 1994 and March 2022 from tertiary-level mood disorder clinics. Sociodemographic and clinical variables concerning psychiatric and medical comorbidities, family history, illness course, suicidal behavior, and response to treatment were compared between the bipolar disorder types. RESULTS: Bipolar II disorder patients were more frequently women, older, married or widowed. Bipolar II disorder was associated with later "bipolar" presentation, higher age at first (hypo)mania and treatment, less frequent referral after a single episode, and more episodes before lithium treatment. A higher proportion of first-degree relatives of bipolar II patients were affected by major depression and anxiety disorders. The course of bipolar II disorder was typically characterized by depressive onset, early depressive episodes, multiple depressive recurrences, and depressive predominant polarity; less often by (hypo)mania or (hypo)mania-depression cycles at onset or during the early course. The lifetime clinical course was more frequently rated as chronic fluctuating than episodic. More patients with bipolar II disorder had a history of rapid cycling and/or high number of episodes. Mood stabilizers and antipsychotics were prescribed less frequently during the early course of bipolar II disorder, while antidepressants were more common. We found no differences in global functioning, lifetime suicide attempts, family history of suicide, age at onset of mood disorders and depressive episodes, and lithium response. CONCLUSIONS: Differences between bipolar I and II disorders are not limited to the severity of (hypo)manic syndromes but include patterns of clinical course and family history. Caution in the use of potentially mood-destabilizing agents is warranted during the early course of bipolar II disorder.

Familial traits of bipolar disorder: A systematic review and meta-analysis.

BACKGROUND: Genetic studies of bipolar disorder (BD) have shown varied results, which is in part because of the heterogeneity of the disorder. Identifying clinical phenotypes of BD could reduce variability and benefit research. Since BD has a robust genetic component, studies can investigate clinical traits that cluster in families to identify phenotypes with a probable genetic basis. METHODS: We conducted a systematic review of the current literature on familial clinical traits of BD. Text screening and data extraction were performed independently by two reviewers, and random effects meta-analysis was used. RESULTS: Of 1117 unique records, 16 studies met inclusion criteria. These studies indicated 14 potentially familial traits of BD: age of onset (OR: 4.50; 95% CI: [3.25, 6.22]), bipolar type (OR: 2.05 [1.50, 2.79]), lithium response (OR: 3.71 [1.28, 10.82]), polarity at onset (OR: 1.17 [1.03, 1.34]), psychotic features (OR: 2.20 [1.51, 3.20]), mood-incongruent psychosis (OR: 2.52 [1.66, 3.83]), puerperal psychosis (OR: 6.54 [2.55, 16.77]), rapid cycling (OR: 4.95 [0.96, 25.40]), suicide attempt (OR: 1.04 [0.65, 1.67]), alcoholism (OR: 1.53 [1.09, 2.16]), obsessive-compulsive disorder (OR: 3.10 [1.31; 7.09]), panic disorder (OR: 2.69 [1.12; 6.48]), social anxiety disorder (OR: 1.00 [0.39, 2.55]), and specific phobia (OR: 1.94 [0.95; 3.96]). For most traits, tests of heterogeneity were significant and publication bias was likely. CONCLUSION: The results of our review and meta-analysis highlight the lack of studies investigating familial clinical traits of BD, despite the need to address heterogeneity. The large degree of variability between studies must be reduced for future research.

Secondary outcomes and qualitative findings of an open-label feasibility trial of lisdexamfetamine dimesylate for adults with bulimia nervosa.

BACKGROUND: There is emerging evidence that stimulants warrant further investigation as a treatment for bulimia nervosa (BN) including a recent open-label feasibility trial examining the use of lisdexamfetamine dimestylate (LDX) for BN. The current report presents the secondary outcomes and qualitative interview results from that feasibility trial. These outcomes explore several purported mechanisms that may explain how stimulants affect symptoms of BN: appetite, impulsivity, obsessive and compulsive symptoms, eating disorder psychopathology/impairment and reward-based decision-making. METHODS: Twenty-three participants with BN received LDX for eight weeks. Questionnaires assessing appetite, impulsivity, obsessive and compulsive symptoms, eating disorder psychopathology and impairment were administered at baseline and post-treatment. Participants also completed a two-step reinforcement learning task to assess their decision-making. Semi-structured interviews took place at baseline, week 5, and follow-up. RESULTS: Reductions in hunger, food-related impulsivity, obsessive and compulsive features, eating disorder psychopathology and impairment were found. However, reward learning, as far as it is assessed by the task, did not seem to contribute to the effect of LDX on BN symptoms. Qualitative analysis suggested four themes: (1) reprieve from the eating disorder, (2) improvement in function and quality of life, (3) renewed hope for recovery, and (4) ability to normalize eating. CONCLUSIONS: This report suggests several potential mechanisms by which LDX may reduce symptoms of binging and purging in those with BN. Importantly, due to the open-label design, we are unable to attribute findings to the medication. Instead, our results should be interpreted as hypothesis generating to inform future studies such as adequately powered randomized controlled trials. Trial registration NCT03397446.

Recent research suggests that stimulant medications could be a potential treatment for bulimia nervosa (BN). Participants in this study took lisdexamfetamine dimesylate (LDX) for 8 weeks while their eating disorder symptoms and medical status were carefully monitored. As part of this study, twenty-three participants with BN completed several interviews, questionnaires and computer tasks at the start and end of treatment which were delivered to help researchers learn more about the how LDX impacts people with BN. Scores on questionnaires measuring different aspects of the eating disorder improved over time. Participants' performance on the computer task which measures a type of decision making did not change during treatment. Interviews exploring participants' experience taking LDX found four common themes: reprieve from the eating disorder, improvement in function and quality of life, renewed hope for recovery, and ability to normalize eating. This report suggests several potential ways LDX may reduce symptoms of binging and purging in those with BN. Importantly, due to the size and type of study, we cannot conclude that changes observed were a direct result of the medication. Instead, our results should be used to form new questions that can be explored by larger studies with controlled designs.

Augmenting Mental Health Support for Patients Accessing Different Degrees of Formal Psychiatric Care through a Supportive Text Messaging Program: Protocol for a Randomized Controlled Trial.

Patients feel more vulnerable when accessing community mental health programs for the first time or after being discharged from psychiatric inpatient units. Long wait times for follow-up appointments, shortage of mental health professionals, lack of service integration, and scarcity of tailored support can weaken their connection to the health care system. As a result, patients can present low adherence, dissatisfaction with treatment, and recurrent hospitalizations. Finding solutions to avoid unnecessary high-cost services and providing tailored and cost-effective mental health interventions may reduce the health system burden and augment patient support. We propose implementing an add-on, supportive text messaging service (Text4Support), developed using cognitive-behavioural therapy (CBT) principles to augment mental health support for patients attending to or being discharged from psychiatric care in Nova Scotia, Canada. This randomized controlled trial aims to investigate the effectiveness of Text4Support in improving mental health outcomes and overall mental well-being compared with usual care. We also will examine the intervention's impact on health services utilization and patient satisfaction. The results from this study will provide evidence on stepped and technology-based mental health care, which will contribute to generating new knowledge about mental health innovations in various clinical contexts, which is not only helpful for the local context but to other jurisdictions in Canada and abroad that are seeking to improve their health care.

A critical evaluation of dynamical systems models of bipolar disorder.

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.

What we learn about bipolar disorder from large-scale neuroimaging: Findings and future directions from the ENIGMA Bipolar Disorder Working Group.

MRI-derived brain measures offer a link between genes, the environment and behavior and have been widely studied in bipolar disorder (BD). However, many neuroimaging studies of BD have been underpowered, leading to varied results and uncertainty regarding effects. The Enhancing Neuro Imaging Genetics through Meta-Analysis (ENIGMA) Bipolar Disorder Working Group was formed in 2012 to empower discoveries, generate consensus findings and inform future hypothesis-driven studies of BD. Through this effort, over 150 researchers from 20 countries and 55 institutions pool data and resources to produce the largest neuroimaging studies of BD ever conducted. The ENIGMA Bipolar Disorder Working Group applies standardized processing and analysis techniques to empower large-scale meta- and mega-analyses of multimodal brain MRI and improve the replicability of studies relating brain variation to clinical and genetic data. Initial BD Working Group studies reveal widespread patterns of lower cortical thickness, subcortical volume and disrupted white matter integrity associated with BD. Findings also include mapping brain alterations of common medications like lithium, symptom patterns and clinical risk profiles and have provided further insights into the pathophysiological mechanisms of BD. Here we discuss key findings from the BD working group, its ongoing projects and future directions for large-scale, collaborative studies of mental illness.

The potential role of stimulants in treating eating disorders.

BACKGROUND: Many individuals with eating disorders remain symptomatic after a course of psychotherapy and pharmacotherapy; therefore, the development of innovative treatments is essential. METHOD: To learn more about the current evidence for treating eating disorders with stimulants, we searched for original articles and reviews published up to April 29, 2021 in PubMed and MEDLINE using the following search terms: eating disorders, anorexia, bulimia, binge eating, stimulants, amphetamine, lisdexamfetamine, methylphenidate, and phentermine. RESULTS: We propose that stimulant medications represent a novel avenue for future research based on the following: (a) the relationship between eating disorders and attention deficit/hyperactivity disorder (ADHD); (b) a neurobiological rationale; and (c) the current (but limited) evidence for stimulants as treatments for some eating disorders. Despite the possible benefits of such medications, there are also risks to consider such as medication misuse, adverse cardiovascular events, and reduction of appetite and pathological weight loss. With those risks in mind, we propose several directions for future research including: (a) randomized controlled trials to study stimulant treatment in those with bulimia nervosa (with guidance on strategies to mitigate risk); (b) examining stimulant treatment in conjunction with psychotherapy; (c) investigating the impact of stimulants on "loss of control" eating in youth with ADHD; and (d) exploring relevant neurobiological mechanisms. We also propose specific directions for exploring mediators and moderators in future clinical trials. DISCUSSION: Although this line of investigation may be viewed as controversial by some in the field, we believe that the topic warrants careful consideration for future research.

A scoping review and comparison of approaches for measuring genetic heterogeneity in psychiatric disorders.

An improved understanding of genetic etiological heterogeneity in a psychiatric condition may help us (a) isolate a neurophysiological 'final common pathway' by identifying its upstream genetic origins and (b) facilitate characterization of the condition's phenotypic variation. This review aims to identify existing genetic heterogeneity measurements in the psychiatric literature and provides a conceptual review of their mechanisms, limitations, and assumptions. The Scopus database was searched for studies that quantified genetic heterogeneity or correlation of psychiatric phenotypes with human genetic data. Ninety studies were included. Eighty-seven reports quantified genetic correlation, five applied genomic structural equation modelling, three evaluated departure from the Hardy-Weinberg equilibrium at one or more loci, and two applied a novel approach known as MiXeR. We found no study that rigorously measured genetic etiological heterogeneity across a large number of markers. Developing such approaches may help better characterize the biological diversity of psychopathology.

The costs and benefits of intensive day treatment programs and outpatient treatments for eating disorders: An idea worth researching.

Outpatient care (e.g., individual, group, or self-help therapies) and day treatment programs (DTPs) are common and effective treatments for adults with eating disorders. Compared to outpatient care, DTPs have additional expenses and could have unintended iatrogenic effects (e.g., may create an overly protective environment that undermines self-efficacy). However, these potential downsides may be offset if DTPs are shown to have advantages over outpatient care. To explore this question, our team conducted a scoping review that aimed to synthesize the existing body of adult eating disorder literature (a) comparing outcomes for DTPs to outpatient care, and (b) examining the use of DTPs as a higher level of care in a stepped care model. Only four studies met the predefined search criteria. The limited results suggest that the treatments have similar effects and that outpatient care is more cost-effective. Furthermore, no studies explored the use of DTPs as a higher level of care in a stepped care model (despite international guidelines recommending this approach). Given the clear dearth of literature on this clinically relevant topic, we have provided specific avenues for further research.

A feasibility study evaluating lisdexamfetamine dimesylate for the treatment of adults with bulimia nervosa.

OBJECTIVE: This study examined the feasibility, safety, and potential efficacy of lisdexamfetamine (LDX) as a treatment for adults with bulimia nervosa (BN). METHOD: An open-label 8-week feasibility study was conducted in participants with BN. Enrollment rate, dropout rate, safety outcomes, and eating disorder symptom change were examined. RESULTS: Eighteen of 23 participants completed the study per protocol. There was no participant-initiated dropout due to adverse drug reactions and no severe and unexpected adverse drug reactions. An average increase in heart rate of 12.1 beats/min was observed. There was a mean weight reduction of 2.1 kg and one participant was withdrawn for clinically significant weight loss. In the intent-to-treat sample, there were reductions in objective binge episodes and compensatory behaviors from Baseline to Post/End-of-Treatment (mean difference = -29.83, 95% confidence interval: -43.38 to -16.27; and mean difference = -33.78, 95% confidence interval: -48.74 to -18.82, respectively). DISCUSSION: Results of this study indicate that a randomized controlled trial would be feasible with close monitoring of certain safety parameters (especially over a longer time period as long-term safety is unknown). However, the results should not be used as evidence for clinicians to prescribe LDX to individuals with BN before its efficacy and safety are properly tested. TRIAL REGISTRATION NUMBER: NCT03397446.

Prediction of lithium response using genomic data.

Predicting lithium response prior to treatment could both expedite therapy and avoid exposure to side effects. Since lithium responsiveness may be heritable, its predictability based on genomic data is of interest. We thus evaluate the degree to which lithium response can be predicted with a machine learning (ML) approach using genomic data. Using the largest existing genomic dataset in the lithium response literature (n = 2210 across 14 international sites; 29% responders), we evaluated the degree to which lithium response could be predicted based on 47,465 genotyped single nucleotide polymorphisms using a supervised ML approach. Under appropriate cross-validation procedures, lithium response could be predicted to above-chance levels in two constituent sites (Halifax, Cohen's kappa 0.15, 95% confidence interval, CI [0.07, 0.24]; and Würzburg, kappa 0.2 [0.1, 0.3]). Variants with shared importance in these models showed over-representation of postsynaptic membrane related genes. Lithium response was not predictable in the pooled dataset (kappa 0.02 [- 0.01, 0.04]), although non-trivial performance was achieved within a restricted dataset including only those patients followed prospectively (kappa 0.09 [0.04, 0.14]). Genomic classification of lithium response remains a promising but difficult task. Classification performance could potentially be improved by further harmonization of data collection procedures.

Exemplar scoring identifies genetically separable phenotypes of lithium responsive bipolar disorder.

Predicting lithium response (LiR) in bipolar disorder (BD) may inform treatment planning, but phenotypic heterogeneity complicates discovery of genomic markers. We hypothesized that patients with "exemplary phenotypes"-those whose clinical features are reliably associated with LiR and non-response (LiNR)-are more genetically separable than those with less exemplary phenotypes. Using clinical data collected from people with BD (n = 1266 across 7 centers; 34.7% responders), we computed a "clinical exemplar score," which measures the degree to which a subject's clinical phenotype is reliably predictive of LiR/LiNR. For patients whose genotypes were available (n = 321), we evaluated whether a subgroup of responders/non-responders with the top 25% of clinical exemplar scores (the "best clinical exemplars") were more accurately classified based on genetic data, compared to a subgroup with the lowest 25% of clinical exemplar scores (the "poor clinical exemplars"). On average, the best clinical exemplars of LiR had a later illness onset, completely episodic clinical course, absence of rapid cycling and psychosis, and few psychiatric comorbidities. The best clinical exemplars of LiR and LiNR were genetically separable with an area under the receiver operating characteristic curve of 0.88 (IQR [0.83, 0.98]), compared to 0.66 [0.61, 0.80] (p = 0.0032) among poor clinical exemplars. Variants in the Alzheimer's amyloid-secretase pathway, along with G-protein-coupled receptor, muscarinic acetylcholine, and histamine H1R signaling pathways were informative predictors. This study must be replicated on larger samples and extended to predict response to other mood stabilizers.

Representational Rényi Heterogeneity.

A discrete system's heterogeneity is measured by the Rényi heterogeneity family of indices (also known as Hill numbers or Hannah-Kay indices), whose units are the numbers equivalent. Unfortunately, numbers equivalent heterogeneity measures for non-categorical data require a priori (A) categorical partitioning and (B) pairwise distance measurement on the observable data space, thereby precluding application to problems with ill-defined categories or where semantically relevant features must be learned as abstractions from some data. We thus introduce representational Rényi heterogeneity (RRH), which transforms an observable domain onto a latent space upon which the Rényi heterogeneity is both tractable and semantically relevant. This method requires neither a priori binning nor definition of a distance function on the observable space. We show that RRH can generalize existing biodiversity and economic equality indices. Compared with existing indices on a beta-mixture distribution, we show that RRH responds more appropriately to changes in mixture component separation and weighting. Finally, we demonstrate the measurement of RRH in a set of natural images, with respect to abstract representations learned by a deep neural network. The RRH approach will further enable heterogeneity measurement in disciplines whose data do not easily conform to the assumptions of existing indices.