Dr Danilo Bzdok
email: danilo.bzdok@rwth-aachen.de

B T Thomas Yeo

The Future of Data Analysis in the Neurosciences

Keywords: systems biology, epistemology, hypothesis testing, high-dimensional statistics, machine learning, sample complexity

Neuroscience is undergoing faster changes than ever before. Over 100 years our field qualitatively described and invasively manipulated single or few organisms to gain anatomical, physiological, and pharmacological insights. In the last 10 years neuroscience spawned quantitative big-sample datasets on microanatomy, synaptic connections, optogenetic brain-behavior assays, and high-level cognition. While growing data availability and information granularity have been amply discussed, we direct attention to a routinely neglected question: How will the unprecedented data richness shape data analysis practices? Statistical reasoning is becoming more central to distill neurobiological knowledge from healthy and pathological brain recordings. We believe that large-scale data analysis will use more models that are non-parametric, generative, mixing frequentist and Bayesian aspects, and grounded in different statistical inferences.

Introduction

During most of neuroscience history, new insights were "inferred" with little or no reliance on statistics. Qualitative, often anecdotal reports have documented impairments after brain lesion [START_REF] Harlow | Passage of an iron rod through the head[END_REF], microscopical inspection of stained tissue [START_REF] Brodmann | Vergleichende Lokalisationslehre der Großhirnrinde[END_REF], electrical stimulation during neurosurgery [START_REF] Penfield | The Brain? S record of auditory and visual experience[END_REF], targeted pharmacological intervention [START_REF] Clark | Principles of psychopharmacology: a textbook for physicians, medical students, and behavioral scientists[END_REF], and brain connections using neuron-transportable dyes [START_REF] Mesulam | Tetramethyl benzidine for horseradish peroxidase neurohistochemistry: a non-carcinogenic blue reaction product with superior sensitivity for visualizing neural afferents and efferents[END_REF]. Connectivity analysis by axonal tracing studies in monkeys exemplifies biologically justified "inference" with many discoveries since the 60s [START_REF] Köbbert | Current concepts in neuroanatomical tracing[END_REF]. A colored tracer substance is injected in vivo into source region A, uptaken by local neuronal receptors, and automatically transported in axons to target region B. This observation in a single monkey allows extrapolating a monosynaptical connection between region A and B to the entire monkey species [START_REF] Mesulam | The evolving landscape of human cortical connectivity: facts and inferences[END_REF]. Instead, later brain-imaging technology propelled the dataintensive characterization of the mammalian brain and today readily quantifies axonal connections, myeloarchitectonic distributions, cytoarchitectonic borders, neurotransmitter receptors, and oscillatory coupling [START_REF] Frackowiak | The future of human cerebral cartography: a novel approach[END_REF][START_REF] Kandel | Neuroscience thinks big (and collaboratively)[END_REF][START_REF] Amunts | BigBrain: an ultrahigh-resolution 3D human brain model[END_REF][START_REF] Van Essen | The Human Connectome Project: a data acquisition perspective[END_REF]. Following many new opportunities to generate digitized brain data, uncertainties about neurobiological phenomena henceforth required assessment in the statistical arena.

In the quantitative sciences, the invention and application of statistical tools has always been dictated by changing contexts and domain questions [START_REF] Efron | Computer-Age Statistical Inference[END_REF]. As this appears to be less appreciated in current neuroscience, the present paper reviews how "optimal" statistical choices are likely to change due to the progressively increasing granularity of digitized brain data. Massive data collection is a game changer in neuroscience [START_REF] Kandel | Neuroscience thinks big (and collaboratively)[END_REF][START_REF] Poldrack | Making big data open: data sharing in neuroimaging[END_REF], and many other public and scientific areas [START_REF] House Of Common | The big data dilemma[END_REF][START_REF] Jordan | Frontiers in Massive Data Analysis[END_REF][START_REF] Manyika | Big data: The next frontier for innovation, competition, and productivity[END_REF]. There is an always-bigger interest in and pressure for data sharing, open access, and building "big-data" repositories [START_REF] Frackowiak | The future of human cerebral cartography: a novel approach[END_REF][START_REF] Randlett | Whole-brain activity mapping onto a zebrafish brain atlas[END_REF][START_REF] Lichtman | The big data challenges of connectomics[END_REF]. For instance, UK Biobank is a longitudinal population study dedicated to the genetic and environmental influence on mental and other disorders. 500,000 enrolled volunteers undergo an impressive battery of clinical diagnostics from brain scans to bone density with a >25 year follow-up.

Targeted reanalysis of such national and international data collections will soon become the new normal in neuroscience and medicine. Statistical scalability to high-dimensional data will therefore be inspected from the independent perspectives of i) parametric versus nonparametric, ii) discriminative versus generative, and iii) frequentist versus Bayesian statistical models as well as iv) classical hypothesis testing and out-of-sample generalization.

Towards adaptive models?

Today, parametric models are still the obvious choice in neuroscience. For instance, many big-sample studies (i.e., data from hundreds of animals or humans) currently apply the same parametric models as previous small-sample studies (i.e., a few dozen animals or humans).

However, concentrating on sample size, parametric analyses such as Student's t-test, ANOVA, and Pearson's linear correlation on brain data from many hundred individuals may not yield an improved quality of statistical insight on a neurobiological phenomenon that could not already be achieved with a dozen participants (Box 1). An important caveat relies in their systematic inability to grow in complexity no matter how much data is collected and analyzed [START_REF] Ghahramani | Probabilistic machine learning and artificial intelligence[END_REF]. In any classification, a linear parametric classifier will always make predictions based on a linear decision boundary between classes, whereas a non-parametric classifier can learn a non-linear boundary whose shape grows more complex with more data.

Analogously, parametric independent component analysis (ICA) and principal component analysis (PCA) require presetting the component number, yet the form and number of clusters in brain data is not known to neuroscientists. Such finite mixture models may give way to infinite mixture models that reframe the cluster number as a function of data availability and hence gradually yield more clusters without bound. Similarly, classical hidden Markov models may get upgraded to infinite hidden Markov models and we might see more applications of non-parametric decision trees and nearest neighbors in future neuroscientific studies. Finally, it is currently debated whether increasingly used "deep" neural network algorithms with many non-linear hidden layers are more accurately viewed as parametric or non-parametric.

A crucial statistical property for data-rich neuroscience will be the strength of nonparametric models to automatically increase the number of model parameters. Many interesting phenomena in the brain are likely to be very complex. Fortunately, probability theory recently proposed stochastic processes that instantiate random variables ranging over unlimited function spaces. Gaussian Processes (GP) are an important non-parametric example for specifying distributions on unknown functions to place minimal structural assumptions or constraints on possible interactions [START_REF] Rasmussen | Gaussian processes for machine learning[END_REF]. Although GPs instantiate infinite dimensional objects, indexing a subset of random variables in this infinite collection is ensured to yield multivariate Gaussian distributions. Instead of fitting one parameter to each variable to predict a behavior or clinical outcome with linear regression, GPs learn a nonparametric distribution of non-linear functions to explain brain-behavior associations. This is likely to leverage predictive regression and classification performance in large-sample studies in neuroscience. For instance, natural scaling to the high-dimensional scenario was demonstrated by a GP regression model that explained 70% of known missing heritable variability in yeast phenotypes [START_REF] Sharp | Explaining Missing Heritability Using Gaussian Process Regression[END_REF]. Such emergent insight from complex non-additive interactions between gene loci is difficult to infer by genome-wide association studies (GWAS) that use parametric models to explain small fractions of total heritable variation.

Unfortunately, the computational cost of parametric alternatives scales exponentially with interaction order and the necessary correction for the multiple tested hypotheses becomes challenging.

As another non-parametric family, kernel-based models can provide statistical advantages by mapping brain variables to a latent variable space [START_REF] Hofmann | Kernel methods in machine learning[END_REF]. Kernels promise effective modality fusion to incorporate heterogeneous sources given that, mathematically, kernel addition equates with concatenation of variable spaces. Such genuine multi-modal analysis can perform conjoint inference on behavioral indices, brain connectivity and function phenotypes. Non-parametric prediction in classification and regression is thus performed based on dot products on the vectors of the inner product space without requiring an explicit mapping from the actual brain variables (i.e., "kernel trick"). In this virtual variable space the added dimensions enable linear separability of complex neurobiological effects that are not linearly separable in the original variables. Kernelization of statistical estimators inherits enriched transformation of the input data that only grow linearly in dimension with increasing sample size. Such purposeful increase of input dimensionality and model complexity is useful for small to intermediate, but probably not extremely large datasets because the kernel matrix can grow to terabytes scales. Disadvantages of kernels include the inability to assess contributions of individual variables and to distinguish informative and noise variables. Finally, kernel-based statistics relate to psychology in that learning pairwise similarity measures between environmental stimuli is an aspect of intelligent behavior.

Radial basis kernels were argued to offer psychological and neurobiological plausibility as computational model for human decision-making [START_REF] Jäkel | Does cognitive science need kernels?[END_REF].

In sum, neurobiology is high-dimensional in nature and thus difficult to understand for human intuition. By expressing brain phenomena in some finite-dimensional space, parametric models are more interpretable, easier to implement, and faster to estimate. They are often the best choice in data scarcity, yet they are further away from neurobiological ground truth. Exclusive reliance on parametric analysis of brain recordings may keep neuroscientists from extracting structured knowledge on emergence properties that can only be grasped in data richness [START_REF] Jordan | Frontiers in Massive Data Analysis[END_REF][START_REF] Ghahramani | Probabilistic machine learning and artificial intelligence[END_REF][START_REF] Halevy | The Unreasonable Effectiveness of Data[END_REF]. Even if more complex models do not always result in greater insight [START_REF] Eliasmith | A large-scale model of the functioning brain[END_REF], non-parametric approaches are naturally prepared to capture complex relationships. This is because the complexity of statistical structure and hence extracted neurobiological insight grow boundlessly with the amount of fitted data [START_REF] Ghahramani | Probabilistic machine learning and artificial intelligence[END_REF][START_REF] Jordan | Machine learning: Trends, perspectives, and prospects[END_REF]. This is likely to be important for modern neuroscience and personalized medicine.

Towards models that incorporate biological structure?

The fact that more interpretable models typically require more data samples explains why, in neuroscience, the less data-hungry discriminative models have been ubiquitious in smallsample studies, while the popularity of more interpretable generative models is steadily increasing with recent data availability (Box 2). For instance, hidden Markov models have recently been applied to high-dimensional temporal data of magnetencephalographic recordings [START_REF] Baker | Fast transient networks in spontaneous human brain activity[END_REF]. These generative models simultaneously inferred the spatial topography of the major brain networks subserving environment responses and their cross-talk dynamics without anatomical assumptions. The model-immanent estimation of the joint distributions between the voxel inputs has identified 100-200ms windows of coherent spatiotemporally states and their exact functional coupling occurring faster than previously thought (i.e., both are hidden variables). As another example, neuroscientists often conceptualize behavioral tasks as recruiting multiple neural processes supported by multiple brain regions. This century-old notion [START_REF] Walton | Contribution to the study of the cortical sensory areas[END_REF] was lacking a formal mathematical model. The conceptual premise was recently encoded with a generative model [START_REF] Yeo | Functional Specialization and Flexibility in Human Association Cortex[END_REF]. Applying the model to 10,449 experiments across 83 behavioral tasks revealed heterogeneity in the degree of functional specialization within association cortices to execute diverse tasks by flexible brain regions integration across specialized networks [START_REF] Yeo | Functional Specialization and Flexibility in Human Association Cortex[END_REF][START_REF] Bertolero | The modular and integrative functional architecture of the human brain[END_REF]. As a clinical example, dynamic causal modeling for model-induced variable spaces and ensuing group classification by learning algorithms were combined to infer neurobiologically interpretable manifolds underlying aphasia [START_REF] Brodersen | Generative embedding for model-based classification of fMRI data[END_REF]. Their model identified the directed interregional connectivity strengths that distinguish aphasic and neurotypical individuals. This generative modeling approach was more explanatory than brute-force prediction based on activity extent or linear correlations.

The influence exerted from right planum temporale and right Heschl's gyrus on their left counterparts turned out as candidate mechanisms underlying communication impairment and formally discarded other plausible auditory processing schemes. Such generative approaches may refine diagnosis and treatment of spectrum disorders in neurology and psychiatry.

Moreover, generative approaches to fitting biological data have successfully reverseengineered i) human facial variation related to gender and ethnicity based on genetic information alone [START_REF] Claes | Modeling 3D facial shape from DNA[END_REF], ii) the complexity gradient in the ventral-visual processing hierarchy which confirmed encoding of low-/mid-/high-level facets in V1, V2, and V4 mostly known from animals [START_REF] Güçlü | Deep neural networks reveal a gradient in the complexity of neural representations across the ventral stream[END_REF], iii) structural content from naturalistic static images by revealing an explicit inverse mapping between occipital activity and retinotopic stimulation [START_REF] Thirion | Inverse retinotopy: inferring the visual content of images from brain activation patterns[END_REF], iv) video scenes from naturalistic movies enabled by revealing importance of separately modeling slow and fast visual movement [START_REF] Nishimoto | Reconstructing visual experiences from brain activity evoked by natural movies[END_REF], and v) brain atlases of semantic specificity from naturalistic speech reinforcing the intuition that various different semantic features are widely distributed in the cortex [START_REF] Huth | Natural speech reveals the semantic maps that tile human cerebral cortex[END_REF]. Finally, it is becoming increasingly clear that neuralnetwork models constitute a class of computational architectures [START_REF] Schmidhuber | Deep learning in neural networks: An overview[END_REF] that lend themselves particularly well to representation and manifold learning tasks [START_REF] Lecun | Deep learning[END_REF]. In particular, autoencoders are modern neural-network models that have been formally shown to generalize commonly employed representation discovery techniques, such as ICA and PCA as well as clustering models [START_REF] Baldi | Neural networks and principal component analysis: Learning from examples without local minima[END_REF][START_REF] Le | ICA with reconstruction cost for efficient overcomplete feature learning[END_REF]. Autoencoder architectures therefore have the generative potential to extract local (brain-area-like) and global (brain-network-like) representation to abandon hand-crafted "feature engineering" made difficult by missing ground truth [START_REF] Bzdok | Semi-Supervised Factored Logistic Regression for High-Dimensional Neuroimaging Data[END_REF].

Neurobiologically valid representations are revealed as sets of predictive patterns that together explain psychological tasks and disease processes without being constrained to functional specialization into distinct regions or functional integration by brain networks.

The predictive validity of the discovered functional compartments thus extends the interpretational spectrum offered by explained-variance approaches like ICA and PCA.

In sum, neuroscientists can afford more generative models as brain data become abundant.

In contrast to discriminative models, their naturally higher interpretability could open the black box of the neural processing architectures underlying behavior and its disturbances.

Generative models enable more detailed understanding by exposing the low-dimensional manifolds embedded within high-dimensional brain data. However, "the more detailed and biologically realistic a model, the greater the challenges of parameter estimation and the danger of overfitting" [START_REF] Stephan | Translational perspectives for computational neuroimaging[END_REF]. A crucial next step in neuroscience might lie in extracting actual pathophysiological mechanisms from brain measurements in mental disorders. In contrast, applying discriminative approaches on patients grouped by the diagnostic manuals DSM or ICD could recapitulate categories that are neither neurobiologically valid nor clinically predictive [START_REF] Hyman | Can neuroscience be integrated into the DSM-V?[END_REF]. Finally, discriminative models may be less potent to characterize the neural mechanisms of information processing up to the ultimate goal of recovering subjective mental experience from brain recordings [START_REF] Brodersen | Generative embedding for model-based classification of fMRI data[END_REF][START_REF] Lake | Human-level concept learning through probabilistic program induction[END_REF][START_REF] Bzdok | Formal models of the network co-occurrence underlying mental operations[END_REF][START_REF] Yanga | Altered global brain signal in schizophrenia[END_REF].

Towards integration of traditional modeling regimes?

After the 19th century was driven by Bayesian and the 20th century by frequentist statistics [START_REF] Efron | Modern science and the Bayesian-frequentist controversy[END_REF], one might wonder about their relative contributions in the 21st century. These two attitudes towards quantitative investigation are the most common distinction in statistics in general [START_REF] Freedman | Some issues in the foundation of statistics[END_REF] and in neuroscience in particular [START_REF] Friston | Classical and Bayesian inference in neuroimaging: applications[END_REF][START_REF] Stephan | Bayesian model selection for group studies[END_REF], yet orthogonal to the parametric/non-parametric and discriminative/generative perspectives on statistical models [START_REF] Freedman | Some issues in the foundation of statistics[END_REF]. Overall, the many desirable properties of Bayesian modeling and interpretation are contrasted by the computationally feasibility of frequentist models in high-dimensional data (Box 3). In neuroscience the speed-accuracy tradeoff in Bayesian posterior inference has been advantageously rebalanced using variational Bayes for limited computational budgets on a number of previous occasions. This includes Bayesian time-series analysis [START_REF] Penny | Variational Bayesian inference for fMRI time series[END_REF], model selection for group analysis [START_REF] Stephan | Bayesian model selection for group studies[END_REF] and mixed-effects classification for imbalanced groups [START_REF] Brodersen | Variational Bayesian mixed-effects inference for classification studies[END_REF].

Bayesian inference is an appealing framework by its intimate relationship to properties of firing in neuronal populations [START_REF] Ma | Bayesian inference with probabilistic population codes[END_REF] and the learning human mind [START_REF] Lake | Human-level concept learning through probabilistic program induction[END_REF]. Additionally, it is important for currently increasing efforts in transdiagnostic clinical neuroscience [START_REF] Goodkind | Identification of a Common Neurobiological Substrate for Mental Illness[END_REF] that Bayesian hierarchical models could elegantly handle class imbalances with very unequal group sizes. Moreover, inferring hierarchies of statistical relationships, a natural way of inducing parsimony, is difficult or impossible in a frequentist model selection regime and much more feasible in a Bayesian model averaging regime.

In particular, recent advances in non-parametric Bayesian methods [START_REF] Orbanz | Bayesian nonparametric models[END_REF] combined with extensive datasets promise progress in longstanding problems in cognitive and clinical neuroscience. As a key problem in cognition, neuroscientists have not agreed on a description system of mental operations (called 'taxonomy' or 'ontology') that would canonically motivate and operationalize their experiments [START_REF] Barrett | The Future of Psychology: Connecting Mind to Brain[END_REF][START_REF] Tenenbaum | How to grow a mind: Statistics, structure, and abstraction[END_REF]. As a key problem in medicine, partly shared neurobiological endophenotypes are today believed to contribute to the pathophysiology of various neurological and psychiatric diagnoses (called 'nosology') despite drastically different clinical exophenotypes [START_REF] Brodersen | Generative embedding for model-based classification of fMRI data[END_REF][START_REF] Hyman | Can neuroscience be integrated into the DSM-V?[END_REF][START_REF] Stephan | Charting the landscape of priority problems in psychiatry, part 2: pathogenesis and aetiology[END_REF]. As an interesting observation, both these neuroscientific challenges can be statistically recast as latent factor problems [cf. 59]. The same class of statistical models could both identify the unnamed building blocks underlying human cognition and the unknown neurobiological mechanisms underlying diverse mental disorders. For instance, hierarchical Bayesian models were recently borrowed from the topic modeling domain to estimate a latent cognitive ontology [START_REF] Bertolero | The modular and integrative functional architecture of the human brain[END_REF] and neurobiological subtypes in Alzheimer's disease [START_REF] Zhang | Bayesian model reveals latent atrophy factors with dissociable cognitive trajectories in Alzheimer's disease[END_REF]. Further, formal posterior inference in non-parametric Bayesian models could gracefully handle complexity in the brain by estimating the number of latent factors in cognition and disease using Chinese Restaurant Processes [START_REF] Kemp | Learning systems of concepts with an infinite relational model[END_REF], relative implications of latent causes in neurobiological observations using Indian Buffet Process [START_REF] Ghahramani | Infinite latent feature models and the Indian buffet process[END_REF], as well as induce hierarchies of cognitive primitives and disease endophenotypes using Hierarchical Dirichlet Processes [START_REF] Teh | Sharing Clusters Among Related Groups: Hierarchical Dirichlet Processes[END_REF]. Cluster detection in the nonparametric Bayesian regime exchanges the neurobiologically implausible winner-takes-all property of traditional clustering by allowing each observation to participate in all clusters.

Crucially, the non-parametric aspect of these Dirichlet Process models allows the number of inferred clusters to grow organically with increasing sample size.

In sum, the scalability of model estimation in the data-rich scenario is calibrated between frequentist numerical optimization and Bayesian numerical integration. Ingredients from both statistical regimes can be advantageously combined by adjusting the modeling goal [65, chapter 5, 66]. Many frequentist methods reliant on gradient-based optimization can be recast as Bayesian integration problems, for instance using Hamiltonian MCMC methods.

Conversely, integration problems can be turned into optimization problems, for instance using variational Bayes approximations. High-dimensional data have been authoritatively argued to motivate novel blends between simplier-to-use frequentist and more integrative Bayesian modeling aspects [START_REF] Efron | Modern science and the Bayesian-frequentist controversy[END_REF]. Indeed, the continuously increasing number of observations and number of parameters to estimate have earlier been predicted to complicate logical consistency in statistical theory [START_REF] Efron | Controversies in the foundations of statistics[END_REF]. We therefore believe that the recent emergence of extensive datasets in neuroscience could analogously prompt emergence of more frequentist-Bayesian hybrid approaches.

Towards diversification of statistical inference?

Drawing statistical inference on regional brain responses during controlled experiments has largely hinged on classical null-hypothesis rejection, but is increasingly flanked by out-ofsample prediction based on cross-validation [START_REF] Bzdok | Classical Statistics and Statistical Learning in Imaging Neuroscience[END_REF][START_REF] Yarkoni | Choosing prediction over explanation in psychology: Lessons from machine learning[END_REF]. Both inferential regimes provide formal justifications for deriving neurobiological knowledge using mathematical models (Box 4).

Classical inference measures the statistical significance associated with a relationship between typically few variables given a prespecified model. Generalization inference assesses the robustness of patterns between typically many variables in the data [START_REF] Hastie | The Elements of Statistical Learning[END_REF]. One might think that these differences in deriving formal conclusions on brain recordings are mostly of technical relevance but there is an often-overlooked misconception that models with high explanatory power always exhibit high predictive power [START_REF] Lo | Why significant variables aren't automatically good predictors[END_REF][START_REF] Wu | Genome-wide association analysis by lasso penalized logistic regression[END_REF]. A neurobiological effect assessed to be statistically significant by a p-value may sometimes not yield successful predictability based on cross-validation, and vice versa. Their theoretical differences are also practically manifested in the high-dimensional setting where classical inference needs to address the multiple comparisons problem and generalization inference involves tackling the curse of dimensionality [START_REF] Domingos | A Few Useful Things to Know about Machine Learning[END_REF][START_REF] Friston | Bayesian decoding of brain images[END_REF].

Consequently, care needs to be taken when combing both inferential regimes in data analysis [START_REF] Bzdok | Classical Statistics and Statistical Learning in Imaging Neuroscience[END_REF][START_REF] Yarkoni | Choosing prediction over explanation in psychology: Lessons from machine learning[END_REF]. Say, a neuroscientist wants to predict Alzheimer diagnosis from the >100,000 brain location per brain scan by sparse L1-penalized logistic regression (i.e., classification with minimal necessary variables) using cross-validation but performs preliminary dimensionality reduction to the most important 10,000 voxels by ANOVAmediated variable ranking using classical inference. In this case, it is not permitted to conduct an ordinary significance test (i.e., classical inference) on the obtained sparse model coefficients (obtained from generalization inference) because it would involve recasting an originally high-dimensional variable-selection as a univariate setting [START_REF] Haufe | On the interpretation of weight vectors of linear models in multivariate neuroimaging[END_REF]. The ANOVA test would ignore the fact that the sparse logistic regression had already reduced the variables to the most important ones [START_REF] Wu | Genome-wide association analysis by lasso penalized logistic regression[END_REF]. Very recently proposed methods for so-called post-selection inference allow replacing naive by selection-adjusted p-values for a set of variables previously chosen to be meaningful predictors [START_REF] Taylor | Statistical learning and selective inference[END_REF]. This and similar clashes between inferential regimes will probably increase in neuroscientific data analysis.

However, classical inference and generalization inference have also been advantageously joined towards a same neuroscientific goal. In a first example, cross-validated classification algorithms estimated the relative contribution of all macroscopical brain networks in a battery of psychological tasks, while ANOVA was used to find smaller subsets of most important networks for each task [START_REF] Bzdok | Formal models of the network co-occurrence underlying mental operations[END_REF]. In a second example, Latent Dirichlet Allocation [START_REF] Teh | Hierarchical Bayesian nonparametric models with applications[END_REF] was used to find a nested hierarchy of volume atrophy endophenotypes in Alzheimer's disease, while these were cross-validated by distinct trajectories in memory and executive function decline using classical hypothesis testing [START_REF] Zhang | Bayesian model reveals latent atrophy factors with dissociable cognitive trajectories in Alzheimer's disease[END_REF]. More generally, a neurobiological question, such as "Are region A and B significantly connected?", can also be cross-validated by an independent method that can make the same observation, such as significant structural and functional connectivities [START_REF] Eickhoff | Connectivity-based parcellation: Critique and implications[END_REF]. The previous two examples performed crossvalidation in a different kind of data, in contrast to the typical practice of performing crossvalidation in unseen data of the same kind. This is important because fMRI, EEG, MEG, fNIRS, and other neuroscientific methods measure biological phenomena only indirectly. Out-ofsample generalization to a different modality (e.g., behavior, genetics, microbiomics) increases confidence that the findings reflect neurobiological reality. Combining different inferential regimes can therefore corroborate neurobiological findings.

In sum, the leap from quantitative brain measurements to neurobiological knowledge is secured by statistical inference. There is not one but several different types of inference that can ask a same question with different mathematical foundations requiring differently nuanced neuroscientific interpretation. Historically, classical inference was invented for problems with small samples that can be addressed by plausible, handpicked models with a small number of parameters [cf. 79]. Some authors therefore emphasize that "one should never use sum of squared errors, p-values, R 2 statistics, or other classical measures of model fit on the training data as evidence of a good model fit in the high-dimensional setting." [80, p. 247, their emphasis]. P-values and other classical guarantees may lose their ability to evaluate model fit in high-dimensional data. Instead, out-of-sample generalization by successful cross-validation to independent data will be increasingly used given natural tuning to problems with more variables and larger samples [cf. 24]. Envisioning a future of precision medicine, only cross-validated predictive models can obtain answers from a single data point [START_REF] Stephan | Translational perspectives for computational neuroimaging[END_REF]. Finally, data richness will increasingly require preliminary dimensionality-reduction and feature-engineering procedures, including k-means clustering and ICA decomposition, that do not themselves perform statistical inference. A back and forth between dimensionalityreducing data transformations, inductive pattern extrapolation and deductive hypothesis testing of the discovered candidate effects will become valid routines in neuroscience.

Concluding Remarks and Future Perspectives

After astronomy, particle physics, and genetics, massive data is the new reality in neuroscience and medicine. Rich datasets can extend the spectrum of possible findings and permissible conclusions about the brain. Yet, the unavoidable impact on data analysis practices is currently shunned. While sample size and information granularity are increasing progressively, the prompted shift in statistical choices may be categorical. Neuroscientists need to extend their modeling instincts towards neurobiological insight with gradually increasing quality of insight as data accumulate [START_REF] Efron | Computer-Age Statistical Inference[END_REF] and towards prediction on the singleindividual level [START_REF] Roberts | The predictive capacity of personal genome sequencing[END_REF]. It is difficult to overstate the importance of closing training gaps in nextgeneration PhD curricula that will include machine-learning, computer programming, distributed multi-core processing, and advanced visualization [START_REF] Akil | Neuroscience Training for the 21st Century[END_REF]. In a nutshell, neuroscience is entering the era of large-scale data collection, curation, and collaboration with a pressing need for statistical models tailored for high-dimensional inference. These will frequently lie beyond the scope of the statistical repertoire cherished today. Analyzing extensive multi-modal datasets with inappropriate statistical models would be a waste of public financial resources and our limited scientific efforts.

TEXT BOX 1: Parametric and non-parametric models

This statistical distinction takes place at the equilibrium between imposing specific assumptions (i.e., parametric) and not assuming a certain functional form (i.e., nonparametric) to encourage discovery of relevant structure driven by the brain data themselves [START_REF] Ghahramani | Probabilistic machine learning and artificial intelligence[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. to common misunderstanding, both classes of quantitative modeling involve parameters. "Non-parametric" is defined in three different flavors: The first, perhaps most widespread meaning implies those statistical estimators that do not make explicit assumptions about the probability distribution (e.g., Gaussian) from which the data have arisen. As a second, more intuitive explanation, non-parametric models do not assume that the structure of the statistical model is fixed. The last, most formal definition emphasizes that the number of model parameters increases explicitly or implicitly with the number of available data points (e.g., number of brain recordings). In contrast, the number of model parameters are fixed in parametric models and do not vary with sample size (Fig. 1). In its most extreme manifestation, effectively fitting an arbitrary functional form can result in non-parametric models larger in memory usage than the actual input data themselves. Flexible non-parametric models include decision trees like random forests, nearest-neighbor estimators, kernel support vector machines or kernel principal component analysis, and hierarchical clustering, as well as bootstrapping and other resampling procedures. More rigid parametric models include methods from classical statistics based on maximum likelihood estimation, Gaussian mixture models, support vector machines, principal component analysis, and k-means clustering, but also modern penalized regression variants like Lasso, ElasticNet, and RidgeRegression.

It is an advantage of parametric modeling to expresses the data compactly in a few model parameters, which increases interpretability, requires smaller datasets, has higher statistical power, and incurs lower computational load. Additionally, if the parametric assumptions hold "true" in nature, non-parametric approaches can be less powerful than their parametric counterparts. Although parametric and non-parametric models require assumptions, nonparametric models have the advantage to be more robust about estimating properties of still poorly known phenomena in the brain if the modeling assumptions are partly false. Due to their robustness, non-parametric methods may prevent improper use and misunderstanding. Most importantly, even if the number of parameters in parametric models can be manually increased, only non-parametric models have an inherent ability to automatically scale with the complexity of the available data resources. Non-parametric nearest-neighbor regression can grow from a rough step-function decision boundary (k=1) to an always-smoother fit (k=9) by incorporating the data in more complex ways. Non-parametric models therefore outperform parametric alternatives in many datarich scenarios. Reused with permission from [START_REF] James | An introduction to statistical learning[END_REF].

TEXT BOX 2: Discriminative and generative models

Discriminative models are common choices when best-possible prediction of neurobiological phenomena is the primary aim (e.g., based on behavioral phenotype, age, performance or clinical scores), which does not require recovery of the explicit relationships between input variables (Fig. 2). Generative models also aim at successful extrapolation of statistical relationships to new data but simultaneously extract special neurobiological structure from brain data by indirectly inferring the target variable from hidden variables [START_REF] Ng | On discriminative vs. generative classifiers: A comparison of logistic regression and naive bayes[END_REF]. Typical discriminative models include linear regression, support vector machines, decision-tree algorithms, and logistic regression, while generative models include hidden Markov models, modern neural network algorithms, dictionary learning methods, and many non-parametric statistical models [START_REF] Teh | Hierarchical Bayesian nonparametric models with applications[END_REF]. As an important advantage in brain science, generative modeling approaches are more expressive in extracting representations in input data that are "economical to describe but allow the input to be reconstructed accurately" [START_REF] Hinton | The "Wake-Sleep" algorithm for unsupervised neural networks[END_REF].

More formally, discriminative models focus on a target variable y, whereas generative models are concerned with the joint distributions between observed and unobserved variables [START_REF] Bishop | Generative or discriminative? getting the best of both worlds[END_REF]. Discriminative models find a direct mapping function by predicting a categorical or continuous target variable y by estimating P(y|x) without capturing special properties in the data x. Generative models estimate relevant properties in the data by P(y|x) from P(x|y) and P(y). Generative models can hence produce synthetic, never observed examples x~ for targets y by sampling from the estimated joint distributions P(x,y). The strength of generative models to conjointly realize predictive modeling and representation learning is paid by requiring more input data, more computational resources, and possibly more model parameters to fit. Consequently, with too few data points available for model estimation, discriminative models will usually achieve better prediction performance in new data because generative models suffer from higher model variance. Finally, generative models make an assumption about the distribution of the input data and their performance therefore suffers more if this model is inaccurate. Discriminative models are generally more robust to assumptions about the data-generating process. In practice, statistical models span a continuum between the extreme poles of frequentism and Bayesianism. They exhibit many unexpected similarities [START_REF] Murphy | Machine learning: a probabilistic perspective[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. For instance, the bootstrap is a frequentist method for population-level inference of confidence intervals and non-parametric null-hypothesis testing [START_REF] Efron | Bootstrap methods: another look at the jackknife[END_REF] that also lends itself to Bayesian interpretations [START_REF] Hastie | The Elements of Statistical Learning[END_REF]. Frequentist practitioners are mostly concerned about model overfitting, while Bayesian practitioners are more concerned about the pertinence and number of domain-informed priors.

Important for data-intensive brain science, the frequentist-Bayesian tradeoff impacts the computational budget required for model estimation (Fig. 3). Generally, the more one adheres to frequentist instead of Bayesian ideology, the less computationally expensive and the less technically involved are the statistical analyses. It is a widespread opinion that Bayesian models do not scale well to the data-rich setting, although there is little work on the behavior of Bayesian methods in high dimensions [START_REF] Jordan | A message from the President: The era of big data[END_REF][START_REF] Yang | On the computational complexity of highdimensional Bayesian variable selection[END_REF]. While the pure frequentist approach computes maximum likelihood estimation, the pure Bayesian approach achieves inference by full posterior probability distributions computed by asymptotically exact MCMC.

Given their computational cost, MCMCs have mainly been used for small-scale problems.

The practical applicability of Bayesian methods has been greatly enhanced through the development of approximate inference algorithms such as variational Bayes and expectation propagation [START_REF] Jordan | An introduction to variational methods for graphical models[END_REF]. Nevertheless, exact MCMC and its accelerated approximate variants tend to suffer from i) uncertainty when convergence is reached, ii) difficult-to-control "randomwalk" behavior, and iii) limited scaling to the high-dimensional setting [START_REF] Mackay | Information theory, inference and learning algorithms[END_REF]. Consequently, the intractability of Bayesian posterior integrals motivated a rich spectrum of model hybrids [START_REF] Efron | Modern science and the Bayesian-frequentist controversy[END_REF]. There is an increasing trend towards incorporating appealing Bayesian aspects into computationally cheap frequentist approaches [e.g., 92].

Figure 3 (in Box 3): Different shades of Bayesian inference

There is no unique Bayesian formulation to perform statistical inference, but there is a variety of them. For instance, type-II maximum likelihood or empirical Bayes has genuine frequentist properties, does not specify a prior distribution before visiting the data, and is often used in non-Bayesian modeling. Generally, the more integrals to be solved or approximated, the higher the computational budget needed for model estimation. Reused with permission from [START_REF] Murphy | Machine learning: a probabilistic perspective[END_REF].

BOX 4: Null-hypothesis testing and out-of-sample generalization

Statistical inference can be generally defined as the extraction of new knowledge from parameters in mathematical models fitted to data [START_REF] Jordan | Frontiers in Massive Data Analysis[END_REF]. In classical inference, invented almost 100 years ago [START_REF] Fisher | Studies in crop variation. II. The manurial response of different potato varieties[END_REF], the neuroscientist articulates two mutually exclusive hypotheses by domain-informed judgment with the agenda to disprove the null hypothesis embraced by the research community. A p-value is then computed that denotes the conditional probability of obtaining an equal or more extreme test statistic provided that the null hypothesis is correct at the conventional significance threshold alpha=0.05 [START_REF] Wasserstein | The ASA's statement on p-values: context, process, and purpose[END_REF]. For instance, statistical significance of the Student's t-test indicates a difference between two means with a 5% chance after sampling twice from the same population. Adopting deductive reasoning and a mostly retrospective viewpoint [START_REF] Goodman | Toward evidence-based medical statistics. 1: The P value fallacy[END_REF], state-of-the-art hypotheses are continuously replaced by always more pertinent hypotheses using verification and falsification in a Darwinian process. The classical framework of null-hypothesis falsification to infer new knowledge is still the go-to choice in many branches of neuroscience.

Considering the data-rich scenario, it is an important problem that p-values intrinsically become better (i.e., lower) as the sample size increases [START_REF] Berkson | Some difficulties of interpretation encountered in the application of the chisquare test[END_REF]. In some fields, it is therefore now mandatory to report effect sizes in addition or instead of p-values [START_REF] Wasserstein | The ASA's statement on p-values: context, process, and purpose[END_REF].

In contrast, generalization inference emerged much more recently in basic statistics [START_REF] Breiman | Submodel selection and evaluation in regression. The X-random case[END_REF] and tests whether complex patterns extrapolate to independent data using cross-validation.

Formally, this inferential regime is mathematically justified by the so-called Vapnik-Chervonenkis (VC) dimensions from statistical learning theory, conceptually related to the degrees of freedom from classical statistics [START_REF] Abu-Mostafa | Learning from data[END_REF]. In generalization inference on highdimensional patterns, the VC dimensions formally express the circumstances under which a class of functions is able to learn from a finite amount of data to successfully predict a given neurobiological phenomenon in unseen data [START_REF] Vapnik | Statistical Learning Theory[END_REF]. Thus, the VC dimensions provide a probabilistic measure of whether a certain model is able to learn a distinction given a

Figure 1 (

 1 Figure 1 (in Box 1): Prediction based on parametric versus non-parametric regression

Figure 2 (

 2 Figure 2 (in Box 2): Class-conditional densities can be more complex than class posteriors

  dataset. This inductive reasoning to learn a general principle from examples adopting a mostly prospective viewpoint contrasts the deductive, retrospective logic of null-hypothesis testing. In practice, cross-validation is frequently used to quantify the out-of-sample performance by an unbiased estimate of a model's capacity to generalize to data samples acquired in the future [100]. Model assessment is done by training on a bigger subset of the available data (i.e., training set) and subsequently applying the trained model to the smaller remaining part of data (i.e., test set).


Pierre-Henri Ducrot
email: pierre-henri.ducrot@ims-bordeaux.fr

Isabelle Dufour

Cédric Ayela

Fabrice Mathieu

Liviu Nicu

Nearby electronic scheme for actuation and detection of resonance frequency and quality factor of organic piezoelectric MEMS resonators

Keywords: MEMS, resonator, electrical sensing technique, electrical compensation I

This work deals with an acquiring system dedicated to the measurement of the resonance frequency and the quality factor of organic piezoelectric resonators. These resonators consist of a free-standing cantilever and its nonreleased counterpart used to compensate electrical signals that are not due to mechanical behavior. The system uses a capacitive half-bridge with two AC voltages in opposite phases, associated with an IQ demodulator. The magnitudes of the two AC voltages can be set independently from one another, which enables a powerful compensation of parasitic capacitance. Thanks to this dedicated system, measurements of the resonance frequency of organic piezoelectric resonators both in air and liquid media have been performed. Several methods to determine accurately the resonance frequency have been tested and compared in terms of measurement noise.

INTRODUCTION

Piezoelectricity is a phenomenon that is widely used in Micro-Electro-Mechanical Systems (MEMS) because it enables, at the same time, the actuation and the detection of a motional structure with only one structural layer [START_REF] Ramadan | A review of piezoelectric polymers as functional materials for electromechanical transducers[END_REF]. In order to apply an electric field in a piezoelectric material, the latter must lie between two electrodes, which leads to a capacitance in parallel with the RLC circuit that represents the mechanical behavior of the resonator. In certain cases, this capacitance is not an issue for the resonance measurement and a simple charge or voltage amplifier circuit is efficient enough [START_REF] Doll | Design of piezoresitive versus piezoelectric contact mode scanning probes[END_REF]. But, this capacitance may be a difficulty for the electrical detection of the resonance with spectrum measurements. Other methods exist such as either a Phase-Locked Loop (PLL) system or an oscillator configuration that tracks the resonance frequency [START_REF] Ferrari | Thick-film resonant piezo-layers as new gravimetricensors[END_REF][START_REF] Manzaneque | Piezoelectric MEMS resonator-based oscillator for density and viscosity sensing[END_REF]. The drawback of such electronic configuration is that only the resonance frequency is determined and not the global resonance spectrum.

A common way to avoid the influence of the geometrical capacitance is based on a differential measurement between two identical structures: one is free to move while the other one is fixed. The present work introduces a compact electronic card that enables a differential measurement using a capacitive halfbridge principle and an IQ modulator. Moreover, it is possible to monitor the resonance frequency and the quality factor in real time thanks to a custom software.

II. ELECTRONIC CARD

A. Principle

The electronic card can generate two AC voltages in opposite phases. The magnitude of each signal can be set independently, which is an important characteristic. The measurement principle can be seen in Fig. 1. The two AC voltages supply both the free and the fixed structures. The middle point of the capacitive half-bridge is sent to a charge amplifier. The charge amplifier output signal is then sent to an IQ modulator system that gives the modulus and the phase of the resulting signal via a dedicated software connected to the card.

B. Results

The resonators used in this work are composed of a PEN (Polyethylene naphthalate) substrate, in addition to bottom and top electrodes made of aluminum on both sides of a PVDF-TrFE (polyvinylidene fluoride-trifluoroethylene) layer, which is a piezoelectric polymer [START_REF] Ducrot | Optimization of PVDF-TrFE processing conditions for the fabrication of organic MEMS resonators[END_REF]. In order to visualize the benefits offered by the dedicated electronic scheme, three measurements have been done in air: one without any compensation, one with compensation by the same AC voltage magnitudes for the free and the fixed structures and one with compensation with tuned voltage magnitudes (Fig. 2). It is clear that without any compensation it is really hard to observe the resonance with either the modulus or the phase spectra. The results become clearer with compensation by the same voltage magnitudes but it still does not look like a classical resonance spectrum, which means that the two structures are not exactly electrically identical. Nevertheless, with the tuned voltage magnitudes for each structure, it becomes possible to have a good compensation, resulting in comprehensive and usable modulus and phase signals for both quality factor and resonance frequency measurements. Using this electronic scheme and these organic piezoelectric resonators, measurements in liquids are also possible (Fig. 3). This is of particular interest for biological applications for example. The quality factor for the resonator presented in Fig. 3 is around 22 in air and 10 in water.

III. DETERMINATION OF THE RESONANCE FREQUENCY

Once the acquisition of the resonance spectrum is done, the resonance frequency must be quantified. Several posttreatment methods have been compared in terms of measurement noise (standard deviation of the resonance frequency estimation). The first method simply gives the frequency that corresponds to the modulus maximum value. The second method consists in fitting the modulus peak with a parabola and determining the maximum value. As the parabola rapidly differs from the modulus, only data points close to the resonance must be taken into account for the fit [START_REF] Boudjiet | New characterization methods for monitoring small resonant frequency variation, experimental tests in the case of hydrogen detection with uncoated silicon microcantilever-based sensors[END_REF]. The third method consists in fitting linearly the phase signal and determining the frequency that corresponds to a -90° phase [START_REF] Boudjiet | New characterization methods for monitoring small resonant frequency variation, experimental tests in the case of hydrogen detection with uncoated silicon microcantilever-based sensors[END_REF]. For the same reasons as previously, this fit is available only near the resonance. The last method is a treatment that determines the coupling that must be taken out in order to have a second order band-pass system [START_REF] Niedermayer | Methods for the robust measurement of the resonance frequency and quality factor of significantly damped resonating devices[END_REF]. Series of about fifty measurements have been recorded in air in order to calculate the mean value and the standard deviation of the resonance frequency. The results are summarized in Tab. 1. According to the results, the parabolic fit of the modulus and the linear fit of the phase give the lowest noise (260ppm which is quite good considering the low quality factor and the simplicity of the method). The developed system composed of organic piezoelectric resonators and a dedicated electronic setup enables the accurate measurement and monitoring of both the resonance frequency and the quality factor, which is of particular interest for future mass sensing assays in air and liquid media.

IV. CONCLUSION

A compact and efficient electronic card is presented in this work. This card combines two AC voltage sources in opposite phases and an IQ modulator. This system enables the accurate compensation of the geometrical capacitance of organic piezoelectric resonators, which makes resonance frequency and quality factor measurements in liquids possible through the acquisition of the resonance spectrum. In order to determine the resonance frequency, post-treatment methods have been compared to find the best one in terms of noise level. The results show that a simple fit of the modulus by a parabola is efficient.

Fig. 1 .

 1 Fig. 1. Representation of the measurement principle.

Fig. 2 .

 2 Fig. 2. Measurement of the resonance (a) without any compensation, (b) with compensation with the same AC voltage magnitudes for the free and the fixed structures and (c) with compensation with tuned voltage magnitudes.

Fig. 3 .

 3 Fig. 3. Measurement of the resonance of a structure (a) in air and (b) in water.

TABLE I

 I

	.	COMPARATION OF THE METHODS TO DETERMINE THE
		RESONANCE FREQUENCY
			Resonance	Standard
	Method	Frequency	Deviation
			[kHz]	[Hz]
	Parabolic fit of the modulus	114.7	30
	Linear fit of the phase	114.3	34
	Band-pass system	114.4	59
	Maximum	114.6	110

ACKNOWLEDGMENT

The authors acknowledge financial support by the CNRS and the University of Bordeaux (project MEMOCELL).


SIMPLE FABRICATION PROCESS OF ORGANIC PIEZOELECTRIC RESONATORS: APPLICATION TO VISCOSITY MEASUREMENTS

FABRICATION

A rapid and extremely low-cost fabrication process has been developed for the elaboration of piezoelectric organic resonators [START_REF] Ducrot | Optimization of PVDF-TrFE processing conditions for the fabrication of organic MEMS resonators[END_REF]. Piezoelectricity has been chosen as it enables the integration of both actuation and read-out at the same time thanks to the reversibility of the piezoelectric effect. Thus, polyvinylidene fluoride-trifluoroethylene (PVDF-TrFE), which is a piezoelectric polymer, is used as it presents the best piezoelectric properties [START_REF] Ramadan | A review of piezoelectric polymers as functional materials for electromechanical transducers[END_REF]. Fig. 1 presents the different steps for the elaboration of the resonators. Polyethylene naphtalate (PEN) serves as substrate with a thickness of 25 µm. Then, the bottom electrode of aluminum (300 nm) is evaporated through a shadow mask. After that, the PVDF-TrFE layer is spin-coated (4 µm) and annealed at 140 °C, which increases the PVDF-TrFE crystallinity and so its piezoelectric efficiency. The top electrode is evaporated in the same condition as the bottom one. Finally, the shape of a cantilever is defined by xurography [START_REF] Lemaire | Fastfabrication process for low environmental impact microsystems[END_REF] and the resulting device is glued on a glass slide, leaving the cantilever part free to move. The typical dimensions of the cantilever are 2 mm in length and 1 mm in width. Actually, two cantilevers are fabricated on the same substrate: one is free to move while the other one is unreleased. This architecture is used in the electric scheme for the resonance spectrum measurements.

APPLICATION

In this work, the organic resonators are used as viscosity sensors in liquid media. In that case, a polydimethylsiloxane (PDMS) layer (10 µm) has been added by spin-coating before the cutting step to passivate the electrodes. In order to measure the resonance spectrum, a dedicated electronic scheme is developed and presented Fig. 2. The two cantilevers (the free one and the unreleased one) constitute a capacitive half-bridge. The signal from the common electrode is sent to a charge amplifier and then goes to an IQ demodulator that gives the modulus and phase spectra. This strategy enables a good compensation of the parasitic capacitance due to the dielectric layer (PVDF-TrFE) lying between two conductors (the electrodes). Thanks to this electronics, measurements

(a) (b) (c) (e) (d) (f)
in liquid media become possible.

Fig. 2: Electronic scheme principle with a picture of a resonator.

In order to determine the viscosity without knowledge of the mass density, the resonance frequency and the quality factor must be measured and six parameters (m 1 , m 2 , m 3 , c 1 , c 2 and c 3 ) must be calculated [START_REF] Heinisch | Reduced order models for resonant viscosity and mass density sensors[END_REF]:

    2 1 2 3 1 2 2 r r f m m m f       ( 1 )   1 2 3 1 2 2 r r Q f c c c f       ( 2 )
with f r the resonance frequency, Q the quality factor,  the viscosity and  the mass density of the liquid. It means that a calibration with three liquids whose mass densities and viscosities are known is necessary to determine these coefficients. Then, the viscosity can be analytically determined for liquids with unknown mass densities. So, measurements of the resonance frequency and the quality factor have been made in water-glycerol mixtures for different concentrations of glycerol. In order to validate the concept, the first measurements are made with a vibrometer for the first out-of-plane mode and the results are presented in Fig. 3. The viscosities determined with the resonators are compared with the ones measured with a viscometer. The results with the resonators are consistent with the viscometer and prove the efficiency of organic resonators for measurements in liquid media. Afterwards, measurements have been made with the electronic scheme but for a resonance mode that is a combination of flexion and torsion, as it is the one that presents the best electrical response. However, due to the electrical coupling that is not entirely compensated, the measurements of the quality factor are misrepresented and it is impossible to determine the viscosity without knowing the mass density. But, knowing the mass density and using (1), the viscosity has been calculated and the results are also consistent (Fig. 3).

CONCLUSION

Here is presented a simple and low-cost fabrication process for the elaboration of organic piezoelectric resonators. Their performances have been demonstrated through measurements of viscosities in liquid media. However, improvement of the electrical measurements must be achieved in order to totally compensate the coupling and thus have a significant quantification of the quality factor, which is necessary to determine the viscosity independently from the knowledge of the mass density. Nevertheless, this work is promising for further biological and/or chemical applications.

P1

  .H. Ducrot 1 , I. Dufour 1 , F. Mathieu 2 , L. Nicu 2 , M. Heinisch 3 , B. Jakoby 3 , C. Ayela 1 Laboratoire de l'Intégration du Matériau au Système UMR 5218, CNRS, Université de Bordeaux, France 2 Laboratoire d'Analyse et d'Architecture des Systèmes UPR 8001, CNRS, Université de Toulouse, France 3 Institute for Microelectronics and Microsensors, Johannes Kepler University, Linz, Austria pierre-henri.ducrot@ims-bordeaux.fr Silicon-based Micro Electro Mechanical Systems (MEMS) success is incontestable as they are widely present in new technologies. However, technologies based on organic electronics are emerging because they are interesting in terms of fabrication costs, with a large panel of functional properties available, and are very attractive for flexible devices [1]. In this context, organic piezoelectric resonators based on a simple low-cost fabrication process are presented in this work for viscosity measurements. The results show the potential of organic resonators in liquid media for further biological and chemical sensing.

Fig. 1 :

 1 Fig. 1: Fabrication steps of the organic resonators. (a) PEN substrate, (b) evaporation of the bottom electrodes of aluminum, (c) spin-coating of the PVDF-TrFE layer, (d) evaporation of the top electrodes of aluminum, (e) cutting of the shape, (f) the device with the free cantilever and its unreleased counterpart.

Fig. 3 :

 3 Fig. 3: Comparison of the viscosity measurements with a viscometer and with a piezoelectric MEMS resonator with the vibrometer for the first flexural mode and with the electronic scheme for another mode.

Proceedings of the 13 th International Workshop on Nanomechanical Sensing NMC 2016 Delft, The Netherlands

ACKNOWLEDGMENTS

The authors acknowledge financial support by the CNRS and the University of Bordeaux (project MEMOCELL, joint call IdeX Bordeaux-CNRS PEPS 2015).