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When foraging in dynamic and uncertain environments, animals can benefit from basing their decisions on smart inferences about hidden properties of the world. Typical theoretical approaches to understand the strategies that animals use in such settings combine Bayesian inference and value iteration to derive optimal behavioral policies that maximize total reward given changing beliefs about the environment. However, specifying these beliefs requires infinite numerical precision; with limited resources, this problem can no longer be separated into optimizing inference and optimizing action selections. To understand the space of behavioral policies in this constrained setting, we enumerate and evaluate all possible behavioral programs that can be constructed from just a handful of states. We show that only a small fraction of the top-performing programs can be constructed by approximating Bayesian inference; the remaining programs are structurally or even functionally distinct from Bayesian. To assess structural and functional relationships among all programs, we developed novel tree embedding algorithms; these embeddings, which are capable of extracting different relational structures within the program space, reveal that nearly all good programs are closely connected through single algorithmic “mutations”. We demonstrate how one can use such relational structures to efficiently search for good solutions via an evolutionary algorithm. Moreover, these embeddings reveal that the diversity of non-Bayesian behaviors originates from a handful of key mutations that broaden the functional repertoire within the space of good programs. The fact that this diversity of behaviors does not significantly compromise performance suggests a novel approach for studying how these strategies generalize across tasks.

Animals exhibit remarkable behavioral flexibility, robustly performing demanding tasks —such as searching for food or avoiding predators— in a variety of different contextual and environmental conditions. However, the demands that detecting and adjusting to changes in the environment place on a sensory system often differ from the demands associated with performing a specific behavioral task, even when both objectives rely on the same sensory modality. This necessitates neural encoding strategies that can dynamically balance these conflicting needs. Here, we develop a theoretical framework that explains how this balance can be achieved, and we use this framework to study tradeoffs in speed, performance, and information transmission that arise as a consequence of efficient coding in dynamic environments. This work generalizes current theories of efficient neural coding to dynamic environments, and thereby provides a unifying perspective on adaptive neural dynamics across different sensory systems, environments, and tasks.

To flexibly navigate, many animals rely on internal spatial representations that persist when the animal is standing still in darkness, and update accurately by integrating the animal’s movements in the absence of localizing sensory cues. Theories of mammalian head direction cells have proposed that these dynamics can be realized in a special class of networks that maintain a localized bump of activity via structured recurrent connectivity, and that shift this bump of activity via angular velocity input. Although there are many different variants of these so-called ring attractor networks, they all rely on large numbers of neurons to generate representations that persist in the absence of input and accurately integrate angular velocity input. Surprisingly, in the fly, Drosophila melanogaster, a head direction representation is maintained by a much smaller number of neurons whose dynamics and connectivity resemble those of a ring attractor network. These findings challenge our understanding of ring attractors and their putative implementation in neural circuits. Here, we analyzed failures of angular velocity integration that emerge in small attractor networks with only a few computational units. Motivated by the peak performance of the fly head direction system in darkness, we mathematically derived conditions under which small networks, even with as few as 4 neurons, achieve the performance of much larger networks. The resulting description reveals that by appropriately tuning the network connectivity, the network can maintain persistent representations over the continuum of head directions, and it can accurately integrate angular velocity inputs. We then analytically determined how performance degrades as the connectivity deviates from this optimally-tuned setting, and we find a trade-off between network size and the tuning precision needed to achieve persistence and accurate integration. This work shows how even small networks can accurately track an animal’s movements to guide navigation, and it informs our understanding of the functional capabilities of discrete systems more broadly.

To interpret the sensory environment, the brain combines ambiguous sensory measurements with context-specific prior experience. But environmental contexts can change abruptly and unpredictably, resulting in uncertainty about the current context. Here we address two questions: how should context-specific prior knowledge optimally guide the interpretation of sensory stimuli in changing environments, and do human decision-making strategies resemble this optimum? We probe these questions with a task in which subjects report the orientation of ambiguous visual stimuli that were drawn from three dynamically switching distributions, representing different environmental contexts. We derive predictions for an ideal Bayesian observer that leverages the statistical structure of the task to maximize decision accuracy and show that its decisions are biased by task context. The magnitude of this decision bias is not a fixed property of the sensory measurement but depends on the observer’s belief about the current context. The model therefore predicts that decision bias will grow with the reliability of the context cue, the stability of the environment, and with the number of trials since the last context switch. Analysis of human choice data validates all three predictions, providing evidence that the brain continuously updates probabilistic representations of the environment to best interpret an uncertain, ever-changing world.SIGNIFICANCEThe brain relies on prior knowledge to make perceptual inferences when sensory information is ambiguous. However, when the environmental context changes, the appropriate prior knowledge often changes with it. Here, we develop a Bayesian observer model to investigate how to make optimal perceptual inferences when sensory information and environmental context are both uncertain. The behavioral signature of this strategy is a context-appropriate decision bias whose strength grows with the reliability of the context cue, the stability of the environment, and with the number of decisions since the most recent change in context. We identified exactly this pattern in the behavior of human subjects performing a dynamic orientation discrimination task. Together, our results suggest that the brain continuously updates probabilistic representations of the environment to make perceptual decisions in the face of uncertainty over both sensory and contextual information.

Many animals rely on a representation of head direction for flexible, goal-directed navigation. In insects, a compass-like head direction representation is maintained in a conserved brain region called the central complex. This head direction representation is updated by self-motion information and by tethering to sensory cues in the surroundings through a plasticity mechanism. However, under natural settings, some of these sensory cues may temporarily disappear—for example, when clouds hide the sun—and prominent landmarks at different distances from the insect may move across the animal’s field of view during translation, creating potential conflicts for a neural compass. We used two-photon calcium imaging in head-fixed Drosophila behaving in virtual reality to monitor the fly’s compass during navigation in immersive naturalistic environments with approachable local landmarks. We found that the fly’s compass remains stable even in these settings by tethering to available global cues, likely preserving the animal’s ability to perform compass-driven behaviors such as maintaining a constant heading.

Flexible behaviors over long timescales are thought to engage recurrent neural networks in deep brain regions, which are experimentally challenging to study. In insects, recurrent circuit dynamics in a brain region called the central complex (CX) enable directed locomotion, sleep, and context- and experience-dependent spatial navigation. We describe the first complete electron microscopy-based connectome of the Drosophila CX, including all its neurons and circuits at synaptic resolution. We identified new CX neuron types, novel sensory and motor pathways, and network motifs that likely enable the CX to extract the fly’s head direction, maintain it with attractor dynamics, and combine it with other sensorimotor information to perform vector-based navigational computations. We also identified numerous pathways that may facilitate the selection of CX-driven behavioral patterns by context and internal state. The CX connectome provides a comprehensive blueprint necessary for a detailed understanding of network dynamics underlying sleep, flexible navigation, and state-dependent action selection.

Internal representations are thought to support the generation of flexible, long-timescale behavioral patterns in both animals and artificial agents. Here, we present a novel conceptual framework for how Drosophila use their internal representation of head direction to maintain preferred headings in their surroundings, and how they learn to modify these preferences in the presence of selective thermal reinforcement. To develop the framework, we analyzed flies' behavior in a classical operant visual learning paradigm and found that they use stochastically generated fixations and directed turns to express their heading preferences. Symmetries in the visual scene used in the paradigm allowed us to expose how flies' probabilistic behavior in this setting is tethered to their head direction representation. We describe how flies' ability to quickly adapt their behavior to the rules of their environment may rest on a behavioral policy whose parameters are flexible but whose form is genetically encoded in the structure of their circuits. Many of the mechanisms we outline may also be relevant for rapidly adaptive behavior driven by internal representations in other animals, including mammals.

Neurons throughout the sensory pathway adapt their responses depending on the statistical structure of the sensory environment. This adaptation is thought to be efficient, resulting in neuronal codes that maximize information about the stimulus. Contrast gain control is a form of efficient adaptation in the auditory cortex and is believed to be crucial for enhancing the detection of signals embedded in background noise. However, it is unclear whether the dynamics of contrast gain control reflect efficient adaptation, and how they inform behavioral perception in noise. Here, we trained mice to detect a target presented in background noise shortly after a change in the contrast of the background. The observed changes in cortical gain and behavioral detection followed the dynamics of a normative model of efficient contrast gain control; specifically, target detection and sensitivity to target level improved in low contrast backgrounds relative to high contrast backgrounds. Additionally, the time course of target detectability adapted asymmetrically depending on contrast, decreasing rapidly after a transition to high contrast, and increasing slowly after a transition to low contrast. Auditory cortex was required for detection of targets in background noise and cortical neuronal responses exhibited the patterns of target detectability observed during behavior and in the normative model. Furthermore, variability in cortical gain predicted behavioral performance beyond the effect of stimulus-driven gain control. Combined, our results demonstrate that dynamic gain adaptation supports efficient coding in auditory cortex and predicts behavioral performance in a target-in-background detection task.

Many animals rely on an internal heading representation when navigating in varied environments1-10. How this representation is linked to the sensory cues that define different surroundings is unclear. In the fly brain, heading is represented by 'compass' neurons that innervate a ring-shaped structure known as the ellipsoid body3,11,12. Each compass neuron receives inputs from 'ring' neurons that are selective for particular visual features13-16; this combination provides an ideal substrate for the extraction of directional information from a visual scene. Here we combine two-photon calcium imaging and optogenetics in tethered flying flies with circuit modelling, and show how the correlated activity of compass and visual neurons drives plasticity17-22, which flexibly transforms two-dimensional visual cues into a stable heading representation. We also describe how this plasticity enables the fly to convert a partial heading representation, established from orienting within part of a novel setting, into a complete heading representation. Our results provide mechanistic insight into the memory-related computations that are essential for flexible navigation in varied surroundings.

Flexible behaviors over long timescales are thought to engage recurrent neural networks in deep brain regions, which are experimentally challenging to study. In insects, recurrent circuit dynamics in a brain region called the central complex (CX) enable directed locomotion, sleep, and context- and experience-dependent spatial navigation. We describe the first complete electron-microscopy-based connectome of theDrosophilaCX, including all its neurons and circuits at synaptic resolution. We identified new CX neuron types, novel sensory and motor pathways, and network motifs that likely enable the CX to extract the fly’s head-direction, maintain it with attractor dynamics, and combine it with other sensorimotor information to perform vector-based navigational computations. We also identified numerous pathways that may facilitate the selection of CX-driven behavioral patterns by context and internal state. The CX connectome provides a comprehensive blueprint necessary for a detailed understanding of network dynamics underlying sleep, flexible navigation, and state-dependent action selection.

Many animals orient using visual cues, but how a single cue is selected from among many is poorly understood. Here we show that Drosophila ring neurons-central brain neurons implicated in navigation-display visual stimulus selection. Using in vivo two-color two-photon imaging with genetically encoded calcium indicators, we demonstrate that individual ring neurons inherit simple-cell-like receptive fields from their upstream partners. Stimuli in the contralateral visual field suppressed responses to ipsilateral stimuli in both populations. Suppression strength depended on when and where the contralateral stimulus was presented, an effect stronger in ring neurons than in their upstream inputs. This history-dependent effect on the temporal structure of visual responses, which was well modeled by a simple biphasic filter, may determine how visual references are selected for the fly's internal compass. Our approach highlights how two-color calcium imaging can help identify and localize the origins of sensory transformations across synaptically connected neural populations.

Previously, in (Hermundstad et al., 2014), we showed that when sampling is limiting, the efficient coding principle leads to a "variance is salience" hypothesis, and that this hypothesis accounts for visual sensitivity to binary image statistics. Here, using extensive new psychophysical data and image analysis, we show that this hypothesis accounts for visual sensitivity to a large set of grayscale image statistics at a striking level of detail, and also identify the limits of the prediction. We define a 66-dimensional space of local grayscale light-intensity correlations, and measure the relevance of each direction to natural scenes. The "variance is salience" hypothesis predicts that two-point correlations are most salient, and predicts their relative salience. We tested these predictions in a texture-segregation task using un-natural, synthetic textures. As predicted, correlations beyond second order are not salient, and predicted thresholds for over 300 second-order correlations match psychophysical thresholds closely (median fractional error, 33 pages, 12 figures

The ability to adapt to changes in stimulus statistics is a hallmark of sensory systems. Here, we developed a theoretical framework that can account for the dynamics of adaptation from an information processing perspective. We use this framework to optimize and analyze adaptive sensory codes, and we show that codes optimized for stationary environments can suffer from prolonged periods of poor performance when the environment changes. To mitigate the adversarial effects of these environmental changes, sensory systems must navigate tradeoffs between the ability to accurately encode incoming stimuli and the ability to rapidly detect and adapt to changes in the distribution of these stimuli. We derive families of codes that balance these objectives, and we demonstrate their close match to experimentally observed neural dynamics during mean and variance adaptation. Our results provide a unifying perspective on adaptation across a range of sensory systems, environments, and sensory tasks.

Behavior relies on the ability of sensory systems to infer properties of the environment from incoming stimuli. The accuracy of inference depends on the fidelity with which behaviorally-relevant properties of stimuli are encoded in neural responses. High-fidelity encodings can be metabolically costly, but low-fidelity encodings can cause errors in inference. Here, we discuss general principles that underlie the tradeoff between encoding cost and inference error. We then derive adaptive encoding schemes that dynamically navigate this tradeoff. These optimal encodings tend to increase the fidelity of the neural representation following a change in the stimulus distribution, and reduce fidelity for stimuli that originate from a known distribution. We predict dynamical signatures of such encoding schemes and demonstrate how known phenomena, such as burst coding and firing rate adaptation, can be understood as hallmarks of optimal coding for accurate inference.

Magnetic resonance imaging enables the noninvasive mapping of both anatomical white matter connectivity and dynamic patterns of neural activity in the human brain. We examine the relationship between the structural properties of white matter streamlines (structural connectivity) and the functional properties of correlations in neural activity (functional connectivity) within 84 healthy human subjects both at rest and during the performance of attention- and memory-demanding tasks. We show that structural properties, including the length, number, and spatial location of white matter streamlines, are indicative of and can be inferred from the strength of resting-state and task-based functional correlations between brain regions. These results, which are both representative of the entire set of subjects and consistently observed within individual subjects, uncover robust links between structural and functional connectivity in the human brain.

Animals smelling in the real world use a small number of receptors to sense a vast number of natural molecular mixtures, and proceed to learn arbitrary associations between odors and valences. Here, we propose a new interpretation of how the architecture of olfactory circuits is adapted to meet these immense complementary challenges. First, the diffuse binding of receptors to many molecules compresses a vast odor space into a tiny receptor space, while preserving similarity. Next, lateral interactions “densify” and decorrelate the response, enhancing robustness to noise. Finally, disordered projections from the periphery to the central brain reconfigure the densely packed information into a format suitable for flexible learning of associations and valences. We test our theory empirically using data from Drosophila. Our theory suggests that the neural processing of olfactory information differs from the other senses in its fundamental use of disorder.

The anatomical connectivity of the human brain supports diverse patterns of correlated neural activity that are thought to underlie cognitive function. In a manner sensitive to underlying structural brain architecture, we examine the extent to which such patterns of correlated activity systematically vary across cognitive states. Anatomical white matter connectivity is compared with functional correlations in neural activity measured via blood oxygen level dependent (BOLD) signals. Functional connectivity is separately measured at rest, during an attention task, and during a memory task. We assess these structural and functional measures within previously-identified resting-state functional networks, denoted task-positive and task-negative networks, that have been independently shown to be strongly anticorrelated at rest but also involve regions of the brain that routinely increase and decrease in activity during task-driven processes. We find that the density of anatomical connections within and between task-positive and task-negative networks is differentially related to strong, task-dependent correlations in neural activity. The space mapped out by the observed structure-function relationships is used to define a quantitative measure of separation between resting, attention, and memory states. We find that the degree of separation between states is related to both general measures of behavioral performance and relative differences in task-specific measures of attention versus memory performance. These findings suggest that the observed separation between cognitive states reflects underlying organizational principles of human brain structure and function., Author Summary Human cognitive function is thought to be supported by patterns of correlated neural activity. While recent work has shown that such functional correlations are differentially supported by specific properties of anatomical brain connectivity, the extent to which brain anatomy shapes cognition is not understood. In this study, we develop new network-based approaches for relating anatomical connectivity, correlations in neural activity (functional connectivity), and behavioral task performance. We use noninvasive neuroimaging techniques to measure whole-brain connectivity in 71 subjects across three cognitive states: at rest, during an attention task, and during a memory task. By associating anatomical and functional connectivity with known functional brain networks, we show that the relative strength of inter- versus intra-network connectivity distinguishes between resting, attention, and memory states. When compared across subjects, we further show that the observed relationship between brain anatomy and function is predictive of individual differences in attention and memory task performance.

The statistical regularities of natural scenes are a starting point for understanding the characteristics of early visual processing, e.g. the center-surround architecture of retinal ganglion cell receptive fields. Can this matching between natural signal statistics and neural processing mechanisms be extended beyond the sensory periphery? Our recent work [1] revealed that human sensitivity to fourth-order correlations in synthetic textures, known to arise in cortex, is closely related to the structure of fourth-order spatial correlations in natural scenes. This leads us to propose a specific organizing principle: The perceptual salience of visual textures increases with the variance, or unpredictability, of the corresponding correlations over the ensemble of natural scenes. To test this principle, we examined the statistical regularities of binarized natural images as characterized by correlations between adjacent pixels within a 2 × 2 square. Local binary textures can be described by four types of second-order correlations between pixels arranged in vertical (βV), horizontal (βH) and diagonal (β\ and β/) configurations, four types of third-order correlations between pixel triplets (θ1,2,3,4), and one type of fourth-order correlation between pixel quadruplets (α). We measured the values of these correlations in a large ensemble of image patches, and we compared the results to psychophysical experiments that measure human sensitivity to synthetic visual textures. Both the ordering and magnitude of natural image variances was found to match perceptual sensitivities to synthetic textures generated with the corresponding correlations (Figure ​(Figure1a).1a). Furthermore, the principal components of the full 9D space of image statistics match the principal components of the corresponding space of sensitivities (Figure 1b-e). These results suggest that central neural mechanisms are efficiently tuned to the higher-order statistics of natural scenes. Figure 1 (a) Variances in image statistics match perceptual sensitivities. Principal components of image statistics match principal components of human sensitivities, as shown by the fractional contribution of (b) βHV, (c) β/\, (d) θ, and ...

Noninvasive MRI technology enables the mapping of white matter connectivity between cortical regions in the human brain. Graph theoretic analysis has shown the organization of this structural brain network facilitates certain functional properties, such as robust and efficient transfer of information between brain regions. What are the generative mechanisms of such a network? To address this, we analyze a broad range of synthetic networks produced by simple growth rules, and we compare these networks both qualitatively and quantitatively to the brain. We consider a growth model in which a set of three parameters govern the tendency of nodes to form connections with one another. These generated networks are analyzed in terms of direct measures of connectivity, such as the length and number of network connections, as well as higher-order graph metrics, such as hierarchy and clustering, that relate to efficient information transfer and resistance to random errors. We find that different sets of input parameters can greatly affect the final structure of the network, with only some subsets of parameter values creating brain-like networks. This shows that very similar growth rules can produce a diverse set of network structures. In ongoing work, we are investigating the role of known biophysical constraints, such as bilateral symmetry and wiring cost, on network growth. This study lends insight into the biophysical mechanisms that govern the development of human brain networks and ultimately shape robust and efficient network function.

Structural configuration drives functionality in a range of different natural and artificial information processing systems. In this study, we use feedforward neural networks to evaluate the impact of structural variations on the ability of a network to learn and retain representations of external information. Performance is evaluated by statistically analyzing the error in the representations produced by parallel and layered networks during supervised, sequential function approximation. By varying the initial network state and the time given to learn the information, we identify tradeoffs between configurations that optimize for the best versus worst case scenarios and for the production of accurate versus retainable and generalizable representations of information. We show that these tradeoffs are maintained in larger networks and for variations in the information presented to the networks. By characterizing the curvature, depth, and participation of network connections about local error landscape minima, we find that variations in landscape structure give rise to the observed tradeoffs in performance. Consistently deep, narrow minima enable parallel networks to produce highly accurate solutions at the cost of more frequent failure in retention and generalizability. In contrast, variability in the depth and curvature of local minima enables layered networks to produce coarse but generalizable solutions at the cost of hindering consistent accuracy. Identifying structural drivers of functional performance is crucial for understanding both successes and limitations of information processing systems.

Recent experimental and computational studies have identified relationships between architecture and functional performance in information processing systems ranging from natural neuronal ensembles [1,2] to artificial neural networks [3,4]. While these systems can vary greatly in their size and complexity, they share certain structural features, such as parallel and layered motifs [5]. Quantifying how these features influence functionality is a first step toward understanding the behavior of both natural and artificial information processing systems. Of particular interest is the impact of structural architecture on the ability of the system to balance stability with flexibility, for example in memory versus learning. In this study, we use neural networks as model information processing systems to examine tradeoffs in learning and memory processes arising from variations in structural organization. We compare the performance of parallel and layered structures during sequential function approximation, a task that requires networks to produce, retain, and dynamically adapt representations of external information. We measure network performance over a range of learning conditions by statistically analyzing the error in these representations while varying the initial network state, the structure of the external information, and the time allowed for learning. By characterizing local error landscape curvature, we can directly relate the functional performance of the system to its underlying architecture. Across a range of both parallel and layered system architectures, we find that variations in error landscape curvature give rise to tradeoffs between the ability of these networks to learn new versus retain old information, maximize success versus minimize failure, and produce specific versus generalizable representations of information. In particular, parallel networks generate smooth error landscapes with deep, narrow minima. Therefore, given sufficient time and through the adjustment of a large number of connection weights, parallel networks can find highly specific representations of the external information. Although accurate, however, these representations are difficult to generalize. In contrast, layered networks generate rough error landscapes with a variety of local minima, allowing them to quickly find coarse representations by adjusting a fewer number of weights. Although less accurate, these representations are more easily adaptable. We have conducted a detailed analysis of network performance over a range of parallel and layered architectures, thereby isolating learning and memory tradeoffs that arise from underlying structural complexity. A thorough understanding of small network systems is crucial for predicting the behavior of larger systems in which statistical studies of performance would not be possible. In particular, these results may provide insight into the behavior of composite systems, such as cortical layers composed of structurally distinct columns [6] or modular divide-and-conquer networks [7], which share features of both parallel and layered architectures. Additionally, the existence of tradeoffs inherent to a range of network structures may help explain the variability of architectural motifs observed in large-scale biological [5] and technical [3] systems. Identifying the structural mechanisms that impact performance has implications for understanding a wide variety of both natural and artificial learning systems.

The performance of information processing systems, from artificial neural networks to natural neuronal ensembles, depends heavily on the underlying system architecture. In this study, we compare the performance of parallel and layered network architectures during sequential tasks that require both acquisition and retention of information, thereby identifying tradeoffs between learning and memory processes. During the task of supervised, sequential function approximation, networks produce and adapt representations of external information. Performance is evaluated by statistically analyzing the error in these representations while varying the initial network state, the structure of the external information, and the time given to learn the information. We link performance to complexity in network architecture by characterizing local error landscape curvature. We find that variations in error landscape structure give rise to tradeoffs in performance; these include the ability of the network to maximize accuracy versus minimize inaccuracy and produce specific versus generalizable representations of information. Parallel networks generate smooth error landscapes with deep, narrow minima, enabling them to find highly specific representations given sufficient time. While accurate, however, these representations are difficult to generalize. In contrast, layered networks generate rough error landscapes with a variety of local minima, allowing them to quickly find coarse representations. Although less accurate, these representations are easily adaptable. The presence of measurable performance tradeoffs in both layered and parallel networks has implications for understanding the behavior of a wide variety of natural and artificial learning systems., Author Summary Information processing systems, such as natural biological networks and artificial computational networks, exhibit a strong interdependence between structural organization and functional performance. However, the extent to which variations in structure impact performance is not well understood, particularly in systems whose functionality must be simultaneously flexible and stable. By statistically analyzing the behavior of network systems during flexible learning and stable memory processes, we quantify the impact of structural variations on the ability of the network to learn, modify, and retain representations of information. Across a range of architectures drawn from both natural and artificial systems, we show that these networks face tradeoffs between the ability to learn and retain information, and the observed behavior varies depending on the initial network state and the time given to process information. Furthermore, we analyze the difficulty with which different network architectures produce accurate versus generalizable representations of information, thereby identifying the structural mechanisms that give rise to functional tradeoffs between learning and memory.

[1] We quantify the energy dissipated to heat and to local disorder in a sheared layer of granular fault gouge. Local disorder is modeled using shear transformation zone theory, a continuum model of nonaffine deformation in amorphous solids that resolves spontaneous localization of strain. Strain localization decreases the total energy dissipated during slip. In addition, a fraction of this energy is dissipated to increasing local disorder as the material is sheared, thereby decreasing the amount of energy dissipated as thermal heat. We quantify the heat dissipated per unit area as a function of total slip in the presence and absence of strain localization and test the parameter dependence of these calculations. We find that less heat is dissipated per unit area compared to results obtained using a traditional heuristic energy partition.

The N -body problem in a double well requires new features for quantum information processing, macroscopic quantum superposition, and other fundamental studies of quantum many body physics in ultracold atoms. One needs (a) tilt, and (b) to go beyond the single-particle ground state in each well, i.e., to two or more energy levels. For (a), we show that a small potential difference between the wells, or tilt, causes the decoherence of Schrodinger cat states. However, these states reappear when the tilt can be compensated by atom-atom interactions; these tilted cat states constitute partial cats that are protected from decoherence by the many body wavefunction. For (b), we provide explicit criteria for when two energy levels are needed to describe the state space. For typical experimental parameters, two levels are indeed required for creation of cat states., Comment: 4 pages, 3 figures