Can We Identify Non-Stationary Dynamics of Trial-to-Trial Variability?

Identifying sources of the apparent variability in non-stationary scenarios is a fundamental problem in many biological data analysis settings. For instance, neurophysiological responses to the same task often vary from each repetition of the same experiment (trial) to the next. The origin and functional role of this observed variability is one of the fundamental questions in neuroscience. The nature of such trial-to-trial dynamics however remains largely elusive to current data analysis approaches. A range of strategies have been proposed in modalities such as electro-encephalography but gaining a fundamental insight into latent sources of trial-to-trial variability in neural recordings is still a major challenge. In this paper, we present a proof-of-concept study to the analysis of trial-to-trial variability dynamics founded on non-autonomous dynamical systems. At this initial stage, we evaluate the capacity of a simple statistic based on the behaviour of trajectories in classification settings, the trajectory coherence, in order to identify trial-to-trial dynamics. First, we derive the conditions leading to observable changes in datasets generated by a compact dynamical system (the Duffing equation). This canonical system plays the role of a ubiquitous model of non-stationary supervised classification problems. Second, we estimate the coherence of class-trajectories in empirically reconstructed space of system states. We show how this analysis can discern variations attributable to non-autonomous deterministic processes from stochastic fluctuations. The analyses are benchmarked using simulated and two different real datasets which have been shown to exhibit attractor dynamics. As an illustrative example, we focused on the analysis of the rat''s frontal cortex ensemble dynamics during a decision-making task. Results suggest that, in line with recent hypotheses, rather than internal noise, it is the deterministic trend which most likely underlies the observed trial-to-trial variability. Thus, the empirical tool developed within this study potentially allows us to infer the source of variability in in-vivo neural recordings.     

Neuronal precision and the limits for acoustic signal recognition in a small neuronal network

Recognition of acoustic signals may be impeded by two factors: extrinsic noise, which degrades sounds before they arrive at the receiver’s ears, and intrinsic neuronal noise, which reveals itself in the trial-to-trial variability of the responses to identical sounds. Here we analyzed how these two noise sources affect the recognition of acoustic signals from potential mates in grasshoppers. By progressively corrupting the envelope of a female song, we determined the critical degradation level at which males failed to recognize a courtship call in behavioral experiments. Using the same stimuli, we recorded intracellularly from auditory neurons at three different processing levels, and quantified the corresponding changes in spike train patterns by a spike train metric, which assigns a distance between spike trains. Unexpectedly, for most neurons, intrinsic variability accounted for the main part of the metric distance between spike trains, even at the strongest degradation levels. At consecutive levels of processing, intrinsic variability increased, while the sensitivity to external noise decreased. We followed two approaches to determine critical degradation levels from spike train dissimilarities, and compared the results with the limits of signal recognition measured in behaving animals.     

Axonal Noise as a Source of Synaptic Variability

Post-synaptic potential (PSP) variability is typically attributed to mechanisms inside synapses, yet recent advances in experimental methods and biophysical understanding have led us to reconsider the role of axons as highly reliable transmission channels. We show that in many thin axons of our brain, the action potential (AP) waveform and thus the Ca⁺⁺ signal controlling vesicle release at synapses will be significantly affected by the inherent variability of ion channel gating. We investigate how and to what extent fluctuations in the AP waveform explain observed PSP variability. Using both biophysical theory and stochastic simulations of central and peripheral nervous system axons from vertebrates and invertebrates, we show that channel noise in thin axons (<1 µm diameter) causes random fluctuations in AP waveforms. AP height and width, both experimentally characterised parameters of post-synaptic response amplitude, vary e.g. by up to 20 mV and 0.5 ms while a single AP propagates in C-fibre axons. We show how AP height and width variabilities increase with a ¾ power-law as diameter decreases and translate these fluctuations into post-synaptic response variability using biophysical data and models of synaptic transmission. We find for example that for mammalian unmyelinated axons with 0.2 µm diameter (matching cerebellar parallel fibres) axonal noise alone can explain half of the PSP variability in cerebellar synapses. We conclude that axonal variability may have considerable impact on synaptic response variability. Thus, in many experimental frameworks investigating synaptic transmission through paired-cell recordings or extracellular stimulation of presynaptic neurons, causes of variability may have been confounded. We thereby show how bottom-up aggregation of molecular noise sources contributes to our understanding of variability observed at higher levels of biological organisation.     

A generative model for measuring latent timing structure in motor sequences

Motor variability often reflects a mixture of different neural and peripheral sources operating over a range of timescales. We present a statistical model of sequence timing that can be used to measure three distinct components of timing variability: global tempo changes that are spread across the sequence, such as might stem from neuromodulatory sources with widespread influence; fast, uncorrelated timing noise, stemming from noisy components within the neural system; and timing jitter that does not alter the timing of subsequent elements, such as might be caused by variation in the motor periphery or by measurement error. In addition to quantifying the variability contributed by each of these latent factors in the data, the approach assigns maximum likelihood estimates of each factor on a trial-to-trial basis. We applied the model to adult zebra finch song, a temporally complex behavior with rich structure on multiple timescales. We find that individual song vocalizations (syllables) contain roughly equal amounts of variability in each of the three components while overall song length is dominated by global tempo changes. Across our sample of syllables, both global and independent variability scale with average length while timing jitter does not, a pattern consistent with the Wing and Kristofferson (1973) model of sequence timing. We also find significant day-to-day drift in all three timing sources, but a circadian pattern in tempo only. In tests using artificially generated data, the model successfully separates out the different components with small error. The approach provides a general framework for extracting distinct sources of timing variability within action sequences, and can be applied to neural and behavioral data from a wide array of systems.     

Do audible and ultrasonic sounds of intensities common in animal facilities affect the autonomic nervous system of rodents?

In animal facilities, noises, often poorly controlled, occur over a wide range of frequencies and intensities. Evidence demonstrates that audible noise and ultrasound have deleterious effects on rodent physiology, but it is not known how they affect the autonomic nervous system (ANS). This study exposed 3 unrestrained, male, Sprague-Dawley rats daily to a 15-min white noise regime (90 dB), a quiet regime, or a 15-min ultrasound regime (90 dB at 4 frequencies in the range 20 to 40 kHz)-each for several weeks—and used radiotelemetry to monitor their cardiovascular responses. Exposure to audible noise increased heart rate and mean arterial pressure. Spectral analysis of HR variability showed diminished stimulation of the parasympathetic nervous system, shifting the sympathovagal balance. However, ultrasound, at the frequencies used, did not reproducibly affect cardiovascular parameters. The preliminary data obtained from this study indicate that audible noise, but not ultrasound (delivered using the same protocol), affects the ANS. Because the cardiovascular, respiratory, renal, and gastrointestinal systems are under autonomic control, such noise could have wide-ranging effects on animal physiology.     

Interindividual variability in social insects – proximate causes and ultimate consequences

Individuals within social groups often show consistent differences in behaviour across time and context. Such interindividual differences and the evolutionary challenge they present have recently generated considerable interest. Social insects provide some of the most familiar and spectacular examples of social groups with large interindividual differences. Investigating these within‐group differences has a long research tradition, and behavioural variability among the workers of a colony is increasingly regarded as fundamental for a key feature of social insects: division of labour. The goal of this review is to illustrate what we know about both the proximate mechanisms underlying behavioural variability among the workers of a colony and its ultimate consequences; and to highlight the many open questions in this research field. We begin by reviewing the literature on mechanisms that potentially introduce, maintain, and adjust the behavioural differentiation among workers. We highlight the fact that so far, most studies have focused on behavioural variability based on genetic variability, provided by e.g. multiple mating of the queen, while other mechanisms that may be responsible for the behavioural differentiation among workers have been largely neglected. These include maturational, nutritional and environmental influences. We further discuss how feedback provided by the social environment and learning and experience of adult workers provides potent and little‐explored sources of differentiation. In a second part, we address what is known about the potential benefits and costs of increased behavioural variability within the workers of a colony. We argue that all studies documenting a benefit of variability so far have done so by manipulating genetic variability, and that a direct test of the effect of behavioural variability on colony productivity has yet to be provided. We emphasize that the costs associated with interindividual variability have been largely overlooked, and that a better knowledge of the cost/benefit balance of behavioural variability is crucial for our understanding of the evolution of the mechanisms underlying the social organization of insect societies. We conclude by highlighting what we believe to be promising but little‐explored avenues for future research on how within‐colony variability has evolved and is maintained. We emphasize the need for comparative studies and point out that, so far, most studies on interindividual variability have focused on variability in individual response thresholds, while the significance of variability in other parameters of individual response, such as probability and intensity of the response, has been largely overlooked. We propose that these parameters have important consequences for the colony response. Much more research is needed to understand if and how interindividual variability is modulated in order to benefit division of labour, homeostasis and ultimately colony fitness in social insects.     

Feedforward origins of response variability underlying contrast invariant orientation tuning in cat visual cortex

Contrast invariant orientation tuning in simple cells of the visual cortex depends critically on contrast dependent trial-to-trial variability in their membrane potential responses. This observation raises the question of whether this variability originates from within the cortical circuit or the feedforward inputs from the lateral geniculate nucleus (LGN). To distinguish between these two sources of variability, we first measured membrane potential responses while inactivating the surrounding cortex, and found that response variability was nearly unaffected. We then studied variability in the LGN, including contrast dependence, and the trial-to-trial correlation in responses between nearby neurons. Variability decreased significantly with contrast, whereas correlation changed little. When these experimentally measured parameters of variability were applied to a feedforward model of simple cells that included realistic mechanisms of synaptic integration, contrast-dependent, orientation independent variability emerged in the membrane potential responses. Analogous mechanisms might contribute to the stimulus dependence and propagation of variability throughout the neocortex.     

Role of uncertainty in sensorimotor control

Neural signals are corrupted by noise and this places limits on information processing. We review the processes involved in goal-directed movements and how neural noise and uncertainty determine aspects of our behaviour. First, noise in sensory signals limits perception. We show that, when localizing our hand, the central nervous system (CNS) integrates visual and proprioceptive information, each with different noise properties, in a way that minimizes the uncertainty in the overall estimate. Second, noise in motor commands leads to inaccurate movements. We review an optimal-control framework, known as 'task optimization in the presence of signal-dependent noise', which assumes that movements are planned so as to minimize the deleterious consequences of noise and thereby minimize inaccuracy. Third, during movement, sensory and motor signals have to be integrated to allow estimation of the body's state. Models are presented that show how these signals are optimally combined. Finally, we review how the CNS deals with noise at the neural and network levels. In all of these processes, the CNS carries out the tasks in such a way that the detrimental effects of noise are minimized. This shows that it is important to consider effects at the neural level in order to understand performance at the behavioural level.     

Average Is Optimal: An Inverted-U Relationship between Trial-to-Trial Brain Activity and Behavioral Performance

It is well known that even under identical task conditions, there is a tremendous amount of trial-to-trial variability in both brain activity and behavioral output. Thus far the vast majority of event-related potential (ERP) studies investigating the relationship between trial-to-trial fluctuations in brain activity and behavioral performance have only tested a monotonic relationship between them. However, it was recently found that across-trial variability can correlate with behavioral performance independent of trial-averaged activity. This finding predicts a U- or inverted-U- shaped relationship between trial-to-trial brain activity and behavioral output, depending on whether larger brain variability is associated with better or worse behavior, respectively. Using a visual stimulus detection task, we provide evidence from human electrocorticography (ECoG) for an inverted-U brain-behavior relationship: When the raw fluctuation in broadband ECoG activity is closer to the across-trial mean, hit rate is higher and reaction times faster. Importantly, we show that this relationship is present not only in the post-stimulus task-evoked brain activity, but also in the pre-stimulus spontaneous brain activity, suggesting anticipatory brain dynamics. Our findings are consistent with the presence of stochastic noise in the brain. They further support attractor network theories, which postulate that the brain settles into a more confined state space under task performance, and proximity to the targeted trajectory is associated with better performance.     

New treatments in neurorehabilitation founded on basic research

Recent discoveries about how the central nervous system responds to injury and how patients reacquire lost behaviours by training have yielded promising new therapies for neurorehabilitation. Until recently, this field had been largely static, but the current melding of basic behavioural science with neuroscience promises entirely new approaches to improving behavioural, perceptual and cognitive capabilities after neurological damage. Studies of phenomena such as cortical reorganization after a lesion, central nervous system repair, and the substantial enhancement of extremity use and linguistic function by behavioural therapy, support this emerging view. The ongoing changes in rehabilitation strategies might well amount to an impending paradigm shift in this field.     

The importance of invertebrates when considering the impacts of anthropogenic noise

Anthropogenic noise is now recognized as a major global pollutant. Rapidly burgeoning research has identified impacts on individual behaviour and physiology through to community disruption. To date, however, there has been an almost exclusive focus on vertebrates. Not only does their central role in food webs and in fulfilling ecosystem services make imperative our understanding of how invertebrates are impacted by all aspects of environmental change, but also many of their inherent characteristics provide opportunities to overcome common issues with the current anthropogenic noise literature. Here, we begin by explaining why invertebrates are likely to be affected by anthropogenic noise, briefly reviewing their capacity for hearing and providing evidence that they are capable of evolutionary adaptation and behavioural plasticity in response to natural noise sources. We then discuss the importance of quantifying accurately and fully both auditory ability and noise content, emphasizing considerations of direct relevance to how invertebrates detect sounds. We showcase how studying invertebrates can help with the behavioural bias in the literature, the difficulties in drawing strong, ecologically valid conclusions and the need for studies on fitness impacts. Finally, we suggest avenues of future research using invertebrates that would advance our understanding of the impact of anthropogenic noise.     

Immune response inhibits associative learning in insects

In vertebrates, it is well established that there are many intricate interactions between the immune system and the nervous system, and vice versa. Regarding insects, until now little has been known about the link between these two systems. Here, we present behavioural evidence indicating a link between the immune system and the nervous system in insects. We show that otherwise non-infected honeybees whose immune systems are challenged by a non-pathogenic immunogenic elicitor lipopolysaccharide (LPS) have reduced abilities to associate an odour with sugar reward in a classical conditioning paradigm. The cost of an immune response therefore not only affects survival of the host, as previously shown, but also everyday behaviour and memory formation.     

Dissociating Variability and Effort as Determinants of Coordination

When coordinating movements, the nervous system often has to decide how to distribute work across a number of redundant effectors. Here, we show that humans solve this problem by trying to minimize both the variability of motor output and the effort involved. In previous studies that investigated the temporal shape of movements, these two selective pressures, despite having very different theoretical implications, could not be distinguished; because noise in the motor system increases with the motor commands, minimization of effort or variability leads to very similar predictions. When multiple effectors with different noise and effort characteristics have to be combined, however, these two cost terms can be dissociated. Here, we measure the importance of variability and effort in coordination by studying how humans share force production between two fingers. To capture variability, we identified the coefficient of variation of the index and little fingers. For effort, we used the sum of squared forces and the sum of squared forces normalized by the maximum strength of each effector. These terms were then used to predict the optimal force distribution for a task in which participants had to produce a target total force of 4–16 N, by pressing onto two isometric transducers using different combinations of fingers. By comparing the predicted distribution across fingers to the actual distribution chosen by participants, we were able to estimate the relative importance of variability and effort of 17, with the unnormalized effort being most important. Our results indicate that the nervous system uses multi-effector redundancy to minimize both the variability of the produced output and effort, although effort costs clearly outweighed variability costs.     

Cross-population coupling of neural activity based on Gaussian process current source densities

Because local field potentials (LFPs) arise from multiple sources in different spatial locations, they do not easily reveal coordinated activity across neural populations on a trial-to-trial basis. As we show here, however, once disparate source signals are decoupled, their trial-to-trial fluctuations become more accessible, and cross-population correlations become more apparent. To decouple sources we introduce a general framework for estimation of current source densities (CSDs). In this framework, the set of LFPs result from noise being added to the transform of the CSD by a biophysical forward model, while the CSD is considered to be the sum of a zero-mean, stationary, spatiotemporal Gaussian process, having fast and slow components, and a mean function, which is the sum of multiple time-varying functions distributed across space, each varying across trials. We derived biophysical forward models relevant to the data we analyzed. In simulation studies this approach improved identification of source signals compared to existing CSD estimation methods. Using data recorded from primate auditory cortex, we analyzed trial-to-trial fluctuations in both steady-state and task-evoked signals. We found cortical layer-specific phase coupling between two probes and showed that the same analysis applied directly to LFPs did not recover these patterns. We also found task-evoked CSDs to be correlated across probes, at specific cortical depths. Using data from Neuropixels probes in mouse visual areas, we again found evidence for depth-specific phase coupling of primary visual cortex and lateromedial area based on the CSDs.     

Environmentally‐induced noise dampens and reddens with increasing trophic level in a complex food web

Stochastic variability of key abiotic factors including temperature, precipitation and the availability of light and nutrients greatly influences species’ ecological function and evolutionary fate. Despite such influence, ecologists have typically ignored the effect of abiotic stochasticity on the structure and dynamics of ecological networks. Here we help to fill that gap by advancing the theory of how abiotic stochasticity, in the form of environmental noise, affects the population dynamics of species within food webs. We do this by analysing an allometric trophic network model of Lake Constance subjected to positive (red), negative (blue), and non‐autocorrelated (white) abiotic temporal variability (noise) introduced into the carrying capacity of basal species. We found that, irrespective of the colour of the introduced noise, the temporal variability of the species biomass within the network both reddens (i.e. its positive autocorrelation increases) and dampens (i.e. the magnitude of variation decreases) as the environmental noise is propagated through the food web by its feeding interactions from the bottom to the top. The reddening reflects a buffering of the noise‐induced population variability by complex food web dynamics such that non‐autocorrelated oscillations of noise‐free deterministic dynamics become positively autocorrelated. Our research helps explain frequently observed red variability of natural populations by suggesting that ecological processing of environmental noise through food webs with a range of species’ body sizes reddens population variability in nature.     

Normalized Index of Synergy for Evaluating the Coordination of Motor Commands

Humans perform various motor tasks by coordinating the redundant motor elements in their bodies. The coordination of motor outputs is produced by motor commands, as well properties of the musculoskeletal system. The aim of this study was to dissociate the coordination of motor commands from motor outputs. First, we conducted simulation experiments where the total elbow torque was generated by a model of a simple human right and left elbow with redundant muscles. The results demonstrated that muscle tension with signal-dependent noise formed a coordinated structure of trial-to-trial variability of muscle tension. Therefore, the removal of signal-dependent noise effects was required to evaluate the coordination of motor commands. We proposed a method to evaluate the coordination of motor commands, which removed signal-dependent noise from the measured variability of muscle tension. We used uncontrolled manifold analysis to calculate a normalized index of synergy. Simulation experiments confirmed that the proposed method could appropriately represent the coordinated structure of the variability of motor commands. We also conducted experiments in which subjects performed the same task as in the simulation experiments. The normalized index of synergy revealed that the subjects coordinated their motor commands to achieve the task. Finally, the normalized index of synergy was applied to a motor learning task to determine the utility of the proposed method. We hypothesized that a large part of the change in the coordination of motor outputs through learning was because of changes in motor commands. In a motor learning task, subjects tracked a target trajectory of the total torque. The change in the coordination of muscle tension through learning was dominated by that of motor commands, which supported the hypothesis. We conclude that the normalized index of synergy can be used to evaluate the coordination of motor commands independently from the properties of the musculoskeletal system.     

Activation of Parallel Fiber Feedback by Spatially Diffuse Stimuli Reduces Signal and Noise Correlations via Independent Mechanisms in a Cerebellum-Like Structure

Correlations between the activities of neighboring neurons are observed ubiquitously across systems and species and are dynamically regulated by several factors such as the stimulus'' spatiotemporal extent as well as by the brain''s internal state. Using the electrosensory system of gymnotiform weakly electric fish, we recorded the activities of pyramidal cell pairs within the electrosensory lateral line lobe (ELL) under spatially localized and diffuse stimulation. We found that both signal and noise correlations were markedly reduced (>40%) under the latter stimulation. Through a network model incorporating key anatomical features of the ELL, we reveal how activation of diffuse parallel fiber feedback from granule cells by spatially diffuse stimulation can explain both the reduction in signal as well as the reduction in noise correlations seen experimentally through independent mechanisms. First, we show that burst-timing dependent plasticity, which leads to a negative image of the stimulus and thereby reduces single neuron responses, decreases signal but not noise correlations. Second, we show trial-to-trial variability in the responses of single granule cells to sensory input reduces noise but not signal correlations. Thus, our model predicts that the same feedback pathway can simultaneously reduce both signal and noise correlations through independent mechanisms. To test this prediction experimentally, we pharmacologically inactivated parallel fiber feedback onto ELL pyramidal cells. In agreement with modeling predictions, we found that inactivation increased both signal and noise correlations but that there was no significant relationship between magnitude of the increase in signal correlations and the magnitude of the increase in noise correlations. The mechanisms reported in this study are expected to be generally applicable to the cerebellum as well as other cerebellum-like structures. We further discuss the implications of such decorrelation on the neural coding strategies used by the electrosensory and by other systems to process natural stimuli.     

Sources of Variability in a Synthetic Gene Oscillator

Synthetic gene oscillators are small, engineered genetic circuits that produce periodic variations in target protein expression. Like other gene circuits, synthetic gene oscillators are noisy and exhibit fluctuations in amplitude and period. Understanding the origins of such variability is key to building predictive models that can guide the rational design of synthetic circuits. Here, we developed a method for determining the impact of different sources of noise in genetic oscillators by measuring the variability in oscillation amplitude and correlations between sister cells. We first used a combination of microfluidic devices and time-lapse fluorescence microscopy to track oscillations in cell lineages across many generations. We found that oscillation amplitude exhibited high cell-to-cell variability, while sister cells remained strongly correlated for many minutes after cell division. To understand how such variability arises, we constructed a computational model that identified the impact of various noise sources across the lineage of an initial cell. When each source of noise was appropriately tuned the model reproduced the experimentally observed amplitude variability and correlations, and accurately predicted outcomes under novel experimental conditions. Our combination of computational modeling and time-lapse data analysis provides a general way to examine the sources of variability in dynamic gene circuits.     

In vivo conditions induce faithful encoding of stimuli by reducing nonlinear synchronization in vestibular sensory neurons

Previous studies have shown that neurons within the vestibular nuclei (VN) can faithfully encode the time course of sensory input through changes in firing rate in vivo. However, studies performed in vitro have shown that these same VN neurons often display nonlinear synchronization (i.e. phase locking) in their spiking activity to the local maxima of sensory input, thereby severely limiting their capacity for faithful encoding of said input through changes in firing rate. We investigated this apparent discrepancy by studying the effects of in vivo conditions on VN neuron activity in vitro using a simple, physiologically based, model of cellular dynamics. We found that membrane potential oscillations were evoked both in response to step and zap current injection for a wide range of channel conductance values. These oscillations gave rise to a resonance in the spiking activity that causes synchronization to sinusoidal current injection at frequencies below 25 Hz. We hypothesized that the apparent discrepancy between VN response dynamics measured in in vitro conditions (i.e., consistent with our modeling results) and the dynamics measured in vivo conditions could be explained by an increase in trial-to-trial variability under in vivo vs. in vitro conditions. Accordingly, we mimicked more physiologically realistic conditions in our model by introducing a noise current to match the levels of resting discharge variability seen in vivo as quantified by the coefficient of variation (CV). While low noise intensities corresponding to CV values in the range 0.04-0.24 only eliminated synchronization for low (<8 Hz) frequency stimulation but not high (>12 Hz) frequency stimulation, higher noise intensities corresponding to CV values in the range 0.5-0.7 almost completely eliminated synchronization for all frequencies. Our results thus predict that, under natural (i.e. in vivo) conditions, the vestibular system uses increased variability to promote fidelity of encoding by single neurons. This prediction can be tested experimentally in vitro.