Abstract Flexible motor control requires that movements adapt to changing temporal contexts. Here, we test whether flexible timing is driven by context-dependent encoding across cortico-cerebellar circuits and how neural dynamics within these circuits enable accurate performance. To overcome the signal-to-noise limitations of conventional electro- and magnetoencephalography, we recorded whole-head neural dynamics using optically pumped magnetometer arrays (OPM-MEG). Participants learned a context-dependent task, executing manual button presses at time intervals 800 ms (T1) and 1,600 ms (T2), respectively, to avoid a periocular air puff, which was associated with an implicit conditioned eyeblink. We found that the motor cortex contralateral to the hand and bilateral cerebellar lobule VI dynamically encoded these intervals through beta-band (13-30 Hz) event-related desynchronisation (ERD) that precisely scaled with T1 and T2. The cerebellar trial-by-trial latency of the beta-band ERD predicted the timing of explicit manual actions. Finally, partial directed coherence revealed that baseline bidirectional beta-band coupling across the network transiently weakened from the ipsilateral cerebellum to the contralateral motor cortex during finger movement execution. Our findings show that cortico-cerebellar coupling functions as a gating mechanism and suggest that cerebellar circuits modulate cortical motor activity for flexible, accurate motor timing. Significance statement How human cortico-cerebellar networks flexibly encode context-dependent timing remains poorly understood due to signal-to-noise constraints in non-invasive neural recordings. By combining a task coupling explicit manual and implicit eyeblink responses with wearable optically pumped magnetometers (OPM-MEG), we overcame these depth-sensitivity constraints to resolve human cortico-cerebellar dynamics. Beta-band (13-30 Hz) event-related desynchronisation in the contralateral cortical motor and bilateral cerebellar lobule VI was scaled to the anticipated response timings, with single-trial cerebellar latencies predicting explicit manual response timing. Crucially, directional functional coupling analysis revealed that bidirectional beta-band coupling transiently reduces from cerebellum to motor cortex during movement execution. This approach establishes a blueprint for non-invasively mapping cortico-cerebellar network dynamics, providing a framework to study circuit-level dysfunctions in disorders affecting the cerebellum.

The brain must infer the state of the external world despite the inherent uncertainty of its sensory inputs and internal processes. Under conditions of heightened uncertainty, it increasingly relies on prior knowledge, derived from accumulated experience with the regularities and statistical structures of the environment. This principle has been formalized by Bayesian inference theories, which are supported by substantial evidence from both behavioral and neuroscience studies. However, direct evidence for the existence of prior knowledge in the brain, and for the encoding of environmental statistics by neural circuits, remains limited. Here we show that cerebellar circuits learn the prior probability distribution of temporal variables during eyeblink conditioning in mice and encode these representations in Purkinje cell simple and complex spike signaling. We further demonstrate that Purkinje cells are involved in eliciting predictive motor behaviors, such as the conditioned eyeblink response, that also reflect the statistics of the experimentally imposed prior distribution of the stimulus. Computational modeling of these results indicates the juxtaposition of counteracting long-term plasticity mechanisms by which cerebellar Purkinje cells could acquire prior knowledge that is shaped by the statistics of different probability distributions. Our results suggest that the cerebellar circuitry may be uniquely poised to learn the probability of events in the world and internalize these as prior knowledge. These findings advance understanding of how neural computations could implement Bayesian inference.

One is seldom aware of the anticipatory and preemptive feats that the eyeblink system achieves in daily life but it frequently protects the eye from projectiles gone awry and insects on apparent collision courses. This poor awareness is why predictive eyeblinks are considered a form of implicit learning. In motor neuroscience, implicit learning is considered to be slow and, eyeblink conditioning, in particular, is believed to be a rigid and inflexible cerebellar-dependent behavior. In cognitive neuroscience, however, implicit and automatic processes are thought to be rapidly acquired. Here we show that the eyeblink system is, in fact, capable of remarkable cognitive flexibility and can learn on more rapid timescales than previously expected. In a task where we yoked contextual learning of predictive eyeblinks and manual responses in humans, well-timed eyeblink responses flexibly adjusted to external context on each trial. The temporal precision of the predictive eyeblinks exceeded that of manual response times. Learning of the well-timed eyeblink responses was also more rapid than that for the manual response times. This pattern persevered with the use of a cognitive strategy, which seemed to accelerate both types of learning. These results suggest that behaviors associated with the cerebellar cortex that were previously believed to be inflexible and largely implicit, can demonstrate rapid and precise context-dependent temporal control.

We possess the ability to anticipate and preempt occurrences under familiar circumstances, which suggests a reliance on prior experience with regularities in our environment, especially when observations become more uncertain. We know little about the neural mechanisms that can learn the probabilities of events in the environment and use this prior experience to guide actions. To examine this, we studied a rudimentary predictive behavior, eyeblink conditioning, and asked whether a simple effector like the eyelid could adapt its movements to varying probabilities of environmental events that reflect different degrees of uncertainty. We found that predictive eyeblink behavior systematically changed almost all its properties according to the temporal statistics of stimulus probability distributions. We also found that the activity of cerebellar Purkinje cells and putative molecular layer interneurons changed concomitantly with temporal statistics of the stimuli and with behavior. Targeted optogenetic perturbation of Purkinje cells during critical time windows severely attenuated the predictive behavior but left reflexive eyeblinks intact. Furthermore, we discovered a novel Purkinje cell complex spike signal coinciding with the onset of the earliest probable time interval in prior distributions with high uncertainty. This signal could not be explained as a motor or sensory correlate and appears to be anticipatory in nature. Theoretical modeling results pointed to a possible synaptic mechanism for how Purkinje cells could encode prior experience of environmental statistics in their activity through the juxtaposition of long-term depression and potentiation dynamics. ### Competing Interest Statement The authors have declared no competing interest.

One is seldom aware of the anticipatory and preemptive feats that the eyeblink systems achieves in daily life but it frequently protects the eye from projectiles gone awry and insects on apparent collision courses. This poor awareness is why predictive eyeblinks are considered a form of implicit learning. In motor neuroscience, implicit learning is considered to be slow and, eyeblink conditioning, in particular, is believed to be a rigid and inflexible cerebellar-dependent behavior. In cognitive neuroscience, however, implicit and automatic processes are thought to be rapidly acquired. Here we show that the eyeblink system is, in fact, capable of remarkable cognitive flexibility and can learn on more rapid timescales than previously expected. In a task where we yoked contextual learning of predictive eyeblinks and manual responses in humans, well-timed eyeblink responses flexibly adjusted to external context on each trial. The temporal precision of the predictive eyeblinks exceeded that of manual response times. Learning of the well-timed eyeblink responses was also more rapid than that for the manual response times. This pattern persevered with the use of a cognitive strategy, which seemed to accelerate both types of learning. These results suggest that behaviors associated with the cerebellar cortex that were previously believed to be inflexible and largely implicit, can demonstrate rapid and precise context-dependent temporal control. ### Competing Interest Statement The authors have declared no competing interest.

Local feedforward and recurrent connectivity are rife in the frontal areas of the cerebral cortex, which gives rise to rich heterogeneous dynamics observed in such areas. Recently, similar local connectivity motifs have been discovered among Purkinje and molecular layer interneurons of the cerebellar cortex, however, task-related activity in these neurons has often been associated with relatively simple facilitation and suppression dynamics. Here, we show that the rodent cerebellar cortex supports heterogeneity in task-related neuronal activity at a scale similar to the cerebral cortex. We provide a computational model that inculcates recent anatomical insights into local microcircuit motifs to show the putative basis for such heterogeneity. We also use cell-type specific chronic viral lesions to establish the involvement of cerebellar lobules in associative learning behaviors. Functional heterogeneity in neuronal profiles may not merely be the remit of the associative cerebral cortex, similar principles may be at play in subcortical areas, even those with seemingly crystalline and homogenous cytoarchitectures like the cerebellum.

Artificial Neural Networks (ANNs) trained on specific cognitive tasks have re-emerged as a useful tool to study the brain. However, ANNs would better aid cognitive neuroscience if a given network could be easily trained on a wide range of tasks for which neural recordings are available. Moreover, unintentional divergent implementations of cognitive tasks can produce variable results, which limits their interpretability. Towards this goal, we present NeuroGym, an open-source Python package that provides a large collection of customizable neuroscience tasks to test and compare network models. Building upon the OpenAI Gym toolbox, NeuroGym tasks (1) are written in a high-level flexible Python framework; (2) possess a shared interface tailored to common needs of neuroscience tasks that facilitates their design and usage; (3) support the training of ANNs using both Reinforcement and Supervised Learning techniques. The toolbox allows easy assembly of new tasks by modifying existing ones in a hierarchical and modular fashion. These design features make it straightforward to take a network designed for one task and train it on many other tasks. NeuroGym is a community-driven effort that contributes to a rapidly evolving open ecosystem of neural network development, data analysis, and model-data comparison.

Bayesian inference has emerged as a general framework that captures how organisms make decisions under uncertainty. Recent experimental findings reveal disparate mechanisms for how the brain generates behaviors predicted by normative Bayesian theories. Here, we identify two broad classes of neural implementations for Bayesian inference: a modular class, where each probabilistic component of Bayesian computation is independently encoded and a transform class, where uncertain measurements are converted to Bayesian estimates through latent processes. Many recent experimental neuroscience findings studying probabilistic inference broadly fall into these classes. We identify potential avenues for synthesis across these two classes and the disparities that, at present, cannot be reconciled. We conclude that to distinguish among implementation hypotheses for Bayesian inference, we require greater engagement among theoretical and experimental neuroscientists in an effort that spans different scales of analysis, circuits, tasks, and species.

Statistical regularities in the environment create prior beliefs that we rely on to optimize our behavior when sensory information is uncertain. Bayesian theory formalizes how prior beliefs can be leveraged and has had a major impact on models of perception, sensorimotor function, and cognition. However, it is not known how recurrent interactions among neurons mediate Bayesian integration. By using a time-interval reproduction task in monkeys, we found that prior statistics warp neural representations in the frontal cortex, allowing the mapping of sensory inputs to motor outputs to incorporate prior statistics in accordance with Bayesian inference. Analysis of recurrent neural network models performing the task revealed that this warping was enabled by a low-dimensional curved manifold and allowed us to further probe the potential causal underpinnings of this computational strategy. These results uncover a simple and general principle whereby prior beliefs exert their influence on behavior by sculpting cortical latent dynamics.

Neural mechanisms that support flexible sensorimotor computations are not well understood. In a dynamical system whose state is determined by interactions among neurons, computations can be rapidly reconfigured by controlling the system's inputs and initial conditions. To investigate whether the brain employs such control mechanisms, we recorded from the dorsomedial frontal cortex of monkeys trained to measure and produce time intervals in two sensorimotor contexts. The geometry of neural trajectories during the production epoch was consistent with a mechanism wherein the measured interval and sensorimotor context exerted control over cortical dynamics by adjusting the system's initial condition and input, respectively. These adjustments, in turn, set the speed at which activity evolved in the production epoch, allowing the animal to flexibly produce different time intervals. These results provide evidence that the language of dynamical systems can be used to parsimoniously link brain activity to sensorimotor computations.

Knowledge about the statistical regularities of the world is essential for cognitive and sensorimotor function. In the domain of timing, prior statistics are crucial for optimal prediction, adaptation and planning. Where and how the nervous system encodes temporal statistics is, however, not known. Based on physiological and anatomical evidence for cerebellar learning, we develop a computational model that demonstrates how the cerebellum could learn prior distributions of time intervals and support Bayesian temporal estimation. The model shows that salient features observed in human Bayesian time interval estimates can be readily captured by learning in the cerebellar cortex and circuit level computations in the cerebellar deep nuclei. We test human behavior in two cerebellar timing tasks and find prior-dependent biases in timing that are consistent with the predictions of the cerebellar model.

A hallmark of higher brain function is the ability to rapidly and flexibly adjust behavioral responses based on internal and external cues. Here, we examine the computational principles that allow decisions and actions to unfold flexibly in time. We adopt a dynamical systems perspective and outline how temporal flexibility in such a system can be achieved through manipulations of inputs and initial conditions. We then review evidence from experiments in nonhuman primates that support this interpretation. Finally, we explore the broader utility and limitations of the dynamical systems perspective as a general framework for addressing open questions related to the temporal control of movements, as well as in the domains of learning and sequence generation.

Musicians can perform at different tempos, speakers can control the cadence of their speech, and children can flexibly vary their temporal expectations of events. To understand the neural basis of such flexibility, we recorded from the medial frontal cortex of nonhuman primates trained to produce different time intervals with different effectors. Neural responses were heterogeneous, nonlinear, and complex, and they exhibited a remarkable form of temporal invariance: firing rate profiles were temporally scaled to match the produced intervals. Recording from downstream neurons in the caudate and from thalamic neurons projecting to the medial frontal cortex indicated that this phenomenon originates within cortical networks. Recurrent neural network models trained to perform the task revealed that temporal scaling emerges from nonlinearities in the network and that the degree of scaling is controlled by the strength of external input. These findings demonstrate a simple and general mechanism for conferring temporal flexibility upon sensorimotor and cognitive functions.

Musicians can perform at different tempos, speakers can control the cadence of their speech, and children can flexibly vary their temporal expectations of events. To understand the neural basis of such flexible timing, we recorded from the medial frontal cortex of primates trained to produce different time intervals with different effectors. The activity of neurons was heterogeneous, nonlinear and complex. However, responses were unified under a remarkable form of invariance: firing rate profiles were temporally stretched for longer intervals and compressed for short ones. At the network level, this phenomenon was evident by flexible changes in the speed with which the population activity traced an invariant trajectory. To identify the origin of speed control, we recorded from both downstream caudate neurons and thalamic neurons projecting to the medial frontal cortex. Speed adjustments were a prominent feature in the caudate but not in the thalamus suggesting that this phenomenon originates within cortical networks. To understand the underlying mechanisms, we created recurrent neural network models at different levels of complexity that could explain flexible timing with speed control. Analysis of the models revealed that the key to flexible speed control was the action of an external input upon the nonlinearities of individual neurons whose recurrent interactions set the network’s relaxation dynamics. These findings demonstrate a simple and general mechanism for conferring temporal flexibility upon sensorimotor and cognitive functions.

Recent studies demonstrate that biases found in human behavior can be explained by rational agents that make incorrect generative-model assumptions. While predicting a sequence of uncorrelated events, humans are biased towards overestimating its serial correlation. We demonstrate how such biases may also be the consequence of considering noisy observations over limited timescales of previous observations. We use the Kalman filter (KF) to study the upper-bound on human prediction performance. We investigate how the brain could estimate the necessary parameters for the KF based on the only source of information available to it, previous observations. We develop a variant of the KF model (dual memory) that obtains estimates of the KF parameters and its state over limited timescales of previous observations. The dual memory model predicts that the serial correlation should be veridical in responses for observations that are correlated in time and should be overestimated for uncorrelated ones. Second, the extent of overestimated correlation in the responses should be robust to varying noise-levels. Third, the overestimated correlation should persist regardless of whether previous observations are shown or not, if the same world-model is used. To test these hypotheses we performed an experiment where human observers were asked to predict time series, each with varying autocorrelations and noise-levels. One group was provided brief feedback whereas another was provided the history of observations. We found that the behavior of the participants was consistent with all three predictions. Further, we found a strong agreement between predictions of the dual memory model and previous empirical reports of bias in human forecasts of time series. We conclude that a markovian state-estimation model that would otherwise be optimal in predicting time series, displays the same biases in its predictions as humans do if it obtains parametric information over limited timescales of noisy observations. Meeting abstract presented at VSS 2016.

We often encounter pairs of variables in the world whose mutual relationship can be described by a function. After training, human responses closely correspond to these functional relationships. Here we study how humans predict unobserved segments of a function that they have been trained on and we compare how human predictions differ to those made by various function-learning models in the literature. Participants' performance was best predicted by the polynomial functions that generated the observations. Further, participants were able to explicitly report the correct generating function in most cases upon a post-experiment survey. This suggests that humans can abstract functions. To understand how they do so, we modeled human learning using an hierarchical Bayesian framework organized at two levels of abstraction: function learning and parameter learning, and used it to understand the time course of participants' learning as we surreptitiously changed the generating function over time. This Bayesian model selection framework allowed us to analyze the time course of function learning and parameter learning in relative isolation. We found that participants acquired new functions as they changed and even when parameter learning was not completely accurate, the probability that the correct function was learned remained high. Most importantly, we found that humans selected the simplest-fitting function with the highest probability and that they acquired simpler functions faster than more complex ones. Both aspects of this behavior, extent and rate of selection, present evidence that human function learning obeys the Occam's razor principle.

We investigate how humans discover hidden dependencies among variables in the visual environment over time. We first perform a visuo-motor experiment to establish that it is possible for humans to learn hidden models of varying complexity over time. Participants perform an experiment in which there is a hidden relationship between the value of an observed variable (location of a visual cue) and the required value of the response variable (interception time to a target). This relationship between the location and the time of a target represents models of different complexities (Constant, Linear, Quadratic) that suddenly switch over the course of the experiment. Given the data, we infer which model was being used to generate the responses at different stages of the experiment and simultaneously control for the different number of parameters in each model. We use Bayesian model selection to determine the posterior probability of each model given the data. We find that participants were able to correctly detect whether the hidden relationship in the stimuli followed a constant, linear or quadratic model. When the model that was used to generate the stimuli changed, participants were able to follow the change. In summary, participants constantly monitored the world relationship between the location and the time of a visual event and exhibited a preference for the simplest model that adequately explained the observed data. Meeting abstract presented at VSS 2013

Recent work has shown that humans can learn or detect complex dependencies among variables. Even learning a simple dependency involves the identification of an underlying model and the learning of its parameters. This process represents learning a structured problem. We are interested in an empirical assessment of some of the factors that enable humans to learn such a dependency over time. More specifically, we look at how the statistics of the presentation of samples from a given structure influence learning. Participants engage in an experimental task where they are required to predict the timing of a target. At the outset, they are oblivious to the existence of a relationship between the position of a stimulus and the required temporal response to intercept it. Different groups of participants are either presented with a Random Walk where consecutive stimuli were correlated or with stimuli that were uncorrelated over time. We find that the structural relationship implicit in the task is only learned in the conditions where the stimuli are independently drawn. This leads us to believe that humans require rich and independent sampling to learn hidden structures among variables.