Mathematical models of learning and what can be learned from them.

Learning is a multi-faceted phenomenon of critical importance and hence attracted a great deal of research, both experimental and theoretical. In this review, we will consider some of the paradigmatic examples of learning and discuss the common themes in theoretical learning research, such as levels of modeling and their corresponding relation to experimental observations and mathematical ideas common to different types of learning.

Forgetting dynamics for items of different categories.

How the dynamic evolution of forgetting changes for different material types is unexplored. By using a common experimental paradigm with stimuli of different types, we were able to directly cross-examine the emerging dynamics and found that even though the presentation sets differ minimally by design, the obtained curves appear to fall on a discrete spectrum. We also show that the resulting curves do not depend on physical time but rather on the number of items shown. All measured curves were compatible with our previously developed mathematical model, hinting to a potential common underlying mechanism of forgetting.

Cross-fixation interactions of orientations suggest high-to-low-level decoding in visual working memory.

Sensory encoding (how stimuli evoke sensory responses) is known to progress from low- to high-level features. Decoding (how responses lead to perception) is less understood but is often assumed to follow the same hierarchy. Accordingly, orientation decoding must occur in low-level areas such as V1, without cross-fixation interactions. However, a study, Ding, Cueva, Tsodyks, and Qian (2017), provided evidence against the assumption and proposed that visual decoding may often follow a high-to-low-level hierarchy in working memory, where higher-to-lower-level constraints introduce interactions among lower-level features. If two orientations on opposite sides of the fixation are both task relevant and enter working memory, then they should interact with each other. We indeed found the predicted cross-fixation interactions (repulsion and correlation) between orientations. Control experiments and analyses ruled out alternative explanations such as reporting bias and adaptation across trials on the same side of the fixation. Moreover, we explained the data using a retrospective high-to-low-level Bayesian decoding framework.

Effects of order on memory of event times.

Memorizing time of an event may employ two processes (1) encoding of the absolute time of events within an episode, (2) encoding of its relative order. Here we study interaction between these two processes. We performed experiments in which one or several items were presented, after which participants were asked to report the time of occurrence of items. When a single item was presented, the distribution of reported times was quite wide. When two or three items were presented, the relative order among them strongly affected the reported time of each of them. Bayesian theory that takes into account the memory for the events order is compatible with the experimental data, in particular in terms of the effect of order on absolute time reports. Our results suggest that people do not deduce order from memorized time, instead people's memory for absolute time of events relies critically on memorized order of the events.

Biases and Variability from Costly Bayesian Inference.

When humans infer underlying probabilities from stochastic observations, they exhibit biases and variability that cannot be explained on the basis of sound, Bayesian manipulations of probability. This is especially salient when beliefs are updated as a function of sequential observations. We introduce a theoretical framework in which biases and variability emerge from a trade-off between Bayesian inference and the cognitive cost of carrying out probabilistic computations. We consider two forms of the cost: a precision cost and an unpredictability cost; these penalize beliefs that are less entropic and less deterministic, respectively. We apply our framework to the case of a Bernoulli variable: the bias of a coin is inferred from a sequence of coin flips. Theoretical predictions are qualitatively different depending on the form of the cost. A precision cost induces overestimation of small probabilities, on average, and a limited memory of past observations, and, consequently, a fluctuating bias. An unpredictability cost induces underestimation of small probabilities and a fixed bias that remains appreciable even for nearly unbiased observations. The case of a fair (equiprobable) coin, however, is singular, with non-trivial and slow fluctuations in the inferred bias. The proposed framework of costly Bayesian inference illustrates the richness of a 'resource-rational' (or 'bounded-rational') picture of seemingly irrational human cognition.

Multiscale representation of very large environments in the hippocampus of flying bats.

Hippocampal place cells encode the animal's location. Place cells were traditionally studied in small environments, and nothing is known about large ethologically relevant spatial scales. We wirelessly recorded from hippocampal dorsal CA1 neurons of wild-born bats flying in a long tunnel (200 meters). The size of place fields ranged from 0.6 to 32 meters. Individual place cells exhibited multiple fields and a multiscale representation: Place fields of the same neuron differed up to 20-fold in size. This multiscale coding was observed from the first day of exposure to the environment, and also in laboratory-born bats that never experienced large environments. Theoretical decoding analysis showed that the multiscale code allows representation of very large environments with much higher precision than that of other codes. Together, by increasing the spatial scale, we discovered a neural code that is radically different from classical place codes.

Retroactive interference model of forgetting.

Memory and forgetting constitute two sides of the same coin, and although the first has been extensively investigated, the latter is often overlooked. A possible approach to better understand forgetting is to develop phenomenological models that implement its putative mechanisms in the most elementary way possible, and then experimentally test the theoretical predictions of these models. One such mechanism proposed in previous studies is retrograde interference, stating that a memory can be erased due to subsequently acquired memories. In the current contribution, we hypothesize that retrograde erasure is controlled by the relevant "importance" measures such that more important memories eliminate less important ones acquired earlier. We show that some versions of the resulting mathematical model are broadly compatible with the previously reported power-law forgetting time course and match well the results of our recognition experiments with long, randomly assembled streams of words.

Fundamental Law of Memory Recall.

Human memory appears to be fragile and unpredictable. Free recall of random lists of words is a standard paradigm used to probe episodic memory. We proposed an associative search process that can be reduced to a deterministic walk on random graphs defined by the structure of memory representations. The corresponding graph model can be solved analytically, resulting in a novel parameter-free prediction for the average number of memory items recalled (R) out of M items in memory: R=sqrt[3πM/2]. This prediction was verified with a specially designed experimental protocol combining large-scale crowd-sourced free recall and recognition experiments with randomly assembled lists of words or common facts. Our results show that human memory can be described by universal laws derived from first principles.

Emergence of hierarchical organization in memory for random material.

Structured information is easier to remember and recall than random one. In real life, information exhibits multi-level hierarchical organization, such as clauses, sentences, episodes and narratives in language. Here we show that multi-level grouping emerges even when participants perform memory recall experiments with random sets of words. To quantitatively probe brain mechanisms involved in memory structuring, we consider an experimental protocol where participants perform 'final free recall' (FFR) of several random lists of words each of which was first presented and recalled individually. We observe a hierarchy of grouping organizations of FFR, most notably many participants sequentially recalled relatively long chunks of words from each list before recalling words from another list. Moreover, participants who exhibited strongest organization during FFR achieved highest levels of performance. Based on these results, we develop a hierarchical model of memory recall that is broadly compatible with our findings. Our study shows how highly controlled memory experiments with random and meaningless material, when combined with simple models, can be used to quantitatively probe the way meaningful information can efficiently be organized and processed in the brain.

Optimal dynamic coding by mixed-dimensionality neurons in the head-direction system of bats.

Ethologically relevant stimuli are often multidimensional. In many brain systems, neurons with "pure" tuning to one stimulus dimension are found along with "conjunctive" neurons that encode several dimensions, forming an apparently redundant representation. Here we show using theoretical analysis that a mixed-dimensionality code can efficiently represent a stimulus in different behavioral regimes: encoding by conjunctive cells is more robust when the stimulus changes quickly, whereas on long timescales pure cells represent the stimulus more efficiently with fewer neurons. We tested our predictions experimentally in the bat head-direction system and found that many head-direction cells switched their tuning dynamically from pure to conjunctive representation as a function of angular velocity-confirming our theoretical prediction. More broadly, our results suggest that optimal dimensionality depends on population size and on the time available for decoding-which might explain why mixed-dimensionality representations are common in sensory, motor, and higher cognitive systems across species.

Stabilizing patterns in time: Neural network approach.

Recurrent and feedback networks are capable of holding dynamic memories. Nonetheless, training a network for that task is challenging. In order to do so, one should face non-linear propagation of errors in the system. Small deviations from the desired dynamics due to error or inherent noise might have a dramatic effect in the future. A method to cope with these difficulties is thus needed. In this work we focus on recurrent networks with linear activation functions and binary output unit. We characterize its ability to reproduce a temporal sequence of actions over its output unit. We suggest casting the temporal learning problem to a perceptron problem. In the discrete case a finite margin appears, providing the network, to some extent, robustness to noise, for which it performs perfectly (i.e. producing a desired sequence for an arbitrary number of cycles flawlessly). In the continuous case the margin approaches zero when the output unit changes its state, hence the network is only able to reproduce the sequence with slight jitters. Numerical simulation suggest that in the discrete time case, the longest sequence that can be learned scales, at best, as square root of the network size. A dramatic effect occurs when learning several short sequences in parallel, that is, their total length substantially exceeds the length of the longest single sequence the network can learn. This model easily generalizes to an arbitrary number of output units, which boost its performance. This effect is demonstrated by considering two practical examples for sequence learning. This work suggests a way to overcome stability problems for training recurrent networks and further quantifies the performance of a network under the specific learning scheme.

Visual perception as retrospective Bayesian decoding from high- to low-level features.

When a stimulus is presented, its encoding is known to progress from low- to high-level features. How these features are decoded to produce perception is less clear, and most models assume that decoding follows the same low- to high-level hierarchy of encoding. There are also theories arguing for global precedence, reversed hierarchy, or bidirectional processing, but they are descriptive without quantitative comparison with human perception. Moreover, observers often inspect different parts of a scene sequentially to form overall perception, suggesting that perceptual decoding requires working memory, yet few models consider how working-memory properties may affect decoding hierarchy. We probed decoding hierarchy by comparing absolute judgments of single orientations and relative/ordinal judgments between two sequentially presented orientations. We found that lower-level, absolute judgments failed to account for higher-level, relative/ordinal judgments. However, when ordinal judgment was used to retrospectively decode memory representations of absolute orientations, striking aspects of absolute judgments, including the correlation and forward/backward aftereffects between two reported orientations in a trial, were explained. We propose that the brain prioritizes decoding of higher-level features because they are more behaviorally relevant, and more invariant and categorical, and thus easier to specify and maintain in noisy working memory, and that more reliable higher-level decoding constrains less reliable lower-level decoding.

Memory States and Transitions between Them in Attractor Neural Networks.

Human memory is capable of retrieving similar memories to a just retrieved one. This associative ability is at the base of our everyday processing of information. Current models of memory have not been able to underpin the mechanism that the brain could use in order to actively exploit similarities between memories. The current idea is that to induce transitions in attractor neural networks, it is necessary to extinguish the current memory. We introduce a novel mechanism capable of inducing transitions between memories where similarities between memories are actively exploited by the neural dynamics to retrieve a new memory. Populations of neurons that are selective for multiple memories play a crucial role in this mechanism by becoming attractors on their own. The mechanism is based on the ability of the neural network to control the excitation-inhibition balance.

A theory of working memory without consciousness or sustained activity.

Working memory and conscious perception are thought to share similar brain mechanisms, yet recent reports of non-conscious working memory challenge this view. Combining visual masking with magnetoencephalography, we investigate the reality of non-conscious working memory and dissect its neural mechanisms. In a spatial delayed-response task, participants reported the location of a subjectively unseen target above chance-level after several seconds. Conscious perception and conscious working memory were characterized by similar signatures: a sustained desynchronization in the alpha/beta band over frontal cortex, and a decodable representation of target location in posterior sensors. During non-conscious working memory, such activity vanished. Our findings contradict models that identify working memory with sustained neural firing, but are compatible with recent proposals of 'activity-silent' working memory. We present a theoretical framework and simulations showing how slowly decaying synaptic changes allow cell assemblies to go dormant during the delay, yet be retrieved above chance-level after several seconds.

Theta-paced flickering between place-cell maps in the hippocampus: A model based on short-term synaptic plasticity.

Hippocampal place cells represent different environments with distinct neural activity patterns. Following an abrupt switch between two familiar configurations of visual cues defining two environments, the hippocampal neural activity pattern switches almost immediately to the corresponding representation. Surprisingly, during a transient period following the switch to the new environment, occasional fast transitions between the two activity patterns (flickering) were observed (Jezek, Henriksen, Treves, Moser, & Moser, ). Here we show that an attractor neural network model of place cells with connections endowed with short-term synaptic plasticity can account for this phenomenon. A memory trace of the recent history of network activity is maintained in the state of the synapses, allowing the network to temporarily reactivate the representation of the previous environment in the absence of the corresponding sensory cues. The model predicts that the number of flickering events depends on the amplitude of the ongoing theta rhythm and the distance between the current position of the animal and its position at the time of cue switching. We test these predictions with new analysis of experimental data. These results suggest a potential role of short-term synaptic plasticity in recruiting the activity of different cell assemblies and in shaping hippocampal activity of behaving animals.

Feature Detection in Visual Cortex during Different Functional States.

Cortical activity exhibits distinct characteristics in different functional states. In awake behaving animals it shows less synchrony, while in rest or sleeping state cortical activity is most synchronous. Previous studies showed that switching between functional states can change the efficiency of flowing sensory information. Switching between functional states can be triggered by releasing neuromodulators which affect neurotransmitter release probability and depolarization of cortical neurons. In this work we focus on studying primary visual area V1, by using firing rate ring model with short-term synaptic depression (STD). We show that reconstruction of visual features from V1 activity depends on the functional state, with best precision achieved at the state with intermediate release probability. We suggest that this regime corresponds to the state of maximal visual attention.

Synaptic Correlates of Working Memory Capacity.

Psychological studies indicate that human ability to keep information in readily accessible working memory is limited to four items for most people. This extremely low capacity severely limits execution of many cognitive tasks, but its neuronal underpinnings remain unclear. Here we show that in the framework of synaptic theory of working memory, capacity can be analytically estimated to scale with characteristic time of short-term synaptic depression relative to synaptic current time constant. The number of items in working memory can be regulated by external excitation, enabling the system to be tuned to the desired load and to clear the working memory of currently held items to make room for new ones.

Practice makes perfect in memory recall.

A large variability in performance is observed when participants recall briefly presented lists of words. The sources of such variability are not known. Our analysis of a large data set of free recall revealed a small fraction of participants that reached an extremely high performance, including many trials with the recall of complete lists. Moreover, some of them developed a number of consistent input-position-dependent recall strategies, in particular recalling words consecutively ("chaining") or in groups of consecutively presented words ("chunking"). The time course of acquisition and particular choice of positional grouping were variable among participants. Our results show that acquiring positional strategies plays a crucial role in improvement of recall performance.

A unifying principle underlying the extracellular field potential spectral responses in the human cortex.

Electrophysiological mass potentials show complex spectral changes upon neuronal activation. However, it is unknown to what extent these complex band-limited changes are interrelated or, alternatively, reflect separate neuronal processes. To address this question, intracranial electrocorticograms (ECoG) responses were recorded in patients engaged in visuomotor tasks. We found that in the 10- to 100-Hz frequency range there was a significant reduction in the exponent χ of the 1/f(χ) component of the spectrum associated with neuronal activation. In a minority of electrodes showing particularly high activations the exponent reduction was associated with specific band-limited power modulations: emergence of a high gamma (80-100 Hz) and a decrease in the alpha (9-12 Hz) peaks. Importantly, the peaks' height was correlated with the 1/f(χ) exponent on activation. Control simulation ruled out the possibility that the change in 1/f(χ) exponent was a consequence of the analysis procedure. These results reveal a new global, cross-frequency (10-100 Hz) neuronal process reflected in a significant reduction of the power spectrum slope of the ECoG signal.

Effects of long-term representations on free recall of unrelated words.

Human memory stores vast amounts of information. Yet recalling this information is often challenging when specific cues are lacking. Here we consider an associative model of retrieval where each recalled item triggers the recall of the next item based on the similarity between their long-term neuronal representations. The model predicts that different items stored in memory have different probability to be recalled depending on the size of their representation. Moreover, items with high recall probability tend to be recalled earlier and suppress other items. We performed an analysis of a large data set on free recall and found a highly specific pattern of statistical dependencies predicted by the model, in particular negative correlations between the number of words recalled and their average recall probability. Taken together, experimental and modeling results presented here reveal complex interactions between memory items during recall that severely constrain recall capacity.