Humans use log-compressed number lines to represent different quantities, including elapsed time, traveled distance, numerosity, sound frequency, etc. Inspired by recent cognitive science and computational neuroscience work, we developed a neural network that learns to construct log-compressed number lines. The network computes a discrete approximation of a real-domain Laplace transform using an RNN with analytically derived weights giving rise to a log-compressed timeline of the past. The network learns to extract latent variables from the input and uses them for global modulation of the recurrent weights turning a timeline into a number line over relevant dimensions. The number line representation greatly simplifies learning on a set of problems that require learning associations in different spaces - problems that humans can typically solve easily. This approach illustrates how combining deep learning with cognitive models can result in systems that learn to represent latent variables in a brain-like manner and exhibit human-like behavior manifested through Weber-Fechner law.

Human learners can readily understand speech, or a melody, when it is presented slower or faster than usual. This paper presents a deep CNN (SITHCon) that uses a logarithmically compressed temporal representation at each level. Because rescaling the time of the input results in a translation of $\log$ time, and because the output of the convolution is invariant to translations, this network can generalize to out-of-sample data that are temporal rescalings of a learned pattern. We compare the performance of SITHCon to a Temporal Convolution Network (TCN) on classification and regression problems with both univariate and multivariate time series. We find that SITHCon, unlike TCN, generalizes robustly over rescalings of about an order of magnitude. Moreover, we show that the network can generalize over exponentially large scales without retraining the weights simply by extending the range of the logarithmically-compressed temporal memory.

Extracting temporal relationships over a range of scales is a hallmark ofhuman perception and cognition---and thus it is a critical feature of machinelearning applied to real-world problems. Neural networks are either plaguedby the exploding/vanishing gradient problem in recurrent neural networks(RNNs) or must adjust their parameters to learn the relevant time scales(e.g., in LSTMs). This paper introduces DeepSITH, a deep network comprisingbiologically-inspired Scale-Invariant Temporal History (SITH) modules inseries with dense connections between layers. Each SITH module is simply aset of time cells coding what happened when with a geometrically-spaced set oftime lags. The dense connections between layers change the definition of whatfrom one layer to the next. The geometric series of time lags implies thatthe network codes time on a logarithmic scale, enabling DeepSITH network tolearn problems requiring memory over a wide range of time scales. We compareDeepSITH to LSTMs and other recent RNNs on several time series prediction anddecoding tasks. DeepSITH achieves results comparable to state-of-the-artperformance on these problems and continues to perform well even as the delaysare increased.

Natural learners must compute an estimate of future outcomes that follow from a stimulus in continuous time. Widely used reinforcement learning algorithms discretize continuous time and estimate either transition functions from one step to the next (model-based algorithms) or a scalar value of exponentially discounted future reward using the Bellman equation (model-free algorithms). An important drawback of model-based algorithms is that computational cost grows linearly with the amount of time to be simulated. An important drawback of model-free algorithms is the need to select a timescale required for exponential discounting. We present a computational mechanism, developed based on work in psychology and neuroscience, for computing a scale-invariant timeline of future outcomes. This mechanism efficiently computes an estimate of inputs as a function of future time on a logarithmically compressed scale and can be used to generate a scale-invariant power-law-discounted estimate of expected future reward. The representation of future time retains information about what will happen when. The entire timeline can be constructed in a single parallel operation that generates concrete behavioral and neural predictions. This computational mechanism could be incorporated into future reinforcement learning algorithms.

The successor representation was introduced into reinforcement learning by Dayan ( 1993 ) as a means of facilitating generalization between states with similar successors. Although reinforcement learning in general has been used extensively as a model of psychological and neural processes, the psychological validity of the successor representation has yet to be explored. An interesting possibility is that the successor representation can be used not only for reinforcement learning but for episodic learning as well. Our main contribution is to show that a variant of the temporal context model (TCM; Howard & Kahana, 2002 ), an influential model of episodic memory, can be understood as directly estimating the successor representation using the temporal difference learning algorithm (Sutton & Barto, 1998 ). This insight leads to a generalization of TCM and new experimental predictions. In addition to casting a new normative light on TCM, this equivalence suggests a previously unexplored point of contact between different learning systems.

We develop a probabilistic model of human memory performance in free recall experiments. In these experiments, a subject first studies a list of words and then tries to recall them. To model these data, we draw on both previous psychological research and statistical topic models of text documents. We assume that memories are formed by assimilating the semantic meaning of studied words (represented as a distribution over topics) into a slowly changing latent context (represented in the same space). During recall, this context is reinstated and used as a cue for retrieving studied words. By conceptualizing memory retrieval as a dynamic latent variable model, we are able to use Bayesian inference to represent uncertainty and reason about the cognitive processes underlying memory. We present a particle filter algorithm for performing approximate posterior inference, and evaluate our model on the prediction of recalled words in experimental data. By specifying the model hierarchically, we are also able to capture inter-subject variability.