To act upon the world, creatures must change continuous variables such as muscle length or chemical concentration. In contrast, decision making is an inherently discrete process, involving the selection among alternative courses of action. In this article, we consider the interface between the discrete and continuous processes that translate our decisions into movement in a Newtonian world—and how movement informs our decisions. We do so by appealing to active inference, with a special focus on the oculomotor system. Within this exemplar system, we argue that the superior colliculus is well placed to act as a discrete-continuous interface. Interestingly, when the neuronal computations within the superior colliculus are formulated in terms of active inference, we find that many aspects of its neuroanatomy emerge from the computations it must perform in this role.

We formulate an equivalence between machine learning and the formulation of statistical data assimilation as used widely in physical and biological sciences. The correspondence is that layer number in a feedforward artificial network setting is the analog of time in the data assimilation setting. This connection has been noted in the machine learning literature. We add a perspective that expands on how methods from statistical physics and aspects of Lagrangian and Hamiltonian dynamics play a role in how networks can be trained and designed. Within the discussion of this equivalence, we show that adding more layers (making the network deeper) is analogous to adding temporal resolution in a data assimilation framework. Extending this equivalence to recurrent networks is also discussed. We explore how one can find a candidate for the global minimum of the cost functions in the machine learning context using a method from data assimilation. Calculations on simple models from both sides of the equivalence are reported. Also discussed is a framework in which the time or layer label is taken to be continuous, providing a differential equation, the Euler-Lagrange equation and its boundary conditions, as a necessary condition for a minimum of the cost function. This shows that the problem being solved is a two-point boundary value problem familiar in the discussion of variational methods. The use of continuous layers is denoted “deepest learning.” These problems respect a symplectic symmetry in continuous layer phase space. Both Lagrangian versions and Hamiltonian versions of these problems are presented. Their well-studied implementation in a discrete time/layer, while respecting the symplectic structure, is addressed. The Hamiltonian version provides a direct rationale for backpropagation as a solution method for a certain two-point boundary value problem.

Researchers building spiking neural networks face the challenge of improving the biological plausibility of their model networks while maintaining the ability to quantitatively characterize network behavior. In this work, we extend the theory behind the neural engineering framework (NEF), a method of building spiking dynamical networks, to permit the use of a broad class of synapse models while maintaining prescribed dynamics up to a given order. This theory improves our understanding of how low-level synaptic properties alter the accuracy of high-level computations in spiking dynamical networks. For completeness, we provide characterizations for both continuous-time (i.e., analog) and discrete-time (i.e., digital) simulations. We demonstrate the utility of these extensions by mapping an optimal delay line onto various spiking dynamical networks using higher-order models of the synapse. We show that these networks nonlinearly encode rolling windows of input history, using a scale invariant representation, with accuracy depending on the frequency content of the input signal. Finally, we reveal that these methods provide a novel explanation of time cell responses during a delay task, which have been observed throughout hippocampus, striatum, and cortex.

Much attention has been paid to the question of how Bayesian integration of information could be implemented by a simple neural mechanism. We show that population vectors based on point-process inputs combine evidence in a form that closely resembles Bayesian inference, with each input spike carrying information about the tuning of the input neuron. We also show that population vectors can combine information relatively accurately in the presence of noisy synaptic encoding of tuning curves.

Knowledge of synaptic input is crucial for understanding synaptic integration and ultimately neural function. However, in vivo , the rates at which synaptic inputs arrive are high, so that it is typically impossible to detect single events. We show here that it is nevertheless possible to extract the properties of the events and, in particular, to extract the event rate, the synaptic time constants, and the properties of the event size distribution from in vivo voltage-clamp recordings. Applied to cerebellar interneurons, our method reveals that the synaptic input rate increases from 600 Hz during rest to 1000 Hz during locomotion, while the amplitude and shape of the synaptic events are unaffected by this state change. This method thus complements existing methods to measure neural function in vivo .

A primary goal of many neuroimaging studies that use magnetic resonance imaging (MRI) is to deduce the structure-function relationships in the human brain using data from the three major neuro-MRI modalities: high-resolution anatomical, diffusion tensor imaging, and functional MRI. To date, the general procedure for analyzing these data is to combine the results derived independently from each of these modalities. In this article, we develop a new theoretical and computational approach for combining these different MRI modalities into a powerful and versatile framework that combines our recently developed methods for morphological shape analysis and segmentation, simultaneous local diffusion estimation and global tractography, and nonlinear and nongaussian spatial-temporal activation pattern classification and ranking, as well as our fast and accurate approach for nonlinear registration between modalities. This joint analysis method is capable of extracting new levels of information that is not achievable from any of those single modalities alone. A theoretical probabilistic framework based on a reformulation of prior information and available interdependencies between modalities through a joint coupling matrix and an efficient computational implementation allows construction of quantitative functional, structural, and effective brain connectivity modes and parcellation. This new method provides an overall increase of resolution, accuracy, level of detail, and information content and has the potential to be instrumental in the clinical adaptation of neuro-MRI modalities, which, when jointly analyzed, provide a more comprehensive view of a subject’s structure-function relations, while the current standard, wherein single-modality methods are analyzed separately, leaves a critical gap in an integrated view of a subject’s neuorphysiological state. As one example of this increased sensitivity, we demonstrate that the jointly estimated structural and functional dependencies of mode power follow the same power law decay with the same exponent.

Controlling the flow and routing of data is a fundamental problem in many distributed networks, including transportation systems, integrated circuits, and the Internet. In the brain, synaptic plasticity rules have been discovered that regulate network activity in response to environmental inputs, which enable circuits to be stable yet flexible. Here, we develop a new neuro-inspired model for network flow control that depends only on modifying edge weights in an activity-dependent manner. We show how two fundamental plasticity rules, long-term potentiation and long-term depression, can be cast as a distributed gradient descent algorithm for regulating traffic flow in engineered networks. We then characterize, both by simulation and analytically, how different forms of edge-weight-update rules affect network routing efficiency and robustness. We find a close correspondence between certain classes of synaptic weight update rules derived experimentally in the brain and rules commonly used in engineering, suggesting common principles to both.

Many previous proposals for adversarial training of deep neural nets have included directly modifying the gradient, training on a mix of original and adversarial examples, using contractive penalties, and approximately optimizing constrained adversarial objective functions. In this article, we show that these proposals are actually all instances of optimizing a general, regularized objective we call DataGrad. Our proposed DataGrad framework, which can be viewed as a deep extension of the layerwise contractive autoencoder penalty, cleanly simplifies prior work and easily allows extensions such as adversarial training with multitask cues. In our experiments, we find that the deep gradient regularization of DataGrad (which also has L1 and L2 flavors of regularization) outperforms alternative forms of regularization, including classical L1, L2, and multitask, on both the original data set and adversarial sets. Furthermore, we find that combining multitask optimization with DataGrad adversarial training results in the most robust performance.

We show that Langevin Markov chain Monte Carlo inference in an energy-based model with latent variables has the property that the early steps of inference, starting from a stationary point, correspond to propagating error gradients into internal layers, similar to backpropagation. The backpropagated error is with respect to output units that have received an outside driving force pushing them away from the stationary point. Backpropagated error gradients correspond to temporal derivatives with respect to the activation of hidden units. These lead to a weight update proportional to the product of the presynaptic firing rate and the temporal rate of change of the postsynaptic firing rate. Simulations and a theoretical argument suggest that this rate-based update rule is consistent with those associated with spike-timing-dependent plasticity. The ideas presented in this article could be an element of a theory for explaining how brains perform credit assignment in deep hierarchies as efficiently as backpropagation does, with neural computation corresponding to both approximate inference in continuous-valued latent variables and error backpropagation, at the same time.

Networks have become instrumental in deciphering how information is processed and transferred within systems in almost every scientific field today. Nearly all network analyses, however, have relied on humans to devise structural features of networks believed to be most discriminative for an application. We present a framework for comparing and classifying networks without human-crafted features using deep learning. After training, autoencoders contain hidden units that encode a robust structural vocabulary for succinctly describing graphs. We use this feature vocabulary to tackle several network mining problems and find improved predictive performance versus many popular features used today. These problems include uncovering growth mechanisms driving the evolution of networks, predicting protein network fragility, and identifying environmental niches for metabolic networks. Deep learning offers a principled approach for mining complex networks and tackling graph-theoretic problems.

This article describes a process theory based on active inference and belief propagation. Starting from the premise that all neuronal processing (and action selection) can be explained by maximizing Bayesian model evidence—or minimizing variational free energy—we ask whether neuronal responses can be described as a gradient descent on variational free energy. Using a standard (Markov decision process) generative model, we derive the neuronal dynamics implicit in this description and reproduce a remarkable range of well-characterized neuronal phenomena. These include repetition suppression, mismatch negativity, violation responses, place-cell activity, phase precession, theta sequences, theta-gamma coupling, evidence accumulation, race-to-bound dynamics, and transfer of dopamine responses. Furthermore, the (approximately Bayes’ optimal) behavior prescribed by these dynamics has a degree of face validity, providing a formal explanation for reward seeking, context learning, and epistemic foraging. Technically, the fact that a gradient descent appears to be a valid description of neuronal activity means that variational free energy is a Lyapunov function for neuronal dynamics, which therefore conform to Hamilton’s principle of least action.

Linear-nonlinear (LN) models and their extensions have proven successful in describing transformations from stimuli to spiking responses of neurons in early stages of sensory hierarchies. Neural responses at later stages are highly nonlinear and have generally been better characterized in terms of their decoding performance on prespecified tasks. Here we develop a biologically plausible decoding model for classification tasks, that we refer to as neural quadratic discriminant analysis (nQDA). Specifically, we reformulate an optimal quadratic classifier as an LN-LN computation, analogous to “subunit” encoding models that have been used to describe responses in retina and primary visual cortex. We propose a physiological mechanism by which the parameters of the nQDA classifier could be optimized, using a supervised variant of a Hebbian learning rule. As an example of its applicability, we show that nQDA provides a better account than many comparable alternatives for the transformation between neural representations in two high-level brain areas recorded as monkeys performed a visual delayed-match-to-sample task

Neuromorphic engineering combines the architectural and computational principles of systems neuroscience with semiconductor electronics, with the aim of building efficient and compact devices that mimic the synaptic and neural machinery of the brain. The energy consumptions promised by neuromorphic engineering are extremely low, comparable to those of the nervous system. Until now, however, the neuromorphic approach has been restricted to relatively simple circuits and specialized functions, thereby obfuscating a direct comparison of their energy consumption to that used by conventional von Neumann digital machines solving real-world tasks. Here we show that a recent technology developed by IBM can be leveraged to realize neuromorphic circuits that operate as classifiers of complex real-world stimuli. Specifically, we provide a set of general prescriptions to enable the practical implementation of neural architectures that compete with state-of-the-art classifiers. We also show that the energy consumption of these architectures, realized on the IBM chip, is typically two or more orders of magnitude lower than that of conventional digital machines implementing classifiers with comparable performance. Moreover, the spike-based dynamics display a trade-off between integration time and accuracy, which naturally translates into algorithms that can be flexibly deployed for either fast and approximate classifications, or more accurate classifications at the mere expense of longer running times and higher energy costs. This work finally proves that the neuromorphic approach can be efficiently used in real-world applications and has significant advantages over conventional digital devices when energy consumption is considered.

Large-scale data collection efforts to map the brain are underway at multiple spatial and temporal scales, but all face fundamental problems posed by high-dimensional data and intersubject variability. Even seemingly simple problems, such as identifying a neuron/brain region across animals/subjects, become exponentially more difficult in high dimensions, such as recognizing dozens of neurons/brain regions simultaneously. We present a framework and tools for functional neurocartography—the large-scale mapping of neural activity during behavioral states. Using a voltage-sensitive dye (VSD), we imaged the multifunctional responses of hundreds of leech neurons during several behaviors to identify and functionally map homologous neurons. We extracted simple features from each of these behaviors and combined them with anatomical features to create a rich medium-dimensional feature space. This enabled us to use machine learning techniques and visualizations to characterize and account for intersubject variability, piece together a canonical atlas of neural activity, and identify two behavioral networks. We identified 39 neurons (18 pairs, 3 unpaired) as part of a canonical swim network and 17 neurons (8 pairs, 1 unpaired) involved in a partially overlapping preparatory network. All neurons in the preparatory network rapidly depolarized at the onsets of each behavior, suggesting that it is part of a dedicated rapid-response network. This network is likely mediated by the S cell, and we referenced VSD recordings to an activity atlas to identify multiple cells of interest simultaneously in real time for further experiments. We targeted and electrophysiologically verified several neurons in the swim network and further showed that the S cell is presynaptic to multiple neurons in the preparatory network. This study illustrates the basic framework to map neural activity in high dimensions with large-scale recordings and how to extract the rich information necessary to perform analyses in light of intersubject variability.

In this article, we deal with the problem of inferring causal directions when the data are on discrete domain. By considering the distribution of the cause and the conditional distribution mapping cause to effect as independent random variables, we propose to infer the causal direction by comparing the distance correlation between and with the distance correlation between and . We infer that X causes Y if the dependence coefficient between and is smaller. Experiments are performed to show the performance of the proposed method.

A neural network model is presented of novelty detection in the CA1 subdomain of the hippocampal formation from the perspective of information flow. This computational model is restricted on several levels by both anatomical information about hippocampal circuitry and behavioral data from studies done in rats. Several studies report that the CA1 area broadcasts a generalized novelty signal in response to changes in the environment. Using the neural engineering framework developed by Eliasmith et al., a spiking neural network architecture is created that is able to compare high-dimensional vectors, symbolizing semantic information, according to the semantic pointer hypothesis. This model then computes the similarity between the vectors, as both direct inputs and a recalled memory from a long-term memory network by performing the dot-product operation in a novelty neural network architecture. The developed CA1 model agrees with available neuroanatomical data, as well as the presented behavioral data, and so it is a biologically realistic model of novelty detection in the hippocampus, which can provide a feasible explanation for experimentally observed dynamics.

In many multivariate time series, the correlation structure is nonstationary, that is, it changes over time. The correlation structure may also change as a function of other cofactors, for example, the identity of the subject in biomedical data. A fundamental approach for the analysis of such data is to estimate the correlation structure (connectivities) separately in short time windows or for different subjects and use existing machine learning methods, such as principal component analysis (PCA), to summarize or visualize the changes in connectivity. However, the visualization of such a straightforward PCA is problematic because the ensuing connectivity patterns are much more complex objects than, say, spatial patterns. Here, we develop a new framework for analyzing variability in connectivities using the PCA approach as the starting point. First, we show how to analyze and visualize the principal components of connectivity matrices by a tailor-made rank-two matrix approximation in which we use the outer product of two orthogonal vectors. This leads to a new kind of transformation of eigenvectors that is particularly suited for this purpose and often enables interpretation of the principal component as connectivity between two groups of variables. Second, we show how to incorporate the orthogonality and the rank-two constraint in the estimation of PCA itself to improve the results. We further provide an interpretation of these methods in terms of estimation of a probabilistic generative model related to blind separation of dependent sources. Experiments on brain imaging data give very promising results.

Common representation learning (CRL), wherein different descriptions (or views) of the data are embedded in a common subspace, has been receiving a lot of attention recently. Two popular paradigms here are canonical correlation analysis (CCA)–based approaches and autoencoder (AE)–based approaches. CCA-based approaches learn a joint representation by maximizing correlation of the views when projected to the common subspace. AE-based methods learn a common representation by minimizing the error of reconstructing the two views. Each of these approaches has its own advantages and disadvantages. For example, while CCA-based approaches outperform AE-based approaches for the task of transfer learning, they are not as scalable as the latter. In this work, we propose an AE-based approach, correlational neural network (CorrNet), that explicitly maximizes correlation among the views when projected to the common subspace. Through a series of experiments, we demonstrate that the proposed CorrNet is better than AE and CCA with respect to its ability to learn correlated common representations. We employ CorrNet for several cross-language tasks and show that the representations learned using it perform better than the ones learned using other state-of-the-art approaches.

Inhibition-stabilized networks (ISNs) are neural architectures with strong positive feedback among pyramidal neurons balanced by strong negative feedback from inhibitory interneurons, a circuit element found in the hippocampus and the primary visual cortex. In their working regime, ISNs produce damped oscillations in the -range in response to inputs to the inhibitory population. In order to understand the properties of interconnected ISNs, we investigated periodic forcing of ISNs. We show that ISNs can be excited over a range of frequencies and derive properties of the resonance peaks. In particular, we studied the phase-locked solutions, the torus solutions, and the resonance peaks. Periodically forced ISNs respond with (possibly multistable) phase-locked activity, whereas networks with sustained intrinsic oscillations respond more dynamically to periodic inputs with tori. Hence, the dynamics are surprisingly rich, and phase effects alone do not adequately describe the network response. This strengthens the importance of phase-amplitude coupling as opposed to phase-phase coupling in providing multiple frequencies for multiplexing and routing information.

There has been significant research over the past two decades in developing new platforms for spiking neural computation. Current neural computers are primarily developed to mimic biology. They use neural networks, which can be trained to perform specific tasks to mainly solve pattern recognition problems. These machines can do more than simulate biology; they allow us to rethink our current paradigm of computation. The ultimate goal is to develop brain-inspired general purpose computation architectures that can breach the current bottleneck introduced by the von Neumann architecture. This work proposes a new framework for such a machine. We show that the use of neuron-like units with precise timing representation, synaptic diversity, and temporal delays allows us to set a complete, scalable compact computation framework. The framework provides both linear and nonlinear operations, allowing us to represent and solve any function. We show usability in solving real use cases from simple differential equations to sets of nonlinear differential equations leading to chaotic attractors.