Computational aspects of feedback in neural circuits

It has previously been shown that generic cortical microcircuit models can perform complex real-time computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate the computational capability of such circuits in the more realistic case where not only readout neurons, but in addition a few neurons within the circuit, have been trained for specific tasks. This is essentially equivalent to the case where the output of trained readout neurons is fed back into the circuit. We show that this new model overcomes the limitation of a rapidly fading memory. In fact, we prove that in the idealized case without noise it can carry out any conceivable digital or analog computation on time-varying inputs. But even with noise, the resulting computational model can perform a large class of biologically relevant real-time computations that require a nonfading memory. We demonstrate these computational implications of feedback both theoretically, and through computer simulations of detailed cortical microcircuit models that are subject to noise and have complex inherent dynamics. We show that the application of simple learning procedures (such as linear regression or perceptron learning) to a few neurons enables such circuits to represent time over behaviorally relevant long time spans, to integrate evidence from incoming spike trains over longer periods of time, and to process new information contained in such spike trains in diverse ways according to the current internal state of the circuit. In particular we show that such generic cortical microcircuits with feedback provide a new model for working memory that is consistent with a large set of biological constraints. Although this article examines primarily the computational role of feedback in circuits of neurons, the mathematical principles on which its analysis is based apply to a variety of dynamical systems. Hence they may also throw new light on the computational role of feedback in other complex biological dynamical systems, such as, for example, genetic regulatory networks.     

Probabilistic inference in general graphical models through sampling in stochastic networks of spiking neurons

An important open problem of computational neuroscience is the generic organization of computations in networks of neurons in the brain. We show here through rigorous theoretical analysis that inherent stochastic features of spiking neurons, in combination with simple nonlinear computational operations in specific network motifs and dendritic arbors, enable networks of spiking neurons to carry out probabilistic inference through sampling in general graphical models. In particular, it enables them to carry out probabilistic inference in Bayesian networks with converging arrows ("explaining away") and with undirected loops, that occur in many real-world tasks. Ubiquitous stochastic features of networks of spiking neurons, such as trial-to-trial variability and spontaneous activity, are necessary ingredients of the underlying computational organization. We demonstrate through computer simulations that this approach can be scaled up to neural emulations of probabilistic inference in fairly large graphical models, yielding some of the most complex computations that have been carried out so far in networks of spiking neurons.     

Neural dynamics as sampling: a model for stochastic computation in recurrent networks of spiking neurons

The organization of computations in networks of spiking neurons in the brain is still largely unknown, in particular in view of the inherently stochastic features of their firing activity and the experimentally observed trial-to-trial variability of neural systems in the brain. In principle there exists a powerful computational framework for stochastic computations, probabilistic inference by sampling, which can explain a large number of macroscopic experimental data in neuroscience and cognitive science. But it has turned out to be surprisingly difficult to create a link between these abstract models for stochastic computations and more detailed models of the dynamics of networks of spiking neurons. Here we create such a link and show that under some conditions the stochastic firing activity of networks of spiking neurons can be interpreted as probabilistic inference via Markov chain Monte Carlo (MCMC) sampling. Since common methods for MCMC sampling in distributed systems, such as Gibbs sampling, are inconsistent with the dynamics of spiking neurons, we introduce a different approach based on non-reversible Markov chains that is able to reflect inherent temporal processes of spiking neuronal activity through a suitable choice of random variables. We propose a neural network model and show by a rigorous theoretical analysis that its neural activity implements MCMC sampling of a given distribution, both for the case of discrete and continuous time. This provides a step towards closing the gap between abstract functional models of cortical computation and more detailed models of networks of spiking neurons.     

Ensembles of Spiking Neurons with Noise Support Optimal Probabilistic Inference in a Dynamically Changing Environment

It has recently been shown that networks of spiking neurons with noise can emulate simple forms of probabilistic inference through “neural sampling”, i.e., by treating spikes as samples from a probability distribution of network states that is encoded in the network. Deficiencies of the existing model are its reliance on single neurons for sampling from each random variable, and the resulting limitation in representing quickly varying probabilistic information. We show that both deficiencies can be overcome by moving to a biologically more realistic encoding of each salient random variable through the stochastic firing activity of an ensemble of neurons. The resulting model demonstrates that networks of spiking neurons with noise can easily track and carry out basic computational operations on rapidly varying probability distributions, such as the odds of getting rewarded for a specific behavior. We demonstrate the viability of this new approach towards neural coding and computation, which makes use of the inherent parallelism of generic neural circuits, by showing that this model can explain experimentally observed firing activity of cortical neurons for a variety of tasks that require rapid temporal integration of sensory information.     

Integrated mechanisms of anticipation and rate-of-change computations in cortical circuits

Local neocortical circuits are characterized by stereotypical physiological and structural features that subserve generic computational operations. These basic computations of the cortical microcircuit emerge through the interplay of neuronal connectivity, cellular intrinsic properties, and synaptic plasticity dynamics. How these interacting mechanisms generate specific computational operations in the cortical circuit remains largely unknown. Here, we identify the neurophysiological basis of both the rate of change and anticipation computations on synaptic inputs in a cortical circuit. Through biophysically realistic computer simulations and neuronal recordings, we show that the rate-of-change computation is operated robustly in cortical networks through the combination of two ubiquitous brain mechanisms: short-term synaptic depression and spike-frequency adaptation. We then show how this rate-of-change circuit can be embedded in a convergently connected network to anticipate temporally incoming synaptic inputs, in quantitative agreement with experimental findings on anticipatory responses to moving stimuli in the primary visual cortex. Given the robustness of the mechanism and the widespread nature of the physiological machinery involved, we suggest that rate-of-change computation and temporal anticipation are principal, hard-wired functions of neural information processing in the cortical microcircuit.     

Learning Universal Computations with Spikes

Providing the neurobiological basis of information processing in higher animals, spiking neural networks must be able to learn a variety of complicated computations, including the generation of appropriate, possibly delayed reactions to inputs and the self-sustained generation of complex activity patterns, e.g. for locomotion. Many such computations require previous building of intrinsic world models. Here we show how spiking neural networks may solve these different tasks. Firstly, we derive constraints under which classes of spiking neural networks lend themselves to substrates of powerful general purpose computing. The networks contain dendritic or synaptic nonlinearities and have a constrained connectivity. We then combine such networks with learning rules for outputs or recurrent connections. We show that this allows to learn even difficult benchmark tasks such as the self-sustained generation of desired low-dimensional chaotic dynamics or memory-dependent computations. Furthermore, we show how spiking networks can build models of external world systems and use the acquired knowledge to control them.     

Bursting dynamics remarkably improve the performance of neural networks on liquid computing

Burst firings are functionally important behaviors displayed by neural circuits, which plays a primary role in reliable transmission of electrical signals for neuronal communication. However, with respect to the computational capability of neural networks, most of relevant studies are based on the spiking dynamics of individual neurons, while burst firing is seldom considered. In this paper, we carry out a comprehensive study to compare the performance of spiking and bursting dynamics on the capability of liquid computing, which is an effective approach for intelligent computation of neural networks. The results show that neural networks with bursting dynamic have much better computational performance than those with spiking dynamics, especially for complex computational tasks. Further analysis demonstrate that the fast firing pattern of bursting dynamics can obviously enhance the efficiency of synaptic integration from pre-neurons both temporally and spatially. This indicates that bursting dynamic can significantly enhance the complexity of network activity, implying its high efficiency in information processing.     

Training Excitatory-Inhibitory Recurrent Neural Networks for Cognitive Tasks: A Simple and Flexible Framework

The ability to simultaneously record from large numbers of neurons in behaving animals has ushered in a new era for the study of the neural circuit mechanisms underlying cognitive functions. One promising approach to uncovering the dynamical and computational principles governing population responses is to analyze model recurrent neural networks (RNNs) that have been optimized to perform the same tasks as behaving animals. Because the optimization of network parameters specifies the desired output but not the manner in which to achieve this output, “trained” networks serve as a source of mechanistic hypotheses and a testing ground for data analyses that link neural computation to behavior. Complete access to the activity and connectivity of the circuit, and the ability to manipulate them arbitrarily, make trained networks a convenient proxy for biological circuits and a valuable platform for theoretical investigation. However, existing RNNs lack basic biological features such as the distinction between excitatory and inhibitory units (Dale’s principle), which are essential if RNNs are to provide insights into the operation of biological circuits. Moreover, trained networks can achieve the same behavioral performance but differ substantially in their structure and dynamics, highlighting the need for a simple and flexible framework for the exploratory training of RNNs. Here, we describe a framework for gradient descent-based training of excitatory-inhibitory RNNs that can incorporate a variety of biological knowledge. We provide an implementation based on the machine learning library Theano, whose automatic differentiation capabilities facilitate modifications and extensions. We validate this framework by applying it to well-known experimental paradigms such as perceptual decision-making, context-dependent integration, multisensory integration, parametric working memory, and motor sequence generation. Our results demonstrate the wide range of neural activity patterns and behavior that can be modeled, and suggest a unified setting in which diverse cognitive computations and mechanisms can be studied.     

A Spiking Neural Network Model of Model-Free Reinforcement Learning with High-Dimensional Sensory Input and Perceptual Ambiguity

A theoretical framework of reinforcement learning plays an important role in understanding action selection in animals. Spiking neural networks provide a theoretically grounded means to test computational hypotheses on neurally plausible algorithms of reinforcement learning through numerical simulation. However, most of these models cannot handle observations which are noisy, or occurred in the past, even though these are inevitable and constraining features of learning in real environments. This class of problem is formally known as partially observable reinforcement learning (PORL) problems. It provides a generalization of reinforcement learning to partially observable domains. In addition, observations in the real world tend to be rich and high-dimensional. In this work, we use a spiking neural network model to approximate the free energy of a restricted Boltzmann machine and apply it to the solution of PORL problems with high-dimensional observations. Our spiking network model solves maze tasks with perceptually ambiguous high-dimensional observations without knowledge of the true environment. An extended model with working memory also solves history-dependent tasks. The way spiking neural networks handle PORL problems may provide a glimpse into the underlying laws of neural information processing which can only be discovered through such a top-down approach.     

Timing as an intrinsic property of neural networks: evidence from in vivo and in vitro experiments

The discrimination and production of temporal patterns on the scale of hundreds of milliseconds are critical to sensory and motor processing. Indeed, most complex behaviours, such as speech comprehension and production, would be impossible in the absence of sophisticated timing mechanisms. Despite the importance of timing to human learning and cognition, little is known about the underlying mechanisms, in particular whether timing relies on specialized dedicated circuits and mechanisms or on general and intrinsic properties of neurons and neural circuits. Here, we review experimental data describing timing and interval-selective neurons in vivo and in vitro. We also review theoretical models of timing, focusing primarily on the state-dependent network model, which proposes that timing in the subsecond range relies on the inherent time-dependent properties of neurons and the active neural dynamics within recurrent circuits. Within this framework, time is naturally encoded in populations of neurons whose pattern of activity is dynamically changing in time. Together, we argue that current experimental and theoretical studies provide sufficient evidence to conclude that at least some forms of temporal processing reflect intrinsic computations based on local neural network dynamics.     

A network of spiking neurons that can represent interval timing: mean field analysis

Despite the vital importance of our ability to accurately process and encode temporal information, the underlying neural mechanisms are largely unknown. We have previously described a theoretical framework that explains how temporal representations, similar to those reported in the visual cortex, can form in locally recurrent cortical networks as a function of reward modulated synaptic plasticity. This framework allows networks of both linear and spiking neurons to learn the temporal interval between a stimulus and paired reward signal presented during training. Here we use a mean field approach to analyze the dynamics of non-linear stochastic spiking neurons in a network trained to encode specific time intervals. This analysis explains how recurrent excitatory feedback allows a network structure to encode temporal representations.     

Stochastic Computations in Cortical Microcircuit Models

Experimental data from neuroscience suggest that a substantial amount of knowledge is stored in the brain in the form of probability distributions over network states and trajectories of network states. We provide a theoretical foundation for this hypothesis by showing that even very detailed models for cortical microcircuits, with data-based diverse nonlinear neurons and synapses, have a stationary distribution of network states and trajectories of network states to which they converge exponentially fast from any initial state. We demonstrate that this convergence holds in spite of the non-reversibility of the stochastic dynamics of cortical microcircuits. We further show that, in the presence of background network oscillations, separate stationary distributions emerge for different phases of the oscillation, in accordance with experimentally reported phase-specific codes. We complement these theoretical results by computer simulations that investigate resulting computation times for typical probabilistic inference tasks on these internally stored distributions, such as marginalization or marginal maximum-a-posteriori estimation. Furthermore, we show that the inherent stochastic dynamics of generic cortical microcircuits enables them to quickly generate approximate solutions to difficult constraint satisfaction problems, where stored knowledge and current inputs jointly constrain possible solutions. This provides a powerful new computing paradigm for networks of spiking neurons, that also throws new light on how networks of neurons in the brain could carry out complex computational tasks such as prediction, imagination, memory recall and problem solving.     

Basic mechanisms for graded persistent activity: discrete attractors,continuous attractors,and dynamic representations

Persistent neural activity is observed in many systems, and is thought to be a neural substrate for holding memories over time delays of a few seconds. Recent work has addressed two issues. First, how can networks of neurons robustly hold such an active memory? Computer systems obtain significant robustness to noise by approximating analogue quantities with discrete digital representations. In a similar manner, theoretical models of persistent activity in spiking neurons have shown that the most robust and stable way to store the short-term memory of a continuous parameter is to approximate it with a discrete representation. This general idea applies very broadly to mechanisms that range from biochemical networks to single cells and to large circuits of neurons. Second, why is it commonly observed that persistent activity in the cortex can be strongly time-varying? This observation is almost ubiquitous, and therefore must be taken into account in our models and our understanding of how short-term memories are held in the cortex.     

Optimizing interneuron circuits for compartment-specific feedback inhibition

Cortical circuits process information by rich recurrent interactions between excitatory neurons and inhibitory interneurons. One of the prime functions of interneurons is to stabilize the circuit by feedback inhibition, but the level of specificity on which inhibitory feedback operates is not fully resolved. We hypothesized that inhibitory circuits could enable separate feedback control loops for different synaptic input streams, by means of specific feedback inhibition to different neuronal compartments. To investigate this hypothesis, we adopted an optimization approach. Leveraging recent advances in training spiking network models, we optimized the connectivity and short-term plasticity of interneuron circuits for compartment-specific feedback inhibition onto pyramidal neurons. Over the course of the optimization, the interneurons diversified into two classes that resembled parvalbumin (PV) and somatostatin (SST) expressing interneurons. Using simulations and mathematical analyses, we show that the resulting circuit can be understood as a neural decoder that inverts the nonlinear biophysical computations performed within the pyramidal cells. Our model provides a proof of concept for studying structure-function relations in cortical circuits by a combination of gradient-based optimization and biologically plausible phenomenological models.     

Connectomic constraints on computation in feedforward networks of spiking neurons

Several efforts are currently underway to decipher the connectome or parts thereof in a variety of organisms. Ascertaining the detailed physiological properties of all the neurons in these connectomes, however, is out of the scope of such projects. It is therefore unclear to what extent knowledge of the connectome alone will advance a mechanistic understanding of computation occurring in these neural circuits, especially when the high-level function of the said circuit is unknown. We consider, here, the question of how the wiring diagram of neurons imposes constraints on what neural circuits can compute, when we cannot assume detailed information on the physiological response properties of the neurons. We call such constraints—that arise by virtue of the connectome—connectomic constraints on computation. For feedforward networks equipped with neurons that obey a deterministic spiking neuron model which satisfies a small number of properties, we ask if just by knowing the architecture of a network, we can rule out computations that it could be doing, no matter what response properties each of its neurons may have. We show results of this form, for certain classes of network architectures. On the other hand, we also prove that with the limited set of properties assumed for our model neurons, there are fundamental limits to the constraints imposed by network structure. Thus, our theory suggests that while connectomic constraints might restrict the computational ability of certain classes of network architectures, we may require more elaborate information on the properties of neurons in the network, before we can discern such results for other classes of networks.     

Spike-based population coding and working memory

Compelling behavioral evidence suggests that humans can make optimal decisions despite the uncertainty inherent in perceptual or motor tasks. A key question in neuroscience is how populations of spiking neurons can implement such probabilistic computations. In this article, we develop a comprehensive framework for optimal, spike-based sensory integration and working memory in a dynamic environment. We propose that probability distributions are inferred spike-per-spike in recurrently connected networks of integrate-and-fire neurons. As a result, these networks can combine sensory cues optimally, track the state of a time-varying stimulus and memorize accumulated evidence over periods much longer than the time constant of single neurons. Importantly, we propose that population responses and persistent working memory states represent entire probability distributions and not only single stimulus values. These memories are reflected by sustained, asynchronous patterns of activity which make relevant information available to downstream neurons within their short time window of integration. Model neurons act as predictive encoders, only firing spikes which account for new information that has not yet been signaled. Thus, spike times signal deterministically a prediction error, contrary to rate codes in which spike times are considered to be random samples of an underlying firing rate. As a consequence of this coding scheme, a multitude of spike patterns can reliably encode the same information. This results in weakly correlated, Poisson-like spike trains that are sensitive to initial conditions but robust to even high levels of external neural noise. This spike train variability reproduces the one observed in cortical sensory spike trains, but cannot be equated to noise. On the contrary, it is a consequence of optimal spike-based inference. In contrast, we show that rate-based models perform poorly when implemented with stochastically spiking neurons.     

Driving reservoir models with oscillations: a solution to the extreme structural sensitivity of chaotic networks

A large body of experimental and theoretical work on neural coding suggests that the information stored in brain circuits is represented by time-varying patterns of neural activity. Reservoir computing, where the activity of a recurrently connected pool of neurons is read by one or more units that provide an output response, successfully exploits this type of neural activity. However, the question of system robustness to small structural perturbations, such as failing neurons and synapses, has been largely overlooked. This contrasts with well-studied dynamical perturbations that lead to divergent network activity in the presence of chaos, as is the case for many reservoir networks. Here, we distinguish between two types of structural network perturbations, namely local (e.g., individual synaptic or neuronal failure) and global (e.g., network-wide fluctuations). Surprisingly, we show that while global perturbations have a limited impact on the ability of reservoir models to perform various tasks, local perturbations can produce drastic effects. To address this limitation, we introduce a new architecture where the reservoir is driven by a layer of oscillators that generate stable and repeatable trajectories. This model outperforms previous implementations while being resistant to relatively large local and global perturbations. This finding has implications for the design of reservoir models that capture the capacity of brain circuits to perform cognitively and behaviorally relevant tasks while remaining robust to various forms of perturbations. Further, our work proposes a novel role for neuronal oscillations found in cortical circuits, where they may serve as a collection of inputs from which a network can robustly generate complex dynamics and implement rich computations.     

Neural circuit models for computations in early visual cortex

Some cortical circuit models study the mechanisms of the transforms from visual inputs to neural responses. They model neural properties such as feature tunings, pattern sensitivities, and how they depend on intracortical connections and contextual inputs. Other cortical circuit models are more concerned with computational goals of the transform from visual inputs to neural responses, or the roles of the neural responses in the visual behavior. The appropriate complexity of a cortical circuit model depends on the question asked. Modeling neural circuits of many interacting hypercolumns is a necessary challenge, which is providing insights to cortical computations, such as visual saliency computation, and linking physiology with global visual cognitive behavior such as bottom-up attentional selection.     

Encoding time in neural dynamic regimes with distinct computational tradeoffs

Converging evidence suggests the brain encodes time in dynamic patterns of neural activity, including neural sequences, ramping activity, and complex dynamics. Most temporal tasks, however, require more than just encoding time, and can have distinct computational requirements including the need to exhibit temporal scaling, generalize to novel contexts, or robustness to noise. It is not known how neural circuits can encode time and satisfy distinct computational requirements, nor is it known whether similar patterns of neural activity at the population level can exhibit dramatically different computational or generalization properties. To begin to answer these questions, we trained RNNs on two timing tasks based on behavioral studies. The tasks had different input structures but required producing identically timed output patterns. Using a novel framework we quantified whether RNNs encoded two intervals using either of three different timing strategies: scaling, absolute, or stimulus-specific dynamics. We found that similar neural dynamic patterns at the level of single intervals, could exhibit fundamentally different properties, including, generalization, the connectivity structure of the trained networks, and the contribution of excitatory and inhibitory neurons. Critically, depending on the task structure RNNs were better suited for generalization or robustness to noise. Further analysis revealed different connection patterns underlying the different regimes. Our results predict that apparently similar neural dynamic patterns at the population level (e.g., neural sequences) can exhibit fundamentally different computational properties in regards to their ability to generalize to novel stimuli and their robustness to noise—and that these differences are associated with differences in network connectivity and distinct contributions of excitatory and inhibitory neurons. We also predict that the task structure used in different experimental studies accounts for some of the experimentally observed variability in how networks encode time.