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The Introduction to the Theory of Neural Computation by Hertz, Krogh and Palmer ( The book fulfills its mission as an introduction for neural network novices. Introduction to Neural Networks. R. Beale & T. Jackson IOP Publishing, Former recommended book. An Introduction to the Theory of. Neural Computation. Introduction to the Theory of Neural Computation. American Journal of Physics 62, (); · John Hertz, Anders Krogh.

Experimental work supports this idea.

Highly distributed representations of tactile information have been described in the cortex Nicolelis et al. In the visual cortex, sensory stimuli recruit intrinsically generated cortical ensembles Miller et al.

Representation of motor programs via cell ensembles has been described Hommel, Moreover, the auditory cortex is dominated by broad scale dynamics in which a complete representation of sounds emerges only at a global scale Bathellier et al.

Accordingly, multineuronal recording studies Fujisawa et al. Therefore, they may be the building blocks used in cortical processing. However, important questions concerning integration of cortical activity remain unresolved. It is still unknown how these ensembles are present in the cortex and how spatially distributed cells can functionally contribute to unified stimulus codification.

Moreover, although cell ensembles have long been thought to be associated with brain rhythms Harris et al. Rhythmic Neuronal Activity in the Brain Excitation and inhibition play a key role in the generation of rhythmic activity Steriade et al. However, the mechanisms underlying oscillations and synchrony are still not well understood. Furthermore, current theories about their computational role are incomplete Thiele and Stoner, ; Roelfsema et al.

It remains unclear what function they play in neural processing.

They may contribute to discretize the computation. Discrete Neural Computation The brain receives a constant flow of analog sensory information from the environment. Successful interaction with the world depends on accurate processing of that information.

Therefore, the challenging task that the brain faces is to rapidly extract changing relevant features from the environment in order to respond adequately. It requires a precise detection of changes sampling the flow of sensory information to compare its contents. Consequently, a primary function of the brain could be to discretize the continuous flow of information, compare those sampled units and extract relevant information from that computation Figure 1.

In sum, although the brain receives a constant flow of analog sensory information from the environment, sensory stimuli changes will be sampled in a discrete manner. Discrete neural processing.

Successful interaction with the world depends on accurate processing of rapidly changing stimuli. Of particular interest are relevant events, such as the appearance of a predator.

15-386 Neural Computation

Discrete samples are taken separately by brief temporal periods. That computation allows the prey to know if the predator is approaching or leaving and to take appropriate action. Scientists have long theorized that our cognition operates discontinuously within a framework of discrete cycles Pitts and McCulloch, ; Harter, ; Allport, ; Varela et al.

Accordingly, neural systems could undergo oscillatory activity patterns.

These oscillations could divide the neural processing into a series of discrete computational events. There are relevant experimental data demonstrating this discrete processing. Discrete computations are well described in visual perception VanRullen et al.

One example is microsaccadic eye movements by which the visual system acquires fine spatial detail Ko et al. Accordingly, vision is interrupted and sensory processing is discretized, separating it into distinct epochs. Recent evidence has been shown for discrete perceptual sampling in the somatosensory domain Baumgarten et al. They demonstrated that somatosensory perception operates in a discrete mode, with sensory input being sampled by discrete perceptual cycles.

Hypothesis and Theory ARTICLE

Memory Lundqvist et al. It is known that oscillatory neuronal activity in the frontal eye field reflects the successive cycles of a sequential attentional exploration process during visual search Buschman and Miller, However, how the brain, especially the cerebral cortex, performs this computation is unknown.

Cortical Neural Computation by Discrete Results Hypothesis: Functional Spatio-Temporal Units of Computation One of the most challenging problems we face in neuroscience is to understand how the cortex performs computations. Here we suggest that complex neural computations underlying cortical processing could be temporally discrete. But how does the cortex perform this computation? We propose that cortical processing is produced by the computation of discrete emergent functional units that we have called Discrete Results Discrete Results Hypothesis.

As we describe in the next sections, this novel concept has the ability to match the spatial and temporal aspects of cortical processing.

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We discuss the possible neural underpinnings of these spatio-temporal computational units and describe the empirical evidence supporting our hypothesis. Our Discrete Results Hypothesis suggests that the computational principle of the cortex lies in the precise temporal coordination of spikes of spatially distributed neurons. It is necessary to divide the temporal and spatial dimension of that proposal for a better clarification.

However, it is unclear how they emerge, with which neurons, what and how the relation is between the members, what spatial and temporal extension they have and what exactly an ensemble functionally means. All PCs organized by that particular synchronized inhibitory network form part of that Ensemble.

It means that the Ensemble is formed by all the PCs whose firing could be transiently constrained by that specific synchronized inhibitory network Figure 2.

The members and spatial extension of the Ensemble is determined by that inhibitory network. Moreover, individual PCs could participate in different emergent Ensembles. Spatial functional units of cortical processing.

All PCs organized by that particular inhibitory network form part of that Ensemble. Individual PCs could participate in different Ensembles. The members and spatial extension of the Ensemble is determined by the inhibitory network. Red lines represent mutual connections between FS inhibitory cells in the network.

These neurons innervate strategically the PCs blue lines extending a blanket of precise inhibition onto them. These emergent clusters of PCs form functional spatial units of cortical computation. However, that spatial aspect must be complemented with a temporal one.

Experimental evidence indicates that the exact time point at which a spike occurs plays an important role in information processing Markram et al. Moreover, the precise timing of neuronal spiking is vital for coding of information Singer and Gray, Therefore, this temporal precision is likely to be crucial for cortical computation. However, the functional significance remains unclear. The Discrete Results hypothesis suggests that cortical processing is produced by a highly ordered temporal organization.

Spike timing of PCs in a particular Ensemble is constrained by an inhibitory network generating a precise structured firing. We defined it as the accurate spike timing organization resulting from the precise temporal suppression of PCs spikes in the Ensemble.

Spikes from PCs occur independently but organized inside the temporal structure.

We propose that this Temporal Structure of Spikes is very important in the processing, coding and transfer of information in the cerebral cortex. The temporally structured firing activity enables information to be processed and coded in a way that downstream networks can compute.

Accordingly, the existence of this specific temporal structure implies that failures in that precision of spikes will result in processing dysfunctions. An example of the importance of the temporal structure is the spike-timing-dependent plasticity Caporale and Dan, This temporal structure is not fixed.

It can be dynamically adjusted for example by sensory input or by top-down influence to meet the finest processing resolution depending on perceptual, task or attentional demands. Furthermore, the structure could be adjusted by neuromodulators.

Accordingly, variations in the temporal structure will produce changes in the rate and temporal precision of PCs firing in the Ensemble. PCs spikes latencies and synchronization between them will vary accordingly. Consequently, changes in that temporal precision will code different content.

Temporal structure of spikes.

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Spatiotemporal dynamics of continuum neural fields. Journal of Physics A: Mathematical and Theoretical, 45 3 , Statistische Mechanik. Cambridge University Press, Cambridge, new. Mathematical Foundations of Neuroscience -. Isaacson and Massimo Scanziani.

How inhibition shapes cortical activity. Neuron, 72 2 , Instantaneous correlation of excitation and inhibition during ongoing and sensory-evoked activities. Nature neuroscience, 11 5 , The variable discharge of cortical neurons: implications for connectivity, computation, and information coding.

The Journal of neuroscience : the official journal of the Society for Neuroscience, 18 10 , Chaotic balanced state in a model of cortical circuits. Neural computation, 10 6 , Chaos in neuronal networks with balanced excitatory and inhibitory activity.

Science New York, N. An Introduction to the Ising Model. Rather, this book is invaluable for the student who wishes to begin a rigorous study of neural networks, or the researcher who wishes to have a solid understanding of the properties and limitations of the main classes of neural networks. As such, ITNC is ideal for a graduate or advanced undergraduate introductory course on neural networks.

Neural networks are studied primarily for one of two purposes: Models that fall under the applied category we use this label loosely are the focus of the text by Hertz, Krogh, and Palmer, and the subject of our review. In the context of applied neural networks, the last decade has been dominated by two classes of models: These two classes of models differ in their properties and applications.

The roots of the differences between these and other models can frequently be traced to the different backgrounds of the scientists who developed them, who typically came either from engineering or from physics. As it is, the text is well suited for a onesemester course focusing on perceptrons and associators.

Instructors interested in covering additional topics in more detail would have to supplement this book with other texts or articles. Given its textbook style, the book could be made more useful by including exercises, and perhaps by including special sections on how one might program some of the models described in the text. Perhaps the authors will consider these possibilities if they decide to prepare a further edition of their book. An extended review Weigend, discusses more recent developments in supervised feedforward neural networks, some of which are part of the helpful trend towards analyzing the networks as a class of statistical likelihood models.

We conclude our review with a brief summary of the contents of each chapter. Chapter 1 provides a brief review of the main characteristics of neurons in the brain, and how these might be related to artificial neurons.

After a brief review of the history of research in neural networks, the chapter concludes with a discussion of some important practical issues regarding neural networks research.

Chapter 2 outlines the problem of associative memory describes the binary, discrete symmetric autoassociator as described by Hopfield, and analyzes its storage capacity.

This chapter includes an illuminatingdescription of the similarity between the simple networks of McCulloch and Pitts, and the Ising spin model from statistical mechanics. The chapter summarizes in simple terms how mean field theory can be used to analyze the behavior of a collection of simple, interacting elements, be they atoms in a lattice-like material or binary neurons in a fully connected network. Chapter 3 describes several modifications of Hopfield's autoassociator, including the extension to continuous units.

The chapter also briefly discusses hardware implementations, and application of these networks to the generation of temporal sequences of patterns.

Chapter 4 illustrates several applications of this class of models, focusing on optimization problems. The chapter begins by describing how one might construct a meaningful energy function for a simple problem, and how the handcrafted energy function can be used to derive a network structure to solve the problem. The chapter then describes how the continuous autoassociator can solve the classical traveling salesman problem. After one more example, the chapter closes with a description of applications in image processing.

Chapter 5 moves to the class of supervised, error-based, feedforward networks, beginning with the simple perceptron. The authors do a nice job of summarizing simple concepts of classification, and then describing in an intuitive fashion the workings of the simple perceptron.

After a stand-alone section proving perceptron convergence when inputs are linearly separable , the chapter describes extensions to include nonlinear or stochastic units. Chapter 6 extends the analysis of Chapter 5 to the realm of multilayer perceptrons and back propagation. Palmer gradient descent on the error surface in weight space. Section 6. The chapter concludes with an overview of methods for modifying the network architecture to improve performance. Chapter 7 focuses on a variety of recurrent or feedback networks, including the Boltzmann Machine an interesting cross between a multilayer perceptron and an autoassociator , recurrent back propagation, and other models for learning time sequences.

The closing section describes reinforcement learning models. It is unclear why this class of model appears in this chapter, as the "feedback" here is simply a signal indicating how the network is performing. This section could easily be extended and turned into an independent chapter.In spite of its condensed format, the chapter does a reasonable job of summarizing some of the main computational points of relevant models based on competitive learning, such as adaptive resonance theory and self-organizing feature maps.

However, despite its importance, how cortical computations are performed and the underlying neural mechanisms remain unclear. Moreover, PCs cannot discharge when they are shunted by strong inhibition.

The chapter provides a useful discussion of how Hebbian networks are related to Principal Component Analysis, a technique for extracting information about the dimensions along which some input data exhibit maximum variance. It means that the Ensemble is formed by all the PCs whose firing could be transiently constrained by that specific synchronized inhibitory network Figure 2.

High-order social interactions in groups of mice.

Introduction to set theory. Memory Lundqvist et al.

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