. In a number of systems including wind detection in the cricket, visual motion perception and coding of arm movement direction in the monkey and place cell response to position in the rat hippocampus, firing rates in a population of tuned neurons are correlated with a vector quantity. We examine and compare several methods that allow the coded vector to be reconstructed from measured firing rates. In cases where the neuronal tuning curves resemble cosines, linear reconstruction methods work as well as more complex statistical methods requiring more detailed information about the responses of the coding neurons. We present a new linear method, the optimal linear estimator (OLE), that on average provides the best possible linear reconstruction. This method is compared with the more familiar vector method and shown to produce more accurate reconstructions using far fewer recorded neurons. Introduction To determine how information is represented by nervous systems, we need to understand ...

We present a novel analytical approach for studying neural encoding. As a first step we model a neural sensory system as a communication channel. Using the method of typical sequence in this context, we show that a coding scheme is an almost bijective relation between equivalence classes of stimulus/response pairs. The analysis allows a quantitative determination of the type of information encoded in neural activity patterns and, at the same time, identification of the code with which that information is represented. Due to the high dimensionality of the sets involved, such a relation is extremely difficult to quantify. To circumvent this problem, and to use whatever limited data set is available most efficiently, we use another technique from information theory— quantization. We quantize the neural responses to a reproduction set of small finite size. Amongmany possible quantizations, we choose one which preserves as much of the informativeness of the original stimulus/response relation as possible, through the use of an information-based distortion function. This method allows us to study coarse but highly informative approximations of a coding scheme model, and then to refine them automatically when more data become available.

This is the second in a series of papers which attempt to recast classical single-neuron biophysics in information theoretical terms. Classical cable theory focuses on analyzing the voltage or current attenuation of a synaptic signal as it propagates from its dendritic input location to the spike initiation zone. On the other hand, we are interested in analyzing the amount of information lost about the signal in this process due to the presence of various noise sources distributed throughout the neuronal membrane. We use a stochastic version of the linear one-dimensional cable equation to derive closedform expressions for the second-order moments of the fluctuations of the membrane potential associated with different membrane current noise sources: thermal noise, noise due to the random opening and closing of sodium and potassium channels and noise due to the presence of "spontaneous" synaptic input. We consider two different scenarios. In the signal estimation paradigm, the time-cour...

critical test of our understanding of sensory coding, however, is to take an opposite approach: to reconstruct sensory inputs from recorded neuronal responses. The decoding approach can provide an objective assessment of what and how much information is available in the neuronal responses. Although the f unction of the brain is not necessarily to reconstruct sensory inputs faithfully, these studies may lead to new insights into the f unctions of neuronal circuits in sensory processing (Rieke et al., 1997). The decoding approach has been used to study several sensory systems (Bialek et al., 1991; Theunissen and Miller, 1991; Rieke et al., 1993, 1997; Roddey and Jacobs, 1996; Warland et al., 1997; Dan et al., 1998). Most of these studies aimed to reconstruct temporal signals from the response of a single neuron (Bialek et al., 1991; Rieke et al., 1993, 1995; Roddey and Jacobs, 1996) or a small number of neurons (Warland et al., 1997). An important challenge in understanding the mammalia

Introduction Biological neural networks are large systems of complex elements interacting through a complex array of connections. Individual neurons express a large number of active conductances (Connors et al., 1982; Adams & Gavin, 1986; Llin'as, 1988; McCormick, 1990; Hille, 1992) and exhibit a wide variety of dynamic behaviors on time scales ranging from milliseconds to many minutes (Llin'as, 1988; Harris-Warrick & Marder, 1991; Churchland & Sejnowski, 1992; Turrigiano et al., 1994). Neurons in cortical circuits are typically coupled to thousands of other neurons (Stevens, 1989) and very little is known about the strengths of these synapses (although see Rosenmund et al., 1993; Hessler et al., 1993; Smetters & Nelson, 1993). The complex firing patterns of large neuronal populations are difficult to describe let alone understand. There is little point in accurately modeling each membrane potential in a large neural

This paper shows how adaptive systems can learn to add an optimal amount of noise to some nonlinear feedback systems. Noise can improve the signal-to-noise ratio of many nonlinear dynamical systems. This "stochastic resonance" effect occurs in a wide range of physical and biological systems. The SR effect may also occur in engineering systems in signal processing, communications, and control. The noise energy can enhance the faint periodic signals or faint broadband signals that force the dynamical systems. Most SR studies assume full knowledge of a system's dynamics and its noise and signal structure. Fuzzy and other adaptive systems can learn to induce SR based only on samples from the process. These samples can tune a fuzzy system's if-then rules so that the fuzzy system approximates the dynamical system and its noise response. The paper derives the SR optimality conditions that any stochastic learning system should try to achieve. The adaptive system learns the SR effect as the sys...

Multivariate time series analysis is extensively used in neurophysiology with the aim of studying the relationship between simultaneously recorded signals. Recently, advances on information theory and nonlinear dynamical systems theory have allowed the study of various types of synchronization from time series. In this work, we first describe the multivariate linear methods most commonly used in neurophysiology and show that they can be extended to assess the existence of nonlinear interdependences between signals. We then review the concepts of entropy and mutual information followed by a detailed description of nonlinear methods based on the concepts of phase synchronization, generalized synchronization and event synchronization. In all cases, we show how to apply these methods to study different kinds of neurophysiological data. Finally, we illustrate the use of multivariate surrogate data test for the assessment of the strength (strong or weak) and the type (linear or nonlinear) of interdependence between neurophysiological signals.

Abstract—Electrical noise can help pulse-train signal detection at the nanolevel. Experiments on a single-walled carbon nanotube transistor confirmed that a threshold exhibited stochastic resonance (SR) for finite-variance and infinite-variance noise: small amounts of noise enhanced the nanotube detector’s performance. The experiments used a carbon nanotube field-effect transistor to detect noisy subthreshold electrical signals. Two new SR hypothesis tests in the Appendix also confirmed the SR effect in the nanotube transistor. Three measures of detector performance showed the SR effect: Shannon’s mutual information, the normalized correlation measure, and an inverted bit error rate compared the input and output discrete-time random sequences. The nanotube detector had a threshold-like input–output characteristic in its gate effect. It produced little current for subthreshold digital input voltages that fed the transistor’s gate. Three types of synchronized

We discuss an analytical approach through which the neural symbols and corresponding stimulus space of a neuron or neural ensemble can be discovered simultaneously and quantitatively, making few assumptions about the nature of the code or relevant features. The basis for this approach is to conceptualize a neural coding scheme as a collection of stimulus-response classes akin to a dictionary or 'codebook', with each class corresponding to a spike pattern 'codeword' and its corresponding stimulus feature in the codebook. The neural codebook is derived by quantizing the neural responses into a small reproduction set, and optimizing the quantization to minimize an information-based distortion function. We apply this approach to the analysis of coding in sensory interneurons of a simple invertebrate sensory system. For a simple sensory characteristic (tuning curve), we demonstrate a case for which the classical definition of tuning does not describe adequately the performance of the studied cell. Considering a more involved sensory operation (sensory discrimination), we also show that, for some cells in this system, a significant amount of information is encoded in patterns of spikes that would not be discovered through analyses based on linear stimulus-response measures.

We study the performance of the maximum likelihood (ML) method in population decoding as a function of the population size. Assuming uncorrelated noise in neural responses, the ML performance, quantified by the expected square difference between the estimated and the actual quantity, follows closely the optimal Cramer--Rao bound, provided that the population size is sufficiently large. However, when the population size decreases below a certain threshold, the performance of the ML method undergoes a rapid deterioration, experiencing a large deviation from the optimal bound. We explain the cause of such threshold behaviour, and present a phenomenological approach for estimating the threshold population size, which is found to be linearly proportional to the inverse of the square of the system's signal-to-noise ratio. If the ML method is used by neural systems, we expect the number of neurons involved in population coding to be above this threshold.