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Workshop 3 Abstracts and Lecture Materials:
Chalk Talks Discussion 1: Liam Paninski,
Mike DeWeese, Stefano Panzeri Discussion 2: Bijoy Ghosh,
Sonja Gruen, Hemant Bokil, Garrett Stanley, Thomas Gedeon Discussion 5: Nandini Singh,
Chris Machens, Prasun Roy
Author: Shun-ichi
Amari , RIKEN Brain Science Institute Streaming Video: Real
Media
1. Orthogonal decomposition of a firing patter into firing rates, pairwise correlations and higher-order interactions of neural firing in a population. 2. Synfiring and higher-order interactions in a population of neurons. 3. Fisher information and encoding/decoding accuracy in a neural field. 4. Algebraic singularities when multiple targets are presented in a neural field, and their resolution by synfiring References 1. Nakahara, H., & Amari, S. (2002). Information-geometric measure for neural spikes. Neural Computation, 14, 2269-2316. 2. Wu, S., Nakahara, H., & Amari, S. (2001). Population coding with correlation and an unfaithful model. Neural Computation, 13, 775-797. 3. Wu, S., Amari, S., & Nakahara, H. (2002). Population coding and decoding in a neural field: a computational study. Neural Computation, 14, 999-1026. 4. Amari, S., Nakahara, H., Wu, S., & Sakai, Y. (2003). Synchronous
firing and higher-order interactions in a neuron pool. Neural
Computation, 15 (to appear).
Author: Charles
Anderson, Washington Univ. School of Medicine Charles H. Anderson Abstracts: PDF
Presentation Materials: PPT1
PDF1 PPT2
PDF2 PPT3
PDF3 Author: David
Arathorn Presentation Materials: PPT Streaming Video: Real
Media
Refs. Author: Michael
J. Black, Brown University Streaming Video: Real
Media Building a direct, artificial, connection between the brain and the world, requires answers to the following questions 1. What "signals" can we measure from the brain? From what regions? With what technology? 2. How is information represented (or encoded) in the brain? 3. What algorithms can we use to infer (or decode) the internal "state" of the brain? 4. How can we build practical interfaces that exploit the available technology? This talk will summarize our work on developing neural prostheses and will provide preliminary answers to the above questions with a focus on the problem of modeling and decoding motor cortical activity. Recent work has shown that simple linear models can be used to approximate the firing rates of a population of cells in primary motor cortex as a function of the position, velocity, and acceleration of the hand. In particular, I will describe a real-time Kalman filter for inferring hand motion from the firing rates of a population of cells recorded with a chronically implanted microelectrode array. I will also describe non-linear generalizations of this model including Generalized Linear Models (GLM), and Generalized Additive Models (GAM). Non-linear decoding is achieved using a recursive Bayesian estimator known as the "particle filter". I will illustrate these ideas by showing recent results with direct neural control of smooth 2D cursor motion. This is joint work with John Donoghue, Elie Bienenstock, Yun Gao, Mijail Serruya, and Wei Wu. Web page: http://www.cs.brown.edu/people/black/ Donoghue Lab home page: http://donoghue.neuro.brown.edu/ Overview of neural prosthetics project: http://www.cs.brown.edu/people/black/Papers/capriOverviewDraft.pdf Kalman filter decoding paper: http://www.cs.brown.edu/people/black/Papers/nips02draft.pdf Particle filtering paper: http://www.cs.brown.edu/people/black/Papers/NIPS14.pdf
Author: Hermant
Bokil , California Institute of Technology I will present a brief discussion of a method for decoding recorded neural activity, both spikes and local field potentials, and report results of its performance on a comprehensive collection of recordings from area LIP of macaque monkeys performing a memory saccade task. Special emphasis will be given to comparing decodes from spikes and LFPs. Finally, I will present a real-time implementation of the method.
Author: Alexander
Borst , Max-Planck-Institute of Neurobiology, Martinsried, Germany Streaming Video: Real
Media Recent theoretical advances allow for the determination of the
information rate inherent in the spike trains of nerve cells. However,
up to now, the dependence of the information rate on stimulus parameters
has not been studied in any neuron in a systematic way. Here, I
investigate the information carried by the spike trains of H1, a
motion-sensitive visual interneuron of the blowfly (Calliphora vicina)
using a moving grating as a stimulus. One might expect that, up
to a certain limit, the information rate becomes the larger the
richer the stimulus entropy. This, however, is not the case: Increasing
either the dynamic range of the stimulus or the maximum velocity
has little or no influence at all on the information rate. In contrast,
the information rate steeply increases when the size or the contrast
of the stimulus is enlarged. It appears that, regardless of the
stimulus entropy, the neuron covers the stimulus with its whole
response repertoir, with the information rate being limited by the
noise of the stimulus and the neural hardware. Author: Naama
Brenner, Technion, Israel Institute of Technology Streaming Video: Real
Media Neural codes are highly adaptive and context dependent. Some results will be reviewed indicating the functional aspects of adaptive coding in sensory systems. Information theory can help in providing a quantitative understanding of these aspects. From a mechanistic point of view, maintaining an adaptive code requires both space and time flexibility of neural responses. Experiments will be described on random networks, indicating that some features of sensory adaptation arise from neural network structure with no anatomy.
1. Brenner, N., Bialek, W., & de Ruyter van Steveninck, R. (2000). Adaptive rescaling maximizes information transmission. Neuron, 26(3), 695-702. 2. Fairhall, A., Lewen, G., Blalek, W., & de Ruyter van Steveninck,
R. (2001). Efficiency and ambiguity in an adaptive neural code.
Nature (London, U. K.), 412(6849), 787-792. Author:
Emery Brown , Harvard
Medical School Streaming Video: Real
Media
Author:
Sharon Crook , University
of Maine When we consider the representation of information in sensory systems, it is important to understand how the different aspects of patterns of neural activity affect the transmission of information to the next layer of processing. We will discuss the role that temporal patterns play in evoking a response in the postsynaptic cell. In particular, we will outline conditions where the arrival time of a spike plays a significant role in a postsynaptic cell that is selective for certain frequencies. Author: Yang
Dan , Berkely University of California Streaming Video: Real
Media A major challenge in studying sensory processing is to understand the meaning of the neural messages encoded in the spiking activity of neurons. In the visual cortex, the majority of neurons have nonlinear response properties, making it difficult to characterize their stimulus-response relationships. I will discuss two nonlinear methods to analyze the input-response relationship of these cortical neurons: training of artificial neural networks with the back-propagation algorithm and the second-order Wiener Kernel analysis. Both methods can capture much of the input-response transformation in the classical receptive fields of the cortical complex cells.
Author: Michael
Deweese , Cold Spring Harbor Laboratory Cortical neurons are usually thought to operate in a highly unreliablemanner: A neuron can signal the same stimulus with a variable number of action potentials. Here we describe a novel mode in which each neuron generates exactly 0 or 1 action potentials, but not more, in response to a stimulus. We used cell-attached recording, which ensured single-unit isolation, to record responses in rat auditory cortex to brief tone pips. Surprisingly, the majority of neurons exhibited binary behavior; several dramatic examples consisted of exactly one spike on 100% of trials, with no trial-to-trial variability in spike count. Many neurons were tuned to stimulus frequency. These binary units allow for some forms of dendritic computation that are not possible with conventional Poisson coding units, and are consistent with a model of cortical processing in which synchronous packets of spikes propagate stably from one neuronal population to the next. Author: Alex
Dimitrov, Center for Computational Biology, Montana State University We present an analytical approach through which the relevant stimulus
space and the corresponding neural symbols 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. This approach
uses tools from Rate Distortion Theory for the analysis of neural
coding schemes. Its success prompted us to consider the general
framework of signal quantization with minimal distortion as a model
for the functioning of early sensory processing. Evidence from behavioural
and neuroanatomical data suggested that symmetries in the sensory
environment need to be taken into account as well. We suggest two
approaches - implicit and explicit - which can incorporate the symmetries
in the quantization model. Author: Chris
Eliasmith, University of Waterloo Streaming Video: Real Media
Charles Anderson and I have recently proposed a unified framework
for generating large-scale neurally plausible models that relies
on integrating recent advances in neural coding with modern control
theory (in our book 'Neural Engineering'). I will briefly describe
this framework, including our approach to unifying population and
temporal coding for scalar, vector, and function representation. Author: David
J. Field, Cornell University Streaming Video: Real
Media Over the last 15 years, a range of insights into visual coding
have developed out of a deeper understanding of the statistics of
the natural environment. The structure arising from correlations
in pixel values as well as the sparse edge related structure of
natural scenes have helped to provide an account of the processing
of information along the visual pathway from retinae to cortex.
However, the statistical dependencies in natural images occur at
all levels of analysis. One can not assume that any method would
be capable of finding descriptions where the units of description
are independent. Independent components are simply impossible with
most natural environments. Then how does one handle redundancy when
independence is either not possible or impractical given the number
of neurons? One insight may come from the lateral connections between
oriented neurons in primary visual cortex. Here, we find conditions
where small collections of neurons appear to be representing the
redundant structure (e.g., the continuity of edges), rather than
single neurons. Do insights from these modes of representation provide
insights into higher levels of representing redundancy? This talk
will probe some of the possible limits of what we can learn by understanding
the redundancy of the natural world. Author: Thomas
Gedeon, Montana State University Author: Bijoy
Ghosh , Washington University in St. Louis Presentation
Materials: PPT
Author: Sonja
Gruen, Free University Berlin, Inst. Biology, Neurobiology Common to most correlation analysis techniques for neuronal spiking
activity are assumptions of stationarity with respect to various
parameters. However, experimental data may fail to be compatible
with these assumptions. This failure can lead to falsely assigned
significant outcomes. Here we study the effect of non-stationarity
of spike rate across trials in a model based approach. Using a two
rate-state model where rates are drawn independently for trials
and neurons, we show in detail that non-stationarity across trials
induces apparent co-variation of spike rates identified as the generator
of false positives. This finding has specific implications for the
'shuffle predictor'. Within the framework developed for our model,
co-variation of spike rates and the mechanism by which the shuffle
predictor leads to wrong interpretation of the data can be discussed.
Corrections for the influence of non-stationarity across trials
by improvements of the predictor are presented [1]. Author: John
Hertz , Nordita, Copenhagen, Denmark [work with Barry Richmond (LN-NIMH), Pauline Ruffiot (Nordita and
Univ Joseph Fourier, Grenoble) Cristina Ursta (Nordita, Niels Bohr
Institute, and West Univ, Timisoara), Gustaf Sterner (KTH Stockholm),
Mandana Ahmadi (Ahvaz Univ, Iran) and Alexander Lerchner (DTU)] Author: Don Johnson
, Rice University Streaming Video: Real
Media To determine whether or not neural populations work in concert
to code information has defied conventional analysis. New techniques
using information theory principles seem to hold the best promise.
Using them requires defining a baseline performance against which
to judge population coding. Using an information processing theoretic
approach, we show that the conventional baseline is misleading.
We show that stimulus-induced dependence alone is sufficient to
encode information perfectly, and we propose that this standard
should serve as the baseline. When using this baseline, we show
that cooperative populations, which exhibit both stimulus- and connection-induced
dependence, can only perform better than the baseline for relatively
small population sizes. Author: Robert
Kass, Carnegie Mellon University Streaming Video: Real
Media My main aim in this presentation is to motivate the use of probability
models in the statistical analysis of neuronal data. Probability
models offer efficiency, flexibility, and the ability to make formal
statistical inferences. I illustrate by considering estimation of
instantaneous firing rate, variation in firing rate across many
neurons, decoding for movement prediction, within-trial firing rate
(non-Poisson spiking), and correlated spiking across pairs of neurons. Author: Peter
Latham , UCLA Streaming Video: Real Media Correlations among action potentials, both within spike trains from single neurons and across spike trains from multiple neurons, are ubiquitous. They are observed in many species, from the common house fly to the primate. The role of these correlations is unclear and has long been the subject of debate. Do correlations carry extra information -- information that can't be extracted from the uncorrelated responses -- or don't they? Part of the reason this question has been hard to answer is that it's not clear how to separate correlated from uncorrelated responses. Here we sidestep this issue, and instead rephrase the question as follows: Is it possible to extract all the information from a set of neuronal responses without any knowledge of the correlational structure? If the answer is "yes", then correlations are not important; otherwise, they are. This provides us with a rigorous method for assessing the role of correlations. We provide several examples to clarify the method, and then compare it to other approaches.
:
Author: Tai Sing Lee , Computer Science Department
and Center for the Neural Basis of Cognition, Carnegie Mellon University Streaming Video: Real
Media The receptive fields of simple cells in the primary visual cortex have been modeled in terms of Gabor wavelets, and derived theoretically from efficient coding principles. In this talk, first, I will report findings of a neurophysiological experiment that demonstrate signals with naturalistic power spectrum provide not only a more efficient but a more accurate means for identifying the kernels (receptive fields) of V1 neurons. The reason is that the neurons have been tuned to functionbest in the regime of natural stimuli rather than in other regimes.Second, I will report findings from another experiment that showsthat different stages of the neural responses in V1 are actually codingdifferent aspects of the visual scenes. While the early stage ofthe responses to a static image reflects the filtering propertiesof the neurons, the later stage of the response reflect the outcomeof perceptual inference, which is in turn influenced by top-downfeedback of the prior statistical experience of the animals intheir environment.
Author: Christian
Machens , Cold Spring Harbor Laboratory Recent work has shown that neurons are often optimized towards certain statistical properties of an animal's natural environment. Usually, these conclusions have been drawn by a combined analysis of natural stimuli and neural response properties. Alternatively, the stimulus statistics that a given system ``expects'' might be extracted directly from the system in online experiments. We demonstrate the feasibility of this idea in electrophysiological experiments on locust auditory receptor neurons. Using a recently developed algorithm (Phys. Rev. Lett. 88:228104), we adapt the parameters of an initial stimulus ensemble so as to maximize the mutual information between stimulus and neural response. We show that the concept of optimality cannot be treated in isolation but rather depends on further assumptions about the system. Here, we present the optimal stimulus ensemble for the case of a rate code and a spike timing code. [joint work with Tim Gollisch, Olga Kolesnikova, and Andreas V. M. Herz]
Author: Bruno
Olshausen, Center for Neuroscience, UC Davis Streaming Video: Real
Media The images that fall upon our retinae contain certain statistical
regularities over space and time. In this talk I will discuss a
method for modeling this structure based upon sparse coding in time.
When adapted to time-varying natural images, the model develops
a set of space-time receptive fields similar to those of simple-cells
in the primary visual cortex. A continuous image sequence is thus
re-represented in terms of a set of punctate, spike-like events
in time. The suggestion is that *both* the receptive fields of V1
neurons and the spiking nature of neural activity go hand in hand---i.e.,
they are part of a coordinated strategy for producing sparse representations
of sensory data. Author: Liam
Paninski , Center for Neural Science It is well-known that the firing rate of neurons in primary motor
cortex (MI) is correlated with hand position and velocity. To our
knowledge, all previous models of this tuning a) are linear in position
or velocity, b) are ``static'' in the sense that the temporal dynamics
of the encoding process are not modelled independently of general
behavioral state, and/or c) do not incorporate the effects of interneuronal
interactions on a given cell's firing rate. Here we introduce a
simple model for MI tuning that does not suffer from any of these
three limitations, and show that this model explains the firing
rate of MI cells better than any previous model. Our two main results
are that 1) the firing rate of most MI cells is in fact a nonlinear
function of the dynamic hand position signal (not just of position
or velocity), and 2) the state of the MI neural network, as measured
by simultaneous recording of multiple isolated units, has a significant
effect on the firing rate of MI cells, in that one can better predict
the firing rate of a given cell after observing the network state
and the hand position signal together, rather than the hand position
signal alone. Author: Stefano
Panzeri, UMIST, Dept. of Optometry & Neuroscience, PO BOX
88, Manchester M60 1QD , UK In this talk I will briefly present a new method to quantify the
impact of correlated firing on the information transmitted by neuronal
populations. This new method [1] considers in an exact way the effects
of high order spike train statistics, with no approximation involved,
and it generalizes our previous work that was valid for short time
windows and small populations [2,3]. The new technique permits to
quantify the information transmitted if each cell were to convey
fully independent information separately from the information available
in presence of synergy-redundancy effects. Synergy-redundancy effects
are shown to arise from three possiblecontributions: a redundant
contribution due to similarities in the meanresponse profiles of
different cells; a stimulus independent correlationalcontribution
term that reflects interactions between the distribution ofrates
of individual cells and the average level of cross-correlation,
and asynergistic stimulus-dependent correlational contribution quantifying
theinformation content of changes of correlations with stimulus.
The latterstimulus-dependent correlational term is shown to be equal
to the measurerecently proposed by Nirenberg et al [4] to quantify
the information lost bydecoders that ignore correlations [1,5].
I will finally presentapplications of this method to data simultaneously
recorded fromsomatosensory and visual cortices, and demonstrate
that our formalism can beused in experimental situations to provide
precise constraints on the roleof correlations in encoding and decoding. Author: Rajesh
Rao , Dept of Computer Science and Engineering & Neurobiology
and Behavior Program University of Washington, Seattle Streaming Video: Real
Media There has been considerable interest in Bayesian networks and probabilisitic
"graphical models" in the artificial intelligence community
in recent years. Simultaneously, a large number of human psychophysical
results have been successfully explained using Bayesian and other
probabilistic models. A central question that is yet to be resolved
is how such models can be implemented neurally. In this talk, I
will show how a network architecture commonly used to model the
cerebral cortex can implement probabilistic (Bayesian) inference
for an arbitrary Markov model. The suggested approach is illustrated
using a visual motion detection task. The simulation results show
that the model network exhibits direction selectivity and correctly
computes the posterior probabilities for motion direction. When
used to solve the well-known random dots motion discrimination task,
the model generates responses that mimic the activities of evidence-accumulating
neurons in cortical areas LIP and FEF. In addition, the model predicts
reaction time distributions that are similar to those obtained in
human psychophysical experiments that manipulate the prior probabilities
of targets and task urgency.
Author: Barry
Richmond, Laboratory of Neuropsychology, National Institute
of Mental Health Streaming Video: Real
Media In the brain, spike trains are generated in time, and presumably
also interpreted as they unfold. Recent work suggests that in several
areas of the monkey brain, individual spike times carry information
because they reflect underlying rate variation. Constructing a model
based on this stochastic structure allows us to apply order statistics
to decode spike trains instant by instant, as spikes arrive or do
not. Order statistics are time-consuming to compute in the general
case. We demonstrate that data from neurons in V1 are well-fit by
a mixture of Poisson processes; in this special case, our computations
are substantially faster. In these data, spike timing contributed
information beyond that available from spike count throughout the
trial. At the end of the trial, a decoder based on the mixture of
Poissons model correctly decoded about three times as many trials
as expected by chance, compared to about twice as many as expected
by chance using spike count only. If our model perfectly described
the spike trains, and enough data were available to estimate model
parameters, then our Bayesian decoder would be optimal. For 4/5
of the sets of stimulus-elicited responses, the observed spike trains
were consistent with the mixture of Poissons model. Most of the
error in estimating stimulus probabilities is due to not having
enough data to specify the parameters of the model rather than to
misspecification of the model itself. Author: Dr.
P.K. Roy , National Brain
Research Centre We elucidate a new approach for analyzing neuronal systems using
non-classical (non-equilibrium) information theory. Specifically,
we consider the irreversible processes behind neural information
transmission, using the tools of non-equilibrium, nonclassical thermodynamics.
This contrasts to other current studies in neuroscience, which use
the Shannon model of information theory, based on the classical
equilibrium thermodynamic model of Boltzman. Based on this new approach,
we are delineating an experimental and theoretical infrastructure
aimed at elucidating the control, communication and computation
processes in neural 1. Avramescu, A. (1980). Coherent informational energy and entropy. J. Documentation, 36, 293-312. 2. Baddeley, R., & Hancock, P. (2000). Information theory and the brain. C.U.P., Cambridge. Chap. 3. Buckingham, M. (1995). Noise in electronic devices and systems. New York: Halstead Press.. 4. Cooper, R., Girish, B., & Miller, J. (2000). Assessing the performance of neural encoding models in presence of noise. J. Comp Neurosci, 8, 95-112. 5. Nicolis, G., & Prigogine, I. (1987). Self-organization in non-equilibrium systems. Wiley, N.Y. 6. Nicolis, J. (1994). Role of chaos in information processing, In E. Basar, et al, eds., Synergetics of brain. Springer, New York. 7. Roy, P. et al. (2000). A neurocybernetic and biothermodynamic study. J. Intelligent Systems, 10, 57-104. 8. Roy, P., Miller, J., et al. (2001). Control analysis of neuronal information: Electrophysiological experiment and non-equilibrium information theory. Springer Lecture notes Artificial Intelligence, 2275, 191-203. 9. Sarpeshkar, R. (1998). Analog vs. digital: Extrapolating from electronics to neurobiology. Neural Computation, 10, 1601-38. 10. Theunissen, F., Roddey, Miller, J., et al. (1996). Information
theoretic analysis of dynamical encoding by four primary interneurons
in cricket. J. Neurophysiol., 75, 1345-1364. Author: Simon
Schultz, Center for Neural Science and Courant Institute of
Mathematical Sciences, New York University Streaming Video: Real
Media The elemental symbol manipulated by cortical neurons is the spike,
or action potential. Spikes are not independent, however, and interactions
between them - whether spikes from a different cell, or from the
same cell but at a different time - may affect the way in which
information is coded. We have developed procedures for separating
out the contribution of interactions to the Shannon information
content of the spike trains. In this talk I will discuss the application
of information theory to a number of experiments which have lead
to insight about how interactions between spikes affect the neural
coding of visual information. The first experiment concerns how
information quantities change over the course of development of
the visual system. The second concerns the effect of correlations
in the spiking activity of pairs of suitably related V1 neurons
- do these correlations result in synergistic or redundant pooling
of information across cells? In the third experiment we examine
the dynamic responses of cells in an extrastriate visual area (MT),
looking for synergistic and redundant interactions between spikes.
In all of these cases, we see that the spike trains cannot be approximated
by Poisson processes - the amount of information represented depends
upon correlations between the spikes. Author: Nandini
Chatterjee Singh, National Brain Research Center, India The talk will begin by describing the new National Research Center
(NBRC). This is a new Institute which has been set up in India and
the first of its kind for the development of Neuroscience Research.
Next I shall talk briefly about the Modulation Spectrum which analyses
the joint spectral and temporal characteristics of sound. I shall
indicate how the modulation spectrum is set up and the interesting
characteristics of natural sounds it has revealed to us. Author: Martin
Stetter, Siemens AG, CT IC 4 How does our brain generate the tremendous diversity of human precognitive and cognitive phenomena and how does it deal with the combinatorial complexity of our environment? Distributed representations of different aspects of our environment - for example orientation, color, motion, shape and spatial relationships in the visual modality, context-dependent working memory, conflict monitoring and many others - can help to avoid the combinatorial explosion, but they raise the binding problem. What else can we gain from a distributed representation in global brain activity patterns?
Author: Jonathan
D. Victor, Cornell Streaming Video: Real
Media A striking feature of the activity of cortical neurons is that the spike trains are irregular, and responses to repeated presentations of the same stimulus can be quite variable. Moreover, neighboring neurons have generally similar response properties, but the variability is largely independent. It is often assumed that an inhomogeneous Poisson process is an adequate description cortical response variability. Were this the case, then individual spikes, at best, serve as estimators of a firing rate, and decoding of a population of similar neurons is optimally done by a population average. We test these ideas with two kinds of experiments carried out on clusters of neurons in the primary visual cortex of the macaque monkey. In the first set of experiments, we record responses to repeated presentations of pseudorandom (m-sequence) patterns, which allows for both an analysis of average response properties and a direct test of the inhomogeneous Poisson hypothesis. In the second set of experiments, we record responses to more traditional visual stimuli (spatial grating patterns), and analyze these responses via a metric-space approach. The latter approach provides a means to formalize and test a wide variety of coding hypotheses, especially as they relate to temporal representation of information. The experiments are complementary, and converge on the conclusion that spike trains are more than estimators of a firing rate, and that the detailed pattern of neural activity within individual spike trains and across neurons cannot be ignored. References and Web Links 1. Lab web page 2. Background material on the spike metric method 3. Review article on temporal aspects of early visual processing 4. Selected publications on neural coding http://www-users.med.cornell.edu/~jdvicto/vipu96.html 6. Victor, J.D., & Purpura, K.P. (1997). Metric-space analysis
of spike trains: theory, algorithms, and application. Network,
8, 127-164. 7. Reich, D.S., Mechler, F., & Victor, J.D. (2001). Independent
and redundant information in nearby cortical neurons. Science,
294, 2566-2568. 8. Mechler, F., Reich, D. S., & Victor, J.D. (2002). Detection
and discrimination of relative spatial phase by V1 neurons. J.
Neurosci., 22, 6129-6157. 9. Victor, J.D. (in press). Binless strategies for estimation of
information from neural data. Phys. Rev. E.
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