One of the most challenging and fascinating problems in science is deciphering how
neural systems encode information. Mathematical models are critical in any effort to
determine how neural systems represent and transmit information. Such mathematical
approaches span a wide range, which may be divided approximately into two kinds of
research activities. The first kind of activity uses detailed biophysical models (e.g.,
Hodgkin–Huxley and its variants) of individual neurons, detailed biophysical
models of networks of neurons, or artificial neural network models to study emergent
behaviors of neural systems. The second kind of activity develops signal-processing
algorithms to analyze the ever-growing volumes of data collected in neuroscience
experiments.
In an ideal scientific investigation there is a direct link between experiments and
theoretical modeling. The theoretical models make predictions that can be used to guide
experiments; the experiments provide data that allow for refinement (or rejection) of
theoretical models. The growing complexity of neuroscience experiments makes use of
appropriate data analysis methods crucial for establishing how reliably specific system
properties can be identified from experimental measurements. Thus, careful data analysis
is an essential complement to theoretical modeling. It allows validation of theoretical
model predictions and provides biologically relevant constraints and parameter values for
further analytic and simulation studies(see figure 1).
Figure 1. Neuroscience data: dynamic and multivariate.
Neural spike train data have special features that present new, exciting challenges for
signal processing research. For this reason in 2000 and 2001 we organized two
workshops entitled `Information and Statistical Structure in Spike Trains' as part of the
Neural Information Processing meetings for those years. The papers in this special issue are a
compilation of some
of the work presented at those two workshops. All focus in some
way on the question of how to decipher the neural codes. In this regard, the broad range
of topics addressed is a sample of the breadth of issues required to make the link from
experiments to theory, and of the theoretical considerations that can make
experimental predictions. The data analysis methods divide into three types: data
pre-processing (spike sorting), statistical modeling of neural spike train data, and algorithms
for calculating information with explicit and implicit models. The system-dependent or
theoretical models use simulations to gain physiologic and mechanistic insight.
The recent advent of the capability to record with multiple electrode arrays the
simultaneous spiking activity of many neurons (>100) has made it possible to study
information encoding by ensembles rather than by just single neurons. Simultaneous
recording of multiple neurons is now a standard tool in neuroscience research. Often, the
first data-analysis problem confronting the investigator is that of discerning the
discharges of individual neurons within the multichannel, noise-contaminated signals that
result from tetrode recordings. Nguyen, Frank and Brown's contribution, `An
application of reversible-jump MCMC to spike classification of multi-unit extracellular
recordings', presents a new approach to this problem. The core of the approach is a
Bayesian Markov chain Monte Carlo procedure that simultaneously estimates the
correlation structure of the noise (the variability of the individual spike waveforms) and
the number of distinct waveforms. In this way the authors determine the number of
neurons being simultaneously recorded while assigning each action potential to its
source neuron.
Kass, Ventura and Cai (`Statistical smoothing of neuronal data') take a careful look at
the notion of firing rate. Though seemingly a simple problem, estimation of the firing rate
(and, using this estimate to determine quantities such as the time of the peak response) is
far from straightforward. As they show, straightforward approaches based on the
post-stimulus histogram and raster plot—visualization tools that are ubiquitous in the
exploratory analysis of neural data—are rather inefficient, and smoothing techniques
based on adaptive splines offer substantial advantages.
Information theory methods are perhaps the most widely used techniques for analyzing
neural data. In `An exact method to quantify the information transmitted by different
mechanisms of correlational coding', Pola, Theil, Hoffman and Panzeri extend their
previous work on exact measures of information encoded by an individual neuron to
exact measures of information encoded by a population of neurons. Their analysis
technique allows a decomposition of the information structure in terms of the mean
neural response, the correlation among the neurons and stimulus-induced changes in
correlation among the neurons.
Dimitrov, Miller and co-workers (`Analysis of neural coding using quantization with an
information-based distortion measure') present another approach for the estimation of
information transmitted by spike trains, and apply this approach to the cricket cercal (air
velocity sensation) system. Rather than calculate information directly by estimation of
joint input–output probabilities, their approach applies rate distortion theory to identify a
`codebook' that minimally distorts the information available in the stimulus. One
advantage of this approach is that it does not assume that the neural code resembles a rate
code. More importantly, along with the estimate of information that the procedure yields,
the codebook provides insight into how information is represented.
Understanding the distinction between single spikes and spikes that belong to bursts is
crucial for characterizing neural encoding schemes. In `Information encoding and
computation with spikes and bursts', Kepecs and Lisman use simulation studies based on
the Hodgkin–Huxley model combined with principal components analysis (PCA) and
discriminant analysis to address this question. Specifically, the authors simulate a
Hodgkin–Huxley model of a bursting neuron stimulated by stochastic inputs and study
the relation between spiking patterns and stimulus features using the PCA discriminant
analysis applied to a form of the spike-triggered average covariance matrix. The authors
suggest a way to identify the distinct stimulus features that spikes and bursts encode.
Dynamic synapses (synapses whose efficacy increases or decreases in a manner that
depends on their recent activity) clearly are differentially
affected by isolated spikes and by
spikes in bursts. Pantic, Torres and Kappen (`Coincidence detection with dynamic synapses') shows that
dynamic changes in synaptic efficacy not only affect the way that spike trains from a
single input are processed, but can also play an important role in the way that spike trains
from convergent inputs interact. Via a computational study of idealized integrate-and-fire
and Hodgkin-Huxley neurons, this work demonstrates that synaptic dynamics
substantially improve the range over which neurons can act as coincidence detectors.
Although action potentials are the primary way in which neurons communicate, a crucial
part of understanding this communication process lies in understanding the intricacies of
the subthreshold processes that lead to the generation of action potentials. In `Influence
of subthreshold nonlinearities on signal-to-noise ratio and timing precision for small
signals in neurons: minimal model analysis', Svirskis and Rinzel use an elementary
integrate-and-fire model to examine the different roles that subthreshold voltages and
time-dependent conductances play in signal integration and the production of action
potentials. The key to their analysis is the role that a non-inactivating low-threshold
outward current can play in increasing the precision of small signal integrations. Svirskis
and Rinzel provide several examples to illustrate the importance of the subthreshold
feedback mechanism.
Defining what a neuron encodes (i.e., its receptive field properties) is a basic question in
neuroscience. Many experiments in neuroscience allow investigators to study this
question by
providing a statistical
characterization of the response of the neuron to a given
stimulus. In `Likelihood approaches to sensory coding in auditory cortex', Jenison and
Reale use likelihood methods to study the problem of sound localization based on the
ensemble response recorded from primary auditory cortex. They do this by using an inverse
Gaussian probability density to model the neural response latency as a function of
multiple acoustic parameters. The interesting feature of this approach is the use of
likelihood methods based on formal probability model to carry out the analysis. There are
several advantages to the likelihood approach. The parameter estimates obtained
using the likelihood approach have several optimality properties such as consistency
(converging to the true value as the sample size increases), having an asymptotic
Gaussian distribution and providing a straightforward way to compute confidence
intervals that are as short as possible. This gives a quantitative description of the
relative importance of direction, azimuth and sound amplitude in inducing auditory
neural responses and hence useful insight into how the neurons use this information for
sound localization.
Most sensory systems are faced with the problem of providing useful signals over a wide
dynamic range of the input, within the constraints of a relatively narrow range of outputs.
In vision, this problem is particularly acute: the retinal output, which consists of spike
trains that rarely exceed 200 impulses/s, can signal contrasts as low as one part in 300,
over a 1010-fold operating range of intensities—performance that can only be
accomplished through gain controls. Lesica, Boloori and Stanley (`Adaptive encoding in
the visual pathway') present a promising approach to the analysis of such gain controls,
through a procedure that adaptively tracks response characteristics. Although described in
the context of data from the visual system, the approach is a general one, and is
particularly useful in the challenging situation in which the timescale of the adaptive
mechanisms and the timescales of the signals being encoded are not well separated.
In sum, the availability of new experimental techniques, analytic and computational
strategies, and the hardware with which to implement them has led to a surge of activity
at the confluence of experimental, computational and theoretical neuroscience. The
papers in this special issue provide windows into the range and vigor of this current research.