Optimization in Reproducing Kernel Hilbert Spaces of Spike Trains

Network: Computation in Neural Systems, 1997

We present the mathematical basis of a new approach to the analysis of temporal coding. The foundation of the approach is the construction of several families of novel distances (metrics) between neuronal impulse trains. In contrast to most previous approaches to the analysis of temporal coding, the present approach does not attempt to embed impulse trains in a vector space, and does not assume a Euclidean notion of distance. Rather, the proposed metrics formalize physiologically based hypotheses for those aspects of the firing pattern that might be stimulus dependent, and make essential use of the point-process nature of neural discharges. We show that these families of metrics endow the space of impulse trains with related but inequivalent topological structures. We demonstrate how these metrics can be used to determine whether a set of observed responses has a stimulus-dependent temporal structure without a vector-space embedding. We show how multidimensional scaling can be used to assess the similarity of these metrics to Euclidean distances. For two of these families of metrics (one based on spike times and one based on spike intervals), we present highly efficient computational algorithms for calculating the distances. We illustrate these ideas by application to artificial data sets and to recordings from auditory and visual cortex. †

Information Science and Statistics, 2010

This paper presents a general framework based on reproducing kernel Hilbert spaces (RKHS) to mathematically describe and manipulate spike trains. The main idea is the definition of inner products to allow spike train signal processing from basic principles while incorporating their statistical description as point processes. Moreover, because many inner products can be formulated, a particular definition can be crafted to best fit an application. These ideas are illustrated by the definition of a number of spike train inner products. To further elicit the advantages of the RKHS framework, a family of these inner products, called the cross-intensity (CI) kernels, is further analyzed in detail. This particular inner product family encapsulates the statistical description from conditional intensity functions of spike trains. The problem of their estimation is also addressed. The simplest of the spike train kernels in this family provides an interesting perspective to other works presented in the literature, as will be illustrated in terms of spike train distance measures. Finally, as an application example, the presented RKHS framework is used to derive from simple principles a clustering algorithm for spike trains.

Computational intelligence and neuroscience, 2014

Neural Computation, 2012

Exploratory tools that are sensitive to arbitrary statistical variations in spike train observations open up the possibility of novel neuroscientific discoveries. Developing such tools, however, is difficult due to the lack of Euclidean structure of the spike train space, and an experimenter usually prefers simpler tools that only capture limited statistical features of the spike train such as mean spike count or mean firing rate. We explore strictly positive definite kernels on the space of spike trains to offer both a structural representation of this space and a platform for developing statistical measures that explore features beyond count or rate. We apply these kernels to construct measures of divergence between two point processes, and use them for hypothesis testing, that is, to observe if two sets of spike trains originate from the same underlying probability law. Although there exist positive definite spike train kernels in the literature, we establish that these kernels are not strictly definite, and thus, do not induce measures of divergence. We discuss the properties of both these existing non-strict kernels and the novel strict kernels in terms of their computational complexity, choice of free parameters, and performance on both synthetic and real data through kernel principal component analysis and hypothesis testing.

PLoS ONE, 2011

Background: Conventional methods for spike train analysis are predominantly based on the rate function. Additionally, many experiments have utilized a temporal coding mechanism. Several techniques have been used for analyzing these two sources of information separately, but using both sources in a single framework remains a challenging problem. Here, an innovative technique is proposed for spike train analysis that considers both rate and temporal information. Methodology/Principal Findings: Point process modeling approach is used to estimate the stimulus conditional distribution, based on observation of repeated trials. The extended Kalman filter is applied for estimation of the parameters in a parametric model. The marked point process strategy is used in order to extend this model from a single neuron to an entire neuronal population. Each spike train is transformed into a binary vector and then projected from the observation space onto the likelihood space. This projection generates a newly structured space that integrates temporal and rate information, thus improving performance of distribution-based classifiers. In this space, the stimulus-specific information is used as a distance metric between two stimuli. To illustrate the advantages of the proposed technique, spiking activity of inferior temporal cortex neurons in the macaque monkey are analyzed in both the observation and likelihood spaces. Based on goodness-of-fit, performance of the estimation method is demonstrated and the results are subsequently compared with the firing rate-based framework. Conclusions/Significance: From both rate and temporal information integration and improvement in the neural discrimination of stimuli, it may be concluded that the likelihood space generates a more accurate representation of stimulus space. Further, an understanding of the neuronal mechanism devoted to visual object categorization may be addressed in this framework as well.

International Journal of Non-Linear Mechanics, 2009

A method for characterizing and identifying firing patterns of neural spike trains is presented. Based on the time evolution of a neural spike train, the counting process is constructed as a time-dependent stair-like function. Three characteristic variables defined at sequential moments, including two formal derivatives and the integration of the counting process, are introduced to reflect the temporal patterns of a spike train. The reconstruction of a spike train with these variables verify the validity of this method. And a model of cold receptor is used as an example to investigate the temporal patterns under different temperature conditions. The most important contribution of our method is that it not only can reflect the features of spike train patterns clearly by using their geometrical properties, but also it can reflect the trait of time, especially the change of bursting of spike train. So it is a useful complementarity to conventional method of averaging.

Neuron, 2016

As information flows through the brain, neuronal firing progresses from encoding the world as sensed by the animal to driving the motor output of subsequent behavior. One of the more tractable goals of quantitative neuroscience is to develop predictive models that relate the sensory or motor streams with neuronal firing. Here we review and contrast analytical tools used to accomplish this task. We focus on classes of models in which the external variable is compared with one or more feature vectors to extract a low-dimensional representation, the history of spiking and other variables are potentially incorporated, and these factors are nonlinearly transformed to predict the occurrences of spikes. We illustrate these techniques in application to datasets of different degrees of complexity. In particular, we address the fitting of models in the presence of strong correlations in the external variable, as occurs in natural sensory stimuli and in movement. Spectral correlation between predicted and measured spike trains is introduced to contrast the relative success of different methods.

Computational Intelligence and Neuroscience, 2010

Journal of Neuroscience Methods, 2008

We propose an efficient algorithm to compute the smoothed correlogram for the detection of temporal relationship between two spike trains. Unlike the conventional histogram-based correlogram estimations, the proposed algorithm operates on continuous time and does not bin either the spike train nor the correlogram. Hence it can be more precise in detecting the effective delay between two recording sites. Moreover, it can take advantage of the higher temporal resolution of the spike times provided by the current recording methods. The Laplacian kernel for smoothing enables efficient computation of the algorithm. We also provide the basic statistics of the estimator and a guideline for choosing the kernel size. This new technique is demonstrated by estimating the effective delays in a neuronal network from synthetic data and recordings of dissociated cortical tissue.

Within the field of Neurological research, the processing and analysis of spike train data is a critical task. This data is obtained from electrodes planted at specific locations within a sample which return electrical information under a range of operations. The data sets collected by such analysis are substantial in size, as they represent voltage measurements sampled at 10 -20 kHz. In addition, multiple electrodes will frequently by used, which dramatically increases the size of the collected data sets. In order to manage these data sets, scientists are frequently forced to make compromises in terms of the quantity of data they can store at any given time and also the analysis routines that can be applied to this data. This paper presents early work from the CARMEN (http://www.carmen.org.uk) project which aims to improve the analysis of neurological data by providing shared data repositories and services that can be used by Neuroscientists.