Neurons can have rapidly changing spike train statistics dictated by the underlying network excitability or behavioural state of an animal. To estimate the time course of such state dynamics from single-or multiple neuronrecordings, we have developed an algorithm that maximizes the likelihood of observed spike trains by optimizing the state lifetimes and the state-conditional interspike-interval (ISI) distributions. Our nonparametric algorithmis free of time-binning and spike-counting problems and has the computational complexity of a Mixed-state Markov Model operating on a state sequence of length equal to the total number ofrecorded spikes. As an example, we fit a two-state model to paired recordings of premotor neurons in the sleeping songbird. We find that the two state-conditional ISI functions are highly similar to the ones measured duringwaking and singing, respectively.

We propose a new linear method for dimension reduction to identify non-Gaussian components in high dimensional data. Our method, NGCA (non-Gaussian component analysis), uses a very general semi-parametric framework. In contrast to existing projection methods we define what is uninteresting (Gaussian): by projecting out uninterestingness, we can estimate therelevant non-Gaussian subspace. We show that the estimation error of finding the non-Gaussian components tends to zero at a parametric rate.Once NGCA components are identified and extracted, various tasks can be applied in the data analysis process, like data visualization, clustering, denoising or classification. A numerical study demonstrates the usefulness of our method.

We initiate the study of learning from multiple sources of limited data, each of which may be corrupted at a different rate. We develop a complete theoryof which data sources should be used for two fundamental problems: estimating the bias of a coin, and learning a classifier in the presence of label noise. In both cases, efficient algorithms are provided for computing the optimal subset of data.

We investigate whether our Visual system selects cues that guide search in an optimal manner.

In multi-agent systems, individuals within a group coordinate and interact to achieve a goal.

The network topology of neurons in the brain exhibits an abundance of feedback connections, but the computational function of these feedback connections is largely unknown. We present a computational theory that characterizes the gain in computational power achieved through feedback in dynamical systems with fading memory. It implies that many such systems acquire through feedback universal computational capabilities for analog computing with a non-fading memory. In particular, we show that feedback enables such systems to process time-varying input streams in diverse ways according to rules that are implemented through internal states of the dynamical system. In contrast to previous attractor-based computational models for neural networks, these flexible internal states are high-dimensional attractors of the circuit dynamics, that still allow the circuit state to absorb new information from online input streams. In this way one arrives at novel models for working memory, integration of evidence, and reward expectation in cortical circuits. We show that they are applicable to circuits of conductance-based Hodgkin-Huxley (HH) neurons with high levels of noise that reflect experimental data on invivo conditions.

Predictive state representations (PSRs) are a method of modeling dynamical systemsusing only observable data, such as actions and observations, to describe their model. PSRs use predictions about the outcome of future teststo summarize the system state.

We determine the asymptotic limit of the function computed by support vector machines (SVM) and related algorithms that minimize a regularized empiricalconvex loss function in the reproducing kernel Hilbert space of the Gaussian RBF kernel, in the situation where the number of examples tends to infinity, the bandwidth of the Gaussian kernel tends to 0, and the regularization parameter is held fixed.

Active learning is the problem in supervised learning to design the locations oftraining input points so that the generalization error is minimized. Existing active learning methods often assume that the model used for learning is correctly specified, i.e., the learning target function can be expressed bythe model at hand. In many practical situations, however, this assumption may not be fulfilled. In this paper, we first show that the existing activelearning method can be theoretically justified under slightly weaker condition: the model does not have to be correctly specified, but slightly misspecified models are also allowed. However, it turns out that the weakened condition is still restrictive in practice.

We considered a gamma distribution of interspike intervals as a statistical modelfor neuronal spike generation. The model parameters consist of a time-dependent firing rate and a shape parameter that characterizes spiking irregularities of individual neurons. Because the environment changes with time, observed data are generated from the time-dependent firing rate, which is an unknown function. A statistical model with an unknown function is called a semiparametric model, which is one of the unsolved problem in statistics and is generally very difficult to solve. We used a novel method of estimating functions in information geometry to estimate the shape parameter without estimating the unknown function. We analytically obtained an optimal estimating function for the shape parameter independent of the functional form of the firing rate. This estimation is efficient without Fisher information loss and better than maximum likelihood estimation.