International Journal of Neural Systems

We consider principal curves and surfaces in the context of multivariate regression modelling. For predictor spaces featuring complex dependency patterns between the involved variables, the intrinsic dimensionality of the data tends to be very small due to the high redundancy induced by the dependencies. In situations of this type, it is useful to approximate the high-dimensional predictor space through a low-dimensional manifold (i.e., a curve or a surface), and use the projections onto the manifold as compressed predictors in the regression problem. In the case that the intrinsic dimensionality of the predictor space equals one, we use the local principal curve algorithm for the the compression step. We provide a novel algorithm which extends this idea to local principal surfaces, thus covering cases of an intrinsic dimensionality equal to two, which is in principle extendible to manifolds of arbitrary dimension. We motivate and apply the novel techniques using astrophysical and oceanographic data examples.

We consider the problem of how to recover the state and parameter values of typical model neurons, such as Hindmarsh-Rose, FitzHugh-Nagumo, Morris-Lecar, from in-vitro measurements of membrane potentials. In control theory, in terms of observer design, model neurons qualify as locally observable. However, unlike most models traditionally addressed in control theory, no parameter-independent diffeomorphism exists, such that the original model equations can be transformed into adaptive canonic observer form. For a large class of model neurons, however, state and parameter reconstruction is possible nevertheless. We propose a method which, subject to mild conditions on the richness of the measured signal, allows model parameters and state variables to be reconstructed up to an equivalence class.

The determination of principal curves relies on the arc-length as a global index to describe the middle of the data distribution. With a non-constant data distribution, however, curves that are constructed by the approach introduced in reference¹³ may not reflect the middle of data distribution, as demonstrated in this article. This is particularly so for curve segments that have a large curvature and a high data density. To overcome this problem, the paper revisits the projection of the samples onto the curve by incorporating Riemannian distances. This analysis suggests estimating the density value of each sample relative to its neighbors and utilize this value to compute the projection index for the curve. The use of density values, in turn, allows penalizing distances between samples along with the arc-length. In a similar fashion to conventional principal curves algorithms, for example proposed by Hastie and Stuetzle¹⁴ and Tibshirani,²⁹ the incorporation of Riemannian distances gives rise to an iterative algorithm that includes a projection and a self-consistent step. Application studies to simulated and experimental data sets shows that the proposed modification has the potential to outperform existing algorithms in areas of high curvature under an non-constant data distribution.

We present several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen's self-organizing maps, a class of artificial neural networks. On several examples we show advantages of using non-linear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and non-linear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of high-throughput data in molecular biology, from analysis of dynamical systems.

We discuss synchronization in networks of Hindmarsh-Rose neurons that are interconnected via gap junctions, also known as electrical synapses. We present theoretical results for interactions without time-delay. These results are supported by experiments with a setup consisting of sixteen electronic equivalents of the Hindmarsh-Rose neuron. We show experimental results of networks where time-delay on the interaction is taken into account. We discuss in particular the influence of the network topology on the synchronization.