Elsevier

Neurocomputing

Volumes 65–66, June 2005, Pages 103-109
Neurocomputing

A direct, interval-based method for reconstructing stimuli from noise-robust tuning curves

https://doi.org/10.1016/j.neucom.2004.10.061Get rights and content

Abstract

We present a method for reconstructing stimuli from a tuning curve, completing a tuning curve estimation method published earlier. Stimuli are reconstructed by dividing the stimulus space into intervals and providing boundaries for the probabilities with which they contain a given stimulus. The endpoints of these intervals are calculated as zeros of polynomials of high degree using the efficient direct method of Dixon polynomial resultants. Repeated measurements refine the decomposition, allowing for more accurate statements about the associated probabilities. Our method employs no special assumptions about the noise distribution, removing tractability problems of Bayesian or Maximum Likelihood estimation.

Introduction

The relationship between sensory stimuli presented to an animal and their representations in the nervous system may be expressed by a neural tuning curve. We assume that an abstract, ideal tuning curve f(ϕ) exists, where ϕ is a stimulus parameter which is to be approximated by a constructed tuning curve given measured data. In earlier publications [1], [2], a noise-robust method achieving this was presented which uses data from all measured stimulus conditions at once. The method exploits the fact that for a number of different approximation criteria, approximation theory specifies sets of polynomials as optimally approximating functions to any continuous function. For instance, a linear combination of 1,cos(kϕ),sin(kϕ), (k is a positive integer) optimally approximates any continuous ideal tuning curve by minimizing the area enclosed by the two functions [4]. Evaluating the polynomials specified by the approximation criterion at measured stimulus conditions allows to determine rows of a matrix Xϕ so that, given a vector of observations Y, a matrix equation Y=Xϕβ+ɛ may be solved for a coefficient vector β of the approximating tuning curve [1], [2]. In this contribution, we propose a method for reconstructing stimuli from such a tuning curve.

Section snippets

An interval-based method for reconstruction

The constructed tuning curve f is a polynomial (Fig. 1). In each stimulus condition ϕ, there is a certain variability σ^2(ϕ) in the firing of the cell due to its stochasticity, for which we will give an exact definition later. Thus, all the stimuli which are mapped by the tuning curve to an interval of responses around the value of the tuning curve are stimuli which might have given the recorded response. Let a sample of n2 responses to a stimulus parameter ϕ of unknown value be given. The

Discussion

We briefly discuss the relationship between our method and traditional methods for reconstructing stimuli, such as Bayesian or maximum likelihood analysis. Our method uses a reconstruction based on the intervals which are mapped to an interval of uncertainty around the value of the tuning curve at a particular location ϕ. It may be sufficient to employ interval estimation rather than point estimation: if the reconstructed intervals are sufficiently small, there will be no behavioural difference

Axel Etzold born in 1974, got his diploma in Mathematics in 2000 from the University of Giessen (Germany). He is currently a Ph.D. student in Prof. Dr. Schwegler's and Dr. Eurich's group at Bremen university. His research interests include signal processing and encoding in neural populations, dynamical systems modeling and statistical data analysis methods.

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Axel Etzold born in 1974, got his diploma in Mathematics in 2000 from the University of Giessen (Germany). He is currently a Ph.D. student in Prof. Dr. Schwegler's and Dr. Eurich's group at Bremen university. His research interests include signal processing and encoding in neural populations, dynamical systems modeling and statistical data analysis methods.

Christian W. Eurich born in 1965, got his Ph.D. in Theoretical Physics in 1995 from the University of Bremen (Germany). As a postdoc, he worked in the Departments of Mathematics and Neurology at the University of Chicago, and he was guest researcher at the Max-Planck Institut für Strömungsforschung in Göttingen and at the RIKEN Brain Institute in Tokyo. In 2001, he held a professorship for Cognitive Neuroinformatics at the University of Osnabrück. Currently, Christian Eurich is Group Leader at the Institute for Theoretical Neurophysics at the University of Bremen. His research interests include signal processing and encoding in neural populations, neural dynamics, and motor control problems such as balancing tasks and postural sway.

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