Kageo Akizuki

On some properties of stochastic information processes in neurons and neuron populations - Mathematical model approach

ZusammenfassungDie in der Nervenimpuls-Sequenz enthaltene Information und ihre Verarbeitung durch neurale Einheiten werden besprochen. Bei diesen Problemen richtet sich unsere Aufmerksamkeit auf die stochastischen Eigenschaften der Neuronen und der Neuronenpopulationen. Diese Abhandlung hat drei Themen, und zwar das Neuron des spontanen Typs, das Neuron des erzwungenen Typs und die wechselseitig sich hemmenden Paare.1.Das Neuron des spontanen Typs produziert Spikes ohne daß erregende Eingänge vorhanden sind. Das mathematische Modell enthält die folgenden Annahmen: Das Neuronenpotential (NP) folgt in seiner Schwankung dem Ornstein-Uhlenbeck-Prozeß, weil es nicht so völlig zufällig verläuft wie das eines Wiener-Prozesses, sondern eine Attraktion zum Restwert aufweist. Die Schwelle besitzt variable Exponentialwerte und das NP eine konstante untere Grenze. Wenn das NP die Schwelle erreicht, feuert das Neuron, und das N P wird auf eine bestimmte Position zurückgesetzt. Nach einer Erregung stellt sich eine absolute Refraktärperiode ein. Bei der Diskussion der stochastischen Eigenschaften der Neuronen sind die Funktion der Dichte der Übergangswahrscheinlichkeit und die Funktion der Dichte der Erstdurchgangszeit wichtige quantitative Größen, die durch die Kolmogorowschen Gleichungen beherrscht werden. Obwohl diese leicht aufzustellen sind, können wir nur in seltenen Fällen eine analytische Lösung in der Zeitdimension bekommen. Überdies erklären die Lösungen nur einfache Eigenschaften. Daher wird die numerische Analyse angewendet, womit sich eine ganze Anzahl von guten Ergebnissen erhalten und behandeln lassen.2.Das Neuron des erzwungenen Typs erhält die Eingangs-Spike-Sequenzen, von denen wir annehmen können, daß sie auf dem Poissonschen Prozeß beruhen. Andere Annahmen und Methoden sind nahezu gleich, wenn man von der Diffusionsapproximation des stochastischen Prozesses absieht. In diesem Fall begegnen wir dem unhomogenen Prozeß, der auf der Pulsfrequenzmodulation beruht, dessen Dichte der Erstdurchgangszeit eine multimodale Verteilung zeigt. Auch eine numerische Analyse wird durchgeführt, und die Intervallverteilung der Ausgangs-Spike-Sequenzen wird im Fall der periodischen Modulation behandelt.3.Zwei Typen der sich wechselseitig hemmenden Paare werden erörtert. Der erste Typ besitzt zwei erregende Eingänge, die voneinander unabhängig sind. Demgegenüber besitzt der zweite Typ einen allgemeinen, erregenden Eingang, der aber auf zwei Wegen erfolgt, wobei einer von beiden eine Zeitverzögerung enthält. Es zeigt sich, daß die Neuronendynamik die gleiche ist wie bei dem Neuron des erzwungenen Typs. Von den Eingangsgrößen kann angenommen werden, daß sie dem Poissonschen Prozeß folgen, und daß eine Inhibition auftritt, wenn das Begleitneuron feuert. In diesem Fall läßt sich die Wahrscheinlichkeitsdichte nicht erhalten. Daher wird eine Computersimulation versucht, aus der hervorgeht, daß der stochastische Rhythmus vorliegt, wohingegen die Eingangs-Spike-Sequenzen vom temporalhomogenen Poisson-Typ sind. Es wird auch der Fall von unhomogenen Eingangsgrößen erörtert.AbstractThe information in the nervous spike trains and its processing by neural units are discussed. In these problems, our attention is focused on the stochastic properties of neurons and neuron populations. There are three subjects in this paper, which are the spontaneous type neuron, the forced type neuron and the reciprocal inhibitory pairs.1.The spontaneous type neuron produces spikes without excitatory inputs. The mathematical model has the following assumptions. The neuron potential (NP) has the fluctuation and obeys the Ornstein-Uhlenbeck process, because the N P is not so perfectly random as that of the Wiener process but has an attraction to the rest value. The threshold varies exponentially and the NP has the constant lower limit. When the NP reaches the threshold, the neuron fires and the NP is reset to a certain position. After a firing, an absolute refractory period exists. In discussing the stochastic properties of neurons, the transition probability density function and the first passage time density function are the important quantities, which are governed by the Kolmogorovs equations. Although they can be set up easily, we can rarely obtain the analytical solutions in time domain. Moreover, they cover only simple properties. Hence the numerical analysis is performed and a good deal of fair results are obtained and discussed.2.The forced type neuron has input pulse trains which are assumed to be based on the Poisson process. Other assumptions and methods are almost the same as above except the diffusion approximation of the stochastic process. In this case, we encounter the inhomogeneous process due to the pulse-frequency-modulation, whose first passage time density reveals the multimodal distribution. The numerical analysis is also tried, and the output spike interval density is further discussed in the case of the periodic modulation.3.Two types of reciprocal inhibitory pairs are discussed. The first type has two excitatory driving inputs which are mutually independent. The second type has one common excitatory input but it advances in two ways, one of which has a time lag. The neuron dynamics is the same as that of the forced type neuron and each neuron has an identical structure. The inputs are assumed to be based on the Poisson process and the inhibition occurs when the companion neuron fires. In this case, the equations of the probability density functions are not obtained. Hence the computer simulation is tried and it is observed that the stochastic rhythm emerges in spite of the temporally homogeneous inputs. Furthermore, the case of inhomogeneous inputs is discussed.

Electric field analysis in the Earth considering attenuation of electromagnetic waves propagated in lossy media

It is pointed out that, in the field of oil well drilling, EM-MWD (electromagnetic measurement while drilling) offers many advantages. The EM-MWD system can transmit measured data from the well bottom to the surface at high speed using electromagnetic waves. Developing the EM-MWD technology requires analysis of the electric field around a drill string. A novel computer simulation method has been developed which considers attenuation of electromagnetic waves propagated in lossy media, the Earth, using features of analysis models. The simulation method can be applied to waveform simulation. This method has been verified by field experiments using a borehole of 500-m depth. >

On Stochastic Dynamics of a Neuron and a Kind of Neuron Group

Abstract Mathematical models of the information processing by neural netlets are discussed. Our attention is focused on the stochastic properties of neurons and pulse-frequency-modulation (PFM). There are three subjects in this paper, which are the spontaneous type neuron, the forced type neuron and the reciprocal inhibitory pairs. Their behaviours are numerically analysed and illustrated.

Diffusive representation of N-th order fractional brownian motion

Abstract This paper describes an effort to give a different representation of a newly Introduced n-th order fractional Brownian motion (n-fBm). The new representation is called diffusive representation which has been successfully applied to the 1/f α fractional noise. Thus this paper generalizes such representation to cover also n-fBm which is an extension to the ordinary fBm, due to the fact that the spectral properties of n-fBm cover larger range of parameter α. Different from 1/f α case, the solution involves finite part concept of theory of distribution. The advantage of the proposed method on synthesizing sample of n-fBm is presented.

Application of Multiple Tree Stochastic Theory on Estimating Signal over Network

Abstract A new multi-scale stochastic theory called the Multiple Tree (MT) theory, will be presented in this paper. The MT construction is a concatenate of several single trees where these trees are connected each other through a Gaussian random vector. As a family of multi-scale stochastic system, the MT is also closely related to the fractal phenomena. The nature of the MT is suitable to the computer network environment. In this paper such advantage will be exploited in estimating 1/f signal given ill-posed observation over the network.

Parameter Estimation of Fractional Brownian Motion Processes: Wavelet Packets Based

Abstract Correlation structure of wavelet packets will be derived first. Then it will be proved that for fractional Brownian motion (fBm) processes, correlation coefficients will decrease exponentially across the wavelet packets scales-in other words almost KL-expansion. Based on the derived theorem, it is possible to estimate the parameters of fBm processes. Flexibility of the wavelet packet structure permits us to choose the bases accordingly. From simulation results it can be concluded that by utilizing wavelet packets it is possible to get better estimate with fewer computation than the previously announced wavelet based estimation method.

A Chaos Control Method by DFC Using State Prediction

The Delayed Feedback Control method (DFC) proposed by Pyragas applies an input based on the difference between the current state of the system, which is generating chaos orbits, and the -time delayed state, and stabilizes the chaos orbit into a target. In DFC, the information about a position in the state space is unnecessary if the period of the unstable periodic orbit to stabilize is known. There exists the fault that DFC cannot stabilize the unstable periodic orbit when a linearlized system around the periodic point has an odd number property. There is the chaos control method using the prediction of the -time future state (PDFC) proposed by Ushio et al. as the method to compensate this fault. Then, we propose a method such as improving the fault of the DFC. Namely, we combine DFC and PDFC with parameter W, which indicates the balance of both methods, not to lose each advantage. Therefore, we stabilize the state into the periodic orbit, and ask for the ranges of Wand gain K using Jury` method, and determine the quasi-optimum pair of (W, K) using a genetic algorithm. Finally, we apply the proposed method to a discrete-time chaotic system, and show the efficiency through some examples of numerical experiments.

STOCHASTIC PROCESS ON MULTIWAVELET

Abstract This paper describes a multi-scale stochastic model defined on a multiwavelet structure. Previously such stochastic models are based either on a single or multiple binary tree as well as on an ordinary wavelet structure. The proposed model lies on a tree structure consists of several sets of data coefficients as a result of multiwavelet transformation. Each data sets is linked only through initial data set at root node and conditionally independent given this initial state. Multiwavelet possesses several interesting properties like simultaneously short support, symmetry and orthogonality. The effect of these properties on the proposed model is shown through simulation of smoothing process of a certain fractal signal. It will be demonstrated that several improvements over previously announced results are obtained.

Estimation of order and delay time of the process in a closed-loop control system

This paper proposes a procedure for estimating the order and delay time of the process in a closed-loop control system from its input and output data. There are identifiability conditions that assure uniqueness and unbiasedness of the estimates, but they assume that the order and delay time of the process are known. Therefore, it is of primary importance to estimate the order and delay time of the process in the identification of a closed-loop system. In this paper, the authors first investigate the influence of the closed loop on identification, then propose a procedure for estimating the order and delay time, taking into account their consequences.

Prediction of Air Pollution Based on Observed Data

Abstract Some methods for predicting air pollutant concentration values based on observed data are presented and compared in this paper. Although many observation stations for air pollution have been set up in Japan, their observed data are usually not enough to construct the complete physical model of air pollution. Therefore an important interest lies in considering prediction of air pollutant concentration values based on the data observed at small numbers of stations. The authors show that it is suitable to construct AR model for prediction from these data. Two kinds of preprocessing of observed data are shown and the results of prediction are compared. The predicted values by these methods do not show good agreements with the observed data, in the case of high concentration values. The authors discussed the possibility of predicting high concentration values from the observed data by using pattren classification of the obser the morning, and a prediction method with help of a physical model is also presented.