arXiv subjects

Kaushik Majumdar

Publications and source records attributed to Kaushik Majumdar.

18 recordsLinked to original sources

arXiv2018

A Geometric Analysis of Time Series Leading to Information Encoding and a New Entropy Measure

A time series is uniquely represented by its geometric shape, which also carries information. A time series can be modelled as the trajectory of a particle moving in a force field with one degree of freedom. The force acting on the particle shapes the trajectory of its motion, which is made up of elementary shapes of infinitesimal neighborhoods of points in the trajectory. It has been proved that an infinitesimal neighborhood of a point in a continuous time series can have at least 29 different shapes or configurations. So information can be encoded in it in at least 29 different ways. A 3-point neighborhood (the smallest) in a discrete time series can have precisely 13 different shapes or configurations. In other words, a discrete time series can be expressed as a string of 13 symbols. Across diverse real as well as simulated data sets it has been observed that 6 of them occur more frequently and the remaining 7 occur less frequently. Based on frequency distribution of 13 configurations or 13 different ways of information encoding a novel entropy measure, called semantic entropy (E), has been defined. Following notion of power in Newtonian mechanics of the moving particle whose trajectory is the time series, a notion of information power (P) has been introduced for time series. E/P turned out to be an important indicator of synchronous behaviour of time series as observed in epileptic EEG signals.

cs.IT↗

arXiv2016

Semantic Information Encoding in One Dimensional Time Domain Signals

A one dimensional time domain analog signal s(t) can be visualized as a trajectory of a moving particle in a force field with one degree of freedom. Then the power of the particle at point t is P(s(t)) = s"(t)s'(t), which is the rate at which kinetic energy is dissipated (assuming the mass of the particle is unit) by the particle in order to create the trajectory or give shape to the signal. Assuming meaning of the signal or the semantic information is in its shape, we can say that P(s(t)) is the rate at which kinetic energy of the particle is dissipated to encode semantic information in s(t) at t. After s(t) is digitized (to make it s[n]) the discrete form P(s[n]) is valid. Considering the sign changes of P(s[n]) it has been shown that in the smallest neighborhood of n, in which n is the middle point, semantic information in s[n] can be encoded in 13 distinct ways. This list is exhaustive. A deterministic finite automaton (DFA) has been designed which can accept any finite length digital signal and therefore collection of all finite length digital signals forms a regular language. The DFA has been generalized to a weighted finite state transducer (WFST), which has been used to identify action potentials in a spike train and also to distinguish two speakers when uttering the same phoneme. It has been shown that in any analog signal semantic information can be encoded at a point in the form of the shape of its infinitesimal neighborhood in 17 distinct ways. The list is exhaustive. A new entropy measure called semantic entropy has been introduced. It has been shown that a signal s(t) is traceable on a piece of paper or in an oscilloscope, only if s"(t) exists on all but at most a finite number of points within any finite interval. This is an essential condition for a signal to be the trajectory of a moving particle.

cs.IT↗

arXiv2016

An Investigation into the Mathematical Nature of Electrophysiological Signals with Applications

In this work we have proposed a rigorous mathematical definition for the one dimensional time domain electrophysiological signals and established its relationship with two of the three Dirichlet's conditions. We have argues that any such signal can be represented as the trajectory of a particle moving in a force field with one degree of freedom. At point on the trajectory, that is, on the signal, the kinetic energy dissipated by the particle embeds semantic information into the trajectory or the signal in terms of giving its shape. We have shown that the rate of kinetic energy dissipation operator or the power operator P is of importance in shape analysis of the signal by considering its sign changes. Operating the P-operator on digital signals we have mathematically proved that its sign change can induce 13 different shapes to a three successive point configuration. In other words, semantic information at each point in a digital signal can be embedded by a syllable of 13 different letters. We have shown some preliminary applications.

q-bio.NC↗

arXiv2016

Comparison of feature extraction and dimensionality reduction methods for single channel extracellular spike sorting

Spikes in the membrane electrical potentials of neurons play a major role in the functioning of nervous systems of animals. Obtaining the spikes from different neurons has been a challenging problem for decades. Several schemes have been proposed for spike sorting to isolate the spikes of individual neurons from electrical recordings in extracellular media. However, there is much scope for improvement in the accuracies obtained using the prevailing methods of spike sorting. To determine more effective spike sorting strategies using well known methods, we compared different types of signal features and techniques for dimensionality reduction in feature space. We tried to determine an optimum or near optimum feature extraction and dimensionality reduction methods and an optimum or near optimum number of features for spike sorting. We assessed relative performance of well known methods on simulated recordings specially designed for development and benchmarking of spike sorting schemes, with varying number of spike classes and the well established method of $k$-means clustering of selected features. We found that almost all well known methods performed quite well. Nevertheless, from spike waveforms of 64 samples, sampled at 24 kHz, using principal component analysis (PCA) to select around 46 to 55 features led to the better spike sorting performance than most other methods (Wilcoxon signed rank sum test, $p < 0.001$).

q-bio.QM↗

arXiv2016

A Novel Matrix Representation of Discrete Biomedical Signals

In this work we propose a novel symmetric square matrix representation of one or more digital signals of finite equal length. For appropriate window length and sliding paradigm this matrix contains useful information about the signals in a two dimensional image form. Then this representation can be treated either as an algebraic matrix or as a geometric image. We have shown applications of both on human multichannel intracranial electroencephalogram (iEEG). In the first application we have shown that for certain patients the highest eigenvalue of the matrix obtained from the epileptic focal channels goes up during a seizure. The focus of this paper is on an application of the second concept, by which we have come up with an automatic seizure detection algorithm on a publicly available benchmark data. Except for delay in detection in all other aspects the new algorithm outperformed the detection performance based on a support vector machine based algorithm. We have also indicated how this sparse random matrix representation of brain electrical signals can encode the activities of the brain.

q-bio.QM↗

arXiv2015

A Peak Synchronization Measure for Multiple Signals

Peaks signify important events in a signal. In a pair of signals how peaks are occurring with mutual correspondence may offer us significant insights into the mutual interdependence between the two signals based on important events. In this work we proposed a novel synchronization measure between two signals, called peak synchronization, which measures the simultaneity of occurrence of peaks in the signals. We subsequently generalized it to more than two signals. We showed that our measure of synchronization is largely independent of the underlying parameter values. A time complexity analysis of the algorithm has also been presented. We applied the measure on intracranial EEG signals of epileptic patients and found that the enhanced synchronization during an epileptic seizure can be modeled better by the new peak synchronization measure than the classical amplitude correlation method.

stat.ME↗

arXiv2014

Identifying features in spike trains using binless similarity measures

Neurons in the central nervous system communicate with each other with the help of series of Action Potentials, or spike trains. Various studies have shown that neurons encode information in different features of spike trains, such as the fine temporal structure, mean firing rate, synchrony etc. An important step in understanding the encoding of information by neurons, is to obtain a reliable measure of correlation between different spike trains. In this paper, two new binless similarity measures for spike trains are proposed. The performance of the new measures are compared with some existing measures in their ability to detect important features of spike trains, such as their firing rate, sensitivity to bursts and common periods of silence and detecting synchronous activity.

q-bio.NC↗

arXiv2013

Synchronization Implies Seizure or Seizure Implies Synchronization?

Epileptic seizures are considered as abnormally hypersynchronous neuronal activities of the brain. Do hypersynchronous neuronal activities in a brain region lead to seizure or the hypersynchronous activities take place due to the progression of the seizure? We have examined the ECoG signals of 21 epileptic patients consisting of 87 focal-onset seizures by three different measures namely, phase synchronization, amplitude correlation and simultaneous occurrence of peaks and troughs. Each of the measures indicates that for a majority of the focal-onset seizures, synchronization or correlation or simultaneity occurs towards the end of the seizure or even after the offset rather than at the onset or in the beginning or during the progression of the seizure. We also have outlined how extracellular acidosis caused due to the seizure in the focal zone can induce synchrony in the seizure generating network. This implies synchronization is an effect rather than the cause of a significant number of pharmacologically intractable focal-onset seizures. Since all the seizures that we have tested belong to the pharmacologically intractable class, their termination through more coherent neuronal activities may lead to new and effective ways of discovery and testing of drugs.

q-bio.NC↗

arXiv2012

A Mathematical model for Astrocytes mediated LTP at Single Hippocampal Synapses

Many contemporary studies have shown that astrocytes play a significant role in modulating both short and long form of synaptic plasticity. There are very few experimental models which elucidate the role of astrocyte over Long-term Potentiation (LTP). Recently, Perea & Araque (2007) demonstrated a role of astrocytes in induction of LTP at single hippocampal synapses. They suggested a purely pre-synaptic basis for induction of this N-methyl-D- Aspartate (NMDA) Receptor-independent LTP. Also, the mechanisms underlying this pre-synaptic induction were not investigated. Here, in this article, we propose a mathematical model for astrocyte modulated LTP which successfully emulates the experimental findings of Perea & Araque (2007). Our study suggests the role of retrograde messengers, possibly Nitric Oxide (NO), for this pre-synaptically modulated LTP.

q-bio.NC↗

arXiv2012

A Mathematical Model of Tripartite Synapse: Astrocyte Induced Synaptic Plasticity

In this paper we present a biologically detailed mathematical model of tripartite synapses, where astrocytes modulate short-term synaptic plasticity. The model consists of a pre-synaptic bouton, a post-synaptic dendritic spine-head, a synaptic cleft and a peri-synaptic astrocyte controlling Ca2+ dynamics inside the synaptic bouton. This in turn controls glutamate release dynamics in the cleft. As a consequence of this, glutamate concentration in the cleft has been modeled, in which glutamate reuptake by astrocytes has also been incorporated. Finally, dendritic spine-head dynamics has been modeled. As an application, this model clearly shows synaptic potentiation in the hippocampal region, i.e., astrocyte Ca2+ mediates synaptic plasticity, which is in conformity with the majority of the recent findings (Perea & Araque, 2007; Henneberger et al., 2010; Navarrete et al., 2012).

q-bio.NC↗

arXiv2009

Differential Operator in Seizure Detection

Differential operators can detect significant changes in signals. This has been utilized to enhance the contrast of the seizure signatures in depth EEG or ECoG. We have actually taken normalized exponential of absolute value of single or double derivative of epileptic ECoG. Variance operation has been performed to automatically detect seizures. A novel method for determining the duration of seizure has also been proposed. Since all operations take only linear time, the whole method is extremely fast. Seven novel parameters have been introduced whose patient specific thresholding brings down the rate of false detection to a bare minimum. Results of implementation on the ECoG data of four epileptic patients have been reported with an ROC curve analysis. High value of the area under the ROC curve indicates excellent detection performance.

q-bio.QM↗

arXiv2009

Fourier Uniformity: An Useful Tool for Analyzing EEG Signals with An Application to Source Localization

If two signals are phase synchronous then the respective Fourier component at each spectral band should exhibit certain properties. In a pair of artificially generated phase synchronous signals the phase difference at each frequency band changes very slowly over the subsequent frequency bands. This has been called Fourier uniformity in this paper and a measure of it has been proposed. An usefulness of this measure has been outlined in the case of cortical source localization of scalp EEG.

q-bio.NC↗

arXiv2008

An FFT based measure of phase synchronization

In this paper phase of a signal has been viewed from a different angle. According to this view a signal can have countably infinitely many phases, one associated with each Fourier component. In other words each frequency has a phase associated with it. It has been shown that if two signals are phase synchronous then the difference between phases at a given component changes very slowly across the subsequent components. This leads to an FFT based phase synchronization measuring algorithm between any two signals. The algorithm does not take any more time than the FFT itself. Mathematical motivations as well as some results of implementation of the algorithm on artificially generated signals and real EEG signals have been presented.

q-bio.NC↗

arXiv2007

Outline of a novel architecture for cortical computation

In this paper a novel architecture for cortical computation has been proposed. This architecture is composed of computing paths consisting of neurons and synapses only. These paths have been decomposed into lateral, longitudinal and vertical components. Cortical computation has then been decomposed into lateral computation (LaC), longitudinal computation (LoC) and vertical computation (VeC). It has been shown that various loop structures in the cortical circuit play important roles in cortical computation as well as in memory storage and retrieval, keeping in conformity with the molecular basis of short and long term memory. A new learning scheme for the brain has also been proposed and how it is implemented within the proposed architecture has been explained. A number of mathematical results about the architecture have been proposed, many of which without proof.

q-bio.NC↗

arXiv2007

A structural and a functional aspect of stable information processing by the brain

In this paper a model of neural circuit in the brain has been proposed which is composed of cyclic sub-circuits. A big loop has been defined to be consisting of a feed forward path from the sensory neurons to the highest processing area of the brain and feed back paths from that region back up to close to the same sensory neurons. It has been mathematically shown how some smaller cycles can amplify signal. A big loop processes information by contrast and amplify principle. It has been assumed that the spike train coming out of a firing neuron encodes all the information produced by it as output. This information over a period of time can be extracted by a Fourier transform. The Fourier coefficients arranged in a vector form will uniquely represent the neural spike train over a period of time. The information emanating out of all the neurons in a given neural circuit over a period of time will be represented by a collection of points in a multidimensional vector space. This cluster of points represents the functional or behavioral form of the neural circuit. It has been proposed that a particular cluster of vectors as the representation of a new behavior is chosen by the brain interactively with respect to the memory stored in that circuit and the synaptic plasticity of the circuit. It has been proposed that in this situation a Coulomb force like expression governs the dynamics of functioning of the circuit and stability of the system is reached at the minimum of all the minima of a potential function derived from the force like expression. The calculations have been done with respect to a pseudometric defined in a multidimensional vector space.

q-bio.NC↗

arXiv2007

Behavioral response to strong aversive stimuli: A neurodynamical model

In this paper a theoretical model of functioning of a neural circuit during a behavioral response has been proposed. A neural circuit can be thought of as a directed multigraph whose each vertex is a neuron and each edge is a synapse. It has been assumed in this paper that the behavior of such circuits is manifested through the collective behavior of neurons belonging to that circuit. Behavioral information of each neuron is contained in the coefficients of the fast Fourier transform (FFT) over the output spike train. Those coefficients form a vector in a multidimensional vector space. Behavioral dynamics of a neuronal network in response to strong aversive stimuli has been studied in a vector space in which a suitable pseudometric has been defined. The neurodynamical model of network behavior has been formulated in terms of existing memory, synaptic plasticity and feelings. The model has an analogy in classical electrostatics, by which the notion of force and potential energy has been introduced. Since the model takes input from each neuron in a network and produces a behavior as the output, it would be extremely difficult or may even be impossible to implement. But with the help of the model a possible explanation for an hitherto unexplained neurological observation in human brain has been offered. The model is compatible with a recent model of sequential behavioral dynamics. The model is based on electrophysiology, but its relevance to hemodynamics has been outlined.

q-bio.NC↗

arXiv2006

A new measure of phase synchronization for a pair of time series and seizure focus localization

Defining and measuring phase synchronization in a pair of nonlinear time series are highly nontrivial. This can be done with the help of Fourier transform, when it exists, for a pair of stored (hence stationary) signals. In a time series instantaneous phase is often defined with the help of Hilbert transform. In this paper phase of a time series has been defined with the help of Fourier transform. This gives rise to a deterministic method to detect phase synchronization in its most general form between a pair of time series. Since this is a stricter method than the statistical methods based on instantaneous phase, this can be used for lateralization and source localization of epileptic seizures with greater accuracy. Based on this method a novel measure of phase synchronization, called syn function, has been defined, which is capable of quantifying neural phase synchronization and asynchronization as important parameters of epileptic seizure dynamics. It has been shown that such a strict measure of phase synchronization has potential application in seizure focus localization from scalp electroencephalogram (EEG) data, without any knowledge of electrical conductivity of the head.

nlin.CD↗

arXiv2003

An investigation into the vortex formation in a turbulent fluid with an application in tropical storm generation

The present work discusses about a possible physical interpretation of the occurrence of turbulence in a dynamic fluid with mathematical modeling and computer simulation. Here turbulence is defined to be a phenomenon of random velocity field in the space-time continuum accompanied by chaotic occurrence of vortices. This interpretation is independent of the Navier-Stokes equations. I have reasoned how individual fluid percels are disintegrated with increasing Reynolds number (Re) (or increasing velocity or decreasing viscosity or both) leading to creation of smaller parcels with arbitrary speeds in arbitrary directions, which destroys the laminar structure of the fluid flow. I have modeled the occurrence of a vortex as a result of collision among fluid jets under certain conditions. Chaotic occurrence of such vortices further randomizes the velocity field. These together ultimately lead to turbulence. I have also shown an application of vortex formation in a dynamic fluid in atmospheric science., where it has been shown how an initial dusturbing cyclonic vortex is created by collision between two linear wind jets under certain conditions, which under favorable conditions, may mature into a severe tropical storm. Then a three dimensional mathematical modeling of the vortex (assuming that it is going to become a matured storm) has been proposed with computer simulations. This helps us to understand the mystery of origin of cyclonic and anticyclonic vortices in atmosphere and some of their observed asymmetries.

nlin.CD↗