Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
This paper focuses on a one person game called Indian policeman's dilemma (IPD). It represents the internal conflict between emotion and profession of a typical Indian police officer. We have 'split' the game to be played independently by different personality modules of the same player. Each module then appears as an independent individual player of the game. None of the players knows the exact payoff values of any of the others. Only greater than or less than type of inequalities among the payoff values across the players are to be inferred probabilistically. There are two Nash equilibrium (NE) points in this game signifying two completely opposing behavior by the policeman involved. With the help of the probabilistic inequalities probable propensities of the different behaviors have been determined. The model underscores the need for new surveys and data generation. A design of one such survey to measure professionalism of the police personnel has been outlined.

Abstract:
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.