Necessery and Sufficient Condition for Extention of Convolution Semigroup

Let and be real - valued continuous functions and ( ptf,(qtg,p and q be cons-tants, where denotes a set of positive rational numbers. Convolution of +Q+Q()fpt, and , denoted by ()q,tg()()qtgp,∗tf, is defined by ()()()()∫−=∗tdvqvtgpvfqtgptf0,,,,, where denotes convolution operation, provided that the integral exists. From the definition of convolution, we introduce a new relation as follows ∗ ()()()()qt,pgqp,tfq,tgp,tf++=∗or , where denotes an ordinary addition. The new relation is called extension of con-volution semigroup. Objective of the study is to discover the necessary and sufficient condition for the new relation. The study is based on Laplace transformable functions. Convolution Theorem in Laplace transform is used to verify the new relation. It is im-possible to achieve the new relation directly since most of the transforms are rational polynomial functions. Furthermore, any transform in terms of exponential function is different from one another. However, we overcome the problem by (a) Identity property under convolution such that ()()(tftδtf=∗, where()tf is a real - valued continuous function, which has Laplace transform and ()tδ is the delta function and it is the identity function under convolution. The Laplace transform of delta function ()tδ is 1. (b) Under certain condition, the delta function ()tδ is a convolution semigroup such that ()()()qp,tδq,tδp,tδ+=∗. (c) Delta function can be replaced by other function under certain condition. ()tδ With (a), (b) and (c), we discover the following results: Proposition 1 Let ()()tpfptfε=, and ()()tqgqtg=, for such that 0≥t()()tgtf≠ε and ()()1lim0==∫∫⊂→dttgdttfRRIεεε. Then ()()() ,,,qptfqtgptf+=∗if and only if ()[]0≠tfLε and ()[]1=tgL, or ()()() ,,,qptgqtgptf+=∗ if and only if ()[]1=tfLε and ()[], where pand are constants with q+Q111=+qp and is an interval of the point with neighborhood. εIε Proposition 2 Let and ()tfε()tg be given real - valued functions with ()()0==tgtfε for . Let 0<t()()ptfptf−=ε, and ()()qtgqtg−=,. ()()tgtf≠ε and ()()1lim0==∫∫⊂→dttgdttfRRIεεε. Then ()()() ,,,qptfqtgptf+=∗if and only if ()[]0≠tfLε and ()[]1=tgL, or ()()() ,,,qptgqtgptf+=∗if and only if ()[]1=tfLε and ()[]0≠tgL, where p and q are constants and is an interval of the point with neighborhood. +QεIε Proposition 1 is called scale form of the functions f and g, while Proposition 2 is called shift form of the functions f and g. The extension of convolution semigroup is formed by a non - impulsive and an impulsive function such that the non - impulsive function is an approximation of the impulsive function under certain condition, where all functions in this study are both real - valued continuous and of exponential order. The study has shown that it is not necessary depend on the same function in order to get the new relation. This study is only true for the conditions described by Propositions 1and 2

Rate Dynamics of Leaky Integrate-and-Fire Neurons with Strong Synapses

Firing-rate models provide a practical tool for studying the dynamics of trial- or population-averaged neuronal signals. A wealth of theoretical and experimental studies has been dedicated to the derivation or extraction of such models by investigating the firing-rate response characteristics of ensembles of neurons. The majority of these studies assumes that neurons receive input spikes at a high rate through weak synapses (diffusion approximation). For many biological neural systems, however, this assumption cannot be justified. So far, it is unclear how time-varying presynaptic firing rates are transmitted by a population of neurons if the diffusion assumption is dropped. Here, we numerically investigate the stationary and non-stationary firing-rate response properties of leaky integrate-and-fire neurons receiving input spikes through excitatory synapses with alpha-function shaped postsynaptic currents for strong synaptic weights. Input spike trains are modeled by inhomogeneous Poisson point processes with sinusoidal rate. Average rates, modulation amplitudes, and phases of the period-averaged spike responses are measured for a broad range of stimulus, synapse, and neuron parameters. Across wide parameter regions, the resulting transfer functions can be approximated by a linear first-order low-pass filter. Below a critical synaptic weight, the cutoff frequencies are approximately constant and determined by the synaptic time constants. Only for synapses with unrealistically strong weights are the cutoff frequencies significantly increased. To account for stimuli with larger modulation depths, we combine the measured linear transfer function with the nonlinear response characteristics obtained for stationary inputs. The resulting linear–nonlinear model accurately predicts the population response for a variety of non-sinusoidal stimuli

Estimation of Thalamocortical and Intracortical Network Models from Joint Thalamic Single-Electrode and Cortical Laminar-Electrode Recordings in the Rat Barrel System

A new method is presented for extraction of population firing-rate models for both thalamocortical and intracortical signal transfer based on stimulus-evoked data from simultaneous thalamic single-electrode and cortical recordings using linear (laminar) multielectrodes in the rat barrel system. Time-dependent population firing rates for granular (layer 4), supragranular (layer 2/3), and infragranular (layer 5) populations in a barrel column and the thalamic population in the homologous barreloid are extracted from the high-frequency portion (multi-unit activity; MUA) of the recorded extracellular signals. These extracted firing rates are in turn used to identify population firing-rate models formulated as integral equations with exponentially decaying coupling kernels, allowing for straightforward transformation to the more common firing-rate formulation in terms of differential equations. Optimal model structures and model parameters are identified by minimizing the deviation between model firing rates and the experimentally extracted population firing rates. For the thalamocortical transfer, the experimental data favor a model with fast feedforward excitation from thalamus to the layer-4 laminar population combined with a slower inhibitory process due to feedforward and/or recurrent connections and mixed linear-parabolic activation functions. The extracted firing rates of the various cortical laminar populations are found to exhibit strong temporal correlations for the present experimental paradigm, and simple feedforward population firing-rate models combined with linear or mixed linear-parabolic activation function are found to provide excellent fits to the data. The identified thalamocortical and intracortical network models are thus found to be qualitatively very different. While the thalamocortical circuit is optimally stimulated by rapid changes in the thalamic firing rate, the intracortical circuits are low-pass and respond most strongly to slowly varying inputs from the cortical layer-4 population

Matematiske aspekter ved lokalisert aktivitet i nevrofeltmodeller

Neural field models assume the form of integral and integro-differential equations, and describe non-linear interactions between neuron populations. Such models reduce the dimensionality and complexity of the microscopic neural-network dynamics and allow for mathematical treatment, efficient simulation and intuitive understanding. Since the seminal studies byWilson and Cowan (1973) and Amari (1977) neural field models have been used to describe phenomena like persistent neuronal activity, waves and pattern formation in the cortex. In the present thesis we focus on mathematical aspects of localized activity which is described by stationary solutions of a neural field model, so called bumps. While neural field models represent a considerable simplification of the neural dynamics in a large network, they are often studied under further simplifying assumptions, e.g., approximating the firing-rate function with a unit step function. In some cases these assumptions may not change essential features of the model, but in other cases they may cause some properties of the model to vary significantly or even break down. The work presented in the thesis aims at studying properties of bump solutions in one- and two-population models relaxing on the common simplifications. Numerical approaches used in mathematical neuroscience sometimes lack mathematical justification. This may lead to numerical instabilities, ill-conditioning or even divergence. Moreover, there are some methods which have not been used in neuroscience community but might be beneficial. We have initiated a work in this direction by studying advantages and disadvantages of a wavelet-Galerkin algorithm applied to a simplified framework of a one-population neural field model. We also focus on rigorous justification of iteration methods for constructing bumps. We use the theory of monotone operators in ordered Banach spaces, the theory of Sobolev spaces in unbounded domains, degree theory, and other functional analytical methods, which are still not very well developed in neuroscience, for analysis of the models.Nevrofeltmodeller formuleres som integral og integro-differensiallikninger. De beskriver ikke-lineære vekselvirkninger mellom populasjoner av nevroner. Slike modeller reduserer dimensjonalitet og kompleksitet til den mikroskopiske nevrale nettverksdynamikken og tillater matematisk behandling, effektiv simulering og intuitiv forståelse. Siden pionerarbeidene til Wilson og Cowan (1973) og Amari (1977), har nevrofeltmodeller blitt brukt til å beskrive fenomener som vedvarende nevroaktivitet, bølger og mønsterdannelse i hjernebarken. I denne avhandlingen vil vi fokusere på matematiske aspekter ved lokalisert aktivitet som beskrives ved stasjonære løsninger til nevrofeltmodeller, såkalte bumps. Mens nevrofeltmodeller innebærer en betydelig forenkling av den nevrale dynamikken i et større nettverk, så blir de ofte studert ved å gjøre forenklende tilleggsantakelser, som for eksempel å approksimere fyringratefunksjonen med en Heaviside-funksjon. I noen tilfeller vil disse forenklingene ikke endre vesentlige trekk ved modellen, mens i andre tilfeller kan de forårsake at modellegenskapene endres betydelig eller at de bryter sammen. Arbeidene presentert i denne avhandlingen har som mål å studere egenskapene til bump-løsninger i en- og to-populasjonsmodeller når en lemper på de vanlige antakelsene. Numeriske teknikker som brukes i matematisk nevrovitenskap mangler i noen tilfeller matematisk begrunnelse. Dette kan lede til numeriske instabiliteter, dårlig kondisjonering, og til og med divergens. I tillegg finnes det metoder som ikke er blitt brukt i nevrovitenskap, men som kunne være fordelaktige å bruke. Vi har startet et arbeid i denne retningen ved å studere fordeler og ulemper ved en wavelet-Galerkin algoritme anvendt på et forenklet rammeverk for en en-populasjons nevrofelt modell. Vi fokuserer også på rigorøs begrunnelse for iterasjonsmetoder for konstruksjon av bumps. Vi bruker teorien for monotone operatorer i ordnede Banachrom, teorien for Sobolevrom for ubegrensede domener, gradteori, og andre funksjonalanalytiske metoder, som for tiden ikke er vel utviklet i nevrovitenskap, for analyse av modellene

Development of the multi-scale dynamic modeling techniques for multi-modal brain recordings

Fundamental principles underlying computation in multi-scale brain networks illustrate how multiple brain areas and their coordinated activity give rise to complex cognitive functions. Whereas brain activity has been studied in the micro- to meso-scale in studying the connections between the dynamical patterns and the behaviors, the investigation of the neural population dynamics is mainly limited by the single-scale analysis. My goal is to develop a multi-scale dynamical model for the collective activity of neuronal populations. First, I introduced a bio-inspired deep learning approach, termed NeuroBondGraph Network (NBGNet), to capture cross-scale dynamics that can infer and map the neural data from multiple scales (Chapter 2). The NBGNet not only exhibits more than an 11-fold improvement in reconstruction accuracy, but also predicts synchronous neural activity and preserves correlated low-dimensional latent dynamics. I also show that the NBGNet robustly predicts held-out data across a long time scale (two weeks) without retraining. The effective connectivity defined from the presented model agrees with the established neuroanatomical hierarchy of motor control in the literature. I then introduced the m̲ulti-s̲cale n̲eural d̲ynamics n̲eural o̲rdinary d̲ifferential e̲quation (msDyNODE) to uncover multiscale brain communications governing cognitive behaviors (Chapter 3). I demonstrated that msDyNODE successfully captured multiscale activity using both simulations and electrophysiological experiments. The msDyNODE-derived casual interactions between recording channels and scales not only aligned well with the abstraction of the hierarchical anatomy of the mammalian nervous system but also exhibited behavioral dependences. This work offers a new approach for in depth mechanistic studies on the brain where simultaneous acquisition of multi-scale neural activity is available. While the traditional neuronal tracing method serves as gold standard to assess the neural connectivity, toxicity and the limited efficiency prevent it from serving as a reliable tool to validate the multi-scale connectivity. It has been demonstrated that dendritic spines increase or become bigger after long-term potentiation while decrease or become smaller after long-term depression. Thus, we hypothesized the existence of excitatory and inhibitory connectivity can be revealed by the dynamics of dendritic spines. Since the dendritic spines can be smaller than the Abbe diffraction limit (~200 nm), we need an in-vivo super-resolution microscopy to achieve our goal. As a super-resolution imaging method, stimulated emission depletion (STED) microscopy has unraveled fine intracellular structures and provided insights into nanoscale organizations in cells. Although image resolution can be further enhanced by continuously increasing the STED-beam power, the resulting photodamage and phototoxicity are major issues for real-world applications of STED microscopy. In Chapter 4, I demonstrated that, with 50% less STED-beam power, the STED image resolution can be improved up to 1.45-fold using the separation of photons by a lifetime tuning (SPLIT) scheme combined with a deep learning-based phasor analysis algorithm termed flimGANE (f̲luorescence l̲ifetime i̲m̲aging based on a g̲enerative a̲dversarial n̲e̲twork). This work offers a new approach for STED imaging in situations where only a limited photon budget is available. In summary, I emphasized that this is the first time we are able to infer the nonlinear multiscale interactions in the brain networks by our multi-scale dynamic modeling approach. This opens a window of opportunity that multi-scale connectivity can provide a mechanistic understanding of brain computations underlying behaviors or mental states.Biomedical Engineerin

Differential/Difference Equations

The study of oscillatory phenomena is an important part of the theory of differential equations. Oscillations naturally occur in virtually every area of applied science including, e.g., mechanics, electrical, radio engineering, and vibrotechnics. This Special Issue includes 19 high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology. This Special Issue brought together mathematicians with physicists, engineers, as well as other scientists. Topics covered in this issue: Oscillation theory; Differential/difference equations; Partial differential equations; Dynamical systems; Fractional calculus; Delays; Mathematical modeling and oscillations