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This is a structured process of self-appraisal, reflection, and planning, which will enable you to chart your personal, academic and professional development throughout your time at university. Effectively, the threshold response of a single neuron is smeared out to yield a sigmoid when averaged over the population. Signals arriving at neurons of type a stimulate neurotransmitter release at synapses.

This is followed by propagation of voltage changes along dendrites and soma charging, with dynamics that spread the temporal profile of the signals. The total cell body potential can thus be written. The neural populations included are cortical excitatory e and inhibitory i neurons, the thalamic reticular nucleus r , thalamic relay neurons s that project to the cortex, thalamic interneurons j , and noncorticothalamic neurons responsible for external inputs n. In the present case, external inputs are visual, the relevant relay nucleus is the lateral geniculate nucleus LGN , and its projections are to the primary visual cortex V1.

Physiologically based corticothalamic model in which the arrows represent excitatory effects and the circles depict inhibitory ones. The populations are cortical excitatory e and inhibitory i neurons, the thalamic reticular nucleus r , thalamic relay neurons s that project to the cortex, and non-corticothalamic neurons responsible for external inputs n. Gray boxes depict the lateral geniculate nucleus LGN left , the thalamic reticular nucleus TRN middle , and primary visual cortex right.

The use of a single form of D ab corresponds to the approximation that the mean dendritic dynamics can be described by a single pair of time constants. In the cortex, the number of synapses is closely proportional to the numbers of source and target neurons Robinson et al. Physiologically estimated model parameters for normal adults in the alert, eyes-open state Robinson et al. The NFT equations are nonlinear in general, and highly nonlinear phenomena like epileptic seizures have been studied with them Breakspear et al. Normal brain states have been shown to correspond to spatially uniform steady states of corticothalamic NFT, and are obtained by setting all time and space derivatives to zero Robinson et al.

Stable steady state solutions are interpreted as representing the baseline of normal activity, which yields firing rates in accord with experiment Robinson et al. Time dependent brain activity is then represented by linear perturbations from the steady states—an approximation that has reproduced a host of experimental phenomena, including evoked responses Robinson et al.

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In this section, we calculate the linear transfer functions of stimuli to corticothalamic populations. The low firing rate of steady-states have been identified with normal states of brain activity Robinson et al.

This yields the variations of order 1 mV , or slightly larger, which corresponds to approximately a 2-fold firing rate bound. Detailed analysis of the model with respect to these parameters can be found in Robinson et al.

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Operation with D ab on both sides of Equation 8 , plus use of Equation 4 , yield. The gain G ab is the response in neurons a due to unit input from neurons b ; i. A transfer function is the ratio of the output of a system to its input in the linear regime. Either the Laplace or Fourier transform can be used to determine transfer functions, but the former is more widely used in engineering control theory, particularly to analyze responses to impulses.

To derive transfer functions one may apply the Laplace transform to both sides of Equation 9 to transform it from time t to complex frequency s. The unilateral Laplace transform is defined by Ogata and Yang Alternatively, one may use the continuous Fourier transform F , which is equivalent to evaluating the bilateral Laplace operator with imaginary argument. Before we calculate system transfer functions, we note that the operator in Equation 4 has the Laplace transform.

## Neural Field Theory of Corticothalamic Prediction With Control Systems Analysis

The firing rate in the spatial-Fourier, temporal-Laplace domain henceforth in the reticular nucleus is. Multiplying Equations 19 and 28 yields the overall transfer function to cortex from retina:. More detailed analysis of the model with respect to these parameters can be found in Robinson et al. Only the the spatially-uniform effects of perturbations, i. Low frequencies are passed, while high frequencies are attenuated, and for input signals with small frequency, each transfer function represents an amplifier with constant gain.

At higher frequencies, pronounced resonances at 9 and 18 Hz are present in all transfer functions, which can be associated with the alpha and beta peaks in the brain's wake state. The functions become less resonant as the signal gets further away from retina to the cortex: the thalamic functions T sn and T rn have higher amplitude and wider bandwidth than the cortical ones T en and T in. Estimated dimensionless synaptic gains for normal adults in the alert, eyes-open state Robinson et al.

The transfer function fully describes the linear system properties and enables us to investigate its response to any external signal. Setting the denominator of the transfer function to zero yields the characteristic equation of the system, whose roots are its eigenvalues and mark the poles; these poles determine the basic modes into which the system response can be decomposed. Furthermore, all corticothalamic transfer functions calculated above, have some or all of their poles basic modes in common, which is a direct result of the interconnectedness of the system.

Roots of the numerator of the transfer function are the zeros of the system; signals at these frequencies are not transfered through the system. In this section, we decompose the transfer functions into elementary modes whose behaviors we associate with data filters whose control system properties are well understood. Only the spatially-uniform effects of perturbations, i. The corticothalamic transfer functions, are ratios of exponential polynomials of s.

We approximate each transfer function T ab s by a rational function, whose properties can be interpreted in terms of data filters Ogata and Yang, ; Kwakernaak and Sivan, , with. Therefore, when the p j are all distinct we do not consider degenerate roots here , one has the partial fraction decomposition. The smaller n is, the simpler the description of the system, although accuracy is lost if n is made too small. In subsequent sections, we seek the smallest n that retains the main dynamics.

Generally, this leads to the most heavily damped modes poles with the largest negative real parts being discarded. Once we have a few-pole approximation of the system transfer function, we examine it from a control-systems perspective to determine its predictive properties. A general transfer function will have one or more pairs of complex conjugate poles in the Laplace domain, in addition to one or more real poles. Therefore, each pair of conjugate poles generates a real response mode.

We also pair up real poles in the next part of the analysis to conveniently treat both cases together as second order filters whose functions are well known. This filter can be rewritten as a standard second-order filter, as. Schematic of stages in the predictor given by Equation The peak magnitude of G a J s satisfies. Dependence of filter properties on parameters. By transforming Equation 49 into the time domain one then finds. The above convolution plus prediction processes can be interpreted as a Proportional-Integral-Derivative PID control scheme used in engineering control systems Ogata and Yang, ; Kwakernaak and Sivan, Specifically, in the time domain Equation 46 becomes.

In this equation the integral I smooths the input signal to reduce the effects of noise. Hence, each pair of poles in Equation 40 yields a partial transfer function that can be interpreted as a PID controller. We now examine each corticothalamic transfer function from Section 2. We find that a pole approximation of T en is accurate to within a root-mean-square rms fractional error of 0. These results show that the 6-pole approximation retains the main features of the dynamics and is sufficient for analyzing its effects; in most cases, it exhibits slightly shifted versions of the least-damped poles of the pole approximation.

Similar observations apply to the other transfer functions T in , T rn , and T sn whose 6-pole approximations are accurate to within an rms fractional error of 0. The pole maps and the frequency responses for these approximations are plotted in Figures 5C—H. We used the Control System Toolbox of Matlab a to carry out these approximations. Poles and magnitudes of rational approximations to the transfer functions T an. A Poles of pole black squares and 6-pole red crosses approximations of T en. B Magnitude of pole black and 6-pole approximations of T en vs. C Same as A for T in. D Same as B for T in.

E Same as A for T rn. F Same as B for T rn. G Same as A for T sn. H Same as B for T sn. We thus write.

The impulse response takes the form. Parameters obtained for low, alpha, and beta filters of corticothalamic transfer functions T an using filter identification method developed in Section 3. Thus, poles closest to the positive half of the s-plane dominate the response. Both filters share the same main features because each comprises a complex conjugate pair of poles. Therefore, we focus on the alpha filter's properties. The frequency response of G n a l p h a has a resonance with. Regarding the temporal response, we must consider the nature of the residues at the poles as well as the pole locations.

An alpha rhythm 7. Magnitudes of transfer functions T an and their low-, alpha-, and beta-frequency parts vs. Note that the total magnitude is not the sum of the magnitudes of the three parts because of phase difference between them. The beta responses exhibit resonances in the beta band Both alpha and beta filters have higher peaks in thalamus than cortex. Furthermore, in every population except s , the damping rate of alpha filters are approximately half of the beta filters', meaning the alpha waves live longer than beta waves in these populations.

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But, the result shows that alpha and beta waves would last for the same time in the response of population s where their damping rates are relatively close. This suggests that beta waves live longer in the LGN response to stimuli than in the rest of the system. In contrast, the population r has the opposite relation, which means its beta filter predicts further in advance. Aside from the issue of how to interpret corticothalamic dynamics in terms of data filters per se , there is the central question of how well these filters enable the system to predict its complex input signals out to some horizon in the future.

The corticothalamic system can only respond significantly to signals out to approximately the flicker fusion frequency of around 20 Hz. Hence, as a particularly severe test of its prediction capabilities, we simulate the system response to white noise, bandwidth limited to 30 Hz, with total power P n.

We are interested in time-varying, but spatially unstructured stimuli. The perturbation analysis then corresponds to presenting the entire filed of view with a stimulus that consists of a sequence of luminances that fluctuates according to a small amplitude perturbation around a base level of intensity steady-state.

This makes it possible to apply boundary output feedbacks which, in the linear case, lead to uniform exponential stability. These hypotheses are shown to hold for a beam equation. They are only sufficient conditions: they do not hold for a wave equation which posseses [sic] however the same regularity and stability properties. This problem has been solved recently, by Dubins in the no-cusp case, and by Reeds and Shepp otherwise. We propose a new solution based on the minimum principle of Pontryagin.

Our approach simplifies the proofs and makes clear the global or local nature of the results. Bounded extremal problems in Hardy spaces for the conjugate Beltrami equation in simply-connected domains 1 edition published in in English and held by 2 WorldCat member libraries worldwide. A note on shortest paths in the plane subject to a constraint on the derivative of the curvature by J.

A finite-dimensional model of the arm, corresponding to a discretization in space, is first considered. The arm is then modeled by a classical partial differential equation, no finite dimensional approximation being made in the control analysis. In both cases, stability of the closed-loop system is proven, using a Lyapunov method and applying Lasalle's principle. The practical significance of the proposed results is commented upon and illustrated by simulation experiments. We give examples of Hilbert-Schmidt composition operators on the Dirichlet spaces. We study the composition operators on the Dirichlet spaces belong to Schatten class and the link with the size of contact points of its symbol with the unit circle.

Approximation and representation of functions on the sphere : applications to inverse problems in geodesy and medical imaging by Ana-Maria Nicu Book 1 edition published in in English and held by 1 WorldCat member library worldwide This work concerns the representation and approximation of functions on a sphere with applications to source localization inverse problems in geodesy and medical imaging.

The inverse problem IP consists in recovering a density inside the ball Earth, human brain from partially known data on the surface.

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Chapter 2 gives the mathematical background used along the thesis. The resolution of the inverse problem IP involves the resolution of two steps : the transmission data problem TP and the density recovery DR problem. For this purpose, in chapter 3, we give an efficient method to build the appropriate Slepian basis on which we express the data. This is set up by using Gauss-Legendre quadrature. The transmission data problem chapter 4 consists in estimating the data spherical harmonic expansion over the whose sphere from noisy measurements expressed in Slepian basis.