Complex spatial patterning, common in the brain as well as in other biological systems, can emerge as a result of dynamic interactions that occur locally within developing structures. In the rodent somatosensory cortex, groups of neurons called "barrels" correspond to individual whiskers on the contralateral face. Barrels themselves often contain subbarrels organized into one of a few characteristic patterns. Here we demonstrate that similar patterns can be simulated by means of local growth-promoting and growth-retarding interactions within the circular domains of single barrels. The model correctly predicts that larger barrels contain more spatially complex subbarrel patterns, suggesting that the development of barrels and of the patterns within them may be understood in terms of some relatively simple dynamic processes. We also simulate the full nonlinear equations to demonstrate the predictive value of our linear analysis. Finally, we show that the pattern formation is robust with respect to the geometry of the barrel by simulating patterns on a realistically shaped barrel domain. This work shows how simple pattern forming mechanisms can explain neural wiring both qualitatively and quantitatively even in complex and irregular domains. © 2009 Ermentrout et al.

Neurons in the cortex exhibit a number of patterns that correlate with working memory. Specifically, averaged across trials of working memory tasks, neurons exhibit different firing rate patterns during the delay of those tasks. These patterns include: 1) persistent fixed-frequency elevated rates above baseline, 2) elevated rates that decay throughout the tasks memory period, 3) rates that accelerate throughout the delay, and 4) patterns of inhibited firing (below baseline) analogous to each of the preceding excitatory patterns. Persistent elevated rate patterns are believed to be the neural correlate of working memory retention and preparation for execution of behavioral/motor responses as required in working memory tasks. Models have proposed that such activity corresponds to stable attractors in cortical neural networks with fixed synaptic weights. However, the variability in patterned behavior and the firing statistics of real neurons across the entire range of those behaviors across and within trials of working memory tasks are typical not reproduced. Here we examine the effect of dynamic synapses and network architectures with multiple cortical areas on the states and dynamics of working memory networks. The analysis indicates that the multiple pattern types exhibited by cells in working memory networks are inherent in networks with dynamic synapses, and that the variability and firing statistics in such networks with distributed architectures agree with that observed in the cortex. © 2009 Verduzco-Flores et al.

The modulation of the sensitivity, or gain, of neural responses to input is an important component of neural computation. It has been shown that divisive gain modulation of neural responses can result from a stochastic shunting from balanced (mixed excitation and inhibition) background activity. This gain control scheme was developed and explored with static inputs, where the membrane and spike train statistics were stationary in time. However, input statistics, such as the firing rates of pre-synaptic neurons, are often dynamic, varying on timescales comparable to typical membrane time constants. Using a population density approach for integrate-and-fire neurons with dynamic and temporally rich inputs, we find that the same fluctuation-induced divisive gain modulation is operative for dynamic inputs driving nonequilibrium responses. Moreover, the degree of divisive scaling of the dynamic response is quantitatively the same as the steady-state responses- thus, gain modulation via balanced conductance fluctuations generalizes in a straight-forward way to a dynamic setting.

Numerical evidence is provided to show that the optimal exercise boundary for American put options with continuous dividend rate d is convex for values d less than or equal to r, where r is the risk-free rate. For d greater than r, the boundary is not convex. As d increases beyond r, the non-convex region moves away from expiry and increases in size. A front-fixing method has been used to transform the American put problem into a nonlinear parabolic differential equation posed on a fixed domain. Explicit and implicit finite-difference methods are used to simulate the problem numerically. As a test, both the explicit and implicit method has been compared and the finite-difference methods give stable results.

The preBotzinger complex located at the ventrolateral medulla in the brainstem is believedto have an important role in generating the respiratory rhythm in mammals, specially theinspiratory process [56]. Keeping this in mind, we will study a small network of such cellsby means of a minimal model suggested and experimentally tested by Butera et al [6, 7]. Athorough analysis of the Butera model was done for two very small networks of pre-Botzingercells: a self coupled single cell and a network of two coupled cells [5]. In order to understandthe role of coupling and heterogeneity in these two particular networks we reduce the selfcoupled single cell network to a one dimensional map using a similar approach as in [37].Using this one dimensional map, some analytical conditions for switching from one regime toanother are determined and numerical results are shown. Using the same idea as for the selfcoupled single cell case, two identical coupled cells are reduced to a two dimensional iteratedmap which is a composition of many one dimensional maps. Using the form of these maps,mechanisms for the transition between previously observed regimes [5] are determined andlinear analysis is performed for a particular set of parameters.Introducing heterogeneity on the network of two coupled identical cells, for a fixed levelof synaptic input, shows that depending on the level of the synaptic input some differentbehaviors arise which were not previously observed in a network of homogenous cells [5].These results suggest that introducing heterogeneity can increase the range in the parameterspace for which cells are bursting. This is desirable, since from experiments it is observedthat bursting is associated with the inspiratory rhythm of respiration.

The following work is divided into three chapters. In the first chapter, we extend the classical definition of Lebesgue function spaces to include values of p < 0. If (Ω,Σ, μ) is a finite, non-atomic measure space, μ a positive measure, then we denote by M(μ) the space of equivalence classes of Σ-measurable functions. For all p > 0, L−p(µ) is the set M(μ) together with a complete, translation invariant metric, d−p, defined using the decreasing rearrangement of functions ƒ ∈ M(μ). Defined as such, we can extend the inclusion Lq (μ) ⊂ Lp (μ) to all real numbers p and q, with p < q. Furthermore, L−p(μ) can be equipped with an F-norm defined by ||f|| = d−p(f, 0). The second chapter deals with the theory of Hilbert frames. We prove several inequalities relating the Schatten norm of the frame operator, S, to the p-norms of the frame elements, ƒj. This is done first in finite dimensional Hilbert spaces, then extended to infinite dimensions using a truncated frame operator for finite subsets of the frame. In the final section of this chapter, we construct a frame for which the averaged 1-norm of the associated Gram matrix exhibits an optimal growth rate. In the paper Generalized Roundness and Negative Type, Lennard, Tonge, and Westonshow that the geometric notion of generalized roundness in a metric space is equivalent tothat of negative type. Using this equivalent characterization, along with classical embedding theorems, the authors prove that for p > 2, L p fails to have generalized roundness q for any q > 0. It is noted, as a consequence, that the Schatten class C p, for p > 2, has maximal generalized roundness 0. In the third chapter, we prove that this result remains true for p in the interval (0, 2).

Cesaro averaging is used in conjunction with Hardy space andHilbert space theory to realize certain types of convergence.In Chapter 1, we study certain Hardy-type sequence spaces, which are analogues ofell-infinity and c_0, respectively. We show that the Mazurproduct is not onto, which provides a new solution of Mazur's Problem 8 in the Scottish Book. We present corollaries for spaces defined via weighted ell-p seminorms and for c_0.In Chapter 2, we study the application of Cesaro operators onBessel sequences to realize a weak version of frame reconstructionin Hilbert space. Conditions for reconstruction via Markuschevichbases that are certain linear combinations of orthonormal basisvectors are given.

An S-space is any topological space which is hereditarily separable but not Lindelof. An L-space, on the other hand, is hereditarily Lindelof but not separable. For almost a century, determining the necessary and suffcient conditions for the existence of these two kinds of spaces has been a fruitful area of research at the boundary of topology and axiomatic set theory. For most of that time, the twoproblems were imagined to be dual; that is, it was believed that the same setsof conditions that required or precluded one type would suffice for the other aswell. This, however, is not the case. When Todorcevic proved in 1981 that itis consistent, under ZFC, for no S-spaces to exist, everyone expected a similarresult to follow for L-spaces as well. Justin Tatch Moore surprised everyonewhen, in 2005, he constructed an L-space in ZFC. This paper summarizes andcontextualizes that result, along with several others in the field.

Development of accurate and efficient numerical methods is an important task for many research areas. This work presents the numerical study of the Discontinuous Galerkin Finite Element (DG) methods in space and various ODE solvers in time applied to 1D parabolic equation. In particular, we study the numerical convergence and computational efficiency of the Backward Euler (BE) in time and high order DG in space methods vs. the numerical convergence and the computational efficiency of the DG in time and space methods.

In many modern approaches to solving Monge's mass transport problem (that is, optimal transport with respect to linear costs) in various metric spaces, one attempts to reduce the problem to one dimension by decomposing the measures along so-called transport (geodesic) rays. Certain key Lipschitz estimates on geodesics are needed in order provide such a decomposition. Herein these estimates for the (three dimensional, sub-Riemannian) Heisenberg Group are provided as a step towards solving Monge's problem in this metric space.