A Maximum Entropy Test for Evaluating Higher-Order Correlations in Spike Counts

Evaluating the importance of higher-order correlations of neural spike counts has been notoriously hard. A large number of samples are typically required in order to estimate higher-order correlations and resulting information theoretic quantities. In typical electrophysiology data sets with many experimental conditions, however, the number of samples in each condition is rather small. Here we describe a method that allows to quantify evidence for higher-order correlations in exactly these cases. We construct a family of reference distributions: maximum entropy distributions, which are constrained only by marginals and by linear correlations as quantified by the Pearson correlation coefficient. We devise a Monte Carlo goodness-of-fit test, which tests - for a given divergence measure of interest - whether the experimental data lead to the rejection of the null hypothesis that it was generated by one of the reference distributions. Applying our test to artificial data shows that the effects of higher-order correlations on these divergence measures can be detected even when the number of samples is small. Subsequently, we apply our method to spike count data which were recorded with multielectrode arrays from the primary visual cortex of anesthetized cat during an adaptation experiment. Using mutual information as a divergence measure we find that there are spike count bin sizes at which the maximum entropy hypothesis can be rejected for a substantial number of neuronal pairs. These results demonstrate that higher-order correlations can matter when estimating information theoretic quantities in V1. They also show that our test is able to detect their presence in typical in-vivo data sets, where the number of samples is too small to estimate higher-order correlations directly

Singularly perturbed boundary value problems in case of exchange of stablities

We consider a mixed boundary value problem for a system of two second order nonlinear differential equations where one equation is singularly perturbed. We assume that the associated equation has two intersecting families of equilibria. This property excludes the application of standard results. By means of the method of upper and lower solutions we prove the existence of a solution of the boundary value problem and determine its asymptotic behavior with respect to the small parameter. The results can be used to study differential systems modelling bimolecular reactions with fast reaction rates

Singularly perturbed reaction-diffusion systems in case of exchange of stabilities

We study singularly perturbed elliptic and parabolic differential equations under the assumption that the associated equation has intersecting families of equilibria (exchange of stabilities). We prove by means of the method of asymptotic lower and upper solutions that the asymptotic behavior with respect to the small parameter changes near the curve of exchange of stabilities. The application of that result to systems modelling fast bimolecular reactions in a heterogeneous environment implies a transition layer (jumping behavior) of the reaction rate. This behavior has to be taken into account for identification problems in reaction systems

Singularly perturbed elliptic problems in the case of exchange of stabilities

We consider the singularly perturbed boundary value problem (E_\ve) \, \ve^ 2 \Delta u = f(u,x,\ve) {for} x \in D, \, \frac{\partial u}{\partial n} - \lambda(x) u =0 \quad \mbox{for} \quad x \in \Gamma where $D \subset R^ 2$ is an open bounded simply connected region with smooth boundary $\Gamma$, \ve is a small positive parameter and $\partial/\partial n$ is the derivative along the inner normal of $\Gamma$. We assume that the degenerate problem $(E_0) \quad f(u,x,0) =0$ has two solutions $\varphi_1(x)$ and $\varphi_2(x)$ intersecting in an smooth Jordan curve ${ \cal C}$ located in $D$ such that $f_u(\varphi_i(x),x,0)$ changes its sign on ${\cal C}$ for $i=1,2$ (exchange of stabilities). By means of the method of asymptotic lower and upper solutions we prove that for sufficiently small \ve, problem (E_\ve) has at least one solution u(x,\ve) satisfying \alpha(x,\ve) \le u(x,\ve) \le \beta(x,\ve) where the upper and lower solutions \beta(x,\ve) and \alpha(x,\ve) respectively fulfil \beta(x,\ve) - \alpha(x,\ve) = O(\sqrt{\ve}) for $x$ in a $\delta$-neighborhood of ${\cal C}$ where $\delta$ is any fixed positive number sufficiently small, while \beta(x,\ve) - \alpha(x,\ve) = O(\ve) for $x \in \overline{D}\backslash D_\delta$. Applying this result to a special reaction system in a nonhomogeneous medium we prove that the reaction rate exhibits a spatial jumping behavior

Singularly perturbed partly dissipative reaction-diffusion systems in case of exchange of stabilities

We consider the singularly perturbed partly dissipative reaction-diffusion system ε2 (∂u ⁄ ∂t - ∂2u ⁄ ∂x2 = g(u,v,x,t,ε), ∂v ⁄ ∂t = ƒ(u,v,x,t,ε) under the condition that the degenerate equation g(u,v,t,0) = 0 has two solutions u = φi(v,x,t), i = 1,2, that intersect (exchange of stabilities). Our main result concerns existence and asymptotic behavior in ε of the solution of the initial boundary value problem under consideration. The proof is based on the method of asymptotic lower and upper solutions

Singularly perturbed boundary value problems for systems of Tichonov's type in case of exchange of stabilities

We consider a system of ordinary differential equations consisting of a singularly perturbed scalar differential equation of second order and a scalar differential equation of first or second order and study a Neuman-Cauchy or a Neuman-Dirichlet problem. We assume that the degenerate equation has two intersecting solutions such that the standard theory for systems of Tichonov's type cannot be applied. We introduce the notation of a composed stable solution. By means of the technique of ordered lower and upper solutions we prove the existence of a solution of our problems near the composed stable solution for sufficiently small ε and determine its asymptotic behavior in ε

Existence and asymptotic stability of a periodic solution with boundary layers of reaction-diffusion equations with singularly perturbed Neumann boundary conditions

We consider singularly perturbed reaction-diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x,t,\ve) with boundary layers and derive conditions for their asymptotic stability The boundary layer part of u(x,t,\ve) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order \ve. Another peculiarity of our problem is that - in contrast to the case of Dirichlet boundary conditions - it may have several asymptotically stable time-periodic solutions, where these solutions differ only in the desribtion of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solution