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Mathematical Biosciences and Engineering: MBE

Massimiliano Tamborrino
The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here, we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passage time density are not available, we propose to approximate the curved boundary by means of a continuous two-piecewise linear threshold. Explicit expressions for the first passage time density towards the new boundary are provided...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Andrey Olypher, Jean Vaillant
Information processing in neuronal networks in certain important cases can be considered as maps of binary vectors, where ones (spikes) and zeros (no spikes) of input neurons are transformed into spikes and no spikes of output neurons. A simple but fundamental characteristic of such a map is how it transforms distances between input vectors into distances between output vectors. We advanced earlier known results by finding an exact solution to this problem for McCulloch-Pitts neurons. The obtained explicit formulas allow for detailed analysis of how the network connectivity and neuronal excitability affect the transformation of distances in neurons...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Frederik Riis Mikkelsen
Determining excitability thresholds in neuronal models is of high interest due to its applicability in separating spiking from non-spiking phases of neuronal membrane potential processes. However, excitability thresholds are known to depend on various auxiliary variables, including any conductance or gating variables. Such dependences pose as a double-edged sword; they are natural consequences of the complexity of the model, but proves difficult to apply in practice, since gating variables are rarely measured...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Marie Levakova
It is studied what level of a continuous-valued signal is optimally estimable on the basis of first-spike latency neuronal data. When a spontaneous neuronal activity is present, the first spike after the stimulus onset may be caused either by the stimulus itself, or it may be a result of the prevailing spontaneous activity. Under certain regularity conditions, Fisher information is the inverse of the variance of the best estimator. It can be considered as a function of the signal intensity and then indicates accuracy of the estimation for each signal level...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Achilleas Koutsou, Jacob Kanev, Maria Economidou, Chris Christodoulou
The operational mode of a neuron (i.e., whether a neuron is an integrator or a coincidence detector) is in part determined by the degree of synchrony in the firing of its pre-synaptic neural population. More specifically, it is determined by the degree of synchrony that causes the neuron to fire. In this paper, we investigate the relationship between the input and the operational mode. We compare the response-relevant input synchrony, which measures the operational mode and can be determined using a membrane potential slope-based measure [7], with the spike time distance of the spike trains driving the neuron, which measures spike train synchrony and can be determined using the multivariate SPIKE-distance metric [10]...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Lubomir Kostal, Shigeru Shinomoto
Recently, it has been suggested that certain neurons with Poissonian spiking statistics may communicate by discontinuously switching between two levels of firing intensity. Such a situation resembles in many ways the optimal information transmission protocol for the continuous-time Poisson channel known from information theory. In this contribution we employ the classical information-theoretic results to analyze the efficiency of such a transmission from different perspectives, emphasising the neurobiological viewpoint...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Sven Blankenburg, Benjamin Lindner
Experimentally it is known that some neurons encode preferentially information about low-frequency (slow) components of a time-dependent stimulus while others prefer intermediate or high-frequency (fast) components. Accordingly, neurons can be categorized as low-pass, band-pass or high-pass information filters. Mechanisms of information filtering at the cellular and the network levels have been suggested. Here we propose yet another mechanism, based on noise shaping due to spontaneous non-renewal spiking statistics...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Roberta Sirovich, Luisa Testa
A new definition of firing time is given in the framework of Integrate and Fire neuronal models. The classical absorption condition at the threshold is relaxed and the firing time is defined as the first time the membrane potential process lies above a fixed depolarisation level for a sufficiently long time. The mathematical properties of the new firing time are investigated both for the Perfect Integrator and the Leaky Integrator. In the latter case, a simulation study is presented to complete the analysis where analytical results are not yet achieved...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Shinsuke Koyama, Ryota Kobayashi
Fluctuation scaling has been observed universally in a wide variety of phenomena. In time series that describe sequences of events, fluctuation scaling is expressed as power function relationships between the mean and variance of either inter-event intervals or counting statistics, depending on measurement variables. In this article, fluctuation scaling has been formulated for a series of events in which scaling laws in the inter-event intervals and counting statistics were related. We have considered the first-passage time of an Ornstein-Uhlenbeck process and used a conductance-based neuron model with excitatory and inhibitory synaptic inputs to demonstrate the emergence of fluctuation scaling with various exponents, depending on the input regimes and the ratio between excitation and inhibition...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Giuseppe D'Onofrio, Enrica Pirozzi
Two different stochastic processes are used to model the evolution of the membrane voltage of a neuron exposed to a time-varying input signal. The first process is an inhomogeneous Ornstein-Uhlenbeck process and its first passage time through a constant threshold is used to model the first spike time after the signal onset. The second process is a Gauss-Markov process identified by a particular mean function dependent on the first passage time of the first process. It is shown that the second process is also of a diffusion type...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora
A model is proposed to describe the spike-frequency adaptation observed in many neuronal systems. We assume that adaptation is mainly due to a calcium-activated potassium current, and we consider two coupled stochastic differential equations for which an analytical approach combined with simulation techniques and numerical methods allow to obtain both qualitative and quantitative results about asymptotic mean firing rate, mean calcium concentration and the firing probability density. A related algorithm, based on the Hazard Rate Method, is also devised and described...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Susanne Ditlevsen, Petr Lansky
This Special Issue of Mathematical Biosciences and Engineering contains 11 selected papers presented at the Neural Coding 2014 workshop. The workshop was held in the royal city of Versailles in France, October 6-10, 2014. This was the 11th of a series of international workshops on this subject, the first held in Prague (1995), then Versailles (1997), Osaka (1999), Plymouth (2001), Aulla (2003), Marburg (2005), Montevideo (2007), Tainan (2009), Limassol (2010), and again in Prague (2012). Also selected papers from Prague were published as a special issue of Mathematical Biosciences and Engineering and in this way a tradition was started...
June 1, 2016: Mathematical Biosciences and Engineering: MBE
Jinhu Xu, Yicang Zhou
A within-host viral infection model with both virus-to-cell and cell-to-cell transmissions and time delay in immune response is investigated. Mathematical analysis shows that delay may destabilize the infected steady state and lead to Hopf bifurcation. Moreover, the direction of the Hopf bifurcation and the stability of the periodic solutions are investigated by normal form and center manifold theory. Numerical simulations are done to explore the rich dynamics, including stability switches, Hopf bifurcations, and chaotic oscillations...
April 1, 2016: Mathematical Biosciences and Engineering: MBE
Marcelo E de Oliveira, Luiz M G Neto
In this work, we present and investigate a multiscale model to simulate 3D growth of glioblastomas (GBMs) that incorporates features of the tumor microenvironment and derives macroscopic growth laws from microscopic tissue structure information. We propose a normalized version of the Shannon entropy as an alternative measure of the directional anisotropy for an estimation of the diffusivity tensor in cases where the latter is unknown. In our formulation, the tumor aggressiveness and morphological behavior is tissue-type dependent, i...
April 1, 2016: Mathematical Biosciences and Engineering: MBE
Agnieszka Bartomiejczyk, Henryk Leszczynski
We prove a weak maximum principle for structured population models with dynamic boundary conditions. We establish existence and positivity of solutions of these models and investigate the asymptotic behaviour of solutions. In particular, we analyse so called size profile.
April 1, 2016: Mathematical Biosciences and Engineering: MBE
Christian Engwer, Markus Knappitsch, Christina Surulescu
Glioma is a broad class of brain and spinal cord tumors arising from glia cells, which are the main brain cells that can develop into neoplasms. They are highly invasive and lead to irregular tumor margins which are not precisely identifiable by medical imaging, thus rendering a precise enough resection very difficult. The understanding of glioma spread patterns is hence essential for both radiological therapy as well as surgical treatment. In this paper we propose a multiscale model for glioma growth including interactions of the cells with the underlying tissue network, along with proliferative effects...
April 1, 2016: Mathematical Biosciences and Engineering: MBE
Steady Mushayabasa, Drew Posny, Jin Wang
We propose a new mathematical modeling framework to investigate the transmission and spread of foot-and-mouth disease. Our models incorporate relevant biological and ecological factors, vaccination effects, and seasonal impacts during the complex interaction among susceptible, vaccinated, exposed, infected, carrier, and recovered animals. We conduct both epidemic and endemic analysis, with a focus on the threshold dynamics characterized by the basic reproduction numbers. In addition, numerical simulation results are presented to demonstrate the analytical findings...
April 1, 2016: Mathematical Biosciences and Engineering: MBE
Louis D Bergsman, James M Hyman, Carrie A Manore
We develop a mathematical model for transmission of West Nile virus (WNV) that incorporates resident and migratory host avian populations and a mosquito vector population. We provide a detailed analysis of the model's basic reproductive number and demonstrate how the exposed infected, but not infectious, state for the bird population can be approximated by a reduced model. We use the model to investigate the interplay of WNV in both resident and migratory bird hosts. The resident host parameters correspond to the American Crow (Corvus brachyrhynchos), a competent host with a high death rate due to disease, and migratory host parameters to the American Robin (Turdus migratorius), a competent host with low WNV death rates...
April 1, 2016: Mathematical Biosciences and Engineering: MBE
Le Liu, Lihong Huang, Jianhong Wu
Motsch and Tadmor considered an extended Cucker-Smale model to investigate the flocking behavior of self-organized systems of interacting species. In this extended model, a cone of the vision was introduced so that outside the cone the influence of one agent on the other is lost and hence the corresponding influence function takes the value zero. This creates a problem to apply the Motsch-Tadmor and Cucker-Smale method to prove the flocking property of the system. Here, we examine the variation of the velocity angles between two arbitrary agents, and obtain a monotonicity property for the maximum cone of velocity angles...
April 1, 2016: Mathematical Biosciences and Engineering: MBE
Daniel Wetzel
We study steady states in a reaction-diffusion system for a benthic bacteria-nutrient model in a marine sediment over 1D and 2D domains by using Landau reductions and numerical path following methods. We point out how the system reacts to changes of the strength of food supply and ingestion. We find that the system has a stable homogeneous steady state for relatively large rates of food supply and ingestion, while this state becomes unstable if one of these rates decreases and Turing patterns such as hexagons and stripes start to exist...
April 1, 2016: Mathematical Biosciences and Engineering: MBE
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