From Synaptic Interactions to Collective Dynamics in Random Neuronal Networks Models: Critical Role of Eigenvectors and Transient Behavior.

The study of neuronal interactions is at the center of several big collaborative neuroscience projects (including the Human Connectome Project, the Blue Brain Project, and the Brainome) that attempt to obtain a detailed map of the entire brain. Under certain constraints, mathematical theory can advance predictions of the expected neural dynamics based solely on the statistical properties of the synaptic interaction matrix. This work explores the application of free random variables to the study of large synaptic interaction matrices. Besides recovering in a straightforward way known results on eigenspectra in types of models of neural networks proposed by Rajan and Abbott (2006), we extend them to heavy-tailed distributions of interactions. More important, we analytically derive the behavior of eigenvector overlaps, which determine the stability of the spectra. We observe that on imposing the neuronal excitation/inhibition balance, despite the eigenvalues remaining unchanged, their stability dramatically decreases due to the strong nonorthogonality of associated eigenvectors. This leads us to the conclusion that understanding the temporal evolution of asymmetric neural networks requires considering the entangled dynamics of both eigenvectors and eigenvalues, which might bear consequences for learning and memory processes in these models. Considering the success of free random variables theory in a wide variety of disciplines, we hope that the results presented here foster the additional application of these ideas in the area of brain sciences.

Distribution of DNA fragment sizes after irradiation with ions.

Ionizing radiation is responsible for production of double-strand breaks (DSBs) in a DNA structure. In contrast to sparsely ionizing radiation, densely ionizing radiation produces DSBs that are non-randomly distributed along the DNA molecule and can form clusters of various size. The paper discusses minimalistic models that describe observable patterns of fragment length in DNA segments irradiated with heavy ions and applies the formalism to interpret the recent experimental data collected by use of atomic force microscope (AFM).

Distribution of double-strand breaks induced by ionizing radiation at the level of single DNA molecules examined by atomic force microscopy.

Double-strand breaks (DSBs) are the most critical radiation-induced lesions, because they result in the fragmentation of the DNA molecule and because a single unrepaired DSB may lead to cell death. We present the results of radiation-induced fragmentation of plasmid DNA analyzed by atomic force microscopy (AFM) to allow the visualization of individual DNA molecules. Linear PhiX174 plasmid DNA was exposed to a wide range of doses of low-LET X rays and high-LET carbon, nickel and uranium ions. The induced DNA fragments were detected and measured based on the recorded AFM images and fragment length distributions were derived for each radiation type and dose. The results show a dose- and radiation type-dependent DNA fragmentation with a significantly larger fraction of short fragments produced by high-LET radiation compared to X rays. This can be considered as experimental evidence of DSB clustering due to inhomogeneous energy deposition at the level of the plasmid DNA molecule. Additionally, the experimentally derived fragment profiles were compared and found to be in agreement with the prediction of a model simulating the fragmentation of DNA molecules induced by radiation.

Transport in a Lévy ratchet: group velocity and distribution spread.

We consider the motion of an overdamped particle in a periodic potential lacking spatial symmetry under the influence of symmetric, white, Lévy noise, being a minimal setup for a "Lévy ratchet." Due to the nonthermal character of the Lévy noise, the particle exhibits a motion with a preferred direction even in the absence of whatever additional time-dependent forces. The examination of the Lévy ratchet has to be based on the characteristics of directionality which are different from typically used measures such as mean current and the dispersion of particle positions, since these become inappropriate when the moments of the noise diverge. To overcome this problem, we discuss robust measures of directionality of transport such as the position of the median of the particle displacement distribution characterizing the group velocity and the interquantile distance giving the measure of the distribution width. Moreover, we analyze the behavior of splitting probabilities for leaving an interval of a given length, unveiling qualitative differences between the noises with Lévy indices below and above unity.

Stationary states in Langevin dynamics under asymmetric Lévy noises.

Properties of systems driven by white non-Gaussian noises can be very different from these of systems driven by the white Gaussian noise. We investigate stationary probability densities for systems driven by alpha-stable Lévy-type noises, which provide natural extension to the Gaussian noise having, however, a new property, namely a possibility of being asymmetric. Stationary probability densities are examined for a particle moving in parabolic, quartic, and in generic double well potential models subjected to the action of alpha-stable noises. Relevant solutions are constructed by methods of stochastic dynamics. In situations where analytical results are known they are compared with numerical results. Furthermore, the problem of estimation of the parameters of stationary densities is investigated.

Cell cycle arrest and aberration yield in normal human fibroblasts. II: Effects of 11 MeV u-1 C ions and 9.9 MeV u-1 Ni ions.

PURPOSE: To investigate further the relationship between high linear energy transfer (LET) induced cell cycle arrests and the yield of chromosome aberrations observable in normal human fibroblasts at the first post-irradiation mitosis. MATERIALS AND METHODS: Normal human fibroblasts (AG01,522C) were exposed in G0/G1 to either 11 MeV u(-1) C ions (LET = 153.5 keV microm(-1)) or 9.9 MeV u(-1) Ni ions (LET = 2,455 keV microm(-1)), subcultured in medium containing 5-Bromo-2'-deoxyuridine (BrdU) and at multiple time-points post-irradiation the yield of chromosomal damage, the mitotic index and the cumulative BrdU-labelling index were determined. Furthermore, a mathematical approach was used to analyse the entire cell population. RESULTS: Following high LET exposure normal fibroblasts suffer a transient delay into S-phase and into mitosis as well as a prolonged, probably permanent cell cycle arrest in the initial G0/G1-phase. Cells that reach the first mitosis at early times carried less aberrations than those collected at later times indicating a relationship between cell cycle delay and the number of aberrations. However, with respect to the whole cell population, only a few aberrant fibroblasts are able to progress to the first mitosis. For all endpoints studied the relative biological effectiveness (RBE) of C ions is in the range of 2 - 4, while for Ni ions RBE < 1 is estimated. In contrast, when compared on a per particle basis Ni ions with the higher ionization density were found to be more effective. CONCLUSIONS: Detailed analysis of the data demonstrates that the number of fibroblasts at risk for neoplastic transformation is significantly reduced by a chronic cell cycle arrest in the initial G0/G1-phase and, for the first time, the LET-dependence of this effect has been shown.

Escape driven by alpha-stable white noises.

We explore the archetype problem of an escape dynamics occurring in a symmetric double well potential when the Brownian particle is driven by white Lévy noise in a dynamical regime where inertial effects can safely be neglected. The behavior of escaping trajectories from one well to another is investigated by pointing to the special character that underpins the noise-induced discontinuity which is caused by the generalized Brownian paths that jump beyond the barrier location without actually hitting it. This fact implies that the boundary conditions for the mean first passage time (MFPT) are no longer determined by the well-known local boundary conditions that characterize the case with normal diffusion. By numerically implementing properly the set up boundary conditions, we investigate the survival probability and the average escape time as a function of the corresponding Lévy white noise parameters. Depending on the value of the skewness beta of the Lévy noise, the escape can either become enhanced or suppressed: a negative asymmetry parameter beta typically yields a decrease for the escape rate while the rate itself depicts a non-monotonic behavior as a function of the stability index alpha that characterizes the jump length distribution of Lévy noise, exhibiting a marked discontinuity at alpha=1. We find that the typical factor of 2 that characterizes for normal diffusion the ratio between the MFPT for well-bottom-to-well-bottom and well-bottom-to-barrier-top no longer holds true. For sufficiently high barriers the survival probabilities assume an exponential behavior versus time. Distinct non-exponential deviations occur, however, for low barrier heights.

Lévy-Brownian motion on finite intervals: Mean first passage time analysis.

We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by Lévy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulas when the stability index alpha approaches 2, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise.

Correlation between mitotic delay and aberration burden, and their role for the analysis of chromosomal damage.

The aim was to investigate further the relationship between radiation-induced mitotic delay and the expression of chromosome damage in V79 cells. Recently published data on the time-course of chromosome aberrations in V79 first-cycle metaphases after exposure to 10.4 MeV u(-1) Ar ions (LET = 1226 keV microm(-1)) were supplemented and reanalysed. A statistical analysis of the distribution of aberrations among cells was performed. Furthermore, cells were grouped into subpopulations carrying 0, 1 -2, 3-4, 5- 6 and 7 or more aberrations. Then, based on the mitotic index, the flux of each subgroup through the first mitosis was determined and the average entrance time to mitosis was estimated. For comparison, the flux of aberrant V79 cells generated by X-irradiation was analysed. Analysis of the Ar ion data revealed that the flux of each subpopulation through the first mitosis is strongly affected by its aberration burden, i.e. a positive correlation between the mitotic delay and the number of aberrations carried by a cell was observed. The distribution of aberrations among cells could be well described by Neyman-type A statistics; the corresponding fit parameters also reflect the damage-dependent mitotic delay. Interestingly, comparison of the flux of Ar ion and X-ray-irradiated V79 cells through mitosis revealed (1) that a direct correlation exists between the number of aberrations carried by a cell and its average entrance time to mitosis, and (2) that this effect is independent of the linear energy transfer. The role of these observations for radiation cytogenetics is discussed.

Diffusion of DNA molecules in gel at high electric fields.

Problems related to the gel electrophoretic migration of polymers can be investigated by models based on a Brownian-type ratchet where a particle can undergo a net transport on a potential energy surface that is externally driven to fluctuate between several distinct states. Here we describe the method of polymer transport and separation by means of the cellular automata technique. Numerical simulations of the polymer reptation in the model system allow us to understand the band-broadening processes in the gel electrophoresis experiments. They indicate also possible ways of fine-tuning the parameters in designing desired resolution of the experiments.

Cell cycle arrest and aberration yield in normal human fibroblasts. I. Effects of X-rays and 195 MeV u(-1) C ions.

PURPOSE: To examine the relationship between cell proliferation and the expression of chromosomal damage in normal human skin fibroblasts after X-ray and particle irradiation. MATERIALS AND METHODS: Confluent G0/G1 AG1522B cells were exposed to X-rays or 195MeV u(-1) C ions with a linear energy transfer of 16.6 keV microm(-1) in the dose range 1-4 Gy. Directly after irradiation, cells were reseeded at a low density in medium containing 5-bromo-2'-deoxyuridine. At multiple time points post-irradiation, the cumulative BrdU-labelling index, mitotic index and aberration frequency were measured. Based on these data, the total amount of damage induced within the entire cell population was estimated by means of mathematical analysis. RESULTS: Both types of radiation exposure exert a pronounced effect on the cell cycle progression of fibroblasts. They result in delayed entry of cells into S-phase and into the first mitosis, and cause a dramatic reduction in mitotic activity. Measurement of chromosomal damage in first-cycle cells at multiple time points post-irradiation shows that the frequencies of aberrant cells and aberrations increase with time up to twofold for the lower doses. However, for the higher doses, this effect is less pronounced or even disappears. When the data for the whole cell population are analysed, it becomes evident that only a few damaged fibroblasts can progress to the first mitosis, a response attributable at least in part to a long-term arrest of injured cells in the initial G0/G1-phase. As observed in other investigations, the effectiveness of 195 MeV u(-1) C ions was similar or slightly higher than X-rays for all endpoints studied leading to a relative biological effectiveness in the range 1.0-1.4. CONCLUSIONS: Cell cycle arrests affect the aberration yield observable in normal human fibroblasts at mitosis. The data obtained for the cell population as a whole reveal that injured cells are rapidly removed from the mitotically active population through a chronic cell cycle arrest, which is consistent with other studies that indicate that this response is a specific strategy of fibroblasts to minimize the fixation and propagation of genetic alterations.

Analysis of Ar-ion and X-ray-induced chromatin breakage and repair in V79 plateau-phase cells by the premature chromosome condensation technique.

PURPOSE: The premature chromosome condensation technique has been used to compare chromatin breakage and repair in noncycling V79 cells following high and low LET radiation. MATERIALS AND METHODS: Plateau-phase V79 cells were exposed to graded doses of low energy Ar ions (LET 1233 keV/microm) and X-rays. Cells were fused to mitotic V79 cells immediately after exposure to examine initial chromatin breakage or after various time intervals of post-irradiation incubation to investigate the kinetics of chromatin break rejoining as well as the fraction of unrejoined fragments. RESULTS AND CONCLUSIONS: For both radiation qualities an average initial number of about 2.4 excess PCC fragments per cell per Gy was found increasing linearly with dose. The distributions of PCC chromosomes plus excess fragments among cells followed Poisson statistics after X-ray irradiation, while an overdispersion of the frequencies was observed after Ar-irradiation indicating that a single particle traversal through a cell nucleus can produce multiple chromatin lesions. Moreover, for both radiation types the rejoining of excess fragments has been examined. Both data sets could be fitted well to first-order kinetics with a single component. Despite similar rates of rejoining cellular repair was noticeably less effective for Ar ions than for X-rays. While after 10 h of post-irradiation incubation 60% of Ar ion induced excess fragments remained unrejoined, only 14% of X-ray-induced lesions were not rejoined. Furthermore, comparison of the residual number of excess PCC fragments with recently published data on the yield of chromosome aberrations in first post-irradiation metaphases shows that for both radiation types more aberrations are detected in interphase than in metaphase cells. Yet, for comparable doses this difference is more pronounced for Ar ions indicating that scoring of high LET induced aberrations in metaphase cells might result in a significant underestimation of the produced damage.

Mathematical models of radiation-induced mitotic delay: time course analysis and statistics of lesions.

Detailed investigations of high and low LET radiation induced chromosome aberrations in various mammalian cell lines have shown that the registered yield of aberrations depends on cell cycle progression delays. The effect of radiation on the cell kinetics can be analyzed in terms of kinetic growth models. The method yields the number of aberrant cells and the number of aberrations as totals obtained after integration over given time-interval.

High-LET-induced chromosomal damage: time-dependent expression.

Chromosome aberrations are routinely analysed in metaphase cells at one sampling time post-irradiation. Yet, accumulating evidence shows that radiation-induced cycle perturbations and mitotic delay influence the yield of aberrations detectable in mitosis. In extended time-course studies a drastic increase in the number of aberrations with sampling time has been observed after particle irradiation, while after the exposure to sparsely ionizing radiation a less pronounced effect has been found. This difference in the time-course of chromosomal damage is particularly important for the determination of accurate RBE values. As will be discussed, meaningful RBE values for particles can only be obtained, if cells are analysed at multiple sampling times and the complete time-course of aberrations is considered. Otherwise, particle-induced damage will be over- or underestimated. Moreover, depending on the cell system chosen for the analysis, factors like the loss of damaged cells due to apoptosis or a permanent cell cycle arrest complicate the determination of accurate RBE values based on chromosome data.

High-LET-induced chromosome aberrations in V79 cells analysed in first and second post-irradiation metaphases.

PURPOSE: As an extension of previous studies, the time-course of high-LET-induced chromosomal damage was investigated in first- and second-cycle V79 Chinese hamster cells. MATERIALS AND METHODS: Cells were exposed in G1 to 10.4 MeV/u Ar ions (LET = 1226 keV/microm) and chromosomal damage was measured at 2h sampling intervals between 10 h and 34 h after irradiation. To distinguish between cells in different post-irradiation cycles, the fluorescence-plus-Giemsa technique was applied. RESULTS: For first- and second-generation cells, the number of aberrant metaphases and aberrations per metaphase were found to increase markedly with sampling time, demonstrating that cell cycle progression was delayed according to the number of lesions carried by the cell. To account for the time-dependent expression of chromosomal damage a mathematical approach was used based on the integrated flux of aberrant cells entering mitosis. Moreover, the analysis of Ar ion-induced chromosome lesions confirmed that high-LET radiation results in specific changes in the spectrum of aberration types. In particular, an increased rate of chromatid-type aberrations as well as a high frequency of chromosomal breaks was found, although the cells were exposed in G1. CONCLUSIONS: Due to the fact that cells collected at one sampling time are not representative of the entire population, the complete time-course of chromosomal damage has to be taken into account for the determination of a meaningful RBE value. Otherwise, the analysis of chromosomal damage can result in a pronounced over- or underestimation of the RBE depending on the subpopulation of cells entering mitosis at that particular sampling time.

Description of interacting channel gating using a stochastic Markovian model.

Single-channel recordings from membrane patches frequently exhibit multiple conductance levels. In some preparations, the steady-state probabilities of observing these levels do not follow a binomial distribution. This behavior has been reported in sodium channels, potassium channels, acetylcholine receptor channels and gap junction channels. A non-binomial distribution suggests interaction of the channels or the presence of channels or the presence of channels with different open probabilities. However, the current trace sometimes exhibits single transitions spanning several levels. Since the probability of simultaneous transitions of independent channels is infinitesimally small, such observations strongly suggest a cooperative gating behavior. We present a Markov model to describe the cooperative gating of channels using only the all-points current amplitude histograms for the probability of observing the various conductance levels. We investigate the steady-state (or equilibrium) properties of a system of N channels and provide a scheme to express all the probabilities in terms of just two parameters. The main feature of our model is that lateral interaction of channels gives rise to cooperative gating. Another useful feature is the introduction of the language of graph theory which can potentially provide a different avenue to study ion channel kinetics. We write down explicit expressions for systems of two, three and four channels and provide a procedure to describe the system of N channels.