117 research outputs found
1D analysis of 2D isotropic random walks
Many stochastic systems in physics and biology are investigated by recording
the two-dimensional (2D) positions of a moving test particle in regular time
intervals. The resulting sample trajectories are then used to induce the
properties of the underlying stochastic process. Often, it can be assumed a
priori that the underlying discrete-time random walk model is independent from
absolute position (homogeneity), direction (isotropy) and time (stationarity),
as well as ergodic. In this article we first review some common statistical
methods for analyzing 2D trajectories, based on quantities with built-in
rotational invariance. We then discuss an alternative approach in which the
two-dimensional trajectories are reduced to one dimension by projection onto an
arbitrary axis and rotational averaging. Each step of the resulting 1D
trajectory is further factorized into sign and magnitude. The statistical
properties of the signs and magnitudes are mathematically related to those of
the step lengths and turning angles of the original 2D trajectories,
demonstrating that no essential information is lost by this data reduction. The
resulting binary sequence of signs lends itself for a pattern counting
analysis, revealing temporal properties of the random process that are not
easily deduced from conventional measures such as the velocity autocorrelation
function. In order to highlight this simplified 1D description, we apply it to
a 2D random walk with restricted turning angles (RTA model), defined by a
finite-variance distribution of step length and a narrow turning angle
distribution , assuming that the lengths and directions of the steps
are independent
Principles of efficient chemotactic pursuit
In chemotaxis, cells are modulating their migration patterns in response to
concentration gradients of a guiding substance. Immune cells are believed to
use such chemotactic sensing for remotely detecting and homing in on pathogens.
Considering that an immune cells may encounter a multitude of targets with
vastly different migration properties, ranging from immobile to highly mobile,
it is not clear which strategies of chemotactic pursuit are simultaneously
efficient and versatile. We takle this problem theoretically and define a
tunable response function that maps temporal or spatial concentration gradients
to migration behavior. The seven free parameters of this response function are
optimized numerically with the objective of maximizing search efficiency
against a wide spectrum of target cell properties. Finally, we reverse-engineer
the best-performing parameter sets to uncover the principles of efficient
chemotactic pursuit under different biologically realistic boundary conditions.
Remarkably, the numerical optimization rediscovers chemotactic strategies that
are well-known in biological systems, such as the gradient-dependent swimming
and tumbling modes of E.coli. Some of our results may also be useful for the
design of chemotaxis experiments and for the development of algorithms that
automatically detect and quantify goal oriented behavior in measured immune
cell trajectories
Scaling properties of correlated random walks
Many stochastic time series can be modelled by discrete random walks in which
a step of random sign but constant length is performed after each
time interval . In correlated discrete time random walks (CDTRWs),
the probability for two successive steps having the same sign is unequal
1/2. The resulting probability distribution that a
displacement is observed after a lagtime is known
analytically for arbitrary persistence parameters . In this short note we
show how a CDTRW with parameters can be mapped onto
another CDTRW with rescaled parameters , for arbitrary scaling parameters , so that both walks
have the same displacement distributions on long time
scales. The nonlinear scaling functions and and
derived explicitely. This scaling method can be used to model time series
measured at discrete sample intervals but actually corresponding to
continuum processes with variations occuring on a much shorter time scale
Bayesian inference of time varying parameters in autoregressive processes
In the autoregressive process of first order AR(1), a homogeneous correlated
time series is recursively constructed as , using random Gaussian deviates and fixed values for
the correlation coefficient and for the noise amplitude . To model
temporally heterogeneous time series, the coefficients and can
be regarded as time-dependend variables by themselves, leading to the
time-varying autoregressive processes TVAR(1). We assume here that the time
series is known and attempt to infer the temporal evolution of the
'superstatistical' parameters and . We present a sequential
Bayesian method of inference, which is conceptually related to the Hidden
Markov model, but takes into account the direct statistical dependence of
successively measured variables . The method requires almost no prior
knowledge about the temporal dynamics of and and can handle
gradual and abrupt changes of these superparameters simultaneously. We compare
our method with a Maximum Likelihood estimate based on a sliding window and
show that it is superior for a wide range of window sizes
Inferring long-range interactions between immune and tumor cells -- pitfalls and (partial) solutions
Upcoming immunotherapies for cancer treatment rely on the ability of the
immune system to detect and eliminate tumors in the body. A highly simplified
version of this process can be studied in a Petri dish: starting with a random
distribution of immune and tumor cells, it can be observed in detail how
individual immune cells migrate towards nearby tumor cells, establish contact,
and attack. Nevertheless, it remains unclear whether the immune cells find
their targets by chance, or if they approach them 'on purpose', using remote
sensing mechanisms such as chemotaxis. In this work, we present methods to
infer the strength and range of long-range cell-cell interactions from
time-lapse recorded cell trajectories, using a maximum likelihood method to fit
the model parameters. First, we model the interactions as a distance-dependent
'force' that attracts immune cells towards their nearest tumor cell. While this
approach correctly recovers the interaction parameters of simulated cells with
constant migration properties, it detects spurious interactions in the case of
independent cells that spontaneously change their migration behavior over time.
We therefore use an alternative approach that models the interactions by
distance-dependent probabilities for positive and negative turning angles of
the migrating immune cell. We demonstrate that the latter approach finds the
correct interaction parameters even with temporally switching cell migration
Detecting long-range attraction between migrating cells based on p-value distributions
Immune cells have evolved to recognize and eliminate pathogens, and the
efficiency of this process can be measured in a Petri dish. Yet, even if the
cells are time-lapse recorded and tracked with high resolution, it is difficult
to judge whether the immune cells find their targets by mere chance, or if they
approach them in a goal-directed way, perhaps using remote sensing mechanisms
such as chemotaxis. To answer this question, we assign to each step of an
immune cell a 'p-value', the probability that a move, at least as
target-directed as observed, can be explained with target-independent migration
behavior. The resulting distribution of p-values is compared to the
distribution of a reference system with randomized target positions. By using
simulated data, based on various chemotactic search mechanisms, we demonstrate
that our method can reliably distinguish between blind migration and
target-directed 'hunting' behavior
Adaptive stochastic resonance based on output autocorrelations
Successful detection of weak signals is a universal challenge for numerous
technical and biological systems and crucially limits signal transduction and
transmission. Stochastic resonance (SR) has been identified to have the
potential to tackle this problem, namely to enable non-linear systems to detect
small, otherwise sub-threshold signals by means of added non-zero noise. This
has been demonstrated within a wide range of systems in physical, technological
and biological contexts. Based on its ubiquitous importance, numerous
theoretical and technical approaches aim at an optimization of signal
transduction based on SR. Several quantities like mutual information,
signal-to-noise-ratio, or the cross-correlation between input stimulus and
resulting detector response have been used to determine optimal noise
intensities for SR. The fundamental shortcoming with all these measures is that
knowledge of the signal to be detected is required to compute them. This
dilemma prevents the use of adaptive SR procedures in any application where the
signal to be detected is unknown. We here show that the autocorrelation
function (AC) of the detector response fundamentally overcomes this drawback.
For a simplified model system, the equivalence of the output AC with the
measures mentioned above is proven analytically. In addition, we test our
approach numerically for a variety of systems comprising different input
signals and different types of detectors. The results indicate a strong
similarity between mutual information and output AC in terms of the optimal
noise intensity for SR. Hence, using the output AC to adaptively vary the
amount of added noise in order to maximize information transmission via SR
might be a fundamental processing principle in nature, in particular within
neural systems which could be implemented in future technical applications
Stochastic resonance in three-neuron motifs
Stochastic resonance is a non-linear phenomenon, in which the sensitivity of
signal detectors can be enhanced by adding random noise to the detector input.
Here, we demonstrate that noise can also improve the information flux in
recurrent neural networks. In particular, we show for the case of three-neuron
motifs that the mutual information between successive network states can be
maximized by adding a suitable amount of noise to the neuron inputs. This
striking result suggests that noise in the brain may not be a problem that
needs to be suppressed, but indeed a resource that is dynamically regulated in
order to optimize information processing
Analysis of structure and dynamics in three-neuron motifs
In neural networks with identical neurons, the matrix of connection weights
completely describes the network structure and thereby determines how it is
processing information. However, due to the non-linearity of these systems, it
is not clear if similar microscopic connection structures also imply similar
functional properties, or if a network is impacted more by macroscopic
structural quantities, such as the ratio of excitatory and inhibitory
connections (balance), or the ratio of non-zero connections (density). To
clarify these questions, we focus on motifs of three binary neurons with
discrete ternary connection strengths, an important class of network building
blocks that can be analyzed exhaustively. We develop new, permutation-invariant
metrics to quantify the structural and functional distance between two given
network motifs. We then use multidimensional scaling to identify and visualize
clusters of motifs with similar structural and functional properties. Our
comprehensive analysis reveals that the function of a neural network is only
weakly correlated with its microscopic structure, but depends strongly on the
balance of the connections
Reconstructing fiber networks from confocal image stacks
We present a numerically efficient method to reconstruct a disordered network
of thin biopolymers, such as collagen gels, from three-dimensional (3D) image
stacks recorded with a confocal microscope. Our method is based on a template
matching algorithm that simultaneously performs a binarization and
skeletonization of the network. The size and intensity pattern of the template
is automatically adapted to the input data so that the method is scale
invariant and generic. Furthermore, the template matching threshold is
iteratively optimized to ensure that the final skeletonized network obeys a
universal property of voxelized random line networks, namely, solid-phase
voxels have most likely three solid-phase neighbors in a
neighborhood. This optimization criterion makes our method free of user-defined
parameters and the output exceptionally robust against imaging noise
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