212 research outputs found
An Efficient Coding Theory for a Dynamic Trajectory Predicts non-Uniform Allocation of Grid Cells to Modules in the Entorhinal Cortex
Grid cells in the entorhinal cortex encode the position of an animal in its
environment using spatially periodic tuning curves of varying periodicity.
Recent experiments established that these cells are functionally organized in
discrete modules with uniform grid spacing. Here we develop a theory for
efficient coding of position, which takes into account the temporal statistics
of the animal's motion. The theory predicts a sharp decrease of module
population sizes with grid spacing, in agreement with the trends seen in the
experimental data. We identify a simple scheme for readout of the grid cell
code by neural circuitry, that can match in accuracy the optimal Bayesian
decoder of the spikes. This readout scheme requires persistence over varying
timescales, ranging from ~1ms to ~1s, depending on the grid cell module. Our
results suggest that the brain employs an efficient representation of position
which takes advantage of the spatiotemporal statistics of the encoded variable,
in similarity to the principles that govern early sensory coding.Comment: 23 pages, 5 figures. Supplemental Information available from the
authors on request. A previous version of this work appeared in abstract form
(Program No. 727.02. 2015 Neuroscience Meeting Planner. Chicago, IL: Society
for Neuroscience, 2015. Online.
Inter-Particle Distribution Functions for One-Species Diffusion-Limited Annihilation, A+A->0
Diffusion-limited annihilation, , and coalescence, , may
both be exactly analyzed in one dimension. While the concentrations of
particles in the two processes bear a simple relation, the inter-particle
distribution functions (IPDF) exhibit remarkable differences. However, the IPDF
is known exactly only for the coalescence process. We obtain the IPDF for the
annihilation process, based on the Glauber spin approach and assuming that the
IPDF's of nearest-particle pairs are statistically independent. This assumption
is supported by computer simulations. Our analysis sheds further light on the
relationship between the annihilation and the coalescence models.Comment: 15 pages, plain TeX, 3 figures - available upon request (snail mail
Target annihilation by diffusing particles in inhomogeneous geometries
The survival probability of immobile targets, annihilated by a population of
random walkers on inhomogeneous discrete structures, such as disordered solids,
glasses, fractals, polymer networks and gels, is analytically investigated. It
is shown that, while it cannot in general be related to the number of distinct
visited points, as in the case of homogeneous lattices, in the case of bounded
coordination numbers its asymptotic behaviour at large times can still be
expressed in terms of the spectral dimension , and its exact
analytical expression is given. The results show that the asymptotic survival
probability is site independent on recurrent structures (),
while on transient structures () it can strongly depend on the
target position, and such a dependence is explicitly calculated.Comment: To appear in Physical Review E - Rapid Communication
Transport and Scaling in Quenched 2D and 3D L\'evy Quasicrystals
We consider correlated L\'evy walks on a class of two- and three-dimensional
deterministic self-similar structures, with correlation between steps induced
by the geometrical distribution of regions, featuring different diffusion
properties. We introduce a geometric parameter , playing a role
analogous to the exponent characterizing the step-length distribution in random
systems. By a {\it single-long jump} approximation, we analytically determine
the long-time asymptotic behavior of the moments of the probability
distribution, as a function of and of the dynamic exponent
associated to the scaling length of the process. We show that our scaling
analysis also applies to experimentally relevant quantities such as escape-time
and transmission probabilities.
Extensive numerical simulations corroborate our results which, in general,
are different from those pertaining to uncorrelated L\'evy-walks models.Comment: 10 pages, 11 figures; some concepts rephrased to improve on clarity;
a few references added; symbols and line styles in some figures changed to
improve on visibilit
Survival probabilities in time-dependent random walks
We analyze the dynamics of random walks in which the jumping probabilities
are periodic {\it time-dependent} functions. In particular, we determine the
survival probability of biased walkers who are drifted towards an absorbing
boundary. The typical life-time of the walkers is found to decrease with an
increment of the oscillation amplitude of the jumping probabilities. We discuss
the applicability of the results in the context of complex adaptive systems.Comment: 4 pages, 3 figure
Survival Probabilities of History-Dependent Random Walks
We analyze the dynamics of random walks with long-term memory (binary chains
with long-range correlations) in the presence of an absorbing boundary. An
analytically solvable model is presented, in which a dynamical phase-transition
occurs when the correlation strength parameter \mu reaches a critical value
\mu_c. For strong positive correlations, \mu > \mu_c, the survival probability
is asymptotically finite, whereas for \mu < \mu_c it decays as a power-law in
time (chain length).Comment: 3 pages, 2 figure
L\'evy-type diffusion on one-dimensional directed Cantor Graphs
L\'evy-type walks with correlated jumps, induced by the topology of the
medium, are studied on a class of one-dimensional deterministic graphs built
from generalized Cantor and Smith-Volterra-Cantor sets. The particle performs a
standard random walk on the sets but is also allowed to move ballistically
throughout the empty regions. Using scaling relations and the mapping onto the
electric network problem, we obtain the exact values of the scaling exponents
for the asymptotic return probability, the resistivity and the mean square
displacement as a function of the topological parameters of the sets.
Interestingly, the systems undergoes a transition from superdiffusive to
diffusive behavior as a function of the filling of the fractal. The
deterministic topology also allows us to discuss the importance of the choice
of the initial condition. In particular, we demonstrate that local and average
measurements can display different asymptotic behavior. The analytic results
are compared with the numerical solution of the master equation of the process.Comment: 9 pages, 9 figure
Phase-Transition in Binary Sequences with Long-Range Correlations
Motivated by novel results in the theory of correlated sequences, we analyze
the dynamics of random walks with long-term memory (binary chains with
long-range correlations). In our model, the probability for a unit bit in a
binary string depends on the fraction of unities preceding it. We show that the
system undergoes a dynamical phase-transition from normal diffusion, in which
the variance D_L scales as the string's length L, into a super-diffusion phase
(D_L ~ L^{1+|alpha|}), when the correlation strength exceeds a critical value.
We demonstrate the generality of our results with respect to alternative
models, and discuss their applicability to various data, such as coarse-grained
DNA sequences, written texts, and financial data.Comment: 4 pages, 4 figure
Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents
A number of results for reactions involving subdiffusive species all with the
same anomalous exponent gamma have recently appeared in the literature and can
often be understood in terms of a subordination principle whereby time t in
ordinary diffusion is replaced by t^gamma. However, very few results are known
for reactions involving different species characterized by different anomalous
diffusion exponents. Here we study the reaction dynamics of a (sub)diffusive
particle surrounded by a sea of (sub)diffusive traps in one dimension. We find
rigorous results for the asymptotic survival probability of the particle in
most cases, with the exception of the case of a particle that diffuses normally
while the anomalous diffusion exponent of the traps is smaller than 2/3.Comment: To appear in Phys. Rev.
Coherent control of correlated nanodevices: A hybrid time-dependent numerical renormalization-group approach to periodic switching
The time-dependent numerical renormalization-group approach (TD-NRG),
originally devised for tracking the real-time dynamics of quantum-impurity
systems following a single quantum quench, is extended to multiple switching
events. This generalization of the TD-NRG encompasses the possibility of
periodic switching, allowing for coherent control of strongly correlated
systems by an external time-dependent field. To this end, we have embedded the
TD-NRG in a hybrid framework that combines the outstanding capabilities of the
numerical renormalization group to systematically construct the effective
low-energy Hamiltonian of the system with the prowess of complementary
approaches for calculating the real-time dynamics derived from this
Hamiltonian. We demonstrate the power of our approach by hybridizing the TD-NRG
with the Chebyshev expansion technique in order to investigate periodic
switching in the interacting resonant-level model. Although the interacting
model shares the same low-energy fixed point as its noninteracting counterpart,
we surprisingly find the gradual emergence of damped oscillations as the
interaction strength is increased. Focusing on a single quantum quench and
using a strong-coupling analysis, we reveal the origin of these
interaction-induced oscillations and provide an analytical estimate for their
frequency. The latter agrees well with the numerical results.Comment: 20 pager, Revtex, 10 figures, submitted to Physical Review
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