135 research outputs found
Energy Dissipation Via Coupling With a Finite Chaotic Environment
We study the flow of energy between a harmonic oscillator (HO) and an
external environment consisting of N two-degrees of freedom non-linear
oscillators, ranging from integrable to chaotic according to a control
parameter. The coupling between the HO and the environment is bilinear in the
coordinates and scales with system size with the inverse square root of N. We
study the conditions for energy dissipation and thermalization as a function of
N and of the dynamical regime of the non-linear oscillators. The study is
classical and based on single realization of the dynamics, as opposed to
ensemble averages over many realizations. We find that dissipation occurs in
the chaotic regime for a fairly small N, leading to the thermalization of the
HO and environment a Boltzmann distribution of energies for a well defined
temperature. We develop a simple analytical treatment, based on the linear
response theory, that justifies the coupling scaling and reproduces the
numerical simulations when the environment is in the chaotic regime.Comment: 7 pages, 10 figure
Modular structure in C. elegans neural network and its response to external localized stimuli
Synchronization plays a key role in information processing in neuronal
networks. Response of specific groups of neurons are triggered by external
stimuli, such as visual, tactile or olfactory inputs. Neurons, however, can be
divided into several categories, such as by physical location, functional role
or topological clustering properties. Here we study the response of the
electric junction C. elegans network to external stimuli using the partially
forced Kuramoto model and applying the force to specific groups of neurons.
Stimuli were applied to topological modules, obtained by the ModuLand
procedure, to a ganglion, specified by its anatomical localization, and to the
functional group composed of all sensory neurons. We found that topological
modules do not contain purely anatomical groups or functional classes,
corroborating previous results, and that stimulating different classes of
neurons lead to very different responses, measured in terms of synchronization
and phase velocity correlations. In all cases, however, the modular structure
hindered full synchronization, protecting the system from seizures. More
importantly, the responses to stimuli applied to topological and functional
modules showed pronounced patterns of correlation or anti-correlation with
other modules that were not observed when the stimulus was applied to ganglia.Comment: 23 pages, 6 figure
Generalized frustration in the multidimensional Kuramoto model
The Kuramoto model was recently extended to arbitrary dimensions by
reinterpreting the oscillators as particles moving on the surface of unit
spheres in a D-dimensional space. Each particle is then represented by a
D-dimensional unit vector. For the particles move on the unit circle and
the vectors can be described by a single phase, recovering the original
Kuramoto model. This multidimensional description can be further extended by
promoting the coupling constant between the particles to a matrix that acts on
the unit vectors, representing a type of generalized frustration. In a recent
paper we have analyzed in detail the role of the coupling matrix for .
Here we extend this analysis to arbitrary dimensions, presenting a study of
synchronous states and their stability. We show that when the natural
frequencies of the particles are set to zero, the system converges either to a
stationary synchronized state with well defined phase, or to an effective
two-dimensional dynamics, where the synchronized particles rotate on the
sphere. The stability of these states depend on the eigenvalues and
eigenvectors of the coupling matrix. When the natural frequencies are not zero,
synchronization depends on whether is even or odd. In even dimensions the
transition to synchronization is continuous and rotating states are replaced by
active states, where the order parameter rotates while its module oscillates.
If is odd the phase transition is discontinuous and active states are
suppressed, occurring only for a restricted class of coupling matrices.Comment: 23 pages, 3 figure
On the numerical integration of the multidimensional Kuramoto model}
The Kuramoto model, describing the synchronization dynamics of coupled
oscillators, has been generalized in many ways over the past years. One recent
extension of the model replaces the oscillators, originally characterized by a
single phase, by particles with internal phases, represented by a point
on the surface of the unit D-sphere. Particles are then more easily represented
by -dimensional unit vectors than by spherical angles. However,
numerical integration of the state equations should ensure that the propagated
vectors remain unit, avoiding artifacts that change their norm. Euler's method,
for instance, displaces the particles in the direction of the tangent velocity
field, moving them away from the sphere. The state vectors, therefore, must be
manually projected back onto the surface after each integration step. As
discussed in \cite{lee2023improved}, this procedure produces a small rotation
of the particles over the sphere, but with the wrong angle. Although higher
order integration methods, such as forth order Runge-Kutta, do provide accurate
results even requiring projection of vectors back to the sphere, integration
can be safely performed by doing a sequence of direct small rotations, as
dictated by the equations of motion. This keeps the particles on the sphere at
all times and ensure exact norm preservation, as proposed in
\cite{lee2023improved}. In this work I propose an alternative way to do such
integration by rotations in 3D that can be generalized to more dimensions using
Cayley-Hamilton's theorem. Explicit formulas are provided for 2, 3 and 4
dimensions.Comment: 15 pages, 3 figure
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