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 $D=2$ 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 $D=2$. 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 $D$ 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 $D$ 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 $D-1$ internal phases, represented by a point on the surface of the unit D-sphere. Particles are then more easily represented by $D$-dimensional unit vectors than by $D-1$ 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