Effect of Cauchy noise on a network of quadratic integrate-and-fire neurons with non-Cauchy heterogeneities

We analyze the dynamics of large networks of pulse-coupled quadratic integrate-and-fire neurons driven by Cauchy noise and non-Cauchy heterogeneous inputs. Two types of heterogeneities defined by families of $q$-Gaussian and flat distributions are considered. Both families are parametrized by an integer $n$, so that as $n$ increases, the first family tends to a normal distribution, and the second tends to a uniform distribution. For both families, exact systems of mean-field equations are derived and their bifurcation analysis is carried out. We show that noise and heterogeneity can have qualitatively different effects on the collective dynamics of neurons.Comment: 9 pages, 5 figure

Stability of linear and non-linear lambda and tripod systems in the presence of amplitude damping

We present the stability analysis of the dark states in the adiabatic passage for the linear and non-linear lambda and tripod systems in the presence of amplitude damping (losses). We perform an analytic evaluation of the real parts of eigenvalues of the corresponding Jacobians, the non-zero eigenvalues of which are found from the quadratic characteristic equations, as well as by the corresponding numerical simulations. For non-linear systems, we evaluate the Jacobians at the dark states. Similarly to the linear systems, here we also find the non-zero eigenvalues from the characteristic quadratic equations. We reveal a common property of all the considered systems showing that the evolution of the real parts of eigenvalues can be split into three stages. In each of them the evolution of the stimulated Raman adiabatic passage (STIRAP) is characterized by different effective dimension. This results in a possible adiabatic reduction of one or two degrees of freedom.Comment: 30 pages, 12 figure

Continuous pole placement method for time-delayed feedback controlled systems

Continuous pole placement method is adapted to time-periodic states of systems with time delay. The method is applied for finding an optimal control matrix in the problem of stabilization of unstable periodic orbits of dynamical systems via time-delayed feedback control algorithm. The optimal control matrix ensures the fastest approach of a perturbed system to the stabilized orbit. An application of the pole placement method to systems with time delay meets a fundamental problem, since the number of the Floquet exponents is infinity, while the number of control parameters is finite. Nevertheless, we show that several leading Floquet exponents can be efficiently controlled. The method is numerically demonstrated for the Lorenz system, which until recently has been considered as a system inaccessible for the standard time-delayed feedback control due to the odd-number limitation. The proposed optimization method is also adapted for an extended time-delayed feedback control algorithm and numerically demonstrated for the Rössler system

Adaptive search for the optimal feedback gain of time-delayed feedback controlled systems in the presence of noise

We propose two adaptive algorithms for the time-delayed feedback control method to tune the feedback gain to an optimal value in the presence of noise. By the optimal value we mean the value of the feedback gain that minimizes the mean square of the control signal. The first algorithm is model independent; it uses trial values of the feedback gain and defines the optimal value by the least-squares polynomial fitting. The second algorithm is based on the gradient descent method and requires the knowledge of the system equations. Here any initial value of the feedback gain is continuously adjusted towards the optimal value without any trials. The efficacy of the algorithms is demonstrated with different specific models, namely, a simple linear map, the Rössler system and the normal form of the subcritical Hopf bifurcation