Geometric Analysis of Synchronization in Neuronal Networks with Global Inhibition and Coupling Delays

We study synaptically coupled neuronal networks to identify the role of coupling delays in network's synchronized behaviors. We consider a network of excitable, relaxation oscillator neurons where two distinct populations, one excitatory and one inhibitory, are coupled and interact with each other. The excitatory population is uncoupled, while the inhibitory population is tightly coupled. A geometric singular perturbation analysis yields existence and stability conditions for synchronization states under different firing patterns between the two populations, along with formulas for the periods of such synchronous solutions. Our results demonstrate that the presence of coupling delays in the network promotes synchronization. Numerical simulations are conducted to supplement and validate analytical results. We show the results carry over to a model for spindle sleep rhythms in thalamocortical networks, one of the biological systems which motivated our study. The analysis helps to explain how coupling delays in either excitatory or inhibitory synapses contribute to producing synchronized rhythms.Comment: 43 pages, 12 figure

Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling

Small lattices of $N$ nearest neighbor coupled excitable FitzHugh-Nagumo systems, with time-delayed coupling are studied, and compared with systems of FitzHugh-Nagumo oscillators with the same delayed coupling. Bifurcations of equilibria in N=2 case are studied analytically, and it is then numerically confirmed that the same bifurcations are relevant for the dynamics in the case $N>2$. Bifurcations found include inverse and direct Hopf and fold limit cycle bifurcations. Typical dynamics for different small time-lags and coupling intensities could be excitable with a single globally stable equilibrium, asymptotic oscillatory with symmetric limit cycle, bi-stable with stable equilibrium and a symmetric limit cycle, and again coherent oscillatory but non-symmetric and phase-shifted. For an intermediate range of time-lags inverse sub-critical Hopf and fold limit cycle bifurcations lead to the phenomenon of oscillator death. The phenomenon does not occur in the case of FitzHugh-Nagumo oscillators with the same type of coupling.Comment: accepted by Phys.Rev.

Phaselocked patterns and amplitude death in a ring of delay coupled limit cycle oscillators

We study the existence and stability of phaselocked patterns and amplitude death states in a closed chain of delay coupled identical limit cycle oscillators that are near a supercritical Hopf bifurcation. The coupling is limited to nearest neighbors and is linear. We analyze a model set of discrete dynamical equations using the method of plane waves. The resultant dispersion relation, which is valid for any arbitrary number of oscillators, displays important differences from similar relations obtained from continuum models. We discuss the general characteristics of the equilibrium states including their dependencies on various system parameters. We next carry out a detailed linear stability investigation of these states in order to delineate their actual existence regions and to determine their parametric dependence on time delay. Time delay is found to expand the range of possible phaselocked patterns and to contribute favorably toward their stability. The amplitude death state is studied in the parameter space of time delay and coupling strength. It is shown that death island regions can exist for any number of oscillators N in the presence of finite time delay. A particularly interesting result is that the size of an island is independent of N when N is even but is a decreasing function of N when N is odd.Comment: 23 pages, 12 figures (3 of the figures in PNG format, separately from TeX); minor additions; typos correcte

Synchronization of coupled neural oscillators with heterogeneous delays

We investigate the effects of heterogeneous delays in the coupling of two excitable neural systems. Depending upon the coupling strengths and the time delays in the mutual and self-coupling, the compound system exhibits different types of synchronized oscillations of variable period. We analyze this synchronization based on the interplay of the different time delays and support the numerical results by analytical findings. In addition, we elaborate on bursting-like dynamics with two competing timescales on the basis of the autocorrelation function.Comment: 18 pages, 14 figure

Subthreshold dynamics of the neural membrane potential driven by stochastic synaptic input

In the cerebral cortex, neurons are subject to a continuous bombardment of synaptic inputs originating from the network's background activity. This leads to ongoing, mostly subthreshold membrane dynamics that depends on the statistics of the background activity and of the synapses made on a neuron. Subthreshold membrane polarization is, in turn, a potent modulator of neural responses. The present paper analyzes the subthreshold dynamics of the neural membrane potential driven by synaptic inputs of stationary statistics. Synaptic inputs are considered in linear interaction. The analysis identifies regimes of input statistics which give rise to stationary, fluctuating, oscillatory, and unstable dynamics. In particular, I show that (i) mere noise inputs can drive the membrane potential into sustained, quasiperiodic oscillations (noise-driven oscillations), in the absence of a stimulus-derived, intraneural, or network pacemaker; (ii) adding hyperpolarizing to depolarizing synaptic input can increase neural activity (hyperpolarization-induced activity), in the absence of hyperpolarization-activated currents

A STUDY ON DYNAMIC SYSTEMS RESPONSE OF THE PERFORMANCE CHARACTERISTICS OF SOME MAJOR BIOPHYSICAL SYSTEMS

Dynamic responses of biophysical systems - performance characteristic

Spatiotemporal dynamics on small-world neuronal networks: The roles of two types of time-delayed coupling

We investigate temporal coherence and spatial synchronization on small-world networks consisting of noisy Terman-Wang (TW) excitable neurons in dependence on two types of time-delayed coupling: $\{x_j(t-\tau)-x_i (t)\}$ and $\{x_j(t-\tau)-x_i(t-\tau)\}$. For the former case, we show that time delay in the coupling can dramatically enhance temporal coherence and spatial synchrony of the noise-induced spike trains. In addition, if the delay time $\tau$ is tuned to nearly match the intrinsic spike period of the neuronal network, the system dynamics reaches a most ordered state, which is both periodic in time and nearly synchronized in space, demonstrating an interesting resonance phenomenon with delay. For the latter case, however, we can not achieve a similar spatiotemporal ordered state, but the neuronal dynamics exhibits interesting synchronization transition with time delay from zigzag fronts of excitations to dynamic clustering anti-phase synchronization (APS), and further to clustered chimera states which have spatially distributed anti-phase coherence separated by incoherence. Furthermore, we also show how these findings are influenced by the change of the noise intensity and the rewiring probability. Finally, qualitative analysis is given to illustrate the numerical results.Comment: 17 pages, 9 figure

A continuum of weakly coupled oscillatory McKean neurons

The McKean model of a neuron possesses a one dimensional fast voltage-like variable and a slow recovery variable. A recent geometric analysis of the singularly perturbed system has allowed an explicit construction of its phase response curve [S Coombes 2001 Phase-locking in networks of synaptically coupled McKean relaxation oscillators, Physica D, Vol 160, 173-188]. Here we use tools from coupled oscillator theory to study weakly coupled networks of McKean neurons. Using numerical techniques we show that the McKean system has traveling wave phase-locked solutions consistent with that of a network of more biophysically detailed Hodgkin-Huxley neurons

Relationship between Mathematical Parameters of Modified Van der Pol Oscillator Model and ECG Morphological Features

The mathematical model describes the electrical and mechanical activity of the cardiac conduction system thought set of differential equations. By changing the value of parameters included in these equations, it is possible to change the amplitude and the period of ECG waves. Although this model is a powerful tool for modeling the electrical activity of the heart, its use is often limited to those familiar with the differential equations that describe the system. The purpose of this work is to provide a system that allows generating an ECG signal using Ryzhii model without knowing the details of differential equations. First, we provide the relationships between the ECG wave features and the model parameters; then we generalize them through fitting neural networks. Finally, putting in series fitting neural network and heart model, we provide a system that allows generating a synthetic signal by setting as input only the morphological ECG feature. We computed numerical simulation in Simulink environment and implemented the fitting neural networks in Matlab. Results show that non-linear trends characterize the correlation functions between ECG morphological features and model parameters and that the fitting neural networks can generalized this trend by providing the model parameters given in input the respectively ECG feature