Exponential transient propagating oscillations in a ring of spiking neurons with unidirectional slow inhibitory synaptic coupling

Dynamics of coexisting rotating waves in unidirectional rings of bistable Duffing oscillators

We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies.

Candelabrum cells are ubiquitous cerebellar cortex interneurons with specialized circuit properties

To understand how the cerebellar cortex transforms mossy fiber (MF) inputs into Purkinje cell (PC) outputs, it is vital to delineate the elements of this circuit. Candelabrum cells (CCs) are enigmatic interneurons of the cerebellar cortex that have been identified based on their morphology, but their electrophysiological properties, synaptic connections and function remain unknown. Here, we clarify these properties using electrophysiology, single-nucleus RNA sequencing, in situ hybridization and serial electron microscopy in mice. We find that CCs are the most abundant PC layer interneuron. They are GABAergic, molecularly distinct and present in all cerebellar lobules. Their high resistance renders CC firing highly sensitive to synaptic inputs. CCs are excited by MFs and granule cells and are strongly inhibited by PCs. CCs in turn primarily inhibit molecular layer interneurons, which leads to PC disinhibition. Thus, inputs, outputs and local signals converge onto CCs to allow them to assume a unique role in controlling cerebellar output.

Lag synchronization of coupled time-delayed FitzHugh-Nagumo neural networks via feedback control

Synchronization plays a significant role in information transfer and decision-making by neurons and brain neural networks. The development of control strategies for synchronizing a network of chaotic neurons with time delays, different direction-dependent coupling (unidirectional and bidirectional), and noise, particularly under external disturbances, is an essential and very challenging task. Researchers have extensively studied the synchronization mechanism of two coupled time-delayed neurons with bidirectional coupling and without incorporating the effect of noise, but not for time-delayed neural networks. To overcome these limitations, this study investigates the synchronization problem in a network of coupled FitzHugh-Nagumo (FHN) neurons by incorporating time delays, different direction-dependent coupling (unidirectional and bidirectional), noise, and ionic and external disturbances in the mathematical models. More specifically, this study investigates the synchronization of time-delayed unidirectional and bidirectional ring-structured FHN neuronal systems with and without external noise. Different gap junctions and delay parameters are used to incorporate time-delay dynamics in both neuronal networks. We also investigate the influence of the time delays between connected neurons on synchronization conditions. Further, to ensure the synchronization of the time-delayed FHN neuronal networks, different adaptive control laws are proposed for both unidirectional and bidirectional neuronal networks. In addition, necessary and sufficient conditions to achieve synchronization are provided by employing the Lyapunov stability theory. The results of numerical simulations conducted for different-sized multiple networks of time-delayed FHN neurons verify the effectiveness of the proposed adaptive control schemes.

Dynamic modes in a network of five oscillators with inhibitory all-to-all pulse coupling

The dynamic modes of five almost identical oscillators with pulsatile inhibitory coupling with time delay have been studied theoretically. The models of the Belousov-Zhabotinsky reaction and phase oscillators with all-to-all coupling have been considered. In the parametric plane C_inh-τ, where C_inh is the coupling strength and τ is the time delay between a spike in one oscillator and pulsed perturbations of all other oscillators, three main regimes have been found: regular modes, when each oscillator gives only one spike during the global period T, C (complex) modes, when the number of pulses of different oscillators is different, and OS (oscillations-suppression) modes, when at least one oscillator is suppressed. The regular modes consist of several cluster modes and are found at relatively small C_inh. The C and OS modes observed at larger C_inh intertwine in the C_inh-τ plane. In a relatively narrow range of C_inh, the dynamics of the C modes are very sensitive to small changes in C_inh and τ, as well as to the initial conditions, which are the characteristic features of the chaos. On the other hand, the dynamics of the C modes are periodic (but with different periods) and well reproducible. The number of different C modes is enormously large. At still larger C_inh, the C modes lose sensitivity to small changes in the parameters and finally vanish, while the OS modes survive.

Effects of self-coupling and asymmetric output on metastable dynamical transient firing patterns in arrays of neurons with bidirectional inhibitory coupling

Metastable dynamical transient patterns in arrays of bidirectionally coupled neurons with self-coupling and asymmetric output were studied. First, an array of asymmetric sigmoidal neurons with symmetric inhibitory bidirectional coupling and self-coupling was considered and the bifurcations of its steady solutions were shown. Metastable dynamical transient spatially nonuniform states existed in the presence of a pair of spatially symmetric stable solutions as well as unstable spatially nonuniform solutions in a restricted range of the output gain of a neuron. The duration of the transients increased exponentially with the number of neurons up to the maximum number at which the spatially nonuniform steady solutions were stabilized. The range of the output gain for which they existed reduced as asymmetry in a sigmoidal output function of a neuron increased, while the existence range expanded as the strength of inhibitory self-coupling increased. Next, arrays of spiking neuron models with slow synaptic inhibitory bidirectional coupling and self-coupling were considered with computer simulation. In an array of Class 1 Hindmarsh-Rose type models, in which each neuron showed a graded firing rate, metastable dynamical transient firing patterns were observed in the presence of inhibitory self-coupling. This agreed with the condition for the existence of metastable dynamical transients in an array of sigmoidal neurons. In an array of Class 2 Bonhoeffer-van der Pol models, in which each neuron had a clear threshold between firing and resting, long-lasting transient firing patterns with bursting and irregular motion were observed.

Synchronization of coupled different chaotic FitzHugh-Nagumo neurons with unknown parameters under communication-direction-dependent coupling

This paper investigates the chaotic behavior and synchronization of two different coupled chaotic FitzHugh-Nagumo (FHN) neurons with unknown parameters under external electrical stimulation (EES). The coupled FHN neurons of different parameters admit unidirectional and bidirectional gap junctions in the medium between them. Dynamical properties, such as the increase in synchronization error as a consequence of the deviation of neuronal parameters for unlike neurons, the effect of difference in coupling strengths caused by the unidirectional gap junctions, and the impact of large time-delay due to separation of neurons, are studied in exploring the behavior of the coupled system. A novel integral-based nonlinear adaptive control scheme, to cope with the infeasibility of the recovery variable, for synchronization of two coupled delayed chaotic FHN neurons of different and unknown parameters under uncertain EES is derived. Further, to guarantee robust synchronization of different neurons against disturbances, the proposed control methodology is modified to achieve the uniformly ultimately bounded synchronization. The parametric estimation errors can be reduced by selecting suitable control parameters. The effectiveness of the proposed control scheme is illustrated via numerical simulations.

Effects of asymmetric coupling and self-coupling on metastable dynamical transient rotating waves in a ring of sigmoidal neurons

Transient rotating waves in a ring of sigmoidal neurons with asymmetric bidirectional coupling and self-coupling were studied. When a pair of stable steady states and an unstable traveling wave coexisted, rotating waves propagating in a ring were generated in transients. The pinning (propagation failure) of the traveling wave occurred in the presence of asymmetric coupling and self-coupling, and its conditions were obtained. A kinematical equation for the propagation of wave fronts of the traveling and rotating waves was then derived for a large output gain of neurons. The kinematical equation showed that the duration of transient rotating waves increases exponentially with the number of neurons as that in a ring of unidirectionally coupled neurons (metastable dynamical transients). However, the exponential growth rate depended on the asymmetry of bidirectional coupling and the strength of self-coupling. The rate was equal to the propagation time of the traveling wave (a reciprocal of the propagation speed), and it increased near pinned regions. Then transient rotating waves could show metastable dynamics (extremely long duration) even in a ring of a small number of neurons.

Metastable dynamical patterns and their stabilization in arrays of bidirectionally coupled sigmoidal neurons

Transient patterns in a bistable ring of bidirectionally coupled sigmoidal neurons were studied. When the system had a pair of spatially uniform steady solutions, the instability of unstable spatially nonuniform steady solutions decreased exponentially with the number of neurons because of the symmetry of the system. As a result, transient spatially nonuniform patterns showed dynamical metastability: Their duration increased exponentially with the number of neurons and the duration of randomly generated patterns obeyed a power-law distribution. However, these metastable dynamical patterns were easily stabilized in the presence of small variations in coupling strength. Metastable rotating waves and their pinning in the presence of asymmetry in the direction of coupling and the disappearance of metastable dynamical patterns due to asymmetry in the output function of a neuron were also examined. Further, in a two-dimensional array of neurons with nearest-neighbor coupling, intrinsically one-dimensional patterns were dominant in transients, and self-excitation in these neurons affected the metastable dynamical patterns.