The synchronization of bursting Hindmarsh-Rose neurons coupled by a time-delayed fast threshold modulation synapse was studied. It is shown that there is a domain of the coupling parameter and nonzero time-lag values such that the bursting neurons are exactly synchronized. Furthermore, and contrary to the case of electrical synapses, such synchronous bursting is stochastically stable.

We consider the coaction of two distinct noise sources on the activation process of a single and two interacting excitable units, which are mathematically described by the Fitzhugh-Nagumo equations. We determine the most probable activation paths around which the corresponding stochastic trajectories are clustered. The key point lies in introducing appropriate boundary conditions that are relevant for a class II excitable unit, which can be immediately generalized also to scenarios involving two coupled units. We analyze the effects of the two noise sources on the statistical features of the activation process, in particular demonstrating how these are modified due to the linear/nonlinear form of interactions. Universal properties of the activation process are qualitatively discussed in the light of a stochastic bifurcation that underlies the transition from a stochastically stable fixed point to continuous oscillations.

Spontaneous formation of clusters of synchronized spiking in a structureless ensemble of equal stochastically perturbed excitable neurons with delayed coupling is demonstrated for the first time. The effect is a consequence of a subtle interplay between interaction delays, noise and the excitable character of a single neuron. Dependence of the cluster properties on the time-lag, noise intensity and the synaptic strength is investigated.Collective behavior in large ensembles of physiological and inorganic systems can be reduced to that of coupled oscillators engaged in various synchronization phenomena. In terms of macroscopic coherent rhythms, it may either be the case where all the units are recruited into a giant component or the case of cluster states characterized by exact or in-phase intra-subset and lag inter-subset synchronization. The spontaneous onset of cluster states is of particular interest to neuroscience [1] for the conjectured role in information encoding, as well as for participating in motor coordination or accompanying some neurological disorders. The approach to clustering has mostly relied on modeling neurons as autonomous oscillators, treating separately the question of whether the proposed mechanisms may be robust under noise [2] and transmission delays [3]. We explore a new mechanism which rests on the excitable character of neuronal dynamics and mutual adjustment between noise and time delay to yield the self-organization into functional modules within an otherwise unstructured network.For the instantaneous couplings, the research on populations of excitable neurons has covered pattern formation due to local inhomogeneity [4], or has invoked a scenario where noise enacts a control parameter tuning the dynamics of ensemble averages between the three generic global regimes [5]. Distinct from the layout with complex connection topologies, here it is demonstrated how coupling delays do alter the latter landscape in a significant fashion, giving rise to an effect one may dub the cluster forming time-delay-induced coherence resonance. In part, the strategy to analyze global dynamics rests on deriving the mean-field (MF) approximation for the exact system. The likely gain from the MF treatment is at least twofold: except for allowing one to extrapolate what occurs in the thermodynamic limit N → ∞, it may serve as an auxiliary means to discriminate between the effects of noise and time delay. Unexpectedly, the MF model undergoes a global bifurcation at certain parameter values where the exact system shows an onset of cluster states.Network dynamics and the tools to analyze it -We focus on an N -size population of all-to-all diffusively coupled Fitzhugh-Nagumo neurons, whose local dynamics is set bywhere the activator variables x i embody the membrane potentials, while the recovery variables y i mimic the action of the K + membrane gating channels. c denotes the synaptic strength and τ stands for the coupling delay, both parameters for simplicity assumed homogeneous across the ensemble. The √ 2Dd...

The analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations composed of N stochastic excitable units each is performed by studying an approximate system, obtained by replacing each population with the corresponding mean-field model. In the exact system, one has the units within an ensemble communicating via the time-delayed linear couplings, whereas the interensemble terms involve the nonlinear time-delayed interaction mediated by the appropriate global variables. The aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system, governed by a set of 4N stochastic delay-differential equations for the microscopic dynamics, can accurately be reproduced by a flow containing just four deterministic delay-differential equations which describe the evolution of the mean-field based variables. In particular, the considered issues include determining the parameter domains where the stationary state is stable, the scenarios for the onset, and the time-delay induced suppression of the collective mode, as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state. We show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations, like those with symmetrical or asymmetrical interpopulation couplings.

Objective: Research indicates moderate-to-limited integration of the risk-need-responsivity (RNR) principles in probation case planning. Efforts to improve implementation are important targets for research, policy, and practice. This study examined the ability of two juvenile probation departments to implement RNR principles with fidelity following a comprehensive implementation protocol that included RNRrelated policies, creation of a service matrix for criminogenic need-to-service matching, and extensive staff training. Hypotheses: The researchers anticipated fidelity to the risk and need principles would be stronger than previous studies. Method: This implementation study involved secondary data analysis of services received over 10 months for 254 adolescent offenders (76.80% male, 72.40% White, M age ϭ 16.13 years) from two probation departments following adoption of the Youth Level of Service/Case Management Inventory. Results: Probation departments evidenced strong fidelity to the risk principle, such that higher risk youth were assigned more services with higher intensity. Fidelity to the need principle was moderate at best (an average 24.61% to 29.38% need-to-service match) and varied by criminogenic need, overall risk level, and the operational definition of criminogenic need. Conclusions: Comprehensive implementation practices are associated with strong fidelity to the risk principle, but it may take longer for probation departments to achieve strong fidelity to the need principle. Researchers should identify more feasible methods for implementing the need principle and strive for a consensus on methods for measuring need-to-service match that are also consistent with probation policies. Public Significance StatementProbation departments can achieve fidelity to some aspects of evidence-based case planning practices for youth following use of a comprehensive implementation protocol for these practices. Adherence to criminogenic risk reduction practices (need-to-service matching) is an important aspect of case planning where fidelity is limited, and which may require longer periods of implementation (e.g., 3 years) to achieve. Researchers should use implementation science methods to create and test decision-making supports that could improve fidelity to risk reduction practices.

Mean field approximation of a large collection of FitzHugh-Nagumo excitable neurons with noise and all-to-all coupling with explicit timedelays, modelled by N ≫ 1 stochastic delay-differential equations is derived. The resulting approximation contains only two deterministic delay-differential equations but provides excellent predictions concerning the stability and bifurcations of the averaged global variables of the exact large system.

We consider the approximations behind the typical mean-field model derived for a class of systems made up of type II excitable units influenced by noise and coupling delays. The formulation of the two approximations, referred to as the Gaussian and the quasi-independence approximation, as well as the fashion in which their validity is verified, are adapted to reflect the essential properties of the underlying system. It is demonstrated that the failure of the mean-field model associated with the breakdown of the quasi-independence approximation can be predicted by the noise-induced bistability in the dynamics of the mean-field system. As for the Gaussian approximation, its violation is related to the increase of noise intensity, but the actual condition for failure can be cast in qualitative, rather than quantitative terms. We also discuss how the fulfillment of the mean-field approximations affects the statistics of the first return times for the local and global variables, further exploring the link between the fulfillment of the quasi-independence approximation and certain forms of synchronization between the individual units.

The influence of white noise on the dynamics of a delayed electrically coupled pair of Hidmarsh-Rose bursting neurons is studied. In particular a simple method to predict the intensity of noise that can destabilize the quiescent state is proposed and compared with numerical computations. Furthermore, it is demonstrated that quite small noise completely destroys the exact synchronization of bursting dynamics, and a qualitative explanation of this effect is discussed.