Local cortical circuits appear highly non-random, but the underlying connectivity rule remains elusive. Here, we analyze experimental data observed in layer 5 of rat neocortex and suggest a model for connectivity from which emerge essential observed non-random features of both wiring and weighting. These features include lognormal distributions of synaptic connection strength, anatomical clustering, and strong correlations between clustering and connection strength. Our model predicts that cortical microcircuits contain large groups of densely connected neurons which we call clusters. We show that such a cluster contains about one fifth of all excitatory neurons of a circuit which are very densely connected with stronger than average synapses. We demonstrate that such clustering plays an important role in the network dynamics, namely, it creates bistable neural spiking in small cortical circuits. Furthermore, introducing local clustering in large-scale networks leads to the emergence of various patterns of persistent local activity in an ongoing network activity. Thus, our results may bridge a gap between anatomical structure and persistent activity observed during working memory and other cognitive processes.

Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in recent years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. At the bifurcation point numerous regimes with non-equal interspike intervals emerge. We show that the number of the emerging, so-called "jittering" regimes grows exponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the "multi-jitter" bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phasereduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit. PACS numbers: 87.19.ll, 05.45.Xt, 87.19.lr, 89.75 Interaction via pulse-like signals is important in neuron populations [1][2][3], biological [4,5], optical and optoelectronic systems [6]. Often, time delays are inevitable in such systems as a consequence of the finite speed of pulse propagation [7]. In this letter we demonstrate that the pulsatile and delayed nature of interactions may lead to novel and unusual phenomena in a large class of systems. In particular, we explore oscillatory systems with pulsatile delayed feedback which exhibit periodic regular spiking (RS). We show that this RS regime may destabilize via a scenario in which a variety of higher-periodical regimes with non-equal interspike intervals (ISIs) emerge simultaneously. The number of the emergent, so-called "jittering" regimes grows exponentially as the delay increases. Therefore we adopt the term "multi-jitter" bifurcation.Usually, the simultaneous emergence of many different regimes is a sign of degeneracy and it is expected to occur generically only when additional symmetries are present [2,8].However, for the class of systems treated here no such symmetry is apparent. Nevertheless, the phenomenon can be reliably observed when just a single parameter, for example the delay, is varied. This means that the observed bifurcation has codimension one [9]. In addition to the theoretical analysis of a simple paradigmatic model, we provide numerical evidence for the occurrence of the multi-jitter bifurcation in a realistic neuronal model, as well as an experimental confirmation in an electronic circuit.As a universal and simplest oscillatory spiking model in the absence of the feedback, we consider the phase oscillator dϕ/dt = ω, where ϕ ∈ R ( mod 1), and ω = 1 without loss of generality. When the oscillator reaches ϕ = 1 at some moment t, the phase is reset to zero and the oscillator produces a pulse signal. If this signal is sent into a delayed feedback loop [ Fig. 1(a)] the emitted pulses affect the oscillator after a delay τ at the time instant t * = t+τ . When the pulse is received, the phase of the oscillator undergoes an instantaneous shift by an amount ∆ϕ = Z(ϕ(t * − 0)), where Z(ϕ) is the phase resetting curve (PRC).Thus, the dynamics of the oscillator can be describ...

A new measure to characterize the stability of complex dynamical systems against large perturbations is suggested, the stability threshold (ST). It quantifies the magnitude of the weakest perturbation capable of disrupting the system and switch it to an undesired dynamical regime. In the phase space, the ST corresponds to the 'thinnest site' of the attraction basin and therefore indicates the most 'dangerous' direction of perturbations. We introduce a computational algorithm for quantification of the ST and demonstrate that the suggested approach is effective and provides important insights. The generality of the obtained results defines their vast potential for application in such fields as engineering, neuroscience, power grids, Earth science and many others where the robustness of complex systems is studied. OPEN ACCESS RECEIVED

Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous "jittering" regimes with nonequal interspike intervals (ISIs). Each of these regimes corresponds to a periodic solution of the system with a period roughly proportional to the delay. The number of different "jittering" solutions emerging at the bifurcation point increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how a periodic solution exhibiting several distinct ISIs can imply the existence of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback.

Phase response curve (PRC) is an extremely useful tool for studying the response of oscillatory systems, e.g. neurons, to sparse or weak stimulation. Here we develop a framework for studying the response to a series of pulses which are frequent or/and strong so that the standard PRC fails. We show that in this case, the phase shift caused by each pulse depends on the history of several previous pulses. We call the corresponding function which measures this shift the phase response function (PRF). As a result of the introduction of the PRF, a variety of oscillatory systems with pulse interaction can be reduced to phase systems. The main assumption of the classical PRC model, i.e. that the effect of the stimulus vanishes before the next one arrives, is no longer a restriction in our approach. However, as a result of the phase reduction, the system acquires memory, which is not just a technical nuisance but an intrinsic property relevant to strong stimulation. We illustrate the PRF approach by its application to various systems, such as Morris-Lecar, Hodgkin-Huxley neuron models, and others. We show that the PRF allows predicting the dynamics of forced and coupled oscillators even when the PRC fails. Thus, the PRF provides an effective tool that may be used for simulation of neural, chemical, optic oscillators, etc.A variety of physical, chemical, biological, and other systems exhibit periodic behaviors.The state of such a system can be naturally determined by its phase [1], that is, the single variable indicating the position of the system within its cycle. The concept of the phase proved to be exceptionally useful for the study of driven and coupled oscillators [1][2][3].In order to describe the response of oscillators to an external force or coupling the so- called phase response curve (PRC) is widely used. The PRC defines the oscillator's response to a single short stimulus (pulse). The PRC can be calculated numerically or measured experimentally for oscillatory systems of different origin [4]. These properties made it a useful tool for the study of forced or coupled oscillators [5][6][7][8][9][10][11], and it is especially effective in neuroscience where the interactions are mediated by pulses. If the pulse arrivals are separated by sufficiently long time intervals, the transient caused by a pulse vanishes before the next one comes. From the theoretical point of view, it means that the system returns to the vicinity of its stable limit cycle before the next pulse arrives, see Fig. 1(a). In this case the effect of each pulse can be described by the classical PRC Z(ϕ), which determines the resulting phase shift given that the pulse arrived at the phase ϕ. Another case when the PRCs are useful is when the forcing is continuous in time but weak ( Fig. 1(b)). In this case the system remains close to the limit cycle, and the phase dynamics can be described by the so-called infinitesimal phase response curve [12].Therefore, the PRC-based approach is applicable for either weak or sparse stimulation.However, in many re...

We consider a network of randomly coupled rate-based neurons influenced by external and internal noise. We derive a second-order stochastic mean-field model for the network dynamics and use it to analyze the stability and bifurcations in the thermodynamic limit, as well as to study the fluctuations due to the finite-size effect. It is demonstrated that the two types of noise have substantially different impact on the network dynamics. While both sources of noise give rise to stochastic fluctuations in case of the finite-size network, only the external noise affects the stationary activity levels of the network in the thermodynamic limit. We compare the theoretical predictions with the direct simulation results and show that they agree for large enough network sizes and for parameter domains sufficiently away from bifurcations.

The dynamic regimes in networks of four almost identical spike oscillators with pulsatile coupling via inhibitor are systematically studied. We used two models to describe individual oscillators: a phase-oscillator model and a model for the Belousov-Zhabotinsky reaction. A time delay τ between a spike in one oscillator and the spike-induced inhibitory perturbation of other oscillators is introduced. Diagrams of all rhythms found for three different types of connectivities (unidirectional on a ring, mutual on a ring, and all-to-all) are built in the plane C(inh)-τ, where C(inh) is the coupling strength. It is shown analytically and numerically that only four regular rhythms are stable for unidirectional coupling: walk (phase shift between spikes of neighbouring oscillators equals the quarter of the global period T), walk-reverse (the same as walk but consecutive spikes take place in the direction opposite to the direction of connectivity), anti-phase (any two neighbouring oscillators are anti-phase), and in-phase oscillations. In the case of mutual on the ring coupling, an additional in-phase-anti-phase mode emerges. For all-to-all coupling, two new asymmetrical patterns (two-cluster and three-cluster modes) have been found. More complex rhythms are observed at large C(inh), when some oscillators are suppressed completely or generate smaller number of spikes than others.

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