Browsing by Author "Franović, Igor"

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    A non-autonomous mega-extreme multistable chaotic system
    Ahmadi, Atefeh; Parthasarathy, Sriram; Natiq, Hayder; Jafari, Sajad; Franović, Igor; Rajagopal, Karthikeyan
    Megastable and extreme multistable systems comprise two major new branches of multistable systems. So far, they have been studied separately in various chaotic systems. Nevertheless, to the best of our knowledge, no chaotic system has so far been reported that possesses both types of multistability. This paper introduces the first three-dimensional non-autonomous chaotic system that displays megastability and extreme multistability, jointly called mega-extreme multistability. Our model shows extreme multistability for a variation of an initial condition associated with one system variable and megastability concerning another variable. The different types of coexisting attractors are characterized by the corresponding phase portraits and first return maps, as well as by constructing the appropriate bifurcation diagrams, calculating the Lyapunov spectra, the Kaplan-Yorke dimension and the connecting curves, and by determining the corresponding basins of attraction. The system is explicitly shown to be dissipative, with the dissipation being state-dependent. We demonstrate the feasibility and applicability of our model by designing and simulating an appropriate analog circuit.
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    Bumps, chimera states, and Turing patterns in systems of coupled active rotators
    Franović, Igor; Omel'chenko, Oleh E.; Wolfrum, Matthias
    Self-organized coherence-incoherence patterns, called chimera states, have first been reported in systems of Kuramoto oscillators. For coupled excitable units, similar patterns where coherent units are at rest are called bump states. Here, we study bumps in an array of active rotators coupled by nonlocal attraction and global repulsion. We demonstrate how they can emerge in a supercritical scenario from completely coherent Turing patterns: a single incoherent unit appears in a homoclinic bifurcation, undergoing subsequent transitions to quasiperiodic and chaotic behavior, which eventually transforms into extensive chaos with many incoherent units. We present different types of transitions and explain the formation of coherence-incoherence patterns according to the classical paradigm of short-range activation and long-range inhibition.
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    Clustering promotes switching dynamics in networks of noisy neurons
    Franović, Igor; Klinshov, Vladimir
    Macroscopic variability is an emergent property of neural networks, typically manifested in spontaneous switching between the episodes of elevated neuronal activity and the quiescent episodes. We investigate the conditions that facilitate switching dynamics, focusing on the interplay between the different sources of noise and heterogeneity of the network topology. We consider clustered networks of rate-based neurons subjected to external and intrinsic noise and derive an effective model where the network dynamics is described by a set of coupled second-order stochastic mean-field systems representing each of the clusters. The model provides an insight into the different contributions to effective macroscopic noise and qualitatively indicates the parameter domains where switching dynamics may occur. By analyzing the mean-field model in the thermodynamic limit, we demonstrate that clustering promotes multistability, which gives rise to switching dynamics in a considerably wider parameter region compared to the case of a non-clustered network with sparse random connection topology.
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    Collective Activity Bursting in a Population of Excitable Units Adaptively Coupled to a Pool of Resources
    Franović, Igor; Eydam, Sebastian; Yanchuk, Serhiy; Berner, Rico
    We study the collective dynamics in a population of excitable units (neurons) adaptively interacting with a pool of resources. The resource pool is influenced by the average activity of the population, whereas the feedback from the resources to the population is comprised of components acting homogeneously or inhomogeneously on individual units of the population. Moreover, the resource pool dynamics is assumed to be slow and has an oscillatory degree of freedom. We show that the feedback loop between the population and the resources can give rise to collective activity bursting in the population. To explain the mechanisms behind this emergent phenomenon, we combine the Ott-Antonsen reduction for the collective dynamics of the population and singular perturbation theory to obtain a reduced system describing the interaction between the population mean field and the resources.
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    Control of Seizure-like Dynamics in Neuronal Populations with Excitability Adaptation Related to Ketogenic Diet
    Eydam, Sebastian; Franović, Igor; Kang, Louis
    We consider a heterogeneous, globally coupled population of excitatory quadratic integrate-and-fire neurons with excitability adaptation due to a metabolic feedback associated with ketogenic diet, a form of therapy for epilepsy. Bifurcation analysis of a three-dimensional mean-field system derived in the framework of next-generation neural mass models allows us to explain the scenarios and suggest control strategies for the transitions between the neurophysiologically desired asynchronous states and the synchronous, seizure-like states featuring collective oscillations. We reveal two qualitatively different scenarios for the onset of synchrony. For weaker couplings, a bistability region between the lower- and the higher-activity asynchronous states unfolds from the cusp point, and the collective oscillations emerge via a supercritical Hopf bifurcation. For stronger couplings, one finds seven co-dimension two bifurcation points, including pairs of Bogdanov–Takens and generalized Hopf points, such that both lower- and higher-activity asynchronous states undergo transitions to collective oscillations, with hysteresis and jump-like behavior observed in vicinity of subcritical Hopf bifurcations. We demonstrate three control mechanisms for switching between asynchronous and synchronous states, involving parametric perturbation of the adenosine triphosphate (ATP) production rate, external stimulation currents, or pulse-like ATP shocks, and indicate a potential therapeutic advantage of hysteretic scenarios.
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    Disordered configurations of the Glauber model in two-dimensional networks
    Bačić, Iva; Franović, Igor; Perc, Matjaž
    We analyze the ordering efficiency and the structure of disordered configurations for the zero-temperature Glauber model on Watts-Strogatz networks obtained by rewiring 2D regular square lattices. In the small-world regime, the dynamics fails to reach the ordered state in the thermodynamic limit. Due to the interplay of the perturbed regular topology and the energy neutral stochastic state transitions, the stationary state consists of two intertwined domains, manifested as multiclustered states on the original lattice. Moreover, for intermediate rewiring probabilities, one finds an additional source of disorder due to the low connectivity degree, which gives rise to small isolated droplets of spins. We also examine the ordering process in paradigmatic two-layer networks with heterogeneous rewiring probabilities. Comparing the cases of a multiplex network and the corresponding network with random inter-layer connectivity, we demonstrate that the character of the final state qualitatively depends on the type of inter-layer connections.
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    Dynamics of a stochastic excitable system with slowly adapting feedback
    Franović, Igor; Yanchuk, Serhiy; Eydam, Sebastian; Bačić, Iva; Wolfrum, Matthias
    We study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noise-induced spiking, noise-perturbed oscillations, or stochastic bursting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance or effectively control the features of the stochastic bursting. The setup can be considered a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influence of a nonlinear control mechanism. We employ a multiple timescale approach that combines the classical adiabatic elimination with averaging of rapid oscillations and stochastic averaging of noise-induced fluctuations by a corresponding stationary Fokker–Planck equation. This allows us to perform a numerical bifurcation analysis of a reduced slow system and to determine the parameter regions associated with different types of dynamics. In particular, we demonstrate the existence of a region of bistability, where the noise-induced switching between a stationary and an oscillatory regime gives rise to stochastic bursting.
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    EFFECT of colored noise on the generation of seismic fault MOVEMENT: Analogy with spring-block model DYNAMICS
    Kostić, Srđan; Vasović, Nebojša; Todorović, Kristina; Franović, Igor
    In present paper authors examined the effect of colored noise on the onset of seismic fault motion. For this purpose, they analyze the dynamics of spring-block model, with 10 all-to all coupled blocks. This spring-block model is considered as a collection of fault patches (with the increased rock friction), which are separated by the material bridges (more petrified parts of the fault). In the first phase of research, authors confirm the presence of autocorrelation in the background of seismic noise, using the measurement of real fault movement, and the recorded ground shaking before and after an earthquake. In the second stage of the research, authors firstly develop a mean-field model, which accurately enough describes the dynamics of a starting block model, with the introduced delayed interaction among the blocks, while colored noise is assumed to be generated by Ornstein-Uhlenbeck process. The results of the analysis indicate the existence of three different dynamical regimes, which correspond to three regimes of fault motion: steady stationary state, aseismic creep and seismic fault motion. The effect of colored noise lies in the possibility of generating the seismic fault motion even for small values of correlation time. Moreover, it is shown that the tight connection between the blocks, i.e. fault patches prevent the occurrence of seismic fault motion.
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    Extending Dynamic Memory of Spiking Neuron Networks
    Klinshov, Vladimir; Kovalchuk, Andrey; Soloviev, Igor; Maslennikov, Oleg; Franović, Igor; Perc, Matjaž
    Explaining the mechanisms of dynamic memory, that allows for a temporary storage of information at the timescale of seconds despite the neuronal firing at the millisecond scale, is an important challenge not only for neuroscience, but also for computation in neuromorphic artificial networks. We demonstrate the potential origin of such longer timescales by comparing the spontaneous activity in excitatory neural networks with sparse random, regular and small-world connection topologies. We derive a mean-field model based on a self-consistent approach and white noise approximation to analyze the transient and long-term collective network dynamics. While the long-term dynamics is typically irregular and weakly correlated independent of the network architecture, especially long timescales are revealed for the transient activity comprised of switching fronts in regular and small-world networks with a small rewiring probability. Analyzing the dynamic memory of networks in performing a simple computational delay task within the framework of reservoir computing, we show that an optimal performance on average is reached for a regular connection topology if the input is appropriately structured, but certain instances of small-world networks may strongly deviate from configuration averages and outperform all the other considered network architectures.
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    Inverse stochastic resonance in a system of excitable active rotators with adaptive coupling
    Bačić, Iva; Klinshov, Vladimir; Nekorkin, Vladimir; Perc, Matjaž; Franović, Igor
    Inverse stochastic resonance is a phenomenon where an oscillating system influenced by noise exhibits a minimal oscillation frequency at an intermediate noise level. We demonstrate a novel generic scenario for such an effect in a multi-timescale system, considering an example of emergent oscillations in two adaptively coupled active rotators with excitable local dynamics. The impact of plasticity turns out to be twofold. First, at the level of multiscale dynamics, one finds a range of intermediate adaptivity rates that give rise to multistability between the limit cycle attractors and the stable equilibria, a condition necessary for the onset of the effect. Second, applying the fast-slow analysis, we show that the plasticity also plays a facilitatory role on a more subtle level, guiding the fast flow dynamics to parameter domains where the stable equilibria become focuses rather than nodes, which effectively enhances the influence of noise. The described scenario persists for different plasticity rules, underlying its robustness in the light of potential applications to neuroscience and other types of cell dynamics.
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    Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units
    Eydam, Sebastian; Franović, Igor; Wolfrum, Matthias
    We study a system of two identical FitzHugh-Nagumo units with a mutual linear coupling in the fast variables. While an attractive coupling always leads to synchronous behavior, a repulsive coupling can give rise to dynamical regimes with alternating spiking order, called leap-frogging. We analyze various types of periodic and chaotic leap-frogging regimes, using numerical path-following methods to investigate their emergence and stability, as well as to obtain the complex bifurcation scenario which organizes their appearance in parameter space. In particular, we show that the stability region of the simplest periodic leap-frog pattern has the shape of a locking cone pointing to the canard transition of the uncoupled system. We also discuss the role of the timescale separation in the coupled FitzHugh-Nagumo system and the relation of the leap-frog solutions to the theory of mixed-mode oscillations in multiple timescale systems.
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    Patched patterns and emergence of chaotic interfaces in arrays of nonlocally coupled excitable systems
    Franović, Igor; Eydam, Sebastian
    We disclose a new class of patterns, called patched patterns, in arrays of non-locally coupled excitable units with attractive and repulsive interactions. The self-organization process involves the formation of two types of patches, majority and minority ones, characterized by uniform average spiking frequencies. Patched patterns may be temporally periodic, quasiperiodic, or chaotic, whereby chaotic patterns may further develop interfaces comprised of units with average frequencies in between those of majority and minority patches. Using chaos and bifurcation theory, we demonstrate that chaos typically emerges via a torus breakup and identify the secondary bifurcation that gives rise to chaotic interfaces. It is shown that the maximal Lyapunov exponent of chaotic patched patterns does not decay, but rather converges to a finite value with system size. Patched patterns with a smaller wavenumber may exhibit diffusive motion of chaotic interfaces, similar to that of the incoherent part of chimeras.
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    Phase-sensitive excitability of a limit cycle
    Franović, Igor; Omel’chenko, Oleh E.; Wolfrum, Matthias
    The classical notion of excitability refers to an equilibrium state that shows under the influence of perturbations a nonlinear threshold-like behavior. Here, we extend this concept by demonstrating how periodic orbits can exhibit a specific form of excitable behavior where the nonlinear threshold-like response appears only after perturbations applied within a certain part of the periodic orbit, i.e., the excitability happens to be phase-sensitive. As a paradigmatic example of this concept, we employ the classical FitzHugh-Nagumo system. The relaxation oscillations, appearing in the oscillatory regime of this system, turn out to exhibit a phase-sensitive nonlinear threshold-like response to perturbations, which can be explained by the nonlinear behavior in the vicinity of the canard trajectory. Triggering the phase-sensitive excitability of the relaxation oscillations by noise, we find a characteristic non-monotone dependence of the mean spiking rate of the relaxation oscillation on the noise level. We explain this non-monotone dependence as a result of an interplay of two competing effects of the increasing noise: the growing efficiency of the excitation and the degradation of the nonlinear response.
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    Rate chaos and memory lifetime in spiking neural networks
    Klinshov, Vladimir V.; Kovalchuk, Andrey V.; Franović, Igor; Perc, Matjaž; Svetec, Milan
    Rate chaos is a collective state of a neural network characterized by slow irregular fluctuations of firing rates of individual neurons. We study a sparsely connected network of spiking neurons which demonstrates three different scenarios for the emergence of rate chaos, based either on increasing the synaptic strength, increasing the synaptic integration time, or clustering of the excitatory synaptic connections. Although all the scenarios lead to collective dynamics with similar statistical features, it turns out that the implications for the computational capability of the network in performing a simple delay task are strongly dependent on the particular scenario. Namely, only the scenario involving slow dynamics of synapses results in an appreciable extension of the network's dynamic memory. In other cases, the dynamic memory remains short despite the emergence of long timescales in the neuronal spike trains. (c) 2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
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    Scale-free avalanches in arrays of FitzHugh–Nagumo oscillators
    Contreras, Max; Medeiros, Everton; Zakharova, Anna; Hövel, Philipp; Franović, Igor
    The activity in the brain cortex remarkably shows a simultaneous presence of robust collective oscillations and neuronal avalanches, where intermittent bursts of pseudo-synchronous spiking are interspersed with long periods of quiescence. The mechanisms allowing for such coexistence are still a matter of an intensive debate. Here, we demonstrate that avalanche activity patterns can emerge in a rather simple model of an array of diffusively coupled neural oscillators with multiple timescale local dynamics in the vicinity of a canard transition. The avalanches coexist with the fully synchronous state where the units perform relaxation oscillations. We show that the mechanism behind the avalanches is based on an inhibitory effect of interactions, which may quench the spiking of units due to an interplay with the maximal canard. The avalanche activity bears certain heralds of criticality, including scale-invariant distributions of event sizes. Furthermore, the system shows increased sensitivity to perturbations, manifested as critical slowing down and reduced resilience.
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    Two paradigmatic scenarios for inverse stochastic resonance
    Bačić, Iva; Franović, Igor
    Inverse stochastic resonance comprises a nonlinear response of an oscillatory system to noise where the frequency of noise-perturbed oscillations becomes minimal at an intermediate noise level. We demonstrate two generic scenarios for inverse stochastic resonance by considering a paradigmatic model of two adaptively coupled stochastic active rotators whose local dynamics is close to a bifurcation threshold. In the first scenario, shown for the two rotators in the excitable regime, inverse stochastic resonance emerges due to a biased switching between the oscillatory and the quasi-stationary metastable states derived from the attractors of the noiseless system. In the second scenario, illustrated for the rotators in the oscillatory regime, inverse stochastic resonance arises due to a trapping effect associated with a noise-enhanced stability of an unstable fixed point. The details of the mechanisms behind the resonant effect are explained in terms of slow–fast analysis of the corresponding noiseless systems.
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    Two scenarios for the onset and suppression of collective oscillations in heterogeneous populations of active rotators
    Klinshov, Vladimir; Franović, Igor
    We consider the macroscopic regimes and the scenarios for the onset and the suppression of collective oscillations in a heterogeneous population of active rotators composed of excitable or oscillatory elements. We analyze the system in the continuum limit within the framework of Ott-Antonsen reduction method, determining the states with a constant mean field and their stability boundaries in terms of the characteristics of the rotators' frequency distribution. The system is established to display three macroscopic regimes, namely the homogeneous stationary state, where all the units lie at the resting state, the global oscillatory state, characterized by the partially synchronized local oscillations, and the heterogeneous stationary state, which includes a mixture of resting and asynchronously oscillating units. The transitions between the characteristic domains are found to involve a complex bifurcation structure, organized around three codimension-two bifurcation points: A Bogdanov-Takens point, a cusp point, and a fold-homoclinic point. Apart from the monostable domains, our study also reveals two domains admitting bistable behavior, manifested as coexistence between the two stationary solutions or between a stationary and a periodic solution. It is shown that the collective mode may emerge via two generic scenarios, guided by a saddle-node of infinite period or the Hopf bifurcation, such that the transition from the homogeneous to the heterogeneous stationary state under increasing diversity may follow the classical paradigm, but may also be hysteretic. We demonstrate that the basic bifurcation structure holds qualitatively in the presence of small noise or small coupling delay, with the boundaries of the characteristic domains shifted compared to the noiseless and the delay-free case.
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    Unbalanced clustering and solitary states in coupled excitable systems
    Franović, Igor; Eydam, Sebastian; Semenova, Nadezhda; Zakharova, Anna
    We discover the mechanisms of emergence and the link between two types of symmetry-broken states, the unbalanced periodic two-cluster states and solitary states, in coupled excitable systems with attractive and repulsive interactions. The prevalent solitary states in non-locally coupled arrays, whose self-organization is based on successive (order preserving) spiking of units, derive their dynamical features from the corresponding unbalanced cluster states in globally coupled networks. Apart from the states with successive spiking, we also find cluster and solitary states where the interplay of excitability and local multiscale dynamics gives rise to so-called leap-frog activity patterns with an alternating order of spiking between the units. We show that the noise affects the system dynamics by suppressing the multistability of cluster states and by inducing pattern homogenization, transforming solitary states into patterns of patched synchrony.