Pith. sign in

REVIEW 3 major objections 4 minor 2 cited by

Chaotic Dynamics and Fractal Geometry in Ring Lattice Systems of Nonchaotic Rulkov Neurons

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that electrically coupling 30 individually nonchaotic Rulkov neurons in a ring produces chaotic spiking, synchronized chaotic bursting, and synchronized hyperchaos, with strange attractors that can occupy up to 45 of the…

desk verdict Qualitatively plausible and a genuine extension of Ref. [40], but the headline '45 of 60 dimensions' claim rests on 1000-step Lyapunov spectra and an under-specified box-counting check. read the letter →

arxiv 2412.12134 v3 pith:7CSB6SCG submitted 2024-12-06 nlin.CD math.DSq-bio.NC

Brandon B. Le This is my paper · ORCID
classification nlin.CDmath.DSq-bio.NC MSC 37D4537C4537M2537N25
keywords neuronaldynamicsnonchaoticRulkovmodelhigh-dimensionalsystemschaoticLyapunovexponentsstrangeattractorsfractaldimensionKaplan-Yorkeconjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Individually nonchaotic Rulkov neurons, which spike regularly when uncoupled, become chaotic when wired into a ring with electrical coupling. The paper shows that varying the coupling strength drives the ring through distinct regimes—unsynchronized chaotic spiking, synchronized chaotic bursting, and synchronized hyperchaos—even though every isolated neuron behaves regularly. Using the full Lyapunov spectrum and the Kaplan–Yorke conjecture, the author estimates the fractal dimension of the resulting attractors and finds that they occupy as many as 45 of the 60 dimensions of state space, with Lyapunov-dimension estimates matching box-counting estimates to within a few percent. The significance is that high-dimensional chaos and fractal geometry emerge from a lattice of simple, individually regular units, making the system a tractable model for collective behavior in neuronal networks.

What carries the argument

The load-bearing object is the piecewise nonchaotic Rulkov map, where the fast variable evolves by $f(x,y;\alpha)$, and the electrical ring coupling enters as $C_i = \frac{g}{2}(x_{i-1}+x_{i+1}-2x_i)$ added to both the fast and slow update of each neuron. From this, the paper constructs the $60\times60$ piecewise Jacobian of the ring and feeds it into the QR-factorization algorithm for the Lyapunov spectrum. The Kaplan–Yorke dimension $d_L = \kappa + \frac{1}{|\lambda_{\kappa+1}|}\sum_{i=1}^\kappa \lambda_i$, where $\kappa$ is the largest index with cumulative Lyapunov sum nonnegative, then converts the spectrum into an estimate of the attractor's fractal dimension, bypassing box-counting, which would require on the order of $10^{36}$ points in 60-dimensional space.

What would settle it

Recompute the full Lyapunov spectrum for the homogeneous ring at $g = 0.1$, $0.25$, $0.6$, and $0.9$ using orbits of length 100,000 with the first 10,000 steps discarded as transient, then compare the Kaplan–Yorke dimensions with those in Table 1; a shift beyond a few percent would show the convergence assumption fails.

Watch

Extended reading notes

Core claim

The central discovery is that electrical coupling alone is enough to make a ring of nonchaotic Rulkov neurons chaotic, and that the strange attractors produced are genuinely high-dimensional. For a homogeneous ring with $\sigma=-0.5$ and $\alpha=4.5$, coupling $g=0.05$ already yields $\lambda_1 \approx 0.0491$; $g=0.25$ yields synchronized chaotic bursting; and $g=1$ yields synchronized hyperchaos with 11 positive Lyapunov exponents out of 60. The author computes all 60 Lyapunov exponents via QR factorization of the piecewise ring Jacobian and uses the Kaplan–Yorke formula to obtain Lyapunov dimensions $d_L$. In the homogeneous regime, $d_L$ reaches values close to 45 at moderate coupling and is validated against direct box-counting estimates at four coupling strengths with errors between 0.04% and 2.65%. A striking result is that the maximal Lyapunov exponent and the attractor dimension do not track each other: synchronized hyperchaos has the largest $\lambda_1$ but a smaller dimension than the weaker, unsynchronized chaotic spiking regime, because synchronization concentrates expansion in few directions.

Load-bearing premise

The reported chaos depends on assuming that a single 1000-step run, with no initial transient discarded and no averaging over starting states, already captures the long-term behavior at every coupling strength.

Editorial extensions

If this is right

  • Coupling strength $g$ acts as a single control parameter that moves a homogeneous ring through unsynchronized chaotic spiking, synchronized chaotic bursting, and synchronized hyperchaos.
  • The Kaplan–Yorke dimension gives a practical proxy for the true fractal dimension of high-dimensional neuron-lattice attractors, where box-counting would need roughly $10^{36}$ points.
  • Synchronization lowers an attractor's fractal dimension even while raising the maximal Lyapunov exponent, because it reduces the number of positive Lyapunov exponents.
  • Partially and fully heterogeneous rings are chaotic even at zero coupling, yet their Lyapunov-dimension curves become similar to the homogeneous case as coupling increases.
  • A 60-dimensional ring of simple maps supports attractors spanning about three quarters of the state space, showing that high-dimensional collective chaos needs no chaotic individual units.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the slow variable evolves with $\mu = 0.001$, the reported orbits of length 1000 cover roughly one slow oscillation; recomputing spectra with $10^4$–$10^5$ iterations and a discarded transient would directly test whether the Lyapunov exponents and dimension curves are numerically converged.
  • The same Lyapunov-dimension workflow could be applied to other lattice topologies the author mentions—mesh, torus, all-to-all coupling—to see whether the dimension-versus-coupling pattern (left peak higher than right in homogeneous rings) is generic.
  • If the synchronization-lowers-dimension effect is robust, it suggests a practical design lever for neuromorphic or reservoir-computing hardware: tuning coupling or heterogeneity can expand or contract the effective dimensionality of a fixed-size network without altering the number of neurons.
  • The numerical confirmation of Kaplan–Yorke for this piecewise-smooth system hints that the conjecture may extend to a broader class of hybrid or hysteretic maps, where piecewise Jacobians make analytic dimension estimates difficult.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This manuscript studies a ring of 30 electrically coupled nonchaotic Rulkov neurons (60-dimensional state space) in three parameter regimes: homogeneous, partially heterogeneous in sigma_i, and fully heterogeneous in sigma_i and alpha_i. The author computes maximal Lyapunov exponents as a function of coupling strength g, identifies regimes of chaotic spiking, synchronized chaotic bursting, and synchronized hyperchaos, and estimates attractor dimensions via the Kaplan-Yorke formula applied to the full 60-exponent Lyapunov spectrum. The headline quantitative claim is that strange attractors can occupy up to about 45 of the 60 dimensions, supported by a comparison of Lyapunov dimension to estimated box-counting dimension for four values of g.

Significance. The qualitative phenomenon---that coupling individually regular-spiking Rulkov neurons can produce chaotic and hyperchaotic collective dynamics---is plausible, visually documented, and of interest for map-based neuronal network studies. The paper contributes an explicit Jacobian for the ring system (Appendix A), pseudocode, and a public GitHub repository, and it systematically scans the coupling strength, going beyond the earlier qualitative discussion in Ref. [40]. If the Lyapunov spectra and box-counting estimates were properly converged and fully documented, the dimension trends (including the non-monotonic relation between lambda_1 and d_L) would be a substantial contribution. At present, however, the quantitative pillars supporting the 45/60 claim and the Kaplan-Yorke validation are not numerically secured.

major comments (3)
  1. [Section 2 / Appendix B] The claim that an orbit of length 1000 is sufficiently long for Lyapunov exponent convergence is load-bearing and unsupported. With mu=0.001, the slow variable evolves on a timescale of order 1/mu = 1000 iterations, so the first 1000 iterates from the initial condition (A12) include the transient toward the attractor. No burn-in, no averaging over initial conditions, and no error bars are reported. Since Eq. (A11) estimates each lambda_i as (1/t) sum ln|r_ii^(k)| with t=1000, finite-time fluctuations in the near-zero and negative exponents that determine kappa in Eq. (12) can be comparable to the exponent magnitudes; an error of only a few percent in the 43rd-45th exponents shifts d_L by several units. All points in Figures 3, 6, and 8 and all rows of Table 1 inherit this uncertainty. Please add convergence tests (lambda_1 and d_L versus t for t=10^3, 10^4, 10^5) and ensemble averages over initial conditions, with transients discarded.
  2. [Section 3, Table 1] The box-counting validation of the Kaplan-Yorke approximation is not reproducible as written. The text states that points are sampled by generating many orbits of length 10^7 and that close values of epsilon are chosen, but it does not give the epsilon values, the total number of sampled points, the number of orbits, the box-counting algorithm in 60 dimensions, or uncertainties on d_L and d. For a claimed dimension near 43, a literal box count at resolution epsilon would require N(epsilon) ~ epsilon^(-43) sampled points, which the stated sampling cannot provide unless the attractor is much lower-dimensional at the scales used. Without this information the 5% agreement in Table 1 cannot be assessed, and the conclusion that the Kaplan-Yorke conjecture does hold for this system is not supported. Please report the full estimation protocol and uncertainties, or use a dimension estimator that is feasible and fully specified in high dimensions.
  3. [Section 2, choice of zeta=30] The statement that By using numerical simulations to systematically vary zeta (see Appendix C), it can be found that for zeta>=4, varying zeta has no effect on the qualitative behavior is not supported by the cited appendix, which contains only pseudocode. If the zeta-independence claim is used to generalize from zeta=30 to other ring sizes, please provide the numerical evidence (e.g., lambda_1 or d_L versus zeta); otherwise restrict the conclusions to the zeta=30 system.
minor comments (4)
  1. [Equation (10)] The last row of the displayed iteration function is missing the factor g/2 in the coupling term of the y-update for neuron zeta-1; compare with Eq. (8) and Eq. (A2), where the factor is present. Please correct the displayed function.
  2. [Algorithm A5] The stopping condition if S<=0 is not the exact implementation of Eq. (12), which defines kappa as the largest index with cumulative sum >=0; the two agree generically but differ when a partial sum is exactly zero. Please align the boundary case.
  3. [Title and Abstract] The terminology nonchaotic Rulkov neurons is used for all three regimes, but in the partially and fully heterogeneous cases some individual neurons are chaotic even at g=0 (lambda_1 approx 0.0644 and 0.0469 in Figures 4a and 5a). Please qualify the terminology.
  4. [Section 3, box-counting description] The phrase close values of epsilon are chosen, where the sampled points scale according to their attractor is circular as written; please state explicitly how the scaling region was selected and how many points were used at each epsilon.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the attractor-dimension results are computed from the explicit Jacobian and independently spot-checked by box-counting; self-citations are methodological, not load-bearing.

full rationale

The paper's central claim—attractors occupying up to 45 of the 60 dimensions—is obtained by computing the full Lyapunov spectrum from the ring system's explicit 60-dimensional Jacobian (Appendix A, Eq. (A4)) with the QR method (Appendix B), then applying the Kaplan-Yorke formula (Eq. (13)). No parameter is fitted to the headline dimension, and no equation reduces to the target result by construction. The Kaplan-Yorke dimension is separately compared with box-counting estimates in Table 1, so the use of d_L as an approximation of the fractal dimension has independent, though under-described, numerical support; the extrapolation of that check to all g values is a generalization, not a circular step. The paper's self-citations (Refs. [24], [41], [52], and [61]) are to the author's prior technical derivations and related studies; none supplies a load-bearing premise, forbids alternatives, or smuggles in the conclusion. Concerns that are legitimate but non-circular: the claim that 1000-step single orbits are sufficient for Lyapunov convergence (Section 2 and Appendix B) is asserted without burn-in or ensemble averaging; the claim that dynamics are independent of zeta for zeta >= 4 refers to Appendix C, which contains pseudocode rather than the stated scan; and the Table 1 box-counting procedure omits epsilon values, point counts, and orbit counts. These affect reproducibility and numerical reliability but do not make the derivation circular.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central numerical results rest on the standard Rulkov map and coupling model, plus several assumptions of convergence and conjecture validity. The orbit-length and ζ-invariance assumptions are asserted in the text without supporting numerical evidence.

free parameters (6)
  • Homogeneous α = 4.5
    Hand-chosen to put individual neurons in the nonchaotic spiking regime (Section 2).
  • Homogeneous σ = -0.5
    Hand-chosen; σ is the excitation parameter controlling y dynamics (Section 2).
  • Slow-variable rate µ = 0.001
    Standard value from Rulkov map literature, cited to Ref. [15], to make y slow.
  • Heterogeneous σ_i draws = Uniform(-1.5,-0.5), values in Eq. (A13)
    Randomly drawn per neuron to create partially and fully heterogeneous regimes; interval chosen by hand.
  • Heterogeneous α_i draws = Uniform(4.25,4.75), values in Eq. (A14)
    Randomly drawn per neuron in fully heterogeneous regime; interval chosen by hand.
  • Initial fast variables x_i,0 = Uniform(-1,1), values in Eq. (A12)
    Random initial conditions per neuron; the specific list is given for reproducibility.
assumptions (6)
  • domain assumption The nonchaotic Rulkov map (Eq. 2) is an adequate model of neuronal spiking and bursting dynamics.
    Adopted from Refs. [13,15]; the paper does not question this modeling step.
  • domain assumption Electrical coupling enters only through the fast-variable difference with strength g on both x and y updates (Eqs. 5-6 with β_i = σ_i = 1).
    Coupling model taken from Ref. [15]; the paper sets β_i^c = σ_i^c = 1.
  • domain assumption The Kaplan-Yorke conjecture holds for this system, so the Lyapunov dimension equals the attractor's fractal dimension.
    Invoked in Section 3; the paper attempts a check at four values of g, but the box-counting procedure is under-specified.
  • ad hoc to paper An orbit of length 1000 is sufficient for Lyapunov exponent convergence.
    Asserted without supporting convergence tests in Section 2; load-bearing for all Lyapunov results.
  • ad hoc to paper For ζ ≥ 4 the qualitative behavior does not depend on ring size.
    Asserted in Section 2 with a reference to Appendix C, which contains only pseudocode; no size-sweep data are shown.
  • domain assumption Chaos is defined as a positive maximal Lyapunov exponent (Ref. [44]).
    Standard definition adopted in Section 2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Chaotic Dynamics and Fractal Geometry in Ring Lattice Systems of Nonchaotic Rulkov Neurons." pith.science (2026). https://pith.science/paper/7CSB6SCG

@misc{pith2026241212134,
  author       = {Pith},
  title        = {Pith review of: Chaotic Dynamics and Fractal Geometry in Ring Lattice Systems of Nonchaotic Rulkov Neurons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CSB6SCG}},
  note         = {Machine review of arXiv:2412.12134}
}
read the original abstract

This paper investigates the complex dynamics and fractal attractors that arise in a 60-dimensional ring lattice system of electrically coupled nonchaotic Rulkov neurons. While networks of chaotic Rulkov neurons have been widely studied, systems of nonchaotic Rulkov neurons have not been extensively explored due to the piecewise complexity of the nonchaotic Rulkov map. Here, we find that rich dynamics emerge from the electrical coupling of regular-spiking Rulkov neurons, including chaotic spiking, synchronized chaotic bursting, and synchronized hyperchaos. By systematically varying the electrical coupling strength between neurons, we also uncover general trends in the maximal Lyapunov exponent across the system's dynamical regimes. By means of the Kaplan-Yorke conjecture, we examine the fractal geometry of the ring system's high-dimensional chaotic attractors and find that these attractors can occupy as many as 45 of the 60 dimensions of state space. We further explore how variations in chaotic behavior - quantified by the full Lyapunov spectra - correspond to changes in the attractors' fractal dimensions. This analysis advances our understanding of how complex collective behavior can emerge from the interaction of multiple simple neuron models and highlights the deep interplay between dynamics and geometry in high-dimensional systems.

Figures

Figures reproduced from arXiv: 2412.12134 by the authors.

Figure 1
Figure 1. Visualization of a ring of ζ = 30 Rulkov neurons. Neurons are shown as blue points, and electrical coupling connections are shown in gold. 2. The Model and its Dynamics The nonchaotic Rulkov map is defined by the following iteration function [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Graphs of the fast-variable orbits of the first eight neurons in the homogeneous regime of the ring lattice system, with xi,0 ∈ (−1, 1), yi,0 = −3.25, σi = −0.5, and αi = 4.5. The four coupling strength values show four distinct regimes of behavior: (a) g = 0, λ1 ≈ −0.0938 (uncoupled nonchaotic spiking); (b) g = 0.05, λ1 ≈ 0.0491 (unsynchronized chaotic spiking); (c) g = 0.25, λ1 ≈ 0.0595 (synchronized chaotic burst… view at source ↗
Figure 3
Figure 3. Graph of the maximal Lyapunov exponent λ1 against the electrical coupling strength g for the homogeneous case, with xi,0 ∈ (−1, 1), yi,0 = −3.25, σi = −0.5, and αi = 4.5. The maximal Lyapunov exponent graph shows the four distinct regimes of behavior: the uncoupled regime, unsynchronized chaotic spiking regime, synchronized chaotic bursting regime, and synchronized hyperchaotic regime. The maximal Lyapunov exponents… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Graphs of the fast-variable orbits of the first eight neurons in the partially heteroge￾neous regime of the ring lattice system, with xi,0 ∈ (−1, 1), yi,0 = −3.25, σi ∈ (−1.5, −0.5), and αi = 4.5. The four coupling strength values show four distinct regimes of behavior…
Figure 5
Figure 5. Figure 5: Graphs of the fast-variable orbits of the first eight neurons in the fully heterogeneous regime of the ring lattice system, with xi,0 ∈ (−1, 1), yi,0 = −3.25, σi ∈ (−1.5, −0.5), and αi ∈ (4.25, 4.75). The four coupling strength values show four distinct regimes of beha…
Figure 6
Figure 6. Figure 6: Graphs of the maximal Lyapunov exponent λ1 against the electrical coupling strength g for (a) the partially heterogeneous case, with xi,0 ∈ (−1, 1), yi,0 = −3.25, σi ∈ (−1.5, −0.5), and αi = 4.5, and (b) the fully heterogeneous case, with xi,0 ∈ (−1, 1), yi,0 = −3.25, …
Figure 7
Figure 7. Figure 7: Projections of attractors of the 60-dimensional ring lattice system onto the (x0, y0) plane for the homogeneous case, with xi,0 ∈ (−1, 1), yi,0 = −3.25, σi = −0.5, and αi = 4.5. The attractors are plotted using orbits of length 100,000, and we show attractors with the …
Figure 8
Figure 8. Figure 8: Graphs of the Lyapunov dimension dl against the electrical coupling strength g for the (a) homogeneous case, (b) partially heterogeneous case, and (c) fully heterogeneous case of the ring lattice system of ζ = 30 electrically coupled Rulkov neurons. The Lyapunov dimens…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Hyperchaos and complex dynamical regimes in $N$-dimensional neuron lattices

    nlin.CD 2025-05 conditional novelty 6.0 of 10

    Coupled nonchaotic Rulkov neurons on N-dimensional lattices exhibit dimension-dependent synchronization regimes, including synchronized hyperchaos and one-step lag synchronization at strong coupling.

  2. Ubiquity of Uncertainty in Neuron Systems

    q-bio.NC 2025-07 conditional novelty 4.0 of 10

    Uncertainty exponents and basin entropy show fractal basin boundaries in five coupled neuron-map models, supporting a 'chance synchronization' mechanism for unpredictability.

Reference graph

Works this paper leans on

62 extracted references · 60 canonical work pages · cited by 2 Pith papers

  1. [40]

    Synchronized chaotic intermittent and spiking behavior in coupled map chains

    Osipov, G.V .; Ivanchenko, M.V .; Kurths, J.; Hu, B. Synchronized chaotic intermittent and spiking behavior in coupled map chains. Phys. Rev. E2005,71, 056209

    work page

  2. [1]

    Neural excitability, spiking and bursting.Int

    Izhikevich, E.M. Neural excitability, spiking and bursting.Int. J. Bifurc. Chaos1999,10, 1171–1266

    work page

  3. [2]

    A quantitative description of membrane current and its application to conduction and excitation in nerve.J

    Hodgkin, A.L.; Huxley, A.F. A quantitative description of membrane current and its application to conduction and excitation in nerve.J. Physiol.1952,117, 500–544

    work page 1952

  4. [3]

    Chaos in a three-variable model of an excitable cell.Phys

    Chay, T.R. Chaos in a three-variable model of an excitable cell.Phys. D Nonlinear Phenom.1985,16, 233–242

    work page 1985

  5. [4]

    Mathematical model of an identified stomatogastric ganglion neuron.J

    Buchholtz, F.; Golowasch, J.; Epstein, I.R.; Marder, E. Mathematical model of an identified stomatogastric ganglion neuron.J. Neurophysiol.1992,67, 332–340

    work page 1992

  6. [5]

    Simple model of spiking neurons.IEEE Trans

    Izhikevich, E.M. Simple model of spiking neurons.IEEE Trans. Neural Netw.2003,14, 1569–1572

    work page 2003

  7. [6]

    Impulses and Physiological States in Theoretical Models of Nerve Membrane.Biophys

    FitzHugh, R. Impulses and Physiological States in Theoretical Models of Nerve Membrane.Biophys. J.1961,1, 445–466

    work page 1961

  8. [7]

    A model of neuronal bursting using three coupled first order differential equations.Proc

    Hindmarsh, J.L.; Rose, R.M. A model of neuronal bursting using three coupled first order differential equations.Proc. R. Soc. B 1984,221, 87–102

    work page 1984

Show all 62 references
  1. [8]

    Rinzel, J., AFormal Classification of Bursting Mechanisms in Excitable Systems; Springer: Berlin/Heidelberg, Germany, 1987; pp. 267–281

    1987

  2. [9]

    Classification of Bursting Mappings.Int

    Izhikevich, E.M.; Hoppensteadt, F. Classification of Bursting Mappings.Int. J. Bifurc. Chaos2004,14, 3847–3854

  3. [10]

    Chaotic oscillations in a map-based model of neural activity.Chaos2007,17, 043109

    Courbage, M.; Nekorkin, V .I.; Vdovin, L.V . Chaotic oscillations in a map-based model of neural activity.Chaos2007,17, 043109

  4. [11]

    Synchronization of slow-fast systems.Eur

    Omelchenko, I.; Rosenblum, M.; Pikovsky, A. Synchronization of slow-fast systems.Eur. Phys. J. Spec. Top.2011,191, 3–14

    2011

  5. [12]

    Which model to use for cortical spiking neurons?IEEE Trans

    Izhikevich, E. Which model to use for cortical spiking neurons?IEEE Trans. Neural Netw.2004,15, 1063–1070

    2004

  6. [13]

    Modeling of spiking-bursting neural behavior using two-dimensional map.Phys

    Rulkov, N.F. Modeling of spiking-bursting neural behavior using two-dimensional map.Phys. Rev. E2002,65, 041922

  7. [14]

    Regularization of synchronized chaotic bursts.Phys

    Rulkov, N.F. Regularization of synchronized chaotic bursts.Phys. Rev. Lett.2001,86, 183–186

    2001

  8. [15]

    Map-based models in neuronal dynamics.Phys

    Ibarz, B.; Casado, J.M.; Sanjuán, M.A.F. Map-based models in neuronal dynamics.Phys. Rep.2011,501, 1–74

    2011

  9. [16]

    Bursting as an emergent phenomenon in coupled chaotic maps.Phys

    de Vries, G. Bursting as an emergent phenomenon in coupled chaotic maps.Phys. Rev. E2001,64, 051914

  10. [17]

    Dynamics in a memristive neural network with three discrete heterogeneous neurons and its application.Nonlinear Dyn.2024,113, 5811–5824

    Luo, D.; Wang, C.; Deng, Q.; Sun, Y. Dynamics in a memristive neural network with three discrete heterogeneous neurons and its application.Nonlinear Dyn.2024,113, 5811–5824

    2024

  11. [18]

    Switching bifurcation of a Rulkov neuron system with ReLu-type memristor.Nonlinear Dyn

    Min, F.; Zhai, G.; Yin, S.; Zhong, J. Switching bifurcation of a Rulkov neuron system with ReLu-type memristor.Nonlinear Dyn. 2024,112, 5687–5706

    2024

  12. [19]

    Memristive effects on an improved discrete Rulkov neuron model.Sci

    Bao, H.; Li, K.; Ma, J.; Hua, Z.; Xu, Q.; Bao, B. Memristive effects on an improved discrete Rulkov neuron model.Sci. China Technol. Sci.2023,66, 3153–3163

    2023

  13. [20]

    Bursting synchronization in non-locally coupled maps.Phys

    de Pontes, J.; Viana, R.; Lopes, S.; Batista, C.; Batista, A. Bursting synchronization in non-locally coupled maps.Phys. A: Stat. Mech. Its Appl.2008,387, 4417–4428

    2008

  14. [21]

    Stability and chaos of Rulkov map-based neuron network with electrical synapse.Commun

    Wang, C.; Cao, H. Stability and chaos of Rulkov map-based neuron network with electrical synapse.Commun. Nonlinear Sci. Numer. Simul.2015,20, 536–545

    2015

  15. [22]

    Controlling the bursting size in the two-dimensional Rulkov model.Commun

    López, J.; Coccolo, M.; Capeáns, R.; Sanjuán, M.A. Controlling the bursting size in the two-dimensional Rulkov model.Commun. Nonlinear Sci. Numer. Simul.2023,120,107–184

    2023

  16. [23]

    Symbolic analysis of bursting dynamical regimes of Rulkov neural networks.Neurocomputing 2021,441

    Budzinski, R.; Lopes, S.; Masoller, C. Symbolic analysis of bursting dynamical regimes of Rulkov neural networks.Neurocomputing 2021,441. 44–51

    2021

  17. [24]

    Asymmetric coupling of nonchaotic Rulkov neurons: Fractal attractors, quasimultistability, and final state sensitivity

    Le, B.B. Asymmetric coupling of nonchaotic Rulkov neurons: Fractal attractors, quasimultistability, and final state sensitivity. Phys. Rev. E2025,111, 034201

  18. [25]

    Intermittent evolution routes to the periodic or the chaotic orbits in Rulkov map featured.Chaos2021,31, 093119

    Ge, P .; Cao, H. Intermittent evolution routes to the periodic or the chaotic orbits in Rulkov map featured.Chaos2021,31, 093119

  19. [26]

    Hidden and self-excited firing activities of an improved Rulkov neuron, and its application in information patterns.Nonlinear Dyn.2024,112, 13503–13517

    Njitacke, Z.T.; Takembo, C.N.; Sani, G.; Marwan, N.; Yamapi, R.; Awrejcewicz, J. Hidden and self-excited firing activities of an improved Rulkov neuron, and its application in information patterns.Nonlinear Dyn.2024,112, 13503–13517

    2024

  20. [27]

    Ding, D.; Niu, Y.; Yang, Z.; Wang, J.; Wang, W.; Wang, M.; Jin, F. Extreme multi-stability and microchaos of fractional-order memristive Rulkov neuron model considering magnetic induction and its digital watermarking application.Nonlinear Dyn.2024, 112, 15523–15545. Fractal Fr...

    2024

  21. [28]

    Stability and synchronization of coupled Rulkov map-based neurons with chemical synapses.Commun

    Hu, D.; Cao, H. Stability and synchronization of coupled Rulkov map-based neurons with chemical synapses.Commun. Nonlinear Sci. Numer. Simul.2016,35, 105–122

    2016

  22. [29]

    Synchronization and firing patterns of coupled Rulkov neuronal map.Nonlinear Dyn

    Rakshit, S.; Ray, A.; Bera, B.K.; Ghosh, D. Synchronization and firing patterns of coupled Rulkov neuronal map.Nonlinear Dyn. 2018,94, 785–805

    2018

  23. [30]

    Complete synchronization of coupled Rulkov neuron networks.Nonlinear Dyn.2016,84, 2423–2434

    Sun, H.; Cao, H. Complete synchronization of coupled Rulkov neuron networks.Nonlinear Dyn.2016,84, 2423–2434

    2016

  24. [31]

    Coupling Dependence on Chaos Synchronization Process in a Network of Rulkov Neurons.Int

    Marghoti, G.; Ferrari, F.A.S.; Viana, R.L.; Lopes, S.R.; de Lima Prado, T. Coupling Dependence on Chaos Synchronization Process in a Network of Rulkov Neurons.Int. J. Bifurc. Chaos2023,33, 2350132

  25. [32]

    Memristive Rulkov neuron model with magnetic induction effects.IEEE Trans

    Li, K.; Bao, H.; Li, H.; Ma, J.; Hua, Z.; Bao, B. Memristive Rulkov neuron model with magnetic induction effects.IEEE Trans. Ind. Inform.2021,18, 1726–1736

    2021

  26. [33]

    A multiplier-free Rulkov neuron under memristive electromagnetic induction: Dynamics analysis, energy calculation, and circuit implementation.Chaos2023,33, 083138

    Zhang, S.; Wang, C.; Zhang, H.; Lin, H. A multiplier-free Rulkov neuron under memristive electromagnetic induction: Dynamics analysis, energy calculation, and circuit implementation.Chaos2023,33, 083138

  27. [34]

    Dynamical variability, order-chaos transitions, and bursting Canards in the memristive Rulkov neuron model.Chaos Solitons, Fractals2024,186, 115317

    Bashkirtseva, I.; Ryashko, L. Dynamical variability, order-chaos transitions, and bursting Canards in the memristive Rulkov neuron model.Chaos Solitons, Fractals2024,186, 115317

  28. [35]

    Lag synchronization in an unidirectional ring of memristive neurons.Eur

    Vijayan, V .; Natiq, H.; Momani, S.; Pham, V .T.; Perc, M. Lag synchronization in an unidirectional ring of memristive neurons.Eur. Phys. J. Spec. Top.2025,234, 1011–1022

    2025

  29. [36]

    Deng, Q.; Wang, C.; Yang, G.; Luo, D. Discrete Memristive Delay Feedback Rulkov Neuron Model: Chaotic Dynamics, Hardware Implementation and Application in Secure Communication.IEEE Internet Things J.2025,12, 25559–25567

    2025

  30. [37]

    Enhancing synchronization in chaotic oscillators by induced heterogeneity.Eur

    Banerjee, R.; Bera, B.K.; Ghosh, D.; Dana, S.K. Enhancing synchronization in chaotic oscillators by induced heterogeneity.Eur. Phys. J. Spec. Top.2017,226, 1893–1902

    2017

  31. [38]

    Synchronization in a network of model neurons.Phys

    Jampa, M.P .K.; Sonawane, A.R.; Gade, P .M.; Sinha, S. Synchronization in a network of model neurons.Phys. Rev. E2007,75, 026215

  32. [39]

    Enhancement of neuronal coherence by diversity in coupled Rulkov-map models.Phys

    Chen, H.; Zhang, J.; Liu, J. Enhancement of neuronal coherence by diversity in coupled Rulkov-map models.Phys. A: Stat. Mech. Its Appl.2008,387, 1071–1076

    2008

  33. [41]

    Exploring Geometrical Properties of Chaotic Systems Through an Analysis of the Rulkov Neuron Maps

    Le, B.B.; Gandhi, N.A. Exploring Geometrical Properties of Chaotic Systems Through an Analysis of the Rulkov Neuron Maps. arXiv2024, arXiv:2406.08385

    arXiv

  34. [42]

    Distribution of spiking and bursting in Rulkov’s neuron model.Eur

    Ramírez-Ávila, G.M.; Depickère, S.; Jánosi, I.M.; Gallas, J.A. Distribution of spiking and bursting in Rulkov’s neuron model.Eur. Phys. J. Spec. Top.2022,231, 319–328

    2022

  35. [43]

    Ergodic theory of chaos and strange attractors.Rev

    Eckmann, J.P .; Ruelle, D. Ergodic theory of chaos and strange attractors.Rev. Mod. Phys.1985,57, 617–656

    1985

  36. [44]

    Alligood, K.T.; Sauer, T.D.; Yorke, J.A.Chaos: An Introduction to Dynamical Systems; Springer: New York, NY, USA, 1996

    1996

  37. [45]

    Hyperchaos.Scholarpedia2007,2, 1936

    Letellier, C.; Rossler, O.E. Hyperchaos.Scholarpedia2007,2, 1936

    1936

  38. [46]

    Estimating fractal dimension.J

    Theiler, J. Estimating fractal dimension.J. Opt. Soc. Am. A1990,7, 2073442

  39. [47]

    Chaotic behavior of multidimensional difference equations.Funct

    Kaplan, J.L.; Yorke, J.A. Chaotic behavior of multidimensional difference equations.Funct. Differ. Equ. Approx. Fixed Points1979, 730, 204–227

  40. [48]

    Controlling system dimension: A class of real systems that obey the Kaplan–Yorke conjecture.Proc

    Nichols, J.M.; Todd, M.D.; Seaver, M.; Trickey, S.T.; Pecora, L.M.; Moniz, L. Controlling system dimension: A class of real systems that obey the Kaplan–Yorke conjecture.Proc. Natl. Acad. Sci. USA2003,100, 15299–15303

  41. [49]

    The dimension of chaotic attractors.Phys

    Farmer, J.D.; Ott, E.; Yorke, J.A. The dimension of chaotic attractors.Phys. D Nonlinear Phenom.1983,7, 153–180

    1983

  42. [50]

    Synchronization of chaotic systems.Chaos2015,25, 097611

    Pecora, L.M.; Carroll, T.L. Synchronization of chaotic systems.Chaos2015,25, 097611

  43. [51]

    Low-dimensional chaos in high-dimensional phase space: How does it occur?Chaos Solitons Fractals 2003,15, 219–232

    Lai, Y.C.; Bollt, E.M.; Liu, Z. Low-dimensional chaos in high-dimensional phase space: How does it occur?Chaos Solitons Fractals 2003,15, 219–232

    2003

  44. [52]

    Hyperchaos and complex dynamical regimes in N-dimensional neuron lattices.Eur

    Le, B.B.; Watkins, D. Hyperchaos and complex dynamical regimes in N-dimensional neuron lattices.Eur. Phys. J. Spec. Top.2025, in press

    2025

  45. [53]

    A digital hardware implementation of spiking neural networks with binary FORCE training.Neurocomputing2020,412, 129–142

    Akbarzadeh-Sherbaf, K.; Safari, S.; Vahabie, A.H. A digital hardware implementation of spiking neural networks with binary FORCE training.Neurocomputing2020,412, 129–142

  46. [54]

    Dynamic analysis and implementation of FPGA for a new 4D fractional-order memristive Hopfield neural network.Fractal Fract.2025,9, 115

    Yu, F.; Zhang, S.; Su, D.; Wu, Y.; Gracia, Y.M.; Yin, H. Dynamic analysis and implementation of FPGA for a new 4D fractional-order memristive Hopfield neural network.Fractal Fract.2025,9, 115

    2025

  47. [55]

    FPGA implementation of a complete digital spiking silicon neuron for circuit design and network approach.Sci

    Miao, X.; Ji, X.; Chen, H.; Mayet, A.M.; Zhang, G.; Wang, C.; Sun, J. FPGA implementation of a complete digital spiking silicon neuron for circuit design and network approach.Sci. Rep.2025,15, 8491

    2025

  48. [56]

    Synchronous Behavior of Two Coupled Biological Neurons.Phys

    Elson, R.C.; Selverston, A.I.; Huerta, R.; Rulkov, N.F.; Rabinovich, M.I.; Abarbanel, H.D.I. Synchronous Behavior of Two Coupled Biological Neurons.Phys. Rev. Lett.1998,81, 5692–5695

    1998

  49. [57]

    Synchronized action of synaptically coupled chaotic model neurons.Neural Comput.1996,8, 1567–1602

    Abarbanel, H.D.I.; Huerta, R.; Rabinovich, M.I.; Rulkov, N.F.; Rowat, P .F.; Selverston, A.I. Synchronized action of synaptically coupled chaotic model neurons.Neural Comput.1996,8, 1567–1602

    1996

  50. [58]

    Dynamics of two electrically coupled chaotic neurons: Experimental observations and model analysis.Biol

    Varona, P .; Torres, J.J.; Abarbanel, H.D.I.; Rabinovich, M.I.; Elson, R.C. Dynamics of two electrically coupled chaotic neurons: Experimental observations and model analysis.Biol. Cybern.2001,84, 91–101. Fractal Fract.2025,9, 584 24 of 24

    2001

  51. [59]

    Diverse coupling of neurons to populations in sensory cortex.Nature2015,521, 511–515

    Okun, M.; Steinmetz, N.A.; Cossell, L.; Iacaruso, M.F.; Ko, H.; Barthó, P .; Moore, T.; Hofer, S.B.; Mrsic-Flogel, T.D.; Carandini, M.; et al. Diverse coupling of neurons to populations in sensory cortex.Nature2015,521, 511–515

  52. [60]

    Excitatory synchronization of rat hippocampal interneurons during network activation in vitro

    Pendeliuk, V .S.; Melnick, I.V . Excitatory synchronization of rat hippocampal interneurons during network activation in vitro. Front. Cell. Neurosci.2023,17, 1129991

    2023

  53. [61]

    Describing chaotic systems.arXiv2024, arXiv:2407.07919

    Le, B. Describing chaotic systems.arXiv2024, arXiv:2407.07919

    arXiv

  54. [62]

    Numerical calculation of Lyapunov exponents.Math

    Sandri, M. Numerical calculation of Lyapunov exponents.Math. J.1996,6, 78–84. Disclaimer/Publisher’s Note:The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI ...

    1996

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.