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REVIEW 3 major objections 5 minor 105 references

High-order synchronization in identical neurons with asymmetric pulse coupling

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper shows that two identical leaky integrate-and-fire neurons, when coupled asymmetrically as an excitatory-inhibitory pair, can synchronize in high-order p/q frequency-locked states, and it provides an event-driven map method that…

desk verdict Solid analytical and numerical study showing asymmetric E-I coupling alone produces rich p/q locking in identical LIF neurons; main gaps are reproducibility and the conjectural status of the island-family claims. read the letter →

arxiv 2501.03557 v1 pith:L7VYIW4J submitted 2025-01-07 nlin.AO

Abhay , Gaurav Dar This is my paper · ORCID
classification nlin.AO MSC 34C1537G1592C2037N25
keywords high-ordersynchronizationp/qfrequencylockingleakyintegrate-and-fireneuronsexcitatory-inhibitorypairevent-drivenmapsgrazingbifurcationsmultistabilityFareytree
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

This paper establishes that two leaky integrate-and-fire neurons with identical intrinsic frequencies can lock into high-order p/q frequency ratios, where p spikes of one neuron accompany q spikes of the other, provided the coupling is asymmetric. The asymmetry is the natural one in an excitatory-inhibitory pair: both neurons send equal-strength pulses, but one pulse excites and the other inhibits. The authors show these p/q states organize into Farey-arranged tongues in the (g, alpha) parameter plane, carry their own internal bifurcation structure, and overlap to produce multistability. They introduce an event-driven map method that determines existence and linear stability of any periodic spike sequence and explicitly handles non-smooth grazing bifurcations, without restricting coupling strength or requiring frequency mismatch.

What carries the argument

The central object is the event-driven map of the two-neuron network, which integrates the voltage x_i and the synaptic variables E_i, Q_i between consecutive network spikes, replacing the infinite sum over past alpha pulses with a differential equation for the synaptic current. A periodic spike sequence is found by expressing the state after p+q spikes in terms of p+q unknown interspike intervals, imposing threshold-crossing equations, and then filtering solutions with Condition 1, that reset happens at the first threshold crossing of the firing neuron, and Condition 2, that the non-firing neuron does not reach threshold earlier. Stability is computed from the product of Jacobian matrices over one period, and grazing bifurcations are identified by tangency of the excitatory neuron's extended voltage trajectory to the threshold.

What would settle it

Simulate the full two-neuron ODE system with the same alpha-pulse synapses and first-crossing reset at a parameter point inside the analytically predicted 1/2 tongue while scanning many initial conditions; if no trajectory settles into the {1,2,2} spike sequence with stable linearized map, the central existence claim fails. Equivalently, hardware or biological E-I pairs that show no predicted p/q locking at the corresponding parameters would refute the model's quantitative relevance.

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Extended reading notes

Core claim

A pair of coupled LIF neurons with identical frequencies and asymmetric pulse coupling can exhibit high-order p/q frequency-locked states. In an E-I pair, equal reciprocal synaptic strengths produce asymmetry naturally: the neuron receiving inhibitory current has a non-monotonic voltage trajectory that can cross threshold multiple times, and the physical reset occurs at the first crossing. Using event-driven maps, the authors derive self-consistent equations for the interspike intervals of any scheduled periodic spike sequence, impose first-crossing and ordering conditions to reject unphysical roots, and construct a product Jacobian for linear stability. This yields the existence and stability of irreducible p/q tongues and reducible np/nq islands, including their internal bifurcations, with boundaries set by saddle-node bifurcations and three types of grazing bifurcations, two of which are new to this study.

Load-bearing premise

All quantitative boundaries assume the idealized leaky integrate-and-fire dynamics with instantaneous reset at the first threshold crossing and alpha-pulse synaptic currents; if real synapses or reset rules deviate from these, the exact p/q boundaries would shift.

Editorial extensions

If this is right

  • High-order p/q synchronization does not require two different intrinsic frequencies; coupling asymmetry alone can produce it in identical E-I neuron pairs.
  • The analytical method gives existence and stability of any p/q spike sequence at arbitrary coupling strength, removing unphysical firing-time roots that earlier approaches could produce for inhibitory coupling.
  • p/q frequency-locked regions in this system are not structure-less Arnold tongues; they contain internal bifurcations, spike-exchange boundaries, and multiple coexisting solutions.
  • The apparent abrupt endings of tongues and reducible np/nq islands in simulations are explained by multistability: the analytically computed regions continue smoothly, but other attractors win for the chosen initial condition.
  • At large alpha values, spike sequences obey a Farey arrangement: the sequence for a ratio P/Q is a concatenation of the sequences of its Farey parents.

Reading between the lines

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

  • If the mechanism is generic, other pairs of identical threshold oscillators with sign-asymmetric or directional coupling, not only chemical E-I neurons, should show similar p/q locking; this could be tested in electronic or optomechanical oscillator pairs.
  • The appearance of reducible np/nq regions only as small-alpha islands suggests a period-inflation cascade in parameter space, and mapping the full n-dependence might reveal scaling near the firing-death boundary. This is an editorial inference, not a claim of the paper.
  • Because intra-sequence and inter-sequence multistability involve different spike patterns with the same firing ratio, neural codes could in principle carry information in which p/q spike sequence is selected, not only in the firing rate. This extension is mine, not the paper's.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies a pair of identical leaky integrate-and-fire neurons coupled by asymmetric excitatory-inhibitory pulse coupling, and demonstrates that high-order p/q frequency locking arises even though the uncoupled neurons have identical frequencies. The authors develop an analytical event-driven map method that determines existence and stability of any periodic spike sequence of a p/q frequency-locked state, including conditions (Conditions 1 and 2 in Sec. IV.A) that filter unphysical solutions arising from multiple threshold crossings of the inhibitory neuron. They use this method to compute existence and bifurcation boundaries for representative cases (0/1, 1/2, 2/13, 2/14, and others), showing that the p/q regions have internal bifurcation structure with saddle-node and grazing bifurcations, and that multistability occurs both within a spike sequence and between different p/q states, as well as between reducible np/nq islands. The simulation results for the (g, α) plane show a rich tongue-and-island structure with a devil's staircase and a Farey arrangement of spike sequences at large α.

Significance. If the analytical method is sound, this is a valuable contribution to the theory of pulse-coupled neurons: it provides a parameter-free framework to determine existence and stability of arbitrary periodic spike sequences in a pair of LIF neurons, and it demonstrates that high-order synchronization does not require frequency mismatch, but can be induced by coupling asymmetry alone. The paper also reports a novel Farey arrangement of spike sequences and identifies two new types of grazing bifurcations (Type-2b and Type-3). The analytical method is validated against direct simulations in several representative cases, which gives confidence in the central claim. The discovery of intra-sequence and inter-sequence multistability inside a single p/q tongue is an interesting and relatively unexplored phenomenon.

major comments (3)
  1. [Sec. II, Eq. (10) and Appendix B] The event-driven map contains terms with factors 1/(α−1) and 1/(α−1)^2, e.g., in Hi(n) in Eq. (10) and in κ1, κ2 in Eq. (B15). The stated parameter range in Sec. III is 0 < α ≤ 30, which includes α = 1. For α = 1 the alpha-pulse f(u) = u e^{-u} is perfectly well defined, so the map should have a well-defined limiting form, but as written it diverges. The paper does not discuss this case or provide the limiting expression. Since the analytical method and the simulations rely on these equations across the full (g, α) range, this is a load-bearing gap that must be fixed by either excluding α = 1 explicitly or deriving the appropriate α = 1 form.
  2. [Secs. IV and V; Figs. 12–18] The analytical boundaries and bifurcation diagrams are obtained by solving the system of equations (23) and by 'continuation algorithms', but the paper gives no details of these numerical methods and provides no code or data. Consequently, an independent reader cannot reproduce the reported boundaries, the eigenvalue computations, or the bifurcation classifications. This is a significant reproducibility gap for a paper whose central claim is the validity of the analytical method, and it weakens the verification of the results beyond the few cases that are explicitly compared with simulations.
  3. [Sec. IV.C, Type-2b and Type-3] The definitions of the new grazing bifurcations (Type-2b and Type-3) are given only through schematic figures and verbal conditions, such as ∂x_max/∂c_k = 0 for Type-2b. The paper does not state how these conditions are evaluated in terms of the state variables and parameters, nor how they are coded in the continuation procedure. Without explicit equations defining the tangency condition and the bifurcation manifold, the claimed classification of boundaries in Figs. 14–16 is not fully verifiable. This is especially important because the paper uses these types to explain the disappearance of solutions and the structure of the tongues.
minor comments (5)
  1. [Sec. II, Eq. (3)] There is a typo: the second term should be 2α dE_i/dt, not 2α dE_i/dt^2.
  2. [Sec. III, Fig. 4 and related text] The discussion of reducible np/nq islands and their 'mutual overlaps' is introduced as a conjecture; this is acceptable, but the text should more clearly separate the numerically observed overlaps from analytically established ones, since the analytical evidence is given only for specific families (e.g., n/6n and n/7n).
  3. [Sec. IV.A, Step 4] The notation τ*_{r(m+1)}(m) is defined, but the text does not explicitly state that τ*_{r(m+1)}(m) is the first positive root of Eq. (11) for the neuron that does not fire; adding a sentence clarifying this would help readability.
  4. [Sec. V.B, Fig. 12] The caption says 'yellow color indicates the numerically obtained 1/2 frequency-locked region', but the figure also shows a region 'A' and 'B'; the text explains these, but the caption could more clearly identify the comparison between analytical boundaries and the numerical region.
  5. [General] The paper would benefit from a brief statement about the treatment of the α = 1 case in the simulations, even if it is only to say that this value was avoided or handled by a limiting procedure.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. The event-driven maps in Eqs. (4)-(10) are obtained by direct integration of the stated LIF model (Eqs. (1)-(3)), with the synaptic current convolution converted to a second-order ODE; no target synchronization property is assumed in that reduction. For existence, Sec. IV.A schedules an arbitrary spike sequence, expresses the periodic state (Eqs. (18)-(20)) purely in terms of the undetermined interspike intervals, and then imposes the threshold-crossing equations (Eq. (23)) and physical admissibility via Conditions 1 and 2. This is a root-finding/shooting construction from the model, not a fit: the p/q intervals and spike sequences are outputs, not inputs. For stability, Eq. (34) is the standard product-of-Jacobians monodromy condition computed along the solved periodic trajectory. Simulation results and continuation-computed boundaries (e.g., Figs. 11, 12, 13-18) serve as independent numerical checks of the same model and agree where checked; the conjectures in Secs. III and V are explicitly labeled as conjectures. There are no load-bearing self-citations: references such as [90]-[92] and [94] are prior technical tools by other authors, and the paper does not invoke a self-authored uniqueness theorem to forbid alternatives. The main limitation is that the continuation computations are not accompanied by code or data, which is a reproducibility gap rather than circular reasoning. Therefore score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted to the phenomenon; g and alpha are scanned control parameters. The analysis relies on the exact LIF-alpha-pulse model and the first-crossing reset rule. No new entities are introduced.

assumptions (3)
  • domain assumption The LIF model with alpha-pulse synaptic currents (Eqs. (1)-(3)) exactly describes the two-neuron dynamics.
    All analytical results and simulations are based on this model; the algebraic form of the synaptic kernel determines the event-driven maps and bifurcation structure.
  • domain assumption Neurons reset instantaneously and uniquely at the first threshold crossing (Conditions 1 and 2 in Sec. IV.A).
    This rule defines which solutions are physical and underlies the grazing bifurcation classification; it is not proven from more basic biology but is part of the LIF model specification.
  • domain assumption The voltage of the neuron receiving excitatory input rises monotonically, and the inhibited neuron's extended voltage trajectory can cross threshold multiple times (Sec. II, Sec. IV.C).
    These properties justify the uniqueness of tau*_2 and the existence of Type-1, Type-2a, Type-2b, and Type-3 grazing bifurcations.

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Pith. "Pith review of High-order synchronization in identical neurons with asymmetric pulse coupling." pith.science (2026). https://pith.science/paper/L7VYIW4J

@misc{pith2026250103557,
  author       = {Pith},
  title        = {Pith review of: High-order synchronization in identical neurons with asymmetric pulse coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L7VYIW4J}},
  note         = {Machine review of arXiv:2501.03557}
}
abstract

The phenomenon of high-order ($p/q$) synchronization, induced by \textit{two different} frequencies in the system, is well-known and studied extensively in forced oscillators including neurons and to a lesser extent in coupled oscillators. Their frequencies are locked such that for every $p$ cycles of one oscillator there are $q$ cycles of the other. We demonstrate this phenomenon in a pair of coupled neurons having \textit{identical} frequencies but \textit{asymmetric} coupling. Specifically, we focus on an excitatory(E)-inhibitory(I) neuron pair where such an asymmetry is naturally present even with equal reciprocal synaptic strengths $(g)$ and inverse time constant $(\alpha)$. We thoroughly investigate the asymmetric coupling-induced $p/q$ frequency locking structure in $(g,\alpha)$ parameter space through simulations and analysis. Simulations display quasiperiodicity, devil staircase, a novel Farey arrangement of spike sequences, and presence of reducible and irreducible $p/q$ regions. We introduce an analytical method, based on event-driven maps, to determine the existence and stability of any spike sequence of the two neurons in a $p/q$ frequency-locked state. Specifically, this method successfully deals with non-smooth bifurcations and we could utilize it to obtain solutions for the case of identical E-I neuron pair under arbitrary coupling strength. In contrast to the so-called Arnold tongues, the $p/q$ regions obtained here are not structure-less. Instead they have their own internal bifurcation structure with varying levels of complexity. Intra-sequence and inter-sequence multistability, involving spike sequences of same $p/q$ state, are found. Additionally, multistability also arises by overlap of $p/q$ with $p'/q'$. The boundaries of both reducible and irreducible $p/q$ regions are defined by saddle node and non-smooth grazing bifurcations of various types.

Figures

Figures reproduced from arXiv: 2501.03557 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) Spike pattern of excitatory (inhibitory) neurons for (a) Two cycles of 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) Dominant sixteen frequency-locked [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. (Color online) Magnification of boxed region of Fig. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (14 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Few sequences of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Devil staircase structure for [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Farey tree arrangement of (a) frequency locking Farey tree and (b) spike sequence Farey tree. Curly brackets in the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (Color online) Schematic figures of extended and actual voltage trajectories of excitatory (neuron 1) and inhibitory [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. (Color online) Schematic figures displaying grazing bifurcation of Type-1 in (a)-(c) and Type-3 in (d)-(f) on varying [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. (Color online) Schematic figures of extended voltage trajectories of excitatory neuron displaying grazing bifurcations [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. (Color online) Bifurcation boundary of the only [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. (Color online) Analytically obtained solutions of 1 [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. (Color online) The analytically obtained outer boundaries of the three valid spike sequences of 2 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. (Color online) This figure, analytically obtained, shows the bifurcation structure and number of solutions of the spike [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. (Color online) This figure, analytically obtained, shows the bifurcation structure and number of solutions of the spike [PITH_FULL_IMAGE:figures/full_fig_p018_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. (Color online) This figure, analytically obtained, shows the bifurcation structure and number of solutions of the spike [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. (Color online) Analytically obtained region of ex [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. (Color online) Outer boundary of the 1 [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]

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Works this paper leans on

105 extracted references · 77 canonical work pages

  1. [1]

    (27) Clearly δτ depends only on the state of neuron s

    The perturbation δτ (n) can be obtained by writing the differential of Fs and solving for δτ (n) yielding δτ (n) = − ∂Fs ∂τ !−1 ∂Fs ∂xs δxs(n) + ∂Fs ∂Es δEs(n) + ∂Fs ∂Qs δQs(n) ! . (27) Clearly δτ depends only on the state of neuron s. Re- placing for δτ (n) in Eqs.(24)-(26) δX(n + 1) = DF(n) s(n+1) δX(n), (28) where X = [ X1, X2]T represents the complete...

    work page arXiv

  2. [2]

    This equation needs to be solved numerically for τ ∗ 1 (0) for a given (g, α)

    Condition 2: τ ∗ 1 (0) should be obtained from Eq.(11) which takes the form ˜x+ 1 (0)e−τ ∗ 1 (0) + ζ1(τ ∗ 1 (0), ˜E+ 1 (0), ˜Q+ 1 (0)) = 1 (B13) Substituting ˜τ (0) into Eqs.(B4)-(B6) and then placing those into the above equation yields a tran- scendental equation in τ ∗ 1 (0). This equation needs to be solved numerically for τ ∗ 1 (0) for a given (g, α)...

  3. [3]

    For example, 1 /6 and 1 /7 are Farey parents of 2 /13

    P/Q = (p + p′)/(q + q′) In a Farey arrangement of spike sequences, the spike se- quence of P/Q is given by the concatenation of spike sequences of p/q and p′/q′. For example, 1 /6 and 1 /7 are Farey parents of 2 /13. Spike sequences of 1 /6 and 1/7 are {1, 26} and {1, 27} respectively. In our simu- lations, at large α values (for instance at A in Fig.4) 2...

  4. [4]

    Hence, ˜τ (0) = τ ∗ 2 (0) irrespective of ( g, α), satisfying con- dition 1

    Condition 1: Starting from ˜ x+ 2 (0) = 0, the un- derlying voltage trajectory of neuron 2 rises mono- tonically, crossing the threshold only once. Hence, ˜τ (0) = τ ∗ 2 (0) irrespective of ( g, α), satisfying con- dition 1

  5. [5]

    Simonet, M

    J. Simonet, M. Warden, and E. Brun, Physical Review E 50, 3383–3391 (1994)

    1994

  6. [6]

    S.-B. Shim, M. Imboden, and P. Mohanty, Science 316, 95–99 (2007)

    2007

  7. [7]

    Vance and J

    W. Vance and J. Ross, The Journal of Chemical Physics 91, 7654–7670 (1989)

    1989

  8. [8]

    D. G. Aronson, R. P. McGehee, I. G. Kevrekidis, and R. Aris, Physical Review A 33, 2190–2192 (1986)

    1986

Show all 105 references
  1. [9]

    Nakata, K

    S. Nakata, K. Miyazaki, S. Izuhara, H. Yamaoka, and D. Tanaka, The Journal of Physical Chemistry A 113, 6876–6879 (2009)

    2009

  2. [10]

    J. E. Flaherty and F. C. Hoppensteadt, Studies in Ap- plied Mathematics 58, 5–15 (1978)

    1978

  3. [12]

    Van der Pol and J

    B. Van der Pol and J. Van der Mark, Nature 120, 363–364 (1927)

    1927

  4. [13]

    Mettin, U

    R. Mettin, U. Parlitz, and W. Lauterborn, Interna- tional Journal of Bifurcation and Chaos 03, 1529–1555 (1993)

    1993

  5. [14]

    Behta, G

    E. Behta, G. H. Goldsztein, and L. English, Physica D: Nonlinear Phenomena 454, 133849 (2023)

    2023

  6. [15]

    M. Levy, A. Molzon, J.-H. Lee, J.-w. Kim, J. Cheon, and D. Bozovic, Scientific Reports 6 (2016), 10.1038/srep39116

    2016 doi

  7. [16]

    Several bifurcations and spike exchange shape these regions. Saddle-node, grazing bifurcations of Type-1, Type-2b and spike exchanges are found on inner bound- aries of all the three spike sequences while Type-2a graz- ing bifurcation is absent in one of the spike sequences ({...

  8. [17]

    J. Buck, E. Buck, J. F. Case, and F. E. Hanson, Journal of Comparative Physiology 144, 287–298 (1981)

    1981

  9. [18]

    M. R. Guevara, A. Shrier, and L. Glass, American Journal of Physiology-Heart and Circulatory Physiol- ogy 254, H1–H10 (1988)

    1988

  10. [19]

    Glass, M

    L. Glass, M. R. Guevara, A. Shrier, and R. Perez, Phys- ica D: Nonlinear Phenomena 7, 89–101 (1983)

    1983

  11. [21]

    Alinejad, D.-P

    H. Alinejad, D.-P. Yang, and P. Robinson, Physica D: Nonlinear Phenomena 402, 132231 (2020)

    2020

  12. [23]

    Pantaleone, American Journal of Physics 70, 992–1000 (2002)

    J. Pantaleone, American Journal of Physics 70, 992–1000 (2002)

    2002

  13. [24]

    H. M. Oliveira and L. V. Melo, Scientific Reports 5 (2015), 10.1038/srep11548

    2015 doi

  14. [25]

    C. Lu, M. Kim, Y. Yang, Y. S. Gui, and C.-M. Hu, Jour- nal of Applied Physics 134 (2023), 10.1063/5.0179159

    2023 doi

  15. [26]

    Srikanth, S

    S. Srikanth, S. A. Pawar, K. Manoj, and R. I. Sujith, Chaos: An Interdisciplinary Journal of Nonlinear Sci- ence 32 (2022), 10.1063/5.0085273

    2022 doi

  16. [28]

    M. H. Matheny, M. Grau, L. G. Villanueva, R. B. Kara- balin, M. Cross, and M. L. Roukes, Physical Review Letters 112 (2014), 10.1103/physrevlett.112.014101

    2014 doi

  17. [29]

    Singh, S

    H. Singh, S. Bhuktare, A. Bose, A. Fukushima, K. Yakushiji, S. Yuasa, H. Kubota, and A. A. Tulapurkar, Physical Review Applied 11 (2019), 10.1103/physrevapplied.11.054028

    2019 doi

  18. [30]

    S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nature 437, 389–392 (2005)

    2005

  19. [31]

    Nakata, T

    S. Nakata, T. Miyata, N. Ojima, and K. Yoshikawa, Physica D: Nonlinear Phenomena 115, 313–320 (1998)

    1998

  20. [32]

    J. C. Neu, SIAM Journal on Applied Mathematics 37, 307–315 (1979)

    1979

  21. [33]

    Wickramasinghe and I

    M. Wickramasinghe and I. Z. Kiss, Physical Review E 88 (2013), 10.1103/physreve.88.062911

    2013 doi

  22. [34]

    Marek and I

    M. Marek and I. Stuchl, Biophysical Chemistry 3, 241–248 (1975)

    1975

  23. [35]

    Tajima, H

    S. Tajima, H. Singh, S. Nakabayashi, T. Singla, and P. Parmananda, Journal of Electroanalytical Chemistry 769, 16–20 (2016)

    2016

  24. [36]

    D. S. Cohen and J. C. Neu, Annals of the New York Academy of Sciences 316, 332–337 (1979)

    1979

  25. [37]

    Y. Liu, J. P´ erez-Mercader, and I. Z. Kiss, Chaos: An In- terdisciplinary Journal of Nonlinear Science 32 (2022), 10.1063/5.0096689

    2022 doi

  26. [38]

    Ignatov, M

    M. Ignatov, M. Hansen, M. Ziegler, and H. Kohlstedt, Applied Physics Letters 108 (2016), 10.1063/1.4942832

    2016 doi

  27. [39]

    L. Q. English, Z. Zeng, and D. Mertens, Physical Re- view E 92 (2015), 10.1103/physreve.92.052912

    2015 doi

  28. [40]

    Parihar, N

    A. Parihar, N. Shukla, S. Datta, and A. Ray- chowdhury, Journal of Applied Physics 117 (2015), 10.1063/1.4906783

    2015 doi

  29. [41]

    Paul Asir, K

    M. Paul Asir, K. Sathiyadevi, P. Philominathan, and D. Premraj, Chaos: An Interdisciplinary Journal of Nonlinear Science 32 (2022), 10.1063/5.0099011

    2022 doi

  30. [42]

    Dixit, A

    S. Dixit, A. Sharma, and M. D. Shrimali, Physics Let- ters A 383, 125930 (2019)

    2019

  31. [43]

    A. K. Singh and R. D. S. Yadava, Physica Scripta 94, 075201 (2019)

    2019

  32. [44]

    Ivanchenko, G

    M. Ivanchenko, G. Osipov, V. Shalfeev, and J. Kurths, Physica D: Nonlinear Phenomena 189, 8–30 (2004)

    2004

  33. [45]

    W¨ unsche, S

    H.-J. W¨ unsche, S. Bauer, J. Kreissl, O. Ushakov, N. Ko- rneyev, F. Henneberger, E. Wille, H. Erzgr¨ aber, M. Peil, W. Els¨ aßer, and I. Fischer, Physical Review Letters94 (2005), 10.1103/physrevlett.94.163901

    2005 doi

  34. [46]

    Gholizade-Narm, A

    H. Gholizade-Narm, A. Azemi, M. Khademi, and M. Karimi-Ghartemani, Biomedical Signal Processing and Control 4, 57–66 (2009)

    2009

  35. [47]

    M. Bier, B. M. Bakker, and H. V. Westerhoff, Biophys- ical Journal 78, 1087–1093 (2000)

    2000

  36. [48]

    Montoya, M

    F. Montoya, M. Rivera, J. Escalona, and P. Par- mananda, Physics Letters A 377, 3124–3127 (2013)

    2013

  37. [49]

    D. Ypey, W. VanMeerwijk, C. Ince, and G. Groos, Journal of Theoretical Biology 86, 731–755 (1980)

    1980

  38. [50]

    Schilder and B

    F. Schilder and B. B. Peckham, Journal of Computa- tional Physics 220, 932–951 (2007)

    2007

  39. [51]

    Valkering, C

    T. Valkering, C. Hooijer, and M. Kroon, Physica D: Nonlinear Phenomena 135, 137–153 (2000)

    2000

  40. [52]

    Coombes and P

    S. Coombes and P. C. Bressloff, Physical Review E 60, 2086–2096 (1999)

    1999

  41. [53]

    P. C. Bressloff and S. Coombes, Physical Review Letters 81, 2168–2171 (1998)

    1998

  42. [54]

    G. B. Ermentrout, Journal of Mathematical Biology 12, 327–342 (1981)

    1981

  43. [55]

    L. M. Ward, Trends in Cognitive Sciences 7, 553–559 (2003)

    2003

  44. [56]

    Buzs´ aki and A

    G. Buzs´ aki and A. Draguhn, Science304, 1926 (2004). 26

    2004

  45. [57]

    Hansel, G

    D. Hansel, G. Mato, and C. Meunier, Europhysics Let- ters (EPL) 23, 367–372 (1993)

    1993

  46. [58]

    Rossoni, Y

    E. Rossoni, Y. Chen, M. Ding, and J. Feng, Physical Review E 71 (2005), 10.1103/physreve.71.061904

    2005 doi

  47. [59]

    Ermentrout and D

    G. Ermentrout and D. Terman, Mathematical Founda- tions of Neuroscience , Interdisciplinary Applied Math- ematics (Springer New York, 2010)

    2010

  48. [60]

    Hansel, G

    D. Hansel, G. Mato, and C. Meunier, Neural Compu- tation 7, 307–337 (1995)

    1995

  49. [61]

    Buri´ c and D

    N. Buri´ c and D. Todorovi´ c, Physical Review E 67 (2003), 10.1103/physreve.67.066222

    2003 doi

  50. [62]

    Song and J

    Y. Song and J. Xu, IEEE Transactions on Neural Net- works and Learning Systems 23, 1659–1670 (2012)

    2012

  51. [63]

    Farajzadeh Tehrani and M

    N. Farajzadeh Tehrani and M. Razvan, Mathematical Biosciences 270, 41–56 (2015)

    2015

  52. [64]

    Van Vreeswijk, L

    C. Van Vreeswijk, L. F. Abbott, and G. Bard Ermen- trout, Journal of Computational Neuroscience 1, 313 (1994)

    1994

  53. [65]

    van Vreeswijk, Physical Review E 54, 5522 (1996)

    C. van Vreeswijk, Physical Review E 54, 5522 (1996)

    1996

  54. [66]

    Chartrand, M

    T. Chartrand, M. S. Goldman, and T. J. Lewis, SIAM Journal on Applied Dynamical Systems 18, 1643–1693 (2019)

    2019

  55. [67]

    Ermentrout, Neural Computation 8, 979–1001 (1996)

    B. Ermentrout, Neural Computation 8, 979–1001 (1996)

    1996

  56. [68]

    Pfeuty, G

    B. Pfeuty, G. Mato, D. Golomb, and D. Hansel, The Journal of Neuroscience 23, 6280–6294 (2003)

    2003

  57. [69]

    Ratas and K

    I. Ratas and K. Pyragas, Scientific Reports 12 (2022), 10.1038/s41598-022-24001-2

    2022 doi

  58. [70]

    Jalife, The Journal of Physiology 356, 221–243 (1984)

    J. Jalife, The Journal of Physiology 356, 221–243 (1984)

    1984

  59. [71]

    C. C. Chow, Physica D: Nonlinear Phenomena 118, 343–370 (1998)

    1998

  60. [72]

    Pikovsky, M

    A. Pikovsky, M. Rosenblum, and J. Kurths, Synchro- nization: A Universal Concept in Nonlinear Sciences , Cambridge Nonlinear Science Series (Cambridge Uni- versity Press, 2001)

    2001

  61. [73]

    Thomas, S

    N. Thomas, S. Karthiga, and M. Senthilvelan, The European Physical Journal Plus 136 (2021), 10.1140/epjp/s13360-021-02216-y

    2021 doi

  62. [74]

    Astakhov, A

    S. Astakhov, A. Gulai, N. Fujiwara, and J. Kurths, Chaos: An Interdisciplinary Journal of Nonlinear Sci- ence 26 (2016), 10.1063/1.4940967

    2016 doi

  63. [75]

    A. N. Pisarchik, R. Jaimes-Re´ ategui, and M. A. Garc ´ ıa- Vellisca, Chaos: An Interdisciplinary Journal of Nonlin- ear Science 28 (2018), 10.1063/1.5003091

    2018 doi

  64. [76]

    Tabekoueng Njitacke, I

    Z. Tabekoueng Njitacke, I. Sami Doubla, J. Kengne, and A. Cheukem, Chaos: An Interdisciplinary Journal of Nonlinear Science 30 (2020), 10.1063/1.5132280

    2020 doi

  65. [79]

    Bragard, G

    J. Bragard, G. Vidal, H. Mancini, C. Mendoza, and S. Boccaletti, Chaos: An Interdisciplinary Journal of Nonlinear Science 17 (2007), 10.1063/1.2797378

    2007 doi

  66. [80]

    Pinto and I

    C. Pinto and I. Labouriau, in 2006 IEEE Inter- national Conference on Computational Cybernetics (IEEE, 2006)

    2006

  67. [81]

    Ryu and S

    H. Ryu and S. Ann Campbell, Mathematical Biosciences and Engineering 17, 7931–7957 (2020)

    2020

  68. [83]

    Hilborn, Chaos and Nonlinear Dynamics: An Intro- duction for Scientists and Engineers , Oxford scholar- ship online: Physics module (Oxford University Press, 2000)

    R. Hilborn, Chaos and Nonlinear Dynamics: An Intro- duction for Scientists and Engineers , Oxford scholar- ship online: Physics module (Oxford University Press, 2000)

    2000

  69. [84]

    Banerjee and C

    S. Banerjee and C. Grebogi, Physical Review E 59, 4052–4061 (1999)

    1999

  70. [85]

    H. E. Nusse and J. A. Yorke, Physica D: Nonlinear Phe- nomena 57, 39–57 (1992)

    1992

  71. [86]

    Jain and S

    P. Jain and S. Banerjee, International Journal of Bifur- cation and Chaos 13, 3341–3351 (2003)

    2003

  72. [87]

    Casas, W

    F. Casas, W. Chin, C. Grebogi, and E. Ott, Physical Review E 53, 134–139 (1996)

    1996

  73. [88]

    Makarenkov and J

    O. Makarenkov and J. S. Lamb, Physica D: Nonlinear Phenomena 241, 1826–1844 (2012)

    2012

  74. [89]

    Khajeh Alijani, Physical Review E 80 (2009), 10.1103/physreve.80.051922

    A. Khajeh Alijani, Physical Review E 80 (2009), 10.1103/physreve.80.051922

    2009 doi

  75. [90]

    Coombes, R

    S. Coombes, R. Thul, and K. Wedgwood, Physica D: Nonlinear Phenomena 241, 2042–2057 (2012)

    2012

  76. [91]

    Coombes, M

    S. Coombes, M. . R. Owen, and G. D. Smith, Physical Review E 64 (2001), 10.1103/physreve.64.041914

    2001 doi

  77. [92]

    C. R. Laing and S. Coombes, International Journal of Bifurcation and Chaos 15, 1433–1444 (2005)

    2005

  78. [93]

    Luccioli, S

    S. Luccioli, S. Olmi, A. Politi, and A. Torcini, Physical Review Letters 109 (2012), 10.1103/phys- revlett.109.138103

    2012 doi

  79. [94]

    Tattini, S

    L. Tattini, S. Olmi, and A. Torcini, Chaos: An In- terdisciplinary Journal of Nonlinear Science 22 (2012), 10.1063/1.4723839

    2012 doi

  80. [95]

    Zillmer, R

    R. Zillmer, R. Livi, A. Politi, and A. Torcini, Physical Review E 76 (2007), 10.1103/physreve.76.046102

    2007 doi

  81. [96]

    S. Olmi, A. Politi, and A. Torcini, The Journal of Math- ematical Neuroscience 2 (2012), 10.1186/2190-8567-2- 12

    2012 doi

  82. [97]

    Zillmer, R

    R. Zillmer, R. Livi, A. Politi, and A. Torcini, Physical Review E 74 (2006), 10.1103/physreve.74.036203

    2006 doi

  83. [98]

    Sterratt, B

    D. Sterratt, B. Graham, A. Gillies, and D. Willshaw, Principles of Computational Modelling in Neuroscience (Cambridge University Press, 2011)

    2011

  84. [99]

    S. Olmi, R. Livi, A. Politi, and A. Torcini, Physical Review E 81 (2010), 10.1103/physreve.81.046119

    2010 doi

  85. [100]

    Angulo-Garcia, S

    D. Angulo-Garcia, S. Luccioli, S. Olmi, and A. Torcini, New Journal of Physics 19, 053011 (2017)

    2017

  86. [101]

    Granados, L

    A. Granados, L. Alsed` a, and M. Krupa, SIAM Review 59, 225–292 (2017)

    2017

  87. [102]

    Granados and M

    A. Granados and M. Krupa, Nonlinearity 28, 1163–1192 (2015)

    2015

  88. [103]

    Piecewise-smooth Dynamical Systems (Springer Lon- don, 2008)

    2008

  89. [104]

    Coombes and P

    S. Coombes and P. C. Bressloff, Physical Review E 63 (2001), 10.1103/physreve.63.059901

    2001 doi

  90. [105]

    Reyes, R

    A. Reyes, R. Lujan, A. Rozov, N. Burnashev, P. Somo- gyi, and B. Sakmann, Nature Neuroscience 1, 279–285 (1998)

    1998

  91. [106]

    A. Lin, R. Yang, S. Dorkenwald, A. Matsliah, A. R. Sterling, P. Schlegel, S.-c. Yu, C. E. McKellar, M. Costa, K. Eichler, A. S. Bates, N. Eckstein, J. Funke, G. S. X. E. Jefferis, and M. Murthy, Nature 634, 153–165 (2024)

    2024

  92. [107]

    R. D. Traub, M. A. Whittington, I. M. Stanford, and J. G. R. Jefferys, Nature 383, 621–624 (1996)

    1996

  93. [108]

    Buzs´ aki, Neuron33, 325–340 (2002)

    G. Buzs´ aki, Neuron33, 325–340 (2002)

    2002

  94. [109]

    N´ adasdy, Journal of Physiology-Paris 94, 505–524 (2000)

    Z. N´ adasdy, Journal of Physiology-Paris 94, 505–524 (2000)

    2000

  95. [110]

    M. F. Carr, S. P. Jadhav, and L. M. Frank, Nature 27 Neuroscience 14, 147–153 (2011)

    2011

  96. [111]

    B. W. Knight, The Journal of General Physiology 59, 734–766 (1972)

    1972

  97. [112]

    Doiron, C

    B. Doiron, C. Laing, A. Longtin, and L. Maler, Journal of Computational Neuroscience 12, 5–25 (2002)

    2002

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.