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REVIEW 3 major objections 6 minor 15 references

Devil's staircase inside shrimp-shaped regions reveals periodicity of plateau spikes and bursts

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Pith's one-line read The paper establishes that shrimp-shaped regions of parameter space in a discrete-time cardiac model contain infinitely many periodic attractors ordered as a complete devil's staircase, with early afterdepolarizations mainly periodic and…

desk verdict A solid numerical observation of striped shrimps, but the 'complete devil's staircase' claim outruns the evidence. read the letter →

arxiv 2411.16373 v2 pith:7G3ACRH6 submitted 2024-11-25 physics.bio-ph cond-mat.stat-mechnlin.AOnlin.CDq-bio.CB

Luiz F. B. Caixeta , Matheus H. P. Gonçalves , M. H. R. Tragtenberg , Mauricio Girardi-Schappo This is my paper · ORCID
classification physics.bio-phcond-mat.stat-mechnlin.AOnlin.CDq-bio.CB MSC 37D4537N2537C27 PACS 05.45.-a87.19.Hh
keywords devil'sstaircaseshrimp-shapedregionsinterspikeintervalafterdepolarizationLyapunovexponentslow-fastdynamicscardiacactionpotentialFareysequence
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

The paper studies a three-variable discrete-time map that models cardiac action potentials with slow-fast dynamics. It claims that the shrimp-shaped regions of its parameter space, traditionally associated with a single periodic orbit, actually contain infinitely many periodic attractors arranged as a complete devil's staircase as one parameter is varied. It also claims that early afterdepolarizations (EADs) are mostly periodic, while delayed afterdepolarizations (DADs) are chaotic, and that the maximum negative Lyapunov exponent scales roughly as the inverse of the interspike-interval period near quasiperiodic orbits. If true, these results give a quantitative handle on how periodicity changes as a cardiac plateau loses stability, which is relevant to arrhythmia mechanisms.

What carries the argument

The object carrying the argument is the winding number $w = P/Q$, computed from the interspike interval (ISI) sequence of the map. Each ISI is the time between successive upcrossings of $x = 0$, and for a periodic attractor of period $Q$ with $P$ spikes per period, the average ISI equals $Q/P$, so $w = 1/\langle\text{ISI}\rangle$. Because time is discrete, the timestamps fluctuate by $\pm 1$, and the paper compensates by plotting $\langle\text{ISI}\rangle - 1$ against $1/w$ to match the staircase steps. The maximum period $P$ of the ISI sequence, the maximum Lyapunov exponent $L$ (computed by the Eckmann\textendash Ruelle method), and the box-counting fractal dimension $D_f$ of the staircase in $(x_R, w)$ space are the supporting measurements that identify the staircase as complete and Farey-organized.

What would settle it

Iterate the map long enough to find the exact period $Q$ of an attractor inside a shrimp and compare the Farey label $w = P/Q$ from the ISI sequence; if the labels do not match the actual rotation numbers, the staircase identification fails.

Watch

Extended reading notes

Core claim

The central discovery is that shrimp-shaped regions in the $(T, x_R)$ parameter plane of the map are not isoperiodic: each shrimp is filled with stripes of constant interspike-interval period $P$, and traversing $x_R$ at fixed $T$ crosses these stripes as steps of a complete devil's staircase. The steps are labeled by rational winding numbers $w = P/Q$, where $Q$ is the period of the attractor and $P$ is the number of cycles (spikes) per period, and the labels follow a nonstandard Farey tree so that between any two steps another step exists. Quasiperiodic orbits, with maximum Lyapunov exponent $L = 0$ and diverging $P$, form the zero-measure boundaries between steps. Along the staircase the relation $-L \sim 1/P$ holds, with the plateaus of $1/P$ coinciding with peaks of $-L$. EAD attractors are found inside shrimps and are mostly periodic, while DAD attractors occupy chaotic regions between shrimps.

Load-bearing premise

The winding number $w = P/Q$, measured from the discrete-time upcrossing timestamps of the ISI sequence, is assumed to correctly label the rotation number of each periodic orbit, relying on a uniform $\pm 1$ timestamp fluctuation correction so that the Farey labels are accurate.

Editorial extensions

If this is right

  • Shrimp-shaped regions in slow-fast systems can host infinitely many periodic attractors, not just one, so isoperiodicity is not a defining feature of shrimps.
  • The complete devil's staircase implies that along the chosen parameter axis the quasiperiodic orbits form a set of zero measure, with periodic steps dense, so the transition from cardiac spiking to bursting passes through a dense ladder of periodicities.
  • The scaling $-L \sim 1/P$ offers a practical proxy: measuring the ISI period near a quasiperiodic boundary estimates the Lyapunov exponent without computing Jacobians.
  • The model predicts that shifting the slow-current reversal potential $x_R$ toward EAD behavior produces a Farey-organized sequence of period changes, a signature that could be looked for in biophysically detailed cardiac myocyte models.
  • Since EADs are mostly periodic and DADs are chaotic, the periodicity of the ISI sequence could serve as a distinguishing marker between these two classes of afterdepolarizations.

Reading between the lines

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

  • One could test the staircase labeling in a continuous-time cardiac model: if the rotation number computed from the actual voltage waveform differs systematically from $w = P/Q$, the discrete-time timestamp correction is not universal.
  • The relation $-L \sim 1/P$ may hold for complete devil's staircases in other dissipative maps, not only this model; checking it in a different system with a known staircase would indicate whether the scaling is generic.
  • The fractal dimension near 0.95\textendash 0.98 suggests the staircase is nearly space-filling; this could imply that in a real myocyte, noise would wash out the finest steps, so only the largest steps would be physiologically observable.
  • If chaotic DADs are indeed more arrhythmogenic than periodic EADs, then the position of a parameter region within a shrimp (periodic interior vs. chaotic border) may correlate with clinical risk; this is not tested here.
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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 / 6 minor

Summary. The paper studies a discrete-time three-variable map-based model of cardiac action potentials, interpreted as a single myocyte. It investigates the transition from cardiac spiking (plateau spikes) to bursting, where early and delayed afterdepolarizations (EADs and DADs) appear. The central claims are: (i) inside shrimp-shaped regions of the (T, xR) parameter plane, multiple periodic attractors coexist and form an internal striped structure that, along a one-parameter cut, constitutes a complete devil's staircase of winding numbers w = P/Q; (ii) EADs are mainly periodic (inside shrimps) while DADs are chaotic (between shrimps); and (iii) near quasiperiodic orbits the maximum negative Lyapunov exponent scales as -L ~ 1/P, where P is the period of the interspike-interval sequence. The paper derives the identity <ISI> = Q/P = 1/w from the definitions of spike timestamps and uses it to label staircase steps, exhibits Farey-mediant relations among a few labeled steps, and estimates fractal dimensions Df ≈ 0.95–0.98 for the staircases.

Significance. If the central claim of a complete devil's staircase inside shrimps is correct, it extends the known phenomenology of shrimp-shaped regions beyond isoperiodic and quasiperiodic shrimps, providing a novel link between slow-fast dynamics, cardiac arrhythmia models, and devil's staircases. The paper has notable strengths: the definitions in Sec. III are precise, the identity in Eq. (6) is derived from first principles (though it is an internal consistency check), the numerical methods (Eckmann-Ruelle Lyapunov exponents, box-counting dimension) are standard, and a data availability link is provided. The biological interpretation is speculative but clearly labeled as such. However, the headline claim of a complete (as opposed to harmless or incomplete) devil's staircase is not supported by the finite numerical evidence presented, and the ad hoc <ISI>-1 correction raises questions about the reliability of the winding-number labels. These issues are load-bearing for the paper's main contribution.

major comments (3)
  1. [Section IV D, Fig. 7] The claim of a 'complete devil's staircase' is not established. Completeness means infinitely many steps, dense in the parameter interval, with a zero-measure quasiperiodic complement. The paper exhibits only two or three Farey-mediant relations per shrimp (e.g., w1 = 6/2016, w2 = 3/1007, w3 = 9/3023 in Fig. 7A), and the box-counting dimension Df ≈ 0.95–0.98 is compatible with a harmless staircase with finitely many jumps, whose graph dimension is also 1. The assertion that 'the gaps in the staircases are due only to the finite simulation time' (Section IV D) is not demonstrated; since some labeled periods Q (e.g., 47524 in Fig. 7C) approach or exceed the 2×10^5 iteration horizon, longer simulations could resolve new steps or change the labels. The authors should either provide a quantitative argument for completeness (e.g., scaling of the number of steps with resolution, or an analytic mechanism such as a monotone circle map with a devil's staircase) or weaken the claim to 'devil's-staircase-like structure' with finitely many resolved steps.
  2. [Section IV D, Eq. (6), Fig. 7] The winding-number labels are not independently confirmed because Eq. (6) is an identity derived from the definitions of ISI and w, so plotting <ISI> against 1/w is a consistency check, not a validation. The paper then plots <ISI>−1 instead of <ISI> to match 1/w, stating only that 'we obtained a better match' without deriving the correction. The ±1 timestamp fluctuation in the cosine example in Sec. III A averages to zero, but for the actual map the shift is not shown to be uniform across the staircase; if the correction varies from step to step, the P/Q labels become unreliable and the Farey-mediant construction loses its quantitative anchor. The authors should derive the <ISI>−1 offset from the spike-timestamp conditions or show numerically that the correction is constant over each staircase.
  3. [Section IV E, Eq. (8)] The relation −L ∼ 1/P near quasiperiodic orbits is supported only by Pearson correlations (R = 0.74 to 0.91) between rescaled curves of −L and 1/P, and the paper acknowledges it is not derived from first principles. Because both quantities vary smoothly and systematically along the staircase, a high correlation does not provide strong evidence for an asymptotic power-law relation; moreover, the claim that P diverges at step boundaries is handled numerically by arbitrarily replacing diverging P with 10^6, which is not a test of divergence. The qualitative correlation is plausible, but the scaling claim as stated overreaches the evidence. The authors should either present a more direct test (e.g., collapse of data over multiple resolutions, or a derivation in a limit of the map) or rephrase Eq. (8) as a heuristic observation without the '∼' notation.
minor comments (6)
  1. [Throughout] There are several typos and inconsistencies: 'analogou' (Section IV D), 'Lyuapnov' in the Fig. 4 caption, 'Ex. (6)' instead of 'Eq. (6)' in Fig. 7, and inconsistent accenting of 'Hénon'/'Henón' in the text and references.
  2. [Eq. (7) and surrounding text] The definition of the Farey mediant w3 = P3/Q3 with mP3 = P1 + P2 and mQ3 = Q1 + Q2 is confusing because it suggests P3 and Q3 are not reduced. Please state explicitly that w3 = (P1+P2)/(Q1+Q2) is reduced by a common factor m, and clarify that m is the greatest common divisor or otherwise specify the reduction.
  3. [Section III B] The terms 'sliding' and 'locked' incommensurate phases are introduced without definition or reference. Since these terms are not standard in the nonlinear dynamics literature cited, please define them or give a precise reference.
  4. [Fig. 6G,H caption] The caption refers to a 'harmless staircase going from one shrimp to another', but the text in Section IV D says the <ISI> in between shrimps is 'spurious' due to aperiodic behavior. This inconsistency should be resolved.
  5. [Section IV D] The sentence 'The <ISI> is much easier to measure' should be qualified: Eq. (5) defines <ISI> only for periodic or quasiperiodic orbits, so the staircase analysis applies only to non-chaotic regions. Please state this restriction explicitly where the comparison is made.
  6. [References] Reference [6] is cited as a Research Square preprint while the paper is being published; if a peer-reviewed version exists, it should be cited instead. Also, reference [41] (Rulkov 2007) is listed as arXiv:0708.1173v1, but the paper appears to be a later published version; please check the citation.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the devil's staircase and periodicity claims are direct numerical observations of iterating Eq. (1); prior self-citations supply model context but do not force the headline result.

full rationale

The paper's central claims (periodic EADs, chaotic DADs, internal stripes inside shrimps forming a devil's staircase, and the -L ~ 1/P correlation) are all obtained by iterating the fixed map Eq. (1) and measuring the ISI sequence, Lyapunov exponents, and winding numbers from the same simulations. No parameter is fitted to a target result, and no external benchmark is reverse-engineered. The relation <ISI> = 1/w in Eqs. (5)-(6) is an algebraic identity for a single orbit (w = P/Q and <ISI> = Q/P), so the match shown in Fig. 7 is an internal consistency check rather than a prediction; the -1 offset is explicitly attributed to the +/-1 timestamp fluctuation of the discrete map and is not used to inject a desired Farey label. The Farey mediant relations in Eq. (7) are applied to the measured rational winding-number labels, so the staircase structure is read off the data, not imposed by an ansatz. Eq. (8) is explicitly stated to be an empirical correlation ('we did not derive it from first principles'), supported by a reported Pearson correlation. The paper cites earlier work by the same group (refs. 6, 8, 9, 15) for the model's derivation, the tanh model's previously known staircase, and the slow-fast analysis, but these citations are contextual: the current logistic-map staircase is demonstrated with the paper's own integrations rather than imported from those references. The completeness claim (dense steps, zero-measure complement) goes beyond the finite-time numerical evidence and the Df values are compatible with a harmless staircase, but that is a correctness/overreach concern, not a circularity; the data are not constructed to equal the conclusion by definition. Hence no significant circularity, with a low score reflecting only minor self-citation in the background.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The ledger shows that the paper's central claims are conditional on a hand-picked parameter regime (K = 0.6, delta = lambda = 0.001, H = 0, fixed initial condition) and on an ad hoc extrapolation from finite staircases to a complete devil's staircase. No genuinely new physical entity is introduced; the biological interpretation is an assumed mapping rather than an experimentally validated one. The main quantitative relations are definitional or empirical, not derived from first principles.

free parameters (1)
  • Model parameters K, delta, lambda, H, and initial condition (x(0), y(0), z(0)) = K = 0.6, delta = lambda = 0.001, H = 0, initial condition (1, 1, 1)
    Chosen by hand from prior work; the central shrimp and devil's staircase findings are only demonstrated in this parameter regime. The authors state in Section IV D that increasing delta = lambda would destroy plateaus and could disrupt the findings.
assumptions (5)
  • domain assumption The sigmoid F(u) = u/(1+|u|) preserves the essential dynamics of the tanh version of the model, including plateaus, bursts, EADs and DADs.
    Introduced in Section II as a simplification of the tanh map; all results are obtained with this F, and no proof is given that the staircase structure is independent of this choice.
  • domain assumption The parameter regime delta = lambda = 0.001, H = 0, K = 0.6 with initial condition (1,1,1) is representative of slow-fast cardiac dynamics.
    The adiabatic slow-fast interpretation in Section II and the statement in Section IV D that increasing delta = lambda destroys plateaus and would possibly disrupt the findings make the results conditional on this regime.
  • standard math The maximum period P of the {ISI_n} sequence is a reliable proxy for the attractor's periodicity for periodic orbits.
    Eqs. (4) and (5) derive this relation for periodic orbits; the authors acknowledge P is ill-defined for chaotic attractors but still plot P in chaotic regions. This is a definitional consequence, not an independent empirical fact.
  • ad hoc to paper The finite numerical staircases converge to a complete devil's staircase with infinitely many commensurate steps and a zero-measure quasiperiodic set.
    Section IV D states that gaps in the staircases are due only to finite simulation time; no convergence proof or direct measurement of the quasiperiodic set is provided. This is the main ad hoc assumption behind the central claim.
  • domain assumption The mapped quantities x(t), y(t), z(t) can be interpreted as membrane potential, fast feedback, and slow calcium-like current, making EAD/DAD-like oscillations relevant to cardiac arrhythmias.
    Section II and the Conclusion adopt this biological mapping; parameters are in arbitrary units and no comparison with experimental cardiac data is made.

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Pith. "Pith review of Devil's staircase inside shrimp-shaped regions reveals periodicity of plateau spikes and bursts." pith.science (2026). https://pith.science/paper/7G3ACRH6

@misc{pith2026241116373,
  author       = {Pith},
  title        = {Pith review of: Devil's staircase inside shrimp-shaped regions reveals periodicity of plateau spikes and bursts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7G3ACRH6}},
  note         = {Machine review of arXiv:2411.16373}
}
read the original abstract

Slow-fast dynamics are intrinsically related to complex phenomena and are responsible for many of the homeostatic dynamics that keep biological systems healthy functioning. We study a discrete-time membrane potential model that can generate a diverse set of spiking behavior depending on the choice of slow-fast time scales, from fast spiking to bursting, or plateau action potentials -- also known as cardiac spikes since they are characteristic in heart myocytes. The plateau of cardiac spikes can lose stability, generating early or delayed afterdepolarizations (EADs and DADs, respectively), both of which are related to cardiac arrhythmia. We show the periodicity changes along the transition from the healthy action potentials to these impaired oscillations. We show that while EADs are mainly periodic attractors, DADs usually come with chaos. EADs are found inside shrimp-shaped regions of the parameter space. However, in our system, multiple periodic attractors live within a shrimp-shaped region, giving it an internal structure made of infinite transitions between periodicities forming a complete devil's staircase. Understanding the periodicity of plateau attractors in slow-fast systems could be useful in unveiling the characteristics of heart myocyte behaviors that are linked to cardiac arrhythmias.

Figures

Figures reproduced from arXiv: 2411.16373 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
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Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

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Reference graph

Works this paper leans on

15 extracted references · 5 canonical work pages

  1. [1]

    shrimps

    Devil’s staircase inside shrimp-shaped regions reveals periodicity of plateau1 spikes and bursts2 Luiz F. B. Caixeta, Matheus H. P. Gon¸ calves, M. H. R. Tragtenberg, and Mauricio Girardi-Schappo a)3 Departamento de F ´ ısica - Universidade Federal de Santa Catarina - Florian´ opolis SC - 88040-900 - Brazil4 (Dated: 22 January 2025)5 Slow-fast dynamics ar...

    work page 2025

  2. [3]

    Phase diagram, oscillation modes and interspike interval. A. Phase diagram coloring different ISI distribution profiles (see panel B, right) with shades of gray (see Methods). Notice a dust-like structure along the CS-B transition, which will be further investigated within the region highlighted by the cyan rectangle (detail shown in Fig. 3). From light t...

    work page 1942

  3. [4]

    However, we observed411 that, in our model, this is not true. Since time is discrete412 and attempting to reconcile our findings with the phe-413 nomenology of the H´ enon map4, we plotted the shrimp-414 shaped regions of the phase diagrams coloring the average415 interspike interval rounded to the nearest integer,⌊⟨IS I⟩⌉416 (Fig. 5).417 The rounded aver...

  4. [5]

    Here, we unveil72 This is the author’s peer reviewed, accepted manuscript

    Re-69 cently, this idea was extended to quasiperiodic shrimps 3,70 highlighting distinct dynamics such as torus-bubbling71 transitions and multitori attractors. Here, we unveil72 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE...

    work page doi:10.1063/5.0250342

  5. [6]

    External inputs (synaptic or otherwise) can104 be introduced via the parameter H

    The slow current has a recovery102 timescale 1/δ and a driving timescale 1/λ, with a reversal103 potential xR. External inputs (synaptic or otherwise) can104 be introduced via the parameter H. All parameters and105 variables are given in arbitrary units.106 This map has inversion symmetry, so that changing107 This is the author’s peer reviewed, accepted m...

  6. [7]

    There is no other t between tn and168 tn+1 that obeys both of these conditions

    [the oscillation is rising] , (3) taking k = n for the spike at time tn and k = n+1 for the167 spike at time tn+1. There is no other t between tn and168 tn+1 that obeys both of these conditions. In other words,169 tn and tn+1 can be regarded as the timestamps of con-170 secutive spikes. Repeating this for every spike produces171 the sequence {IS In} illus...

  7. [8]

    The sliding263 case means that x(t) can take any value within a con-264 nected subinterval of this range

    ranged mapped by the sigmoid F (u). The sliding263 case means that x(t) can take any value within a con-264 nected subinterval of this range. A locked attractor exist265 only in a disconnected subinterval of this range.266 The transition between commensurate and incommen-267 surate phases has been studied in the context of spatial268 ordering in magnets a...

  8. [9]

    Isoperiodic shrimps are found only319 in regions where P and Q stay constant as the parameters320 are varied.321 C

    This315 also suggests that shrimps are more general than origi-316 nally thought, and can contain: (a) isoperiodic orbits 4,5;317 (b) infinitely many periodic orbits (our study); or (c)318 quasiperiodic orbits3. Isoperiodic shrimps are found only319 in regions where P and Q stay constant as the parameters320 are varied.321 C. Lyapunov exponents322 The Lya...

Show all 15 references
  1. [10]

    The maximum period of the357 {IS In} reveals that these non-chaotic regions form rings358 of isoperiodic ISI sequences (Fig. 3B). These rings closely359 match the ring patterns in the maximum Lyapunov ex-360 ponent L when L < 0 (Fig. 3A–right). Note, however,361 that the regio...

  2. [11]

    Detail of the ⟨I S I⟩ data pointed by arrows in Fig

    Shrimps’ complete devil’s staircases. Detail of the ⟨I S I⟩ data pointed by arrows in Fig. 6H compared to the inverse winding number, 1 /w = Q/P . Some staircase steps are labeled by w = P/Q to highlight the Farey tree structure of the system. Each panel has xR crossing the sh...

    2016

  3. [12]

    This happens concomitantly with the divergence526 of Q, leaving an irrational w at the boundary between527 periodic phases

    Meanwhile, the maximum ISI524 period P has to diverge for these orbits, since they never525 repeat. This happens concomitantly with the divergence526 of Q, leaving an irrational w at the boundary between527 periodic phases. Each step on the devil’s staircase com-528 prises per...

  4. [13]

    In our model, shrimp-shaped re-611 gions display a fractal structure with internal isoperi-612 odic stripes forming complete devil’s staircases of peri-613 odic attractors

    This contrasts609 with the period-doubling structure of periodic shrimps610 in the H´ enon map4. In our model, shrimp-shaped re-611 gions display a fractal structure with internal isoperi-612 odic stripes forming complete devil’s staircases of peri-613 odic attractors. Thus, o...

  5. [20]

    (3), produces the sequence {IS In} =208 {20, 19, 21, 19, 20, 21, · · · }

    Taking v(t)206 only at integer t ≥ 1, and applying it to the con-207 ditions in Eq. (3), produces the sequence {IS In} =208 {20, 19, 21, 19, 20, 21, · · · }. By construction, we know the209 correct ISI should be equal to Q = 20, but values fluc-210 tuate. Sampling the series l...

  6. [44]

    Therefore, p is the752 probability of being in one of both tails of the Student’s753 t-distribution

    In other words, P(t; ν) is750 the probability of finding some s < t,751 P(t; ν) = 1√νπ Γ ν+1 2 Γ ν 2 Z t −∞ 1 + s2 ν − ν+1 2 ds , where Γ( ·) is the Gamma function. Therefore, p is the752 probability of being in one of both tails of the Student’s753 t-distribution. This essent...

    2001

  7. [1978]

    264–277.816 20R

    pp. 264–277.816 20R. FitzHugh, Thresholds and Plateaus in the Hodgkin-Huxley817 nerve equations , The Journal of General Physiology 43, 867818 (1960).819 21W. Teka, K. Tsaneva-Atanasova, R. Bertram, and J. Tabak,820 From Plateau to Pseudo-Plateau Bursting: Making the Transi-82...

    1960 arXiv

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