Collective dynamics of pulse-coupled threshold-element systems beyond the convergence regime: avalanche exponents, supercritical and inhibitory coupling, attractor measures, and energy-function programming (Herz–Hopfield 1995)

The 1995 Herz–Hopfield letter introduced a solvable class of pulse-coupled threshold-element models that unifies slider-block earthquake dynamics and integrate-and-fire neural networks, and proved that for non-negative couplings with total incoming pulse weight $A \le 1$ and periodic boundary conditions the dynamics rapidly reaches phase-locked periodic attractors. The collective behavior outside this proved regime was left essentially uncharacterized: the avalanche size–frequency exponents (numerically $B = 0.79 \pm 0.05$, and $B = 0.40$–$0.52$ for $\gamma = 0$) lack any analytic theory and the criticality of the dissipative regime is contested; nothing is proved for supercritical coupling $A > 1$ or signed (inhibitory) pulses; the measures of the fine-tuned periodic attractors for $\gamma < 1$ are unquantified; the Lyapunov (energy-landscape) description exists only for the Abelian case $\gamma = 1$ at stroboscopic times and not for the continuous-time leaky integrate-and-fire dynamics; and no controlled mapping connects model parameters to observed seismic recurrence intervals or neural binding time scales. The later literature finds partial and adjacent results but no resolution of any component: the problem of a predictive theory of the collective dynamics of this threshold-element class beyond the convergence regime remains open.

Background

Many natural systems are built from slowly driven threshold elements: a state variable charges gradually until it reaches a trigger, discharges rapidly, and the discharge delivers pulses to neighboring elements. Two physical realizations dominate the literature. In slider-block (stick-slip) models of a fault, blocks coupled by springs accumulate tectonic stress and slip suddenly when a static friction threshold is exceeded; a two-dimensional lattice version with a non-conserving redistribution rule (the Olami–Feder–Christensen model) became the standard cellular automaton for earthquakes and exhibits self-organized criticality-like avalanche statistics — that is, the system spontaneously approaches a scale-free state in which event sizes follow a power-law frequency distribution whose exponent depends on the fraction of stress dissipated in each toppling. In neurobiology, the integrate-and-fire neuron is the simplest spiking unit: its membrane potential charges toward threshold, and upon reaching threshold it emits a short pulse and resets; Mirollo and Strogatz proved that such pulse-coupled units with equal charging curves synchronize. Phase locking without global synchronization was identified early as the borderline phenomenon between full synchrony and self-organized criticality.

Herz and Hopfield (Phys. Rev. Lett. 75, 1222 (1995)) unified these threads in one solvable class. Each site $i$ of a two-dimensional lattice carries a scalar stress (or potential) $F_i$ loaded uniformly, $dF_i/dt = 1$; when $F_i$ reaches the threshold (set to 1), the site is reset to $F_i' = \gamma(F_i - 1)$, keeping a fraction $\gamma$ of the overshoot, while each of its four neighbors receives a pulse of weight $\alpha \le 1/4$ (general interaction matrices $T_{ji}$ are also considered). Initial stresses are drawn uniformly from $[1-w, 1]$. The case $\gamma = 1$ is Abelian — the result of an event does not depend on the update order of simultaneous topplings — and equivalent to Gabrielov's avalanche model, with cyclic oscillations of period $P = 1 - 4\alpha$; the case $\gamma = 0$ recovers the stick-slip model of Feder and Feder at $\alpha = 1/4$. Compared with the Olami–Feder–Christensen update, the reset rule differs only when several sites fail during one event, yet this minute change of local dynamics has pronounced collective consequences.

The letter's central theorem proves rapid phase locking: with periodic boundary conditions and non-negative couplings whose total incoming pulse weight satisfies $A = \sum_i T_{ji} \le 1$ (the nearest-neighbor case being $4\alpha \le 1$), a periodic phase-locked attractor is reached as soon as every element has discharged once, for all $0 \le \gamma \le 1$; neither symmetric couplings nor a fixed update order are required. Outside this regime the paper offers numerical observations only. Below the line $w + 2\alpha = 1$ the size–frequency relation of events follows a power law with exponent $B = 0.79 \pm 0.05$ (at $r = w/\alpha \approx 2.44$), while averages over the first event after reaching an attractor give $B_{\mathrm{first}} = 0.05 \pm 0.03$; for $\gamma = 0$ one finds $B = 0.40 \pm 0.05$ at $\alpha = 0.1$ and $B = 0.52 \pm 0.05$ at $\alpha \approx 1/4$, so the exponent depends on the dissipation parameter $\alpha$, and self-organized-criticality-like behavior appears (at least for small $\gamma$) only in ensemble averages. Open boundaries destroy complete phase locking and produce intermittency, with the main slip frequency shifting from $(1-4\alpha)^{-1}$ to $(1-3\alpha)^{-1}$. For $\gamma = 1$ the dynamics is a downhill motion on an energy landscape: $-F_{\mathrm{tot}}$ (minus the total stress) is a Lyapunov function — a quantity that decreases monotonically along the dynamics and thereby identifies the attractors — when evaluated at stroboscopic times separated by the period $P$; this is what makes it possible to program computational tasks in the coupled system, as developed in the companion PNAS paper of Hopfield and Herz. The letter closes by writing the leaky-integrator extension $dF_i/dt = -F_i/\tau + I_i$ (leakage toward zero plus heterogeneous inputs $I_i$), the standard leaky integrate-and-fire neuron, only as pointing to a potentially rich connection, with no analogue of the energy landscape available for it.

Problem Statement

Develop a predictive, quantitative theory of the collective dynamics of the Herz–Hopfield pulse-coupled threshold-element class — slowly loaded scalar variables $F_i$ (loading rate 1, threshold 1) on a two-dimensional lattice, with initial stresses drawn uniformly from $[1-w, 1]$ so that $r = w/\alpha$ measures the initial stress spread relative to the pulse weight, reset rule $F_i' = \gamma(F_i - 1)$, and pulse redistribution of weight $\alpha \le 1/4$ per neighbor, generalized to interaction matrices with total incoming pulse weight $A = \sum_i T_{ji}$ — valid beyond the regime covered by the source letter's convergence proof, which establishes rapid phase locking to periodic attractors only for non-negative couplings with $A \le 1$ and periodic boundary conditions, for all $0 \le \gamma \le 1$. The theory should determine, with analytic control rather than only numerical estimation in components (i)–(iv): (i) the avalanche size–frequency statistics and exponents — numerically $B = 0.79 \pm 0.05$ at $r = w/\alpha \approx 2.44$ with $B_{\mathrm{first}} = 0.05 \pm 0.03$, and for $\gamma = 0$ the values $B = 0.40 \pm 0.05$ at $\alpha = 0.1$ and $B = 0.52 \pm 0.05$ at $\alpha \approx 1/4$ — as functions of the dissipation parameters $\alpha$ and $\gamma$, the initial-stress spread $r = w/\alpha$, system size, and boundary conditions, including whether the dissipative regime is critical at all; (ii) the collective dynamics when the total incoming pulse weight violates the proof's bound ($A > 1$, where one discharge can trigger cascading further discharges) and for signed, mixed excitatory/inhibitory pulse weights; (iii) the statistics of the fine-tuned periodic attractors for $\gamma < 1$: the measure or basin size of each attractor and the probability of reaching it from generic random initial conditions; (iv) an energy-like (Lyapunov) functional for the continuous-time leaky integrate-and-fire evolution $dF_i/dt = -F_i/\tau + I_i$ — the extension announced in the source letter, in which the leaky-integrator equation replaces only the uniform loading $dF_i/dt = 1$, while the firing threshold, the reset rule $F_i' = \gamma(F_i - 1)$, and the pulse redistribution through the couplings $T_{ji}$ are retained — including leakage and heterogeneous inputs $I_i$, that governs convergence and would enable analytical design (programming) of computational attractors in spiking networks, extending the stroboscopic Lyapunov result available only for the Abelian case $\gamma = 1$; and (v) a controlled mapping from model parameters to observable time scales — for example seismic recurrence intervals of characteristic earthquakes, biologically relevant synchronization and binding (the temporary association of distributed neural activity into a coherent percept, hypothesized to require temporally coordinated firing), and reverberation time scales — together with explicitly stated validity conditions marking where the homogeneous-model predictions cease to apply to real faults and neural circuits. These deliverables jointly constitute one objective: replacing the source paper's partial proofs and numerical exponent estimates by a predictive theory of the same model class. They are linked as characterizations of a single dynamics — what the attractors are and how likely each is to be reached (iii), when phase locking breaks down and which collective states appear in its place (ii), the fluctuation and avalanche statistics of those states (i), the optimization principle governing the dynamics wherever one exists (iv), and the observable consequences of the parameters (v). Each is an independently refutable sub-claim, so a complete treatment of any single component settles that component and leaves the others open.

References

1. Andreas V. M. Herz and John J. Hopfield, 'Earthquake cycles and neural reverberations: Collective oscillations in systems of pulse-coupled threshold elements', Physical Review Letters 75, 1222–1225 (1995). DOI: 10.1103/PhysRevLett.75.1222, https://doi.org/10.1103/PhysRevLett.75.1222
2. John J. Hopfield and Andreas V. M. Herz, 'Rapid local synchronization of action potentials: Toward computation with coupled integrate-and-fire neurons', Proceedings of the National Academy of Sciences USA 92, 6655–6662 (1995). DOI: 10.1073/pnas.92.15.6655, https://doi.org/10.1073/pnas.92.15.6655
3. Zeev Olami, Hans Jacob S. Feder, and Kim Christensen, 'Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes', Physical Review Letters 68, 1244–1247 (1992). DOI: 10.1103/PhysRevLett.68.1244, https://doi.org/10.1103/PhysRevLett.68.1244
4. H. J. S. Feder and J. Feder, 'Self-organized criticality in a stick-slip process', Physical Review Letters 66, 2669–2672 (1991). DOI: 10.1103/PhysRevLett.66.2669, https://doi.org/10.1103/PhysRevLett.66.2669
5. Renato E. Mirollo and Steven H. Strogatz, 'Synchronization of pulse-coupled biological oscillators', SIAM Journal on Applied Mathematics 50, 1645–1662 (1990). DOI: 10.1137/0150098, https://doi.org/10.1137/0150098
6. Andrei Gabrielov, 'Abelian avalanches and Tutte polynomials', Physica A 195, 253–274 (1993). DOI: 10.1016/0378-4371(93)90267-8, https://doi.org/10.1016/0378-4371(93)90267-8
7. Samuele Bottani and Bertrand Delamotte, 'Self-organized-criticality and synchronization in pulse coupled relaxation oscillator systems; the Olami, Feder and Christense and the Feder and Feder model', Physica D 103, 430–441 (1997). DOI: 10.1016/S0167-2789(96)00275-8, https://doi.org/10.1016/S0167-2789(96)00275-8
8. A. Díaz-Guilera, A. Arenas, Á. Corral, and C. J. Pérez, 'Stability of spatio-temporal structures in a lattice model of pulse-coupled oscillators', Physica D 103, 419–429 (1997). DOI: 10.1016/S0167-2789(96)00274-6, https://doi.org/10.1016/S0167-2789(96)00274-6
9. S. T. R. Pinho, C. P. C. Prado, and O. Kinouchi, 'Absence of self-organized criticality in a random-neighbor version of the OFC stick-slip model', Physica A 257, 488–494 (1998). DOI: 10.1016/S0378-4371(98)00181-2, https://doi.org/10.1016/S0378-4371(98)00181-2
10. Josué X. de Carvalho and Carmen P. C. Prado, 'Self-organized criticality in the Olami-Feder-Christensen model', Physical Review Letters 84, 4006–4009 (2000). DOI: 10.1103/PhysRevLett.84.4006, https://doi.org/10.1103/PhysRevLett.84.4006
11. Kim Christensen, Dominic Hamon, Henrik J. Jensen, and Stefano Lise, 'Comment on "Self-Organized Criticality in the Olami-Feder-Christensen Model"', Physical Review Letters 87, 039801 (2001). DOI: 10.1103/PhysRevLett.87.039801, https://doi.org/10.1103/PhysRevLett.87.039801
12. Wulfram Gerstner, J. Leo van Hemmen, and Jack D. Cowan, 'What Matters in Neuronal Locking?', Neural Computation 8, 1653–1676 (1996). DOI: 10.1162/neco.1996.8.8.1653, https://doi.org/10.1162/neco.1996.8.8.1653
13. Bo Cartling, 'Control of computational dynamics of coupled integrate-and-fire neurons', Biological Cybernetics 76, 383–395 (1997). DOI: 10.1007/s004220050352, https://doi.org/10.1007/s004220050352
14. Dezhe Z. Jin, 'Fast convergence of spike sequences to periodic patterns in recurrent networks', Physical Review Letters 89, 208102 (2002). DOI: 10.1103/PhysRevLett.89.208102, https://doi.org/10.1103/PhysRevLett.89.208102
15. Marc Timme, Fred Wolf, and Theo Geisel, 'Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators', Physical Review Letters 89, 258701 (2002). DOI: 10.1103/PhysRevLett.89.258701, https://doi.org/10.1103/PhysRevLett.89.258701
16. Marc Timme, Fred Wolf, and Theo Geisel, 'Prevalence of unstable attractors in networks of pulse-coupled oscillators', Physical Review Letters 89, 154105 (2002). DOI: 10.1103/PhysRevLett.89.154105, https://doi.org/10.1103/PhysRevLett.89.154105
17. J. B. Rundle, K. F. Tiampo, W. Klein, and J. S. Sá Martins, 'Self-organization in leaky threshold systems: The influence of near-mean field dynamics and its implications for earthquakes, neurobiology, and forecasting', Proceedings of the National Academy of Sciences USA 99, 2514–2521 (2002). DOI: 10.1073/pnas.012581899, https://doi.org/10.1073/pnas.012581899
18. K. F. Tiampo, J. B. Rundle, W. Klein, J. S. Sá Martins, and C. D. Ferguson, 'Ergodic dynamics in a natural threshold system', Physical Review Letters 91, 238501 (2003). DOI: 10.1103/PhysRevLett.91.238501, https://doi.org/10.1103/PhysRevLett.91.238501
19. O. Ramos, E. Altshuler, and K. J. Måløy, 'Quasiperiodic events in an earthquake model', Physical Review Letters 96, 098501 (2006). DOI: 10.1103/PhysRevLett.96.098501, https://doi.org/10.1103/PhysRevLett.96.098501
20. Pulin Gong and Cees van Leeuwen, 'Dynamically maintained spike timing sequences in networks of pulse-coupled oscillators with delays', Physical Review Letters 98, 048104 (2007). DOI: 10.1103/PhysRevLett.98.048104, https://doi.org/10.1103/PhysRevLett.98.048104
21. Juan A. Bonachela, Sebastiano de Franciscis, Joaquín J. Torres, and Miguel A. Muñoz, 'Self-organization without conservation: Are neuronal avalanches generically critical?', Journal of Statistical Mechanics P02015 (2010). DOI: 10.1088/1742-5468/2010/02/P02015, arXiv:1001.3256, https://arxiv.org/abs/1001.3256
22. Kevin P. O’Keeffe, P. L. Krapivsky, and Steven H. Strogatz, 'Synchronization as aggregation: Cluster kinetics of pulse-coupled oscillators', Physical Review Letters 115, 064101 (2015). DOI: 10.1103/PhysRevLett.115.064101, https://doi.org/10.1103/PhysRevLett.115.064101
23. E. Paxon Frady and Friedrich T. Sommer, 'Robust computation with rhythmic spike patterns', Proceedings of the National Academy of Sciences USA 116, 18050–18059 (2019). DOI: 10.1073/pnas.1902653116, https://doi.org/10.1073/pnas.1902653116
24. Anirban Das and Anna Levina, 'Critical neuronal models with relaxed timescale separation', Physical Review X 9, 021062 (2019). DOI: 10.1103/PhysRevX.9.021062, arXiv:1808.04196, https://arxiv.org/abs/1808.04196
25. Gangyong Gwon and Young Sul Cho, 'Percolation critical exponents in cluster kinetics of pulse-coupled oscillators', Chaos 33, 033102 (2023). DOI: 10.1063/5.0138880, https://doi.org/10.1063/5.0138880