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Kramer MA, Chu CJ. A General, Noise-Driven Mechanism for the 1/f-Like Behavior of Neural Field Spectra. Neural Comput 2024; 36:1643-1668. [PMID: 39028955 DOI: 10.1162/neco_a_01682] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2023] [Accepted: 03/25/2024] [Indexed: 07/21/2024]
Abstract
Consistent observations across recording modalities, experiments, and neural systems find neural field spectra with 1/f-like scaling, eliciting many alternative theories to explain this universal phenomenon. We show that a general dynamical system with stochastic drive and minimal assumptions generates 1/f-like spectra consistent with the range of values observed in vivo without requiring a specific biological mechanism or collective critical behavior.
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Affiliation(s)
- Mark A Kramer
- Department of Mathematics and Statistics, and Center for Systems Neuroscience, Boston University, Boston, MA 02214, U.S.A.
| | - Catherine J Chu
- Department of Neurology, Massachusetts General Hospital and Harvard Medical School, Boston, MA 02114, U.S.A.
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Pérez-Cervera A, Gutkin B, Thomas PJ, Lindner B. A universal description of stochastic oscillators. Proc Natl Acad Sci U S A 2023; 120:e2303222120. [PMID: 37432992 PMCID: PMC10629544 DOI: 10.1073/pnas.2303222120] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2023] [Accepted: 05/18/2023] [Indexed: 07/13/2023] Open
Abstract
Many systems in physics, chemistry, and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example, linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here, we introduce a nonlinear transformation of stochastic oscillators to a complex-valued function [Formula: see text](x) that greatly simplifies and unifies the mathematical description of the oscillator's spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function [Formula: see text] (x) is the eigenfunction of the Kolmogorov backward operator with the least negative (but nonvanishing) eigenvalue λ1 = μ1 + iω1. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency ω1 and half-width μ1; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around ω1; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.
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Affiliation(s)
- Alberto Pérez-Cervera
- Department of Applied Mathematics, Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Madrid28040, Spain
| | - Boris Gutkin
- Group for Neural Theory, LNC2 INSERM U960, Département d’Etudes Cognitives, Ecole Normale Supérieure - Paris Science Letters University, Paris75005, France
| | - Peter J. Thomas
- Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH44106
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Berlin10115, Germany
- Department of Physics, Humboldt Universität zu Berlin, BerlinD-12489, Germany
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Heteroclinic cycling and extinction in May-Leonard models with demographic stochasticity. J Math Biol 2023; 86:30. [PMID: 36637504 PMCID: PMC9839821 DOI: 10.1007/s00285-022-01859-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2022] [Revised: 09/14/2022] [Accepted: 12/16/2022] [Indexed: 01/14/2023]
Abstract
May and Leonard (SIAM J Appl Math 29:243-253, 1975) introduced a three-species Lotka-Volterra type population model that exhibits heteroclinic cycling. Rather than producing a periodic limit cycle, the trajectory takes longer and longer to complete each "cycle", passing closer and closer to unstable fixed points in which one population dominates and the others approach zero. Aperiodic heteroclinic dynamics have subsequently been studied in ecological systems (side-blotched lizards; colicinogenic Escherichia coli), in the immune system, in neural information processing models ("winnerless competition"), and in models of neural central pattern generators. Yet as May and Leonard observed "Biologically, the behavior (produced by the model) is nonsense. Once it is conceded that the variables represent animals, and therefore cannot fall below unity, it is clear that the system will, after a few cycles, converge on some single population, extinguishing the other two." Here, we explore different ways of introducing discrete stochastic dynamics based on May and Leonard's ODE model, with application to ecological population dynamics, and to a neuromotor central pattern generator system. We study examples of several quantitatively distinct asymptotic behaviors, including total extinction of all species, extinction to a single species, and persistent cyclic dominance with finite mean cycle length.
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Holzhausen K, Ramlow L, Pu S, Thomas PJ, Lindner B. Mean-return-time phase of a stochastic oscillator provides an approximate renewal description for the associated point process. BIOLOGICAL CYBERNETICS 2022; 116:235-251. [PMID: 35166932 PMCID: PMC9068687 DOI: 10.1007/s00422-022-00920-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/19/2021] [Accepted: 01/11/2022] [Indexed: 06/14/2023]
Abstract
Stochastic oscillations can be characterized by a corresponding point process; this is a common practice in computational neuroscience, where oscillations of the membrane voltage under the influence of noise are often analyzed in terms of the interspike interval statistics, specifically the distribution and correlation of intervals between subsequent threshold-crossing times. More generally, crossing times and the corresponding interval sequences can be introduced for different kinds of stochastic oscillators that have been used to model variability of rhythmic activity in biological systems. In this paper we show that if we use the so-called mean-return-time (MRT) phase isochrons (introduced by Schwabedal and Pikovsky) to count the cycles of a stochastic oscillator with Markovian dynamics, the interphase interval sequence does not show any linear correlations, i.e., the corresponding sequence of passage times forms approximately a renewal point process. We first outline the general mathematical argument for this finding and illustrate it numerically for three models of increasing complexity: (i) the isotropic Guckenheimer-Schwabedal-Pikovsky oscillator that displays positive interspike interval (ISI) correlations if rotations are counted by passing the spoke of a wheel; (ii) the adaptive leaky integrate-and-fire model with white Gaussian noise that shows negative interspike interval correlations when spikes are counted in the usual way by the passage of a voltage threshold; (iii) a Hodgkin-Huxley model with channel noise (in the diffusion approximation represented by Gaussian noise) that exhibits weak but statistically significant interspike interval correlations, again for spikes counted when passing a voltage threshold. For all these models, linear correlations between intervals vanish when we count rotations by the passage of an MRT isochron. We finally discuss that the removal of interval correlations does not change the long-term variability and its effect on information transmission, especially in the neural context.
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Affiliation(s)
- Konstantin Holzhausen
- Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, Haus 2, 10115 Berlin, Germany
- Physics Department of Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
| | - Lukas Ramlow
- Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, Haus 2, 10115 Berlin, Germany
- Physics Department of Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
| | - Shusen Pu
- Department of Biomedical Engineering, 5814 Stevenson Center, Vanderbilt University, Nashville, TN 37215 USA
| | - Peter J. Thomas
- Department of Mathematics, Applied Mathematics, and Statistics, 212 Yost Hall, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio USA
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, Haus 2, 10115 Berlin, Germany
- Physics Department of Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
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Holzhausen K, Thomas PJ, Lindner B. Analytical approach to the mean-return-time phase of isotropic stochastic oscillators. Phys Rev E 2022; 105:024202. [PMID: 35291171 DOI: 10.1103/physreve.105.024202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2021] [Accepted: 01/24/2022] [Indexed: 06/14/2023]
Abstract
One notion of phase for stochastic oscillators is based on the mean return-time (MRT): a set of points represents a certain phase if the mean time to return from any point in this set to this set after one rotation is equal to the mean rotation period of the oscillator (irrespective of the starting point). For this so far only algorithmically defined phase, we derive here analytical expressions for the important class of isotropic stochastic oscillators. This allows us to evaluate cases from the literature explicitly and to study the behavior of the MRT phase in the limits of strong noise. We also use the same formalism to show that lines of constant return time variance (instead of constant mean return time) can be defined, and that they in general differ from the MRT isochrons.
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Affiliation(s)
- Konstantin Holzhausen
- Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, Haus 2, 10115 Berlin, Germany
- Physics Department of Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
| | - Peter J Thomas
- Department of Mathematics, Applied Mathematics and Statistics, 212 Yost Hall, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio, USA
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, Haus 2, 10115 Berlin, Germany
- Physics Department of Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
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Pérez-Cervera A, Lindner B, Thomas PJ. Isostables for Stochastic Oscillators. PHYSICAL REVIEW LETTERS 2021; 127:254101. [PMID: 35029447 DOI: 10.1103/physrevlett.127.254101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2021] [Revised: 10/18/2021] [Accepted: 11/04/2021] [Indexed: 05/25/2023]
Abstract
Thomas and Lindner [P. J. Thomas and B. Lindner, Phys. Rev. Lett. 113, 254101 (2014).PRLTAO0031-900710.1103/PhysRevLett.113.254101], defined an asymptotic phase for stochastic oscillators as the angle in the complex plane made by the eigenfunction, having a complex eigenvalue with a least negative real part, of the backward Kolmogorov (or stochastic Koopman) operator. We complete the phase-amplitude description of noisy oscillators by defining the stochastic isostable coordinate as the eigenfunction with the least negative nontrivial real eigenvalue. Our results suggest a framework for stochastic limit cycle dynamics that encompasses noise-induced oscillations.
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Affiliation(s)
- Alberto Pérez-Cervera
- National Research University Higher School of Economics, 109208 Moscow, Russia and Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Philippstraße 13, Haus 2, 10115 Berlin, Germany and Institute of Physics, Humboldt University at Berlin, Newtonstraße 15, D-12489 Berlin, Germany
| | - Peter J Thomas
- Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, Ohio 44106, USA
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