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Bettinger JS, Friston KJ. Conceptual foundations of physiological regulation incorporating the free energy principle and self-organized criticality. Neurosci Biobehav Rev 2023; 155:105459. [PMID: 37956880 DOI: 10.1016/j.neubiorev.2023.105459] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2022] [Revised: 11/02/2023] [Accepted: 11/03/2023] [Indexed: 11/15/2023]
Abstract
Bettinger, J. S., K. J. Friston. Conceptual Foundations of Physiological Regulation incorporating the Free Energy Principle & Self-Organized Criticality. NEUROSCI BIOBEHAV REV 23(x) 144-XXX, 2022. Since the late nineteen-nineties, the concept of homeostasis has been contextualized within a broader class of "allostatic" dynamics characterized by a wider-berth of causal factors including social, psychological and environmental entailments; the fundamental nature of integrated brain-body dynamics; plus the role of anticipatory, top-down constraints supplied by intrinsic regulatory models. Many of these evidentiary factors are integral in original descriptions of homeostasis; subsequently integrated; and/or cite more-general operating principles of self-organization. As a result, the concept of allostasis may be generalized to a larger category of variational systems in biology, engineering and physics in terms of advances in complex systems, statistical mechanics and dynamics involving heterogenous (hierarchical/heterarchical, modular) systems like brain-networks and the internal milieu. This paper offers a three-part treatment. 1) interpret "allostasis" to emphasize a variational and relational foundation of physiological stability; 2) adapt the role of allostasis as "stability through change" to include a "return to stability" and 3) reframe the model of homeostasis with a conceptual model of criticality that licenses the upgrade to variational dynamics.
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Affiliation(s)
- Jesse S Bettinger
- Center for Process Studies, Claremont, CA, United States; The Cobb Institute, Claremont, CA, United States.
| | - Karl J Friston
- Wellcome Centre for Human Neuroimaging, Institute of Neurology, University College London, London WC1N 3AR, UK; Queen Square Institute of Neurology, University College London, London WC1N 3AR, UK; The National Hospital for Neurology and Neurosurgery, London WC1N 3BG, UK
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2
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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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Majidifar S, Hayati M, Malekshahi MR, Abbott D. Low Cost Digital Implementation of Hybrid FitzHugh Nagumo-Morris Lecar Neuron Model Considering Electromagnetic Flux Coupling. IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS 2022; 16:1366-1374. [PMID: 36251896 DOI: 10.1109/tbcas.2022.3214851] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/05/2023]
Abstract
Digital realization of neuron models, especially implementation on a field programmable gate array (FPGA), is one of the key objectives of neuromorphic research, because the effective hardware realization of the biological neural networks plays a crucial role in implementing the behaviors of the brain for future applications. In this paper, a hybrid FitzHugh Nagumo-Morris Lecar (FNML) neuron model with electromagnetic flux coupling is considered, and two multiplierless piecewise linear (PWL) models, which have similar behaviors to the biological neuron, are presented. A comparison between digital implementation results of the original FNML and PWL models illustrates that, the PWL1 model provides a 65% speed-up with an overall saving (in FPGA resources) of 66.2%, and the PWL2 model yields a 71% speed-up with an overall saving of 78.2%.
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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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Rajagopal K, Jafari S, Moroz I, Karthikeyan A, Srinivasan A. Noise induced suppression of spiral waves in a hybrid FitzHugh-Nagumo neuron with discontinuous resetting. CHAOS (WOODBURY, N.Y.) 2021; 31:073117. [PMID: 34340329 DOI: 10.1063/5.0059175] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2021] [Accepted: 06/22/2021] [Indexed: 06/13/2023]
Abstract
A modified FitzHugh-Nagumo neuron model with sigmoid function-based recovery variable is considered with electromagnetic flux coupling. The dynamical properties of the proposed neuron model are investigated, and as the excitation current becomes larger, the number of fixed points decreases to one. The bifurcation plots are investigated to show the chaotic and periodic regimes for various values of excitation current and parameters. A N×N network of the neuron model is constructed to study the wave propagation and wave re-entry phenomena. Investigations are conducted to show that for larger flux coupling values, the spiral waves are suppressed, but for such values of the flux coupling, the individual nodes are driven into periodic regimes. By introducing Gaussian noise as an additional current term, we showed that when noise is introduced for the entire simulation time, the dynamics of the nodes are largely altered while the noise exposure for 200-time units will not alter the dynamics of the nodes completely.
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Affiliation(s)
- Karthikeyan Rajagopal
- Center for Nonlinear Systems, Chennai Institute of Technology, Chennai, Tamil Nadu 600069, India
| | - Sajad Jafari
- Department of Biomedical Engineering, Amirkabir University of Technology, Tehran 1591634311, Iran
| | - Irene Moroz
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Oxford OX1 2JD, United Kingdom
| | - Anitha Karthikeyan
- Department of Electronics and Communication Engineering, Prathyusha Engineering College, Tiruvallur, Tamil Nadu 602025, India
| | - Ashokkumar Srinivasan
- Center for Nonlinear Systems, Chennai Institute of Technology, Chennai, Tamil Nadu 600069, India
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Schleimer JH, Hesse J, Contreras SA, Schreiber S. Firing statistics in the bistable regime of neurons with homoclinic spike generation. Phys Rev E 2021; 103:012407. [PMID: 33601551 DOI: 10.1103/physreve.103.012407] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2019] [Accepted: 11/20/2020] [Indexed: 11/07/2022]
Abstract
Neuronal voltage dynamics of regularly firing neurons typically has one stable attractor: either a fixed point (like in the subthreshold regime) or a limit cycle that defines the tonic firing of action potentials (in the suprathreshold regime). In two of the three spike onset bifurcation sequences that are known to give rise to all-or-none type action potentials, however, the resting-state fixed point and limit cycle spiking can coexist in an intermediate regime, resulting in bistable dynamics. Here, noise can induce switches between the attractors, i.e., between rest and spiking, and thus increase the variability of the spike train compared to neurons with only one stable attractor. Qualitative features of the resulting spike statistics depend on the spike onset bifurcations. This paper focuses on the creation of the spiking limit cycle via the saddle-homoclinic orbit (HOM) bifurcation and derives interspike interval (ISI) densities for a conductance-based neuron model in the bistable regime. The ISI densities of bistable homoclinic neurons are found to be unimodal yet distinct from the inverse Gaussian distribution associated with the saddle-node-on-invariant-cycle bifurcation. It is demonstrated that for the HOM bifurcation the transition between rest and spiking is mainly determined along the downstroke of the action potential-a dynamical feature that is not captured by the commonly used reset neuron models. The deduced spike statistics can help to identify HOM dynamics in experimental data.
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Affiliation(s)
- Jan-Hendrik Schleimer
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany
- Bernstein Center for Computational Neuroscience, 10115 Berlin, Germany
| | - Janina Hesse
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany
- MSH Medical School Hamburg, Am Kaiserkai 1, 20457 Hamburg, Germany
| | - Susana Andrea Contreras
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany
- Bernstein Center for Computational Neuroscience, 10115 Berlin, Germany
| | - Susanne Schreiber
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany
- Bernstein Center for Computational Neuroscience, 10115 Berlin, Germany
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Multiple Firing Patterns in Coupled Hindmarsh-Rose Neurons with a Nonsmooth Memristor. Neural Plast 2020; 2020:8826369. [PMID: 33224191 PMCID: PMC7669342 DOI: 10.1155/2020/8826369] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/02/2020] [Revised: 09/19/2020] [Accepted: 10/16/2020] [Indexed: 11/18/2022] Open
Abstract
A model is introduced by coupling two three-dimensional Hindmarsh-Rose models with the help of a nonsmooth memristor. The firing patterns dependent on the external forcing current are explored, which undergo a process from adding-period to chaos. The stability of equilibrium points of the considered model is investigated via qualitative analysis, from which it can be gained that the model has diversity in the number and stability of equilibrium points for different coupling coefficients. The coexistence of multiple firing patterns relative to initial values is revealed, which means that the referred model can appear various firing patterns with the change of the initial value. Multiple firing patterns of the addressed neuron model induced by different scales are uncovered, which suggests that the discussed model has a multiscale effect for the nonzero initial value.
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Thomas PJ, Lindner B. Phase descriptions of a multidimensional Ornstein-Uhlenbeck process. Phys Rev E 2019; 99:062221. [PMID: 31330649 DOI: 10.1103/physreve.99.062221] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2019] [Indexed: 05/25/2023]
Abstract
Stochastic oscillators play a prominent role in different fields of science. Their simplified description in terms of a phase has been advocated by different authors using distinct phase definitions in the stochastic case. One notion of phase that we put forward previously, the asymptotic phase of a stochastic oscillator, is based on the eigenfunction expansion of its probability density. More specifically, it is given by the complex argument of the eigenfunction of the backward operator corresponding to the least-negative eigenvalue. Formally, besides the "backward" phase, one can also define the "forward" phase as the complex argument of the eigenfunction of the forward Kolomogorov operator corresponding to the least-negative eigenvalue. Until now, the intuition about these phase descriptions has been limited. Here we study these definitions for a process that is analytically tractable, the two-dimensional Ornstein-Uhlenbeck process with complex eigenvalues. For this process, (i) we give explicit expressions for the two phases; (ii) we demonstrate that the isochrons are always the spokes of a wheel but that (iii) the spacing of these isochrons (their angular density) is different for backward and forward phases; (iv) we show that the isochrons of the backward phase are completely determined by the deterministic part of the vector field, whereas the forward phase also depends on the noise matrix; and (v) we demonstrate that the mean progression of the backward phase in time is always uniform, whereas this is not true for the forward phase except in the rotationally symmetric case. We illustrate our analytical results for a number of qualitatively different cases.
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Affiliation(s)
- Peter J Thomas
- Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, Ohio 44106, USA
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Philippstraße 13, Haus 2, 10115 Berlin, Germany and Department of Physics, Humboldt University Berlin, Newtonstraße 15, 12489 Berlin, Germany
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Li Y, Liu X. Noise induced escape in one-population and two-population stochastic neural networks with internal states. CHAOS (WOODBURY, N.Y.) 2019; 29:023137. [PMID: 30823742 DOI: 10.1063/1.5055051] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2018] [Accepted: 01/24/2019] [Indexed: 06/09/2023]
Abstract
In the present paper, the escapes from the basins of fixed points induced by intrinsic noise are investigated in both one- and two-population stochastic hybrid neural networks. In the weak noise limit, the quasipotentials are computed through the application of WKB approximation to the original hybrid system and the results of quasi-steady-state (QSS) diffusion approximation. It is seen that the two results are consistent with each other within the neighborhood of a fixed point and an obvious discrepancy arises in the other area, of which the reason is then explored and revealed. Furthermore, the relationship between the fluctuational paths and the relaxational ones is analyzed, based on which some specific results for the hybrid system is obtained. Besides, for the two-population model, the phenomenon of saddle point avoidance is investigated by using both theoretical and numerical methods. Finally, the topological structure of Lagrangian manifold is analyzed, and its particular features and something analogous to the stochastic differential equation are found according to the accuracy of QSS within the vicinity of the saddle point.
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Affiliation(s)
- Yang Li
- State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing 210016, China
| | - Xianbin Liu
- State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing 210016, China
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10
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Bressloff PC, Maclaurin JN. Stochastic Hybrid Systems in Cellular Neuroscience. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2018; 8:12. [PMID: 30136005 PMCID: PMC6104574 DOI: 10.1186/s13408-018-0067-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/09/2018] [Accepted: 08/05/2018] [Indexed: 06/08/2023]
Abstract
We review recent work on the theory and applications of stochastic hybrid systems in cellular neuroscience. A stochastic hybrid system or piecewise deterministic Markov process involves the coupling between a piecewise deterministic differential equation and a time-homogeneous Markov chain on some discrete space. The latter typically represents some random switching process. We begin by summarizing the basic theory of stochastic hybrid systems, including various approximation schemes in the fast switching (weak noise) limit. In subsequent sections, we consider various applications of stochastic hybrid systems, including stochastic ion channels and membrane voltage fluctuations, stochastic gap junctions and diffusion in randomly switching environments, and intracellular transport in axons and dendrites. Finally, we describe recent work on phase reduction methods for stochastic hybrid limit cycle oscillators.
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Bressloff PC, MacLaurin J. A variational method for analyzing limit cycle oscillations in stochastic hybrid systems. CHAOS (WOODBURY, N.Y.) 2018; 28:063105. [PMID: 29960393 DOI: 10.1063/1.5027077] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Many systems in biology can be modeled through ordinary differential equations, which are piece-wise continuous, and switch between different states according to a Markov jump process known as a stochastic hybrid system or piecewise deterministic Markov process (PDMP). In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we develop a phase reduction method for stochastic hybrid systems that support a stable limit cycle in the deterministic limit. A classic example is the Morris-Lecar model of a neuron, where the switching Markov process is the number of open ion channels and the continuous process is the membrane voltage. We outline a variational principle for the phase reduction, yielding an exact analytic expression for the resulting phase dynamics. We demonstrate that this decomposition is accurate over timescales that are exponential in the switching rate ϵ-1. That is, we show that for a constant C, the probability that the expected time to leave an O(a) neighborhood of the limit cycle is less than T scales as T exp (-Ca/ϵ).
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA
| | - James MacLaurin
- Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA
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de la Cruz R, Perez-Carrasco R, Guerrero P, Alarcon T, Page KM. Minimum Action Path Theory Reveals the Details of Stochastic Transitions Out of Oscillatory States. PHYSICAL REVIEW LETTERS 2018; 120:128102. [PMID: 29694079 DOI: 10.1103/physrevlett.120.128102] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2017] [Revised: 12/13/2017] [Indexed: 06/08/2023]
Abstract
Cell state determination is the outcome of intrinsically stochastic biochemical reactions. Transitions between such states are studied as noise-driven escape problems in the chemical species space. Escape can occur via multiple possible multidimensional paths, with probabilities depending nonlocally on the noise. Here we characterize the escape from an oscillatory biochemical state by minimizing the Freidlin-Wentzell action, deriving from it the stochastic spiral exit path from the limit cycle. We also use the minimized action to infer the escape time probability density function.
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Affiliation(s)
- Roberto de la Cruz
- Centre de Recerca Matemàtica, Edifici C, Campus de Bellaterra, 08193 Bellaterra (Barcelona), Spain and Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Ruben Perez-Carrasco
- Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
| | - Pilar Guerrero
- Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
| | - Tomas Alarcon
- ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
- Centre de Recerca Matemàtica, Edifici C, Campus de Bellaterra, 08193 Bellaterra (Barcelona), Spain
- Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain and Barcelona Graduate School of Mathematics (BGSMath), Barcelona, Spain
| | - Karen M Page
- Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
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