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Ferrara A, Angulo-Garcia D, Torcini A, Olmi S. Population spiking and bursting in next-generation neural masses with spike-frequency adaptation. Phys Rev E 2023; 107:024311. [PMID: 36932567 DOI: 10.1103/physreve.107.024311] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Accepted: 02/03/2023] [Indexed: 06/18/2023]
Abstract
Spike-frequency adaptation (SFA) is a fundamental neuronal mechanism taking into account the fatigue due to spike emissions and the consequent reduction of the firing activity. We have studied the effect of this adaptation mechanism on the macroscopic dynamics of excitatory and inhibitory networks of quadratic integrate-and-fire (QIF) neurons coupled via exponentially decaying post-synaptic potentials. In particular, we have studied the population activities by employing an exact mean-field reduction, which gives rise to next-generation neural mass models. This low-dimensional reduction allows for the derivation of bifurcation diagrams and the identification of the possible macroscopic regimes emerging both in a single and in two identically coupled neural masses. In single populations SFA favors the emergence of population bursts in excitatory networks, while it hinders tonic population spiking for inhibitory ones. The symmetric coupling of two neural masses, in absence of adaptation, leads to the emergence of macroscopic solutions with broken symmetry, namely, chimera-like solutions in the inhibitory case and antiphase population spikes in the excitatory one. The addition of SFA leads to new collective dynamical regimes exhibiting cross-frequency coupling (CFC) among the fast synaptic timescale and the slow adaptation one, ranging from antiphase slow-fast nested oscillations to symmetric and asymmetric bursting phenomena. The analysis of these CFC rhythms in the θ-γ range has revealed that a reduction of SFA leads to an increase of the θ frequency joined to a decrease of the γ one. This is analogous to what has been reported experimentally for the hippocampus and the olfactory cortex of rodents under cholinergic modulation, which is known to reduce SFA.
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Affiliation(s)
- Alberto Ferrara
- Sorbonne Université, INSERM, CNRS, Institut de la Vision, 75012 Paris, France
| | - David Angulo-Garcia
- Departamento de Matemáticas y Estadística, Universidad Nacional de Colombia (UNAL), Cra 27 No. 64-60, 170003, Manizales, Colombia
| | - Alessandro Torcini
- Laboratoire de Physique Théorique et Modélisation, UMR 8089, CY Cergy Paris Université, CNRS, 95302 Cergy-Pontoise, France
- CNR, Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy
- INFN, Sezione di Firenze, via Sansone 1, 50019 Sesto Fiorentino, Italy
| | - Simona Olmi
- CNR, Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy
- INFN, Sezione di Firenze, via Sansone 1, 50019 Sesto Fiorentino, Italy
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Pyragas V, Pyragas K. Mean-field equations for neural populations with q-Gaussian heterogeneities. Phys Rev E 2022; 105:044402. [PMID: 35590671 DOI: 10.1103/physreve.105.044402] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2022] [Accepted: 03/29/2022] [Indexed: 06/15/2023]
Abstract
Describing the collective dynamics of large neural populations using low-dimensional models for averaged variables has long been an attractive task in theoretical neuroscience. Recently developed reduction methods make it possible to derive such models directly from the microscopic dynamics of individual neurons. To simplify the reduction, the Cauchy distribution is usually assumed for heterogeneous network parameters. Here we extend the reduction method for a wider class of heterogeneities defined by the q-Gaussian distribution. The shape of this distribution depends on the Tsallis index q and gradually changes from the Cauchy distribution to the normal Gaussian distribution as this index changes. We derive the mean-field equations for an inhibitory network of quadratic integrate-and-fire neurons with a q-Gaussian-distributed excitability parameter. It is shown that the dynamic modes of the network significantly depend on the form of the distribution determined by the Tsallis index. The results obtained from the mean-field equations are confirmed by numerical simulation of the microscopic model.
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Affiliation(s)
- Viktoras Pyragas
- Center for Physical Sciences and Technology, 10257 Vilnius, Lithuania
| | - Kestutis Pyragas
- Center for Physical Sciences and Technology, 10257 Vilnius, Lithuania
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3
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Laing CR. Chimeras with uniformly distributed heterogeneity: Two coupled populations. Phys Rev E 2022; 105:024306. [PMID: 35291147 DOI: 10.1103/physreve.105.024306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2021] [Accepted: 01/28/2022] [Indexed: 06/14/2023]
Abstract
Chimeras occur in networks of two coupled populations of oscillators when the oscillators in one population synchronize while those in the other are asynchronous. We consider chimeras of this form in networks of planar oscillators for which one parameter associated with the dynamics of an oscillator is randomly chosen from a uniform distribution. A generalization of the previous approach [Laing, Phys. Rev. E 100, 042211 (2019)2470-004510.1103/PhysRevE.100.042211], which dealt with identical oscillators, is used to investigate the existence and stability of chimeras for these heterogeneous networks in the limit of an infinite number of oscillators. In all cases, making the oscillators more heterogeneous destroys the stable chimera in a saddle-node bifurcation. The results help us understand the robustness of chimeras in networks of general oscillators to heterogeneity.
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Affiliation(s)
- Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Private Bag 102-904 North Shore Mail Centre, Auckland, New Zealand
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4
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Pyragas K, Fedaravičius AP, Pyragienė T. Suppression of synchronous spiking in two interacting populations of excitatory and inhibitory quadratic integrate-and-fire neurons. Phys Rev E 2021; 104:014203. [PMID: 34412351 DOI: 10.1103/physreve.104.014203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2021] [Accepted: 06/14/2021] [Indexed: 01/28/2023]
Abstract
Collective oscillations and their suppression by external stimulation are analyzed in a large-scale neural network consisting of two interacting populations of excitatory and inhibitory quadratic integrate-and-fire neurons. In the limit of an infinite number of neurons, the microscopic model of this network can be reduced to an exact low-dimensional system of mean-field equations. Bifurcation analysis of these equations reveals three different dynamic modes in a free network: a stable resting state, a stable limit cycle, and bistability with a coexisting resting state and a limit cycle. We show that in the limit cycle mode, high-frequency stimulation of an inhibitory population can stabilize an unstable resting state and effectively suppress collective oscillations. We also show that in the bistable mode, the dynamics of the network can be switched from a stable limit cycle to a stable resting state by applying an inhibitory pulse to the excitatory population. The results obtained from the mean-field equations are confirmed by numerical simulation of the microscopic model.
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Affiliation(s)
- Kestutis Pyragas
- Department of Fundamental Research, Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| | - Augustinas P Fedaravičius
- Department of Fundamental Research, Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| | - Tatjana Pyragienė
- Department of Fundamental Research, Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
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Jüttner B, Henriksen C, Martens EA. Birth and destruction of collective oscillations in a network of two populations of coupled type 1 neurons. CHAOS (WOODBURY, N.Y.) 2021; 31:023141. [PMID: 33653075 DOI: 10.1063/5.0031630] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2020] [Accepted: 01/12/2021] [Indexed: 06/12/2023]
Abstract
We study the macroscopic dynamics of large networks of excitable type 1 neurons composed of two populations interacting with disparate but symmetric intra- and inter-population coupling strengths. This nonuniform coupling scheme facilitates symmetric equilibria, where both populations display identical firing activity, characterized by either quiescent or spiking behavior, or asymmetric equilibria, where the firing activity of one population exhibits quiescent but the other exhibits spiking behavior. Oscillations in the firing rate are possible if neurons emit pulses with non-zero width but are otherwise quenched. Here, we explore how collective oscillations emerge for two statistically identical neuron populations in the limit of an infinite number of neurons. A detailed analysis reveals how collective oscillations are born and destroyed in various bifurcation scenarios and how they are organized around higher codimension bifurcation points. Since both symmetric and asymmetric equilibria display bistable behavior, a large configuration space with steady and oscillatory behavior is available. Switching between configurations of neural activity is relevant in functional processes such as working memory and the onset of collective oscillations in motor control.
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Affiliation(s)
- Benjamin Jüttner
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Christian Henriksen
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Erik A Martens
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
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Bick C, Goodfellow M, Laing CR, Martens EA. Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2020; 10:9. [PMID: 32462281 PMCID: PMC7253574 DOI: 10.1186/s13408-020-00086-9] [Citation(s) in RCA: 93] [Impact Index Per Article: 23.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2019] [Accepted: 05/07/2020] [Indexed: 05/03/2023]
Abstract
Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.
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Affiliation(s)
- Christian Bick
- Centre for Systems, Dynamics, and Control, University of Exeter, Exeter, UK.
- Department of Mathematics, University of Exeter, Exeter, UK.
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK.
- Mathematical Institute, University of Oxford, Oxford, UK.
- Institute for Advanced Study, Technische Universität München, Garching, Germany.
| | - Marc Goodfellow
- Department of Mathematics, University of Exeter, Exeter, UK
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK
- Living Systems Institute, University of Exeter, Exeter, UK
- Wellcome Trust Centre for Biomedical Modelling and Analysis, University of Exeter, Exeter, UK
| | - Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Auckland, New Zealand
| | - Erik A Martens
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark.
- Department of Biomedical Science, University of Copenhagen, Copenhagen N, Denmark.
- Centre for Translational Neuroscience, University of Copenhagen, Copenhagen N, Denmark.
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Pietras B, Devalle F, Roxin A, Daffertshofer A, Montbrió E. Exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks. Phys Rev E 2020; 100:042412. [PMID: 31771022 DOI: 10.1103/physreve.100.042412] [Citation(s) in RCA: 22] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2019] [Indexed: 01/09/2023]
Abstract
Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models-also known as firing rate models or firing rate equations-to account for electrical synapses. Here, we introduce a firing rate model that exactly describes the mean-field dynamics of heterogeneous populations of quadratic integrate-and-fire (QIF) neurons with both chemical and electrical synapses. The mathematical analysis of the firing rate model reveals a well-established bifurcation scenario for networks with chemical synapses, characterized by a codimension-2 cusp point and persistent states for strong recurrent excitatory coupling. The inclusion of electrical coupling generally implies neuronal synchrony by virtue of a supercritical Hopf bifurcation. This transforms the cusp scenario into a bifurcation scenario characterized by three codimension-2 points (cusp, Takens-Bogdanov, and saddle-node separatrix loop), which greatly reduces the possibility for persistent states. This is generic for heterogeneous QIF networks with both chemical and electrical couplings. Our results agree with several numerical studies on the dynamics of large networks of heterogeneous spiking neurons with electrical and chemical couplings.
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Affiliation(s)
- Bastian Pietras
- Faculty of Behavioural and Movement Sciences, Amsterdam Movement Sciences & Institute of Brain and Behavior Amsterdam, Vrije Universiteit Amsterdam, van der Boechorststraat 9, Amsterdam 1081 BT, The Netherlands.,Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom.,Institute of Mathematics, Technical University Berlin, 10623 Berlin, Germany.,Bernstein Center for Computational Neuroscience Berlin, 10115 Berlin, Germany
| | - Federico Devalle
- Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom.,Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08003 Barcelona, Spain
| | - Alex Roxin
- Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain.,Barcelona Graduate School of Mathematics, 08193 Barcelona, Spain
| | - Andreas Daffertshofer
- Faculty of Behavioural and Movement Sciences, Amsterdam Movement Sciences & Institute of Brain and Behavior Amsterdam, Vrije Universiteit Amsterdam, van der Boechorststraat 9, Amsterdam 1081 BT, The Netherlands
| | - Ernest Montbrió
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08003 Barcelona, Spain
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Ratas I, Pyragas K. Noise-induced macroscopic oscillations in a network of synaptically coupled quadratic integrate-and-fire neurons. Phys Rev E 2019; 100:052211. [PMID: 31869871 DOI: 10.1103/physreve.100.052211] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2019] [Indexed: 05/26/2023]
Abstract
We consider the effect of small independent local noise on a network of quadratic integrate-and-fire neurons, globally coupled via synaptic pulses of finite width. The Fokker-Planck equation for a network of infinite size is reduced to a low-dimensional system of ordinary differential equations using the recently proposed perturbation theory based on circular cumulants. A bifurcation analysis of the reduced equations is performed, and areas in the parameter space, where the noise causes macroscopic oscillations of the network, are determined. The validity of the reduced equations is verified by comparing their solutions with "exact" solutions of the Fokker-Planck equation, as well as with the results of direct simulation of stochastic microscopic dynamics of a finite-size network.
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Affiliation(s)
- Irmantas Ratas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| | - Kestutis Pyragas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
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9
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Laing CR. Dynamics and stability of chimera states in two coupled populations of oscillators. Phys Rev E 2019; 100:042211. [PMID: 31770875 DOI: 10.1103/physreve.100.042211] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2019] [Indexed: 06/10/2023]
Abstract
We consider networks formed from two populations of identical oscillators, with uniform strength all-to-all coupling within populations and also between populations with a different strength. Such systems are known to support chimera states in which oscillators within one population are perfectly synchronized while in the other the oscillators are incoherent and have a different mean frequency from those in the synchronous population. Assuming that the oscillators in the incoherent population always lie on a closed smooth curve C, we derive and analyze the dynamics of the shape of C and the probability density on C for four different types of oscillators. We put some previously derived results on a more rigorous footing and analyze two new systems.
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Affiliation(s)
- Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Auckland 0745, New Zealand
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