Derivation of a neural field model from a network of theta neurons

Mean-field approximation for networks with synchrony-driven adaptive coupling

Synaptic plasticity plays a fundamental role in neuronal dynamics, governing how connections between neurons evolve in response to experience. In this study, we extend a network model of θ-neuron oscillators to include a realistic form of adaptive plasticity. In place of the less tractable spike-timing-dependent plasticity, we employ recently validated phase-difference-dependent plasticity rules, which adjust coupling strengths based on the relative phases of θ-neuron oscillators. We explore two distinct implementations of this plasticity: pairwise updates to individual coupling strengths and global updates applied to the mean coupling strength. We derive a mean-field approximation and assess its accuracy by comparing it to θ-neuron simulations across various stability regimes. The synchrony of the system is quantified using the Kuramoto order parameter. Through bifurcation analysis and the calculation of maximal Lyapunov exponents, we uncover interesting phenomena such as bistability and chaotic dynamics via period-doubling and boundary crisis bifurcations. These behaviors emerge as a direct result of adaptive coupling and are absent in systems without such plasticity.

Low-dimensional model for adaptive networks of spiking neurons

We investigate a large ensemble of quadratic integrate-and-fire neurons with heterogeneous input currents and adaptation variables. Our analysis reveals that, for a specific class of adaptation, termed quadratic spike-frequency adaptation, the high-dimensional system can be exactly reduced to a low-dimensional system of ordinary differential equations, which describes the dynamics of three mean-field variables: the population's firing rate, the mean membrane potential, and a mean adaptation variable. The resulting low-dimensional firing rate equations (FREs) uncover a key generic feature of heterogeneous networks with spike-frequency adaptation: Both the center and width of the distribution of the neurons' firing frequencies are reduced, and this largely promotes the emergence of collective synchronization in the network. Our findings are further supported by the bifurcation analysis of the FREs, which accurately captures the collective dynamics of the spiking neuron network, including phenomena such as collective oscillations, bursting, and macroscopic chaos.

High-Mode Coupling Yields Multicoherent-Phase Phenomena in Nonlocally Coupled Oscillators

Capturing the intricate dynamics of partially coherent patterns in coupled oscillator systems is vibrant and one of the crucial areas of nonlinear sciences. Considering higher-order Fourier modes in the coupling, we discover a novel type of clustered coherent state in phase models, where inside the coherent region oscillators are further split into q dynamically equivalent subgroups with a 2π/q phase difference between two neighboring subgroups, forming a multicoherent-phase (MUP) chimera state. Both a self-consistency analysis and the Ott-Antonsen dimension reduction techniques are used to theoretically derive these solutions, whose stability are further demonstrated by spectral analysis. The universality of MUP effects is demonstrated by generalized twisted states and higher-order spatial swarm chimera states of swarmalators which are beyond pure phase models since oscillators can move in space.

Firing rate models for gamma oscillations in I-I and E-I networks

Firing rate models for describing the mean-field activities of neuronal ensembles can be used effectively to study network function and dynamics, including synchronization and rhythmicity of excitatory-inhibitory populations. However, traditional Wilson-Cowan-like models, even when extended to include an explicit dynamic synaptic activation variable, are found unable to capture some dynamics such as Interneuronal Network Gamma oscillations (ING). Use of an explicit delay is helpful in simulations at the expense of complicating mathematical analysis. We resolve this issue by introducing a dynamic variable, u, that acts as an effective delay in the negative feedback loop between firing rate (r) and synaptic gating of inhibition (s). In effect, u endows synaptic activation with second order dynamics. With linear stability analysis, numerical branch-tracking and simulations, we show that our r-u-s rate model captures some key qualitative features of spiking network models for ING. We also propose an alternative formulation, a v-u-s model, in which mean membrane potential v satisfies an averaged current-balance equation. Furthermore, we extend the framework to E-I networks. With our six-variable v-u-s model, we demonstrate in firing rate models the transition from Pyramidal-Interneuronal Network Gamma (PING) to ING by increasing the external drive to the inhibitory population without adjusting synaptic weights. Having PING and ING available in a single network, without invoking synaptic blockers, is plausible and natural for explaining the emergence and transition of two different types of gamma oscillations.

Activity patterns in ring networks of quadratic integrate-and-fire neurons with synaptic and gap junction coupling

We consider a ring network of quadratic integrate-and-fire neurons with nonlocal synaptic and gap junction coupling. The corresponding neural field model supports solutions such as standing and traveling waves, and also lurching waves. We show that many of these solutions satisfy self-consistency equations which can be used to follow them as parameters are varied. We perform numerical bifurcation analysis of the neural field model, concentrating on the effects of varying gap junction coupling strength. Our methods are generally applicable to a wide variety of networks of quadratic integrate-and-fire neurons.

How synaptic function controls critical transitions in spiking neuron networks: insight from a Kuramoto model reduction

The dynamics of synaptic interactions within spiking neuron networks play a fundamental role in shaping emergent collective behavior. This paper studies a finite-size network of quadratic integrate-and-fire neurons interconnected via a general synaptic function that accounts for synaptic dynamics and time delays. Through asymptotic analysis, we transform this integrate-and-fire network into the Kuramoto-Sakaguchi model, whose parameters are explicitly expressed via synaptic function characteristics. This reduction yields analytical conditions on synaptic activation rates and time delays determining whether the synaptic coupling is attractive or repulsive. Our analysis reveals alternating stability regions for synchronous and partially synchronous firing, dependent on slow synaptic activation and time delay. We also demonstrate that the reduced microscopic model predicts the emergence of synchronization, weakly stable cyclops states, and non-stationary regimes remarkably well in the original integrate-and-fire network and its theta neuron counterpart. Our reduction approach promises to open the door to rigorous analysis of rhythmogenesis in networks with synaptic adaptation and plasticity.

Pulse Shape and Voltage-Dependent Synchronization in Spiking Neuron Networks

Pulse-coupled spiking neural networks are a powerful tool to gain mechanistic insights into how neurons self-organize to produce coherent collective behavior. These networks use simple spiking neuron models, such as the θ-neuron or the quadratic integrate-and-fire (QIF) neuron, that replicate the essential features of real neural dynamics. Interactions between neurons are modeled with infinitely narrow pulses, or spikes, rather than the more complex dynamics of real synapses. To make these networks biologically more plausible, it has been proposed that they must also account for the finite width of the pulses, which can have a significant impact on the network dynamics. However, the derivation and interpretation of these pulses are contradictory, and the impact of the pulse shape on the network dynamics is largely unexplored. Here, I take a comprehensive approach to pulse coupling in networks of QIF and θ-neurons. I argue that narrow pulses activate voltage-dependent synaptic conductances and show how to implement them in QIF neurons such that their effect can last through the phase after the spike. Using an exact low-dimensional description for networks of globally coupled spiking neurons, I prove for instantaneous interactions that collective oscillations emerge due to an effective coupling through the mean voltage. I analyze the impact of the pulse shape by means of a family of smooth pulse functions with arbitrary finite width and symmetric or asymmetric shapes. For symmetric pulses, the resulting voltage coupling is not very effective in synchronizing neurons, but pulses that are slightly skewed to the phase after the spike readily generate collective oscillations. The results unveil a voltage-dependent spike synchronization mechanism at the heart of emergent collective behavior, which is facilitated by pulses of finite width and complementary to traditional synaptic transmission in spiking neuron networks.

Neurophysiological avenues to better conceptualizing adaptive cognition

We delve into the human brain's remarkable capacity for adaptability and sustained cognitive functioning, phenomena traditionally encompassed as executive functions or cognitive control. The neural underpinnings that enable the seamless navigation between transient thoughts without detracting from overarching goals form the core of our article. We discuss the concept of "metacontrol," which builds upon conventional cognitive control theories by proposing a dynamic balancing of processes depending on situational demands. We critically discuss the role of oscillatory processes in electrophysiological activity at different scales and the importance of desynchronization and partial phase synchronization in supporting adaptive behavior including neural noise accounts, transient dynamics, phase-based measures (coordination dynamics) and neural mass modelling. The cognitive processes focused and neurophysiological avenues outlined are integral to understanding diverse psychiatric disorders thereby contributing to a more nuanced comprehension of cognitive control and its neural bases in both health and disease.

Controlling neural activity: LPV modelling of optogenetically actuated Wilson-Cowan model. J Neural Eng 2024;21:036002. [PMID: 38653250 DOI: 10.1088/1741-2552/ad4212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

Objective.This paper aims to bridge the gap between neurophysiology and automatic control methodologies by redefining the Wilson-Cowan (WC) model as a control-oriented linear parameter-varying (LPV) system. A novel approach is presented that allows for the application of a control strategy to modulate and track neural activity.Approach.The WC model is redefined as a control-oriented LPV system in this study. The LPV modelling framework is leveraged to design an LPV controller, which is used to regulate and manipulate neural dynamics.Main results.Promising outcomes, in understanding and controlling neural processes through the synergistic combination of control-oriented modelling and estimation, are obtained in this study. An LPV controller demonstrates to be effective in regulating neural activity.Significance.The presented methodology effectively induces neural patterns, taking into account optogenetic actuation. The combination of control strategies with neurophysiology provides valuable insights into neural dynamics. The proposed approach opens up new possibilities for using control techniques to study and influence brain functions, which can have key implications in neuroscience and medicine. By means of a model-based controller which accounts for non-linearities, noise and uncertainty, neural signals can be induced on brain structures.

Macroscopic behavior of populations of quadratic integrate-and-fire neurons subject to non-Gaussian white noise

We study macroscopic behavior of populations of quadratic integrate-and-fire neurons subject to non-Gaussian noises; we argue that these noises must be α-stable whenever they are delta-correlated (white). For the case of additive-in-voltage noise, we derive the governing equation of the dynamics of the characteristic function of the membrane voltage distribution and construct a linear-in-noise perturbation theory. Specifically for the recurrent network with global synaptic coupling, we theoretically calculate the observables: population-mean membrane voltage and firing rate. The theoretical results are underpinned by the results of numerical simulation for homogeneous and heterogeneous populations. The possibility of the generalization of the pseudocumulant approach to the case of a fractional α is examined for both irrational and fractional rational α. This examination seemingly suggests the pseudocumulant approach or its modifications to be employable only for the integer values of α=1 (Cauchy noise) and 2 (Gaussian noise) within the physically meaningful range (0;2]. Remarkably, the analysis for fractional α indirectly revealed that, for the Gaussian noise, the minimal asymptotically rigorous model reduction must involve three pseudocumulants and the two-pseudocumulant model reduction is an artificial approximation. This explains a surprising gain of accuracy for the three-pseudocumulant models as compared to the two-pseudocumulant ones reported in the literature.

Bridging functional and anatomical neural connectivity through cluster synchronization

The dynamics of the brain results from the complex interplay of several neural populations and is affected by both the individual dynamics of these areas and their connection structure. Hence, a fundamental challenge is to derive models of the brain that reproduce both structural and functional features measured experimentally. Our work combines neuroimaging data, such as dMRI, which provides information on the structure of the anatomical connectomes, and fMRI, which detects patterns of approximate synchronous activity between brain areas. We employ cluster synchronization as a tool to integrate the imaging data of a subject into a coherent model, which reconciles structural and dynamic information. By using data-driven and model-based approaches, we refine the structural connectivity matrix in agreement with experimentally observed clusters of brain areas that display coherent activity. The proposed approach leverages the assumption of homogeneous brain areas; we show the robustness of this approach when heterogeneity between the brain areas is introduced in the form of noise, parameter mismatches, and connection delays. As a proof of concept, we apply this approach to MRI data of a healthy adult at resting state.

Periodic solutions in next generation neural field models

We consider a next generation neural field model which describes the dynamics of a network of theta neurons on a ring. For some parameters the network supports stable time-periodic solutions. Using the fact that the dynamics at each spatial location are described by a complex-valued Riccati equation we derive a self-consistency equation that such periodic solutions must satisfy. We determine the stability of these solutions, and present numerical results to illustrate the usefulness of this technique. The generality of this approach is demonstrated through its application to several other systems involving delays, two-population architecture and networks of Winfree oscillators.

Perspectives on adaptive dynamical systems

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems, such as the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges and give perspectives on future research directions, looking to inspire interdisciplinary approaches.

Constructive role of shot noise in the collective dynamics of neural networks

Finite-size effects may significantly influence the collective dynamics of large populations of neurons. Recently, we have shown that in globally coupled networks these effects can be interpreted as additional common noise term, the so-called shot noise, to the macroscopic dynamics unfolding in the thermodynamic limit. Here, we continue to explore the role of the shot noise in the collective dynamics of globally coupled neural networks. Namely, we study the noise-induced switching between different macroscopic regimes. We show that shot noise can turn attractors of the infinitely large network into metastable states whose lifetimes smoothly depend on the system parameters. A surprising effect is that the shot noise modifies the region where a certain macroscopic regime exists compared to the thermodynamic limit. This may be interpreted as a constructive role of the shot noise since a certain macroscopic state appears in a parameter region where it does not exist in an infinite network.

Exact finite-dimensional description for networks of globally coupled spiking neurons

We consider large networks of globally coupled spiking neurons and derive an exact low-dimensional description of their collective dynamics in the thermodynamic limit. Individual neurons are described by the Ermentrout-Kopell canonical model that can be excitable or tonically spiking and interact with other neurons via pulses. Utilizing the equivalence of the quadratic integrate-and-fire and the theta-neuron formulations, we first derive the dynamical equations in terms of the Kuramoto-Daido order parameters (Fourier modes of the phase distribution) and relate them to two biophysically relevant macroscopic observables, the firing rate and the mean voltage. For neurons driven by Cauchy white noise or for Cauchy-Lorentz distributed input currents, we adapt the results by Cestnik and Pikovsky [Chaos 32, 113126 (2022)1054-150010.1063/5.0106171] and show that for arbitrary initial conditions the collective dynamics reduces to six dimensions. We also prove that in this case the dynamics asymptotically converges to a two-dimensional invariant manifold first discovered by Ott and Antonsen. For identical, noise-free neurons, the dynamics reduces to three dimensions, becoming equivalent to the Watanabe-Strogatz description. We illustrate the exact six-dimensional dynamics outside the invariant manifold by calculating nontrivial basins of different asymptotic regimes in a bistable situation.

Shot noise in next-generation neural mass models for finite-size networks

Neural mass models is a general name for various models describing the collective dynamics of large neural populations in terms of averaged macroscopic variables. Recently, the so-called next-generation neural mass models have attracted a lot of attention due to their ability to account for the degree of synchrony. Being exact in the limit of infinitely large number of neurons, these models provide only an approximate description of finite-size networks. In the present Letter we study finite-size effects in the collective behavior of neural networks and prove that these effects can be captured by appropriately modified neural mass models. Namely, we show that the finite size of the network leads to the emergence of the so-called shot noise appearing as a stochastic term in the neural mass model. The power spectrum of this shot noise contains pronounced peaks, therefore its impact on the collective dynamics might be crucial due to resonance effects.

Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories

Populations of globally coupled phase oscillators are described in the thermodynamic limit by kinetic equations for the distribution densities or, equivalently, by infinite hierarchies of equations for the order parameters. Ott and Antonsen [Chaos 18, 037113 (2008)] have found an invariant finite-dimensional subspace on which the dynamics is described by one complex variable per population. For oscillators with Cauchy distributed frequencies or for those driven by Cauchy white noise, this subspace is weakly stable and, thus, describes the asymptotic dynamics. Here, we report on an exact finite-dimensional reduction of the dynamics outside of the Ott-Antonsen subspace. We show that the evolution from generic initial states can be reduced to that of three complex variables, plus a constant function. For identical noise-free oscillators, this reduction corresponds to the Watanabe-Strogatz system of equations [Watanabe and Strogatz, Phys. Rev. Lett. 70, 2391 (1993)]. We discuss how the reduced system can be used to explore the transient dynamics of perturbed ensembles.

Theta neuron subject to delayed feedback: a prototypical model for self-sustained pulsing

We consider a single theta neuron with delayed self-feedback in the form of a Dirac delta function in time. Because the dynamics of a theta neuron on its own can be solved explicitly—it is either excitable or shows self-pulsations—we are able to derive algebraic expressions for the existence and stability of the periodic solutions that arise in the presence of feedback. These periodic solutions are characterized by one or more equally spaced pulses per delay interval, and there is an increasing amount of multistability with increasing delay time. We present a complete description of where these self-sustained oscillations can be found in parameter space; in particular, we derive explicit expressions for the loci of their saddle-node bifurcations. We conclude that the theta neuron with delayed self-feedback emerges as a prototypical model: it provides an analytical basis for understanding pulsating dynamics observed in other excitable systems subject to delayed self-coupling.

Chimeras on annuli

Chimeras occur in networks of coupled oscillators and are characterized by the coexistence of synchronous and asynchronous groups of oscillators in different parts of the network. We consider a network of nonlocally coupled phase oscillators on an annular domain. The Ott/Antonsen ansatz is used to derive a continuum level description of the oscillators' expected dynamics in terms of a complex-valued order parameter. The equations for this order parameter are numerically analyzed in order to investigate solutions with the same symmetry as the domain and chimeras which are analogous to the "multi-headed" chimeras observed on one-dimensional domains. Such solutions are stable only for domains with widths that are neither too large nor too small. We also study rotating waves with different winding numbers, which are similar to spiral wave chimeras seen in two-dimensional domains. We determine ranges of parameters, such as the size of the domain for which such solutions exist and are stable, and the bifurcations by which they lose stability. All of these bifurcations appear subcritical.

A global bifurcation organizing rhythmic activity in a coupled network

We study a system of coupled phase oscillators near a saddle-node on invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before bifurcation from noncontractibile ones after bifurcation. Both families are stable for the model at hand.

Collective Activity Bursting in a Population of Excitable Units Adaptively Coupled to a Pool of Resources

We study the collective dynamics in a population of excitable units (neurons) adaptively interacting with a pool of resources. The resource pool is influenced by the average activity of the population, whereas the feedback from the resources to the population is comprised of components acting homogeneously or inhomogeneously on individual units of the population. Moreover, the resource pool dynamics is assumed to be slow and has an oscillatory degree of freedom. We show that the feedback loop between the population and the resources can give rise to collective activity bursting in the population. To explain the mechanisms behind this emergent phenomenon, we combine the Ott-Antonsen reduction for the collective dynamics of the population and singular perturbation theory to obtain a reduced system describing the interaction between the population mean field and the resources.

Collective states in a ring network of theta neurons

We consider a ring network of theta neurons with non-local homogeneous coupling. We analyse the corresponding continuum evolution equation, analytically describing all possible steady states and their stability. By considering a number of different parameter sets, we determine the typical bifurcation scenarios of the network, and put on a rigorous footing some previously observed numerical results.

Hierarchy of Exact Low-Dimensional Reductions for Populations of Coupled Oscillators

We consider an ensemble of phase oscillators in the thermodynamic limit, where it is described by a kinetic equation for the phase distribution density. We propose an Ansatz for the circular moments of the distribution (Kuramoto-Daido order parameters) that allows for an exact truncation at an arbitrary number of modes. In the simplest case of one mode, the Ansatz coincides with that of Ott and Antonsen [Chaos 18, 037113 (2008)CHAOEH1054-150010.1063/1.2930766]. Dynamics on the extended manifolds facilitate higher-dimensional behavior such as chaos, which we demonstrate with a simulation of a Josephson junction array. The findings are generalized for oscillators with a Cauchy-Lorentzian distribution of natural frequencies.

Bumps, chimera states, and Turing patterns in systems of coupled active rotators

Self-organized coherence-incoherence patterns, called chimera states, have first been reported in systems of Kuramoto oscillators. For coupled excitable units, similar patterns where coherent units are at rest are called bump states. Here, we study bumps in an array of active rotators coupled by nonlocal attraction and global repulsion. We demonstrate how they can emerge in a supercritical scenario from completely coherent Turing patterns: a single incoherent unit appears in a homoclinic bifurcation, undergoing subsequent transitions to quasiperiodic and chaotic behavior, which eventually transforms into extensive chaos with many incoherent units. We present different types of transitions and explain the formation of coherence-incoherence patterns according to the classical paradigm of short-range activation and long-range inhibition.

Interpolating between bumps and chimeras

A "bump" refers to a group of active neurons surrounded by quiescent ones while a "chimera" refers to a pattern in a network in which some oscillators are synchronized while the remainder are asynchronous. Both types of patterns have been studied intensively but are sometimes conflated due to their similar appearance and existence in similar types of networks. Here, we numerically study a hybrid system that linearly interpolates between a network of theta neurons that supports a bump at one extreme and a network of phase oscillators that supports a chimera at the other extreme. Using the Ott/Antonsen ansatz, we derive the equation describing the hybrid network in the limit of an infinite number of oscillators and perform bifurcation analysis on this equation. We find that neither the bump nor chimera persists over the whole range of parameters, and the hybrid system shows a variety of other states such as spatiotemporal chaos, traveling waves, and modulated traveling waves.

Mean-field models of populations of quadratic integrate-and-fire neurons with noise on the basis of the circular cumulant approach

We develop a circular cumulant representation for the recurrent network of quadratic integrate-and-fire neurons subject to noise. The synaptic coupling is global or macroscopically equivalent to it. We assume a Lorentzian distribution of the parameter controlling whether the isolated individual neuron is periodically spiking or excitable. For the infinite chain of circular cumulant equations, a hierarchy of smallness is identified; on the basis of it, we truncate the chain and suggest several two-cumulant neural mass models. These models allow one to go beyond the Ott-Antonsen Ansatz and describe the effect of noise on hysteretic transitions between macroscopic regimes of a population with inhibitory coupling. The accuracy of two-cumulant models is analyzed in detail.

Reduction Methodology for Fluctuation Driven Population Dynamics

Lorentzian distributions have been largely employed in statistical mechanics to obtain exact results for heterogeneous systems. Analytic continuation of these results is impossible even for slightly deformed Lorentzian distributions due to the divergence of all the moments (cumulants). We have solved this problem by introducing a "pseudocumulants" expansion. This allows us to develop a reduction methodology for heterogeneous spiking neural networks subject to extrinsic and endogenous fluctuations, thus obtaining a unified mean-field formulation encompassing quenched and dynamical sources of disorder.

Effects of degree distributions in random networks of type-I neurons

We consider large networks of theta neurons and use the Ott-Antonsen ansatz to derive degree-based mean-field equations governing the expected dynamics of the networks. Assuming random connectivity, we investigate the effects of varying the widths of the in- and out-degree distributions on the dynamics of excitatory or inhibitory synaptically coupled networks and gap junction coupled networks. For synaptically coupled networks, the dynamics are independent of the out-degree distribution. Broadening the in-degree distribution destroys oscillations in inhibitory networks and decreases the range of bistability in excitatory networks. For gap junction coupled neurons, broadening the degree distribution varies the values of parameters at which there is an onset of collective oscillations. Many of the results are shown to also occur in networks of more realistic neurons.

Reduction of the collective dynamics of neural populations with realistic forms of heterogeneity

Reduction of collective dynamics of large heterogeneous populations to low-dimensional mean-field models is an important task of modern theoretical neuroscience. Such models can be derived from microscopic equations, for example with the help of Ott-Antonsen theory. An often used assumption of the Lorentzian distribution of the unit parameters makes the reduction especially efficient. However, the Lorentzian distribution is often implausible as having undefined moments, and the collective behavior of populations with other distributions needs to be studied. In the present Letter we propose a method which allows efficient reduction for an arbitrary distribution and show how it performs for the Gaussian distribution. We show that a reduced system for several macroscopic complex variables provides an accurate description of a population of thousands of neurons. Using this reduction technique we demonstrate that the population dynamics depends significantly on the form of its parameter distribution. In particular, the dynamics of populations with Lorentzian and Gaussian distributions with the same center and width differ drastically.

Mean-Field Models for EEG/MEG: From Oscillations to Waves

Neural mass models have been used since the 1970s to model the coarse-grained activity of large populations of neurons. They have proven especially fruitful for understanding brain rhythms. However, although motivated by neurobiological considerations they are phenomenological in nature, and cannot hope to recreate some of the rich repertoire of responses seen in real neuronal tissue. Here we consider a simple spiking neuron network model that has recently been shown to admit an exact mean-field description for both synaptic and gap-junction interactions. The mean-field model takes a similar form to a standard neural mass model, with an additional dynamical equation to describe the evolution of within-population synchrony. As well as reviewing the origins of this next generation mass model we discuss its extension to describe an idealised spatially extended planar cortex. To emphasise the usefulness of this model for EEG/MEG modelling we show how it can be used to uncover the role of local gap-junction coupling in shaping large scale synaptic waves.

On the interpretation of Dirac δ pulses in differential equations for phase oscillators

In this note, we discuss the usage of the Dirac δ function in models of phase oscillators with pulsatile inputs. Many authors use a product of the delta function and the phase response curve in the right-hand side of an ordinary differential equation to describe the discontinuous phase dynamics in such systems. We point out that this notation has to be treated with care as it is ambiguous. We argue that the presumably most canonical interpretation does not lead to the intended behavior in many cases.

Traveling waves in non-local pulse-coupled networks

Traveling phase waves are commonly observed in recordings of the cerebral cortex and are believed to organize behavior across different areas of the brain. We use this as motivation to analyze a one-dimensional network of phase oscillators that are nonlocally coupled via the phase response curve (PRC) and the Dirac delta function. Existence of waves is proven and the dispersion relation is computed. Using the theory of distributions enables us to write and solve an associated stability problem. First and second order perturbation theory is applied to get analytic insight and we show that long waves are stable while short waves are unstable. We apply the results to PRCs that come from mitral neurons. We extend the results to smooth pulse-like coupling by reducing the nonlocal equation to a local one and solving the associated boundary value problem.

Exact neural mass model for synaptic-based working memory

A synaptic theory of Working Memory (WM) has been developed in the last decade as a possible alternative to the persistent spiking paradigm. In this context, we have developed a neural mass model able to reproduce exactly the dynamics of heterogeneous spiking neural networks encompassing realistic cellular mechanisms for short-term synaptic plasticity. This population model reproduces the macroscopic dynamics of the network in terms of the firing rate and the mean membrane potential. The latter quantity allows us to gain insight of the Local Field Potential and electroencephalographic signals measured during WM tasks to characterize the brain activity. More specifically synaptic facilitation and depression integrate each other to efficiently mimic WM operations via either synaptic reactivation or persistent activity. Memory access and loading are related to stimulus-locked transient oscillations followed by a steady-state activity in the β-γ band, thus resembling what is observed in the cortex during vibrotactile stimuli in humans and object recognition in monkeys. Memory juggling and competition emerge already by loading only two items. However more items can be stored in WM by considering neural architectures composed of multiple excitatory populations and a common inhibitory pool. Memory capacity depends strongly on the presentation rate of the items and it maximizes for an optimal frequency range. In particular we provide an analytic expression for the maximal memory capacity. Furthermore, the mean membrane potential turns out to be a suitable proxy to measure the memory load, analogously to event driven potentials in experiments on humans. Finally we show that the γ power increases with the number of loaded items, as reported in many experiments, while θ and β power reveal non monotonic behaviours. In particular, β and γ rhythms are crucially sustained by the inhibitory activity, while the θ rhythm is controlled by excitatory synapses.

Working Memory (WM) is the ability to temporarily store and manipulate stimuli representations that are no longer available to the senses. We have developed an innovative coarse-grained population model able to mimic several operations associated to WM. The novelty of the model consists in reproducing exactly the dynamics of spiking neural networks with realistic synaptic plasticity composed of hundreds of thousands of neurons in terms of a few macroscopic variables. These variables give access to experimentally measurable quantities such as local field potentials and electroencephalographic signals. Memory operations are joined to sustained or transient oscillations emerging in different frequency bands, in accordance with experimental results for primate and humans performing WM tasks. We have designed an architecture composed of many excitatory populations and a common inhibitory pool able to store and retain several memory items. The capacity of our multi-item architecture is around 3–5 items, a value similar to the WM capacities measured in many experiments. Furthermore, the maximal capacity is achievable only for presentation rates within an optimal frequency range. Finally, we have defined a measure of the memory load analogous to the event-related potentials employed to test humans’ WM capacity during visual memory tasks.

Exact Mean-Field Theory Explains the Dual Role of Electrical Synapses in Collective Synchronization

Electrical synapses play a major role in setting up neuronal synchronization, but the precise mechanisms whereby these synapses contribute to synchrony are subtle and remain elusive. To investigate these mechanisms mean-field theories for quadratic integrate-and-fire neurons with electrical synapses have been recently put forward. Still, the validity of these theories is controversial since they assume that the neurons produce unrealistic, symmetric spikes, ignoring the well-known impact of spike shape on synchronization. Here, we show that the assumption of symmetric spikes can be relaxed in such theories. The resulting mean-field equations reveal a dual role of electrical synapses: First, they equalize membrane potentials favoring the emergence of synchrony. Second, electrical synapses act as "virtual chemical synapses," which can be either excitatory or inhibitory depending upon the spike shape. Our results offer a precise mathematical explanation of the intricate effect of electrical synapses in collective synchronization. This reconciles previous theoretical and numerical works, and confirms the suitability of recent low-dimensional mean-field theories to investigate electrically coupled neuronal networks.

The Winfree model with non-infinitesimal phase-response curve: Ott-Antonsen theory

A novel generalization of the Winfree model of globally coupled phase oscillators, representing phase reduction under finite coupling, is studied analytically. We consider interactions through a non-infinitesimal (or finite) phase-response curve (PRC), in contrast to the infinitesimal PRC of the original model. For a family of non-infinitesimal PRCs, the global dynamics is captured by one complex-valued ordinary differential equation resorting to the Ott-Antonsen ansatz. The phase diagrams are thereupon obtained for four illustrative cases of non-infinitesimal PRC. Bistability between collective synchronization and full desynchronization is observed in all cases.

The effects of within-neuron degree correlations in networks of spiking neurons

We consider the effects of correlations between the in- and out-degrees of individual neurons on the dynamics of a network of neurons. By using theta neurons, we can derive a set of coupled differential equations for the expected dynamics of neurons with the same in-degree. A Gaussian copula is used to introduce correlations between a neuron's in- and out-degree, and numerical bifurcation analysis is used determine the effects of these correlations on the network's dynamics. For excitatory coupling, we find that inducing positive correlations has a similar effect to increasing the coupling strength between neurons, while for inhibitory coupling it has the opposite effect. We also determine the propensity of various two- and three-neuron motifs to occur as correlations are varied and give a plausible explanation for the observed changes in dynamics.

Theta-Nested Gamma Oscillations in Next Generation Neural Mass Models

Theta-nested gamma oscillations have been reported in many areas of the brain and are believed to represent a fundamental mechanism to transfer information across spatial and temporal scales. In a series of recent experiments in vitro it has been possible to replicate with an optogenetic theta frequency stimulation several features of cross-frequency coupling (CFC) among theta and gamma rhythms observed in behaving animals. In order to reproduce the main findings of these experiments we have considered a new class of neural mass models able to reproduce exactly the macroscopic dynamics of spiking neural networks. In this framework, we have examined two set-ups able to support collective gamma oscillations: namely, the pyramidal interneuronal network gamma (PING) and the interneuronal network gamma (ING). In both set-ups we observe the emergence of theta-nested gamma oscillations by driving the system with a sinusoidal theta-forcing in proximity of a Hopf bifurcation. These mixed rhythms always display phase amplitude coupling. However, two different types of nested oscillations can be identified: one characterized by a perfect phase locking between theta and gamma rhythms, corresponding to an overall periodic behavior; another one where the locking is imperfect and the dynamics is quasi-periodic or even chaotic. From our analysis it emerges that the locked states are more frequent in the ING set-up. In agreement with the experiments, we find theta-nested gamma oscillations for forcing frequencies in the range [1:10] Hz, whose amplitudes grow proportionally to the forcing intensity and which are clearly modulated by the theta phase. Furthermore, analogously to the experiments, the gamma power and the frequency of the gamma-power peak increase with the forcing amplitude. At variance with experimental findings, the gamma-power peak does not shift to higher frequencies by increasing the theta frequency. This effect can be obtained, in our model, only by incrementing, at the same time, also the stimulation power. An effect achieved by increasing the amplitude either of the noise or of the forcing term proportionally to the theta frequency. On the basis of our analysis both the PING and the ING mechanism give rise to theta-nested gamma oscillations with almost identical features.

Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review

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.

Synaptic Diversity Suppresses Complex Collective Behavior in Networks of Theta Neurons

Comprehending how the brain functions requires an understanding of the dynamics of neuronal assemblies. Previous work used a mean-field reduction method to determine the collective dynamics of a large heterogeneous network of uniformly and globally coupled theta neurons, which are a canonical formulation of Type I neurons. However, in modeling neuronal networks, it is unreasonable to assume that the coupling strength between every pair of neurons is identical. The goal in the present work is to analytically examine the collective macroscopic behavior of a network of theta neurons that is more realistic in that it includes heterogeneity in the coupling strength as well as in neuronal excitability. We consider the occurrence of dynamical structures that give rise to complicated dynamics via bifurcations of macroscopic collective quantities, concentrating on two biophysically relevant cases: (1) predominantly excitable neurons with mostly excitatory connections, and (2) predominantly spiking neurons with inhibitory connections. We find that increasing the synaptic diversity moves these dynamical structures to distant extremes of parameter space, leaving simple collective equilibrium states in the physiologically relevant region. We also study the node vs. focus nature of stable macroscopic equilibrium solutions and discuss our results in the context of recent literature.

Collective in-plane magnetization in a two-dimensional XY macrospin system within the framework of generalized Ott-Antonsen theory

The problem of magnetic transitions between the low-temperature (macrospin ordered) phases in two-dimensional XY arrays is addressed. The system is modelled as a plane structure of identical single-domain particles arranged in a square lattice and coupled by the magnetic dipole-dipole interaction; all the particles possess a strong easy-plane magnetic anisotropy. The basic state of the system in the considered temperature range is an antiferromagnetic (AF) stripe structure, where the macrospins (particle magnetic moments) are still involved in thermofluctuational motion: the superparamagnetic blocking T_b temperature is lower than that (T_af) of the AF transition. The description is based on the stochastic equations governing the dynamics of individual magnetic moments, where the interparticle interaction is added in the mean-field approximation. With the technique of a generalized Ott-Antonsen theory, the dynamics equations for the order parameters (including the macroscopic magnetization and the AF order parameter) and the partition function of the system are rigorously obtained and analysed. We show that inside the temperature interval of existence of the AF phase, a static external field tilted to the plane of the array is able to induce first-order phase transitions from AF to ferromagnetic state; the phase diagrams displaying stable and metastable regions of the system are presented. This article is part of the theme issue 'Patterns in soft and biological matters'.

Exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks

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.

Moving bumps in theta neuron networks

We consider large networks of theta neurons on a ring, synaptically coupled with an asymmetric kernel. Such networks support stable "bumps" of activity, which move along the ring if the coupling kernel is asymmetric. We investigate the effects of the kernel asymmetry on the existence, stability, and speed of these moving bumps using continuum equations formally describing infinite networks. Depending on the level of heterogeneity within the network, we find complex sequences of bifurcations as the amount of asymmetry is varied, in strong contrast to the behavior of a classical neural field model.

Bumps and oscillons in networks of spiking neurons

We study localized patterns in an exact mean-field description of a spatially extended network of quadratic integrate-and-fire neurons. We investigate conditions for the existence and stability of localized solutions, so-called bumps, and give an analytic estimate for the parameter range, where these solutions exist in parameter space, when one or more microscopic network parameters are varied. We develop Galerkin methods for the model equations, which enable numerical bifurcation analysis of stationary and time-periodic spatially extended solutions. We study the emergence of patterns composed of multiple bumps, which are arranged in a snake-and-ladder bifurcation structure if a homogeneous or heterogeneous synaptic kernel is suitably chosen. Furthermore, we examine time-periodic, spatially localized solutions (oscillons) in the presence of external forcing, and in autonomous, recurrently coupled excitatory and inhibitory networks. In both cases, we observe period-doubling cascades leading to chaotic oscillations.

Noise-induced macroscopic oscillations in a network of synaptically coupled quadratic integrate-and-fire neurons

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.

Predicting the effects of deep brain stimulation using a reduced coupled oscillator model

Deep brain stimulation (DBS) is known to be an effective treatment for a variety of neurological disorders, including Parkinson's disease and essential tremor (ET). At present, it involves administering a train of pulses with constant frequency via electrodes implanted into the brain. New 'closed-loop' approaches involve delivering stimulation according to the ongoing symptoms or brain activity and have the potential to provide improvements in terms of efficiency, efficacy and reduction of side effects. The success of closed-loop DBS depends on being able to devise a stimulation strategy that minimizes oscillations in neural activity associated with symptoms of motor disorders. A useful stepping stone towards this is to construct a mathematical model, which can describe how the brain oscillations should change when stimulation is applied at a particular state of the system. Our work focuses on the use of coupled oscillators to represent neurons in areas generating pathological oscillations. Using a reduced form of the Kuramoto model, we analyse how a patient should respond to stimulation when neural oscillations have a given phase and amplitude, provided a number of conditions are satisfied. For such patients, we predict that the best stimulation strategy should be phase specific but also that stimulation should have a greater effect if applied when the amplitude of brain oscillations is lower. We compare this surprising prediction with data obtained from ET patients. In light of our predictions, we also propose a new hybrid strategy which effectively combines two of the closed-loop strategies found in the literature, namely phase-locked and adaptive DBS.

Firing rate models for gamma oscillations

Gamma oscillations are readily observed in a variety of brain regions during both waking and sleeping states. Computational models of gamma oscillations typically involve simulations of large networks of synaptically coupled spiking units. These networks can exhibit strongly synchronized gamma behavior, whereby neurons fire in near synchrony on every cycle, or weakly modulated gamma behavior, corresponding to stochastic, sparse firing of the individual units on each cycle of the population gamma rhythm. These spiking models offer valuable biophysical descriptions of gamma oscillations; however, because they involve many individual neuronal units they are limited in their ability to communicate general network-level dynamics. Here we demonstrate that few-variable firing rate models with established synaptic timescales can account for both strongly synchronized and weakly modulated gamma oscillations. These models go beyond the classical formulations of rate models by including at least two dynamic variables per population: firing rate and synaptic activation. The models' flexibility to capture the broad range of gamma behavior depends directly on the timescales that represent recruitment of the excitatory and inhibitory firing rates. In particular, we find that weakly modulated gamma oscillations occur robustly when the recruitment timescale of inhibition is faster than that of excitation. We present our findings by using an extended Wilson-Cowan model and a rate model derived from a network of quadratic integrate-and-fire neurons. These biophysical rate models capture the range of weakly modulated and coherent gamma oscillations observed in spiking network models, while additionally allowing for greater tractability and systems analysis. NEW & NOTEWORTHY Here we develop simple and tractable models of gamma oscillations, a dynamic feature observed throughout much of the brain with significant correlates to behavior and cognitive performance in a variety of experimental contexts. Our models depend on only a few dynamic variables per population, but despite this they qualitatively capture features observed in previous biophysical models of gamma oscillations that involve many individual spiking units.

Network mechanisms underlying the role of oscillations in cognitive tasks

Oscillatory activity robustly correlates with task demands during many cognitive tasks. However, not only are the network mechanisms underlying the generation of these rhythms poorly understood, but it is also still unknown to what extent they may play a functional role, as opposed to being a mere epiphenomenon. Here we study the mechanisms underlying the influence of oscillatory drive on network dynamics related to cognitive processing in simple working memory (WM), and memory recall tasks. Specifically, we investigate how the frequency of oscillatory input interacts with the intrinsic dynamics in networks of recurrently coupled spiking neurons to cause changes of state: the neuronal correlates of the corresponding cognitive process. We find that slow oscillations, in the delta and theta band, are effective in activating network states associated with memory recall. On the other hand, faster oscillations, in the beta range, can serve to clear memory states by resonantly driving transient bouts of spike synchrony which destabilize the activity. We leverage a recently derived set of exact mean-field equations for networks of quadratic integrate-and-fire neurons to systematically study the bifurcation structure in the periodically forced spiking network. Interestingly, we find that the oscillatory signals which are most effective in allowing flexible switching between network states are not smooth, pure sinusoids, but rather burst-like, with a sharp onset. We show that such periodic bursts themselves readily arise spontaneously in networks of excitatory and inhibitory neurons, and that the burst frequency can be tuned via changes in tonic drive. Finally, we show that oscillations in the gamma range can actually stabilize WM states which otherwise would not persist.

A multiple timescales approach to bridging spiking- and population-level dynamics

A rigorous bridge between spiking-level and macroscopic quantities is an on-going and well-developed story for asynchronously firing neurons, but focus has shifted to include neural populations exhibiting varying synchronous dynamics. Recent literature has used the Ott-Antonsen ansatz (2008) to great effect, allowing a rigorous derivation of an order parameter for large oscillator populations. The ansatz has been successfully applied using several models including networks of Kuramoto oscillators, theta models, and integrate-and-fire neurons, along with many types of network topologies. In the present study, we take a converse approach: given the mean field dynamics of slow synapses, we predict the synchronization properties of finite neural populations. The slow synapse assumption is amenable to averaging theory and the method of multiple timescales. Our proposed theory applies to two heterogeneous populations of N excitatory n-dimensional and N inhibitory m-dimensional oscillators with homogeneous synaptic weights. We then demonstrate our theory using two examples. In the first example, we take a network of excitatory and inhibitory theta neurons and consider the case with and without heterogeneous inputs. In the second example, we use Traub models with calcium for the excitatory neurons and Wang-Buzsáki models for the inhibitory neurons. We accurately predict phase drift and phase locking in each example even when the slow synapses exhibit non-trivial mean-field dynamics.

Relationship between the mechanisms of gamma rhythm generation and the magnitude of the macroscopic phase response function in a population of excitatory and inhibitory modified quadratic integrate-and-fire neurons

Gamma oscillations are thought to play an important role in brain function. Interneuron gamma (ING) and pyramidal interneuron gamma (PING) mechanisms have been proposed as generation mechanisms for these oscillations. However, the relation between the generation mechanisms and the dynamical properties of the gamma oscillation are still unclear. Among the dynamical properties of the gamma oscillation, the phase response function (PRF) is important because it encodes the response of the oscillation to inputs. Recently, the PRF for an inhibitory population of modified theta neurons that generate an ING rhythm was computed by the adjoint method applied to the associated Fokker-Planck equation (FPE) for the model. The modified theta model incorporates conductance-based synapses as well as the voltage and current dynamics. Here, we extended this previous work by creating an excitatory-inhibitory (E-I) network using the modified theta model and described the population dynamics with the corresponding FPE. We conducted a bifurcation analysis of the FPE to find parameter regions which generate gamma oscillations. In order to label the oscillatory parameter regions by their generation mechanisms, we defined ING- and PING-type gamma oscillation in a mathematically plausible way based on the driver of the inhibitory population. We labeled the oscillatory parameter regions by these generation mechanisms and derived PRFs via the adjoint method on the FPE in order to investigate the differences in the responses of each type of oscillation to inputs. PRFs for PING and ING mechanisms are derived and compared. We found the amplitude of the PRF for the excitatory population is larger in the PING case than in the ING case. Finally, the E-I population of the modified theta neuron enabled us to analyze the PRFs of PING-type gamma oscillation and the entrainment ability of E and I populations. We found a parameter region in which PRFs of E and I are both purely positive in the case of PING oscillations. The different entrainment abilities of E and I stimulation as governed by the respective PRFs was compared to direct simulations of finite populations of model neurons. We find that it is easier to entrain the gamma rhythm by stimulating the inhibitory population than by stimulating the excitatory population as has been found experimentally.

Dynamics of Noisy Oscillator Populations beyond the Ott-Antonsen Ansatz

We develop an approach for the description of the dynamics of large populations of phase oscillators based on "circular cumulants" instead of the Kuramoto-Daido order parameters. In the thermodynamic limit, these variables yield a simple representation of the Ott-Antonsen invariant solution [E. Ott and T. M. Antonsen, Chaos 18, 037113 (2008)CHAOEH1054-150010.1063/1.2930766] and appear appropriate for constructing perturbation theory on top of the Ott-Antonsen ansatz. We employ this approach to study the impact of small intrinsic noise on the dynamics. As a result, a closed system of equations for the two leading cumulants, describing the dynamics of noisy ensembles, is derived. We exemplify the general theory by presenting the effect of noise on the Kuramoto system and on a chimera state in two symmetrically coupled populations.