Mean-field equations for neural populations with q-Gaussian heterogeneities

Biologically realistic mean field model of spiking neural networks with fast and slow inhibitory synapses

We present a mean field model for a spiking neural network of excitatory and inhibitory neurons with fast GABAA and nonlinear slow GABAB inhibitory conductance-based synapses. This mean field model can predict the spontaneous and evoked response of the network to external stimulation in asynchronous irregular regimes. The model displays theta oscillations for sufficiently strong GABAB conductance. Optogenetic activation of interneurons and an increase of GABAB conductance caused opposite effects on the emergence of gamma oscillations in the model. In agreement with direct numerical simulations of neural networks and experimental data, the mean field model predicts that an increase of GABAB conductance reduces gamma oscillations. Furthermore, the slow dynamics of GABAB synapses regulates the appearance and duration of transient gamma oscillations, namely gamma bursts, in the mean field model. Finally, we show that nonlinear GABAB synapses play a major role to stabilize the network from the emergence of epileptic seizures.

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.

Bursting gamma oscillations in neural mass models

Gamma oscillations (30-120 Hz) in the brain are not periodic cycles, but they typically appear in short-time windows, often called oscillatory bursts. While the origin of this bursting phenomenon is still unclear, some recent studies hypothesize its origin in the external or endogenous noise of neural networks. We demonstrate that an exact neural mass model of excitatory and inhibitory quadratic-integrate and fire-spiking neurons theoretically predicts the emergence of a different regime of intrinsic bursting gamma (IBG) oscillations without any noise source, a phenomenon due to collective chaos. This regime is indeed observed in the direct simulation of spiking neurons, characterized by highly irregular spiking activity. IBG oscillations are distinguished by higher phase-amplitude coupling to slower theta oscillations concerning noise-induced bursting oscillations, thus indicating an increased capacity for information transfer between brain regions. We demonstrate that this phenomenon is present in both globally coupled and sparse networks of spiking neurons. These results propose a new mechanism for gamma oscillatory activity, suggesting deterministic collective chaos as a good candidate for the origin of gamma bursts.

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.

Dynamics of Oscillator Populations Globally Coupled with Distributed Phase Shifts

We consider a population of globally coupled oscillators in which phase shifts in the coupling are random. We show that in the maximally disordered case, where the pairwise shifts are independent identically distributed random variables, the dynamics of a large population reduces to one without randomness in the shifts but with an effective coupling function, which is a convolution of the original coupling function with the distribution of the phase shifts. This result is valid for noisy oscillators and/or in the presence of a distribution of natural frequencies. We argue also, using the property of global asymptotic stability, that this reduction is valid in a partially disordered case, where random phase shifts are attributed to the forced units only. However, the reduction to an effective coupling in the partially disordered noise-free situation may fail if the coupling function is complex enough to ensure the multistability of locked states.

Neural heterogeneity controls computations in spiking neural networks

The brain is composed of complex networks of interacting neurons that express considerable heterogeneity in their physiology and spiking characteristics. How does this neural heterogeneity influence macroscopic neural dynamics, and how might it contribute to neural computation? In this work, we use a mean-field model to investigate computation in heterogeneous neural networks, by studying how the heterogeneity of cell spiking thresholds affects three key computational functions of a neural population: the gating, encoding, and decoding of neural signals. Our results suggest that heterogeneity serves different computational functions in different cell types. In inhibitory interneurons, varying the degree of spike threshold heterogeneity allows them to gate the propagation of neural signals in a reciprocally coupled excitatory population. Whereas homogeneous interneurons impose synchronized dynamics that narrow the dynamic repertoire of the excitatory neurons, heterogeneous interneurons act as an inhibitory offset while preserving excitatory neuron function. Spike threshold heterogeneity also controls the entrainment properties of neural networks to periodic input, thus affecting the temporal gating of synaptic inputs. Among excitatory neurons, heterogeneity increases the dimensionality of neural dynamics, improving the network's capacity to perform decoding tasks. Conversely, homogeneous networks suffer in their capacity for function generation, but excel at encoding signals via multistable dynamic regimes. Drawing from these findings, we propose intra-cell-type heterogeneity as a mechanism for sculpting the computational properties of local circuits of excitatory and inhibitory spiking neurons, permitting the same canonical microcircuit to be tuned for diverse computational tasks.

Exact low-dimensional description for fast neural oscillations with low firing rates

Recently, low-dimensional models of neuronal activity have been exactly derived for large networks of deterministic, quadratic integrate-and-fire (QIF) neurons. Such firing rate models (FRM) describe the emergence of fast collective oscillations (>30 Hz) via the frequency locking of a subset of neurons to the global oscillation frequency. However, the suitability of such models to describe realistic neuronal states is seriously challenged by the fact that during episodes of fast collective oscillations, neuronal discharges are often very irregular and have low firing rates compared to the global oscillation frequency. Here we extend the theory to derive exact FRM for QIF neurons to include noise and show that networks of stochastic neurons displaying irregular discharges at low firing rates during episodes of fast oscillations are governed by exactly the same evolution equations as deterministic networks. Our results reconcile two traditionally confronted views on neuronal synchronization and upgrade the applicability of exact FRM to describe a broad range of biologically realistic neuronal states.

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.

Introduction to Focus Issue: Dynamics of oscillator populations

Even after about 50 years of intensive research, the dynamics of oscillator populations remain one of the most popular topics in nonlinear science. This Focus Issue brings together studies on such diverse aspects of the problem as low-dimensional description, effects of noise and disorder on synchronization transition, control of synchrony, the emergence of chimera states and chaotic regimes, stability of power grids, etc.