Bifurcations of large networks of two-dimensional integrate and fire neurons

Anti-Disturbance of Scale-Free Spiking Neural Network against Impulse Noise

The bio-brain presents robustness function to external stimulus through its self-adaptive regulation and neural information processing. Drawing from the advantages of the bio-brain to investigate the robustness function of a spiking neural network (SNN) is conducive to the advance of brain-like intelligence. However, the current brain-like model is insufficient in biological rationality. In addition, its evaluation method for anti-disturbance performance is inadequate. To explore the self-adaptive regulation performance of a brain-like model with more biological rationality under external noise, a scale-free spiking neural network(SFSNN) is constructed in this study. Then, the anti-disturbance ability of the SFSNN against impulse noise is investigated, and the anti-disturbance mechanism is further discussed. Our simulation results indicate that: (i) our SFSNN has anti-disturbance ability against impulse noise, and the high-clustering SFSNN outperforms the low-clustering SFSNN in terms of anti-disturbance performance. (ii) The neural information processing in the SFSNN under external noise is clarified, which is a dynamic chain effect of the neuron firing, the synaptic weight, and the topological characteristic. (iii) Our discussion hints that an intrinsic factor of the anti-disturbance ability is the synaptic plasticity, and the network topology is a factor that affects the anti-disturbance ability at the level of performance.

Macroscopic dynamics of neural networks with heterogeneous spiking thresholds

Mean-field theory links the physiological properties of individual neurons to the emergent dynamics of neural population activity. These models provide an essential tool for studying brain function at different scales; however, for their application to neural populations on large scale, they need to account for differences between distinct neuron types. The Izhikevich single neuron model can account for a broad range of different neuron types and spiking patterns, thus rendering it an optimal candidate for a mean-field theoretic treatment of brain dynamics in heterogeneous networks. Here we derive the mean-field equations for networks of all-to-all coupled Izhikevich neurons with heterogeneous spiking thresholds. Using methods from bifurcation theory, we examine the conditions under which the mean-field theory accurately predicts the dynamics of the Izhikevich neuron network. To this end, we focus on three important features of the Izhikevich model that are subject here to simplifying assumptions: (i) spike-frequency adaptation, (ii) the spike reset conditions, and (iii) the distribution of single-cell spike thresholds across neurons. Our results indicate that, while the mean-field model is not an exact model of the Izhikevich network dynamics, it faithfully captures its different dynamic regimes and phase transitions. We thus present a mean-field model that can represent different neuron types and spiking dynamics. The model comprises biophysical state variables and parameters, incorporates realistic spike resetting conditions, and accounts for heterogeneity in neural spiking thresholds. These features allow for a broad applicability of the model as well as for a direct comparison to experimental data.

Exact mean-field models for spiking neural networks with adaptation

Networks of spiking neurons with adaption have been shown to be able to reproduce a wide range of neural activities, including the emergent population bursting and spike synchrony that underpin brain disorders and normal function. Exact mean-field models derived from spiking neural networks are extremely valuable, as such models can be used to determine how individual neurons and the network they reside within interact to produce macroscopic network behaviours. In the paper, we derive and analyze a set of exact mean-field equations for the neural network with spike frequency adaptation. Specifically, our model is a network of Izhikevich neurons, where each neuron is modeled by a two dimensional system consisting of a quadratic integrate and fire equation plus an equation which implements spike frequency adaptation. Previous work deriving a mean-field model for this type of network, relied on the assumption of sufficiently slow dynamics of the adaptation variable. However, this approximation did not succeed in establishing an exact correspondence between the macroscopic description and the realistic neural network, especially when the adaptation time constant was not large. The challenge lies in how to achieve a closed set of mean-field equations with the inclusion of the mean-field dynamics of the adaptation variable. We address this problem by using a Lorentzian ansatz combined with the moment closure approach to arrive at a mean-field system in the thermodynamic limit. The resulting macroscopic description is capable of qualitatively and quantitatively describing the collective dynamics of the neural network, including transition between states where the individual neurons exhibit asynchronous tonic firing and synchronous bursting. We extend the approach to a network of two populations of neurons and discuss the accuracy and efficacy of our mean-field approximations by examining all assumptions that are imposed during the derivation. Numerical bifurcation analysis of our mean-field models reveals bifurcations not previously observed in the models, including a novel mechanism for emergence of bursting in the network. We anticipate our results will provide a tractable and reliable tool to investigate the underlying mechanism of brain function and dysfunction from the perspective of computational neuroscience.

A Rate-Reduced Neuron Model for Complex Spiking Behavior

We present a simple rate-reduced neuron model that captures a wide range of complex, biologically plausible, and physiologically relevant spiking behavior. This includes spike-frequency adaptation, postinhibitory rebound, phasic spiking and accommodation, first-spike latency, and inhibition-induced spiking. Furthermore, the model can mimic different neuronal filter properties. It can be used to extend existing neural field models, adding more biological realism and yielding a richer dynamical structure. The model is based on a slight variation of the Rulkov map.

Interaction between synaptic inhibition and glial-potassium dynamics leads to diverse seizure transition modes in biophysical models of human focal seizures

How focal seizures initiate and evolve in human neocortex remains a fundamental problem in neuroscience. Here, we use biophysical neuronal network models of neocortical patches to study how the interaction between inhibition and extracellular potassium ([K (+)] o ) dynamics may contribute to different types of focal seizures. Three main types of propagated focal seizures observed in recent intracortical microelectrode recordings in humans were modelled: seizures characterized by sustained (∼30-60 Hz) gamma local field potential (LFP) oscillations; seizures where the onset in the propagated site consisted of LFP spikes that later evolved into rhythmic (∼2-3 Hz) spike-wave complexes (SWCs); and seizures where a brief stage of low-amplitude fast-oscillation (∼10-20 Hz) LFPs preceded the SWC activity. Our findings are fourfold: (1) The interaction between elevated [K (+)] o (due to abnormal potassium buffering by glial cells) and the strength of synaptic inhibition plays a predominant role in shaping these three types of seizures. (2) Strengthening of inhibition leads to the onset of sustained narrowband gamma seizures. (3) Transition into SWC seizures is obtained either by the weakening of inhibitory synapses, or by a transient strengthening followed by an inhibitory breakdown (e.g. GABA depletion). This reduction or breakdown of inhibition among fast-spiking (FS) inhibitory interneurons increases their spiking activity and leads them eventually into depolarization block. Ictal spike-wave discharges in the model are then sustained solely by pyramidal neurons. (4) FS cell dynamics are also critical for seizures where the evolution into SWC activity is preceded by low-amplitude fast oscillations. Different levels of elevated [K (+)] o were important for transitions into and maintenance of sustained gamma oscillations and SWC discharges. Overall, our modelling study predicts that the interaction between inhibitory interneurons and [K (+)] o glial buffering under abnormal conditions may explain different types of ictal transitions and dynamics during propagated seizures in human focal epilepsy.

Obtaining Arbitrary Prescribed Mean Field Dynamics for Recurrently Coupled Networks of Type-I Spiking Neurons with Analytically Determined Weights

A fundamental question in computational neuroscience is how to connect a network of spiking neurons to produce desired macroscopic or mean field dynamics. One possible approach is through the Neural Engineering Framework (NEF). The NEF approach requires quantities called decoders which are solved through an optimization problem requiring large matrix inversion. Here, we show how a decoder can be obtained analytically for type I and certain type II firing rates as a function of the heterogeneity of its associated neuron. These decoders generate approximants for functions that converge to the desired function in mean-squared error like 1/N, where N is the number of neurons in the network. We refer to these decoders as scale-invariant decoders due to their structure. These decoders generate weights for a network of neurons through the NEF formula for weights. These weights force the spiking network to have arbitrary and prescribed mean field dynamics. The weights generated with scale-invariant decoders all lie on low dimensional hypersurfaces asymptotically. We demonstrate the applicability of these scale-invariant decoders and weight surfaces by constructing networks of spiking theta neurons that replicate the dynamics of various well known dynamical systems such as the neural integrator, Van der Pol system and the Lorenz system. As these decoders are analytically determined and non-unique, the weights are also analytically determined and non-unique. We discuss the implications for measured weights of neuronal networks.

Analytical approximations of the firing rate of an adaptive exponential integrate-and-fire neuron in the presence of synaptic noise

Computational models offer a unique tool for understanding the network-dynamical mechanisms which mediate between physiological and biophysical properties, and behavioral function. A traditional challenge in computational neuroscience is, however, that simple neuronal models which can be studied analytically fail to reproduce the diversity of electrophysiological behaviors seen in real neurons, while detailed neuronal models which do reproduce such diversity are intractable analytically and computationally expensive. A number of intermediate models have been proposed whose aim is to capture the diversity of firing behaviors and spike times of real neurons while entailing the simplest possible mathematical description. One such model is the exponential integrate-and-fire neuron with spike rate adaptation (aEIF) which consists of two differential equations for the membrane potential (V) and an adaptation current (w). Despite its simplicity, it can reproduce a wide variety of physiologically observed spiking patterns, can be fit to physiological recordings quantitatively, and, once done so, is able to predict spike times on traces not used for model fitting. Here we compute the steady-state firing rate of aEIF in the presence of Gaussian synaptic noise, using two approaches. The first approach is based on the 2-dimensional Fokker-Planck equation that describes the (V,w)-probability distribution, which is solved using an expansion in the ratio between the time constants of the two variables. The second is based on the firing rate of the EIF model, which is averaged over the distribution of the w variable. These analytically derived closed-form expressions were tested on simulations from a large variety of model cells quantitatively fitted to in vitro electrophysiological recordings from pyramidal cells and interneurons. Theoretical predictions closely agreed with the firing rate of the simulated cells fed with in-vivo-like synaptic noise.

Simple, biologically-constrained CA1 pyramidal cell models using an intact, whole hippocampus context

The hippocampus is a heavily studied brain structure due to its involvement in learning and memory. Detailed models of excitatory, pyramidal cells in hippocampus have been developed using a range of experimental data. These models have been used to help us understand, for example, the effects of synaptic integration and voltage gated channel densities and distributions on cellular responses. However, these cellular outputs need to be considered from the perspective of the networks in which they are embedded. Using modeling approaches, if cellular representations are too detailed, it quickly becomes computationally unwieldy to explore large network simulations. Thus, simple models are preferable, but at the same time they need to have a clear, experimental basis so as to allow physiologically based understandings to emerge. In this article, we describe the development of simple models of CA1 pyramidal cells, as derived in a well-defined experimental context of an intact, whole hippocampus preparation expressing population oscillations. These models are based on the intrinsic properties and frequency-current profiles of CA1 pyramidal cells, and can be used to build, fully examine, and analyze large networks.