The population firing rate in the presence of GABAergic tonic inhibition in single neurons and application to general anaesthesia

Tonic inhibition has been found experimentally in single neurons and affects the activity of neural populations. This kind of inhibition is supposed to set the background or resting level of neural activity and plays a role in the brains arousal system, e.g. during general anaesthesia. The work shows how to involve tonic inhibition in population rate-coding models by deriving a novel transfer function. The analytical and numerical study of the novel transfer function reveals the impact of tonic inhibition on the population firing rate. Finally, a first application to a recent neural field model for general anaesthesia discusses the origin of the loss of consciousness during anaesthesia.     

Shunting inhibition modulates neuronal gain during synaptic excitation

Neuronal gain control is important for processing information in the brain. Shunting inhibition is not thought to control gain since it shifts input-output relationships during tonic excitation rather than changing their slope. Here we show that tonic inhibition reduces the gain and shifts the offset of cerebellar granule cell input-output relationships during frequency-dependent excitation with synaptic conductance waveforms. Shunting inhibition scales subthreshold voltage, increasing the excitation frequency required to attain a particular firing rate. This reduces gain because frequency-dependent increases in input variability, which couple mean subthreshold voltage to firing rate, boost voltage fluctuations during inhibition. Moreover, synaptic time course and the number of inputs also influence gain changes by setting excitation variability. Our results suggest that shunting inhibition can multiplicatively scale rate-coded information in neurons with high-variability synaptic inputs.     

A neural mass model based on single cell dynamics to model pathophysiology

Neural mass models are successful in modeling brain rhythms as observed in macroscopic measurements such as the electroencephalogram (EEG). While the synaptic current is explicitly modeled in current models, the single cell electrophysiology is not taken into account. To allow for investigations of the effects of channel pathologies, channel blockers and ion concentrations on macroscopic activity, we formulate neural mass equations explicitly incorporating the single cell dynamics by using a bottom-up approach. The mean and variance of the firing rate and synaptic input distributions are modeled. The firing rate curve (F(I)-curve) is used as link between the single cell and macroscopic dynamics. We show that this model accurately reproduces the behavior of two populations of synaptically connected Hodgkin-Huxley neurons, also in non-steady state.     

Mechanisms of Gain Control by Voltage-Gated Channels in Intrinsically-Firing Neurons

Gain modulation is a key feature of neural information processing, but underlying mechanisms remain unclear. In single neurons, gain can be measured as the slope of the current-frequency (input-output) relationship over any given range of inputs. While much work has focused on the control of basal firing rates and spike rate adaptation, gain control has been relatively unstudied. Of the limited studies on gain control, some have examined the roles of synaptic noise and passive somatic currents, but the roles of voltage-gated channels present ubiquitously in neurons have been less explored. Here, we systematically examined the relationship between gain and voltage-gated ion channels in a conductance-based, tonically-active, model neuron. Changes in expression (conductance density) of voltage-gated channels increased (Ca²⁺ channel), reduced (K⁺ channels), or produced little effect (h-type channel) on gain. We found that the gain-controlling ability of channels increased exponentially with the steepness of their activation within the dynamic voltage window (voltage range associated with firing). For depolarization-activated channels, this produced a greater channel current per action potential at higher firing rates. This allowed these channels to modulate gain by contributing to firing preferentially at states of higher excitation. A finer analysis of the current-voltage relationship during tonic firing identified narrow voltage windows at which the gain-modulating channels exerted their effects. As a proof of concept, we show that h-type channels can be tuned to modulate gain by changing the steepness of their activation within the dynamic voltage window. These results show how the impact of an ion channel on gain can be predicted from the relationship between channel kinetics and the membrane potential during firing. This is potentially relevant to understanding input-output scaling in a wide class of neurons found throughout the brain and other nervous systems.     

Intrinsic gain modulation and adaptive neural coding

In many cases, the computation of a neural system can be reduced to a receptive field, or a set of linear filters, and a thresholding function, or gain curve, which determines the firing probability; this is known as a linear/nonlinear model. In some forms of sensory adaptation, these linear filters and gain curve adjust very rapidly to changes in the variance of a randomly varying driving input. An apparently similar but previously unrelated issue is the observation of gain control by background noise in cortical neurons: the slope of the firing rate versus current (f-I) curve changes with the variance of background random input. Here, we show a direct correspondence between these two observations by relating variance-dependent changes in the gain of f-I curves to characteristics of the changing empirical linear/nonlinear model obtained by sampling. In the case that the underlying system is fixed, we derive relationships relating the change of the gain with respect to both mean and variance with the receptive fields derived from reverse correlation on a white noise stimulus. Using two conductance-based model neurons that display distinct gain modulation properties through a simple change in parameters, we show that coding properties of both these models quantitatively satisfy the predicted relationships. Our results describe how both variance-dependent gain modulation and adaptive neural computation result from intrinsic nonlinearity.     

Mechanisms Explaining Transitions between Tonic and Phasic Firing in Neuronal Populations as Predicted by a Low Dimensional Firing Rate Model

Several firing patterns experimentally observed in neural populations have been successfully correlated to animal behavior. Population bursting, hereby regarded as a period of high firing rate followed by a period of quiescence, is typically observed in groups of neurons during behavior. Biophysical membrane-potential models of single cell bursting involve at least three equations. Extending such models to study the collective behavior of neural populations involves thousands of equations and can be very expensive computationally. For this reason, low dimensional population models that capture biophysical aspects of networks are needed. The present paper uses a firing-rate model to study mechanisms that trigger and stop transitions between tonic and phasic population firing. These mechanisms are captured through a two-dimensional system, which can potentially be extended to include interactions between different areas of the nervous system with a small number of equations. The typical behavior of midbrain dopaminergic neurons in the rodent is used as an example to illustrate and interpret our results. The model presented here can be used as a building block to study interactions between networks of neurons. This theoretical approach may help contextualize and understand the factors involved in regulating burst firing in populations and how it may modulate distinct aspects of behavior.     

Modulation of a feeding neural circuit by microinjection of K+ channel expression genes into a single identified neuron in Aplysia kurodai

In Aplysia buccal ganglion expression genes for voltage-dependent K(+) channels (AKv1.1a) were injected into one of four electrically coupled multi-action (MA) neurons that directly inhibit jaw-closing (JC) motor neurons and may cooperatively generate their firing pattern during the feeding response. Following the DNA injection, the firing threshold increased and the spike frequency at the same current decreased in the current-induced excitation of the MA neuron; indicating a decrease in excitability of the MA neuron. This procedure also reduced the firing activity of MA neurons during the feeding-like rhythmic responses induced by the electrical nerve stimulation. Moreover, the firing pattern in JC motor neurons was remarkably changed, suggesting the effective contribution of a single MA neuron or electrically coupled MA neurons to the generation of the firing pattern in the JC motor neurons. This method appears useful for exploring the functional roles of specific neurons in complex neural circuits.     

Modulation of the dynamics of cerebellar Purkinje cells through the interaction of excitatory and inhibitory feedforward pathways

The dynamics of cerebellar neuronal networks is controlled by the underlying building blocks of neurons and synapses between them. For which, the computation of Purkinje cells (PCs), the only output cells of the cerebellar cortex, is implemented through various types of neural pathways interactively routing excitation and inhibition converged to PCs. Such tuning of excitation and inhibition, coming from the gating of specific pathways as well as short-term plasticity (STP) of the synapses, plays a dominant role in controlling the PC dynamics in terms of firing rate and spike timing. PCs receive cascade feedforward inputs from two major neural pathways: the first one is the feedforward excitatory pathway from granule cells (GCs) to PCs; the second one is the feedforward inhibition pathway from GCs, via molecular layer interneurons (MLIs), to PCs. The GC-PC pathway, together with short-term dynamics of excitatory synapses, has been a focus over past decades, whereas recent experimental evidence shows that MLIs also greatly contribute to controlling PC activity. Therefore, it is expected that the diversity of excitation gated by STP of GC-PC synapses, modulated by strong inhibition from MLI-PC synapses, can promote the computation performed by PCs. However, it remains unclear how these two neural pathways are interacted to modulate PC dynamics. Here using a computational model of PC network installed with these two neural pathways, we addressed this question to investigate the change of PC firing dynamics at the level of single cell and network. We show that the nonlinear characteristics of excitatory STP dynamics can significantly modulate PC spiking dynamics mediated by inhibition. The changes in PC firing rate, firing phase, and temporal spike pattern, are strongly modulated by these two factors in different ways. MLIs mainly contribute to variable delays in the postsynaptic action potentials of PCs while modulated by excitation STP. Notably, the diversity of synchronization and pause response in the PC network is governed not only by the balance of excitation and inhibition, but also by the synaptic STP, depending on input burst patterns. Especially, the pause response shown in the PC network can only emerge with the interaction of both pathways. Together with other recent findings, our results show that the interaction of feedforward pathways of excitation and inhibition, incorporated with synaptic short-term dynamics, can dramatically regulate the PC activities that consequently change the network dynamics of the cerebellar circuit.     

Travelling waves and EEG patterns during epileptic seizure: analysis with an integrate-and-fire neural network

Epilepsy is characterized by paradoxical patterns of neural activity. They may cause different types of electroencephalogram (EEG), which dynamically change in shape and frequency content during the temporal evolution of seizure. It is generally assumed that these epileptic patterns may originate in a network of strongly interconnected neurons, when excitation dominates over inhibition. The aim of this work is to use a neural network composed of 50 x 50 integrate-and-fire neurons to analyse which parameter alterations, at the level of synapse topology, may induce network instability and epileptic-like discharges, and to study the corresponding spatio-temporal characteristics of electrical activity in the network. We assume that a small group of central neurons is stimulated by a depolarizing current (epileptic focus) and that neurons are connected via a Mexican-hat topology of synapses. A signal representative of cortical EEG (ECoG) is simulated by summing the membrane potential changes of all neurons. A sensitivity analysis on the parameters describing the synapse topology shows that an increase in the strength and in spatial extension of excitatory vs. inhibitory synapses may cause the occurrence of travelling waves, which propagate along the network. These propagating waves may cause EEG patterns with different shape and frequency, depending on the particular parameter set used during the simulations. The resulting model EEG signals include irregular rhythms with large amplitude and a wide frequency content, low-amplitude high-frequency rapid discharges, isolated or repeated bursts, and low-frequency quasi-sinusoidal patterns. A slow progressive temporal variation in a single parameter may cause the transition from one pattern to another, thus generating a highly non-stationary signal which resembles that observed during ECoG measurements. These results may help to elucidate the mechanisms at the basis of some epileptic discharges, and to relate rapid changes in EEG patterns with the underlying alterations at the network level.     

When inhibition not excitation synchronizes neural firing

Excitatory and inhibitory synaptic coupling can have counter-intuitive effects on the synchronization of neuronal firing. While it might appear that excitatory coupling would lead to synchronization, we show that frequently inhibition rather than excitation synchronizes firing. We study two identical neurons described by integrate-and-fire models, general phase-coupled models or the Hodgkin-Huxley model with mutual, non-instantaneous excitatory or inhibitory synapses between them. We find that if the rise time of the synapse is longer than the duration of an action potential, inhibition not excitation leads to synchronized firing.     

Rhythmicity of Biopotentials: Its Origin and Involvement in Information Processing

Experimental data and mathematical simulation of a neural network were used to develop ideas concerning the origin of the rhythmicity of biopotentials and its involvement in information processing. Baseline slow oscillations—the primate α rhythm, the α-like rhythms of lower animals, the Δ rhythm of humans and animals, secondary components of sensory evoked potentials or responses to direct brain stimulation, and pathological epileptiform potentials—develop as a result of interactions between excitatory and inhibitory postsynaptic potentials. The main inhibitory transmitter in the brain cortex is γ-aminobutyric acid (GABA). EEG activation in the form of a decrease in the amplitude of baseline oscillations and the appearance of the stress rhythm in the θ band upon exposure to new or biologically significant stimuli is associated with a relative decay of inhibitory hyperpolarization processes. The cholinergic and noradrenergic neurotransmitter systems are substantially involved in the rearrangement of the neural activity associated with EEG activation. An enhancement of high-amplitude baseline oscillations and phasic activity of neurons, i.e., alternation of activation and inhibition of firing, which reflects a relative enhancement of hyperpolarization processes, restricts excitation propagation over brain structures and impedes the fixation of new information. As a result of the decay of the inhibitory processes, EEG activation is accompanied by a higher regularity of neuronal firing and a decrease in entropy in the time distribution of firing in the form of tonic or grouped (in the stress rhythm) discharges. The resulting ordered streams of impulses transfer information, control its propagation, and ensure its fixation and reproduction.__________Translated from Fiziologiya Cheloveka, Vol. 31, No. 3, 2005, pp. 59–71.Original Russian Text Copyright © 2005 by Shul’gina.     

A thalamo-cortical neural mass model for the simulation of brain rhythms during sleep

Cortico-thalamic interactions are known to play a pivotal role in many brain phenomena, including sleep, attention, memory consolidation and rhythm generation. Hence, simple mathematical models that can simulate the dialogue between the cortex and the thalamus, at a mesoscopic level, have a great cognitive value. In the present work we describe a neural mass model of a cortico-thalamic module, based on neurophysiological mechanisms. The model includes two thalamic populations (a thalamo-cortical relay cell population, TCR, and its related thalamic reticular nucleus, TRN), and a cortical column consisting of four connected populations (pyramidal neurons, excitatory interneurons, inhibitory interneurons with slow and fast kinetics). Moreover, thalamic neurons exhibit two firing modes: bursting and tonic. Finally, cortical synapses among pyramidal neurons incorporate a disfacilitation mechanism following prolonged activity. Simulations show that the model is able to mimic the different patterns of rhythmic activity in cortical and thalamic neurons (beta and alpha waves, spindles, delta waves, K-complexes, slow sleep waves) and their progressive changes from wakefulness to deep sleep, by just acting on modulatory inputs. Moreover, simulations performed by providing short sensory inputs to the TCR show that brain rhythms during sleep preserve the cortex from external perturbations, still allowing a high cortical activity necessary to drive synaptic plasticity and memory consolidation. In perspective, the present model may be used within larger cortico-thalamic networks, to gain a deeper understanding of mechanisms beneath synaptic changes during sleep, to investigate the specific role of brain rhythms, and to explore cortical synchronization achieved via thalamic influences.     

Information processing in noisy burster models of sensory neurons

Processing of external stimuli by sensory neurons often involves bursting, when epochs of fast firing alternate with intervals of quiescence. In particular, sensory neurons of electroreceptors in paddlefish (Polyodon spathula) undergo bursting when stimulated externally with broad-band noise, but otherwise fire spontaneously in a quasiperiodic tonic manner. We use a simple phenomenological model for noise-induced bursting to quantify analytically, by means of the Kullback entropy and Fisher information, the gain in information transfer and electroreceptor sensitivity for external noisy stimuli. A good agreement between theoretical predictions, numerical simulations and experimental data is shown.     

Computation of the electroencephalogram (EEG) from network models of point neurons

The electroencephalogram (EEG) is a major tool for non-invasively studying brain function and dysfunction. Comparing experimentally recorded EEGs with neural network models is important to better interpret EEGs in terms of neural mechanisms. Most current neural network models use networks of simple point neurons. They capture important properties of cortical dynamics, and are numerically or analytically tractable. However, point neurons cannot generate an EEG, as EEG generation requires spatially separated transmembrane currents. Here, we explored how to compute an accurate approximation of a rodent’s EEG with quantities defined in point-neuron network models. We constructed different approximations (or proxies) of the EEG signal that can be computed from networks of leaky integrate-and-fire (LIF) point neurons, such as firing rates, membrane potentials, and combinations of synaptic currents. We then evaluated how well each proxy reconstructed a ground-truth EEG obtained when the synaptic currents of the LIF model network were fed into a three-dimensional network model of multicompartmental neurons with realistic morphologies. Proxies based on linear combinations of AMPA and GABA currents performed better than proxies based on firing rates or membrane potentials. A new class of proxies, based on an optimized linear combination of time-shifted AMPA and GABA currents, provided the most accurate estimate of the EEG over a wide range of network states. The new linear proxies explained 85–95% of the variance of the ground-truth EEG for a wide range of network configurations including different cell morphologies, distributions of presynaptic inputs, positions of the recording electrode, and spatial extensions of the network. Non-linear EEG proxies using a convolutional neural network (CNN) on synaptic currents increased proxy performance by a further 2–8%. Our proxies can be used to easily calculate a biologically realistic EEG signal directly from point-neuron simulations thus facilitating a quantitative comparison between computational models and experimental EEG recordings.     

Early Cortical Orientation Selectivity: How Fast Inhibition Decodes the Order of Spike Latencies

Following a flashed stimulus, I show that a simple neurophysiological mechanism in the primary visual system can generate orientation selectivity based on the first incoming spikes. A biological model of the lateral geniculate nucleus generates an asynchronous wave of spikes, with the most strongly activated neurons firing first. Geniculate activation leads to both the direct excitation of a cortical pyramidal cell and disynaptic feed-forward inhibition. The mechanism provides automatic gain control, so the cortical neurons respond over a wide range of stimulus contrasts. It also demonstrates the biological plausibility of a new computationally efficient neural code: latency rank order coding.     

Bistable,Irregular Firing and Population Oscillations in a Modular Attractor Memory Network

Attractor neural networks are thought to underlie working memory functions in the cerebral cortex. Several such models have been proposed that successfully reproduce firing properties of neurons recorded from monkeys performing working memory tasks. However, the regular temporal structure of spike trains in these models is often incompatible with experimental data. Here, we show that the in vivo observations of bistable activity with irregular firing at the single cell level can be achieved in a large-scale network model with a modular structure in terms of several connected hypercolumns. Despite high irregularity of individual spike trains, the model shows population oscillations in the beta and gamma band in ground and active states, respectively. Irregular firing typically emerges in a high-conductance regime of balanced excitation and inhibition. Population oscillations can produce such a regime, but in previous models only a non-coding ground state was oscillatory. Due to the modular structure of our network, the oscillatory and irregular firing was maintained also in the active state without fine-tuning. Our model provides a novel mechanistic view of how irregular firing emerges in cortical populations as they go from beta to gamma oscillations during memory retrieval.     

On reverberatory processes in homogeneous neuronal spaces

Reverberating neural activity is strictly defined and examined in continuous and discrete neuronal spaces with homogeneous structure. Reverberations start with a specific population of firing neurons called the initial excitation and spread out in waves of firing and refractory bands of neurons toward the periphery. The necessary and sufficient conditions for having reverberations are obtained for continuous space and discrete one-dimensional space. The excitation fronts of reverberating waves have stable shapes which depend only upon the structure of the neuronal space. The reverberatory processes in high-threshold discrete neuronal spaces show strongly nonlinear properties. Relation between reverberations and nervous functions is discussed.     

Divisive Gain Modulation with Dynamic Stimuli in Integrate-and-Fire Neurons

The modulation of the sensitivity, or gain, of neural responses to input is an important component of neural computation. It has been shown that divisive gain modulation of neural responses can result from a stochastic shunting from balanced (mixed excitation and inhibition) background activity. This gain control scheme was developed and explored with static inputs, where the membrane and spike train statistics were stationary in time. However, input statistics, such as the firing rates of pre-synaptic neurons, are often dynamic, varying on timescales comparable to typical membrane time constants. Using a population density approach for integrate-and-fire neurons with dynamic and temporally rich inputs, we find that the same fluctuation-induced divisive gain modulation is operative for dynamic inputs driving nonequilibrium responses. Moreover, the degree of divisive scaling of the dynamic response is quantitatively the same as the steady-state responses—thus, gain modulation via balanced conductance fluctuations generalizes in a straight-forward way to a dynamic setting.