We investigate the dynamics of large-scale interacting neural populations, composed of conductance based, spiking model neurons with modifiable synaptic connection strengths, which are possibly also subjected to external noisy currents. The network dynamics is controlled by a set of neural population probability distributions ($\mathrm{PPD}$)which are constructed along the same lines as in the Klimontovich approach to the kinetic theory of plasmas. An exact non-closed, nonlinear, system of integro-partial differential equations is derived for the $\mathrm{PPD}$s. As is customary, a closing procedure leads to a mean field limit. The equations we have obtained are of the same type as those which have been recently derived using rigorous techniques of probability theory. The numerical solutions of these so called McKean-Vlasov-Fokker-Planck equations, which are only valid in the limit of infinite size networks, actually shows that the statistical measures as obtained from $\mathrm{PPD}$s are in good agreement with those obtained through direct integration of the stochastic dynamical system for large but finite size networks. Although numerical solutions have been obtained in the case of Fitzhugh-Nagumo model neurons, the theory can be readily applied to systems of Hodgkin-Huxley type model neurons of arbitrary dimension.
Neuronal spiking exhibits an exquisite combination of modulation and robustness properties, rarely matched in artificial systems. We exploit the particular interconnection structure of conductance based models to investigate this remarkable property. We find that much of neuronal modulation and robustness can be explained by separating the total transmembrane current into three different components corresponding to the three time scales of neuronal bursting. Each equivalent current aggregates many ionic contributions into an equivalent voltage-dependent conductance, which defines a key modulat…
We consider a classical space-clamped Hodgkin-Huxley model neuron stimulated by synaptic excitation and inhibition with conductances represented by Ornstein-Uhlenbeck processes. Using numerical solutions of the stochastic model system obtained by an Euler method, it is found that with excitation only there is a critical value of the steady state excitatory conductance for repetitive spiking without noise and for values of the conductance near the critical value small noise has a powerfully inhibitory effect. For a given level of inhibition there is also a critical value of the steady state exc…
Strong inhibitory input to neurons, which occurs in balanced states of neural networks, increases synaptic current fluctuations. This has led to the assumption that inhibition contributes to the high spike-firing irregularity observed in vivo. We used single compartment neuronal models with time-correlated (due to synaptic filtering) and state-dependent (due to reversal potentials) input to demonstrate that inhibitory input acts to decrease membrane potential fluctuations, a result that cannot be achieved with simplified neural input models. To clarify the effects on spike-firing regularity, w…
The highly variable dynamics of neocortical circuits observed in vivo have been hypothesized to represent a signature of ongoing stochastic inference but stand in apparent contrast to the deterministic response of neurons measured in vitro. Based on a propagation of the membrane autocorrelation across spike bursts, we provide an analytical derivation of the neural activation function that holds for a large parameter space, including the high-conductance state. On this basis, we show how an ensemble of leaky integrate-and-fire neurons with conductance-based synapses embedded in a spiking enviro…
Voltage-sensitive dye imaging (VSDi) has revealed fundamental properties of neocortical processing at mesoscopic scales. Since VSDi signals report the average membrane potential, it seems natural to use a mean-field formalism to model such signals. Here, we investigate a mean-field model of networks of Adaptive Exponential (AdEx) integrate-and-fire neurons, with conductance-based synaptic interactions. The AdEx model can capture the spiking response of different cell types, such as regular-spiking (RS) excitatory neurons and fast-spiking (FS) inhibitory neurons. We use a Master Equation formal…
Artificial neural networks (ANNs) have been extensively used for the description of problems arising from biological systems and for constructing neuromorphic computing models. The third generation of ANNs, namely, spiking neural networks (SNNs), inspired by biological neurons enable a more realistic mimicry of the human brain. A large class of the problems from these domains is characterized by the necessity to deal with the combination of neurons, spikes and synapses via integrate-and-fire neuron models. Motivated by important applications of the integrate-and-fire of neurons in neuromorphic…
We present an optimized conductance-based retina microcircuit simulator which transforms light stimuli into a series of graded and spiking action potentials through photo transduction. We use discrete retinal neuron blocks based on a collation of single-compartment models and morphologically realistic formulations, and successfully achieve a biologically real-time simulator. This is done by optimizing the numerical methods employed to solve the system of over 270 nonlinear ordinary differential equations and parameters. Our simulator includes some of the most recent advances in compartmental m…
While spike timing has been shown to carry detailed stimulus information at the sensory periphery, its possible role in network computation is less clear. Most models of computation by neural networks are based on population firing rates. In equivalent spiking implementations, firing is assumed to be random such that averaging across populations of neurons recovers the rate-based approach. Recently, however, Denéve and colleagues have suggested that the spiking behavior of neurons may be fundamental to how neuronal networks compute, with precise spike timing determined by each neuron's contrib…
We investigate the efficient transmission and processing of weak, subthreshold signals in a realistic neural medium in the presence of different levels of the underlying noise. Assuming Hebbian weights for maximal synaptic conductances -- that naturally balances the network with excitatory and inhibitory synapses -- and considering short-term synaptic plasticity affecting such conductances, we found different dynamic phases in the system. This includes a memory phase where population of neurons remain synchronized, an oscillatory phase where transitions between different synchronized populatio…
Serotonergic, noradrenergic and dopaminergic brainstem (including midbrain) neurons, often exhibit spontaneous and fairly regular spiking with frequencies of order a few Hz, though dopaminergic and noradrenergic neurons only exhibit such pacemaker-type activity in vitro or in vivo under special conditions. A large number of ion channel types contribute to such spiking so that detailed modeling of spike generation leads to the requirement of solving very large systems of differential equations. It is useful to have simplified mathematical models of spiking in such neurons so that, for example, …
The mechanisms and impact of correlated, or synchronous, firing among pairs and groups of neurons is under intense investigation throughout the nervous system. A ubiquitous circuit feature that can give rise to such correlations consists of overlapping, or common, inputs to pairs and populations of cells, leading to common spike train responses. Here, we use computational tools to study how the transfer of common input currents into common spike outputs is modulated by the physiology of the recipient cells. We focus on a key conductance - gA, for the A-type potassium current - which drives neu…
The apparent stochasticity of in-vivo neural circuits has long been hypothesized to represent a signature of ongoing stochastic inference in the brain. More recently, a theoretical framework for neural sampling has been proposed, which explains how sample-based inference can be performed by networks of spiking neurons. One particular requirement of this approach is that the neural response function closely follows a logistic curve. Analytical approaches to calculating neural response functions have been the subject of many theoretical studies. In order to make the problem tractable, particular…
High-level brain function such as memory, classification or reasoning can be realized by means of recurrent networks of simplified model neurons. Analog neuromorphic hardware constitutes a fast and energy efficient substrate for the implementation of such neural computing architectures in technical applications and neuroscientific research. The functional performance of neural networks is often critically dependent on the level of correlations in the neural activity. In finite networks, correlations are typically inevitable due to shared presynaptic input. Recent theoretical studies have shown…
Neuronal voltage dynamics of regularly firing neurons typically has one stable attractor: either a fixed point (like in the subthreshold regime) or a limit cycle that defines the tonic firing of action potentials (in the suprathreshold regime). In two of the three spike onset bifurcation sequences that are known to give rise to all-or-none type action potentials, however, the resting-state fixpoint and limit cycle spiking can coexist in an intermediate regime, resulting in bistable dynamics. Here, noise can induce switches between the attractors, i.e., between rest and spiking, and thus increa…
We consider here a single-compartment model of these neurons which is capable of describing many of the known features of spike generation, particularly the slow rhythmic pacemaking activity often observed in these cells in a variety of species. Included in the model are ten kinds of voltage dependent ion channels as well as calcium-dependent potassium current. Calcium dynamics includes buffering and pumping. In sections 3-9, each component is considered in detail and parameters estimated from voltage clamp data where possible. In the next two sections simplified versions of some components ar…
Over the past decade there has been a growing interest in the development of parallel hardware systems for simulating large-scale networks of spiking neurons. Compared to other highly-parallel systems, GPU-accelerated solutions have the advantage of a relatively low cost and a great versatility, thanks also to the possibility of using the CUDA-C/C++ programming languages. NeuronGPU is a GPU library for large-scale simulations of spiking neural network models, written in the C++ and CUDA-C++ programming languages, based on a novel spike-delivery algorithm. This library includes simple LIF (leak…
We investigate different dynamical regimes of neuronal network in the CA3 area of the hippocampus. The proposed neuronal circuit includes two fast- and two slowly-spiking cells which are interconnected by means of dynamical synapses. On the individual level, each neuron is modeled by FitzHugh-Nagumo equations. Three basic rhythmic patterns are observed: gamma-rhythm in which the fast neurons are uniformly spiking, theta-rhythm in which the individual spikes are separated by quiet epochs, and theta/gamma rhythm with repeated patches of spikes. We analyze the influence of asymmetry of synaptic s…
Centre for Theoretical Neuroscience, University of Waterloo（滑铁卢大学理论神经科学中心）
Nonlinear interactions in the dendritic tree play a key role in neural computation. Nevertheless, modeling frameworks aimed at the construction of large-scale, functional spiking neural networks, such as the Neural Engineering Framework, tend to assume a linear superposition of post-synaptic currents. In this paper, we present a series of extensions to the Neural Engineering Framework that facilitate the construction of networks incorporating Dale's principle and nonlinear conductance-based synapses. We apply these extensions to a two-compartment LIF neuron that can be seen as a simple model o…
Many types of neurons exhibit spike rate adaptation, mediated by intrinsic slow $\mathrm{K}^+$-currents, which effectively inhibit neuronal responses. How these adaptation currents change the relationship between in-vivo like fluctuating synaptic input, spike rate output and the spike train statistics, however, is not well understood. In this computational study we show that an adaptation current which primarily depends on the subthreshold membrane voltage changes the neuronal input-output relationship (I-O curve) subtractively, thereby increasing the response threshold. A spike-dependent adap…
A central question in neuroscience is to understand how noisy firing patterns are used to transmit information. Because neural spiking is noisy, spiking patterns are often quantified via pairwise correlations, or the probability that two cells will spike coincidentally, above and beyond their baseline firing rate. One observation frequently made in experiments, is that correlations can increase systematically with firing rate. Theoretical studies have determined that stimulus-dependent correlations that increase with firing rate can have beneficial effects on information coding; however, we st…
Networks in the brain consist of different types of neurons. Here we investigate the influence of neuron diversity on the dynamics, phase space structure and computational capabilities of spiking neural networks. We find that already a single neuron of a different type can qualitatively change the network dynamics and that mixed networks may combine the computational capabilities of ones with a single neuron type. We study inhibitory networks of concave leaky (LIF) and convex "anti-leaky" (XIF) integrate-and-fire neurons that generalize irregularly spiking non-chaotic LIF neuron networks. Endo…
We present an unsupervised, local activation-dependent learning rule for intrinsic plasticity (IP) which affects the composition of ion channel conductances for single neurons in a use-dependent way. We use a single-compartment conductance-based model for medium spiny striatal neurons in order to show the effects of parametrization of individual ion channels on the neuronal activation function. We show that parameter changes within the physiological ranges are sufficient to create an ensemble of neurons with significantly different activation functions. We emphasize that the effects of intrins…
Self-sustained activity in the brain is observed in the absence of external stimuli and contributes to signal propagation, neural coding, and dynamic stability. It also plays an important role in cognitive processes. In this work, by means of studying intracellular recordings from CA1 neurons in rats and results from numerical simulations, we demonstrate that self-sustained activity presents high variability of patterns, such as low neural firing rates and activity in the form of small-bursts in distinct neurons. In our numerical simulations, we consider random networks composed of coupled, ad…
An increasing body of evidence suggests that the trial-to-trial variability of spiking activity in the brain is not mere noise, but rather the reflection of a sampling-based encoding scheme for probabilistic computing. Since the precise statistical properties of neural activity are important in this context, many models assume an ad-hoc source of well-behaved, explicit noise, either on the input or on the output side of single neuron dynamics, most often assuming an independent Poisson process in either case. However, these assumptions are somewhat problematic: neighboring neurons tend to shar…
This paper proposes a neuronal circuitry layout and synaptic plasticity principles that allow the (pyramidal) neuron to act as a "combinatorial switch". Namely, the neuron learns to be more prone to generate spikes given those combinations of firing input neurons for which a previous spiking of the neuron had been followed by a positive global reward signal. The reward signal may be mediated by certain modulatory hormones or neurotransmitters, e.g., the dopamine. More generally, a trial-and-error learning paradigm is suggested in which a global reward signal triggers long-term enhancement or w…
In recent years, there has been increasing interest in developing models and tools to address the complex patterns of connectivity found in brain tissue. Specifically, this is due to a need to understand how emergent properties emerge from these network structures at multiple spatiotemporal scales. We argue that computational models are key tools for elucidating the possible functionalities that can emerge from interactions of heterogeneous neurons connected by complex networks on multi-scale temporal and spatial domains. Here we review several classes of models including spiking neurons, inte…
Synaptic delays modulate population phase and amplitude responses in oscillatory excitatory-inhibitory networks
Institute for Advanced Studies in Basic Sciences (IASBS)（基础科学高等研究院）；Institute for Research in Fundamental Sciences (IPM)（基本科学研究院）；Pasargad Institute for Advanced Innovative Solutions (PIAIS)（帕萨尔加德先进创新解决方案研究所）；The Zapata-Briceño Institute of Neuroscience（萨帕塔-布里斯尼奥神经科学研究所）
In a first step towards the comprehension of neural activity, one should focus on the stability of the various dynamical states. Even the characterization of idealized regimes, such as a perfectly periodic spiking activity, reveals unexpected difficulties. In this paper we discuss a general approach to linear stability of pulse-coupled neural networks for generic phase-response curves and post-synaptic response functions. In particular, we present: (i) a mean-field approach developed under the hypothesis of an infinite network and small synaptic conductances; (ii) a "microscopic" approach whic…
We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the Fitzhugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons initial conditions are drawn independently from the same law that depends only on the population t…
Spontaneous cortical population activity exhibits a multitude of oscillatory patterns, which often display synchrony during slow-wave sleep or under certain anesthetics and stay asynchronous during quiet wakefulness. The mechanisms behind these cortical states and transitions among them are not completely understood. Here we study spontaneous population activity patterns in random networks of spiking neurons of mixed types modeled by Izhikevich equations. Neurons are coupled by conductance-based synapses subject to synaptic noise. We localize the population activity patterns on the parameter d…
