This paper models the dynamics of a large set of interacting neurons within the framework of statistical field theory. We use a method initially developed in the context of statistical field theory [44] and later adapted to complex systems in interaction [45][46]. Our model keeps track of individual interacting neurons dynamics but also preserves some of the features and goals of neural field dynamics, such as indexing a large number of neurons by a space variable. Thus, this paper bridges the scale of individual interacting neurons and the macro-scale modelling of neural field theory.
Recurrent neural networks are powerful tools for understanding and modeling computation and representation by populations of neurons. Continuous-variable or "rate" model networks have been analyzed and applied extensively for these purposes. However, neurons fire action potentials, and the discrete nature of spiking is an important feature of neural circuit dynamics. Despite significant advances, training recurrently connected spiking neural networks remains a challenge. We present a procedure for training recurrently connected spiking networks to generate dynamical patterns autonomously, to p…
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…
Starting from a spectral expansion of the Fokker-Plank equation for the membrane potential density in a network of spiking neurons, a low-dimensional dynamics of the collective firing rate is derived. As a result a $n$-order ordinary differential equation for the network activity can be worked out by taking into account the slowest $n$ modes of the expansion. The resulting low-dimensional dynamics naturally takes into account the strength of the synaptic couplings under the hypothesis of a not too fast changing membrane potential density. By considering only the two slowest modes, the firing r…
We revisit the dynamics of a prototypical model of balanced activity in networks of spiking neutrons. A detailed investigation of the thermodynamic limit for fixed density of connections (massive coupling) shows that, when inhibition prevails, the asymptotic regime is not asynchronous but rather characterized by a self-sustained irregular, macroscopic (collective) dynamics. So long as the connectivity is massive, this regime is found in many different setups: leaky as well as quadratic integrate-and-fire neurons; large and small coupling strength; weak and strong external currents.
We propose a model of an adaptive network of spiking neurons that gives rise to a hypernetwork of its dynamic states at the upper level of description. Left to itself, the network exhibits a sequence of transient clustering which relates to a traffic in the hypernetwork in the form of a random walk. Receiving inputs the system is able to generate reproducible sequences corresponding to stimulus-specific paths in the hypernetwork. We illustrate these basic notions by a simple network of discrete-time spiking neurons together with its FPGA realization and analyze their properties.
Spiking neural network is a type of artificial neural network in which neurons communicate between each other with spikes. Spikes are identical Boolean events characterized by the time of their arrival. A spiking neuron has internal dynamics and responds to the history of inputs as opposed to the current inputs only. Because of such properties a spiking neural network has rich intrinsic capabilities to process spatiotemporal data. However, because the spikes are discontinuous 'yes or no' events, it is not trivial to apply traditional training procedures such as gradient descend to the spiking …
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 d…
École polytechnique fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Can we use spiking neural networks (SNN) as generative models of multi-neuronal recordings, while taking into account that most neurons are unobserved? Modeling the unobserved neurons with large pools of hidden spiking neurons leads to severely underconstrained problems that are hard to tackle with maximum likelihood estimation. In this work, we use coarse-graining and mean-field approximations to derive a bottom-up, neuronally-grounded latent variable model (neuLVM), where the activity of the unobserved neurons is reduced to a low-dimensional mesoscopic description. In contrast to previous la…
A complex interplay of single-neuron properties and the recurrent network structure shapes the activity of cortical neurons. The single-neuron activity statistics differ in general from the respective population statistics, including spectra and, correspondingly, autocorrelation times. We develop a theory for self-consistent second-order single-neuron statistics in block-structured sparse random networks of spiking neurons. In particular, the theory predicts the neuron-level autocorrelation times, also known as intrinsic timescales, of the neuronal activity. The theory is based on an extension…
A long tradition in theoretical neuroscience casts sensory processing in the brain as the process of inferring the maximally consistent interpretations of imperfect sensory input. Recently it has been shown that Gamma-band inhibition can enable neural attractor networks to approximately carry out such a sampling mechanism. In this paper we propose a novel neural network model based on irregular gating inhibition, show analytically how it implements a Monte-Carlo Markov Chain (MCMC) sampler, and describe how it can be used to model networks of both neural attractors as well as of single spiking…
We establish a general linear response relation for spiking neuronal networks, based on chains with unbounded memory. This relation allows us to predict the influence of a weak amplitude time-dependent external stimuli on spatio-temporal spike correlations, from the spontaneous statistics (without stimulus) in a general context where the memory in spike dynamics can extend arbitrarily far in the past. Using this approach, we show how linear response is explicitly related to neuronal dynamics with an example, the gIF model, introduced by M. Rudolph and A. Destexhe. This example illustrates the …
Limitations in processing capabilities and memory of today's computers make spiking neuron-based (human) whole-brain simulations inevitably characterized by a compromise between bio-plausibility and computational cost. It translates into brain models composed of a reduced number of neurons and a simplified neuron's mathematical model, leading to the search for new simulation strategies. Taking advantage of the sparse character of brain-like computation, the event-driven technique could represent a way to carry out efficient simulation of large-scale Spiking Neural Networks (SNN). The recent Le…
Several studies with brain signals suggested that bottom-up and top-down influences are exerted through distinct frequency bands among visual cortical areas. It has been recently shown that theta and gamma rhythms subserve feedforward, whereas the feedback influence is dominated by the alpha-beta rhythm in primates. A few theoretical models for reproducing these effects have been proposed so far. Here we show that a simple but biophysically plausible two-network motif composed of spiking-neuron models and chemical synapses can exhibit feedforward and feedback influences through distinct freque…
We address the problem of learning feedback control where the controller is a network constructed solely of deterministic spiking neurons. In contrast to previous investigations that were based on a spike rate model of the neuron, the control signal here is determined by the precise temporal positions of spikes generated by the output neurons of the network. We model the problem formally as a hybrid dynamical system comprised of a closed loop between a plant and a spiking neuron network. We derive a novel synaptic weight update rule via which the spiking neuron network controller learns to hol…
A main concern in cognitive neuroscience is to decode the overt neural spike train observations and infer latent representations under neural circuits. However, traditional methods entail strong prior on network structure and hardly meet the demand for real spike data. Here we propose a novel neural network approach called Neuron Activation Network that extracts neural information explicitly from single trial neuron population spike trains. Our proposed method consists of a spatiotemporal learning procedure on sensory environment and a message passing mechanism on population graph, followed by…
In machine learning, error back-propagation in multi-layer neural networks (deep learning) has been impressively successful in supervised and reinforcement learning tasks. As a model for learning in the brain, however, deep learning has long been regarded as implausible, since it relies in its basic form on a non-local plasticity rule. To overcome this problem, energy-based models with local contrastive Hebbian learning were proposed and tested on a classification task with networks of rate neurons. We extended this work by implementing and testing such a model with networks of leaky integrate…
Top-down feedback in cortex is critical for guiding sensory processing, which has prominently been formalized in the theory of hierarchical predictive coding (hPC). However, experimental evidence for error units, which are central to the theory, is inconclusive, and it remains unclear how hPC can be implemented with spiking neurons. To address this, we connect hPC to existing work on efficient coding in balanced networks with lateral inhibition, and predictive computation at apical dendrites. Together, this work points to an efficient implementation of hPC with spiking neurons, where predictio…
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…
Spontaneous fluctuations and stimulus response are essential features of neural functioning but how they are connected is poorly understood. I derive fluctuation-dissipation relations (FDR) between the spontaneous spike and voltage correlations and the firing rate susceptibility for i) the leaky integrate-and-fire (IF) model with white noise; ii) an IF model with arbitrary voltage dependence, an adaptation current, and correlated noise. The FDRs can be used to derive correlation statistics or to infer the system's response from observations of its spontaneous activity.
The brain is composed of complex networks of interacting neurons that express considerable heterogeneity in their physiology and spiking characteristics. How does neural heterogeneity affect macroscopic neural dynamics and how does it contribute to neurodynamic functions? In this letter, we address these questions by studying the macroscopic dynamics of networks of heterogeneous Izhikevich neurons. We derive mean-field equations for these networks and examine how heterogeneity in the spiking thresholds of Izhikevich neurons affects the emergent macroscopic dynamics. Our results suggest that th…
We study an abstracted model of neuronal activity via numerical simulation, and report spatiotemporal pattern formation and critical like dynamics. A population of pulse coupled, discretised, relaxation oscillators is simulated over networks with varying edge density and spatial embedded ness. For intermediate edge density and sufficiently strong spatial embeddedness, we observe a novel spatiotemporal pattern in the field of oscillator phases, visually resembling the surface of a frothing liquid. Increasing the edge density results in critical dynamics, with the distribution of neuronal avalan…
Maintaining the ability to fire sparsely is crucial for information encoding in neural networks. Additionally, spiking homeostasis is vital for spiking neural networks with changing numbers of weights and neurons. We discuss a range of network stabilisation approaches, inspired by homeostatic synaptic plasticity mechanisms reported in the brain. These include weight scaling, and weight change as a function of the network's spiking activity. We tested normalisation of the sum of weights for all neurons, and by neuron type. We examined how this approach affects firing rate and performance on clu…
