On computing Boolean functions by a spiking neuron

Computations by spiking neurons are performed using the timing of action potentials. We investigate the computational power of a simple model for such a spiking neuron in the Boolean domain by comparing it with traditional neuron models such as threshold gates (or McCulloch–Pitts neurons) and sigma-pi units (or polynomial threshold gates). In particular, we estimate the number of gates required to simulate a spiking neuron by a disjunction of threshold gates and we establish tight bounds for this threshold number. Furthermore, we analyze the degree of the polynomials that a sigma-pi unit must use for the simulation of a spiking neuron. We show that this degree cannot be bounded by any fixed value. Our results give evidence that the use of continuous time as a computational resource endows single-cell models with substantially larger computational capabilities. This revised version was published online in June 2006 with corrections to the Cover Date.     

单房室脉冲神经元建模方法综述

要通过人工神经网络来模拟神经系统的功能并对实际问题进行求解,构建合适的脉冲神经元模型非常重要。为了使研究者了解此问题的研究进展,对目前的单房室脉冲神经元建模方法进行了综述。根据复杂程度将这些模型分为三类:具有生物可解释性的生理模型,具有脉冲生成机制的非线性模型和具有固定阈值的线性模型。对各类不同建模方法进行了阐述和分析,并讨论了各自的优缺点。     

On the simulation of nonlinear bidimensional spiking neuron models

Bidimensional spiking models are garnering a lot of attention for their simplicity and their ability to reproduce various spiking patterns of cortical neurons and are used particularly for large network simulations. These models describe the dynamics of the membrane potential by a nonlinear differential equation that blows up in finite time, coupled to a second equation for adaptation. Spikes are emitted when the membrane potential blows up or reaches a cutoff θ. The precise simulation of the spike times and of the adaptation variable is critical, for it governs the spike pattern produced and is hard to compute accurately because of the exploding nature of the system at the spike times. We thoroughly study the precision of fixed time-step integration schemes for this type of model and demonstrate that these methods produce systematic errors that are unbounded, as the cutoff value is increased, in the evaluation of the two crucial quantities: the spike time and the value of the adaptation variable at this time. Precise evaluation of these quantities therefore involves very small time steps and long simulation times. In order to achieve a fixed absolute precision in a reasonable computational time, we propose here a new algorithm to simulate these systems based on a variable integration step method that either integrates the original ordinary differential equation or the equation of the orbits in the phase plane, and compare this algorithm with fixed time-step Euler scheme and other more accurate simulation algorithms.     

分段线性脉冲神经元模型的簇放电分析

应用一种二维分段线性脉冲神经元模型的简单计算特性,对其簇放电特性进行了详细研究。发现该模型表现出强迫和内禀两类簇放电模式,并分析了内禀簇放电的产生机理及分岔现象。在实验中,应用该模型对一类具有簇放电特性的皮层神经元进行了模拟。     

Dynamic moment analysis of the extracellular electric field of a biologically realistic spiking neuron

Based on the membrane currents generated by an action potential in a biologically realistic model of a pyramidal, hippocampal cell within rat CA1, we perform a moment expansion of the extracellular field potential. We decompose the potential into both inverse and classical moments and show that this method is a rapid and efficient way to calculate the extracellular field both near and far from the cell body. The action potential gives rise to a large quadrupole moment that contributes to the extracellular field up to distances of almost 1 cm. This method will serve as a starting point in connecting the microscopic generation of electric fields at the level of neurons to macroscopic observables such as the local field potential.     

Multistability in spiking neuron models of delayed recurrent inhibitory loops

We consider the effect of the effective timing of a delayed feedback on the excitatory neuron in a recurrent inhibitory loop, when biological realities of firing and absolute refractory period are incorporated into a phenomenological spiking linear or quadratic integrate-and-fire neuron model. We show that such models are capable of generating a large number of asymptotically stable periodic solutions with predictable patterns of oscillations. We observe that the number of fixed points of the so-called phase resetting map coincides with the number of distinct periods of all stable periodic solutions rather than the number of stable patterns. We demonstrate how configurational information corresponding to these distinct periods can be explored to calculate and predict the number of stable patterns.     

仿生型脉冲神经网络学习算法和网络模型

为解决脉冲神经网络训练困难的问题,基于仿生学思路,提出脉冲神经网络的权值学习算法和结构学习算法,设计一种含有卷积结构的脉冲神经网络模型,搭建适合脉冲神经网络的软件仿真平台。实验结果表明,权值学习算法训练的网络对MNIST数据集识别准确率能够达到84.12%,具备良好的快速收敛能力和低功耗特点;结构学习算法能够自动生成网络结构,具有高度生物相似性。     

Training a single sigmoidal neuron is hard

We first present a brief survey of hardness results for training feedforward neural networks. These results are then completed by the proof that the simplest architecture containing only a single neuron that applies a sigmoidal activation function sigma: kappa --> [alpha, beta], satisfying certain natural axioms (e.g., the standard (logistic) sigmoid or saturated-linear function), to the weighted sum of n inputs is hard to train. In particular, the problem of finding the weights of such a unit that minimize the quadratic training error within (beta - alpha)(2) or its average (over a training set) within 5(beta - alpha)(2)/ (12n) of its infimum proves to be NP-hard. Hence, the well-known backpropagation learning algorithm appears not to be efficient even for one neuron, which has negative consequences in constructive learning.     

Emergence of synchronicity in a self-organizing spiking neuron network: an approach via genetic algorithms

Based on the Theory of Neuronal Group Selection (TNGS), we have investigated the emergence of synchronicity in a network composed of spiking neurons via genetic algorithm. The TNGS establishes that a neuronal group is the most basic unit in the cortical area of the brain and, as a rule, it is not formed by a single neuron, but by a cluster of tightly coupled neural cells which fire and oscillate in synchrony at a predefined frequency. Thus, this paper describes a method of tuning the parameters of the Izhikevich spiking neuron model through genetic algorithm in order to enable the self-organization of the neural network. Computational experiments were performed considering a network composed of neurons of the same type and another composed of neurons of different types.     

On spiking neural P systems

This work deals with several aspects concerning the formal verification of SN P systems and the computing power of some variants. A methodology based on the information given by the transition diagram associated with an SN P system is presented. The analysis of the diagram cycles codifies invariants formulae which enable us to establish the soundness and completeness of the system with respect to the problem it tries to resolve. We also study the universality of asynchronous and sequential SN P systems and the capability these models have to generate certain classes of languages. Further, by making a slight modification to the standard SN P systems, we introduce a new variant of SN P systems with a special I/O mode, called SN P modules, and study their computing power. It is demonstrated that, as string language acceptors and transducers, SN P modules can simulate several types of computing devices such as finite automata, a-finite transducers, and systolic trellis automata.     

On the computational power of circuits of spiking neurons

Complex real-time computations on multi-modal time-varying input streams are carried out by generic cortical microcircuits. Obstacles for the development of adequate theoretical models that could explain the seemingly universal power of cortical microcircuits for real-time computing are the complexity and diversity of their computational units (neurons and synapses), as well as the traditional emphasis on offline computing in almost all theoretical approaches towards neural computation. In this article, we initiate a rigorous mathematical analysis of the real-time computing capabilities of a new generation of models for neural computation, liquid state machines, that can be implemented with—in fact benefit from—diverse computational units. Hence, realistic models for cortical microcircuits represent special instances of such liquid state machines, without any need to simplify or homogenize their diverse computational units. We present proofs of two theorems about the potential computational power of such models for real-time computing, both on analog input streams and for spike trains as inputs.     

On the computational complexity of spiking neural P systems

It is shown here that there is no standard spiking neural P system that simulates Turing machines with less than exponential time and space overheads. The spiking neural P systems considered here have a constant number of neurons that is independent of the input length. Following this, we construct a universal spiking neural P system with exhaustive use of rules that simulates Turing machines in linear time and has only 10 neurons.     

Time-series prediction with single integrate-and-fire neuron

In this paper, a learning algorithm for a single integrate-and-fire neuron (IFN) is proposed and tested for various applications in which a multilayer perceptron neural network is conventionally used. It is found that a single IFN is sufficient for the applications that require a number of neurons in different hidden layers of a conventional neural network. Several benchmark and real-life problems of classification and time-series prediction have been illustrated. It is observed that the inclusion of some more biological phenomenon in an artificial neural network can make it more powerful.     

Computation in a single neuron: Hodgkin and Huxley revisited

A spiking neuron "computes" by transforming a complex dynamical input into a train of action potentials, or spikes. The computation performed by the neuron can be formulated as dimensional reduction, or feature detection, followed by a nonlinear decision function over the low-dimensional space. Generalizations of the reverse correlation technique with white noise input provide a numerical strategy for extracting the relevant low-dimensional features from experimental data, and information theory can be used to evaluate the quality of the low-dimensional approximation. We apply these methods to analyze the simplest biophysically realistic model neuron, the Hodgkin-Huxley (HH) model, using this system to illustrate the general methodological issues. We focus on the features in the stimulus that trigger a spike, explicitly eliminating the effects of interactions between spikes. One can approximate this triggering "feature space" as a two-dimensional linear subspace in the high-dimensional space of input histories, capturing in this way a substantial fraction of the mutual information between inputs and spike time. We find that an even better approximation, however, is to describe the relevant subspace as two dimensional but curved; in this way, we can capture 90% of the mutual information even at high time resolution. Our analysis provides a new understanding of the computational properties of the HH model. While it is common to approximate neural behavior as "integrate and fire," the HH model is not an integrator nor is it well described by a single threshold.     

单神经元自适应PID控制器的性能优化设计

研究了单神经元自适应PID摔制器性能优化问题,阐述了该摔制器的特点、控制律;给出了一种控制灵敏度的快速近似求取方法,实现了PID参数的在线自学习;使单神经元控制器具有可调参数少、易于整定、控制输出平稳、鲁棒件强的独特优点,适用于大滞后且要求平稳控制输出的工业过程。     

Adaptive unsupervised extraction of one component of a linearmixture with a single neuron

Extracting one specific component of a linear mixture is to isolate it due to the observation of several mixtures of all the components. This is done in an unsupervised way, based on the sole knowledge that the components are independent. The classical solution is independent component analysis which extracts the components all at the same time. In this paper, given at least as many sensors as components, we propose a simpler approach which independently extracts each component with one neuron. The weights of the neuron are optimized by minimizing an even polynomial of its output. The corresponding adaptive algorithm is an extended anti-Hebbian rule with very low complexity. It can extract any specific negative kurtosis component. Global stability of the algorithm is investigated as well as steady-state fluctuations. The influence of additive noise is also considered. These theoretical results are thoroughly confirmed by computer simulations.     

A very simple spiking neuron model that allows for modeling of large, complex systems

This letter introduces a biologically inspired very simple spiking neuron model. The model retains only crucial aspects of biological neurons: a network of time-delayed weighted connections to other neurons, a threshold-based generation of action potentials, action potential frequency proportional to stimulus intensity, and interneuron communication that occurs with time-varying potentials that last longer than the associated action potentials. The key difference between this model and existing spiking neuron models is its great simplicity: it is basically a collection of linear and discontinuous functions with no differential equations to solve. The model's ability to operate in a complex network was tested by using it as a basis of a network implementing a hypothetical echolocation system. The system consists of an emitter and two receivers. The outputs of the receivers are connected to a network of spiking neurons (using the proposed model) to form a detection grid that acts as a map of object locations in space. The network uses differences in the arrival times of the signals to determine the azimuthal angle of the source and time of flight to calculate the distance. The activation patterns observed indicate that for a network of spiking neurons, which uses only time delays to determine source locations, the spatial discrimination varies with the number and relative spacing of objects. These results are similar to those observed in animals that use echolocation.     

航空发动机的单神经元双变量解耦控制

针对航空发动机这样的多变量控制对象,要解决的突出问题是输入变量对输出变量的交叉影响,介绍了单神经元进行多变量系统解耦控制的基本方法,采用改进的Hebb学习算法以加速收敛。对某涡喷发动机的数学模型进行了双变量单神经元PID控制仿真研究,结果表明：采用此算法构成的神经网络PID控制对地面模型和高空模型都具有完全解耦、响应速度快、稳态误差小、算法简单的优点;用两个神经元作为双变量控制器,可以使整个飞行包线内的控制器数目明显减少。     

On the power of elementary features in spiking neural P systems

Since their first publication in 2006, spiking neural (SN) P systems have already attracted the attention of a lot of researchers. This might be owing to the fact that this abstract computing device follows basic principles known from spiking neural nets, but its implementation is discrete, using membrane computing background. Among the elementary properties which confer SN P systems their computational power one can count the unbounded fan-in (indegree) and fan-out (outdegree) of each “neuron”, synchronicity of the whole system, the possibility of delaying and/or removing spikes in neurons, the capability of evaluating arbitrary regular expressions in neurons in constant time and some others. In this paper we focus on the power of these elementary features. Particularly, we study the power of the model when some of these features are disabled. Rather surprisingly, even very restricted SN P systems keep their universal computational power. Certain important questions regarding this topic still remain open.