Delay learning and polychronization for reservoir computing

We propose a multi-timescale learning rule for spiking neuron networks, in the line of the recently emerging field of reservoir computing. The reservoir is a network model of spiking neurons, with random topology and driven by STDP (spike-time-dependent plasticity), a temporal Hebbian unsupervised learning mode, biologically observed. The model is further driven by a supervised learning algorithm, based on a margin criterion, that affects the synaptic delays linking the network to the readout neurons, with classification as a goal task. The network processing and the resulting performance can be explained by the concept of polychronization, proposed by Izhikevich [Polychronization: computation with spikes, Neural Comput. 18(2) (2006) 245–282], on physiological grounds. The model emphasizes that polychronization can be used as a tool for exploiting the computational power of synaptic delays and for monitoring the topology and activity of a spiking neuron network.     

Modeling synaptic plasticity in conjuction with the timing of pre- and postsynaptic action potentials

We present a spiking neuron model that allows for an analytic calculation of the correlations between pre- and postsynaptic spikes. The neuron model is a generalization of the integrate-and-fire model and equipped with a probabilistic spike-triggering mechanism. We show that under certain biologically plausible conditions, pre- and postsynaptic spike trains can be described simultaneously as an inhomogeneous Poisson process. Inspired by experimental findings, we develop a model for synaptic long-term plasticity that relies on the relative timing of pre- and post-synaptic action potentials. Being given an input statistics, we compute the stationary synaptic weights that result from the temporal correlations between the pre- and postsynaptic spikes. By means of both analytic calculations and computer simulations, we show that such a mechanism of synaptic plasticity is able to strengthen those input synapses that convey precisely timed spikes at the expense of synapses that deliver spikes with a broad temporal distribution. This may be of vital importance for any kind of information processing based on spiking neurons and temporal coding.     

Effects of spike timing on winner-take-all competition in model cortical circuits

Synaptic interactions in cortical circuits involve strong recurrent excitation between nearby neurons and lateral inhibition that is more widely spread. This architecture is commonly thought to promote a winner-take-all competition, in which a small fraction of neuronal responses is selected for further processing. Here I report that such a competition is remarkably sensitive to the timing of neuronal action potentials. This is shown using simulations of model neurons and synaptic connections representing a patch of cortical tissue. In the simulations, uncorrelated discharge among neuronal units results in patterns of response dominance and suppression, that is, in a winner-take-all competition. Synchronization of firing, however, prevents such competition. These results demonstrate a novel property of recurrent cortical-like circuits, suggesting that the temporal patterning of cortical activity may play an important part in selection among stimuli competing for the control of attention and motor action.     

Accelerating event-driven simulation of spiking neurons with multiple synaptic time constants

The simulation of spiking neural networks (SNNs) is known to be a very time-consuming task. This limits the size of SNN that can be simulated in reasonable time or forces users to overly limit the complexity of the neuron models. This is one of the driving forces behind much of the recent research on event-driven simulation strategies. Although event-driven simulation allows precise and efficient simulation of certain spiking neuron models, it is not straightforward to generalize the technique to more complex neuron models, mostly because the firing time of these neuron models is computationally expensive to evaluate. Most solutions proposed in literature concentrate on algorithms that can solve this problem efficiently. However, these solutions do not scale well when more state variables are involved in the neuron model, which is, for example, the case when multiple synaptic time constants for each neuron are used. In this letter, we show that an exact prediction of the firing time is not required in order to guarantee exact simulation results. Several techniques are presented that try to do the least possible amount of work to predict the firing times. We propose an elegant algorithm for the simulation of leaky integrate-and-fire (LIF) neurons with an arbitrary number of (unconstrained) synaptic time constants, which is able to combine these algorithmic techniques efficiently, resulting in very high simulation speed. Moreover, our algorithm is highly independent of the complexity (i.e., number of synaptic time constants) of the underlying neuron model.     

Population density methods for large-scale modelling of neuronal networks with realistic synaptic kinetics: cutting the dimension down to size

Population density methods provide promising time-saving alternatives to direct Monte Carlo simulations of neuronal network activity, in which one tracks the state of thousands of individual neurons and synapses. A population density method has been found to be roughly a hundred times faster than direct simulation for various test networks of integrate-and-fire model neurons with instantaneous excitatory and inhibitory post-synaptic conductances. In this method, neurons are grouped into large populations of similar neurons. For each population, one calculates the evolution of a probability density function (PDF) which describes the distribution of neurons over state space. The population firing rate is then given by the total flux of probability across the threshold voltage for firing an action potential. Extending the method beyond instantaneous synapses is necessary for obtaining accurate results, because synaptic kinetics play an important role in network dynamics. Embellishments incorporating more realistic synaptic kinetics for the underlying neuron model increase the dimension of the PDF, which was one-dimensional in the instantaneous synapse case. This increase in dimension causes a substantial increase in computation time to find the exact PDF, decreasing the computational speed advantage of the population density method over direct Monte Carlo simulation. We report here on a one-dimensional model of the PDF for neurons with arbitrary synaptic kinetics. The method is more accurate than the mean-field method in the steady state, where the mean-field approximation works best, and also under dynamic-stimulus conditions. The method is much faster than direct simulations. Limitations of the method are demonstrated, and possible improvements are discussed.     

A new synaptic plasticity rule for networks of spiking neurons

In this paper, we describe a new Synaptic Plasticity Activity Rule (SAPR) developed for use in networks of spiking neurons. Such networks can be used for simulations of physiological experiments as well as for other computations like image analysis. Most synaptic plasticity rules use artificially defined functions to modify synaptic connection strengths. In contrast, our rule makes use of the existing postsynaptic potential values to compute the value of adjustment. The network of spiking neurons we consider consists of excitatory and inhibitory neurons. Each neuron is implemented as an integrate-and-fire model that accurately mimics the behavior of biological neurons. To test performance of our new plasticity rule we designed a model of a biologically-inspired signal processing system, and used it for object detection in eye images of diabetic retinopathy patients, and lung images of cystic fibrosis patients. The results show that the network detects the edges of objects within an image, essentially segmenting it. Our ultimate goal, however, is not the development of an image segmentation tool that would be more efficient than nonbiological algorithms, but developing a physiologically correct neural network model that could be applied to a wide range of neurological experiments. We decided to validate the SAPR by using it in a network of spiking neurons for image segmentation because it is easy to visually assess the results. An important thing is that image segmentation is done in an entirely unsupervised way.     

On the combination of top-down and bottom-up methodologies for the design of coordination mechanisms in self-organising systems

In resource-flow systems, e.g. production lines, agents are processing resources by applying capabilities to them in a given order. Such systems benefit from self-organization as they become easier to manage and more robust against failures. In this paper, we demonstrate the conception of a decentralized coordination process for resource-flow systems and its integration into an agent-based software system. This process restores a system’s functionality after a failure by propagating information about the error through the system until a fitting agent is found that is able to perform the required function. The mechanism has been designed by combining a top-down design approach for self-organizing resource-flow system and a systemic development framework for the development of decentralized, distributed coordination processes. Using the latter framework, a process is designed and integrated in a system realization that follows the former conceptual model. Evaluations of convergence as well as performance of the mechanism and the required amount of redundancy of the system are performed by simulations.     

Robust gamma oscillations in networks of inhibitory hippocampal interneurons

Recent experiments suggest that inhibitory networks of interneurons can synchronize the neuronal discharge in in vitro hippocampal slices. Subsequent theoretical work has shown that strong synchronization by mutual inhibition is only moderately robust against neuronal heterogeneities in the current drive, provided by activation of metabotropic glutamate receptors. In vivo neurons display greater variability in the interspike intervals due to the presence of synaptic noise. Noise and heterogeneity affect synchronization properties differently. In this paper we study, using model simulations, how robust synchronization can be in the presence of synaptic noise and neuronal heterogeneity. We find that stochastic weak synchronization (SWS) (i.e. when neurons spike within a short interval from each other, but not necessarily at each period) is produced with at least a minimum amount of noise and that it is much more robust than strong synchronization (i.e. when neurons spike at each period). The statistics produced by the SWS population discharge are consistent with previous experimental data. We also find robust SWS in the gamma-frequency range (20-80 Hz) for a stronger synaptic coupling compared with previous models and for networks with 10-1000 neurons.     

On the performance of voltage stepping for the simulation of adaptive, nonlinear integrate-and-fire neuronal networks

In traditional event-driven strategies, spike timings are analytically given or calculated with arbitrary precision (up to machine precision). Exact computation is possible only for simplified neuron models, mainly the leaky integrate-and-fire model. In a recent paper, Zheng, Tonnelier, and Martinez (2009) introduced an approximate event-driven strategy, named voltage stepping, that allows the generic simulation of nonlinear spiking neurons. Promising results were achieved in the simulation of single quadratic integrate-and-fire neurons. Here, we assess the performance of voltage stepping in network simulations by considering more complex neurons (quadratic integrate-and-fire neurons with adaptation) coupled with multiple synapses. To handle the discrete nature of synaptic interactions, we recast voltage stepping in a general framework, the discrete event system specification. The efficiency of the method is assessed through simulations and comparisons with a modified time-stepping scheme of the Runge-Kutta type. We demonstrated numerically that the original order of voltage stepping is preserved when simulating connected spiking neurons, independent of the network activity and connectivity.     

Dynamics of the firing probability of noisy integrate-and-fire neurons

Cortical neurons in vivo undergo a continuous bombardment due to synaptic activity, which acts as a major source of noise. Here, we investigate the effects of the noise filtering by synapses with various levels of realism on integrate-and-fire neuron dynamics. The noise input is modeled by white (for instantaneous synapses) or colored (for synapses with a finite relaxation time) noise. Analytical results for the modulation of firing probability in response to an oscillatory input current are obtained by expanding a Fokker-Planck equation for small parameters of the problem - when both the amplitude of the modulation is small compared to the background firing rate and the synaptic time constant is small compared to the membrane time constant. We report here the detailed calculations showing that if a synaptic decay time constant is included in the synaptic current model, the firing-rate modulation of the neuron due to an oscillatory input remains finite in the high-frequency limit with no phase lag. In addition, we characterize the low-frequency behavior and the behavior of the high-frequency limit for intermediate decay times. We also characterize the effects of introducing a rise time to the synaptic currents and the presence of several synaptic receptors with different kinetics. In both cases, we determine, using numerical simulations, an effective decay time constant that describes the neuronal response completely.     

Bifurcation analysis in a silicon neuron

In this paper, we describe an analysis of the nonlinear dynamical phenomenon associated with a silicon neuron. Our silicon neuron in Very Large Scale Integration (VLSI) integrates Hodgkin?CHuxley (HH) model formalism, including the membrane voltage dependency of temporal dynamics. Analysis of the bifurcation conditions allow us to identify different regimes in the parameter space that are desirable for biasing our silicon neuron. This approach of studying bifurcations is useful because it is believed that computational properties of neurons are based on the bifurcations exhibited by these dynamical systems in response to some changing stimulus. We describe numerical simulations of the Hopf bifurcation which is characteristic of class 2 excitability in the HH model. We also show experimental measurements of an observed phenomenon in biological neurons and termed excitation block, firing rate and effect of current impulses. Hence, by showing that this silicon neuron has similar bifurcations to a certain class of biological neurons, we can claim that the silicon neuron can also perform similar computations.     

Real-time computing platform for spiking neurons (RT-spike)

A computing platform is described for simulating arbitrary networks of spiking neurons in real time. A hybrid computing scheme is adopted that uses both software and hardware components to manage the tradeoff between flexibility and computational power; the neuron model is implemented in hardware and the network model and the learning are implemented in software. The incremental transition of the software components into hardware is supported. We focus on a spike response model (SRM) for a neuron where the synapses are modeled as input-driven conductances. The temporal dynamics of the synaptic integration process are modeled with a synaptic time constant that results in a gradual injection of charge. This type of model is computationally expensive and is not easily amenable to existing software-based event-driven approaches. As an alternative we have designed an efficient time-based computing architecture in hardware, where the different stages of the neuron model are processed in parallel. Further improvements occur by computing multiple neurons in parallel using multiple processing units. This design is tested using reconfigurable hardware and its scalability and performance evaluated. Our overall goal is to investigate biologically realistic models for the real-time control of robots operating within closed action-perception loops, and so we evaluate the performance of the system on simulating a model of the cerebellum where the emulation of the temporal dynamics of the synaptic integration process is important.     

Analog-digital simulations of full conductance-based networks of spiking neurons with spike timing dependent plasticity

We introduce and test a system for simulating networks of conductance-based neuron models using analog circuits. At the single-cell level, we use custom-designed analog circuits (ASICs) that simulate two types of spiking neurons based on Hodgkin-Huxley like dynamics: "regular spiking" excitatory neurons with spike-frequency adaptation, and "fast spiking" inhibitory neurons. Synaptic interactions are mediated by conductance-based synaptic currents described by kinetic models. Connectivity and plasticity rules are implemented digitally through a real time interface between a computer and a PCI board containing the ASICs. We show a prototype system of a few neurons interconnected with synapses undergoing spike-timing dependent plasticity (STDP), and compare this system with numerical simulations. We use this system to evaluate the effect of parameter dispersion on the behavior of small circuits of neurons. It is shown that, although the exact spike timings are not precisely emulated by the ASIC neurons, the behavior of small networks with STDP matches that of numerical simulations. Thus, this mixed analog-digital architecture provides a valuable tool for real-time simulations of networks of neurons with STDP. They should be useful for any real-time application, such as hybrid systems interfacing network models with biological neurons.     

On the classification capability of sign-constrained perceptrons

The perceptron (also referred to as McCulloch-Pitts neuron, or linear threshold gate) is commonly used as a simplified model for the discrimination and learning capability of a biological neuron. Criteria that tell us when a perceptron can implement (or learn to implement) all possible dichotomies over a given set of input patterns are well known, but only for the idealized case, where one assumes that the sign of a synaptic weight can be switched during learning. We present in this letter an analysis of the classification capability of the biologically more realistic model of a sign-constrained perceptron, where the signs of synaptic weights remain fixed during learning (which is the case for most types of biological synapses). In particular, the VC-dimension of sign-constrained perceptrons is determined, and a necessary and sufficient criterion is provided that tells us when all 2(m) dichotomies over a given set of m patterns can be learned by a sign-constrained perceptron. We also show that uniformity of L(1) norms of input patterns is a sufficient condition for full representation power in the case where all weights are required to be nonnegative. Finally, we exhibit cases where the sign constraint of a perceptron drastically reduces its classification capability. Our theoretical analysis is complemented by computer simulations, which demonstrate in particular that sparse input patterns improve the classification capability of sign-constrained perceptrons.     

Neuronal bases of perceptual learning revealed by a synaptic balance scheme

Our ability to perceive external sensory stimuli improves as we experience the same stimulus repeatedly. This perceptual enhancement, called perceptual learning, has been demonstrated for various sensory systems, such as vision, audition, and somatosensation. I investigated the contribution of lateral excitatory and inhibitory synaptic balance to perceptual learning. I constructed a simple associative neural network model in which sensory features were expressed by the activities of specific cell assemblies. Each neuron is sensitive to a specific sensory feature, and the neurons belonging to the same cell assembly are sensitive to the same feature. During perceptual learning processes, the network was presented repeatedly with a stimulus that was composed of a sensory feature and noise, and the lateral excitatory and inhibitory synaptic connection strengths between neurons were modified according to a pulse-timing-based Hebbian rule. Perceptual learning enhanced the cognitive performance of the network, increasing the signal-to-noise ratio of neuronal activity. I suggest here that the alteration of the synaptic balance may be essential for perceptual learning, especially when the brain tries to adopt the most suitable strategy--signal enhancement, noise reduction, or both--for a given perceptual task.     

Intraneuronal information processing, directional selectivity and memory for spatio-temporal sequences

Interacting intracellular signalling pathways can perform computations on a scale that is slower, but more fine-grained, than the interactions between neurons upon which we normally build our computational models of the brain (Bray D 1995 Nature 376 307-12). What computations might these potentially powerful intraneuronal mechanisms be performing? The answer suggested here is: storage of spatio-temporal sequences of synaptic excitation so that each individual neuron can recognize recurrent patterns that have excited it in the past. The experimental facts about directionally selective neurons in the visual system show that neurons do not integrate separately in space and time, but along straight spatio-temporal trajectories; thus, neurons have some of the capacities required to perform such a task. In the retina, it is suggested that calcium-induced calcium release (CICR) may provide the basis for directional selectivity. In the cortex, if activation mechanisms with different delays could be separately reinforced at individual synapses, then each such Hebbian super-synapse would store a memory trace of the delay between pre- and post-synaptic activity, forming an ideal basis for the memory and response to phase sequences.     

When response variability increases neural network robustness to synaptic noise

Cortical sensory neurons are known to be highly variable, in the sense that responses evoked by identical stimuli often change dramatically from trial to trial. The origin of this variability is uncertain, but it is usually interpreted as detrimental noise that reduces the computational accuracy of neural circuits. Here we investigate the possibility that such response variability might in fact be beneficial, because it may partially compensate for a decrease in accuracy due to stochastic changes in the synaptic strengths of a network. We study the interplay between two kinds of noise, response (or neuronal) noise and synaptic noise, by analyzing their joint influence on the accuracy of neural networks trained to perform various tasks. We find an interesting, generic interaction: when fluctuations in the synaptic connections are proportional to their strengths (multiplicative noise), a certain amount of response noise in the input neurons can significantly improve network performance, compared to the same network without response noise. Performance is enhanced because response noise and multiplicative synaptic noise are in some ways equivalent. So if the algorithm used to find the optimal synaptic weights can take into account the variability of the model neurons, it can also take into account the variability of the synapses. Thus, the connection patterns generated with response noise are typically more resistant to synaptic degradation than those obtained without response noise. As a consequence of this interplay, if multiplicative synaptic noise is present, it is better to have response noise in the network than not to have it. These results are demonstrated analytically for the most basic network consisting of two input neurons and one output neuron performing a simple classification task, but computer simulations show that the phenomenon persists in a wide range of architectures, including recurrent (attractor) networks and sensorimotor networks that perform coordinate transformations. The results suggest that response variability could play an important dynamic role in networks that continuously learn.     

Efficient event-driven simulation of large networks of spiking neurons and dynamical synapses

A simulation procedure is described for making feasible large-scale simulations of recurrent neural networks of spiking neurons and plastic synapses. The procedure is applicable if the dynamic variables of both neurons and synapses evolve deterministically between any two successive spikes. Spikes introduce jumps in these variables, and since spike trains are typically noisy, spikes introduce stochasticity into both dynamics. Since all events in the simulation are guided by the arrival of spikes, at neurons or synapses, we name this procedure event-driven. The procedure is described in detail, and its logic and performance are compared with conventional (synchronous) simulations. The main impact of the new approach is a drastic reduction of the computational load incurred upon introduction of dynamic synaptic efficacies, which vary organically as a function of the activities of the pre- and postsynaptic neurons. In fact, the computational load per neuron in the presence of the synaptic dynamics grows linearly with the number of neurons and is only about 6% more than the load with fixed synapses. Even the latter is handled quite efficiently by the algorithm. We illustrate the operation of the algorithm in a specific case with integrate-and-fire neurons and specific spike-driven synaptic dynamics. Both dynamical elements have been found to be naturally implementable in VLSI. This network is simulated to show the effects on the synaptic structure of the presentation of stimuli, as well as the stability of the generated matrix to the neural activity it induces.     

On the nonlearnability of a single spiking neuron

We study the computational complexity of training a single spiking neuron N with binary coded inputs and output that, in addition to adaptive weights and a threshold, has adjustable synaptic delays. A synchronization technique is introduced so that the results concerning the nonlearnability of spiking neurons with binary delays are generalized to arbitrary real-valued delays. In particular, the consistency problem for N with programmable weights, a threshold, and delays, and its approximation version are proven to be NP-complete. It follows that the spiking neurons with arbitrary synaptic delays are not properly PAC learnable and do not allow robust learning unless RP = NP. In addition, the representation problem for N, a question whether an n-variable Boolean function given in DNF (or as a disjunction of O(n) threshold gates) can be computed by a spiking neuron, is shown to be coNP-hard.     

面向图像识别的多层脉冲神经网络学习算法综述

相较于第1代和第2代神经网络,第3代神经网络的脉冲神经网络是一种更加接近于生物神经网络的模型,因此更具有生物可解释性和低功耗性。基于脉冲神经元模型,脉冲神经网络可以通过脉冲信号的形式模拟生物信号在神经网络中的传播,通过脉冲神经元的膜电位变化来发放脉冲序列,脉冲序列通过时空联合表达不仅传递了空间信息还传递了时间信息。当前面向模式识别任务的脉冲神经网络模型性能还不及深度学习,其中一个重要原因在于脉冲神经网络的学习方法不成熟,深度学习中神经网络的人工神经元是基于实数形式的输出,这使得其可以使用全局性的反向传播算法对深度神经网络的参数进行训练,脉冲序列是二值性的离散输出,这直接导致对脉冲神经网络的训练存在一定困难,如何对脉冲神经网络进行高效训练是一个具有挑战的研究问题。本文首先总结了脉冲神经网络研究领域中的相关学习算法,然后对其中主要的方法:直接监督学习、无监督学习的算法以及ANN2SNN的转换算法进行分析介绍,并对其中代表性的工作进行对比分析,最后基于对当前主流方法的总结,对未来更高效、更仿生的脉冲神经网络参数学习方法进行展望。