Integrate-and-fire Research Articles

Extending integrate-and fire model neurons to account for the effects of weak electric fields in the presence of dendrites

Transcranial brain stimulation techniques have recently sparked a strong interest in understanding the effects of weak electric fields on neuronal network dynamics (e.g. [1,2]). The collective dynamics of large populations of coupled neurons can be efficiently studied using single-compartment (point) model neurons of the integrate-and-fire (IF) type [3], which allow for a systematic model reduction at the population level [4,5], as opposed to multi-compartment Hodgkin-Huxley type models and complex morphologies. However, existing point neuron models cannot adequately reproduce the effects of an electric field on the somatic membrane potential, which are influenced by the presence of dendritic processes [2]. Here, we present an extension for IF type point neuron models to take into account the subthreshold effects of an oscillating weak uniform extracellular field, similar to those generated in the brain by transcranial electrical stimulation [6]. Based on a ball-and-stick neuron model (i.e., a passive finite dendritic cable with a lumped soma at its end) we analytically calculate the somatic membrane polarization induced by a weak extracellular electric field using the cable equation. From this polarization we derive an equivalent input current for leaky IF as well as adaptive nonlinear IF point neurons, which explicitly depends on the (soma+dendrite) neuron model and electric field parameters. The extended point neuron model can well reproduce the relationships between electric field properties (intensity, frequency) and neuronal responses (membrane polarization, sensitivity and phase), as observed by simulations of neuron models with complex morphologies and reported in the experimental literature [7]. Our point neuron model extension is simple to implement and well suited for application in IF based neural networks.

Nonlinear system identification of receptive fields from spiking neuron data

Identification of a linear filter in cascade with a spiking neuron has been previously considered [1] under the assumption that the input of the Hodgkin-Huxley spiking neuron can be measured in addition to the input and output of the circuit. An alternative method to identify a linear filter given the input to the circuit and the spike time sequence, has been proposed [2] for Integrate-and-Fire (IAF) neurons. The method was further extended [3] to nonlinear receptive fields that can be described by a Volterra series. Here we propose a new system identification methodology to identify a more general NARMAX representation of the nonlinear receptive field in cascade with an ideal-IAF neuron, based only on measurements of the input stimulus and the spike time sequence of the IAF neuron. By using an orthogonal forward selection algorithm we are able to derive the NARMAX representation of a scaled version of the nonlinear filter directly from the reconstructed input of an ideal IAF neuron. The method is further extended to leaky-IAF neurons which require estimating an additional spiking neuron parameter. We use the NARMAX methodology to identify recursively the nonlinear filter and the spiking neuron parameter in the presence of noise. Statistical and dynamical model validation tests are used to check if the identified nonlinear filter models are an adequate representation of the underlying nonlinear information processing mechanisms. The performance and robustness to noise of the proposed methods is demonstrated through numerical simulation studies. Specifically, we compared the generalized (higher-order) frequency response functions (GFRFs) of the original and the identified nonlinear filter (Figures 1 A-D) and evaluated, for different noise levels, the SNR of the model predicted output on a validation data set (Figure 1 E). Figure 1 A&B - GFRF functions of the original filter, C&D - Difference between the original GFRFs and the scaled versions for the identified filter, E - SNR between the validation signal and the identified filter output.

Structural stability for the Morris–Lecar neuron model

This paper is concerned with the biological neuron model of Morris–Lecar system of equations (Morris and Lecar, 1981) [17]. We prove the existence and uniqueness of strong solutions. In addition, we prove the continuous dependence of solutions to the leak conductance parameter.

Minimal conductance-based model of auditory coincidence detector neurons.

Sound localization is a fundamental sensory function of a wide variety of animals. The interaural time difference (ITD), an important cue for sound localization, is computed in the auditory brainstem. In our previous modeling study, we introduced a two-compartment Hodgkin-Huxley type model to investigate how cellular and synaptic specializations may contribute to precise ITD computation of the barn owl's auditory coincidence detector neuron. Although our model successfully reproduced fundamental physiological properties observed in vivo, it was unsuitable for mathematical analyses and large scale simulations because of a number of nonlinear variables. In the present study, we reduce our former model into three types of conductance-based integrate-and-fire (IF) models. We test their electrophysiological properties using data from published in vivo and in vitro studies. Their robustness to parameter changes and computational efficiencies are also examined. Our numerical results suggest that the single-compartment active IF model is superior to other reduced models in terms of physiological reproducibility and computational performance. This model will allow future theoretical studies that use more rigorous mathematical analysis and network simulations.

Event-driven contrastive divergence: neural sampling foundations.

In a recent Frontiers in Neuroscience paper (Neftci et al., 2014) we contributed an on-line learning rule, driven by spike-events in an Integrate and Fire (IF) neural network, that emulates the learning performance of Contrastive Divergence (CD) in an equivalent Restricted Boltzmann Machine (RBM) amenable to real-time implementation in spike-based neuromorphic systems. The event-driven CD framework assumes the foundations of neural sampling (Buesing et al., 2011; Maass, 2014) in mapping spike rates of a deterministic IF network onto probabilities of a corresponding stochastic neural network. In Neftci et al. (2014), we used a particular form of neural sampling previously analyzed in Petrovici et al. (2013)1, although this connection was not made sufficiently clear in the published article. The purpose of this letter is to clarify this connection, and to raise the reader's awareness to the existence of various forms of neural sampling. We highlight the differences as well as strong connections across these various forms, and suggest applications of event-driven CD in a more general setting enabled by the broader interpretations of neural sampling.

Quantifying chaotic dynamics from integrate-and-fire processes

Characterizing chaotic dynamics from integrate-and-fire (IF) interspike intervals (ISIs) is relatively easy performed at high firing rates. When the firing rate is low, a correct estimation of Lyapunov exponents (LEs) describing dynamical features of complex oscillations reflected in the IF ISI sequences becomes more complicated. In this work we discuss peculiarities and limitations of quantifying chaotic dynamics from IF point processes. We consider main factors leading to underestimated LEs and demonstrate a way of improving numerical determining of LEs from IF ISI sequences. We show that estimations of the two largest LEs can be performed using around 400 mean periods of chaotic oscillations in the regime of phase-coherent chaos. Application to real data is discussed.

Integrate-and-fire neurons driven by asymmetric dichotomous noise.

We consider a general integrate-and-fire (IF) neuron driven by asymmetric dichotomous noise. In contrast to the Gaussian white noise usually used in the so-called diffusion approximation, this noise is colored, i.e., it exhibits temporal correlations. We give an analytical expression for the stationary voltage distribution of a neuron receiving such noise and derive recursive relations for the moments of the first passage time density, which allow us to calculate the firing rate and the coefficient of variation of interspike intervals. We study how correlations in the input affect the rate and regularity of firing under variation of the model's parameters for leaky and quadratic IF neurons. Further, we consider the limit of small correlation times and find lowest order corrections to the first passage time moments to be proportional to the square root of the correlation time. We show analytically that to this lowest order, correlations always lead to a decrease in firing rate for a leaky IF neuron. All theoretical expressions are compared to simulations of leaky and quadratic IF neurons.

Mean field analysis gives accurate predictions of the behaviour of large networks of sparsely coupled and heterogeneous neurons

Large networks of integrate-and-fire (IF) model neurons are often used to simulate and study the behaviour of biologically realistic networks. However, to fully study the large network behaviour requires an exploration of large regions of a multidimensional parameter space. Such exploration is generally not feasible with large network models, due to the computational time required to simulate a network with biologically significant size. To circumvent these difficulties we use a mean-field approach, based on the work of [1]. We consider a sparsely coupled, excitatory network of 10,000 Izhikevich model neurons [2], with Destexhe-type synapses [3]. The cellular models were fit to hippocampal CA1 pyramidal neurons and have heterogeneous applied currents with a normal distribution. We derived a mean-field system for the network which consists of differential equations for the mean of the adaptation current and the synaptic conductance. As CA1 is an area that displays prominent theta oscillations [4], we used the mean-field system to study how the frequency of bursting depends on various model parameters. Figure ​Figure1A1A shows an example study. These studies were successful in guiding numerical simulations of the large network. When parameter values determined from the mean-field analysis are used in a large network simulation, bursting of the predicted frequency occurs (Figure ​(Figure1B1B). Figure 1 A. Mean-field prediction of the bursting frequency as a function of the unitary synaptic conductance and the mean applied current. B. Simulation of a network of 10,000 neurons with unitary conductance 0.058 nS and mean current 250 pA, showing an oscillation ...

Fixed-point bifurcation analysis in biological models using interval polynomials theory.

The paper proposes a systematic method for fixed-point bifurcation analysis in circadian cells and similar biological models using interval polynomials theory. The stages for performing fixed-point bifurcation analysis in such biological systems comprise (i) the computation of fixed points as functions of the bifurcation parameter and (ii) the evaluation of the type of stability for each fixed point through the computation of the eigenvalues of the Jacobian matrix that is associated with the system's nonlinear dynamics model. Stage (ii) requires the computation of the roots of the characteristic polynomial of the Jacobian matrix. This problem is nontrivial since the coefficients of the characteristic polynomial are functions of the bifurcation parameter and the latter varies within intervals. To obtain a clear view about the values of the roots of the characteristic polynomial and about the stability features they provide to the system, the use of interval polynomials theory and particularly of Kharitonov's stability theorem is proposed. In this approach, the study of the stability of a characteristic polynomial with coefficients that vary in intervals is equivalent to the study of the stability of four polynomials with crisp coefficients computed from the boundaries of the aforementioned intervals. The efficiency of the proposed approach for the analysis of fixed-point bifurcations in nonlinear models of biological neurons is tested through numerical and simulation experiments.

Interplay of intrinsic and synaptic conductances in the generation of high-frequency oscillations in interneuronal networks with irregular spiking.

High-frequency oscillations (above 30 Hz) have been observed in sensory and higher-order brain areas, and are believed to constitute a general hallmark of functional neuronal activation. Fast inhibition in interneuronal networks has been suggested as a general mechanism for the generation of high-frequency oscillations. Certain classes of interneurons exhibit subthreshold oscillations, but the effect of this intrinsic neuronal property on the population rhythm is not completely understood. We study the influence of intrinsic damped subthreshold oscillations in the emergence of collective high-frequency oscillations, and elucidate the dynamical mechanisms that underlie this phenomenon. We simulate neuronal networks composed of either Integrate-and-Fire (IF) or Generalized Integrate-and-Fire (GIF) neurons. The IF model displays purely passive subthreshold dynamics, while the GIF model exhibits subthreshold damped oscillations. Individual neurons receive inhibitory synaptic currents mediated by spiking activity in their neighbors as well as noisy synaptic bombardment, and fire irregularly at a lower rate than population frequency. We identify three factors that affect the influence of single-neuron properties on synchronization mediated by inhibition: i) the firing rate response to the noisy background input, ii) the membrane potential distribution, and iii) the shape of Inhibitory Post-Synaptic Potentials (IPSPs). For hyperpolarizing inhibition, the GIF IPSP profile (factor iii)) exhibits post-inhibitory rebound, which induces a coherent spike-mediated depolarization across cells that greatly facilitates synchronous oscillations. This effect dominates the network dynamics, hence GIF networks display stronger oscillations than IF networks. However, the restorative current in the GIF neuron lowers firing rates and narrows the membrane potential distribution (factors i) and ii), respectively), which tend to decrease synchrony. If inhibition is shunting instead of hyperpolarizing, post-inhibitory rebound is not elicited and factors i) and ii) dominate, yielding lower synchrony in GIF networks than in IF networks.

Synaptic and nonsynaptic plasticity approximating probabilistic inference.

Learning and memory operations in neural circuits are believed to involve molecular cascades of synaptic and nonsynaptic changes that lead to a diverse repertoire of dynamical phenomena at higher levels of processing. Hebbian and homeostatic plasticity, neuromodulation, and intrinsic excitability all conspire to form and maintain memories. But it is still unclear how these seemingly redundant mechanisms could jointly orchestrate learning in a more unified system. To this end, a Hebbian learning rule for spiking neurons inspired by Bayesian statistics is proposed. In this model, synaptic weights and intrinsic currents are adapted on-line upon arrival of single spikes, which initiate a cascade of temporally interacting memory traces that locally estimate probabilities associated with relative neuronal activation levels. Trace dynamics enable synaptic learning to readily demonstrate a spike-timing dependence, stably return to a set-point over long time scales, and remain competitive despite this stability. Beyond unsupervised learning, linking the traces with an external plasticity-modulating signal enables spike-based reinforcement learning. At the postsynaptic neuron, the traces are represented by an activity-dependent ion channel that is shown to regulate the input received by a postsynaptic cell and generate intrinsic graded persistent firing levels. We show how spike-based Hebbian-Bayesian learning can be performed in a simulated inference task using integrate-and-fire (IAF) neurons that are Poisson-firing and background-driven, similar to the preferred regime of cortical neurons. Our results support the view that neurons can represent information in the form of probability distributions, and that probabilistic inference could be a functional by-product of coupled synaptic and nonsynaptic mechanisms operating over several timescales. The model provides a biophysical realization of Bayesian computation by reconciling several observed neural phenomena whose functional effects are only partially understood in concert.

Granger causality network reconstruction of conductance-based integrate-and-fire neuronal systems.

Reconstruction of anatomical connectivity from measured dynamical activities of coupled neurons is one of the fundamental issues in the understanding of structure-function relationship of neuronal circuitry. Many approaches have been developed to address this issue based on either electrical or metabolic data observed in experiment. The Granger causality (GC) analysis remains one of the major approaches to explore the dynamical causal connectivity among individual neurons or neuronal populations. However, it is yet to be clarified how such causal connectivity, i.e., the GC connectivity, can be mapped to the underlying anatomical connectivity in neuronal networks. We perform the GC analysis on the conductance-based integrate-and-fire (IF) neuronal networks to obtain their causal connectivity. Through numerical experiments, we find that the underlying synaptic connectivity amongst individual neurons or subnetworks, can be successfully reconstructed by the GC connectivity constructed from voltage time series. Furthermore, this reconstruction is insensitive to dynamical regimes and can be achieved without perturbing systems and prior knowledge of neuronal model parameters. Surprisingly, the synaptic connectivity can even be reconstructed by merely knowing the raster of systems, i.e., spike timing of neurons. Using spike-triggered correlation techniques, we establish a direct mapping between the causal connectivity and the synaptic connectivity for the conductance-based IF neuronal networks, and show the GC is quadratically related to the coupling strength. The theoretical approach we develop here may provide a framework for examining the validity of the GC analysis in other settings.

Integrate and Fire Pulse Train Automaton for QRS detection

Monitoring heart activity from electrocardiograms (ECG) is crucial to avoid unnecessary fatalities; therefore, detection of QRS complex is fundamental to automated ECG monitoring. Continuous, portable 24/7 ECG monitoring requires wireless technology with constraints on power, bandwidth, area, and resolution. In order to provide continuous remote monitoring of patients and fast transmission of data to medical personnel for instantaneous intervention, we propose a methodology that converts analog inputs into pulses for ultralow power implementation. The signal encoding scheme is the time-based integrate and fire (IF) sampler from which a set of signal descriptors in the pulse domain are proposed. Furthermore, a logical decision rule for QRS detection based on morphological checking is derived. The proposed decision logic depends exclusively on relational and logical operators resulting in ultrafast recognition and can be implemented using combinatorial logic hardware to guarantee power consumption orders of magnitude lower than any microprocessor device. The algorithm was evaluated using the MIT-BIH arrhythmia database and results show that our algorithm performance is comparable to the state-of-the art software-based detection.

Glutamate gated spiking Neuron Model.

Biological neuron models mainly analyze the behavior of neural networks. Neurons are described in terms of firing rates viz an analog signal. The Izhikevich neuron model is an efficient, powerful model of spiking neuron. This model is a reduction of Hodgkin-Huxley model to a two variable system and is capable of producing rich firing patterns for many biological neurons. In this paper, the Regular Spiking (RS) neuron firing pattern is used to simulate the spiking of Glutamate gated postsynaptic membrane. Simulation is done in MATLAB environment for excitatory action of synapses. Analogous simulation of spiking of excitatory postsynaptic membrane potential is obtained.

Non-instantaneous synaptic transmission in spiking neuron networks and equivalence with delay distribution

Synapses in neuron networks filter incoming spikes with a wide variety of time constants, affecting the stability of various collective dynamics [1], the selectivity in transmitting information [2] and the reactivity to suddenly appearing exogenous stimuli [3]. Even for extremely simplified models of spiking neurons and network connectivities, theoretical approaches including non-instantaneous transmission rely on approximations valid for relatively small time scales [3,4] or on the assumption of a quasi-adiabatic dynamical regime [5]. Here, we addressed this issue working out a unified framework in which from small to large synaptic time scales the same approximated theoretical description holds for the firing rate dynamics of spiking neuron networks. Starting from single-neuron stochastic dynamics, we derived a set of ordinary differential equations for the population emission rate relying on the spectral expansion of the associated Fokker-Planck equation, eventually extending to the colored noise case a previous derivation obtained under white noise hypothesis [6]. The resulting population dynamics valid under a low-noise approximation, took into account also finite-size effects. This allowed to compare theoretical power spectra of population emission rates with those estimated from extensive network simulations performed with NEST [7]. Match between theory and simulation was investigated for different integrate-and-fire (IF) neuron models with different levels of complexity, e.g., VLSI IF (VIF [6], a generalization of the Perfect IF model), Leaky IF (LIF), and Exponential IF (EIF), confirming the effectiveness of the reduced theoretical description. Effectiveness which in turn could be further improved relying on higher-order perturbative terms in the fluctuation size of the synaptic current for any synaptic time scale. Moreover, such perturbative approach allowed to avoid the analytical difficulties of dealing with multi-dimensional Fokker-Planck equations with discontinuous boundary conditions. A simplification which helped in highlighting as main result that networks of neurons with non-instantaneous synapses can, under fairly broad assumptions, be described by equivalent networks with suitable distributions of transmission delays. A formal analogy valid from simple VIF models to more realistic point-like simplifications like LIF or EIF neurons.

Non-renewal spiking and neural dynamics - a simple theory of interspike-interval correlations in adapting neurons

There is accumulating evidence that the spiking of many neurons is not a renewal process but is characterized by correlations between interspike intervals (ISIs) [1]. These correlations are crucial for understanding signal processing in single neurons, however, their origin and structure is still poorly understood theoretically. Here, we present a simple theory of correlations in neural oscillators with spike-triggered adaptation currents, which are a major source of non-renewal spiking. These currents mediate spike-frequency adaptation and are commonly believed to result in negative correlations between adjacent ISIs. For such adapting neurons, we show that the serial correlation coefficient (SCC) is fundamentally related to the neuron's phase response curve (PRC). The relation predicts possible correlation patterns that characterize how correlations depend on the lag between ISIs. Different patterns arise from the specific interplay between nonlinear neural dynamics and adaptation dynamics. In particular, the correlation structure can be determined by a single parameter that includes the shape of the PRC as well as the strength and time scale of the adaptation current (Fig.1). For a positive PRC (type I), the SCC is always negative at lag 1. At higher lags, we find either a monotonically decaying (Fig.1A) or oscillating (Fig.1C) behavior of the SCC depending on the strength of adaptation. Similar correlation structures have been observed in different experimental studies (see e.g. [2]). Despite the distinct patterns, the total correlation as expressed by the sum of the SCC over all lags displays a universal value close to -0.5 at high firing rates and strong adaptation. As an example of a type I neuron, we discuss one-dimensional integrate-and-fire (IF) neurons with adaptation (like e.g. the adaptive exponential IF model [3]) operating in the supra-threshold regime. For these models, the different behaviors of the SCC can be explained by qualitatively different structures of the phase plane spanned by the voltage and the adaptation variables. Our theory also predicts that adapting neurons with a partly negative PRC (type II phase resetting) can additionally exhibit non-negative ISI correlations, which is indeed found in a two-dimensional IF model with adaptation (Fig.1D,E). Thus, adapting neurons can show a richer repertoire of correlation patterns than previously thought. Figure 1 Possible patterns of ISI correlations of an adapting neuron (bottom). The correlation pattern is determined by the area under the weighted PRCs (top). Shown are data for a two-dimensional noisy resonate-and-fire model with adaptation (simulations: circles, ...

Modeling intracellular silent oscillations and rhythmic discharge in olfactory bulb mitral cells

In mammals, the main olfactory bulb (MOB) is driven by air flow respiratory input into the nose. This slow (2-10 Hz) rhythmic input can be observed both at the local field potential (LFP) level, single cell discharge pattern level and intracellular slow oscillations of mitral cells (principal cells of the MOB). Using intracellular recordings in freely breathing anesthetized rats, we examined membrane potential slow oscillations (MPSO) in mitral cells and their relationship with their discharge patterns [1]. Interestingly, we found cells which showed no MSPO at resting membrane potential but had nevertheless a respiratory-synchronous discharge. Changing the excitability level of these cells with hyperpolarizing current injection favored the apparition of an MPSO while decreasing the cell discharge. We called such a dynamic silent MPSO. In this study, we use a modeling approach to show how such a silent MPSO can exist in mitral cells and favor the synchronization of cell discharge to the MOB slow rhythmic input. First, using a simple integrate-and-fire (IF) model, we showed that the presence of strong and balanced excitatory and inhibitory inputs with similar oscillatory dynamics led the model to display a silent MPSO. This silent MPSO is characterized by a flat membrane potential in spite of strong synaptic conductance oscillations. Second, assuming that the slow oscillatory input could be decomposed as a mean input and stochastic fluctuation with oscillating amplitudes, we showed that the model could display a slow rhythmic discharge similar to our experimental observations. However, MOB mitral cells have a complex anatomy with (i) an apical dendrite and a tuft receiving both excitatory nose input and local inhibitory regulation, and (ii) long lateral dendrites which receive a large recurrent inhibitory input from the MOB network. We thus wondered how our previous modeling results could generalize in this case. To this aim, we used a previously developed detailed mitral cell model and showed with numerical simulations that silent MPSO could be generated in this model. However, the precise dynamics of the oscillations depended on the main inhibitory input of the mitral cell. If most of the inhibition comes from the tuft, then the silent MPSO can quite easily be generated in the cell, and partial imprecision in the excitation-inhibition balance can be hindered by the apical dendrite filtering. However if a fraction of the inhibition is coming from the lateral dendrites, the silent MPSOs are more difficult to stabilize and the resulting MPSOs have complex shapes resembling some of our intracellular recordings. Overall, we found that mitral cell silent MPSOs with respiratory-patterned discharge, that we observed experimentally, can indeed be a consequence of the precise balance of their slow oscillatory excitatory and inhibitory inputs.

Inferring effective dynamics in large-scale networks of cortical neurons

Identifying a system entails determining effective nonlinear dynamics capable of quantitatively predict its behaviour in time and the repertoire of states under a suited choice of few key parameters [1]. System identification has been used to devise effective ways in controlling and improving the comprehension of both neuronal network and single cell dynamics [2,3]. Here, we extend a recently introduced theory-driven method for the identification of integrate-and-fire (IF) neuron networks [4], to characterize the spontaneous and evoked activity in large-scale networks of cortical neurons in-vitro [5]. The approach assumes that population bursts (PBs) observed in these biological preparations, result from the competitive interplay between a net excitatory synaptic reverberation driving the firing growth, and an activity-dependent inhibition akin to processes underlying spike-frequency adaptation (SFA) phenomena [2]. Mean-field approximation provides a theoretical framework [6] in which these two ingredients are quantitatively described by nonlinearly coupled differential equations for population firing rate and a fatigue variable, not experimentally accessible, determining the excitability level of the network. The fatigue state variable is reconstructed looking for an optimal anti-correlation between firing rates and fatigue level following electrical stimulations, which are periodically delivered to the network to elicit PBs. Inferred fatigue dynamics largely correlates both with failure rate in evoking PBs and with response delay between stimulation times and PB onsets. Nevertheless, spontaneously PBs, those that occur without exogenous stimulations, show a variability which is not fully captured by the identified two dimensional dynamics. Resorting to a principal component analysis, we further refined our system identification, and uncovered an additional hidden degree of freedom acting as an inhibitory force at the population level. This inhibition rapidly reacts to firing changes and expresses characteristic time scales of few tens of milliseconds, much shorter than the few seconds estimated for fatigue dynamics [7]. The composition of these slow and fast activity-dependent inhibitions finally allowed us to infer other dynamical features of the system such as the current-to-rate gain function of neurons involved in PBs.

A novel recovery algorithm of time encoded signals

The integrate-and-fire (IAF) neuron model is a Time Encoding Machine (TEM), which maps analog signals into a sequence of strictly increasing time events. There are a number of reconstruction algorithms that ensure perfect recovery of bandlimited input signals from spike trains [1]. For signals that are not bandlimited, or when their bandwidth is unknown, the algorithms available [2] ensure that the reconstructed signal satisfies a consistency constraint, i.e., the reconstructed stimulus generates the same spike train as the original stimulus. As noted in [2], consistent reconstruction is more relevant and useful for recovering real sensory stimuli because a good estimate of the bandwidth is often not available. The existing algorithms that reconstruct the inputs encoded with TEMs exploit, in some form, the nonuniform sampling theory. Here we propose a novel consistent algorithm for signal reconstruction from spike trains, which involves solving an associated interpolation problem based on uniformly sampled data, and present theoretical results that underpin the proposed reconstruction method. While providing similar accuracy, our algorithm is faster and thus better suited for real-time processing, because our approach does not require recalculating the basis functions used in reconstruction for different sets of spikes. In addition, the new approach provides an alternative framework to study spike processing. To show the performance of our algorithm, we compared it to the best consistent reconstruction method available [2]. The input used was a periodic signal with randomly generated Fourier coefficients. The computation time and reconstruction accuracy of each algorithm were evaluated for 100 randomly generated input sequences. The estimated probability density functions corresponding to each performance index are shown in Figures 1A&B. The average computation time, plotted as a function of the number of spikes (Figure ​(Figure1B),1B), demonstrates that computing time is almost independent on the number of spikes. The simulations were carried out in Matlab on a 3.10 GHz Intel Single Core PC workstation. Figure 1 Reconstruction performance comparison: A - computing time, B - accuracy (SNR), C - computation time as function of the number of spikes used in reconstruction.

CPU-GPU hybrid platform for efficient spiking neural-network simulation

Nowadays, research in computational neuroscience is progressively demanding both detailed biologically-plausible neuron models and, at the same time, the simulation of large-scale neural networks in order to better understand the operation of specific nervous circuits of the central nervous system. To that aim, several neural simulators have been developed during last decades; these simulators have been conceived to either simulate detailed neuron models within small-scale neural networks (NEURON [1] and GENESIS [2]), or to simulate neuron models with low degree of biophysical detail within large-scale neural networks (Brian [3] and NEST [4]). In view of this situation, it would be desirable to go a step further in simulating neural networks and combine fast-and-simple neural models with detailed biologically-plausible neurons within large-scale neural networks. To achieve this goal, we have integrated in our generic neural simulator (EDLUT) [5] two simulation methods (time driven and event driven) with two different hardware processing architectures (CPU and GPU). (For an overview about current types of simulation strategies and algorithms the reader is referred to Brette [6]). It is well known that traditional CPU architectures present good performance when simulating small-scale neural network, however, they still present some drawbacks in case of large-scale simulations. To avoid CPU-based architecture limitations, we have enhanced our simulator efficiency by making it capable of running in a hybrid CPU-GPU platform. In this on-trend CPU-GPU platform, processors can operate conjointly to accomplish a real-time simulation taking full advantage of GPU high performance and CPU versatility. In this work we present how our neural network simulator EDLUT has been upgraded implementing most of those techniques referred in [5] making it capable of being configured in an hybrid event-and-time-driven simulating architecture (results show that using this architecture, up to 50 times higher speed-up values are obtained than using a stand-alone CPU approach). Once relatively-large-scale simulations can be achieved, the detailed biological neuron models to be simulated become fundamental. These neuron model simulations are intrinsically associated to their techniques of numerical integration. In other words, while numerical integration methods such as Euler and Runge-Kutta are computationally efficient and suitable for neuron models with non-stiff dynamics, more complex numerical integration methods, such as backward differentiation formula (BDF), are usually required for more complex neuron models. We have extended EDLUT with the ability of dealing with complex neural models and also with detailed network characteristics such as spike propagation delays. This point has been revealed as a key factor for learning laws based on spike-timing dependent plasticity. To sum up, we present here an event-and-time-driven neural network simulator in a hybrid CPU-GPU platform that can simultaneously use simple or complex integration methods.