Structure, dynamics, coding and optimal biophysical parameters of efficient excitatory-inhibitory spiking networks.

The principle of efficient coding posits that sensory cortical networks are designed to encode maximal sensory information with minimal metabolic cost. Despite the major influence of efficient coding in neuroscience, it has remained unclear whether fundamental empirical properties of neural network activity can be explained solely based on this normative principle. Here, we rigorously derive the structural, coding, biophysical and dynamical properties of excitatory-inhibitory recurrent networks of spiking neurons that emerge directly from imposing that the network minimizes an instantaneous loss function and a time-averaged performance measure enacting efficient coding. The optimal network has biologically-plausible biophysical features, including realistic integrate-and-fire spiking dynamics, spike-triggered adaptation, and a non-stimulus-specific excitatory external input regulating metabolic cost. The efficient network has excitatory-inhibitory recurrent connectivity between neurons with similar stimulus tuning implementing feature-specific competition, similar to that recently found in visual cortex. Networks with unstructured connectivity cannot reach comparable levels of coding efficiency. The optimal biophysical parameters include 4 to 1 ratio of excitatory vs inhibitory neurons and 3 to 1 ratio of mean inhibitory-to-inhibitory vs. excitatory-to-inhibitory connectivity that closely match those of cortical sensory networks. The efficient network has biologically-plausible spiking dynamics, with a tight instantaneous E-I balance that makes them capable to achieve efficient coding of external stimuli varying over multiple time scales. Together, these results explain how efficient coding may be implemented in cortical networks and suggests that key properties of biological neural networks may be accounted for by efficient coding.

A scalable implementation of the recursive least-squares algorithm for training spiking neural networks.

Training spiking recurrent neural networks on neuronal recordings or behavioral tasks has become a popular way to study computations performed by the nervous system. As the size and complexity of neural recordings increase, there is a need for efficient algorithms that can train models in a short period of time using minimal resources. We present optimized CPU and GPU implementations of the recursive least-squares algorithm in spiking neural networks. The GPU implementation can train networks of one million neurons, with 100 million plastic synapses and a billion static synapses, about 1,000 times faster than an unoptimized reference CPU implementation. We demonstrate the code's utility by training a network, in less than an hour, to reproduce the activity of > 66, 000 recorded neurons of a mouse performing a decision-making task. The fast implementation enables a more interactive in-silico study of the dynamics and connectivity underlying multi-area computations. It also admits the possibility to train models as in-vivo experiments are being conducted, thus closing the loop between modeling and experiments.

Rapid prototyping mixed-signal development kit for tactile neural computing.

Intelligent sensor systems are essential for building modern Internet of Things applications. Embedding intelligence within or near sensors provides a strong case for analog neural computing. However, rapid prototyping of analog or mixed signal spiking neural computing is a non-trivial and time-consuming task. We introduce mixed-mode neural computing arrays for near-sensor-intelligent computing implemented with Field-Programmable Analog Arrays (FPAA) and Field-Programmable Gate Arrays (FPGA). The combinations of FPAA and FPGA pipelines ensure rapid prototyping and design optimization before finalizing the on-chip implementations. The proposed approach architecture ensures a scalable neural network testing framework along with sensor integration. The experimental set up of the proposed tactile sensing system in demonstrated. The initial simulations are carried out in SPICE, and the real-time implementation is validated on FPAA and FPGA hardware.

NNMT: Mean-Field Based Analysis Tools for Neuronal Network Models.

Mean-field theory of neuronal networks has led to numerous advances in our analytical and intuitive understanding of their dynamics during the past decades. In order to make mean-field based analysis tools more accessible, we implemented an extensible, easy-to-use open-source Python toolbox that collects a variety of mean-field methods for the leaky integrate-and-fire neuron model. The Neuronal Network Mean-field Toolbox (NNMT) in its current state allows for estimating properties of large neuronal networks, such as firing rates, power spectra, and dynamical stability in mean-field and linear response approximation, without running simulations. In this article, we describe how the toolbox is implemented, show how it is used to reproduce results of previous studies, and discuss different use-cases, such as parameter space explorations, or mapping different network models. Although the initial version of the toolbox focuses on methods for leaky integrate-and-fire neurons, its structure is designed to be open and extensible. It aims to provide a platform for collecting analytical methods for neuronal network model analysis, such that the neuroscientific community can take maximal advantage of them.

The Mean Field Approach for Populations of Spiking Neurons.

Mean field theory is a device to analyze the collective behavior of a dynamical system comprising many interacting particles. The theory allows to reduce the behavior of the system to the properties of a handful of parameters. In neural circuits, these parameters are typically the firing rates of distinct, homogeneous subgroups of neurons. Knowledge of the firing rates under conditions of interest can reveal essential information on both the dynamics of neural circuits and the way they can subserve brain function. The goal of this chapter is to provide an elementary introduction to the mean field approach for populations of spiking neurons. We introduce the general idea in networks of binary neurons, starting from the most basic results and then generalizing to more relevant situations. This allows to derive the mean field equations in a simplified setting. We then derive the mean field equations for populations of integrate-and-fire neurons. An effort is made to derive the main equations of the theory using only elementary methods from calculus and probability theory. The chapter ends with a discussion of the assumptions of the theory and some of the consequences of violating those assumptions. This discussion includes an introduction to balanced and metastable networks and a brief catalogue of successful applications of the mean field approach to the study of neural circuits.

Asynchronous and Coherent Dynamics in Balanced Excitatory-Inhibitory Spiking Networks.

Dynamic excitatory-inhibitory (E-I) balance is a paradigmatic mechanism invoked to explain the irregular low firing activity observed in the cortex. However, we will show that the E-I balance can be at the origin of other regimes observable in the brain. The analysis is performed by combining extensive simulations of sparse E-I networks composed of N spiking neurons with analytical investigations of low dimensional neural mass models. The bifurcation diagrams, derived for the neural mass model, allow us to classify the possible asynchronous and coherent behaviors emerging in balanced E-I networks with structural heterogeneity for any finite in-degree K. Analytic mean-field (MF) results show that both supra and sub-threshold balanced asynchronous regimes are observable in our system in the limit N >> K >> 1. Due to the heterogeneity, the asynchronous states are characterized at the microscopic level by the splitting of the neurons in to three groups: silent, fluctuation, and mean driven. These features are consistent with experimental observations reported for heterogeneous neural circuits. The coherent rhythms observed in our system can range from periodic and quasi-periodic collective oscillations (COs) to coherent chaos. These rhythms are characterized by regular or irregular temporal fluctuations joined to spatial coherence somehow similar to coherent fluctuations observed in the cortex over multiple spatial scales. The COs can emerge due to two different mechanisms. A first mechanism analogous to the pyramidal-interneuron gamma (PING), usually invoked for the emergence of γ-oscillations. The second mechanism is intimately related to the presence of current fluctuations, which sustain COs characterized by an essentially simultaneous bursting of the two populations. We observe period-doubling cascades involving the PING-like COs finally leading to the appearance of coherent chaos. Fluctuation driven COs are usually observable in our system as quasi-periodic collective motions characterized by two incommensurate frequencies. However, for sufficiently strong current fluctuations these collective rhythms can lock. This represents a novel mechanism of frequency locking in neural populations promoted by intrinsic fluctuations. COs are observable for any finite in-degree K, however, their existence in the limit N >> K >> 1 appears as uncertain.

Mapping input noise to escape noise in integrate-and-fire neurons: a level-crossing approach.

Noise in spiking neurons is commonly modeled by a noisy input current or by generating output spikes stochastically with a voltage-dependent hazard rate ("escape noise"). While input noise lends itself to modeling biophysical noise processes, the phenomenological escape noise is mathematically more tractable. Using the level-crossing theory for differentiable Gaussian processes, we derive an approximate mapping between colored input noise and escape noise in leaky integrate-and-fire neurons. This mapping requires the first-passage-time (FPT) density of an overdamped Brownian particle driven by colored noise with respect to an arbitrarily moving boundary. Starting from the Wiener-Rice series for the FPT density, we apply the second-order decoupling approximation of Stratonovich to the case of moving boundaries and derive a simplified hazard-rate representation that is local in time and numerically efficient. This simplification requires the calculation of the non-stationary auto-correlation function of the level-crossing process: For exponentially correlated input noise (Ornstein-Uhlenbeck process), we obtain an exact formula for the zero-lag auto-correlation as a function of noise parameters, mean membrane potential and its speed, as well as an exponential approximation of the full auto-correlation function. The theory well predicts the FPT and interspike interval densities as well as the population activities obtained from simulations with colored input noise and time-dependent stimulus or boundary. The agreement with simulations is strongly enhanced across the sub- and suprathreshold firing regime compared to a first-order decoupling approximation that neglects correlations between level crossings. The second-order approximation also improves upon a previously proposed theory in the subthreshold regime. Depending on a simplicity-accuracy trade-off, all considered approximations represent useful mappings from colored input noise to escape noise, enabling progress in the theory of neuronal population dynamics.

Patient-Specific Network Connectivity Combined With a Next Generation Neural Mass Model to Test Clinical Hypothesis of Seizure Propagation.

Dynamics underlying epileptic seizures span multiple scales in space and time, therefore, understanding seizure mechanisms requires identifying the relations between seizure components within and across these scales, together with the analysis of their dynamical repertoire. In this view, mathematical models have been developed, ranging from single neuron to neural population. In this study, we consider a neural mass model able to exactly reproduce the dynamics of heterogeneous spiking neural networks. We combine mathematical modeling with structural information from non invasive brain imaging, thus building large-scale brain network models to explore emergent dynamics and test the clinical hypothesis. We provide a comprehensive study on the effect of external drives on neuronal networks exhibiting multistability, in order to investigate the role played by the neuroanatomical connectivity matrices in shaping the emergent dynamics. In particular, we systematically investigate the conditions under which the network displays a transition from a low activity regime to a high activity state, which we identify with a seizure-like event. This approach allows us to study the biophysical parameters and variables leading to multiple recruitment events at the network level. We further exploit topological network measures in order to explain the differences and the analogies among the subjects and their brain regions, in showing recruitment events at different parameter values. We demonstrate, along with the example of diffusion-weighted magnetic resonance imaging (dMRI) connectomes of 20 healthy subjects and 15 epileptic patients, that individual variations in structural connectivity, when linked with mathematical dynamic models, have the capacity to explain changes in spatiotemporal organization of brain dynamics, as observed in network-based brain disorders. In particular, for epileptic patients, by means of the integration of the clinical hypotheses on the epileptogenic zone (EZ), i.e., the local network where highly synchronous seizures originate, we have identified the sequence of recruitment events and discussed their links with the topological properties of the specific connectomes. The predictions made on the basis of the implemented set of exact mean-field equations turn out to be in line with the clinical pre-surgical evaluation on recruited secondary networks.

Integrate-and-Fire Neuron Circuit Without External Bias Voltages.

In this study, we propose an integrate-and-fire (I&F) neuron circuit using a p-n-p-n diode that utilizes a latch-up phenomenon and investigate the I&F operation without external bias voltages using mixed-mode technology computer-aided design (TCAD) simulations. The neuron circuit composed of one p-n-p-n diode, three MOSFETs, and a capacitor operates with no external bias lines, and its I&F operation has an energy consumption of 0.59 fJ with an energy efficiency of 96.3% per spike. The presented neuron circuit is superior in terms of structural simplicity, number of external bias lines, and energy efficiency in comparison with that constructed with only MOSFETs. Moreover, the neuron circuit exhibits the features of controlling the firing frequency through the amplitude and time width of the synaptic pulse despite of the reduced number of the components and no external bias lines.

Fast Simulations of Highly-Connected Spiking Cortical Models Using GPUs.

Over the past decade there has been a growing interest in the development of parallel hardware systems for simulating large-scale networks of spiking neurons. Compared to other highly-parallel systems, GPU-accelerated solutions have the advantage of a relatively low cost and a great versatility, thanks also to the possibility of using the CUDA-C/C++ programming languages. NeuronGPU is a GPU library for large-scale simulations of spiking neural network models, written in the C++ and CUDA-C++ programming languages, based on a novel spike-delivery algorithm. This library includes simple LIF (leaky-integrate-and-fire) neuron models as well as several multisynapse AdEx (adaptive-exponential-integrate-and-fire) neuron models with current or conductance based synapses, different types of spike generators, tools for recording spikes, state variables and parameters, and it supports user-definable models. The numerical solution of the differential equations of the dynamics of the AdEx models is performed through a parallel implementation, written in CUDA-C++, of the fifth-order Runge-Kutta method with adaptive step-size control. In this work we evaluate the performance of this library on the simulation of a cortical microcircuit model, based on LIF neurons and current-based synapses, and on balanced networks of excitatory and inhibitory neurons, using AdEx or Izhikevich neuron models and conductance-based or current-based synapses. On these models, we will show that the proposed library achieves state-of-the-art performance in terms of simulation time per second of biological activity. In particular, using a single NVIDIA GeForce RTX 2080 Ti GPU board, the full-scale cortical-microcircuit model, which includes about 77,000 neurons and 3 · 108 connections, can be simulated at a speed very close to real time, while the simulation time of a balanced network of 1,000,000 AdEx neurons with 1,000 connections per neuron was about 70 s per second of biological activity.

Monosynaptic inference via finely-timed spikes.

Observations of finely-timed spike relationships in population recordings have been used to support partial reconstruction of neural microcircuit diagrams. In this approach, fine-timescale components of paired spike train interactions are isolated and subsequently attributed to synaptic parameters. Recent perturbation studies strengthen the case for such an inference, yet the complete set of measurements needed to calibrate statistical models is unavailable. To address this gap, we study features of pairwise spiking in a large-scale in vivo dataset where presynaptic neurons were explicitly decoupled from network activity by juxtacellular stimulation. We then construct biophysical models of paired spike trains to reproduce the observed phenomenology of in vivo monosynaptic interactions, including both fine-timescale spike-spike correlations and firing irregularity. A key characteristic of these models is that the paired neurons are coupled by rapidly-fluctuating background inputs. We quantify a monosynapse's causal effect by comparing the postsynaptic train with its counterfactual, when the monosynapse is removed. Subsequently, we develop statistical techniques for estimating this causal effect from the pre- and post-synaptic spike trains. A particular focus is the justification and application of a nonparametric separation of timescale principle to implement synaptic inference. Using simulated data generated from the biophysical models, we characterize the regimes in which the estimators accurately identify the monosynaptic effect. A secondary goal is to initiate a critical exploration of neurostatistical assumptions in terms of biophysical mechanisms, particularly with regards to the challenging but arguably fundamental issue of fast, unobservable nonstationarities in background dynamics.

Theta-Nested Gamma Oscillations in Next Generation Neural Mass Models.

Theta-nested gamma oscillations have been reported in many areas of the brain and are believed to represent a fundamental mechanism to transfer information across spatial and temporal scales. In a series of recent experiments in vitro it has been possible to replicate with an optogenetic theta frequency stimulation several features of cross-frequency coupling (CFC) among theta and gamma rhythms observed in behaving animals. In order to reproduce the main findings of these experiments we have considered a new class of neural mass models able to reproduce exactly the macroscopic dynamics of spiking neural networks. In this framework, we have examined two set-ups able to support collective gamma oscillations: namely, the pyramidal interneuronal network gamma (PING) and the interneuronal network gamma (ING). In both set-ups we observe the emergence of theta-nested gamma oscillations by driving the system with a sinusoidal theta-forcing in proximity of a Hopf bifurcation. These mixed rhythms always display phase amplitude coupling. However, two different types of nested oscillations can be identified: one characterized by a perfect phase locking between theta and gamma rhythms, corresponding to an overall periodic behavior; another one where the locking is imperfect and the dynamics is quasi-periodic or even chaotic. From our analysis it emerges that the locked states are more frequent in the ING set-up. In agreement with the experiments, we find theta-nested gamma oscillations for forcing frequencies in the range [1:10] Hz, whose amplitudes grow proportionally to the forcing intensity and which are clearly modulated by the theta phase. Furthermore, analogously to the experiments, the gamma power and the frequency of the gamma-power peak increase with the forcing amplitude. At variance with experimental findings, the gamma-power peak does not shift to higher frequencies by increasing the theta frequency. This effect can be obtained, in our model, only by incrementing, at the same time, also the stimulation power. An effect achieved by increasing the amplitude either of the noise or of the forcing term proportionally to the theta frequency. On the basis of our analysis both the PING and the ING mechanism give rise to theta-nested gamma oscillations with almost identical features.

Enabling Spike-Based Backpropagation for Training Deep Neural Network Architectures.

Spiking Neural Networks (SNNs) have recently emerged as a prominent neural computing paradigm. However, the typical shallow SNN architectures have limited capacity for expressing complex representations while training deep SNNs using input spikes has not been successful so far. Diverse methods have been proposed to get around this issue such as converting off-the-shelf trained deep Artificial Neural Networks (ANNs) to SNNs. However, the ANN-SNN conversion scheme fails to capture the temporal dynamics of a spiking system. On the other hand, it is still a difficult problem to directly train deep SNNs using input spike events due to the discontinuous, non-differentiable nature of the spike generation function. To overcome this problem, we propose an approximate derivative method that accounts for the leaky behavior of LIF neurons. This method enables training deep convolutional SNNs directly (with input spike events) using spike-based backpropagation. Our experiments show the effectiveness of the proposed spike-based learning on deep networks (VGG and Residual architectures) by achieving the best classification accuracies in MNIST, SVHN, and CIFAR-10 datasets compared to other SNNs trained with a spike-based learning. Moreover, we analyze sparse event-based computations to demonstrate the efficacy of the proposed SNN training method for inference operation in the spiking domain.

Optimal Localist and Distributed Coding of Spatiotemporal Spike Patterns Through STDP and Coincidence Detection.

Repeating spatiotemporal spike patterns exist and carry information. Here we investigated how a single spiking neuron can optimally respond to one given pattern (localist coding), or to either one of several patterns (distributed coding, i.e., the neuron's response is ambiguous but the identity of the pattern could be inferred from the response of multiple neurons), but not to random inputs. To do so, we extended a theory developed in a previous paper (Masquelier, 2017), which was limited to localist coding. More specifically, we computed analytically the signal-to-noise ratio (SNR) of a multi-pattern-detector neuron, using a threshold-free leaky integrate-and-fire (LIF) neuron model with non-plastic unitary synapses and homogeneous Poisson inputs. Surprisingly, when increasing the number of patterns, the SNR decreases slowly, and remains acceptable for several tens of independent patterns. In addition, we investigated whether spike-timing-dependent plasticity (STDP) could enable a neuron to reach the theoretical optimal SNR. To this aim, we simulated a LIF equipped with STDP, and repeatedly exposed it to multiple input spike patterns, embedded in equally dense Poisson spike trains. The LIF progressively became selective to every repeating pattern with no supervision, and stopped discharging during the Poisson spike trains. Furthermore, tuning certain STDP parameters, the resulting pattern detectors were optimal. Tens of independent patterns could be learned by a single neuron using a low adaptive threshold, in contrast with previous studies, in which higher thresholds led to localist coding only. Taken together these results suggest that coincidence detection and STDP are powerful mechanisms, fully compatible with distributed coding. Yet we acknowledge that our theory is limited to single neurons, and thus also applies to feed-forward networks, but not to recurrent ones.

Automatically Selecting a Suitable Integration Scheme for Systems of Differential Equations in Neuron Models.

On the level of the spiking activity, the integrate-and-fire neuron is one of the most commonly used descriptions of neural activity. A multitude of variants has been proposed to cope with the huge diversity of behaviors observed in biological nerve cells. The main appeal of this class of model is that it can be defined in terms of a hybrid model, where a set of mathematical equations describes the sub-threshold dynamics of the membrane potential and the generation of action potentials is often only added algorithmically without the shape of spikes being part of the equations. In contrast to more detailed biophysical models, this simple description of neuron models allows the routine simulation of large biological neuronal networks on standard hardware widely available in most laboratories these days. The time evolution of the relevant state variables is usually defined by a small set of ordinary differential equations (ODEs). A small number of evolution schemes for the corresponding systems of ODEs are commonly used for many neuron models, and form the basis of the neuron model implementations built into commonly used simulators like Brian, NEST and NEURON. However, an often neglected problem is that the implemented evolution schemes are only rarely selected through a structured process based on numerical criteria. This practice cannot guarantee accurate and stable solutions for the equations and the actual quality of the solution depends largely on the parametrization of the model. In this article, we give an overview of typical equations and state descriptions for the dynamics of the relevant variables in integrate-and-fire models. We then describe a formal mathematical process to automate the design or selection of a suitable evolution scheme for this large class of models. Finally, we present the reference implementation of our symbolic analysis toolbox for ODEs that can guide modelers during the implementation of custom neuron models.

Training Deep Spiking Convolutional Neural Networks With STDP-Based Unsupervised Pre-training Followed by Supervised Fine-Tuning.

Spiking Neural Networks (SNNs) are fast becoming a promising candidate for brain-inspired neuromorphic computing because of their inherent power efficiency and impressive inference accuracy across several cognitive tasks such as image classification and speech recognition. The recent efforts in SNNs have been focused on implementing deeper networks with multiple hidden layers to incorporate exponentially more difficult functional representations. In this paper, we propose a pre-training scheme using biologically plausible unsupervised learning, namely Spike-Timing-Dependent-Plasticity (STDP), in order to better initialize the parameters in multi-layer systems prior to supervised optimization. The multi-layer SNN is comprised of alternating convolutional and pooling layers followed by fully-connected layers, which are populated with leaky integrate-and-fire spiking neurons. We train the deep SNNs in two phases wherein, first, convolutional kernels are pre-trained in a layer-wise manner with unsupervised learning followed by fine-tuning the synaptic weights with spike-based supervised gradient descent backpropagation. Our experiments on digit recognition demonstrate that the STDP-based pre-training with gradient-based optimization provides improved robustness, faster (~2.5 ×) training time and better generalization compared with purely gradient-based training without pre-training.

Spatio-Temporal Backpropagation for Training High-Performance Spiking Neural Networks.

Spiking neural networks (SNNs) are promising in ascertaining brain-like behaviors since spikes are capable of encoding spatio-temporal information. Recent schemes, e.g., pre-training from artificial neural networks (ANNs) or direct training based on backpropagation (BP), make the high-performance supervised training of SNNs possible. However, these methods primarily fasten more attention on its spatial domain information, and the dynamics in temporal domain are attached less significance. Consequently, this might lead to the performance bottleneck, and scores of training techniques shall be additionally required. Another underlying problem is that the spike activity is naturally non-differentiable, raising more difficulties in supervised training of SNNs. In this paper, we propose a spatio-temporal backpropagation (STBP) algorithm for training high-performance SNNs. In order to solve the non-differentiable problem of SNNs, an approximated derivative for spike activity is proposed, being appropriate for gradient descent training. The STBP algorithm combines the layer-by-layer spatial domain (SD) and the timing-dependent temporal domain (TD), and does not require any additional complicated skill. We evaluate this method through adopting both the fully connected and convolutional architecture on the static MNIST dataset, a custom object detection dataset, and the dynamic N-MNIST dataset. Results bespeak that our approach achieves the best accuracy compared with existing state-of-the-art algorithms on spiking networks. This work provides a new perspective to investigate the high-performance SNNs for future brain-like computing paradigm with rich spatio-temporal dynamics.

Coding of time-dependent stimuli in homogeneous and heterogeneous neural populations.

We compare the information transmission of a time-dependent signal by two types of uncoupled neuron populations that differ in their sources of variability: i) a homogeneous population whose units receive independent noise and ii) a deterministic heterogeneous population, where each unit exhibits a different baseline firing rate ('disorder'). Our criterion for making both sources of variability quantitatively comparable is that the interspike-interval distributions are identical for both systems. Numerical simulations using leaky integrate-and-fire neurons unveil that a non-zero amount of both noise or disorder maximizes the encoding efficiency of the homogeneous and heterogeneous system, respectively, as a particular case of suprathreshold stochastic resonance. Our findings thus illustrate that heterogeneity can render similarly profitable effects for neuronal populations as dynamic noise. The optimal noise/disorder depends on the system size and the properties of the stimulus such as its intensity or cutoff frequency. We find that weak stimuli are better encoded by a noiseless heterogeneous population, whereas for strong stimuli a homogeneous population outperforms an equivalent heterogeneous system up to a moderate noise level. Furthermore, we derive analytical expressions of the coherence function for the cases of very strong noise and of vanishing intrinsic noise or heterogeneity, which predict the existence of an optimal noise intensity. Our results show that, depending on the type of signal, noise as well as heterogeneity can enhance the encoding performance of neuronal populations.

STDP Allows Close-to-Optimal Spatiotemporal Spike Pattern Detection by Single Coincidence Detector Neurons.

Repeating spatiotemporal spike patterns exist and carry information. How this information is extracted by downstream neurons is unclear. Here we theoretically investigate to what extent a single cell could detect a given spike pattern and what the optimal parameters to do so are, in particular the membrane time constant τ. Using a leaky integrate-and-fire (LIF) neuron with homogeneous Poisson input, we computed this optimum analytically. We found that a relatively small τ (at most a few tens of ms) is usually optimal, even when the pattern is much longer. This is somewhat counter-intuitive as the resulting detector ignores most of the pattern, due to its fast memory decay. Next, we wondered if spike-timing-dependent plasticity (STDP) could enable a neuron to reach the theoretical optimum. We simulated a LIF equipped with additive STDP, and repeatedly exposed it to a given input spike pattern. As in previous studies, the LIF progressively became selective to the repeating pattern with no supervision, even when the pattern was embedded in Poisson activity. Here we show that, using certain STDP parameters, the resulting pattern detector is optimal. These mechanisms may explain how humans learn repeating sensory sequences. Long sequences could be recognized thanks to coincidence detectors working at a much shorter timescale. This is consistent with the fact that recognition is still possible if a sound sequence is compressed, played backward, or scrambled using 10-ms bins. Coincidence detection is a simple yet powerful mechanism, which could be the main function of neurons in the brain.

Exact analytical results for integrate-and-fire neurons driven by excitatory shot noise.

A neuron receives input from other neurons via electrical pulses, so-called spikes. The pulse-like nature of the input is frequently neglected in analytical studies; instead, the input is usually approximated to be Gaussian. Recent experimental studies have shown, however, that an assumption underlying this approximation is often not met: Individual presynaptic spikes can have a significant effect on a neuron's dynamics. It is thus desirable to explicitly account for the pulse-like nature of neural input, i.e. consider neurons driven by a shot noise - a long-standing problem that is mathematically challenging. In this work, we exploit the fact that excitatory shot noise with exponentially distributed weights can be obtained as a limit case of dichotomous noise, a Markovian two-state process. This allows us to obtain novel exact expressions for the stationary voltage density and the moments of the interspike-interval density of general integrate-and-fire neurons driven by such an input. For the special case of leaky integrate-and-fire neurons, we also give expressions for the power spectrum and the linear response to a signal. We verify and illustrate our expressions by comparison to simulations of leaky-, quadratic- and exponential integrate-and-fire neurons.