Martingales and the characteristic functions of absorption time on bipartite graphs

Evolutionary graph theory investigates how spatial constraints affect processes that model evolutionary selection, e.g. the Moran process. Its principal goals are to find the fixation probability and the conditional distributions of fixation time, and show how they are affected by different graphs that impose spatial constraints. Fixation probabilities have generated significant attention, but much less is known about the conditional time distributions, even for simple graphs. Those conditional time distributions are difficult to calculate, so we consider a close proxy to it: the number of times the mutant population size changes before absorption. We employ martingales to obtain the conditional characteristic functions (CCFs) of that proxy for the Moran process on the complete bipartite graph. We consider the Moran process on the complete bipartite graph as an absorbing random walk in two dimensions. We then extend Wald's martingale approach to sequential analysis from one dimension to two. Our expressions for the CCFs are novel, compact, exact, and their parameter dependence is explicit. We show that our CCFs closely approximate those of absorption time. Martingales provide an elegant framework to solve principal problems of evolutionary graph theory. It should be possible to extend our analysis to more complex graphs than we show here

Neuromorphic Engineering Editors' Pick 2021

This collection showcases well-received spontaneous articles from the past couple of years, which have been specially handpicked by our Chief Editors, Profs. André van Schaik and Bernabé Linares-Barranco. The work presented here highlights the broad diversity of research performed across the section and aims to put a spotlight on the main areas of interest. All research presented here displays strong advances in theory, experiment, and methodology with applications to compelling problems. This collection aims to further support Frontiers’ strong community by recognizing highly deserving authors

Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality

Evolutionary graph theory (EGT) investigates the Moran birth– death process constrained by graphs. Its two principal goals are to find the fixation probability and time for some initial population of mutants on the graph. The fixation probability of graphs has received considerable attention. Less is known about the distribution of fixation time. We derive clean, exact expressions for the full conditional characteristic functions (CCFs) of a close proxy to fixation and extinction times. That proxy is the number of times that the mutant population size changes before fixation or extinction. We derive these CCFs from a product martingale that we identify for an evolutionary graph with any number of partitions. The existence of that martingale only requires that the connections between those partitions are of a certain type. Our results are the first expressions for the CCFs of any proxy to fixation time on a graph with any number of partitions. The parameter dependence of our CCFs is explicit, so we can explore how they depend on graph structure. Martingales are a powerful approach to study principal problems of EGT. Their applicability is invariant to the number of partitions in a graph, so we can study entire families of graphs simultaneously

An optimized multi-layer spiking neural network implementation in FPGA without multipliers

This paper presents an expansion and evaluation of the hardware architecture for the Optimized Deep Event-driven Spiking Neural Network Architecture (ODESA). ODESA is a state-of-the-art, event-driven multi-layer Spiking Neural Network (SNN) architecture that offers an end-to-end, online, and local supervised training method. In previous work, ODESA was successfully implemented on Field-Programmable Gate Array (FPGA) hardware, showcasing its effectiveness in resource-constrained hardware environments. Building upon the previous implementation, this research focuses on optimizing the ODESA network hardware by introducing a novel approach. Specifically, we propose substituting the dot product multipliers in the Neurons with a low-cost shift-register design. This optimization strategy significantly reduces the hardware resources required for implementing a neuron, thereby enabling more complex SNNs to be accommodated within a single FPGA. Additionally, this optimization results in a reduction in power consumption, further enhancing the practicality and efficiency of the hardware implementation. To evaluate the effectiveness of the proposed optimization, extensive experiments and measurements were conducted. The results demonstrate the successful reduction in hardware resource utilization while maintaining the network's functionality and performance. Moreover, the power consumption reduction contributes to the overall energy efficiency of the hardware implementation

Event-based object detection and tracking for space situational awareness

In this work, we present an optical space imaging dataset using a range of event-based neuromorphic vision sensors. The unique method of operation of event-based sensors makes them ideal for space situational awareness (SSA) applications due to the sparseness inherent in space imaging data. These sensors offer significantly lower bandwidth and power requirements making them particularly well suited for use in remote locations and space-based platforms. We present the first publicly-accessible event-based space imaging dataset including recordings using sensors from multiple providers, greatly lowering the barrier to entry for other researchers given the scarcity of such sensors and the expertise required to operate them for SSA applications. The dataset contains both day time and night time recordings, including simultaneous co-collections from different event-based sensors. Recorded at a remote site, and containing 572 labeled targets with a wide range of sizes, trajectories, and signal-to-noise ratios, this real-world event-based dataset represents a challenging detection and tracking task that is not readily solved using previously proposed methods. We propose a highly optimized and robust feature-based detection and tracking method, designed specifically for SSA applications, and implemented via a cascade of increasingly selective event filters. These filters rapidly isolate events associated with space objects, maintaining the high temporal resolution of the sensors. The results from this simple yet highly optimized algorithm on the space imaging dataset demonstrate robust high-speed event-based detection and tracking which can readily be implemented on sensor platforms in space as well as terrestrial environments

Event-based processing of single photon avalanche diode sensors

Single Photon Avalanche Diode sensor arrays operating in direct time of flight mode can perform 3D imaging using pulsed lasers. Operating at high frame rates, SPAD imagers typically generate large volumes of noisy and largely redundant spatio-temporal data. This results in communication bottlenecks and unnecessary data processing. In this work, we propose a neuromorphic processing solution to this problem. By processing the spatio-temporal patterns generated by the SPADs in a local, event-based manner, the proposed 128 imes 128 pixel sensor-processor system reduces the size of output data from the sensor by orders of magnitude while increasing the utility of the output data in the context of challenging recognition tasks. To test the proposed system, the first large scale complex SPAD imaging dataset is captured using an existing 32 imes 32 pixel sensor. The generated dataset consists of 24000 recordings and involves high-speed view-invariant recognition of airplanes with background clutter. The frame-based SPAD imaging dataset is converted via several alternative methods into event-based data streams and processed using the proposed 125 imes 125 receptive field neuromorphic processor as well as a range of feature extractor networks and pooling methods. The output of the proposed event generation methods are then processed by an event-based feature extraction and classification system implemented in FPGA hardware. The event-based processing methods are compared to processing the original frame-based dataset via frame-based but otherwise identical architectures. The results show the event-based methods are superior to the frame-based approach both in terms of classification accuracy and output data-rate

Noise-robust text-dependent speaker identification using cochlear models

One challenging issue in speaker identification (SID) is to achieve noise-robust performance. Humans can accurately identify speakers, even in noisy environments. We can leverage our knowledge of the function and anatomy of the human auditory pathway to design SID systems that achieve better noise-robust performance than conventional approaches. We propose a text-dependent SID system based on a real-time cochlear model called cascade of asymmetric resonators with fast-acting compression (CARFAC). We investigate the SID performance of CARFAC on signals corrupted by noise of various types and levels. We compare its performance with conventional auditory feature generators including mel-frequency cepstrum coefficients, frequency domain linear predictions, as well as another biologically inspired model called the auditory nerve model. We show that CARFAC outperforms other approaches when signals are corrupted by noise. Our results are consistent across datasets, types and levels of noise, different speaking speeds, and back-end classifiers. We show that the noise-robust SID performance of CARFAC is largely due to its nonlinear processing of auditory input signals. Presumably, the human auditory system achieves noise-robust performance via inherent nonlinearities as well

Event-based feature extraction using adaptive selection thresholds

Unsupervised feature extraction algorithms form one of the most important building blocks in machine learning systems. These algorithms are often adapted to the event-based domain to perform online learning in neuromorphic hardware. However, not designed for the purpose, such algorithms typically require significant simplification during implementation to meet hardware constraints, creating trade offs with performance. Furthermore, conventional feature extraction algorithms are not designed to generate useful intermediary signals which are valuable only in the context of neuromorphic hardware limitations. In this work a novel event-based feature extraction method is proposed that focuses on these issues. The algorithm operates via simple adaptive selection thresholds which allow a simpler implementation of network homeostasis than previous works by trading off a small amount of information loss in the form of missed events that fall outside the selection thresholds. The behavior of the selection thresholds and the output of the network as a whole are shown to provide uniquely useful signals indicating network weight convergence without the need to access network weights. A novel heuristic method for network size selection is proposed which makes use of noise events and their feature representations. The use of selection thresholds is shown to produce network activation patterns that predict classification accuracy allowing rapid evaluation and optimization of system parameters without the need to run back-end classifiers. The feature extraction method is tested on both the N-MNIST (Neuromorphic-MNIST) benchmarking dataset and a dataset of airplanes passing through the field of view. Multiple configurations with different classifiers are tested with the results quantifying the resultant performance gains at each processing stage

Real-time event-based unsupervised feature consolidation and tracking for space situational awareness

Earth orbit is a limited natural resource that hosts a vast range of vital space-based systems that support the international community's national, commercial and defence interests. This resource is rapidly becoming depleted with over-crowding in high demand orbital slots and a growing presence of space debris. We propose the Fast Iterative Extraction of Salient targets for Tracking Asynchronously (FIESTA) algorithm as a robust, real-time and reactive approach to optical Space Situational Awareness (SSA) using Event-Based Cameras (EBCs) to detect, localize, and track Resident Space Objects (RSOs) accurately and timely. We address the challenges of the asynchronous nature and high temporal resolution output of the EBC accurately, unsupervised and with few tune-able parameters using concepts established in the neuromorphic and conventional tracking literature. We show this algorithm is capable of highly accurate in-frame RSO velocity estimation and average sub-pixel localization in a simulated test environment to distinguish the capabilities of the EBC and optical setup from the proposed tracking system. This work is a fundamental step toward accurate end-to-end real-time optical event-based SSA, and developing the foundation for robust closed-form tracking evaluated using standardized tracking metrics

A simple sequent calculus for nominal logic

The front end of the human auditory system, the cochlea, converts sound signals from the outside world into neural impulses transmitted along the auditory pathway for further processing. The cochlea senses and separates sound in a nonlinear active fashion, exhibiting remarkable sensitivity and frequency discrimination. Although several electronic models of the cochlea have been proposed and implemented, none of these are able to reproduce all the characteristics of the cochlea, including large dynamic range, large gain and sharp tuning at low sound levels, and low gain and broad tuning at intense sound levels. Here, we implement the 'Cascade of Asymmetric Resonators' (CAR) model of the cochlea on an FPGA. CAR represents the basilar membrane filter in the 'Cascade of Asymmetric Resonators with Fast-Acting Compression' (CAR-FAC) cochlear model. CAR-FAC is a neuromorphic model of hearing based on a pole-zero filter cascade model of auditory filtering. It uses simple nonlinear extensions of conventional digital filter stages that are well suited to FPGA implementations, so that we are able to implement up to 1224 cochlear sections on Virtex-6 FPGA to process sound data in real time. The FPGA implementation of the electronic cochlea described here may be used as a front-end sound analyser for various machine-hearing applications