A time encoding machine is a real-time asynchronous mechanism for encoding amplitude information into a time sequence. We investigate the operating characteristics of a machine consisting of a feedback loop containing an adder, a linear filter, and a noninverting Schmitt trigger. We show that the amplitude information of a bandlimited signal can be perfectly recovered if the difference between any two consecutive values of the time sequence is bounded by the inverse of the Nyquist rate. We also show how to build a nonlinear inverse time decoding machine (TDM) that perfectly recovers the amplitude information from the time sequence. We demonstrate the close relationship between the recovery algorithms for time encoding and irregular sampling. We also show the close relationship between time encoding and a number of nonlinear modulation schemes including FM and asynchronous sigma-delta modulation. We analyze the sensitivity of the time encoding recovery algorithm and demonstrate how to construct a TDM that perfectly recovers the amplitude information from the time sequence and is trigger parameter insensitive. We derive bounds on the error in signal recovery introduced by the quantization of the time sequence. We compare these with the recovery error introduced by the quantization of the amplitude of the bandlimited signal when irregular sampling is employed. Under Nyquist-type rate conditions, quantization of a bandlimited signal in the time and amplitude domains are shown to be largely equivalent methods of information representation.

Perfect recovery and sensitivity analysis of time encoded bandlimited signals

Spiking neural networks for handwritten digit recognition—Supervised learning and network optimization

Methods and systems for encoding and decoding signals using a Multi-input Multi-output Time Encoding Machine (TEM) and Time Decoding Machine are disclosed herein.

Multi-input multi-output time encoding and decoding machines

As advances in integrated circuit (IC) fabrication technology reduce feature sizes to dimensions on the order of nanometers, IC designers are facing many of the problems that evolution has had to overcome in order to perform meaningful and accurate computations in biological neural circuits. In this paper, we explore the current state of IC technology including the many new and exciting opportunities “beyond CMOS.” We review the role of noise in both biological and engineered systems and discuss how “stochastic facilitation” can be used to perform useful and precise computation. We explore nondeterministic methodologies for computation in hardware and introduce the concept of stochastic electronics (SE); a new way to design circuits and increase performance in highly noisy and mismatched fabrication environments. This approach is illustrated with several circuit examples whose results demonstrate its exciting potential.

Stochastic Electronics: A Neuro-Inspired Design Paradigm for Integrated Circuits

We investigate architectures for time encoding and time decoding of visual stimuli such as natural and synthetic video streams (movies, animation). The architecture for time encoding is akin to models of the early visual system. It consists of a bank of filters in cascade with single-input multi-output neural circuits. Neuron firing is based on either a threshold-and-fire or an integrate-and-fire spiking mechanism with feedback. We show that analog information is represented by the neural circuits as projections on a set of band-limited functions determined by the spike sequence. Under Nyquist-type and frame conditions, the encoded signal can be recovered from these projections with arbitrary precision. For the video time encoding machine architecture, we demonstrate that band-limited video streams of finite energy can be faithfully recovered from the spike trains and provide a stable algorithm for perfect recovery. The key condition for recovery calls for the number of neurons in the population to be above a threshold value.

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Video Time Encoding Machines

We investigate a class of finite-dimensional time decoding algorithms that: (1) is insensitive with respect to the time-encoding parameters; (2) is highly efficient and stable; and (3) can be implemented in real time. These algorithms are based on the observation that the recovery of time encoded signals given a finite number of observations has the property that the quality of signal recovery is very high in a reduced time range. We show how to obtain a local representation of the time encoded signal in an efficient and stable manner using a Vandermonde formulation of the recovery algorithm. Once the signal values are obtained from a finite number of possibly overlapping observations, the reduced-range segments are stitched together. The signal obtained by segment stitching is subsequently filtered for improved performance in recovery. Finally, we evaluate the complexity of the algorithms and their computational requirements for real-time implementation.

An Overcomplete Stitching Algorithm for Time Decoding Machines

Time encoding is a real-time asynchronous mechanism of mapping amplitude information into a time sequence. We investigate fast algorithms for the recovery of time encoded bandlimited signals and construct an algorithm that has provably low computational complexity. We also devise a fast algorithm that is parameter-insensitive.

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Fast recovery algorithms for time encoded bandlimited signals

Time encoding is a real-time, asynchronous mechanism for encoding the amplitude information of an analog bandlimited signal into a time sequence, or time codes, based on which the signal can be reconstructed Time codes can be generated by simple nonlinear asynchronous analog circuits with low power consumption The circuits generating the time codes and the procedures carrying out the reconstruction are referred to as Time Encoding Machines (TEMs) and Time Decoding Machines (TDMs), respectively We give an overview of this novel information representation modality

Time Encoding of Bandlimited Signals, an Overview ∗

Time encoding is a novel real-time asynchronous mechanism for encoding amplitude information into a time sequence The analog bandlimited input is fed into a simple nonlinear neuron-like circuit that generates a strictly increasing time sequence based on which the signal can be reconstructed The heart of the reconstruction is solving a system of ill-conditioned linear equations The paper shows that the equations can be manipulated so that the reconstruction becomes feasible using a cellular neural network (CNN) with a banded system matrix In particular, the system is first transformed into a smaller well-conditioned system, and, then, the Lanczos process is used to lay it out into a set of even smaller systems characterized by a set of tridiagonal matrices Each of these systems can directly be solved by CNNs, whereas the preprocessing (transformation and Lanczos algorithm) and simple postprocessing phases can be partly or fully implemented by using the digital capabilities of the CNN universal machine (CNN-UM) Each step of the proposed formulation is confirmed by numerical (digital) simulations

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A time decoding realization with a CNN

Time encoding is a real-time asynchronous mechanism for encoding the amplitude information of an analog bandlimited signal into a time sequence, or time codes, based on which the signal can be reconstructed. Using a Toeplitz formulation we propose an efficient real-time reconstruction procedure. As an illustration, time encoding is carried out by an asynchronous sigma-delta modulator. The proposed method is confirmed by numerical simulations.

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E.K. Simonyi

Papers

An Overcomplete Stitching Algorithm for Time Decoding Machines

Fast recovery algorithms for time encoded bandlimited signals

Time Encoding of Bandlimited Signals, an Overview ∗

A time decoding realization with a CNN

A Toeplitz Formulation of a Real-Time Algorithm for Time Decoding Machines ∗