Channel Identification Machines

Abstract

We present a formal methodology for identifying a channel in a system consisting of a communication channel in cascade with an asynchronous sampler. The channel is modeled as a multidimensional filter, while models of asynchronous samplers are taken from neuroscience and communications and include integrate-and-fire neurons, asynchronous sigma/delta modulators and general oscillators in cascade with zero-crossing detectors. We devise channel identification algorithms that recover a projection of the filter(s) onto a space of input signals loss-free for both scalar and vector-valued test signals. The test signals are modeled as elements of a reproducing kernel Hilbert space (RKHS) with a Dirichlet kernel. Under appropriate limiting conditions on the bandwidth and the order of the test signal space, the filter projection converges to the impulse response of the filter. We show that our results hold for a wide class of RKHSs, including the space of finite-energy bandlimited signals. We also extend our channel identification results to noisy circuits.

1. Introduction

Signal distortions introduced by a communication channel can severely affect the reliability of communication systems. If properly utilized, knowledge of the channel response can lead to a dramatic improvement in the performance of a communication link. In practice, however, information about the channel is rarely available a priori and the channel needs to be identified at the receiver. A number of channel identification methods [1] have been proposed for traditional clock-based systems that rely on the classical sampling theorem [2, 3]. However, there is a growing need to develop channel identification methods for asynchronous nonlinear systems, of which time encoding machines (TEMs) [4] are a prime example.

TEMs naturally arise as models of early sensory systems in neuroscience [5, 6] as well as models of nonlinear samplers in signal processing and analog-to-discrete (A/D) converters in communication systems [4, 6]. Unlike traditional clock-based amplitude-domain devices, TEMs encode analog signals as a strictly increasing sequence of irregularly spaced times (𝑡_𝑘)_𝑘∈ℤ. As such, they are closely related to irregular (amplitude) samplers [4, 7] and, due to their asynchronous nature, are inherently low-power devices [8]. TEMs are also readily amenable to massive parallelization [9]. Furthermore, under certain conditions, TEMs faithfully represent analog signals in the time domain; given the parameters of the TEM and the time sequence at its output, a time decoding machine (TDM) can recover the encoded signal loss-free [4, 5].

A general TEM of interest is shown in Figure 1. An analog multidimensional signal 𝐮 is passed through a channel with memory that models physical communication links. We assume that the effect of this channel on the signal 𝐮 can be described by a linear multidimensional filter. The output of the channel 𝑣 is then mapped, or encoded, by a nonlinear asynchronous sampler into the time sequence (𝑡_𝑘)_𝑘∈ℤ. A few examples of samplers include asynchronous A/D converters such as the one based on an asynchronous sigma/delta modulator (ASDM) [4], nonlinear oscillators such as the van der Pol oscillator in cascade with a zero-crossing detector (ZCD) [6], and spiking neurons such as the integrate-and-fire (IAF) or the threshold-and-fire (TAF) neurons [9]. The above-mentioned asynchronous samplers incorporate the temporal dynamics of spike (pulse) generation and allow one to consider, in particular for neuroscience applications, more biologically plausible nonlinear spike generation (sampling) mechanisms.

Figure 1 

Modeling the channel identification problem. A known multidimensional signal 𝑢(𝑡), 𝑡∈ℝ, is first passed through a communication channel. A nonlinear sampler then maps the output 𝑣 of the channel into an observable time sequence (𝑡𝑘)𝑘∈ℤ.

In this paper, we investigate the following nonlinear identification problem: given both the input signal 𝐮 and the time sequence (𝑡_𝑘)_𝑘∈ℤ at the output of a TEM, what is the channel filter? System identification problems of this kind are key to understanding the nature of neural encoding and processing [10–14], process modeling and control [15], and, more generally, methods for constructing mathematical models of dynamical systems [16].

Identification of the channel from a time sequence is to be contrasted with existing methods for rate-based models in neuroscience (see [10] for an extensive review). In such models the output of the system is taken to be its instantaneous response rate and the nonlinear generation of a time sequence is not explicitly modeled. Furthermore, in order to fit model parameters, identification methods for such models typically require the response rate to be known [17]. This is often difficult in practice since the same experiment needs to be repeated a large number of times to estimate the response rate. Moreover, the use of the same stimulus typically introduces a systematic bias during the identification procedure [10].

The channel identification methodology presented in this paper employs test signals that are neither white nor have stationary statistics (e.g., Gaussian with a fixed mean/variance). This is a radical departure from the widely employed nonlinear system identification methods [10], including the spike-triggered average [18] and the spike-triggered covariance [19] methods. We carry out the channel identification using input signals that belong to reproducing kernel Hilbert spaces (RKHSs), and, in particular, spaces of bandlimited functions, that is, functions that have a finite support in the frequency domain. The latter signals are extensively used to describe sensory stimuli in biological systems and to model signals in communications. We show that for such signals the channel identification problem becomes mathematically tractable. Furthermore, we demonstrate that the choice of the input signal space profoundly effects the type of identification results that can be achieved.

The paper is organized as follows. In Section 2, we introduce three application-driven examples of the system in Figure 1 and formally state the channel identification problem. In Section 3, we present the single-input single-output (SISO) channel identification machine (CIM) for the finite-dimensional input signal space of trigonometric polynomials. Using analytical methods and simulations, we demonstrate that it is possible to identify the projection of the filter onto the input space loss-free and show that the SISO CIM algorithm can recover the original filter with arbitrary precision, provided that both the bandwidth and the order of the input space are sufficiently high. Then, in Section 4, we extend our methodology to multidimensional systems and present multi-input single-output (MISO) CIM algorithms for the identification of vector-valued filters modeling the channel. We generalize our methods to classes of RKHSs of input signals in Section 5.1 and work out in detail channel identification algorithms for infinite-dimensional Paley-Wiener spaces. In Section 5.2 we discuss extensions of our identification results to noisy systems, where additive noise is introduced either by the channel or the asynchronous sampler. Finally, Section 6 concludes our work.

2. The Channel Identification Problem

We investigate a general I/O system comprised of a filter or a bank of filters (i.e., a linear operator) in cascade with an asynchronous (nonlinear) sampler (Figure 1). The I/O circuit belongs to the class of [Filter]-[Asynchronous Sampler] circuits. In general terms, the input to such a system is a vector-valued analog signal 𝐮=[𝑢¹(𝑡),𝑢²(𝑡),…,𝑢^𝑀(𝑡)]^𝑇, 𝑡∈ℝ, 𝑀∈ℕ, and the output is a time sequence (𝑡_𝑘)_𝑘∈ℤ generated by its asynchronous sampling mechanism. In the neural coding literature, such a system is called a time encoding machine (TEM) [4] as it encodes an unknown signal 𝐮 into an observable time sequence (𝑡_𝑘)_𝑘∈ℤ.

2.1. Examples of Asynchronous SISO and MISO Systems

An instance of the TEM in Figure 1 is the SISO [Filter]-[Ideal IAF] neural circuit depicted in Figure 2(a). Here the filter is used to model the aggregate processing of a stimulus performed by the dendritic tree of a sensory neuron. The output of the filter 𝑣 is encoded into the sequence of spike times (𝑡_𝑘)_𝑘∈ℤ by an ideal integrate-and-fire neuron. Identification of dendritic processing in such a circuit is an important problem in systems neuroscience. It was first investigated in [20]. Another instance of the system in Figure 1 is the SISO [Filter]-[Nonlinear Oscillator-ZCD] circuit shown in Figure 2(b). In contrast to the first example, where the input was coupled additively, in this circuit the biased filter output 𝑣 is coupled multiplicatively into a nonlinear oscillator. The zero-crossing detector then generates a time sequence (𝑡_𝑘)_𝑘∈ℤ by extracting zeros from the observable modulated waveform at the output of the oscillator. Called a TEM with multiplicative coupling [6], this circuit is encountered in generalized frequency modulation [21].

Figure 2 

Examples of systems arising in neuroscience and communications. (a) Single-input single-output model of a sensory neuron. (b) Single-input single-output nonlinear oscillator in cascade with a zero-crossing detector. (c) Multi-input single-output analog-to-discrete converter implemented with an asynchronous sigma-delta modulator. 𝑀 liner filters model 𝑀 (different) communication links.

An example of a MISO system is the [Filter]-[ASDM-ZCD] circuit shown in Figure 2(c). Similar circuits arise practically in all modern-day A/D converters and constitute important front-end components of measurement and communication systems. Each signal 𝑢^𝑚(𝑡), 𝑡∈ℝ, 𝑚=1,2,…,𝑀, is transmitted through a communication channel and the effect of the channel on each signal is modeled using a linear filter with an impulse response ℎ^𝑚(𝑡), 𝑡∈ℝ, 𝑚=1,2,…,𝑀. The aggregate channel output 𝑣(𝑡)=∑^𝑀_𝑚=1𝑣^𝑚(𝑡)=∑^𝑀_𝑚=1(𝑢^𝑚∗ℎ^𝑚)(𝑡), where 𝑢^𝑚∗ℎ^𝑚 denotes the convolution of 𝑢^𝑚 with ℎ^𝑚, is additively coupled into an ASDM. Specifically, 𝑣(𝑡) is passed through an integrator and a noninverting Schmitt trigger to produce a binary output 𝑧(𝑡)∈{−𝑏,𝑏}, 𝑡∈ℝ. A zero-crossing detector is then used to extract the sequence of zero-crossing times (𝑡_𝑘)_𝑘∈ℤ from 𝑧(𝑡). Thus, the output of this [Filter]-[ASDM-ZCD] circuit is the time sequence (𝑡_𝑘)_𝑘∈ℤ.

2.2. Modeling the Input Space

We model channel input signals 𝑢=𝑢(𝑡), 𝑡∈ℝ, as elements of the space of trigonometric polynomials ℋ (see Section 5.1 for more general input spaces).

Definition 1. The space of trigonometric polynomials ℋ is a Hilbert space of complex-valued functions 𝑢(𝑡)=1√𝑇^𝐿_𝑙=−𝐿𝑢_𝑙exp𝑗𝑙Ω𝑡𝐿,𝑡∈[0,𝑇], (1) where 𝑢_𝑙∈ℂ, Ω is the bandwidth, 𝐿 is the order and 𝑇=2𝜋𝐿/Ω, endowed with the inner product ⟨⋅,⋅⟩∶ℋ×ℋ→ℂ⟨𝑢,𝑤⟩=^𝑇₀𝑢(𝑡)𝑤(𝑡)𝑑𝑡. (2) Given the inner product in (2), the set of elements 𝑒_𝑙(𝑡)=1√𝑇exp𝑗𝑙Ω𝑡𝐿,𝑙=−𝐿,−𝐿+1,…,𝐿, (3) forms an orthonormal basis in ℋ. Thus, any element 𝑢∈ℋ and any inner product ⟨𝑢,𝑤⟩ can be compactly written as 𝑢=∑^𝐿_𝑙=−𝐿𝑢_𝑙𝑒_𝑙 and ⟨𝑢,𝑤⟩=∑^𝐿_𝑙=−𝐿𝑢_𝑙𝑤_𝑙. Moreover, ℋ is a reproducing kernel Hilbert space (RKHS) with a reproducing kernel (RK) given by 𝐾(𝑠,𝑡)=^𝐿_𝑙=−𝐿𝑒_𝑙(𝑠)𝑒_𝑙(𝑡)=1𝑇^𝐿_𝑙=−𝐿exp𝑗𝑙Ω𝐿(𝑠−𝑡), (4) also known as a Dirichlet kernel [22].

We note that a function 𝑢∈ℋ satisfies 𝑢(0)=𝑢(𝑇). There is a natural connection between functions on an interval of length 𝑇 that take on the same values at interval end-points and functions on ℝ that are 𝑇-periodic: both provide equivalent descriptions of the same mathematical object, namely a function on a circle. By abuse of notation, in what follows 𝑢 will denote both a function defined on an interval of length 𝑇 and a function defined on the entire real line. In the latter case, the function 𝑢 is simultaneously periodic with period 𝑇 and bandlimited with bandwidth Ω, that is, it has a finite spectral support supp(ℱ𝑢)⊆[−Ω,Ω], where ℱ denotes the Fourier transform. In what follows we will assume that 𝑢_𝑙≠0 for all 𝑙=−𝐿,−𝐿+1,…,𝐿, that is, a signal 𝑢∈ℋ contains all 2𝐿+1 frequency components.

2.3. Modeling the Channel and Channel Identification

The channel is modeled as a bank of 𝑀 filters with impulse responses ℎ^𝑚, 𝑚=1,2,…,𝑀. We assume that each filter is linear, causal, BIBO-stable and has a finite temporal support of length 𝑆≤𝑇, that is, it belongs to the space 𝐻={ℎ∈𝕃¹(ℝ)∣supp(ℎ)⊆[0,𝑇]}. Since the length of the filter support is smaller than or equal to the period of an input signal, we effectively require that for a given 𝑆 and a fixed input signal bandwidth Ω, the order 𝐿 of the space ℋ satisfies 𝐿≥𝑆⋅Ω/(2𝜋). The aggregate channel output is given by 𝑣(𝑡)=∑^𝑀_𝑚=1(𝑢^𝑚∗ℎ^𝑚)(𝑡). The asynchronous sampler maps the input signal 𝑣 into the output time sequence (𝑡_𝑘)^𝑛_𝑘=1, where 𝑛 denotes the total number of spikes produced on an interval 𝑡∈[0,𝑇].

Definition 2. A signal 𝐮∈ℋ^𝑀 at the input to a [Filter]-[Asynchronous Sampler] circuit together with the resulting output 𝕋=(𝑡_𝑘)^𝑛_𝑘=1 of that circuit is called an input/output (I/O) pair and is denoted by (𝐮,𝕋).

We are now in a position to define the channel identification problem.

Definition 3. Let (𝐮^𝑖), 𝑖=1,2,…,𝑁, be a set of 𝑁 signals from a test space ℋ^𝑀. A channel identification machine implements an algorithm that estimates the impulse response of the filter from the I/O pairs (𝐮^𝑖,𝕋^𝑖), 𝑖=1,2,…,𝑁, of the [Filter]-[Asynchronous Sampler] circuit.

Remark 4. We note that a CIM recovers the impulse response of the filter based on the knowledge of I/O pairs (𝐮^𝑖,𝕋^𝑖), 𝑖=1,2,…,𝑁, and the sampler circuit. In contrast, a time decoding machine recovers an encoded signal 𝐮 based on the knowledge of the entire TEM circuit (both the channel filter and the sampler) and the output time sequence 𝕋.

3. SISO Channel Identification Machines

As already mentioned, the circuits under investigation consist of a channel and an asynchronous sampler. Throughout this paper, we will assume that the structure and the parameters of the asynchronous sampler are known. We start by formally describing asynchronous channel measurements in Section 3.1. Channel identification algorithms from asynchronous measurements are given in Section 3.2. Examples characterizing the performance of the identification algorithms are discussed in Section 3.3.

3.1. Asynchronous Measurements of the Channel Output

Consider the SISO [Filter]-[Ideal IAF] neural circuit in Figure 2(a). In this circuit, an input signal 𝑢∈ℋ is passed through a filter with an impulse response (or kernel) ℎ∈𝐻 and then encoded by an ideal IAF neuron with a bias 𝑏∈ℝ₊, a capacitance 𝐶∈ℝ₊, and a threshold 𝛿∈ℝ₊. The output of the circuit is a sequence of spike times (𝑡_𝑘)^𝑛_𝑘=1 on the time interval [0,𝑇] that is available to an observer. This neural circuit is an instance of a TEM and its operation can be described by a set of equations ^𝑡𝑘+1_𝑡𝑘(𝑢∗ℎ)(𝑠)𝑑𝑠=𝑞_𝑘,𝑘=1,2,…,𝑛−1, (5) where 𝑞_𝑘=𝐶𝛿−𝑏(𝑡_𝑘+1−𝑡_𝑘). Intuitively, at every spike time 𝑡_𝑘+1 the ideal IAF neuron is providing a measurement 𝑞_𝑘 of the signal 𝑣(𝑡)=(𝑢∗ℎ)(𝑡) on the time interval [𝑡_𝑘,𝑡_𝑘+1).

Definition 5. The mapping of an analog signal 𝑢(𝑡), 𝑡∈ℝ, into an increasing sequence of times (𝑡_𝑘)_𝑘∈ℤ (as in (5)) is called the 𝑡-transform [4].

Definition 6. The operator 𝒫∶𝐻→ℋ given by (𝒫ℎ)(𝑡)=^𝑇₀ℎ(𝑠)𝐾(𝑠,𝑡)𝑑𝑠 (6) is called the projection operator.

Proposition 7 (conditional duality). For all 𝑢∈ℋ, a [Filter]-[Ideal IAF] TEM with a filter kernel ℎ is I/O-equivalent to a [Filter]-[Ideal IAF] TEM with the filter kernel 𝒫ℎ. Furthermore, the CIM algorithm for identifying the filter kernel 𝒫ℎ is equivalent to the TDM algorithm for recovering the input signal 𝒫ℎ encoded by a [Filter]-[Ideal IAF] TEM with the filter kernel 𝑢.

Proof. Since 𝑢∈ℋ, 𝑢(𝑡)=⟨𝑢(⋅),𝐾(⋅,𝑡)⟩ by the reproducing property of the kernel 𝐾(𝑠,𝑡). Hence, (𝑢∗ℎ)(𝑡)=^(𝑎)∫_ℝℎ(𝑤)𝑢(𝑡−𝑤)𝑑𝑤=^(𝑏)∫^𝑇₀ℎ(𝑤)∫^𝑇₀𝑢(𝑧)𝐾(𝑧,𝑡−𝑤)
𝑑𝑧𝑑𝑤=^(𝑐)∫^𝑇₀𝑢(𝑧)∫^𝑇₀ℎ(𝑤)𝐾(𝑤,𝑡−𝑧)𝑑𝑤𝑑𝑧=^(𝑑)∫^𝑇₀𝑢(𝑧)(𝒫ℎ)(𝑡−𝑧)𝑑𝑧⁽=^𝑒)(𝑢∗𝒫ℎ)(𝑡), where (𝑎) follows from the commutativity of convolution, (𝑏) from the reproducing property of the kernel 𝐾 and the assumption that supp(ℎ)⊆[0,𝑇], (𝑐) from the equality 𝐾(𝑧,𝑡−𝑤)=𝐾(𝑤,𝑡−𝑧), (𝑑) from the definition of 𝒫ℎ in (6), and (𝑒) from the definition of convolution for periodic functions [23]. It follows that on the interval 𝑡∈[0,𝑇], (5) can be rewritten as ^𝑡𝑘+1_𝑡𝑘(𝑢∗𝒫ℎ)(𝑠)𝑑𝑠=^{(𝑓)𝑡𝑘+1}_𝑡𝑘(𝒫ℎ∗𝑢)(𝑠)𝑑𝑠=𝑞_𝑘, (7) for all 𝑘=1,2,…,𝑛−1, where (𝑓) comes from the commutativity of convolution. The right-hand side of (7) is the 𝑡-transform of a [Filter]-[Ideal IAF] TEM with an input 𝒫ℎ and a filter that has an impulse response 𝑢. Hence, a TDM can identify 𝒫ℎ, given a filter-output pair (𝑢,𝕋).

The conditional duality between time encoding and channel identification is visualized in Figure 3. First, we note the conditional I/O equivalence between the circuit in Figure 3(a) and the original circuit in Figure 2(a). The equivalence is conditional since 𝒫ℎ is a projection onto a particular space ℋ and the two circuits are I/O-equivalent only for input signals in that space. Second, identifying the filter of the circuit in Figure 3(a) is the same as decoding the signal encoded with the circuit in Figure 3(b). Note that the filter projection 𝒫ℎ is now treated as the input to the [Filter]-[Ideal IAF] circuit and the signal 𝑢 appears as the impulse response of the filter. Effectively, we have transformed the channel identification problem into a time decoding problem and we can use the TDM machinery of [5] to identify the filter projection (𝒫ℎ)(𝑡) on 𝑡∈[0,𝑇].

(a) Conditional I/O equivalence

(b) Duality

Figure 3 

Conditional duality between channel identification and time encoding. (a) For all 𝑢∈ℋ, the [Filter]-[Ideal IAF] circuit with an input-filter pair (𝑢,ℎ) is I/O equivalent to a [Filter]-[Ideal IAF] circuit with an input-filter pair (𝑢,𝒫ℎ). (b) The input-filter pair (𝑢,𝒫ℎ) in channel identification is dual to the (𝒫ℎ,𝑢) pair in time encoding.

3.2. Channel Identification from Asynchronous Measurements

Given the parameters of the asynchronous sampler, the measurements 𝑞_𝑘 of the channel output 𝑣 can be readily computed from spike times (𝑡_𝑘)^𝑛_𝑘=1 using the definition of 𝑞_𝑘 ((5) for the IAF neuron). Furthermore, as we will now show, for a known input signal, these measurements can be reinterpreted as measurements of the channel itself.

Lemma 8. There is a function 𝜙_𝑘(𝑡)=∑^𝐿_𝑙=−𝐿𝜙_𝑙,𝑘𝑒_𝑙(𝑡)∈ℋ, such that the 𝑡-transform of the [Filter]-[Ideal IAF] neuron in (7) can be written as 𝒫ℎ,𝜙_𝑘=𝑞_𝑘, (8) and 𝜙_𝑙,𝑘=√𝑇∫^𝑡𝑘+1_𝑡𝑘𝑢_𝑙𝑒_𝑙(𝑡)𝑑𝑡 for all 𝑙=−𝐿,−𝐿+1,…,𝐿 and 𝑘=1,2,…,𝑛−1.

Proof. The linear functional ℒ_𝑘∶ℋ→ℝ defined by ℒ_𝑘(𝑤)=^𝑡𝑘+1_𝑡𝑘(𝑢∗𝑤)(𝑠)𝑑𝑠, (9) where 𝑤∈ℋ, is bounded. Thus, by the Riesz representation theorem [22], there exists a function 𝜙_𝑘∈ℋ such that ℒ_𝑘(𝑤)=⟨𝑤,𝜙_𝑘⟩, 𝑘=1,2,…,𝑛−1, and 𝑞_𝑘=ℒ_𝑘(𝒫ℎ)=∫^𝑡𝑘+1_𝑡𝑘(𝑢∗𝒫ℎ)(𝑠)𝑑𝑠=⟨𝒫ℎ,𝜙_𝑘⟩. Since 𝜙_𝑘∈ℋ, we have 𝜙_𝑘(𝑡)=∑^𝐿_𝑙=−𝐿𝜙_𝑙,𝑘𝑒_𝑙 for some 𝜙_𝑙,𝑘∈ℂ, 𝑙=−𝐿,−𝐿+1,…,𝐿. To find the latter coefficients, we note that 𝜙_𝑙,𝑘=⟨𝜙_𝑘,𝑒_𝑙⟩=⟨𝑒_𝑙,𝜙_𝑘⟩=ℒ_𝑘(𝑒_𝑙). By definition of ℒ_𝑘 in (9), ℒ_𝑘(𝑒_𝑙)=∫^𝑡𝑘+1_𝑡𝑘(𝑢∗𝑒_𝑙)(𝑡)𝑑𝑡=∫^𝑡𝑘+1_𝑡𝑘∫^𝑇₀∑^𝐿_𝑖=−𝐿𝑢_𝑖𝑒_𝑖(𝑠)𝑒_𝑙(𝑡−𝑠)𝑑𝑠𝑑𝑡=√𝑇∫^𝑡𝑘+1_𝑡𝑘𝑢_𝑙𝑒_𝑙(𝑡)𝑑𝑡.

Since 𝑞_𝑘=∫^𝑡𝑘+1_𝑡𝑘(𝑢∗𝒫ℎ)(𝑠)𝑑𝑠=⟨𝑣,𝒫1_{[𝑡𝑘,𝑡𝑘+1]}⟩, the measurements 𝑞_𝑘 are projections of 𝑣=𝑢∗𝒫ℎ onto 𝒫1_{[𝑡𝑘,𝑡𝑘+1]}, 𝑘=1,2,…,𝑛−1. Assuming that 𝑢 is known and there are enough measurements available, 𝒫ℎ can be obtained by first recovering 𝑣 from these projections and then deconvolving it with 𝑢. However, this two-step procedure does not work when the circuit is not producing enough measurements and one cannot recover 𝑣. A more direct route is suggested by Lemma 8, since the measurements (𝑞_𝑘)^𝑛−1_𝑘=1 can also be interpreted as the projections of 𝒫ℎ onto 𝜙_𝑘, that is, ⟨𝒫ℎ,𝜙_𝑘⟩, 𝑘=1,2,…,𝑛−1. A natural question then is how to identify 𝒫ℎ directly from the latter projections.

Lemma 9. Let 𝑢∈ℋ be the input to a [Filter]-[Ideal IAF] circuit with ℎ∈𝐻. If the number of spikes 𝑛 generated by the neuron in a time interval of length 𝑇 satisfies 𝑛≥2𝐿+2, then the filter projection 𝒫ℎ can be perfectly identified from the I/O pair (𝑢,𝕋) as (𝒫ℎ)(𝑡)=∑^𝐿_𝑙=−𝐿ℎ_𝑙𝑒_𝑙(𝑡), where 𝐡=Φ⁺𝐪 with [𝐪]_𝑘=𝑞_𝑘 and Φ⁺ denotes the Moore-Penrose pseudoinverse of Φ. The matrix Φ is of size (𝑛−1)×(2𝐿+1) and its elements are given by [Φ]_𝑘𝑙=𝑢_𝑙𝑡_𝑘+1−𝑡_𝑘𝑢,𝑙=0,_𝑙𝐿√𝑇𝑒_𝑙𝑡_𝑘+1−𝑒_𝑙𝑡_𝑘𝑗𝑙Ω,𝑙≠0. (10)

Proof. Since 𝒫ℎ∈ℋ, it can be written as (𝒫ℎ)(𝑡)=∑^𝐿_𝑙=−𝐿ℎ_𝑙𝑒_𝑙(𝑡). Then from (8) we have 𝑞_𝑘=𝒫ℎ,𝜙_𝑘=^𝐿_𝑙=−𝐿ℎ_𝑙𝜙_𝑙,𝑘. (11) Writing (11) for all 𝑘=1,2,…,𝑛−1, we obtain 𝐪=Φ𝐡 with [𝐪]_𝑘=𝑞_𝑘, [Φ]_𝑘𝑙=𝜙_𝑙,𝑘 and [𝐡]_𝑙=ℎ_𝑙. This system of linear equations can be solved for 𝐡, provided that the rank 𝑟(Φ) of the matrix Φ satisfies 𝑟(Φ)=2𝐿+1. A necessary condition for the latter is that the number of measurements 𝑞_𝑘 is at least 2𝐿+1, or, equivalently, the number of spikes 𝑛≥2𝐿+2. Under this condition, the solution can be computed as 𝐡=Φ⁺𝐪.

Remark 10. If the signal 𝑢 is fed directly into the neuron, then ∫^𝑡𝑘+1_𝑡𝑘(𝑢∗𝒫ℎ)(𝑡)𝑑𝑡=∫^𝑡𝑘+1_𝑡𝑘𝑢(𝑡)𝑑𝑡, for 𝑘=1,2,…,𝑛−1, that is, (𝒫ℎ)(𝑡)=𝐾(𝑡,0), 𝑡∈ℝ. In other words, if there is no processing on the input signal 𝑢, then the kernel 𝐾(𝑡,0) in ℋ is identified as the filter projection. This is also illustrated in Figure 7.
In order to ensure that the neuron produces at least 2𝐿+1 measurements in a time interval of length 𝑇, it suffices to have 𝑡_𝑘+1−𝑡_𝑘≤𝑇/(2𝐿+2). Since 𝑡_𝑘+1−𝑡_𝑘≤𝐶𝛿/(𝑏−𝑐) for |𝑣(𝑡)|≤𝑐<𝑏, it suffices to have 𝐶𝛿<(𝑏−𝑐)𝑇/(2𝐿+2). Using the definition of 𝑇=2𝜋𝐿/Ω and taking the limit as 𝐿→∞, we obtain the familiar Nyquist-type criterion 𝐶𝛿<𝜋(𝑏−𝑐)/Ω for a bandlimited stimulus 𝑢∈Ξ [4, 20] (see also Section 5.1).
Ideally, we would like to identify the impulse response of the filter ℎ. Note that unlike ℎ∈𝐻, the projection 𝒫ℎ belongs to the space ℋ. Nevertheless, under quite natural conditions on ℎ (see Section 3.4), 𝒫ℎ approximates ℎ arbitrarily closely on 𝑡∈[0,𝑇], provided that both the bandwidth and the order of the signal 𝑢 are sufficiently large (see also Figure 9).

The requirement of Lemma 9 that the number of spikes 𝑛 produced by the system in Figure 2(a) has to satisfy 𝑛≥2𝐿+2 is quite stringent and may be hard to meet in practice, especially if the order 𝐿 of the space ℋ is high. In that case we have the following result.

Theorem 11 (SISO channel identification machine). Let {𝑢^𝑖∣𝑢^𝑖∈ℋ}^𝑁_𝑖=1 be a collection of 𝑁 linearly independent stimuli at the input to a [Filter]-[Ideal IAF] circuit with ℎ∈𝐻. If the total number of spikes 𝑛=∑^𝑁_𝑖=1𝑛^𝑖 generated by the neuron satisfies 𝑛≥2𝐿+𝑁+1, then the filter projection 𝒫ℎ can be perfectly identified from a collection of I/O pairs {(𝑢^𝑖,𝕋^𝑖)}^𝑁_𝑖=1 as (𝒫ℎ)(𝑡)=^𝐿_𝑙=−𝐿ℎ_𝑙𝑒_𝑙(𝑡), (12) where 𝐡=Φ⁺𝐪. Furthermore, Φ=[Φ¹;Φ²;…;Φ^𝑁], 𝐪=[𝐪¹;𝐪²;…;𝐪^𝑁] and [𝐪^𝑖]_𝑘=𝑞^𝑖_𝑘 with each Φ^𝑖 of size (𝑛^𝑖−1)×(2𝐿+1) and 𝐪^𝑖 of size (𝑛^𝑖−1)×1. The elements of matrices Φ^𝑖 are given by Φ^𝑖_𝑘𝑙=𝑢^𝑖_𝑙𝑡^𝑖_𝑘+1−𝑡^𝑖_𝑘𝑢,𝑙=0,_𝑖𝑙𝐿√𝑇𝑒_𝑙𝑡^𝑖_𝑘+1−𝑒_𝑙𝑡^𝑖_𝑘𝑗𝑙Ω,𝑙≠0, (13) for all 𝑘=1,2,…,𝑛−1, 𝑙=−𝐿,−𝐿+1,…,𝐿, and 𝑖=1,2,…,𝑁.

Proof. Since 𝒫ℎ∈ℋ, it can be written as (𝒫ℎ)(𝑡)=∑^𝐿_𝑙=−𝐿ℎ_𝑙𝑒_𝑙(𝑡). Furthermore, since the stimuli are linearly independent, the measurements (𝑞^𝑖_𝑘)^𝑛𝑖−1_𝑘=1 provided by the IAF neuron are distinct. Writing (5) for a stimulus 𝑢^𝑖, we obtain 𝑞^𝑖_𝑘=⟨𝒫ℎ,𝜙^𝑖_𝑘⟩=^𝐿_𝑙=−𝐿ℎ_𝑙𝜙^𝑖_𝑙,𝑘, (14) or 𝐪^𝑖=Φ^𝑖𝐡, with [𝐪^𝑖]_𝑘=𝑞^𝑖_𝑘, [Φ^𝑖]_𝑘𝑙=𝜙^𝑖_𝑙,𝑘 and [𝐡]_𝑙=ℎ_𝑙. Repeating for all 𝑖=1,…,𝑁, we get 𝐪=Φ𝐡 with Φ=[Φ¹;Φ²;…;Φ^𝑁] and 𝐪=[𝐪¹;𝐪²;…;𝐪^𝑁]. This system of linear equations can be solved for 𝐡, provided that the rank 𝑟(Φ) of matrix Φ satisfies 𝑟(Φ)=2𝐿+1. A necessary condition for the latter is that the total number 𝑛=∑^𝑁_𝑖=1𝑛^𝑖 of spikes generated in response to all 𝑁 signals satisfies 𝑛≥2𝐿+𝑁+1. Then, the solution can be computed as 𝐡=Φ⁺𝐪. To find the coefficients 𝜙^𝑖_𝑙,𝑘, we note that 𝜙^𝑖_𝑙,𝑘=ℒ^𝑖_𝑘(𝑒_𝑙) (see Lemma 8). Hence, the result follows.

The time encoding interpretation of the channel identification problem for a SISO [Filter]-[Ideal IAF] circuit is shown in Figure 4(a). The block diagram of the SISO CIM in Theorem 11 is shown in Figure 4(b). Note that the key idea behind the SISO CIM is the introduction of multiple linearly independent test signals 𝑢^𝑖∈ℋ, 𝑖=1,2,...,𝑁. When the [Filter]-[Ideal IAF] circuit is producing very few measurements of 𝒫ℎ in response to any given test signal 𝑢^𝑖, we use more signals to obtain additional measurements. We can do so and identify 𝒫ℎ because 𝒫ℎ∈ℋ is fixed. In contrast, identifying 𝒫ℎ in a two-step deconvolving procedure requires reconstructing at least one 𝑣^𝑖. This is an ill-posed problem since each 𝑣^𝑖 is signal-dependent and has a small number of associated measurements.

(a)

(b)

Figure 4 

SISO CIM algorithm for the [Filter]-[Ideal IAF] circuit. (a) Time encoding interpretation of the channel identification problem. (b) Block diagram of the SISO channel identification machine.

3.3. Examples

We now demonstrate the performance of the identification algorithms in Lemma 9 and Theorem 11. First, we identify a filter in the SISO [Filter]-[Ideal IAF] circuit (Figure 2(a)) from a single I/O pair when this circuit produces a sufficient number of measurements in an interval of length 𝑇. Second, we identify the filter using multiple I/O pairs for the case when the number of measurements produced in response to any given input signal is small. Finally, we consider the SISO [Filter]-[Nonlinear Oscillator-ZCD] circuit with multiplicative coupling (Figure 2(b)) and identify its filter from multiple I/O pairs.

3.3.1. SISO [Filter]-[Ideal IAF] Circuit, Single I/O Pair

We model the dendritic processing filter using the causal linear kernel ℎ(𝑡)=𝑐𝑒^−𝛼𝑡(𝛼𝑡)³−()3!𝛼𝑡⁵[]5!,𝑡∈0,0.1s, (15) with 𝑐=3 and 𝛼=200. The general form of this kernel was suggested in [24] as a plausible approximation to the temporal structure of a visual receptive field. Since the length of the filter support 𝑆=0.1s, we will need to use a signal with a period 𝑇≥0.1s. In Figure 5(a), we apply a signal 𝑢 that is bandlimited to 25Hz and has a period of 𝑇=0.2s, that is, the order of the space 𝐿=𝑇⋅Ω/(2𝜋)=5. The biased output of the filter 𝑣=(𝑢∗ℎ)+𝑏 is then fed into an ideal integrate-and-fire neuron (Figure 5(b)). Here the bias 𝑏 guarantees that the output of the integrator reaches the threshold value in finite time. Whenever the biased filter output is above zero (Figure 5(b)), the membrane potential is increasing (Figure 5(c)). If the membrane potential ∫^𝑡_𝑡𝑘[(𝑢∗ℎ)(𝑠)+𝑏]𝑑𝑠 reaches a threshold 𝛿, a spike is generated by the neuron at a time 𝑡_𝑘+1 and the potential is reset to zero (Figure 5(c)). The resulting spike train (𝑡_𝑘)^𝑛_𝑘=1 at the output of the [Filter]-[Ideal IAF] circuit is shown in Figure 5(d). Note that the circuit generated a total of 𝑛=13 spikes in an interval of length 𝑇=0.2s. According to Theorem 14, we need at least 𝑛=2𝐿+2=12 spikes, corresponding to 2𝐿+1=11 measurements, in order to identify the projection 𝒫ℎ of the filter ℎ loss-free. Hence, for this particular example, it will suffice to use a single I/O pair (𝑢,𝕋).

Figure 5 

Channel identification in a SISO [Filter]-[Ideal IAF] circuit using a single I/O pair. (a) An input signal 𝑢 is bandlimited to 25 Hz. The order of the space is 𝐿=5. (b) The corresponding biased output of the filter 𝑣(𝑡)+𝑏. (c) The filter output in (b) is integrated by the ideal IAF neuron. Whenever the membrane potential reaches a threshold 𝛿, a spike is produced by the neuron and the potential is reset to 0. (d) The neuron generated a total of 13 spikes. (e) The identified impulse response of the filter 𝒫ℎ∗ (red) is shown together with the original filter ℎ (dashed black) and its projection 𝒫ℎ (blue). The MSE between 𝒫ℎ∗ and 𝒫ℎ is −77.5dB. (f)–(h) Fourier amplitude spectra of 𝐾, ℎ and 𝒫ℎ∗. Note that supp(ℱ𝐾)=supp(ℱ𝒫ℎ∗)=[−Ω,Ω] but supp(ℱℎ)⊃[−Ω,Ω]. In other words, 𝒫ℎ∗∈ℋ but ℎ∉ℋ.

In Figure 5(e), we plot the original impulse response of the filter ℎ, the filter projection 𝒫ℎ, and the filter 𝒫ℎ^∗. The latter filter was identified using the algorithm in Theorem 14. Notice that the identified impulse response 𝒫ℎ^∗ (red) is quite different from ℎ (dashed black). In contrast, and as expected, the blue and red curves corresponding, respectively, to 𝒫ℎ and 𝒫ℎ^∗ are indistinguishable. The mean squared error (MSE) between 𝒫ℎ^∗ and 𝒫ℎ amounts to −77.5 dB.

The difference between 𝒫ℎ and ℎ is further evaluated in Figures 5(f)–5(h). By definition of 𝒫ℎ in (6), 𝒫ℎ=ℎ∗𝐾(⋅,0), or ℱ(𝒫ℎ)=ℱ(ℎ)ℱ(𝐾(⋅,0)) since 𝐾=𝐾. Hence both the projection 𝒫ℎ and the identified filter 𝒫ℎ^∗ will contain frequencies that are present in the reproducing kernel 𝐾, or equivalently in the input signal 𝑢. In Figure 5(f) we show the double-sided Fourier amplitude spectrum of 𝐾(𝑡,0). As expected, we see that the kernel is bandlimited to 25Hz and contains 2𝐿+1=11 distinct frequencies. On the other hand, as shown in Figure 5(g), the original filter ℎ is not bandlimited (since it has a finite temporal support). As a result, the input signal 𝑢 explores ℎ in a limited spectrum of [−Ω,Ω]rad/s, effectively projecting ℎ onto the space ℋ with Ω=2𝜋⋅25rad/s and 𝐿=5. The Fourier amplitude spectrum of the identified projection 𝒫ℎ^∗ is shown in Figure 5(h).

3.3.2. SISO [Filter]-[Ideal IAF] Circuit, Multiple I/O Pairs

Next, we identify the projection of ℎ onto the space of functions that are bandlimited to 100 Hz and have the same period 𝑇=0.2s as in the first example. This means that the order 𝐿 of the space of input signals ℋ is 𝐿=𝑇⋅Ω/(2𝜋)=20. In order to identify the projection 𝒫ℎ loss-free, the neuron has to generate at least 2𝐿+1=41 measurements. If the neuron produces about 13 spikes (12 measurements) on an interval of length 𝑇, as in the previous example, a single I/O pair will not suffice. However, we can still recover the projection 𝒫ℎ if we use multiple I/O pairs.

In Figure 6 we illustrate identification of the filter using the algorithm in Theorem 11. A total of 48 spikes were produced by the neuron in response to four different signals 𝑢¹,…,𝑢⁴. Since 48>2𝐿+𝑁+1=45, the MSE between the identified filter 𝒫ℎ^∗ (red) and the projection 𝒫ℎ (blue) is −73.3 dB.

Figure 6 

Channel identification in a SISO [Filter]-[Ideal IAF] circuit using multiple I/O pairs. (a) Input signals 𝑢1,…,𝑢4 are bandlimited to 100 Hz. The order of the space 𝐿=20. (b) Biased output of the filter 𝑣1(𝑡)+𝑏 in response to the stimulus 𝑢1. (c) The filter output in (b) is integrated by an ideal IAF neuron. (d) The neuron generated a total of 48 spikes in response to all 4 input signals. (e) The identified impulse response 𝒫ℎ∗ (red) is shown together with the original filter ℎ (dashed black) and its projection 𝒫ℎ (blue). The MSE between 𝒫ℎ∗ and 𝒫ℎ is −73.3 dB. (f)–(h) Fourier amplitude spectra of 𝐾, ℎ and 𝒫ℎ∗. Note that supp(ℱ𝐾)=[−Ω,Ω]=supp(ℱ𝒫ℎ∗) but supp(ℱℎ)⊃[−Ω,Ω]. In other words, 𝒫ℎ∗∈ℋ but ℎ∉ℋ.

Figure 7 

Channel identification for ℎ(𝑡)=𝛿(𝑡). (a) Input signals 𝑢1,𝑢2 are bandlimited to 50 Hz. The order of the space 𝐿=10. (b) Biased output of the filter 𝑣1(𝑡)+𝑏 in response to the stimulus 𝑢1. (c) The filter output in (b) is integrated by an ideal IAF neuron. (d) The neuron generated a total of 28 spikes in response to 2 input signals. (e) The identified filter 𝒫ℎ∗ (red) is exactly the kernel 𝐾(𝑡,0) for ℋ1Ω,𝐿 with Ω=2𝜋⋅10rad/s and 𝐿=10. Also shown is the original filter ℎ=𝛿 (dashed black) and its projection 𝒫ℎ=𝛿∗𝐾(⋅,0)=𝐾(⋅,0) (blue). The MSE between 𝒫ℎ∗ and 𝒫ℎ is −87.6 dB. (f)–(h) Fourier amplitude spectra of 𝐾, ℎ and 𝒫ℎ∗. As before, 𝒫ℎ∗∈ℋ but ℎ∉ℋ.

3.3.3. SISO [Filter]-[Ideal IAF] Circuit, ℎ(𝑡)=𝛿(𝑡)

Now we consider a special case when the channel does not alter the input signal, that is, when ℎ(𝑡)=𝛿(𝑡), 𝑡∈ℝ, is the Dirac delta function. As explained in Remark 10, the CIM should identify the projection of 𝛿(𝑡) onto ℋ, that is, it should identify the kernel 𝐾(𝑡,0). This is indeed the case as shown in Figure 7.

3.3.4. SISO [Filter]-[Nonlinear Oscillator-ZCD] Circuit, Multiple I/O Pairs

Next we consider a SISO circuit consisting of a channel in cascade with a nonlinear dynamical system that has a stable limit cycle. We assume that the (positive) output of the channel 𝑣(𝑡)+𝑏 is multiplicatively coupled to the dynamical system (Figure 2(b)) so that the circuit is governed by a set of equations 𝑑𝐲=(𝑣(𝑡))𝐟(𝐲).𝑑𝑡+𝑏 (16) A system (16) followed by a zero-crossing detector is an example of a TEM with multiplicative coupling and has been previously investigated in [6]. It can be shown that such a TEM is input/output equivalent to an IAF neuron with a threshold 𝛿 that is equal to the period of the dynamical system on a stable limit cycle [6].

As an example, we consider a [Filter]-[van der Pol - ZCD] TEM with the van der Pol oscillator described by a set of equations 𝑑𝑦₁=(𝜇𝑦𝑑𝑡𝑢∗ℎ+𝑏)₁−13𝑦³₁−𝑦₂,𝑑𝑦₂=()𝑦𝑑𝑡𝑢∗ℎ+𝑏₁, (17) where 𝜇 is the damping coefficient [6]. We assume that 𝑦₁ is the only observable state of the oscillator and without loss of generality we choose the zero phase of the limit cycle to be the peak of 𝑦₁.

In Figure 8, we show the results of a simulation in which a SISO CIM was used to identify the channel. Input signals (Figure 8(a)) were bandlimited to 50 Hz and had a period 𝑇=0.5s, that is, 𝐿=25. In the absence of an input, that is, when 𝑢=0, a constant bias 𝑏=1 (Figure 8(b)) resulted a in period of 34.7 ms on a stable limit cycle (Figure 8(e)). As seen in Figures 8(b) and 8(c), downward/upward deviations of 𝑣¹(𝑡)+𝑏 in response to 𝑢¹ resulted in the slowing-down/speeding-up of the oscillator. In order to identify the filter projection onto a space of order 𝐿=25 loss-free, we used a total of 𝑛=56 zeros at the output of the zero-crossing detector (Figure 8(d)). This is 1 more zero than the rank requirement of 2𝐿+𝑁+1=55 zeros, or equivalently of 2𝐿+1=51 measurements. The MSE between the identified filter 𝒫ℎ^∗ (red) and the projection 𝒫ℎ (blue) is −66.6 dB.

Figure 8 

Channel identification in a SISO [Filter]-[van der Pol-ZCD] circuit using multiple I/O pairs. (a) Input signals 𝑢1,…,𝑢4 are bandlimited to 50 Hz. The order of the space 𝐿=25. (b) Biased output of the filter 𝑣1(𝑡)+𝑏 in response to the stimulus 𝑢1. (c) Downward and upward deviations of 𝑣1(𝑡)+𝑏 from the bias 𝑏 cause the oscillator to slow down and to speed up, respectively. The damping coefficient 𝜇=20. (d) The oscillator produced a total of 56 spikes in response to 4 stimuli. Here spikes correspond to the peaks of the observable state variable 𝑦11. (e) A limit cycle of the van der Pol oscillator for 𝜇=20 is shown in the phase plane. In the absence of channel output, the bias 𝑏 resulted in a constant period of oscillation 𝑇(𝑏)=34.7ms. The red dot denotes the zero phase (spike) of an oscillation. (f) The identified filter 𝒫ℎ∗ (red) is shown together with the original filter ℎ (dashed black) and its projection 𝒫ℎ (blue). The MSE between 𝒫ℎ∗ and 𝒫ℎ is −66.6 dB. (g) Fourier amplitude spectra of ℎ and 𝒫ℎ∗. As before, 𝒫ℎ∗∈ℋ but ℎ∉ℋ.

(a) Time domain

(b) Frequency domain

Figure 9 

Comparison between ℎ and 𝒫ℎ in time and frequency domains. (a) ℎ (red) and its projection 𝒫ℎ (blue) are shown for several values of Ω and 𝐿 in the time domain. Ω=2𝜋⋅20rad/s, 2𝜋⋅50rad/s and 2𝜋⋅100rad/s in the top, middle and bottom row, respectively. The period 𝑇 is fixed at 𝑇=0.2s in the left column and 𝑇=0.5s in the right column. (b) Fourier amplitude spectra of ℎ (red) and 𝒫ℎ (blue) for the same values of Ω and 𝐿 as in (a). Note that the differentiating filter ℎ clearly removes the zero-frequency (dc) coefficient corresponding to 𝑙=0 in all cases.

3.4. Convergence of the SISO CIM Estimate

Recall, that the original problem of interest is that of recovering the impulse response of the filter ℎ. The CIM lets us identify the projection 𝒫ℎ of that filter onto the input space. A natural question to ask is whether 𝒫ℎ converges to ℎ and if so how and under what conditions. We formalize this below.

Proposition 12. If ∫^𝑇₀|ℎ(𝑡)|²𝑑𝑡<∞, then 𝒫ℎ→ℎ in the 𝐿² norm and almost everywhere on 𝑡∈[0,𝑇] with increasing Ω, 𝐿 and fixed 𝑇.

Proof. Let 𝑇=2𝜋𝐿/Ω=const. Then 𝐾(𝑡,0)=(1/𝑇)∑^𝐿_𝑙=−𝐿𝑒^{(𝑗Ω𝑙/𝐿)𝑡}=(1/𝑇)∑^𝐿_𝑙=−𝐿𝑒^{(𝑗2𝜋𝑙/𝑇)𝑡} and by definition of 𝒫ℎ in (6), we have (𝒫ℎ)(𝑡)=^𝑇₀1𝑇^𝐿_𝑙=−𝐿𝑒^{(𝑗2𝜋𝑙/𝑇)(𝑡−𝑠)}ℎ(𝑠)=𝑑𝑠_{𝐿𝑙=−𝐿}1𝑇^𝑇₀ℎ(𝑠)𝑒^{−(𝑗2𝜋𝑙/𝑇)𝑠}𝑑𝑠𝑒^{(𝑗2𝜋𝑙/𝑇)𝑡}=^𝐿_𝑙=−𝐿ℎ(𝑙)𝑒^{(𝑗2𝜋𝑙/𝑇)𝑡}=𝑆^ℎ_𝐿(𝑡), (18) where 𝑆^ℎ_𝐿 is the 𝐿th partial sum of the Fourier series of ℎ and ℎ(𝑙) is the 𝑙th Fourier coefficient. Hence the problem of convergence of 𝒫ℎ to ℎ is the same as that of the convergence of the Fourier series of ℎ. We thus have convergence in the 𝐿² norm and convergence almost everywhere follows from Carleson's theorem [23].

Remark 13. More generally, if ∫^𝑇₀|ℎ(𝑡)|^𝑝𝑑𝑡<∞, 𝑝∈(1,∞), then 𝒫ℎ→ℎ in the 𝐿^𝑝 norm and almost everywhere by Hunt's theorem [23].

It follows from Proposition 12 that 𝒫ℎ approximates ℎ arbitrarily closely (in the 𝐿² norm, or MSE sense), given an appropriate choice of Ω and 𝐿. Since the number of measurements needed to identify the projection 𝒫ℎ increases linearly with 𝐿, a single channel identification problem leads us to consider a countably infinite number of time encoding problems in order to identify the impulse response of the filter with arbitrary precision. To provide further intuition about the relationship between ℎ and 𝒫ℎ, we compare the two in time and frequency domains for multiple values of Ω and 𝐿 in Figure 9.

4. MISO Channel Identification Machines

In this section we consider the identification of a bank of 𝑀 filters with impulse responses ℎ^𝑚, 𝑚=1,2,…,𝑀. We present a MISO CIM algorithm in Section 4.1, followed by an example demonstrating its performance in Section 4.2.

4.1. An Identification Algorithm for MISO Channels

Consider now the MISO ASDM-based circuit in Figure 2(c), where the signal 𝐮=[𝑢¹(𝑡),𝑢²(𝑡),…,𝑢^𝑀(𝑡)]^𝑇, 𝑡∈[0,𝑇], 𝑀∈ℕ, is transformed into the time sequence (𝑡_𝑘)^𝑛_𝑘=1. This circuit is also an instance of a TEM and (assuming 𝑧(𝑡₁)=𝑏) its 𝑡-transform is given by ^𝑡𝑘+1_𝑡𝑘^𝑀_𝑚=1𝑢^𝑚∗ℎ^𝑚(𝑠)𝑑𝑠=⟨𝑣,𝜙_𝑘⟩=𝑞_𝑘, (19) where 𝑣=∑_𝑚(𝑢^𝑚∗ℎ^𝑚)(𝑡), 𝜙_𝑘∈ℋ with 𝜙_𝑘=∑_𝑙𝜙_𝑙,𝑘𝑒_𝑙(𝑡) and 𝑞_𝑘=(−1)^𝑘[2𝐶𝛿−𝑏(𝑡_𝑘+1−𝑡_𝑘)]. One simple way to identify filters ℎ^𝑚, 𝑚=1,2,…,𝑀, is to identify them one by one as in Theorem 11. For instance, this can be achieved by applying signals of the form 𝐮=[0,…,0,𝑢^𝑚,0,…,0] when identifying the filter ℎ^𝑚. In a number of applications, most notably in early olfaction [25], this model of system identification cannot be applied. An alternative procedure that allows to identify all filters at once is given below.

Theorem 14 (MISO channel identification machine). Let {𝐮^𝑖∣𝐮^𝑖∈ℋ^𝑀}^𝑁_𝑖=1 be a collection of 𝑁 linearly-independent vector-valued signals at the input of a MISO [Filter]-[ASDM-ZCD] circuit with filters ℎ^𝑚∈𝐻, 𝑚=1,…,𝑀. The filter projections 𝒫ℎ^𝑚 can be perfectly identified from a collection of I/O pairs {(𝐮^𝑖,𝕋^𝑖)}^𝑁_𝑖=1 as 𝒫ℎ^𝑚(𝑡)=^𝐿_𝑙=−𝐿ℎ^𝑚_𝑙𝑒_𝑙(𝑡), (20)𝑚=1,…,𝑀. Here the coefficients ℎ^𝑚_𝑙 are given by 𝐡=Φ⁺𝐪 with 𝐪=[𝐪¹,𝐪²,…,𝐪^𝑁]^𝑇, [𝐪^𝑖]_𝑘=𝑞^𝑖_𝑘 and 𝐡=[ℎ¹_−𝐿,…,ℎ^𝑀_−𝐿,ℎ¹_−𝐿+1,…,ℎ^𝑀_−𝐿+1,…,ℎ¹_𝐿,…,ℎ^𝑀_𝐿]^𝑇, provided that the matrix Φ has rank 𝑟(Φ)=𝑀(2𝐿+1). The matrix Φ is given by ΦΦ=¹𝟎⋯𝟎𝟎Φ²⋯𝟎⋮⋮⋱⋮𝟎𝟎⋯Φ^𝑁𝐔¹𝐔²⋮𝐔^𝑁𝐔,with_𝑖=𝐮^𝑖_−𝐿𝟎⋯𝟎𝟎𝐮^𝑖_−𝐿+1⋯𝟎⋮⋮⋱⋮𝟎𝟎⋯𝐮^𝑖_𝐿, (21) where 𝐮^𝑖_𝑙=[𝑢_𝑙^𝑖1,𝑢_𝑙^𝑖2,…,𝑢_𝑙^𝑖𝑀], 𝑖=1,2,…,𝑁. Finally, the elements of matrix Φ^𝑖 are given by Φ^𝑖_𝑘𝑙=𝑡^𝑖_𝑘+1−𝑡^𝑖_𝑘𝐿√,𝑙=0,𝑇𝑒_𝑙𝑡^𝑖_𝑘+1−𝑒_𝑙𝑡^𝑖_𝑘𝑗𝑙Ω,𝑙≠0. (22)

Proof. Since 𝒫ℎ^𝑚∈ℋ for all 𝑚=1,…,𝑀, it can be written as (𝒫ℎ^𝑚)(𝑡)=∑^𝐿_𝑙=−𝐿ℎ^𝑚_𝑙𝑒_𝑙(𝑡). Hence, for the 𝑚th component of the stimulus 𝐮^𝑖 we get (𝑢^𝑖𝑚∗ℎ^𝑚)(𝑡)=(𝑢^𝑖𝑚∗𝒫ℎ^𝑚)(𝑡)=√𝑇∑^𝐿_𝑙=−𝐿ℎ^𝑚_𝑙𝑢_𝑙^𝑖𝑚𝑒_𝑙(𝑡) and 𝑣^𝑖(𝑡)=^𝑀_𝑚=1√𝑇^𝐿_𝑙=−𝐿ℎ^𝑚_𝑙𝑢_𝑙^𝑖𝑚𝑒_𝑙(𝑡). (23) Using the definition of 𝜙^𝑖_𝑘=∑^𝐿_𝑙=−𝐿𝜙^𝑖_𝑙,𝑘𝑒_𝑙(𝑡) and substituting (23) into the 𝑡-transform (19), we obtain 𝑞^𝑖_𝑘=𝑣^𝑖,𝜙^𝑖_𝑘=^𝑀𝐿_{𝑚=1𝑙=−𝐿}√𝑇ℎ^𝑚_𝑙𝑢_𝑙^𝑖𝑚𝜙^𝑖_𝑙,𝑘, (24) or 𝐪^𝑖=Φ^𝑖𝐔^𝑖𝐡 with [𝐪^𝑖]_𝑘=𝑞^𝑖_𝑘, [Φ^𝑖]_𝑘𝑙=√𝑇⋅𝜙^𝑖_𝑙,𝑘, 𝐔^𝑖=diag(𝐮^𝑖_−𝐿,…,𝐮^𝑖_𝐿), 𝐮^𝑖_𝑙=[𝑢_𝑙^𝑖1,…,𝑢_𝑙^𝑖𝑀] and 𝐡=[ℎ¹_−𝐿,…,ℎ^𝑀_−𝐿,ℎ¹_−𝐿+1,…,ℎ^𝑀_−𝐿+1,…,ℎ¹_𝐿,…,ℎ^𝑀_𝐿]^𝑇. Repeating for all stimuli 𝐮^𝑖, 𝑖=1,…,𝑁, we obtain 𝐪=Φ𝐡 with Φ as specified in (21). This system of linear equations can be solved for 𝐡, provided that the rank of Φ satisfies the condition 𝑟(Φ)=𝑀(2𝐿+1). To find the coefficients 𝜙^𝑖_𝑙,𝑘, we note that 𝜙^𝑖_𝑙,𝑘=ℒ^𝑖_𝑘(𝑒_𝑙). Hence, the result follows.

The MIMO time-encoding interpretation of the channel identification problem for a MISO [Filter]-[ASDM-ZCD] circuit is shown in Figure 10(a). The block diagram of the MISO CIM in Theorem 14 is shown in Figure 10(b).

(a)

(b)

Figure 10 

MISO CIM algorithm for the [Filter]-[ASDM-ZCD] circuit. (a) Time encoding interpretation of the MISO channel identification problem. (b) Block diagram of the MISO channel identification machine.

Remark 15. From (23), we see that 𝑣^𝑖=∑^𝐿_𝑙=−𝐿𝑣^𝑖_𝑙𝑒_𝑙(𝑡) with 𝑣^𝑖_𝑙=√𝑇∑^𝑀_𝑚=1ℎ^𝑚_𝑙𝑢_𝑙^𝑖𝑚. Writing this for all 𝑖=1,…,𝑁, we obtain 𝐯_𝑙=𝐔_𝑙𝐡_𝑙, where [𝐔_𝑙]_𝑖𝑚=√𝑇𝑢_𝑙^𝑖𝑚, 𝐡_𝑙=[ℎ¹_𝑙,ℎ²_𝑙,…,ℎ^𝑀_𝑙]^𝑇 and 𝐯_𝑙=[𝑣¹_𝑙,𝑣²_𝑙,…,𝑣^𝑁_𝑙]^𝑇. In order to identify the multidimensional channel this system of equations must have a solution for every 𝑙. A necessary condition for the latter is that 𝑁≥𝑀, that is, the number 𝑁 of test signals 𝐮^𝑖 is greater than the number of signal components 𝑀.

Remark 16. The rank condition 𝑟(Φ)=𝑀(2𝐿+1) can be satisfied by increasing the number 𝑁 of input signals 𝐮^𝑖. Specifically, if on average the system is providing 𝜈 measurements in a time interval 𝑡∈[0,𝑇], then the minimum number of test signals is 𝑁=⌈𝑀(2𝐿+1)/𝜈⌉.

4.2. Example: MISO [Filter]-[ASDM-ZCD] Circuit

We now describe simulation results for identifying the channel in a MISO [Filter]-[ASDM - ZCD] circuit of Figure 2(c). We use three different filters: ℎ¹(𝑡)=𝑐𝑒^−𝛼𝑡(𝛼𝑡)³−3!(𝛼𝑡)⁵,ℎ5!₂(𝑡)=ℎ¹,ℎ𝑡−𝛽₃(𝑡)=−ℎ¹(𝑡), (25) with 𝑡∈[0,0.1] s, 𝑐=3 and 𝛼=200 and 𝛽=20ms. All 𝑁=5 signals are bandlimited to 100 Hz and have a period of 𝑇=0.2s, that is, the order of the space 𝐿=20. According to Theorem 14, the ASDM has to generate a total of at least 𝑀(2𝐿+1)+𝑁=128 trigger times in order to identify the projections 𝒫ℎ¹, 𝒫ℎ² and 𝒫ℎ³ loss-free. We use all five triplets 𝐮^𝑖=[𝑢^𝑖1,𝑢^𝑖2,𝑢^𝑖3], 𝑖=1,…,5, to produce 131 trigger times.

A single such triplet 𝐮¹ is shown in Figure 11(a). The corresponding biased aggregate channel output 𝑣¹(𝑡)−𝑧¹(𝑡) is shown in Figure 11(b). Since the Schmitt trigger output 𝑧(𝑡) switches between +𝑏 and −𝑏 (Figure 11(d)), the signal 𝑣¹(𝑡)−𝑧¹(𝑡) is piece-wise continuous. Figure 11(c) shows the integrator output. Note that when 𝑧(𝑡)=−𝑏, the channel output is positively biased and the integrator output ∫^𝑡_𝑡𝑘[𝑣¹(𝑠)−𝑧(𝑠)]𝑑𝑠 is compared against a threshold +𝛿. As soon as that threshold is reached, the Schmitt trigger output switches to 𝑧(𝑡)=𝑏 and the negatively-biased channel output is compared to a threshold −𝛿. Passing the ASDM output 𝑧¹(𝑡) through a zero-crossing device (Figure 11(d)), we obtain a corresponding sequence of trigger times (𝑡¹_𝑘)²²_𝑘=1. The set of all 131 trigger times is shown in Figure 11(e). Three identified filters 𝒫ℎ^1∗, 𝒫ℎ^2∗ and 𝒫ℎ^3∗ are plotted in Figures 11(f)–11(h). The MSE between filter projections and filters recovered by the algorithm in Theorem 14 is on the order of −60 dB.

Figure 11 

Channel identification in a MISO [FIlter]-[ASDM] circuit using multiple I/O pairs. (a) An input triplet signal 𝐮1=[𝑢11,𝑢12,𝑢13] is bandlimited to 100 Hz. The order of the space 𝐿=20. (b) Biased aggregate output of the channel 𝑣1(𝑡)−𝑧1(𝑡) in response to the triplet 𝐮1. (c) Integrator output ∫𝑡𝑡𝑘[𝑣1(𝑠)−𝑧1(𝑠)]𝑑𝑠 (blue) is compared against two thresholds +𝛿 and −𝛿 (dashed red). Trigger times of the noninverting Schmitt trigger are indicated by red dots. (d) The ASDM output 𝑧1(𝑡) (blue) is passed through a zero-crossing detector to produce a sequence of trigger times (𝑡1𝑘)22𝑘=1. (e) A total of 131 trigger times were generated by the ASDM in response to five input triplets. (f)–(h) Identified filters 𝒫ℎ1∗ (red), 𝒫ℎ2∗ (green) and 𝒫ℎ3∗ (blue) are shown together with the original filters ℎ1, ℎ2, ℎ3 (dashed black) and their projections 𝒫ℎ1, 𝒫ℎ2 and 𝒫ℎ3 (black). The MSE achieved by the identification algorithm is less than −60 dB.

5. Generalizations

We shall briefly generalize the results presented in previous sections in two important directions. First, we consider a general class of signal spaces for test signals in Section 5.1. Then we discus channel models with noisy observations in Section 5.2.

5.1. Hilbert Spaces and RKHSs for Input Signals

Until now we have presented channel identification results for a particular space of input signals, namely the space of trigonometric polynomials. The finite-dimensionality of this space and the simplicity of the associated inner product makes it an attractive space to work with when implementing a SISO or a MISO CIM algorithm. However, fundamentally the identification methodology relied on the the geometry of the Hilbert space of test signals [5, 26]; computational tractability was based on kernel representations in an RKHS.

Theorem 17. Let {𝑢^𝑖∣𝑢^𝑖∈ℋ(𝐼)}^𝑁_𝑖=1 be a collection of 𝑁 linearly independent and bounded stimuli at the input of a [Filter]-[Asynchronous Sampler] circuit with a linear processing filter ℎ∈𝐻 and the 𝑡-transform ℒ^𝑖_𝑘(𝒫ℎ)=𝑞^𝑖_𝑘, (26) where ℒ^𝑖_𝑘∶ℋ→ℝ is a bounded linear functional mapping 𝒫ℎ into a measurement 𝑞^𝑖_𝑘. Then there is a set of sampling functions {(𝜙^𝑖_𝑘)_𝑘∈ℤ}^𝑁_𝑖=1, in ℋ such that 𝑞^𝑖_𝑘=𝒫ℎ,𝜙^𝑖_𝑘, (27) for all 𝑘∈ℤ, 𝑖=1,2,…,𝑁. Furthermore, if ℋ is an RKHS with a kernel 𝐾(𝑠,𝑡), 𝑠,𝑡∈𝐼, then 𝜙^𝑖_𝑘(𝑡)=ℒ^𝑖_𝑘(𝐾(⋅,𝑡)). Let the set of representation functions {(𝜓^𝑖_𝑘)_𝑘∈ℤ}^𝑁_𝑖=1, span the Hilbert space ℋ. Then (𝒫ℎ)(𝑡)=^𝑁_{𝑖=1𝑘∈ℤ}ℎ^𝑖_𝑘𝜓^𝑖_𝑘(𝑡). (28) Finally, if {(𝜙^𝑖_𝑘)_𝑘∈ℤ}^𝑁_𝑖=1 and {(𝜓^𝑖_𝑘)_𝑘∈ℤ}^𝑁_𝑖=1 are orthogonal basis or frames for ℋ, then the filter coefficients amount to 𝐡=Φ⁺𝐪, where 𝐡=[𝐡¹,𝐡²,…,𝐡^𝑁]^𝑇 with [𝐡^𝑖]_𝑘=ℎ^𝑖_𝑘, [Φ^𝑖𝑗]_𝑙𝑘=⟨𝜙^𝑖_𝑙,𝜓^𝑗_𝑘⟩ and 𝐪=[𝐪¹,𝐪²,…,𝐪^𝑁]^𝑇 with [𝐪^𝑖]_𝑙=𝑞^𝑖_𝑘  for all 𝑖,𝑗=1,2,…,𝑁, and 𝑘,𝑙∈ℤ.

Proof. By the Riesz representation theorem, since the linear functional ℒ^𝑖_𝑘∶ℋ→ℝ is bounded, there is a set of sampling functions {(𝜙^𝑖_𝑘)_𝑘∈ℤ}^𝑁_𝑖=1 in ℋ such that ℒ^𝑖_𝑘(𝒫ℎ)=⟨𝒫ℎ,𝜙^𝑖_𝑘⟩. If ℋ is an RKHS, a sampling function 𝜙^𝑖_𝑘 can be computed using the reproducing property of the kernel 𝐾 as in 𝜙^𝑖_𝑘(𝑡)=𝜙^𝑖_𝑘,𝐾(⋅,𝑡)≡𝐾(⋅,𝑡),𝜙^𝑖_𝑘=ℒ^𝑖_𝑘(𝐾(⋅,𝑡)). (29) Finally, writing all inner products ⟨𝜙^𝑖_𝑘,𝒫ℎ⟩=𝑞^𝑖_𝑘 yields, with the notation above, a system of linear equations Φ𝐡=𝐪 and the fiter coefficients amount to 𝐡=Φ⁺𝐪.

5.1.1. Example: Paley-Wiener Space

As an example, we consider the Paley-Wiener space which is closely related to the space of trigonometric polynomials. Specifically, the finite-dimensional space ℋ can be thought of as a discretized version of the infinite-dimensional Paley-Wiener space Ξ=𝑢∈𝕃²(ℝ)∣supp(ℱ𝑢)⊆[−Ω,Ω] (30) in the frequency domain. An element 𝑢∈ℋ has a line spectrum at frequencies 𝑙Ω/𝐿, 𝑙=−𝐿,−𝐿+1,…,𝐿. This spectrum becomes dense in [−Ω,Ω] as 𝐿→∞. The space Ξ with the inner product ⟨⋅,⋅⟩∶Ξ×Ξ→ℝ given by ⟨𝑢,𝑤⟩=_ℝ𝑢(𝑡)𝑤(𝑡)𝑑𝑡 (31) is also an RKHS with an RK [22] 𝐾()=(Ω(𝑠,𝑡sin𝑡−𝑠))𝜋(),𝑡−𝑠 (32) with 𝑡,𝑠∈ℝ. Defining the projection of the filter ℎ onto Ξ as (𝒫ℎ)(𝑡)=∫_ℝℎ(𝑠)𝐾(𝑠,𝑡)𝑑𝑠, we find that Lemma 8 still holds with 𝜙_𝑘∈Ξ and we can extend Theorem 11 to the following.

Proposition 18. Let {𝑢^𝑖∣supp(ℱ𝑢^𝑖)=[−Ω,Ω]}^𝑁_𝑖=1 be a collection of 𝑁 linearly independent and bounded stimuli at the input of a [Filter]-[Ideal IAF] neural circuit with a dendritic processing filter ℎ∈𝐻. If ∑^𝑁_𝑗=1(𝑏/𝐶𝛿)>Ω/𝜋, then (𝒫ℎ)(𝑡) can be perfectly identified from the collection of I/O pairs {(𝑢^𝑖,𝕋^𝑖)}^𝑁_𝑖=1 as (𝒫ℎ)(𝑡)=^𝑁_{𝑖=1𝑘∈ℤ}ℎ^𝑖_𝑘𝜓^𝑖_𝑘(𝑡), (33) where 𝜓^𝑖_𝑘(𝑡)=𝐾(𝑡,𝑡^𝑖_𝑘), 𝑖=1,2,…,𝑁, and 𝑘∈ℤ. Finally, 𝐡=Φ⁺𝐪, where 𝐡=[𝐡¹,𝐡²,…,𝐡^𝑁]^𝑇 with [𝐡^𝑖]_𝑘=ℎ^𝑖_𝑘, [Φ^𝑖𝑗]_𝑙𝑘=∫^{𝑡𝑖𝑙+1}_𝑡𝑖𝑙𝑢^𝑖(𝑠−𝑡^𝑗_𝑘)𝑑𝑠 and 𝐪=[𝐪¹,𝐪²,…,𝐪^𝑁]^𝑇 with [𝐪^𝑖]_𝑙=𝐶𝛿−𝑏(𝑡^𝑖_𝑙+1−𝑡^𝑖_𝑙) for all 𝑖,𝑗=1,2,…,𝑁, and 𝑘,𝑙∈ℤ.

Proof. As before, the spikes (𝑡^𝑖_𝑘)_𝑘∈ℤ in response to each test signal 𝑢^𝑖, 𝑖=1,2,…,𝑁, represent distinct measurements 𝑞^𝑖_𝑘=⟨𝜙^𝑖_𝑘,𝒫ℎ⟩ of (𝒫ℎ)(𝑡). Thus we can think of the {(𝑞^𝑖_𝑘)_𝑘∈ℤ}^𝑁_𝑖=1's as projections of 𝒫ℎ onto {(𝜙^𝑖_𝑘)_𝑘∈ℤ}^𝑁_𝑖=1, where 𝜙^𝑖_𝑘(𝑡)=ℒ^𝑖_𝑘(𝐾(⋅,𝑡))=∫^{𝑡𝑖𝑘+1}_𝑡𝑖𝑘∫_ℝ𝑢^𝑖(𝑧)𝐾(𝑠−𝑧,𝑡)𝑑𝑧𝑑𝑠=∫^{𝑡𝑖𝑘+1}_𝑡𝑖𝑘𝑢^𝑖(𝑠−𝑡)𝑑𝑠. Since the signals are linearly independent and bounded [5], it follows that, if ∑^𝑁_𝑖=1(𝑏/𝐶𝛿)>Ω/𝜋 or equivalently if the number of test signals 𝑁>Ω𝐶𝛿/𝜋𝑏, the set of functions {(𝜓^𝑖_𝑘)_𝑘∈ℤ}^𝑁_𝑖=1 with 𝜓^𝑖_𝑘(𝑡)=𝐾(𝑡,𝑡^𝑖_𝑘), is a frame for Ξ [5, 26]. Hence (𝒫ℎ)(𝑡)=^𝑁_{𝑖=1𝑘∈ℤ}ℎ^𝑖_𝑘𝜓^𝑖_𝑘(𝑡). (34) If the set of functions {(𝜙^𝑖_𝑘)_𝑘∈ℤ}^𝑁_𝑖=1 forms a frame for Ξ, we can find the coefficients ℎ^𝑖_𝑘,𝑘∈ℤ, 𝑖=1,2,…,𝑁, by taking the inner product of (34) with each element of {𝜙^𝑖_𝑙(𝑡)}^𝑁_𝑖=1: ⟨𝜙^𝑖_𝑙,𝒫ℎ⟩=∑_𝑘∈ℤℎ¹_𝑘⟨𝜙^𝑖_𝑙,𝜓¹_𝑘⟩+∑_𝑘∈ℤℎ²_𝑘⟨𝜙^𝑖_𝑙,𝜓²_𝑘⟩+⋯+∑_𝑘∈ℤℎ^𝑁_𝑘⟨𝜙^𝑖_𝑙,𝜓^𝑁_𝑘⟩≡𝑞^𝑖_𝑙, for 𝑖=1,2,…,𝑁, 𝑙∈ℤ. Letting [Φ^𝑖𝑗]_𝑙𝑘=⟨𝜙^𝑖_𝑙,𝜓^𝑗_𝑘⟩, we obtain 𝑞^𝑖_𝑙=_𝑘∈ℤΦ^𝑖1_𝑙𝑘ℎ¹_𝑘+_𝑘∈ℤΦ^𝑖2_𝑙𝑘ℎ²_𝑘+⋯+_𝑘∈ℤΦ^𝑖𝑁_𝑙𝑘ℎ^𝑁_𝑘, (35) for 𝑖=1,2,…,𝑁, 𝑙∈ℤ. Writing (35) in matrix form, we have 𝐪=Φ𝐡 with [Φ^𝑖𝑗]_𝑙𝑘=⟨𝜙^𝑖_𝑙,𝜓^𝑗_𝑘⟩=⟨𝜙^𝑖_𝑙(⋅),𝐾(⋅,𝑡^𝑗_𝑘)⟩=𝜙^𝑖_𝑙(𝑡^𝑗_𝑘)=∫^{𝑡𝑖𝑙+1}_𝑡𝑖𝑙𝑢^𝑖(𝑠−𝑡^𝑗_𝑘)𝑑𝑠. Finally, the coefficients ℎ^𝑖_𝑘, 𝑖=1,2,…,𝑁 and 𝑘∈ℤ, amount to 𝐡=Φ⁺𝐪.

Simulation results of a SISO CIM for a Paley-Wiener space of test signals is shown in Figure 12 Input signals 𝑢¹,…,𝑢⁵ were bandlimited to 100 Hz and the circuit generated a total of 38 spikes. The MSE between the identified filter 𝒫ℎ^∗ (red) and the projection 𝒫ℎ (blue) is −71.1 dB.

Figure 12 

Channel identification in a SISO [Filter]-[Ideal IAF] circuit using signals from the Paley-Wiener space Ξ. (a) In contrast to Figure 6, input signals 𝑢𝑖∈Ξ, 𝑖=1,…,5. (b) Biased output of the filter 𝑣1(𝑡)+𝑏 in response to the stimulus 𝑢1. (c) The filter output in (b) is integrated by an ideal IAF neuron. (d) The neuron generated a total of 38 spikes in response to all 5 input signals. (e) The identified impulse response of the filter 𝒫ℎ∗ (red) is shown together with the original filter ℎ (dashed black) and its projection 𝒫ℎ (blue). The MSE between 𝒫ℎ∗ and 𝒫ℎ is −71.1 dB. (f)–(h) Fourier amplitude spectra of 𝐾, ℎ, and 𝒫ℎ∗. In contrast to Figure 6, 𝐾 and 𝒫ℎ∗ do not exhibit a discrete (line) spectrum. Again, 𝒫ℎ∗∈Ξ but ℎ∉Ξ.

5.2. Channels with Noisy Observations

In the derivations above we implicitly assumed that the I/O system was noiseless. In practice, noise is introduced either by the channel or the sampler itself. Here we revisit the 𝑡-transform in (5) and show that the analysis/methodology employed in the previous sections can be extended within an appropriate mathematical setting to I/O systems with noisy measurements.

Recall the 𝑡-transform of an ideal IAF neuron is given by ∫^𝑡𝑘+1_𝑡𝑘(𝑢∗ℎ)(𝑡)𝑑𝑡=⟨𝒫ℎ,𝜙_𝑘⟩=𝑞_𝑘, 𝑘=1,2,…,𝑛−1, where 𝑛 is the number of spikes generated by the neuron in an interval of length 𝑇. The measurements 𝑞_𝑘 were obtained by applying a piece-wise linear operator on the channel output 𝑣=𝑢∗ℎ. If either the channel or the sampler introduce an error, we can model it by adding a noise term 𝜀_𝑘 to the 𝑡-transform [9]: ⟨𝒫ℎ,𝜙_𝑘⟩=𝑞_𝑘+𝜀_𝑘. (36) Here we will assume that 𝜀_𝑘∼𝒩(0,𝜎²), 𝑘=1,2,…,𝑛−1, are i.i.d.

In the presence of noise it is not possible to identify the projection 𝒫ℎ loss-free. However, we can still identify an estimate 𝒫ℎ of 𝒫ℎ that is optimal for an appropriately defined cost function. For example, we can formulate a bi-criterion Tikhonov regularization problem min𝒫ℎ^𝑁_∈ℋ^{𝑖=1𝑛−1𝑘=1}⟨𝒫ℎ,𝜙^𝑖_𝑘⟩−𝑞^𝑖_𝑘²+𝜆𝒫ℎ²_ℋ, (37) where the scalar 𝜆>0 provides a trade-off between the faithfulness of the identified filter projection 𝒫ℎ to measurements (𝑞_𝑘)^𝑛−1_𝑘=1 and its norm ‖𝒫ℎ‖_ℋ.

Theorem 19. Problem (37) can be solved explicitly in analytical form. The optimal solution is achieved by 𝒫ℎ(𝑡)=^𝐿_𝑙=−𝐿ℎ_𝑙𝑒_𝑙(𝑡), (38) with 𝐡=(Φ^𝐻Φ+𝜆𝐈)⁻¹Φ^𝐻𝐪, Φ=[Φ¹;Φ²;…;Φ^𝑁] and Φ^𝑖, 𝑖=1,2,…,𝑁, as defined in (13).

Proof. Since the minimizer 𝒫ℎ is in ℋ, it is of the form given in (38). Substituting this into (37), we obtain min_{𝐡∈ℂ2𝐿+1}‖Φ𝐡−𝐪‖²_𝑅𝑛−1+𝜆‖𝐡‖²_ℂ2𝐿+1, (39) where Φ=[Φ¹;Φ²;…;Φ^𝑁] with Φ^𝑖, 𝑖=1,2,…,𝑁, as defined in (13). This quadratic optimization problem can be solved analytically by expressing the objective as a convex quadratic function 𝐽(𝐡)=𝐡^𝐻Φ^𝐻Φ𝐡−2𝐪^𝐻Φ𝐡+𝐪^𝐻𝐪+𝜆𝐡^𝐻𝐡 with 𝐻 denoting the conjugate transpose. A vector 𝐡 minimizes 𝐽 if and only if ∇𝐽=2(Φ^𝐻Φ+𝜆𝐈)𝐡−2Φ^𝐻𝐪=0, that is, 𝐡=(Φ^𝐻Φ+𝜆𝐈)⁻¹Φ^𝐻𝐪.

Remark 20. In Section 3.2, identification of the projection (𝒫ℎ)(𝑡)=∑^𝐿_𝑙=−𝐿ℎ_𝑙𝑒_𝑙(𝑡) amounted to finding 𝒫ℎ∈ℋ such that the sum of the residuals (⟨𝒫ℎ,𝜙_𝑘⟩−𝑞_𝑘)² was minimized [9]. In other words, we were solving an unconstrained convex optimization problem of the form min^𝑁_{𝒫ℎ∈ℋ𝑖=1}^𝑛−1_𝑘=1𝒫ℎ,𝜙^𝑖_𝑘−𝑞^𝑖_𝑘²⟺min_{𝐡∈ℂ2𝐿+1}‖Φ𝐡−𝐪‖²_ℝ𝑛−1, (40) where 𝐡=[ℎ_−𝐿,…,ℎ_𝐿] and Φ=[Φ¹;Φ²;…;Φ^𝑁] with Φ^𝑖, 𝑖=1,2,…,𝑁, as defined in (13).

5.2.1. Example: Noisy SISO [Filter]-[Ideal IAF] Circuit

In the following example, we assume that noise is added to the measurements (𝑞^𝑖_𝑘)^𝑛−1_𝑘=1, 𝑖=1,2, by the neuron and we model that noise by introducing random thresholds that are normally distributed with a mean 𝛿 and a standard deviation 0.1𝛿, that is, 𝛿_𝑘∼𝒩(𝛿,(0.1𝛿)²)∶∫^{𝑡𝑖𝑘+1}_𝑡𝑖𝑘(𝑢^𝑖∗ℎ)(𝑡)𝑑𝑡=𝐶𝛿_𝑘−𝑏(𝑡^𝑖_𝑘+1−𝑡^𝑖_𝑘)=[𝐶𝛿−𝑏(𝑡^𝑖_𝑘+1−𝑡^𝑖_𝑘)]+𝐶(𝛿_𝑘−𝛿)=𝑞^𝑖_𝑘+𝜀^𝑖_𝑘, where 𝜀^𝑖_𝑘∼𝒩(0,(0.1𝐶𝛿)²). Thus random thresholds result in additive noise 𝜀^𝑖_𝑘∼𝒩(0,(0.1𝐶𝛿)²), 𝑖=1,2.

In Figure 13(a) we show two stimuli that were used to probe the [Filter]-[Ideal IAF] circuit. Both stimuli are bandlimited to 25 Hz and have a period of 𝑇=0.2s, that is, the order of the space is 𝐿=5. The response of the neuron to a biased filter output 𝑣¹(𝑡)+𝑏 (Figure 13(b)) is shown in Figure 13(c). Note the significant deviations in thresholds 𝛿_𝑘 around the mean value of 𝛿=0.05. Although a significant amount of noise is introduced into the system, we can identify an optimal estimate 𝒫ℎ^∗ that is still quite close to the true projection 𝒫ℎ. The MSE of identification is −31.8 dB.

Figure 13 

Noisy channel identification in a SISO [Filter]-[Ideal IAF] circuit using multiple I/O pairs. (a) Input signals 𝑢1,𝑢2 are bandlimited to 25 Hz. The order of the space 𝐿=5. (b) Biased output of the filter 𝑣1(𝑡)+𝑏 in response to the stimulus 𝑢1. (c) Thresholds are random with 𝛿𝑘∼𝒩(𝛿,(0.1𝛿)2). (d) The neuron produced a total of 26 spikes in response to 2 stimuli. (e) The optimal estimate 𝒫ℎ∗ (red) is shown together with the original filter ℎ (dashed black) and its projection 𝒫ℎ (blue). Note that the MSE between 𝒫ℎ∗ and 𝒫ℎ is −31.8 dB. (f)–(h) Fourier amplitude spectra of 𝐾, ℎ and 𝒫ℎ∗. As before, supp(ℱ𝐾)=[−Ω,Ω] = supp(ℱ𝒫ℎ∗) but supp(ℱℎ)⊃[−Ω,Ω]. In other words, 𝒫ℎ∗∈ℋ but ℎ∉ℋ.

6. Conclusion

In this paper we presented a class of channel identification problems arising in the context of communication channels in [Filter]-[Asynchronous Sampler] circuits. Our results are based on a key structural conditional duality result between time decoding and channel identification. The conditional duality result shows that given a class of test signals, the projection of the filter onto the space of input signals can be recovered loss-free. Moreover, the channel identification problem can be converted into a time decoding problem. We considered a number of channel identification problems that arise both in communications and in neuroscience. We presented CIM algorithms that allow one to recover projections of both one-dimensional and multi-dimensional filters in such problems and demonstrated their performance through numerical simulations. Furthermore, we showed that under natural conditions on the impulse response of the filter, the filter projection converges to the original filter almost everywhere and in the mean-squared sense (𝐿² norm), with increasing bandwidth and order of the space. Thus in order to identify the impulse response of the filter with arbitrary precision, we are lead to consider a countably infinite number of time encoding problems. Finally, we generalized our results to a large class of test signal spaces and to channel models with noisy observations.

Acknowledgments

This work was supported in part by the NIH under the grant no. R01 DC008701-05 and in part by AFOSR under grant no. FA9550-12-1-0232. The authors’ names are alphabetically ordered.