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.

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 [

TEMs naturally arise as models of early sensory systems in neuroscience [

A general TEM of interest is shown in Figure

Modeling the channel identification problem. A known multidimensional signal

In this paper, we investigate the following

Identification of the channel from a time sequence is to be contrasted with existing methods for rate-based models in neuroscience (see [

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 [

The paper is organized as follows. In Section

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

An instance of the TEM in Figure

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.

An example of a MISO system is the [Filter]-[ASDM-ZCD] circuit shown in Figure

We model channel input signals

The space of trigonometric polynomials

We note that a function

The channel is modeled as a bank of

A signal

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

Let

We note that a CIM recovers the impulse response of the filter based on the knowledge of I/O pairs

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

Consider the SISO [Filter]-[Ideal IAF] neural circuit in Figure

The mapping of an analog signal

The operator

For all

Since

The conditional duality between time encoding and channel identification is visualized in Figure

Conditional duality between channel identification and time encoding. (a) For all

Conditional I/O equivalence

Duality

Given the parameters of the asynchronous sampler, the measurements

There is a function

The linear functional

Since

Let

Since

If the signal

In order to ensure that the neuron produces at least

Ideally, we would like to identify the impulse response of the filter

The requirement of Lemma

Let

Since

The time encoding interpretation of the channel identification problem for a SISO [Filter]-[Ideal IAF] circuit is shown in Figure

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.

We now demonstrate the performance of the identification algorithms in Lemma

We model the dendritic processing filter using the causal linear kernel

Channel identification in a SISO [Filter]-[Ideal IAF] circuit using a single I/O pair. (a) An input signal

In Figure

The difference between

Next, we identify the projection of

In Figure

Channel identification in a SISO [Filter]-[Ideal IAF] circuit using multiple I/O pairs. (a) Input signals

Channel identification for

Now we consider a special case when the channel does not alter the input signal, that is, when

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

As an example, we consider a [Filter]-[van der Pol - ZCD] TEM with the van der Pol oscillator described by a set of equations

In Figure

Channel identification in a SISO [Filter]-[van der Pol-ZCD] circuit using multiple I/O pairs. (a) Input signals

Comparison between

Time domain

Frequency domain

Recall, that the original problem of interest is that of recovering the impulse response of the filter

If

Let

More generally, if

It follows from Proposition

In this section we consider the identification of a bank of

Consider now the MISO ASDM-based circuit in Figure

Let

Since

The MIMO time-encoding interpretation of the channel identification problem for a MISO [Filter]-[ASDM-ZCD] circuit is shown in Figure

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.

From (

The rank condition

We now describe simulation results for identifying the channel in a MISO [Filter]-[ASDM - ZCD] circuit of Figure

A single such triplet

Channel identification in a MISO [FIlter]-[ASDM] circuit using multiple I/O pairs. (a) An input triplet signal

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

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 [

Let

By the Riesz representation theorem, since the linear functional

As an example, we consider the Paley-Wiener space which is closely related to the space of trigonometric polynomials. Specifically, the finite-dimensional space

Let

As before, the spikes

Simulation results of a SISO CIM for a Paley-Wiener space of test signals is shown in Figure

Channel identification in a SISO [Filter]-[Ideal IAF] circuit using signals from the Paley-Wiener space

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

Recall the

In the presence of noise it is not possible to identify the projection

Problem (

Since the minimizer

In Section

In the following example, we assume that noise is added to the measurements

In Figure

Noisy channel identification in a SISO [Filter]-[Ideal IAF] circuit using multiple I/O pairs. (a) Input signals

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 (

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.