We consider the problem of reconstructing finite energy stimuli encoded with a population of spiking leaky integrate-and-fire neurons. The reconstructed signal satisfies a consistency condition: when passed through the same neuron, it triggers the same spike train as the original stimulus. The recovered stimulus has to also minimize a quadratic smoothness optimality criterion. We formulate the reconstruction as a spline interpolation problem for scalar as well as vector valued stimuli and show that the recovery has a unique solution. We provide explicit reconstruction algorithms for stimuli encoded with single as well as a population of integrate-and-fire neurons. We demonstrate how our reconstruction algorithms can be applied to stimuli encoded with ON-OFF neural circuits with feedback. Finally, we extend the formalism to multi-input multi-output neural circuits and demonstrate that vector-valued finite energy signals can be efficiently encoded by a neural population provided that its size is beyond a threshold value. Examples are given that demonstrate the potential applications of our methodology to systems neuroscience and neuromorphic engineering.

Formal spiking neuron models, such as integrate-and-fire (IAF) neurons, encode information in the time domain [

These results are based on the key insight that neural encoding of a stimulus with a population of LIF neurons is akin to taking a set of measurements on the stimulus. These measurements or encodings can be represented as projections (inner products) of the stimulus on a set of sampling functions. Stimulus recovery therefore calls for the reconstruction of the encoded stimuli from these inner products. These findings have shown that sensory information can be faithfully encoded into the spike trains of a neural ensemble and can serve as a theoretical basis for modeling of sensory systems (e.g., auditory, vision) [

In this paper we investigate the problem of reconstructing scalar and vector stimuli from a population of spike trains on a finite time horizon. The encoding circuits considered are either single-input multi-output or multi-input multi-output (MIMO). The increasing availability of multi-neuron population recordings has led to a paradigm shift towards population-centric approaches of neural coding and processing. Examples of MIMO models in systems neuroscience include extensive investigations of spike train transformations between neuron populations [

The stimuli considered in this paper have finite energy and are defined on a finite time horizon. Even though restricted to finite time intervals, finite energy signals have infinite degrees of freedom. Consequently, the formal stimulus recovery is ill-defined. We cast the stimulus reconstruction problem in the abstract spline theory [

Through the formulation of the interpolation spline problem, the reconstructed signal will give the same measurements as the original one. We show that this leads to a signal recovery that is

A preliminary version of some of the ideas presented here appears in [

Formally, MIMO neural circuits encode

The paper is organized as follows. Section

In this section we formulate and solve the problem of optimal consistent reconstruction for the simple case of a stimulus encoded with a single LIF neuron. We show how the spiking of an LIF neuron can be associated with a series of projections in the general

Let

Assume that the encoder is an LIF neuron, with threshold

The

The inner products or projections

The problem of stimulus reconstruction calls for estimating the signal

A reconstruction

As before, assume that at time 0 the membrane potential of the LIF neuron is set to the resting potential 0. Then the consistency condition above is equivalent with

A consistent reconstruction

The optimal consistent reconstruction

It follows directly from Definitions

An introduction to splines and the general solution to spline interpolation problems is presented in the Appendix

The optimal consistent reconstruction is unique and is given by

The proof follows from Theorem

The representation functions (

By letting

The input to an LIF neuron is a bandlimited signal with bandwidth of

Encoding and reconstruction with a single LIF neuron.

In this section we consider the reconstruction of a stimulus encoded with a population of LIF neurons. We demonstrate that the consistent recovery can be again formulated as a spline interpolation problem and provide the reconstruction algorithm. We also show how the methodology developed in this section can be applied to a simple encoding circuit consisting of two-coupled ON-OFF neurons with feedback.

In what follows we consider a neural encoding circuit consisting of

The

The

Let

The following theorem first appeared in [

Assume that at time 0 the membrane potential of all neurons is at the rest value 0. The optimal consistent reconstruction

The proof is notationally more complex but closely follows the proof of Theorem

We consider an encoding circuit consisting of two interconnected integrate-and-fire neurons with feedback. For simplicity we assume that the IAF neurons are ideal, that is,

Coupled ON-OFF integrate-and-fire neurons.

Let

A simple example consisting of two symmetric neurons with parameters

The input was chosen to be the temporal contrast of an artificial (positive) input photocurrent. With

Recovery of temporal contrast from an ON-OFF IAF neural pair.

In this section we present our model of consistent information representation of

Multiple-Input Multiple-Output time encoding machine.

Let

The

An implicit assumption in writing the

The optimal consistent reconstruction is given by the solution of the following spline interpolation problem:

Assume that at time 0 the membrane potential of all neurons is at the rest value 0. The optimal consistent reconstruction

The proof is based on Theorem

Note that since the signal reconstruction is set up as a spline interpolation problem, the algorithm presented in Theorem

We present the realization of the recovery algorithm for a filtering kernel that induces

Each filter

The representation functions

The entries of (

Note that the entries of

The recovered stimuli using the spikes from 3, 6, and 9 neurons, respectively, are depicted from top to bottom in Figure

Recovery of the 3-dimensional input vector valued signal. In each row the original (blue) and recovered (green) signals are shown for the indicated number of neurons used for recovery. The columns correspond to each component of the input signal.

Figure

SNR as a function of the number of neurons.

The methodology of interpolating splines presented here applies to the deterministic case where the input stimulus and the LIF neurons have low noise levels. It ties in naturally with theoretical results that show that neural encoding of bandlimited signals leads to perfect signal reconstruction if Nyquist-type rate conditions are satisfied [

In neuromorphic engineering applications the noise levels can be kept low. Neuronal spike trains, however, often exhibit strong variability in response to identical inputs due to various noise sources. For stimuli encoded with neural circuits the problem of optimal reconstruction can be formulated as a smoothing spline problem [

The methodology presented here can be applied to the reconstruction of stimuli encoded with neurons that belong to other model classes of interest. An example is provided by neuron models with level crossing spiking mechanisms and feedback that have been investigated in [

The MIMO architecture presented here consists of a linear, time-invariant filtering kernel that is separated from the neural spiking mechanism. By relaxing the time-invariance property and embedding spike-triggered reseting mechanisms at the level of the filtering kernel, more complex transformations can be modeled. Consequently dendritic trees incorporating compartmental neuron models and spike backpropagation [

We assume throughout that stimuli

The solution to the interpolation problem

We restrict ourselves to the case where the operator

If

In order to derive the general form of the interpolation spline, we introduce the notion of reproducing kernel for a Hilbert space with respect to the energy operator

The function

for any functional

any functional

The solution to the spline interpolation problem is given by

For the representation result of (

Let

The reproducing kernel for the Hilbert space

It can be shown that the reproducing kernel can be written in the form

Note that for

This work was supported by NIH Grant R01 DC008701-01 and NSF Grant CCF-06-35252. E. A. Pnevmatikakis was also supported by the Onassis Public Benefit Foundation. The authors would like to thank the reviewers for their suggestions for improving the presentation of this paper.