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Edited by: Klaus R. Pawelzik, University of Bremen, Germany

Reviewed by: Klaus R. Pawelzik, University of Bremen, Germany; Liam Paninsky, Columbia University, USA

*Correspondence:

This is an open-access article subject to an exclusive license agreement between the authors and Frontiers Media SA, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are credited.

Reconstructing stimuli from the spike trains of neurons is an important approach for understanding the neural code. One of the difficulties associated with this task is that signals which are varying continuously in time are encoded into sequences of discrete events or spikes. An important problem is to determine how much information about the continuously varying stimulus can be extracted from the time-points at which spikes were observed, especially if these time-points are subject to some sort of randomness. For the special case of spike trains generated by leaky integrate and fire neurons, noise can be introduced by allowing variations in the threshold every time a spike is released. A simple decoding algorithm previously derived for the noiseless case can be extended to the stochastic case, but turns out to be biased. Here, we review a solution to this problem, by presenting a simple yet efficient algorithm which greatly reduces the bias, and therefore leads to better decoding performance in the stochastic case.

One of the fundamental problems in systems neuroscience is to understand how neural populations encode sensory stimuli into spatio-temporal patterns of action potentials. The relationship between external stimuli and neural spike trains can be analyzed from two different perspectives. In the

In addition to decoding external stimuli from spike trains,

The general problem of decoding stimuli from spike trains can be challenging: First, the mapping of stimuli to the spike trains of single neurons (the

Another complication for decoding is that spike trains of neurons consist of discrete events, whereas sensory signals (or other signals of interest may) change continuously in time. Therefore, the discreteness of neural spikes implies constraints on what properties of the stimulus can be successfully reconstructed. For example, if neurons have low firing rates, then it may be impossible to resolve very high-frequency information in their inputs (Seydnejad and Kitney,

In a recent study (Gerwinn et al.,

Starting from the noiseless encoding of leaky integrate and fire neurons, we review the non-linear decoding scheme presented in Seydnejad and Kitney (

The decoding algorithm for the deterministic case can not simply be used in the stochastic case, as it suffers from a systematic

Given an encoding model, we can aim to “invert” this model for decoding, and thus perform optimal decoding. We assume that the encoding model is known, and, concretely, assume that it is a

The neuron model consists of a membrane potential V_{t}

where _{t}_{t}_{reset} and a spike is released. The solution to this equation can be found to be:

where _{−} is the time of the last reset (spike). Without loss of generality, we assume the reset potential _{reset} to be zero. From this solution, we see that the stimulus _{t}

In addition to the encoding model, we also need a representation or parametrization of the stimulus. We assume that the stimulus can be represented by a weighted superposition of a fixed number of basis functions (Figure

Importantly, the operations that map the stimulus onto the membrane potential (filtering, integration, and reset at the spikes times) can also be applied to each basis function individually. Superimposing these ^{1}

For the case of only two basis functions we have illustrated the constraints defined by two interspike intervals in Figure _{*} that were used to generate the stimulus, therefore have to be at the intersection of these two constraint subspaces. In the case of underdetermined, i.e., uncertain stimulus reconstruction, there are more constraints which could be exploited. As can be seen in Figure

If fewer interspike intervals than basis functions have been observed, the linear system which is defined by the threshold constraints is underdetermined. This means that multiple reconstructions are consistent with the equality constraints, and one can freely choose an estimate within the solution space. One principle for choosing a single one under all possible solutions (for example along the green line in Figure

In the previous section we assumed a fixed threshold which must reached to fire a spike. Thus, there was no noise in the model, and the mapping from stimuli to spikes is deterministic. To make the model more realistic, we assume that the membrane potential is not fixed, but rather varies randomly from spike to spike (Jolivet et al.,

By varying the threshold every time a spike is fired, we have introduced a noise source. This readily leads to a probabilistic interpretation of the encoding task: Fixing a specific stimulus, or equivalently a vector of coefficients, results in a probability distribution over spikes times induced by the varying threshold. After having specified a prior distribution over stimuli, the distribution over spikes times can be inverted by Bayes rule to give a distribution over possible stimuli having produced an observed set of spikes, called the

To analyze the situation, it is instructive to split the stimulus, or rather the coefficient vector which describes the stimulus (see Figure

In Figure

To illustrate the Gaussian approximation, we analyze the equality constraint obtained from the spikes times in some more detail. Mathematically, the equality constraint is given by the equation:

where θ is the threshold and _{i}

So far, we have used the equality constraint to approximate the likelihood of observing the set of spikes, given the stimulus. This in turn enabled us to construct a probability over stimuli conditioned on the observations. Instead of using this approximation, we could have used one of the following alternatives:

The Gaussian factor approximation used previously was obtained by assuming independence of the indirect observation of the modified basis functions and the threshold. Under this assumption, we could replace the actual threshold value by its mean. If we do not assume this independence but still only use the fact that the membrane potential has to reach an unknown threshold whose distribution is known, we can derive another approximation of the likelihood. In this case, the mean stimulus cannot be calculated analytically. However, the most likely stimulus (maximum a posteriori estimate) can still be found by using gradient based optimizers. This procedure results in an algorithm very similar to the one presented by Cunningham et al. (

Instead of approximating the likelihood function, we could try to use the exact form, without neglecting any constraints imposed by the observation of spikes. For the case of a continuously varying threshold, which is equivalent to additive noise to the membrane potential, algorithms for evaluating the likelihood have been presented by Paninski et al. (

The difficulty in computing the likelihood of a spike train for the leaky integrate and fire neuron model arises primarily from the hard threshold used in the model. Therefore, a potential remedy is to approximate the model by one with a probabilistic, “soft” threshold, and for which decoding is easier. In particular, GLMs are possible candidates, as inference over stimuli or parameters can be done efficiently. The corresponding decoding algorithms are either based on the Laplace (saddle point) approximation (Paninski et al.,

A very popular decoding algorithm which does not use the information about an explicit encoding model is the linear decoder (Bialek et al.,

Decoding stimuli from sequences of spike-patterns generated by populations of neurons is an important approach for understanding the neural code. Two challenges associated with this task are that continuous stimuli or inputs are sampled only at discrete points in time, and that the neural response can be noisy. When we incorporate noise, the decoding problem naturally turns into a probabilistic estimation problem: We want to estimate a distribution over stimuli which is consistent with the observed spike trains. In our case, we assumed that the encoding model is known, and given by a leaky integrate and fire neuron model. Thus, decoding the stimuli requires inferring the mapping from spikes to stimuli, while properly accounting for the noise in the observed spike trains.

Previous studies mostly investigated the static case, in which the decoding has to be done on the basis of an estimated rate at one instant of time (Georgopoulos et al.,

Above, we assumed that neurons are influenced only by the stimulus, but not by the activity of other neurons in the population. In other words, we can trivially extend the decoding scheme to the population case, if we assume that the neurons are “conditionally independent given the stimulus.” Importantly, this assumption can be relaxed easily, such that the decoding algorithm can also be applied to populations in which the neurons directly influence the spike-rates of each other. Specifically, we can also allow each spike of a neuron to generate a post-synaptic potential in any other neuron within the modeled population. As long as the form of the caused potential is known, every spike just adds a basis function with known coefficients to the indirect observations. In this way, we could analyze the effect of correlations on the decoding performance (Nirenberg and Latham,

Furthermore, the decoding scheme not only yields a point estimate of the most likely stimulus, but also an estimate of the uncertainty around that stimulus. Having access to this uncertainty allows one to optimize encoding parameters such as receptive fields in order to minimize the reconstruction error or to maximize the mutual information between stimulus and neural population response. In this way it becomes possible to extend unsupervised learning models such as independent component analysis (Bell and Sejnowski,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work is supported by the German Ministry of Education, Science, Research and Technology through the Bernstein award to Matthias Bethge (BMBF, FKZ:01GQ0601), and the Max Planck Society. Jakob H. Macke was supported by the Gatsby Charitable Foundation and the European Commission under the Marie Curie Fellowship Programme. We would like to thank Philipp Berens, Alexander Ecker, Fabian Sinz, and Holly Gerhard for discussions and comments on the manuscript.

^{1}Interspike intervals, not only the time of the last spike are needed, because the start point for the integration (last reset) has to be known as well.

the process of reconstructing the stimulus from the spike times of a real neuron or theoretical neuron model.

A neuron model which describes how a neuron will generate spikes in response to a given stimulus.

Mathematical theorem which outlines the conditions under which a stimulus with limited frequency range can be perfectly reconstructed from discrete measurements.

A formula which specifies how likely a given spike train is given a stimulus. The likelihood depends on both the encoding model and the noise model.

Average difference between the true stimulus and the reconstructed stimulus. A decoder with small bias will usually be preferable to a decoder with large bias.

Prior assumptions or knowledge about the stimuli, which are encoded in a probability distribution over possible stimuli.

A probability distribution which determines which stimuli are consistent with a given spike train: Stimuli which are consistent with the spike train have high posterior probability.

A simple yet powerful decoding algorithm which reconstructs the stimulus as a superposition of fixed waveforms centered at spike times.