# Rate vs. timing (XIX) Spike timing precision and sparse coding

Spike-based theories are sometimes discarded on the basis that spike timing is not reproducible in vivo, in response to the same stimulus. I already argued that, in addition to the fact that this is a controversial statement (because for example this could be due to a lack of control of independent variables such as attentional state), this is not a case for rate-based theories but for stochastic theories.

But I think it also reveals a misunderstanding of the nature of spike-based theories, because in fact, even deterministic spike-based theories may predict irreproducible spike timing. Underlying the argument of noise is the assumption that spikes are produced by applying some operation on the stimulus and then producing the spikes (with some decision threshold). If the timing of these spikes is not reproducible between trials, so the argument goes, then there must be noise inserted at some point in the operation. However, spike-based theories, at least some of them, do not fit this picture. Rather, the hypothesis is that spikes produced by different neurons are coordinated so as to produce some function. But then there is no reason why spikes need to be produced at the same time by the same neurons in all trials in order to produce the same global result. What matters is that spikes are precisely coordinated, which means that the firing of one neuron depends on the previous firing of other neurons. So if one neuron misses a spike, for example, then it will affect the firing of other neurons, precisely so as to make the computation more reliable. In other words, it is implied by the hypothesis of precise spike-based coordination that the firing of a spike by a single neuron should impact the firing of all other neurons, which makes individual firing non-reproducible.

The theory of sparse coding is line with this idea. In this theory, it is postulated that the stimulus can be reconstructed from the firing of neurons. That is, each spike contributes a “kernel” to the reconstruction, at the time of the spike, and all such contributions are added together so that the reconstruction is as close as possible to the original stimulus. Note how this principle is in some way the converse of the previously described principle: the spikes are not described as the result of a function applied to the stimulus, but rather the stimulus is described as a function of the spikes. So spike encoding is defined as an inverse problem. This theory has been rather successful in explaining receptive fields in the visual (Olshausen) and auditory (Lewicki) systems. It is also meant to make sense from the point of view of minimizing energy consumption, as it minimizes the number of spikes required to encode the stimulus with a given precision. There are two interesting points here, regarding our present discussion. First, it appears that spikes are coordinated in the way I just described above: if one spike is missed, then the other spikes should be produced so as to compensate for this loss, which means there is a precise spike-based coordination between neurons. Second, the pattern of spikes is seen as a solution to an inverse problem. This implies that if the problem is degenerate, then there are several solutions that equally good in terms of reconstruction error. Imagine for example that two neurons contribute exactly the same kernel to the reconstruction – which is not useless if one considers the fact that firing rate is limited by the refractory period. Then on one given trial, either of these two neurons may spike. From the observer point of view, this represents a lack of reproducibility. However, this lack of reproducibility is precisely due to the fact that there is a precise spike-based coordination between neurons: to minimize the reconstruction error, just one of the two neurons should be active, and the timing should be precise too.

Sparse coding with spikes also implies that reproducibility should depend on the stimulus. That is, a stimulus that is highly redundant such as a sinusoidal grating makes a degenerate inverse problem, leading to lack of reproducibility of spikes, precisely because of the coordination between spikes; a stimulus that is highly informative such a movie of a natural scene should lead to higher reproducibility of spikes. Therefore, in the sparse coding framework, the spike-based coordination hypothesis predicts, contrary to rate-based theories, that spike time reproducibility should depend on the information content of the stimulus – in the sense that a more predictable stimulus leads more irreproducible spiking. But even when spiking is not reproducible, it is still precise.
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