Misconception #2: “Neural responses are variable in vivo, therefore neural codes can only be based on rates”. Again, this is a category error. Neural variability (assuming this means randomness) is about determinism vs. stochasticity, not about rate vs. timing. There can be stochastic or deterministic spike-based theories.

I will expand on this point, because it is central to many argumentations in favor of rate-based theories. There are two ways to understand the term "variable" and I will first discard the meaning based on temporal variability. Interspike intervals (ISIs) are highly variable in the cortex (Softky and Koch, 1993), and their distribution is close to an exponential (or Gamma) function, as for Poisson processes (possibly with a refractory period). This could be interpreted as the sign that spike trains are realizations of random point processes. This argument is very weak, because the exponential distribution is also the distribution with maximum entropy for a given average rate, which means that maximizing the information content in the timing of spikes of a single train also implies an exponential distribution of ISIs. Temporal variability cannot distinguish between rate-based and spike-based theories.

Therefore the only reasonable variability-based argument in support of the rate-based view is the variability of spike trains across trials. In the cortex (but not so much in some early sensory areas such as the retina and some parts of the auditory brainstem), both the timing and number of spikes produced by a neuron in response to a given stimulus varies from one trial to another. This means that the response of a neuron to a stimulus cannot be described by a deterministic function. In other words, the stimulus-output relationship of neurons is stochastic. This is the only fact that this observation tells us (note that we may also argue that stochasticity only reflects uncertainty on hidden variables). That this stochasticity is entirely captured by an intrinsic time-varying rate signal is pure speculation at this stage. Therefore, the argument of spike train variability is about stochastic vs. deterministic theories, not about rate-based vs. spike-based theories. It only discards deterministic spike-based theories based on absolute spike timing. However, the prevailing spike-based theories are based on relative timing across different neurons (for example synchrony or rank order), not on absolute timing.

In fact, the argument can be returned against rate-based theories. It is often written or implied that rate-based theories take into account biological variability, whereas spike-based theories do not. But actually, quite the opposite is true. Rate-based theories are fundamentally deterministic, and a deterministic description is obtained at the cost of averaging noisy responses over many neurons, or over a long integration time. On the other hand, spike-based theories take into account individual spikes, and therefore do not rely on averaging. In other words, it is not that rate-based descriptions account for more observed variability, it is just that they acknowledge that neural responses are noisy, but they do not account for any variability at all. Accounting for more variability would require stochastic spike-based accounts. This confusion may stem from the fact that spike-based theories are often described in deterministic terms. But as stressed above, rate-based theories are also described in deterministic terms.

Throwing dice can be described by deterministic laws of mechanics. The fact that the outcomes are variable does not invalidate the laws of mechanics. It simply means that noise (or chaos) is involved in the process. Therefore criticizing spike-based theories for not being stochastic is not a fair point, and stochasticity of neural responses cannot be a criterion to distinguish between rate-based and spike-based theories.

To complement the previous post, I will comment on what firing rate means in spike-based theories. First of all, rate is important in spike-based theories. The timing of a spike can only exist if there is a spike. Therefore, the firing rate determines the rate of information in spike-based theories, but it does not determine the content of information.

A related point is energy consumption. The energy consumption of a cell is essentially proportional to the number of spikes it produces (taking into account the cost of synaptic transmission to target neurons) (Attwell and Laughlin, 2001). It seems reasonable to think that the organism tries to avoid any waste of energy, therefore a cell that fires at high rate must be doing something important. In terms of information, it is likely that the amount of information transmitted by a neuron is roughly proportional, or at least correlates with its firing rate.

For these two observations, it follows that, in spike-based theories, firing rate is a necessary correlate of information processing in a neuron. This stands in contrast with rate-based theories, in which rate is the basis of information processing. But both types of theories predict that firing rates correlate with various aspects of stimuli – and therefore that there is information about stimuli in firing rates, at least for an external observer.

This post starts a series on the debate between rate-based and spike-based theories of neural computation and coding. My primary goal is to clarify the concepts. I will start by addressing a few common misconceptions about the debate.

Misconception #1: “Both rate and spike timing are important for coding, so the truth is in between”. This statement, I will argue, is what philosophers would call a “category error”: it is not that only one of the alternatives can be right, it is just that the two alternatives belong to different categories.

Neurons mainly communicate with each other using trains of spikes – at least this is what the rate-timing debate is concerned about. A spike train can be completely characterized by the timing of its spikes. The firing rate, on the other hand, is an abstract definition, that is only valid in a limit, which involves an infinite number of spikes. For example, it can be defined for a single neuron as a temporal average: the inverse of the mean inter-spike interval. It appears that rate is defined from the timing of spikes. Thus these are two different concepts: spike timing is what defines spike trains, whereas rate is an abstract mathematical construction on spike trains. Therefore the rate vs. timing debate is not about which one is right, but about whether rate is a sufficiently good description of neural activity. Spike-based theories do not necessarily claim that rate does not matter, they refute the notion that rate is the essential quantity that matters.

There are different ways to define the firing rate: over time (number of spikes divided by the duration, in the limit of infinite duration), over neurons (average number of spikes in a population of neurons, in the limit of an infinite number of neurons) or over trials (average number of spikes over an infinite number of trials). In the third definition (which might be the prevailing view), the rate is seen as an intrinsic time-varying signal r(t) and spikes are seen as random events occurring at rate r(t). In all these definitions, rate is an abstract quantity defined on the spike trains. Therefore when stating that the neural “code” is based on rates rather than spike timing, what is meant is that the concept of rate captures most of the important details of neural activity and computation, while precise spike timing is essentially meaningless. On the other hand, when stating that spike timing matters, it is not meant that rate is meaningless; it simply means that precise timing information cannot be discarded. Thus, these are not two symmetrical views: the stronger assumptions are on the side of the rate-based view. Now of course each specific spike-based theory makes a number of possibly strong assumptions. But the general idea that the neural “code” is based on individual spikes and not just rates is not based on strong assumptions. The rate-based view is based on an approximation, which may be a good one or a bad one. This is the nature of the rate vs. timing debate.