Philosophy of the Spike: Rate-Based vs. Spike-Based Theories of the Brain

Abstract

Does the brain use a firing rate code or a spike timing code? Considering this controversial question from an epistemological perspective, I argue that progress has been hampered by its problematic phrasing. It takes the perspective of an external observer looking at whether those two observables vary with stimuli, and thereby misses the relevant question: which one has a causal role in neural activity? When rephrased in a more meaningful way, the rate-based view appears as an ad hoc methodological postulate, one that is practical but with virtually no empirical or theoretical support.

Keywords: action potentials; firing rate; information; neural coding; neural computation; neural variability; spike timing.

Figure 1

Definitions of firing rate. (A) Rate as a temporal average (number of spikes n divided by observation time T). (B) Rate as a spatial average over N neurons, on a short time window dt. (C) Rate as a probability of firing, corresponding to an average over N trials for the same neuron.

Figure 2

Neural variability. (A) Responses of a MT neuron to the same stimulus (reproduced from Shadlen and Newsome, 1998). Top: spike trains over repeated trials, with the corresponding firing rate, meant as a firing probability (Figure 1C). Middle: distribution of Interspike intervals (ISIs), with an exponential fit (solid curve). Bottom: variance of spike count as a function of mean count, with the prediction for Poisson processes (dashed). (B) Responses of the same V1 neuron over five trials of the same stimulus, represented as temporal firing rate (adapted from Schölvinck et al., 2015). Left: comparison with the average response (gray curve), showing variability over trials. Right: comparison with a prediction using the responses of other neurons, showing that the variability does not reflect private noise. (C) Responses of a single cortical neuron to a fluctuating current (middle) injected in vitro (reproduced from Mainen and Sejnowski, 1995). Top: superimposed voltage traces. Bottom: spike trains produced in the 25 trials. (D) The Lorentz attractor, consisting of trajectories of a chaotic three-dimensional climate model. Chaos is not randomness, as it implies particular relations between the variables represented by the attractor.

Figure 3

Variability due to degeneracy. (A) Spikes can be seen as the result of a sequence of operations applied on an input signal, followed by spike generation. In this view, variability comes from noise added in the spiking process. (B) The state of a physical system can often be described as a minimum of energy. Symmetries in the energy landscape can imply observed variability, whose magnitude bears no relation with the amount of intrinsic noise. (C) An example of the energy view is spike-based sparse coding. A reconstruction of the signal is obtained from combining filtered spike trains together, and spikes are timed so as to make the reconstruction accurate. (D) If the system is redundant, the reconstruction problem is degenerate, leading to several equally accurate spiking solutions (here obtained by permutation of neurons).

Figure 4

The timescale argument. (A) Responses of a neuron over repeated trials, where the firing rate (PSTH shown below) varies on a fast (left) or slow (right) time scale. (B) Whether the firing rate varies quickly or slowly, the average rate is not generally sufficient to predict the response of a postsynaptic neuron. Here the responses of two neurons are shown over two trials. The postsynaptic neuron responds strongly when the presynaptic spike trains are taken in the same trial (because they are synchronous), but not if the spike trains are shuffled over trials.

Figure 5

The rate-based view. (A) Each neuron is described by a private quantity r(t), its rate, which determines the spike trains through a stochastic process. Crucially, even though only the spike trains are directly observed by the neuron, it is postulated that the rate r(t) only depends on the presynaptic rates r_i(t), and not on the spike trains themselves (f can be a function, dynamical process or filter). (B) One option is that synaptic integration leads to a quantity x(t); (e.g., membrane potential) that only depends on the rates r_i(t) by law of large numbers. As x(t) is deterministic, private noise must be added so as to produce stochastic spike trains. (C) The other option is that there are random fluctuations around the mean x(t) due to the spiking inputs, causing output spikes. For these fluctuations to depend only on the rates, the presynaptic processes must be independent, but we note that we have now introduced correlations between inputs and outputs.

Figure 6

Reduction of a spike-based model to a rate-based model. The spike model defines a relationship between presynaptic and postsynaptic spike trains through a function S. The rate model defines a relationship between rates through a function f. Rates are related to spikes through an observation function R. Reduction is possible when the diagram commutes: the composition of R and S equals the composition of f and R (mathematically, R o S = f o R). This is not generally possible because R is not invertible (many spike trains have the same rate).