# Rate vs. timing (X) Rate theories in spiking network models

According to the rate-based hypothesis, 1) neural activity can be entirely described by the dynamics of underlying firing rates and 2) firing is independent between neurons, conditionally to these rates. This hypothesis can be investigated in models of spiking neural networks by a self-consistency strategy. If all inputs to a neuron are independent Poisson processes, then the output firing rate can be calculated as a function of input rates. Rates in the network are then solutions of a fixed point equation. This has been investigated in random networks in particular by Nicolas Brunel. In a statistically homogeneous network, theory gives the stationary firing rate, which can be compared to numerical simulations. The approach has also been applied to calculate self-sustained oscillations (time-varying firing rates) in such networks. In general, theory works nicely for sparse random networks, in which a pair of neurons is connected with a low probability. Sparseness implies that there are no short cycles in the connectivity graph, so that the fact that the inputs to a neuron and its output are strongly dependent has little impact on the dynamics. Results of simulations diverge from theory when the connection probability increases. This means that the rate-based hypothesis is not true in general. On the contrary, it relies on specific hypotheses.

Real neural networks do not look like random sparse networks, for example they can be strongly connected locally, neurons can be bidirectionally connected or form clusters. Recently, there have been a number of nice theoretical papers on densely connected balanced networks (Renart et al., Science 2010; Litwin-Kumar and Doiron, Nat Neurosci 2012), which a number of people have interpreted as supporting rate-based theories. In such networks, when inhibition precisely counteracts excitation, excitatory correlations (due to shared inputs) are cancelled by the coordination between inhibition and excitation. As a result, there are very weak pairwise correlations between neurons. I hope it is now clear from my previous posts that this is not an argument in favor of rate-based theories. The fact that correlations are small says nothing about whether dynamics can be faithfully described by underlying time-varying rates.

In fact, in such networks, neurons are in a fluctuation-driven regime, meaning that they are highly sensitive to coincidences. What inhibition does is to cancel the correlations due to shared inputs, i.e., the meaningless correlations. But this is precisely what one would want the network to do in spike-based schemes based on stimulus-specific synchrony (detecting coincidences that are unlikely to occur by chance) or on predictive coding (firing when there is a discrepancy between input and prediction). In summary, these studies do not support the idea that rates are an adequate basis for describing network dynamics. They show how it is possible to cancel expected correlations, a useful mechanism in both rate-based and spike-based theories.

Update. These observations highlight the difference between correlation and synchrony. Correlations are meant as temporal averages, for example pairwise cross-correlation. But on a timescale relevant to behavior, temporal averages are irrelevant. What might be relevant are spatial averages. Thus, synchrony is generally meant as the fact that a number of neurons fire at the same time, or a number of spikes arrive at the same time at a postsynaptic neuron. This is a transient event, which may not be repeated. A single event is meaningful if such synchrony (possibly involving many neurons) is unlikely to occur by chance. The terms “by chance” refer to what could be expected given the past history of spiking events. This is precisely what coordinated inhibition may correspond to in the scheme described above: the predicted level of input correlations. In this sense, inhibition can be tuned to cancel the expected correlations, but by definition it cannot cancel coincidences that are not expected. Thus, the effect of such an excitation-inhibition coordination is precisely to enhance the salience of unexpected synchrony.