The input-output function of a neuron population is sometimes described as a linear filter of the input 
, as a linear filter of the input reduced as a function of past activity 
, as a non-linear function of the filtered input 
, or by any of the more recent population encoding frameworks 
. These theories differ in their underlying assumptions. To the best of our knowledge, a closed-form expression that does not assume weak refractoriness or weak adaptation has not been published before.
We have derived self-consistent formulas for the population activity of independent adapting neurons. There are two levels of approximation, EME1 (Eq. 10
) is valid at low coupling between spikes which can be observed in real neurons whenever (i) the interspike intervals are large, (ii) the SAPs have small amplitudes or (iii) both the firing rate is low and the SAPs have small amplitudes. The second level of approximation merges renewal theory with the moment-expansion to give an accurate description on all time-scales. We called this approach the QR theory.
The QR equation captures almost perfectly the population code for time-dependent input even at the high firing rates observed in retinal ganglion cells 
. But for the large interspike intervals and lower population activity levels of in vivo
neurons of the cortex 
, it is possible that the simpler encoding scheme of Eq. 10
is sufficient. Most likely, the appropriate level of approximation will depend on the neural system; cortical sparse coding may be well represented by EME, while neuron populations in the early stages of perception may require QR.
We have focused here on the Spike Response Model with escape noise which is an instantiation of a Generalized Linear Model. The escape noise model, defined as the instantaneous firing rate
given the momentary distance between the (deterministic) membrane potential and threshold should be contrasted with the diffusive noise model where the membrane potential fluctuates because of noisy input. Nevertheless, the two noise models have been linked in the past 
. For example, the interval-distribution of a leaky integrate-and-fire model with diffusive noise and arbitrary input can be well captured by escape noise with instantaneous firing rate
which depends both on the membrane potential and its temporal derivative
. The dependence upon
accounts for the rapid and replicable response that one observes when an integrate-and-fire model with diffusive noise is driven in the supra-threshold regime 
and can, in principle, be included in the framework of the QR theory.
The decoding schemes presented in this paper (Eq. 11
) reveal a fundamental aspect of population coding with adapting neurons. Namely, the ambiguity introduced by the adaptation can be resolved by considering a well-tuned accumulator of past activity. The neural code of adapting populations is ambiguous because the momentary level of activity could be the result of different stimulus histories. We have shown that resolving the ambiguity requires the knowledge of the activity in the past but to a good approximation does not require the knowledge of which neuron was active. At high population activity for neurons with large SAPs, however, the individual timing of the last spike in the spike trains is required to resolve the ambiguity (compare also Fairhall et al.
). Unlike bayesian spike-train decoding 
, we note that in our decoding frameworks the operation requires only knowledge of the population activity history and the single neuron characteristics. The properties of the QR or EME1 decoder can be used to find biophysical correlates of neural decoding such as previously proposed for short term plasticity 
, non-linear dendrites 
or lateral inhibition 
. Note that, a constant percept in spite of spike frequency adaptation does not necessarily mean that neurons use a QR decoder. It depends on the synaptic structure. In an over-representing cortex, a constant percept can be achieved even when the neurons exhibit strong adaptation transients 
Using the results presented here, existing mean-field methods for populations of spiking neurons can readily be adapted to include spike-frequency adaptation. In Methods
we show the QR theory for the interspike interval distribution and the steady-state autocorrelation function () as well as linear filter characterizing the impulse response function (or frequency-dependent gain function) of the population. From the linear filter and the autocorrelation function, we can calculate the signal-to-noise ratio 
and thus the transmitted information 
. The autocorrelation function also gives an estimate of the coefficient of variation 
and clarifies the role of the SAP in quenching the spike count variability 
. The finite-size effects 
is another, more challenging, extension that should be possible.
Steady-state interspike interval distribution and auto-correlation.
The scope of the present investigation was restricted to unconnected neurons. In the mean-field approximation, it is straight-forward to extend the results to several populations of connected neurons 
. For instance, similar to EME1, a network made of inter-connected neurons of
cell-types would correspond to the self-consistent system of equation:
is the scaled post-synaptic potential kernel from cell-type
(following the formalism of Gerstner and Kislter 
is an external driving force, each subpopulation is characterized by its population activity
and its specific spike after potential
. The analogous equation for QR theory is:
Since the SAP is one of the most important parameter for distinguishing between cell classes 
, the approach presented in this paper opens the door to network models that take into account the neuronal cell-types beyond the sign of the synaptic connection. Even within the same class of cells, real neurons have slightly different parameters from one cell to the next 
and it remains to be tested whether we can describe a moderately inhomogeneous population with our theory. Also, further work will be required to see if the decoding methods presented here can be applied to brain-machine interfacing