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The authors have declared that no competing interests exist.

Conceived and designed the experiments: RN WG. Performed the experiments: RN. Analyzed the data: RN. Wrote the paper: RN WG.

The response of a neuron to a time-dependent stimulus, as measured in a Peri-Stimulus-Time-Histogram (PSTH), exhibits an intricate temporal structure that reflects potential temporal coding principles. Here we analyze the encoding and decoding of PSTHs for spiking neurons with arbitrary refractoriness and adaptation. As a modeling framework, we use the spike response model, also known as the generalized linear neuron model. Because of refractoriness, the effect of the most recent spike on the spiking probability a few milliseconds later is very strong. The influence of the last spike needs therefore to be described with high precision, while the rest of the neuronal spiking history merely introduces an average self-inhibition or adaptation that depends on the expected number of past spikes but not on the exact spike timings. Based on these insights, we derive a ‘quasi-renewal equation’ which is shown to yield an excellent description of the firing rate of adapting neurons. We explore the domain of validity of the quasi-renewal equation and compare it with other rate equations for populations of spiking neurons. The problem of decoding the stimulus from the population response (or PSTH) is addressed analogously. We find that for small levels of activity and weak adaptation, a simple accumulator of the past activity is sufficient to decode the original input, but when refractory effects become large decoding becomes a non-linear function of the past activity. The results presented here can be applied to the mean-field analysis of coupled neuron networks, but also to arbitrary point processes with negative self-interaction.

How can information be encoded and decoded in populations of adapting neurons? A quantitative answer to this question requires a mathematical expression relating neuronal activity to the external stimulus, and, conversely, stimulus to neuronal activity. Although widely used equations and models exist for the special problem of relating external stimulus to the action potentials of a single neuron, the analogous problem of relating the external stimulus to the activity of a

Encoding and decoding of information with populations of neurons is a fundamental question of computational neuroscience

What is the function that relates an arbitrary stimulus to the population activity of adapting neurons? We focus on the problem of relating the filtered input

When driven by a step change in the input, the population of neurons coding for this stimulus responds first strongly but then adapts to the stimulus. To cite a few examples, the activity of auditory nerve fibers adapt to pure tones

Synapse- and network-specific mechanisms merge with intrinsic neuronal properties to produce an adapting population response. Here we focus on the intrinsic mechanisms, commonly called spike-frequency adaptation. Spike-frequency adaptation appears in practically all neuron types of the nervous system

Mean-field methods were used to describe: attractors

The results described in the present article are based on two principal insights. The first one is that adaptation reduces the effect of the stimulus primarily as a function of the expected number of spikes in the recent history and only secondarily as a function of the higher moments of the spiking history such as spike-spike correlations. We derive such an expansion of the history moments from the single neuron parameters. The second insight is that the effects of the refractory period are well captured by renewal theory and can be superimposed on the effects of adaptation.

The article is organized as follows: after a description of the population dynamics, we derive a mathematical expression that predicts the momentary value of the population activity from current and past values of the input. Then, we verify that the resulting encoding framework accurately describes the response to input steps. We also study the accuracy of the encoding framework in response to fluctuating stimuli and analyze the problem of decoding. Finally, we compare with simpler theories such as renewal theory and a truncated expansion of the past history moments.

To keep the discussion transparent, we focus on a population of unconnected neurons. Our results can be generalized to coupled populations using standard theoretical methods

How does a population of adapting neurons encode a given stimulating current

Mathematically, we consider a set of spike trains in which spikes are represented by Dirac-pulses centered on the spike time

Since the population activity represents the instantaneous firing probability, it is different from the conditional firing intensity,

Ideally, one could hope to estimate

To see that the function

While refractoriness refers to the interspike-interval distribution and therefore to the dependence upon the

Conceptually, contributions of multiple spikes must accumulate to generate spike frequency adaptation. In the Spike Response Model, this accumulation is written as a convolution:

The effects described by

In a population of neurons, every neuron has a different spiking history defined by its past spike train

We truncate the series expansion resulting from

We note that by removing the integral of

Let us now assess the domain of validity of the QR theory by comparing it with direct simulations of a population of SRM neurons. To describe the single neurons dynamics, we use a set of parameters characteristic of L2–3 pyramidal cells

The response to a step increase in stimulating current is a standard paradigm to assess adaptation in neurons and used here as a qualitative test of our theory. We use three different step amplitudes: weak, medium and strong. The response of a population of, say, 25,000 model neurons to a

(

In contrast to renewal models (i.e., models with refractoriness but no adaptation), we observe in

The QR equation describes well both the damped oscillation and the adapting tail of the population activity response to steps (

Step changes in otherwise constant input are useful for qualitative assessment of the theory but quite far from natural stimuli. Keeping the same SAP as in

(

Decoding the population activity requires solving the QR equation (

(

We will consider two recent theories of population activity from the literature. Both can be seen as extensions of rate models such as the Linear-Nonlinear Poisson model where the activity of a homogeneous population is

To discuss the relation to existing theories, we recall that the instantaneous rate of our model

We compare the prediction of EME1, EME2 and renewal theory with the simulated responses to step inputs (

(

Fluctuating input makes the population respond in peaks of activity separated by periods of quiescence. This effectively reduces the coupling between the spikes and therefore improves the accuracy of EME1. The validity of EME1 for encoding time-dependent stimulus (

Decoding with EME1 is done according to a simple relation:

In summary, the EMEs yield theoretical expressions for the time-dependent as well as steady-state population activity. These expressions are valid in the limit of small coupling between the spikes which corresponds to either large interspike intervals or small SAP. Renewal theory on the other hand is valid when the single-neuron dynamics does not adapt and whenever the refractory effects dominate.

The input-output function of a neuron population is sometimes described as a linear filter of the input

We have derived self-consistent formulas for the population activity of independent adapting neurons. There are two levels of approximation, EME1 (

The QR equation captures almost perfectly the population code for time-dependent input even at the high firing rates observed in retinal ganglion cells

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

The decoding schemes presented in this paper (

Using the results presented here, existing mean-field methods for populations of spiking neurons can readily be adapted to include spike-frequency adaptation. In

(

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

This section is organized in 3 subsections. Subsection A covers the mathematical steps to derive the main theoretical results (

The probability density of a train of

In order to single out the effect of the previous spike, we replace

We can recognize in

A derivation of the renewal equation

First consider the expected value in

At the steady state with a constant input

All simulations were performed on a desktop computer with 4 cores (Intel Core i7, 2.6 GHz, 24 GB RAM) using Matlab (The Mathworks, Natwick, MA). The Matlab codes to numerically solve the self-consistent equations are made available on the author's websites. The algorithmic aspects of the numerical methods are discussed now.

All temporal units in this code are given in milliseconds. Direct simulation of

For all simulations, the baseline current was 10 pA (except for time-dependent current where the mean was specified), the baseline excitability was

Time-dependent input consisted of an Ornstein-Uhlenbeck process which is computed at every time step as:

We consider the QR equation,

The second vector

We can therefore calculate the population activity iteratively at each time bin using

Isolating the input

The structure of EME1 and EME2 allows us to use a nonlinear grid spacing in order to save memory resources. The bins should be small where

To perform the numerical integration, we define the vector

To compute the second order equation, we first build the correlation vector

The first-order expansion (

When assessing the accuracy of the encoding or the decoding, we used the correlation coefficient. The correlation coefficient is the variance-normalized covariance between two random variables

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We would like to thank C. Pozzorini, D. J. Rezende and G. Hennequin for helpful discussions.