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Conceived and designed the experiments: SO NB. Performed the experiments: SO. Analyzed the data: SO. Wrote the paper: SO NB.

The authors have declared that no competing interests exist.

Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an

Deciphering the encoding of information in the brain implies understanding how individual neurons emit action potentials (APs) in response to time-varying stimuli. This task is made difficult by two facts: (i) although the biophysics of AP generation are well understood, the dynamics of the membrane potential in response to a time-varying input are highly complex; (ii) the firing of APs in response to a given stimulus is inherently stochastic as only a fraction of the inputs to a neuron are directly controlled by the stimulus, the remaining being due to the fluctuating activity of the surrounding network. As a result, the input-output transform of individual neurons is often represented with the help of simplified phenomenological models that do not take into account the biophysical details. In this study, we directly relate a class of such phenomenological models, the so called linear-nonlinear models, with more biophysically detailed spiking neuron models. We provide a quantitative mapping between the two classes of models, and show that the linear-nonlinear models provide a good approximation of the input-output transform of spiking neurons, as long as the fluctuating inputs from the surrounding network are not exceedingly weak.

Neurons encode stimuli by emitting trains of actions potentials in response to sensory inputs. To uncover the corresponding neural code, the mapping between sensory inputs and output action potentials needs to be described with the help of a quantitative model

In more detailed models of the neural input-output mapping, membrane potential dynamics play the role of the intermediate between input currents and output action potentials

In this communication, we examine to what extent a linear-nonlinear cascade can quantitatively reproduce the firing rate dynamics of spiking neuron models. To this end, we exploit known analytic results for integrate-and-fire models to obtain parameter-free expressions for the linear filter and static non-linearity. We then compare quantitatively the estimates of instantaneous firing rates obtained from various LN models with results from simulations of spiking neurons. For both the leaky integrate-and-fire (LIF) and exponential integrate-and-fire (EIF) models, in most of parameters space we find a good match between the estimate and the simulation results. In the case of the EIF, we show that a single exponential provides a good approximation for the linear filter, so that the LN cascade reduces to a firing rate model, the time constant of which we compute analytically. We then introduce an

We model a typical setup in which a given stimulus is repeatedly applied to a preparation, and action potentials of a neuron are recorded over many trials. We represent this neuron as a spiking neuron (either integrate-and-fire or conductance based) receiving a time-varying input. Here we consider only the case of input current, but our results could be easily extended to an input conductance. This current is assumed to consist of a sum of two components: an

A: A spiking neuron receives an input current consisting of a signal component that is identical in all trials and a noise component that is uncorrelated from trial to trial. Averaging trains of action potentials across trials gives a time-dependent output firing rate. B: Our aim is to obtain an estimate of the output firing rate by applying to the input signal a linear temporal filter followed by a static non-linearity. C: Illustration in the case of an exponential integrate-and-fire model. From top to bottom: input signal; raster plot of action potentials in a subset of

Our aim is to examine the extent to which the mapping between the input signal and the output firing rate can be approximated by a linear-nonlinear (LN) cascade consisting of two steps: (i) a linear temporal filter applied to the input signal; (ii) a static non-linear function applied to obtain the instantaneous firing rate (see

To limit the available parameter space, we assume that the temporal statistics of the noise input are Gaussian with mean

We wish to approximate the trial-averaged firing rate

The LN approximation of firing rate dynamics becomes exact in two extreme cases: (i) the linear limit of vanishing signal amplitude

For a signal of vanishing amplitude

On the other hand, in the linear limit the firing rate of the spiking neuron is given by

Comparing Eqs. (2) and (3), it is straightforward to identify

In the limit

In the same limit, as the input signal varies slowly, at each point in time the neuron effectively receives a white noise input of mean

The transfer function for the LIF and EIF models receiving white noise is known analytically

Comparing Eqs. (4) and (5) leads to the following identification:

For finite signal amplitude

We start by examining the leaky integrate-and-fire (LIF) model

As we set the linear filter

Background noise strongly modulates the response of the neuron

A: Analytic filter compared with the numerical spike triggered averages of the input signal, for three different amplitudes

For strong background noise,

For weak background noise,

The amplitude of the linear filter approximately scales as the inverse of the standard deviation

It is interesting to compare the analytic linear filter

The analytic derivation of the transfer function

The static non-linearity

To compare our static non-linear function

Once the linear filter and the static non-linearity are determined, we are in position to compare the estimate of the instantaneous firing rate provided by the LN cascade with the actual, numerically determined firing rates for different points in parameter space.

A: Illustration for a given set of parameters (

The degree to which various approximations match the numerical PSTH clearly depends on the parameters of the input signal and background noise. To get a quantitative comparison, we computed the Pearson's correlation coefficient

As

The correlation between the signal and the output increases as the correlation time of the signal is increased (

Although the non-linear filter and static non-linearity depend on the parameters of background noise (see

In summary, the linear-nonlinear model of input-output mapping provides a good approximation of the firing rate dynamics for most of the parameter space, two notable exceptions being the limit of weak background noise (

The exponential integrate-and-fire (EIF) model

The linear filter

A: Comparison between the full filter and the single timescales approximation for three different values of noise amplitude

A: Effect of varying parameters

In the case of the EIF model, as the linear filter

With such an exponential filter, the linear non-linear cascade of Eq. (1) can be rewritten as

To derive an analytic expression for the timescale

To compare quantitatively the full linear filter

As shown in

Illustration for a given set of parameters (

Similarly to the full LN model, the performance of the rate model degrades in the two limits of weak background noise and very large input signal amplitude. The advantage of the rate model over the full LN cascade is its simplicity, which allows for a very efficient and robust implementation.

While the LN and rate models provide good estimates of firing rate dynamics in most of parameter space, their accuracy deteriorates as the amplitude

So far, the linear filter

In the adaptive LN model, the linear filter has to be computed in principle at every timestep by integrating the Fokker-Planck equation, which is computationally cumbersome. Instead, for the EIF model, at every timestep we approximate the instantaneous filter by the corresponding exponential filter (see Eq. 9). We thus obtain an

In this model the timescale

As shown in

So far we examined only models of the integrate-and-fire type, which are one-dimensional in the sense that action potential generation is controlled by a single variable, the membrane potential. In contrast, in biophysically more detailed models, the dynamics of the membrane potential are coupled to the dynamics of a number of ionic conductances, so that these models have higher dimensionality. In spite of this additional complexity, we will show that our results can be easily extended to a standard conductance-based model of Hodgkin-Huxley type, the Wang-Buzsáki model

Studying the dynamics of conductance-based models in the presence of noise is in general very challenging, and the transfer and linear response functions are in general not known analytically. It has however been found that the exponential integrate-and-fire model with appropriately chosen threshold, reset, spike sharpness and refractory period closely reproduces the transfer and linear response functions of the Wang-Buzsáki model

The linear filter and static non-linearity for the Wang-Buzsáki model can thus be directly obtained from the transfer function and linear response function of the EIF model with appropriate parameters (see

A: Illustration for a given set of parameters (

In this study, we examined the ability of phenomenological models to describe the firing rate output of spiking neurons in response to a time-varying input signal that the neurons receive on top of background synaptic noise. The phenomenological models we considered belong to the class of linear-nonlinear cascade models: the firing rate is estimated by first applying a linear filter to the input signal and then correcting for deviations from linearity using a static non-linear function. Instead of using a fitting procedure, the linear filter and static non-linearity were obtained in a parameter-free form by exploiting analytic results valid for particular limits of input signal parameters. This approach allowed us to systematically quantify the accuracy of the phenomenological models by comparing their predictions with results of numerical simulations of spiking neurons.

We found that linear non-linear models provide a quantitatively accurate description of firing rate dynamics of leaky integrate-and-fire, exponential integrate-and-fire and conductance-based models, as long as the background noise is not excessively weak. In the limit of vanishing variance of background noise, the spiking of neurons exhibits locking to the input signal

For the exponential integrate-and-fire and conductance-based models, the linear filter can be accurately approximated by a single exponential in a large range of noise amplitudes, so that the linear-nonlinear model can be reduced to a firing rate model. We obtained a simple analytic expression for the time constant of the rate model, directly relating it to the biophysical parameters of the neuron. The value of the time constant in particular depends on the sharpness of action potential initiation and the baseline firing rate of the neuron.

Interestingly, the EIF model is essentially the only non-linear integrate-and-fire model that can be described by such a simple rate model, since it is the only model in this class whose firing rate response decays as

Finally, we introduced a simple generalization of the rate model in which the time constant depends on the instantaneous firing rate of the neuron. This

Phenomenological firing-rate models (and the closely related neural field models) are basic tools of theoretical neuroscience, and several earlier studies have looked for quantitative mappings between such models and more biophysically detailed, spiking neuron models. To our knowledge, our study is the first to compare extensively across parameter values the output of a phenomenological rate model to the firing rate dynamics of spiking neurons.

The question of how to reduce the firing rate dynamics of populations of spiking neurons to simplified ‘firing rate’ models has been the subject of numerous previous studies. Most reductions however ignore the single cell dynamics and eventually end up with rate equations in which the only time scale is a synaptic time scale (see e.g.

The correspondence between linear-nonlinear cascade models and spiking neuron models has been examined in several earlier works. In

To produce trains of action potentials, the linear-nonlinear cascade model is often supplemented by a third step, a stochastic Poisson process which at every time step generates an action potential with a probability given by the instantaneous firing rate obtained from the cascade. In this study, we have not attempted to compare the full statistics of spike trains produced by such a linear-nonlinear-Poisson model with the statistics of spike trains of integrate-and-fire neurons. Instead we have concentrated on the instantaneous firing rate, which is equivalent to the first-order statistics of spike trains. The instantaneous firing rate provides information about the timing of individual spikes, but does not specify the correlations between successive spikes in a given train. It has been argued that the refractory period and other post-spike effects play an important role in determining precise spike timing

To reproduce faithfully the full statistics of spike trains of spiking neurons, the linear-nonlinear cascade would have to be supplemented with post-spike history filters leading to correct higher order statistics. Several modeling approaches have been developed to include post-spike filters

A large number of studies have exploited linear-nonlinear models to fit experimentally measured data. In the majority of these studies

Exponential integrate-and-fire models have been used to predict individual action-potentials of cortical neurons, however post-spike adaptation currents had to be taken into account

In integrate-and-fire models, action potentials are generated solely from the underlying dynamics of the membrane potential

We studied two different versions of the integrate-and-fire model:

Once the membrane potential crosses the threshold

We used the Wang-Buzsáki model

The activation of the sodium current is assumed instantaneous:

The functions

The maximum conductance densities and reversal potentials are:

As explained in the main text, in this study we assume that the synaptic inputs to the neuron are separated into two groups: (i) inputs that are identical across trials, and which we call the “signal” inputs; (ii) inputs that are uncorrelated from trial to trial, which we call the background noise. In consequence, the total synaptic input

We further assume that both signal and noise inputs consists of a sum of large number of synaptic inputs, each individual synaptic input being of small amplitude. We therefore use the diffusion approximation

For convenience, the mean of the input signal

The background noise

Here we provide the summary of definitions and expressions for the transfer function and linear response functions of integrate-and-fire neurons. For completeness full derivations are provided below.

The transfer function

For the leaky integrate-and-fire neuron receiving background noise uncorrelated in time, the transfer function is given by

For the exponential integrate-and-fire neuron receiving background noise uncorrelated in time, the transfer function can be expressed as

The rate response function

For the LIF receiving a background noise uncorrelated in time, the response function in frequency

For the EIF, no explicit expression is available for

To assess the precision of the firing rates predicted by various models, we have systematically compared the predicted firing rates with results of simulations of the LIF, EIF and Wang-Buzsáki neurons.

The membrane potential dynamics of the neuronal models were simulated using a standard second-order Runge-Kutta algorithm with a time step of

To obtain the predicted firing rates, the original input signal was sampled at intervals of

To compare quantitatively the prediction with the numerical firing rate, we computed the Pearson's correlation coefficient:

The value of the Pearson correlation coefficient

An alternative standard measure of the similarity between

If the means and variances of the two time series are identical, there is a simple relationship between

An advantage of the RMS distance

For a fixed set of parameters, the RMS distance

The value of

For the leaky integrate-and-fire neuron receiving background noise uncorrelated in time, the rate response in frequency

We consider a leaky integrate-and-fire neuron with membrane potential dynamics defined by Eq. (17), receiving an input current of the form

To study the stochastic dynamics of the membrane potential, we look at the probability distribution of the membrane potential

This equation expresses the conservation of probability in time, and can also be written as

The instantaneous firing rate is given by the flux of probability density through the threshold membrane potential

The membrane potential is reset to

As the membrane potential cannot exceed the threshold, for

Eqs. (41–44) are the four boundary conditions for the Fokker-Planck Equation. In addition we will require that

If

For convenience, we now introduce the rescaled notations:

To calculate the linear perturbation of the firing rate arising from a time-varying input current

Keeping only first-order terms, the Fokker-Planck equation becomes

To solve Eq. (50), we take its Fourier transform which yields

In Fourier space, the boundary conditions become

The solution of Eq. (53) can be expressed as

As shown in

The homogeneous equation reads

Two independent solutions of Eq. (58) can be expressed as

The full solution for

This discrepancy is resolved by noting that

In conclusion, we have

Note that the function

We consider here the limit

We are grateful to Evan Schaffer for many discussions and a careful reading of the manuscript. NB thanks Magnus Richardson for discussions on the relationship between EIF and rate models.