Shifting Spike Times or Adding and Deleting Spikes—How Different Types of Noise Shape Signal Transmission in Neural Populations
© S.O. Voronenko et al.; licensee Springer 2015
Received: 15 April 2014
Accepted: 17 November 2014
Published: 12 January 2015
We study a population of spiking neurons which are subject to independent noise processes and a strong common time-dependent input. We show that the response of output spikes to independent noise shapes information transmission of such populations even when information transmission properties of single neurons are left unchanged. In particular, we consider two Poisson models in which independent noise either (i) adds and deletes spikes (AD model) or (ii) shifts spike times (STS model). We show that in both models suprathreshold stochastic resonance (SSR) can be observed, where the information transmitted by a neural population is increased with addition of independent noise. In the AD model, the presence of the SSR effect is robust and independent of the population size or the noise spectral statistics. In the STS model, the information transmission properties of the population are determined by the spectral statistics of the noise, leading to a strongly increased effect of SSR in some regimes, or an absence of SSR in others. Furthermore, we observe a high-pass filtering of information in the STS model that is absent in the AD model. We quantify information transmission by means of the lower bound on the mutual information rate and the spectral coherence function. To this end, we derive the signal–output cross-spectrum, the output power spectrum, and the cross-spectrum of two spike trains for both models analytically.
KeywordsTime-dependent input Population coding Common noise Shifting of spikes Addition and deletion of spikes Mutual information Suprathreshold stochastic resonance
Neurons in the sensory periphery encode information about continuous time-dependent signals in sequences of action potentials. Hereby, upon repeated presentation of a stimulus, the response of the neuron is not perfectly reproducible but exhibits trial-to-trial variability. Processes, leading to such variability, are termed noise and can have various origins [1, 2]. How such noise processes affect the transmission of time-dependent signals in neurons can be studied in the framework of information theory [3, 4]. Within this framework, it has been shown, for instance, that the presence of noise can enhance the transmission of weak (subthreshold) signals in single neurons and neural models [5–7], an effect known as stochastic resonance and also observed outside biology [8, 9]. At the level of neural population coding, noise can also have a beneficial role for the transmission of strong (suprathreshold) signals [10, 11] by means of suprathreshold stochastic resonance (SSR), the mechanism of which is quite distinct from that of conventional stochastic resonance despite the similarity in their naming. Additionally, noise not only impacts the total transmitted information, but it also affects which frequencies of the sensory signal are preferably encoded by a neural system. The suppression of information about the input signal in certain frequency bands can be regarded as a form of information filtering [12–16]. Put differently, we may ask whether the neural system is preferentially encoding slow (low-frequency) components of a signal or fast (high-frequency) components of a signal, which can be quantified by the coherence function, as described below.
This work is organized as follows: First, we describe the methods by which we will study the effect of noise on signal transmission in a population of spiking neurons. Second, we introduce two models where independent noise either adds and deletes spikes, or shifts spike times in the output spike trains. In Sect. 4, we then derive the spectral statistics for the two models. These derivations can be skipped upon the first reading. In Sect. 5, we summarize the derived spectral statistics and proceed to study the effect of independent noise on information filtering and the total transmission of information in neural populations. We conclude with a summary and a discussion of our results in Sect. 6.
2.1 Spike Train Statistics & Ensemble Averages
of the individual output spike trains.
This applies analogously to averages over the processes and .
2.2 Information Transmission & Spectral Statistics
From Eq. (10) we see that for the cross-spectrum of two spike trains, , appears in the denominator of the coherence function and gains significance as N becomes larger. Therefore, an essential theoretical problem is to calculate this cross-spectrum.
As outlined above, the coherence function allows one to estimate the total flow of information through the neural population. However, because enters in a monotonic fashion in Eq. (4), we can also regard the coherence as a frequency-resolved measure of information transfer. Reduction of the coherence in certain frequency bands can be regarded as a form of information filtering, which needs to be distinguished from power filtering. Hence, besides the lower bound , we will also inspect the frequency dependence of the coherence function.
Poisson statistics of spontaneous activity;
high correlations among neurons due to strong common noise input;
encoding of a sensory signal in the time-dependent population rate.
For simplicity, we consider a linear encoding of a weak time-dependent signal. This will allow us to use the lower bound on the mutual information rate as an approximation for the total transmitted information. Note that, although already a single Poisson process can show conventional stochastic resonance , with our linear encoding paradigm we exclude this possibility. In our models, the signal transmission in a single neuron is always degraded by noise.
In our theoretical model, we assume that all neurons fire to zeroth-order in complete synchrony and a weak noise input, which is independent for every neuron, leads to a decorrelation of the output spike trains. For simplicity, we assume that for each neuron the independent noise process and the sensory signal are additive. Both, the sensory signal and the independent noise signals, are modeled by Gaussian processes with unit variance and zero mean.
The considered models can be regarded as inhomogeneous Poisson processes , which are rate-modulated by a common signal and an independent noise . Such processes are examples of a doubly stochastic process  or a Cox process and are a special case of the inhomogeneous Bernoulli process . The simplicity of the considered models will allow us to characterise the information transfer of weak time-dependent signals analytically. Note that the assumptions (1)–(3) made above describe, in good approximation, spiking in specific sensory systems, e.g. in tangential neurons of the fly visual system [20, 39, 40]. The additional modifications that make up the differences between our two models can be regarded as additional operations on the spike trains in the form of thinning (or the opposite of it) and the introduction of an operational time [37, 41].
Before we introduce in detail the two models sketched in Fig. 2, it is worth to note that, for weak stimuli and weak independent noise, these models possess the same signal–output cross-spectrum , the same power spectrum , and the same time-dependent output firing rate. Therefore, for the coherence function and the information rate are identical for both models. The models are mainly distinguished by how independent noise affects the spikes of the output spike trains, which results in different cross-spectra of two spike trains. This setup allows us to study how the response of spikes to noise affects information transmission in neural populations, while keeping all other potential influences on signal transmission unchanged.
3.1 Addition and Deletion Model (AD Model)
where is the Heaviside function (implementing the indicator function) and the second argument of indicates the time-discretized version of the spike train. Here is the midpoint of the time bin where the k th spike of the μ th spike train was generated. In the limit the spike train approximates the sum of δ-functions given by Eq. (1).
Throughout the paper, we will consider the limit , such that we can neglect correction terms like the one in the above equation.
In the left column of Fig. 3, we show how a sensory signal is encoded in the population firing rate of a population of five AD neurons and how the output spike trains of the neurons are modulated by independent noise.
3.2 Spike-Time-Shifting Model (STS Model)
with defined in Eq. (11). For a given spike time , we integrate the right hand side of Eq. (16), until the integral attains the value . The resulting integration boundary is then the k th spike time of the μ th spike train . In general, due to the different independent noise processes , the output spike trains will be different for each neuron. Hereby, each spike train is an inhomogeneous Poisson spike train with a time-dependent firing rate. The procedure described in this section is equivalent to the simulation of a perfect integrate-and-fire neuron with exponentially distributed thresholds . The time t obtained after the transformation of the time axis h in Eq. (16) is also known as operational time [37, 41].
Although we do not model the underlying noise process explicitly, we think of the homogeneous spike trains in Eq. (15) as a result of a common noise process ξ, analogously to the AD model. By the average , we will denote the average over different realizations of the homogeneous Poisson spike trains in Eq. (15).
which in the limit leads to the same mean firing rate as for the AD model Eq. (14) in the limit of .
A simulation of five spike trains of the STS population, driven by a common noise process ξ, a common signal s, and independent noise processes , is shown Fig. 3e. Note that the modulation in Eq. (16) is very distinct from adding jitter to the single spike times, as is considered in [42–44], in that the modulation of the spike times presented here preserves the order of the spikes in each spike train. Other models that incorporate the deletion of spikes in a Poisson spike train  or a combination of deletion and shifting as in the thinning and shifting model [42, 44], differ from the models presented here in that the single spike trains of those models are homogeneous spike trains with constant rates. However, the models in the present paper are designed such that the single spike trains have a prescribed time-dependent firing rate , which still depends on the realization of the signal s and the individual noise η. The cross-correlations between spike trains are a consequence of the different implementations of the time-dependent firing rate and are not prescribed a priori as in [42, 44, 45]. Even if the deletion or shifting of spikes in the thinning and shifting model is performed on a rate-modulated mother process, the resulting process would not be equivalent to the AD model or STS model, in which the addition and deletion of spikes and the shifting of spike times are not independent of the signal realization. In particular, the thinning and shifting model of a population of daughter processes for which the stimulus is solely encoded in the firing rate of the mother process cannot exhibit suprathreshold stochastic resonance.
3.3 Modeling the Common Signal and the Independent Noise Processes
where and are lower and upper cutoff frequencies, respectively. Throughout the paper, we will consider a finite upper cutoff frequency and a non-vanishing lower cutoff frequency. As we will show in our analytical calculation below, the cross-spectrum for two spike trains of the STS model is finite only for . A realization of the common signal s is shown in Fig. 3a and 3d.
Parameters used in numerical simulations
Δt in seconds
T in seconds
Fig. 7a STS
Fig. 7a AD
Fig. 7b STS
Fig. 7b AD
4 Derivation of Spectral Measures
4.1 Input–Output Cross-spectrum
which is equal for both models.
4.2 Cross-spectrum for Two Spike Trains for the AD Model
4.3 Cross-spectrum for Two Spike Trains for the STS Model
We note that the linear term in vanishes due to the zero mean of the Gaussian signal . Equivalently, all higher-order odd terms in in Eq. (37) vanish due to the Gaussian nature of the signal (except for the correction term due to realizations of signal and individual noise that lead to ). From Eqs. (36) and (37) it can be seen that for a vanishing lower cutoff frequency of the independent noise spectrum ( ), the variance diverges and as a consequence of this the cross-correlation between the two spike trains vanishes—only the part that is due to the signal (second term in Eq. (37)) still contributes.
In Fig. 4, the analytical result for the cross-spectrum for two spike trains of the STS model Eq. (38) is compared with simulations. As for the AD model the cross-spectrum of two spike trains is real valued. In contrast to the AD model Eq. (25), the cross-spectrum of two spike trains for the STS model Eq. (38) exhibits a strong decrease at high frequencies, while it approaches the spike train power spectrum Eq. (39) at low frequencies. Note that, although we derived only up to second-order in , the theory fits the simulation results very well even for .
4.4 Single Spike Train Power Spectrum
For and , the power spectrum is flat, as we would expect for homogeneous Poisson spike trains.
5 Information Transmission in Neural Populations
in the analytical calculations to obtain simpler expressions. In the subsequent sections, we will study information transmission in populations of AD neurons and STS neurons.
5.1 AD Population
The linear term in Eq. (44) is always positive. Hence, the population of AD neurons always profits from weak independent noise regardless of the specific choice of model parameters.
5.2 STS Population
where and are defined in Eq. (38). As for the AD model discussed above, we used that signal and noise have equal power-spectra . Due to the frequency dependence of the cross-spectrum , the coherence function also depends strongly on the frequency and exhibits a monotone increase as shown in Fig. 5. Thus, the population of STS neurons can be regarded as a high-pass filter of information, similar to that observed for heterogeneous short-term plasticity  or coding by synchrony [13, 15].
In order to understand the high-pass filter effect in the coherence function as well as the stochastic resonance effect discussed below, we note that the cross-correlations between different spike trains contribute largely to the sum’s output variability, in particular in the absence of intrinsic noise. This output variability is quantified by the output’s power spectrum and appears in the denominator of the coherence function. With individual intrinsic noise, spike times of different neurons are slightly shifted, drastically reducing cross-correlations at high frequencies and thus the amount of the signal-unrelated variability in these frequency bands. Therefore, the coherence function increases with frequency.
The lower bound on the mutual information rate for the STS population is compared with simulations for two sets of parameters in Fig. 6. We observe that for the given parameters the STS model shows a large SSR effect, while the AD model profits only weakly from additional noise.
for which the lower bound on the mutual information rate is equal for both models. From the above equation we can see that whether the STS population or the AD population transmits more information for a given value of independent noise is mainly determined by the noise and signal cutoff frequencies and .
6 Summary and Conclusions
In this paper, we investigated how the effect of noise on the output spikes influences information transmission properties of Poisson neurons. In particular, we considered two populations with strong common input, where in one case weak independent noise added and deleted spikes, while in the other it shifted spikes. In the limit of a weak sensory signal, we analytically derived the spectral statistics of both models and studied information filtering and the emergence of suprathreshold stochastic resonance (SSR). We showed that, even when single neurons of the AD model and STS model cannot be distinguished by their response statistics, the different effects of independent noise on spikes lead to qualitative and quantitative differences in information transmission on a population level.
In the AD model, the presence of the SSR effect is robust—whenever we consider a population with , a small amount of intrinsic noise has a beneficial effect on the signal transmission. In the STS model, the information transmission properties of the population are determined by the cutoff frequencies of the noise. Depending on the specific parameters, one finds a pronounced SSR in some regimes (exceeding the effect in the AD model by far) or no SSR effect in other regimes. Furthermore, we observe a high-pass filtering of information in the STS model that is absent in the AD model.
There are a number of studies that explored theoretically the case of weakly correlated neurons and employed perturbation methods to relate output spike train correlations to input correlations [46–52]. In this paper, we have considered the opposite limit of strongly correlated spike trains that are only weakly decorrelated due to intrinsic noise sources. In this limit, we were not only able to derive comparatively simple expressions for the cross-correlation between two spike trains but were also able to explore analytically the consequences of these correlations for the transmission of time-dependent signals.
The question arises how the specific choice of the output, which is taken to be the sum of individual spike trains, affects the findings discussed above. The most general approach would be to study the multivariate mutual information between the input signal and the population of output spike trains. This quantity is hard to compute numerically and analytically, and its exact calculation is beyond the scope of this study. However, the mutual information between the input signal and the sum of outputs is a lower bound for the full multivariate mutual information, because the summation can only degrade the information content contained in the entire set of the output spike trains. Additionally, for vanishing individual noise, , all output spike trains are identical and the information content of the population does not differ from the information content of the sum of identical spike trains. Therefore, if the mutual information between the input signal and the summed output increases with individual noise, i.e. exhibits suprathreshold stochastic resonance, the full multivariate mutual information increases as well.
The mutual information between the input signal and the summed output has been estimated here by its lower bound . In our setting with a weak signal that is encoded in the firing rate of the Poisson process, we expect that this bound is rather tight. In fact, for a single inhomogeneous Poisson process, the mutual information and its lower bound coincide in leading-order of the signal amplitude .
In this study, we inspected two simple and abstract models for the effect of a weak noise on neural spikes and its consequences on signal transmission by neural populations. We would like to emphasize that the pure limits of an AD model or an STS model approximate the behavior of biophysical neuron models. On one hand, it is plausible that in an excitable neuron model, in which the crossing of a threshold may be aided or prevented by a weak driving, addition and deletion of spikes as in our AD model can be observed. Stochastic oscillators, on the other hand, display a shifting of spike times due to a weak driving, as described by the phase response curve . In between these limits, we expect a combination of both, addition and deletion as well as shifting of spikes. Indeed, such a combination has been observed experimentally . Hence, a generalization of our framework to a Poisson process that includes both effects and allows one to tune gradually between the pure AD and STS models inspected in this paper would be certainly worth additional efforts in a future study.
Appendix A: Mean Firing Rate of the AD and STS Model
A similar estimation leads to the same formula for the STS model.
Appendix B: Probability for Synchronous Spikes in the AD Model
Appendix C: Simplification of the Cross-correlation Function for the STS Model
used in the calculation of the cross-correlation function in Eq. (34).
Appendix D: Variance of the Integrated Independent Noise
Appendix E: Single Spike Train Power Spectrum
where we assume that the spike trains are stationary. Since a single spike train is considered and the average is taken over one independent noise process η, we will drop the subscript employed previously.
The order of the correction term in the above equation is proportional to the square root of the probability that , which has been calculated in Appendix A Eq. (52).
This work was funded by the BMBF (FKZ:01GQ1001A and FKZ:01GQ1001B).
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