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We theoretically investigate the collective response of an ensemble of leaky integrate-and-fire neuron units to a noisy periodic signal by including local spatially correlated noise. By using the linear response theory, we obtained the analytic expression of signal-to-noise ratio (SNR). Numerical simulation results show that the rms amplitude of internal noise can be increased up to an optimal value where the output SNR reaches a maximum value. Due to the existence of the local spatially correlated noise in the units of the ensemble, the SNR gain of the collective ensemble response can exceed unity and can be optimized when the nearest-neighborhood correlation is negative. This nonlinear collective phenomenon of SNR gain amplification in an ensemble of leaky integrate-and-fire neuron units can be related to the array stochastic resonance (SR) phenomenon. Furthermore, we also show that the SNR gain can also be optimized by tuning the number of neuron units, frequency and amplitude of the weak periodic signal. The present study illustrates the potential to utilize the local spatially correlation noise and the number of ensemble units for optimizing the collective response of the neuron to inputs, as well as a guidance in the design of information processing devices to weak signal detection.

The concept of stochastic resonance (SR) was invented in 1981-82 in the rather exotic context of the evolution of the earth’s climate [

Sensory neurons transform a signal from the environment into trains of spikes that propagate to other structures in the nervous system. Since internal and external noise are ubiquitous and unavoidable, many studies involving peripheral sensory systems that have exhibited SR have been carried out. The SR phenomena have received considerable attention because of the surprisingly beneficial effect of noise in the community of neuroscience. In this regard, we should note that SR has been found in various neuron models and related experimental observations have also been reported [

The importance of the study of the SNR gains on signal transduction across ensembles was stressed in [

As far as an ensemble of neuron units is concerned, the existing investigations were mainly focused on independent internal noise of each unit [

It is worth noting that we consider an ensemble of N leaky integrate-and-fire neuron units in the present paper. As a matter of fact, the leaky integrate-and-fire neuron model has become widely accepted as one of the canonical models for the study of neural systems. The model provides a good description of the subthreshold integration of synaptic inputs. Assembling the leaky integrate-and-fire neuron units into arrays, we will show that the collective response of a parallel array to a given noisy signal can be enhanced by the internal array noise. For a noisy signal, the SNR gain is employed and numerically analyzed. The regions of the SNR gain exceeding unity, testify the efficiency of the parallel array assembled by this kind of leaky integrate-and-fire neuron units.

Let us consider an ensemble of N leaky integrate-and-fire (LIF) neuron units receiving the same periodic stimulation S ( t ) , as shown in

τ d V i ( t ) d t = − V i ( t ) + μ + 2 D ξ i ( t ) + S ( t ) (1)

with 1 ≤ i ≤ N . Here the parameter D stands for the intensity of the white Gaussian noise (i.e., ξ i ( t ) ) while μ is a dc component in the noisy synaptic input and τ is the membrane time constant for the subthreshold dynamics. S ( t ) is a common component in the input and the Gaussian white noise ξ i ( t ) models the internal stochastic component for the ith neuron. The dynamics Equation (1) is complemented by the well-known fire and reset rule: whenever the membrane potential reaches a prescribed constant V T , the neuron fires and the potential is kept fixed for an absolute refractory period τ R and then reset to a value V R . In the following we set V T = 1 and V R = 0 . In Equation (1), the single neuron properties are described by − V i ( t ) , μ , and ξ i ( t ) standing for a leakage term, a constant base dc component, and an internal Gaussian white

noise of intensity D, respectively. This intrinsic noise leads to spontaneous activity even in the absence of the external stimuli. The output of the ith LIF neuron is a δ spike train determined by the jth instants of threshold crossing of the ith neuron t i , j

x i ( t ) = ∑ j δ ( t − t i , j ) (2)

The ensemble response y ( t ) is taken as the average of output x i ( t ) as,

y ( t ) = 1 N ∑ i = 1 N x i ( t ) (3)

By using the leaky integrate-and-fire model, some authors [

〈 ξ i ( t ) ξ j ( t + τ ) 〉 = [ δ i , j = i + λ δ i , j = i + 1 ] δ ( τ ) (4)

with 1 ≤ i , j ≤ N , where the first term describes the self-correlation of each neuron unit, and the second term describes the correlation between two nearest-neighborhood neuron units with λ being a tunable nearest-neighborhood correlation coefficient. The function δ ( τ ) means the correlation must be instantaneous. Because response of each neuron is not affected by the local correlation between the nearest-neighborhood neuron units, it is necessary to calculate the spectral statistics R y y ( ω ) = 〈 Y ( ω ) Y * ( ω ) 〉 for the ensemble output spike train, where

Y ( ω ) = 1 T ∫ 0 T d t exp ( i ω t ) ( y ( t ) − r 0 ( D ) ) (5)

is the Fourier transform of the zero average output spike train with r 0 ( D ) being the stationary firing rate at the noise level D. For weak signals, we can adopt the linear response theory. Based on the linear response theory [

Y i ( ω ) = Y i , 0 ( ω , D ) + B ( ω , D ) S ( ω ) (6)

Equation (6) provides a system of equations relating spike train power spectrum 〈 Y i ( ω ) Y i * ( ω ) 〉 , cross spectrum between distinct spike trains 〈 Y i ( ω ) Y j * ( ω ) 〉 ( j ≠ i ) , and cross spectrum between the stimulus and a spike train 〈 Y i ( ω ) S * ( ω ) 〉 with * denotes complex conjugation. We further assume that 〈 Y 0 , i ( ω ) Y 0 , j * ( ω ) 〉 = 〈 Y 0 , i ( ω ) S * ( ω ) 〉 = 〈 Y i ( ω ) ξ j * ( ω ) 〉 = 0 ( i ≠ j ) . We assume the external stimuli ( η ( t ) ) to be Gaussian white noise of intensity D η . For such a stimulus a linear correction of the spectral quantities is not valid anymore, because the variance of the white noise is not small but in fact infinite. Then we replace the spectrum of the transmitted stimulus as

S s t ( ω ) = S 0 ( ω , D ) + | B ( ω , D ) | 2 R s s ( ω ) (7)

with R s s ( ω ) standing for the power spectrum of the input coherent signal. Here we explicitly show the parametric dependence of the power spectrum and the susceptibility on the internal noise level. This is a linear approximation of S 0 , Q = S 0 ( ω , Q ) with Q = D + D η the intensity of the summed internal and external noise sources. If both internal and external noises are white and Gaussian, the single neuron cannot distinguish between both kinds of noise, thus to replace the power spectrum of unperturbed system S 0 ( ω , D ) by S 0 ( ω , Q ) seems to be plausible. This also leads to the firing rate and the susceptibility functions that should be taken at noise intensity Q and not at D anymore. As a matter of fact, an external stimulus treated by the linear response in Equation (7) will never affect the stationary firing rate of the neuron. In contrast to this we expect an increase in firing rate for a neuron that experiences a white noise of total intensity Q = D + D η compared to the unperturbed case ( D η = 0 ). Then the susceptibility B ( ω , D ) at full noise level Q can be replaced by B ( ω , Q ) . For the self-consistent determination of the firing rate we also use the full noise intensity Q instead of D. For the LIF model we can calculate the firing rate by the following expression [

r 0 ( μ , Q ) = ( τ R + π ∫ ( μ − V T ) / 2 Q ( μ − V R ) / 2 Q d z e z 2 e r f c ( z ) ) − 1 (8)

The power spectrum of the unperturbed system S 0 ( ω , Q ) and the susceptibility B ( ω , Q ) [

S 0 ( ω , Q ) = r 0 ( μ , Q ) | D i ω ( μ − V T Q ) | 2 − e 2 β | D i ω ( μ − V R Q ) | 2 | D i ω ( μ − V T Q ) − e i ω t R e β D i ω ( μ − V R Q ) | 2 (9)

B ( ω , Q ) = r 0 ( μ , Q ) i ω Q ( i ω − 1 ) D i ω − 1 ( μ − V T Q ) − e β D i ω − 1 ( μ − V R Q ) D i ω ( μ − V T Q ) − e i ω τ R e β D i ω ( μ − V R Q ) (10)

where β = [ V R 2 − V T 2 + 2 μ ( V T − V R ) ] / 4 Q , and D a ( z ) denotes the parabolic cylinder function [

d V i ( t ) d t = − V i ( t ) + μ + 2 Q ( 1 − | λ | ) ξ i ( t ) + δ [ 2 Q | λ | η ( t ) + S ( t ) ] (11)

where ξ i ( t ) and η ( t ) in the above Equation (11) are mutually independent Gaussian white noises. The terms δ [ 2 Q | λ | η ( t ) + S ( t ) ] are perturbation parts. The response of the (I + 1)th neuron is governed by

d V i + 1 ( t ) d t = − V i + 1 ( t ) + μ + 2 Q ( 1 − | λ | ) ξ i + 1 ( t ) + δ [ 2 Q sgn ( t ) | λ | η ( t ) + S ( t ) ] (12)

where sgn ( x ) being sign function. As far as we know there is not any stochastic neuron model for which an exact expression for this cross spectrum is given. If we know an expression or we measure it from a simulation of or experiment on two uncoupled neurons (i.e., the ith and (I + 1)th neurons), we can use the following relations for a better approximation of the cross spectrum,

S c r o s s ( ω ) = | B ( ω , Q ( 1 − | λ | ) ) | 2 [ 2 λ Q + R s s ( ω ) ] (13)

As mentioned above, R s s ( ω ) is the power spectrum of the input coherent signal. Based on Equations (7) and (13), we separate S s t ( ω ) and S c r o s s ( ω ) into coherent (i.e., S s t c o ( ω ) = | B ( ω , Q ) | 2 R s s ( ω ) , S c r o s s c o ( ω ) = | B ( ω , Q ( 1 − | λ | ) ) | 2 R s s ( ω ) ) and incoherent terms i.e., S s t i n ( ω ) = S 0 ( ω , Q ) , S c r o s s i n ( ω ) = | B ( ω , Q ( 1 − | λ | ) ) | 2 2 λ Q ). We can also calculate a better approximation for the spectrum for the ensemble of the neuron units

R y y ( ω ) = [ 1 − 2 ( N − 1 ) N 2 ] S s t c o ( ω ) + 1 N S s t i n ( ω ) + 2 ( N − 1 ) N 2 S c r o s s c o ( ω ) + 2 ( N − 1 ) N 2 S c r o s s i n ( ω ) (14)

In order to study the transmission of the weak periodic signal induced by the noise we can use the definition of the output signal-to-noise ratio (SNR) in [

R o u t = π 2 A 2 [ ( N 2 − 2 N + 2 ) | B ( ω , Q ) | 2 + 2 ( N − 1 ) | B ( ω , Q ( 1 − | λ | ) ) | 2 ] N S 0 ( ω , Q ) + 4 λ Q ( N − 1 ) | B ( ω , Q ( 1 − | λ | ) ) | 2 (15)

In the same way, the collective input SNR R i n is defined by replacing the numerator and denominator in Equation (15) with the corresponding quantities for the collective inputs. Based on the definitions of output and input SNR, the SNR gain of the ensemble of neuron units can be defined as the ratio of the output SNR of the ensemble to the input SNR for the coherent component, follows as

G = R o u t R i n (16)

The above Equations (15) and (16) can at best provide a generic theory of evaluating SNR of ensemble of neuron units. If the SNR gain G exceeds unity, the interactions of ensemble of neuron units and controllable internal noise provide a specific potentiality for neural signal processing.

The performance of a single leaky integrate-and-fire neuron model was analyzed in detail in many works [^{−3} and 10^{−5} (latter at high noise intensity) for the integration of Equations (11) and (12).

As mentioned in the introduction, SR phenomenon will be exhibited by studying the variations of the output SNR as a function of the additive noise power density D. Based on the definitions of SNR ( R o u t ) and SNR gain (G) in Equations (15) and (16), we will analyze numerically the evolution of output R o u t and G with different parameters of system. If the SNR gain satisfies G > 1 at the optimal internal noise amplitude, then we say that the input signal can be improved or enhanced by the internal noise of the ensemble of neuron units.

Based on the definitions of SNR ( R o u t ) and SNR gain (G) in Equations (15) and (16), we will analyze numerically the evolution of output R o u t and G with different parameters of system. The numerical method for calculating SNR was introduced in [

correlation λ varies from positive correlation ( λ = 0.3 ) to negative correlation λ = − 0.3 , which implies that the negative correlation is optimal for triggering the neuron units and transmission of the weak signal among the three cases of statistically independence ( λ = 0 ), positive correlation ( λ = 0.3 ) and negative correlation ( λ = − 0.3 ). In

In

As shown in

In addition, one can also find in

Up to now, we have demonstrated that SNR gain can be optimized by choosing appropriate nearest-neighborhood correlation λ , input signal frequency ω , as well as the rms amplitude of internal noise ξ i ( t ) . However, these results showed in Figures 2-4 focused on the weak noisy periodic signal with fixed amplitude A = 0.3 in an ensemble of leaky integrate-and-fire neuron units by including the internal noise. When one studies the sensory nervous system, it is inevitable to consider the influences of the complex signals. For examining the external stimulus on the sensory nervous system, the external signal can be coded by means of the amplitude of calcium ion oscillation, and the periodical calcium ion signal further enters into the nucleus to change the responsible activities [

One of the most interesting characters of the input periodic signal with amplitude modulation is that the output SNR R o u t and SNR gain G can be optimized by varying the amplitude A of the input signal. Whether there is an optimal amplitude that produces the largest output SNR and SNR gain. In the following, we will study the influence of the amplitude modulation on the output SNR and SNR gain. We plot in

A = 0.6 (blue “O” line), A = 0.4 (green “,” line), and A = 0.3 (red “å” line).

For a better insight into the effects of the amplitude A of the weak periodic signal S ( t ) and the rms amplitude σ ξ of internal noise ξ i ( t ) on global behavior of the output SNR, the contour map of the output SNR as the function of both the amplitude and rms amplitude σ ξ of internal noise is shown in

Obviously, the curves of both

In conclusion, we have investigated the collective response of an ensemble of leaky integrate-and-fire neuron units to a noisy periodic signal by including the internal spatially correlated noise. By using the linear response theory, we obtained the analytic expression of signal-to-noise ratio (SNR). The present investigation shows that the rms amplitude of internal noise can be increased up to an optimal value where the output SNR reaches a maximum value. This property of noise-enhanced transmission of weak periodic signal can be related to the stochastic resonance phenomenon. The results also show that the local spatial correlation of the internal noise in an ensemble of leaky integrate-and-fire neuron units can influence the collective response. Furthermore, the curve of output SNR versus the RMS amplitude of internal noise has more pronounced peak when the internal noise correlation is negative, which implies that the negative correlation has advantage over positive correlation and zero correlation in transmission of a weak periodic signal in the ensemble of leaky integrate-and-fire neuron units. The improvement of the SNR gain results from the internal noise can be related to the so called array SR phenomenon.

More interestingly, we find that the SNR gain can exceed unity and can be optimized by tuning the number of the neuron units and input signal frequency. For a weak periodic signal, our investigation shows that the larger number of the neuron units can induce more optimal SNR gain whether the nearest-neighborhood correlation is statistically independence (zero correlation), negative or positive. Thus we can conclude that the larger number of the neuron units might be more useful for assisting the ensemble of neurons to process the weak periodic signal. It is worth noting that our results have also demonstrated that the slower periodic signal induce larger SNR gain in comparison to the fast periodic signal, which is similar with previous investigation in SNR gain of a single noisy LIF neuron that transmits subthreshold periodic spike trains [

Furthermore, we found that the SNR gain can exceed unity and can also be optimized by tuning the amplitude of the weak periodic signal. The present study proved the possibility of SNR gain in an ensemble of LIF neuron units with more realistic signal inputs involving modulated amplitude. These results might provide the theoretical mechanism for the investigations in the spontaneously released neurotransmitter trains of inner hair cells and spontaneous auditory signals. The present results fit into the main features of SR of hair cells. While refinements of the model and further experiments may be needed to optimize the model parameters, the present correspondence between theoretical and experimental works [

The research is supported in part by National Natural Science Foundation of China under Grant Nos. 31100752, 11675060, and 91330113, and by Scientific Research Foundation for Distinguished Scholars of NJNU under Grant No. 1243211602A044, by Huazhong Agricultural University Scientific and Technological Self-Innovation Foundation Program No. 2015RC021.

The authors declare no conflicts of interest regarding the publication of this paper.

Feng, T.Q., Chen, Q.R. and Yi, M. (2018) Local Correlated Noise Improvement of Signal-to-Noise Ratio Gain in an Ensemble of Noisy Neuron. Journal of Intelligent Learning Systems and Applications, 10, 104-119. https://doi.org/10.4236/jilsa.2018.103007