External noise suppression by intrinsic noise in a neuron

Results in Physics 15 (2019) 102615

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Results in Physics journal homepage: www.elsevier.com/locate/rinp

External noise suppression by intrinsic noise in a neuron a

Shen Tao , Long Zhangcai a b

a,1

, Chen Bo

b,⁎

T

School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China School of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Noise Neuron Signal transmission

Noises are usually being considered as a detrimental factor in signal processing. The usual method to deal with noise and enhance output signal is reducing noises from different sources. In this research, we demonstrated in a neuronal model that noise in output of a neuron driven by signal with external noise can be suppressed by internal noise of the neuron. This work revealed a new positive role of noise in noisy signal processing.

Introduction Stochastic fluctuations exist vastly in biological systems, such as gene expression, enzymatic processes [1] and information processing and transmission in neuron systems. In neuron systems, the system itself and the signal to be processed are both with noises. The noises come from the omnipresent thermal motion [41] and fluctuation in neural action, such as ion channel gating [2,3], ion pumping, ion concentrating [4] and synaptic transmitting [5]. The role of noise in neural signal processing has been being a focus of research [41,4,6,7]. In recent decades, it has been found that appropriate noise may benefit the signal processing of neurons by stochastic resonance (SR) [8–10], a phenomena in which optimal noise can make signal transmitted best. Stochastic resonance has also been found in other nonlinear systems, such as optical systems [11], electromagnetic systems [12], chemical systems [13], biological systems [14], quantum system [15] and nonlinear signal detectors [16]. In the stochastic resonance phenomenon, the benefit of noise for the signal transmission in nonlinear system is due to that the noise added to the signal or nonlinear system helping weak signal overcome the threshold [17,18], or strong signal leave out of saturation of nonlinear system [19]. Due to independent noises in different elements of signal transmission array induced diversity expression of information, suprathreshold signal can also be enhanced by noise [20]. In fact, noise does not exists only in signal to be transferred, or only in nonlinear system in which the signal to be transferred in. Both the transmission system and the signal to be transmitted could be noisy simultaneously. In these cases, noises from different sources have been considered unable to play a beneficial factor in signal transmission. To reduce noise in the output and enhance output signal, the usual method is to reduce noises from different sources. It has never been

expected that noise can be suppressed by another noise, until Vilar [21] found that in a system which relaxes fast to a stationary state, system noise in some cases can be depressed by external noise. In gene expression models with both intrinsic and extrinsic fluctuations, it have been found that intrinsic noise can speed up response time and the correlations of extrinsic fluctuations can combine constructively [42], Keizer et al. developed a linear-noise approximation method to research intrinsic and extrinsic noises’ interaction which can result in either amplification or destruction of one or both noises [43]. However, there are not plenty researches focused on the effects of intrinsic and extrinsic noises and their interaction of neuron systems. In this work, we demonstrated in a neuron model that noise in output of a neuron driven by signal with external noise can be depressed by intrinsic noise of the neuron and the output signal can be augmented significantly at the same time. This suggests a new possible noisy signal processing mechanism in neural system that utilizes internal noise as a beneficial factor to suppress noise from external input and enhance output signal. Model In this research, we adopted the widely used Integrate-and-Fire (I-F) neuron model [6,9,17,22]. In this model, if the membrane potential v is lower than the threshold h , the membrane potential follows the dynamical equation below:

dv v = − + s (t ), dt τ

(1)

where τ is the time constant of the neuron, s (t ) is the input of the neuron. When the potential v crosses the threshold h from below, the

⁎

Corresponding author. E-mail addresses: [email protected] (Z. Long), [email protected] (B. Chen). 1 Joint first author. https://doi.org/10.1016/j.rinp.2019.102615 Received 8 July 2019; Received in revised form 22 August 2019; Accepted 23 August 2019 Available online 30 August 2019 2211-3797/ © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).

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T. Shen, et al.

neuron fires a spike and v resets to the resting potential vr . In this research, vr = 0 . The input signal s (t ) is a noisy sinusoidal signal with DC current and the noise in this signal is considered as the external noise of this neuron:

s (t ) = A + B cosω0 t + σs ξ (t ),

added to input signal, or internal noise, which can be considered a noise added to the neuron threshold [30,31], alone will be a detrimental factor to signal transmission. Our concern is due to mode lock phenomenon in a neuron. In some situation, the output firing train of a neuron driven by a periodic signal can form a pattern that in every M signal periods there are N output firing spikes [32,33]. This is called M:N mode lock [34]. In mode lock state, neuron output is more regular, and the output signal intensity is augmented. Whether a neuron enters mode lock state depends on parameters including neuronal threshold [32–34]. Fig. 2A shows that as threshold changes, neuronal output enters locked modes with different M:N ratios. Noise added to the threshold as internal neuronal noise can also introduce mode locks similar to those introduced by different neuronal threshold [33]. Fig. 2B (solid line) shows this situation (numerically simulated). This means that an internal noise may be a beneficial factor to enhance signal transmission in a neuron due to its induced mode lock. But previous researches have demonstrated that internal noise has never been a beneficial factor to the transmission of suprathreshold neuron signal without external noise [17], and the same result we show in this work. This is due to that when internal noise introduces mode lock, it introduces background noise at the same time. An exceptional phenomena, we found, is that when input signal is added with external noise and output is randomized, internal noise can still introduce mode lock. Fig. 2B (dash line) demonstrates this situation. It can be seen that due to external noise some mode lock is broken [30], but in a smaller range of intensity, internal noise, presented as threshold fluctuation, can still introduce mode lock. The newly observed phenomena means that the original random output spike train can be transferred into regular pattern by internal noise and the background noise can be depressed by internal noise at the same time. This suggests that when input signal is with external noise, internal noise may be a beneficial factor to depress noise and enhance neuron output signal, though for suprathreshold signal without noise, internal noise cannot produce positive effect to enhance output signal.

(2)

where ω0 is angular frequency of input signal, ξ (t ) is white Gaussian noise with unit intensity, σs is intensity of the external noise. The internal noise is represented by fluctuation of neuron threshold [23–26] and the neuron threshold is described in −

(3)

h = h + σT X (t ). −

In Eq. (3), h is the mean of the fluctuant threshold, X (t ) is an Ornstein-Uhlenbeck process (OUP) with dynamical equation dX = −λXdt + εdW , where λ is the inverse of the correlation time of the OUP, dW is the increment of a standard Wiener process W [31]. Coefficient ε is employed to ensure that X (t ) have unit variance. When potential v reaches threshold h , the neuron fires a spike, and the membrane potential v resets to 0, simultaneously the threshold resets to −

h . Fig. 1 demonstrates this process. In this model, without internal and external noises, potential v determined by Eq. (1) is: vd (t ) = C0exp ⎛− ⎝

t − t0 Bτ τ 2ωB ⎞ + Aτ + cosω0 t + 2 2 sinω0 t , τ ⎠ τ 2ω2 + 1 τ ω +1 (4) Bτ cosω0 t0 τ 2ω02 + 1

τ 2ω0 B sinω0 t0⎞ τ 2ω02 + 1

− where C0 = ⎛v0 − Aτ − and v0 (v0 < h (t0 ) ⎠ ⎝ is the membrane potential of the neuron at initial time t0 . The maximum of vd (t ) is

vmax = Aτ

Bτ τ 2ω02 + 1

.

When vmax < h , the neuron cannot fire spikes solely without noise, and external noise in input signal, or internal noise, originated from fluctuant current through ion channels and appearing as a threshold fluctuation of neuron [23–26], can trigger fire and help the signal transmission through the neuron [9,10,27,28]. When vmax > h , that is, a suprathreshold situation, the signal will be transmitted through the neuron without the help of noise, and noise will be a detrimental factor as usual. This is the typical phenomenon observed and studied previously in classical stochastic resonance of nonlinear system including neurons [10,17,28,29]. We are now concerned with weather internal noise can be a beneficial factor to suppress noise from input signal and enhance output signal, even in suprathreshold situation where either external noise

First-passage time with internal and external noise To check the exact effect of noise, we use stochastic analysis method to solve the response of noisy I-F neuron driven by signal with external noise. In biological systems with stochastic fluctuations, such as stochastic chemical kinetical systems or noisy neuron systems, the FokkerPlanck Equation (FPE) is a widely used mathematical tool to describe the variance of stochastic dynamical systems using a probabilistic language, and many adequate methods can be applied for solving the equation [1]. Since the dynamical equation of the I-F model is a linear ODE, the membrane potential v (t ) can be considered as the summation of neuron response to determined input A + B cosω0 t and that to noise input σs ξ (t ) . The former, represented in vd (t ) , is the solution of Eq. (1) driven only by A + B cosω0 t , which has been given in Eq. (4). The later, represented in vs (t ) , is the solution of Eq. (1) with only noise input driving. The Eq. (1) with only noise input is as following

dvs 1 = − vs + σs ξ (t ) dt τ This noise driven equation has a corresponding FPE

∂ρ (vs , t ) ∂ v ∂2 ⎛ ρ (vs , t ) ⎞ + D 2 ρ (vs , t ) = ∂t ∂t ⎝ τ ∂vs ⎠ Solving the FPE, we can get ρ (vs , t ) , the probability density function of vs (t ) . With v (t ) = vd (t ) + vs (t ) , replacing vs in ρ (vs , t ) byv (t ) − vd (t ) , we get the probability density function (pdf) of the potential v of a neuron driven by input s (t ) = A + B cosω0 t + σs ξ (t ) before firing:

Fig. 1. Demonstration of membrane potential and threshold trajectories in the I-F neuron with both internal noise and external noise. The dashed line is the −

mean threshold h = 0.15, the other parameter values are τ = 0.02 , λ = 500 , σT = 0.02 , ε = 22.4 , A = 10 , B = 10 , ω0 = 100π , θ = 0 , σs = 0.1. 2

Results in Physics 15 (2019) 102615

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−

Fig. 2. Mode lock introduced by suitable neural threshold, or noise added to neural threshold as internal noise.ISI is average inter-spike interval time of neuron output, T is period of input signal. (A) Suitable neural threshold induced mode lock for input without external noise, S (t ) = 10 + 10cos100πt . (B) Noise added to neural threshold as internal noise induced mode lock for input without external noise, S (t ) = 10 + 10cos100πt (solid line), and for input with external noise, −

s (t ) = 10 + 10cos100πt + σs ξ (t ) (dash line). Other parameter values are τ = 0.02 , h = 0.15, λ = 50000 , ε = 223.6 . The result is produced by numerical simulation.

ρ (v, t|v0, t0) =

1 (v − vd ⎞⎟. exp ⎛⎜− 2πτD (1 − exp(−2(t − t0)/ τ )) 2 τD (1 exp( − −2(t − t0)/ τ )) ⎠ ⎝ (5)

∞

h

∫−∞ g1 (h) ∫−∞ f1 (v) dvdh.

The probability that the I-F system fires between time t0 and t1 is

P1 = F1 − F0 = 1 − q1. For a determined v , the probability that the threshold h is higher than the v at t1 is

∫v

1 exp 2πDT (1 − exp(−2λ (t − t0 ))) −

g1 (h) dhdv =

F1 = F0 q1.

η (h, t|h 0 , t0)

∞

g1 (h) dh.

So, the distribution of v (t ) , which does not overpass threshold at t1

−

⎛ (h − h − (h 0 − h )exp(−λ (t − t0 )))2 − ⎜ 2DT (1 − exp(−2λ (t − t0 ))) ⎝

∞

The probability that the system has not triggered a fire until time t1is

In Eq. (5), v0 is the membrane potential of the neuron at the initial time t0 , D = σs2/2 . Use distribution of OUP and Eq. (3), the pdf of the threshold h can be obtained

=

∞

∫−∞ f1 (v) ∫v

q1 = )2

⎞ . ⎟ ⎠

is: (6)

f1 (v )

In Eq. (6), h 0 is the threshold of the neuron at the initial time t0 , DT = σT2/2 . With the distributions of the potential v and threshold h , our next task is to get the inter-spike time interval (ISI) distribution of the neuron, that is, the distribution of first-passage time. There are different approaches to solve first-passage problem in literature [31], but these approaches, which only deal with either external noise, or internal noise, are unable to treat the problem in this research, which involves both internal and external noises at the same time. Here, for the first time, we introduce a discrete method to solve this problem. We discretize time into a discrete series:

∫v

∞

g1 (h) dh.

In the same reason, the distribution of threshold h (t ) , which is above the potential v (t ) at t1 is:

g1 (h)

h

∫−∞ f1 (v) dv.

Under the condition that the system has not triggered a fire until time t1, the conditional distribution of v (t ) at time t2 is ∞

∫−∞ ⎡⎣f1 (v1 ) ∫v

f2 (v ) =

∞

1

g1 (h) dh⎤ ρ (v, t2 |v1, t1 ) dv1/ F1, ⎦

and the conditional distribution of h (t ) at time t2 is

tn = nΔt , n = 0, 1, 2, ⋯.

g2 (h) =

∞

h1

∫−∞ ⎡⎣g1 (h1 ) ∫−∞ f1 (v) dv⎤⎦ η (h, t2 |h1, t1 ) dh1/F1.

At the initial time t0 , the initial membrane v (t0) = v0 = 0 is lower than the threshold h 0 . The probability that the system has not triggered a fire until time t0 is

The conditional probability that the system does not fire at time t2 under the condition that the system had not triggered a fire until t1 is

F0 = 1.

q2 =

At t1 moment, the distribution of v (t ) is

∞

∫−∞ f2 (v) ∫v

∞

g2 (h) dhdv.

Then, we can get F2 , the probability that the system has not triggered a fire until t2

f1 (v ) = ρ (v, t1 |0,t0),

F2 = F1 q2 .

and the distribution of h (t ) is

The probability that the I-F system fires between time t1and t2 is

g1 (h) = η (h, t|h 0 , t0).

P2 = F2 − F1 = F1 (1 − q2).

Under the condition that the system has not triggered a fire until timet0 , the probability that the system does not fire at time t1 is

In this approach, the recursive formula of conditional distribution of 3

Results in Physics 15 (2019) 102615

T. Shen, et al.

spectrum density and signal-to-noise ratio of the neuron output through spike train analysis. Neuronal output firing train in a time length of L can be considered as the sum of a series of δ functions

v (t ) under the condition that the system has not trigger afire until time tn − 1 is fn (v ) =

∞

∫−∞ ⎡⎣fn−1 (vn−1 ) ∫v

∞

n−1

gn − 1 (h) dh⎤ ρ (v, tn |vn − 1, tn − 1 ) dvn − 1/ Fn − 1. ⎦ (7)

S (t ) =

where tn stands for the firing moment of the n th spike and N the number of spikes in the period of L . According to Ref. [36], the one sided power spectral density (PSD) S (ω) of the firing train S (t ) can be ~ obtained by the Fourier transform S (ω) of S (t ) and its conjugation ~* S (ω) , ~ ~ L 2S (ω)·S (ω) 2 L S (ω) = = ∫ S (t ) e−iωt dt ∫ S (t ) eiωt dt L L0 (13) 0

The conditional distribution of the threshold is

gn (h) =

∞

hn − 1

∫−∞ ⎡⎣gn−1 (hn−1 ) ∫−∞

fn − 1 (v ) dv⎤ η (h, tn |hn − 1, tn − 1 ) dhn − 1/ Fn − 1, ⎦ (8)

and the conditional probability that the system does not fire at time tn is

qn =

∞

∫−∞ fn (v) ∫v

∞

gn (h) dhdv.

N

∑n=1 δ (t − tn),

and can be expressed as (see Eqs. (14) and (15) in Ref. [36])

(9)

The probability that the system has not triggered a fire until time tn

N 1+ L

(

S (ω) =

is

Fn = Fn − 1 qn .

∞

The probability of the I-F system fires between time tn − 1 and tn is

hN (t ) =

(11)

1 N

∫0

∞

)

hN (t ) eiωt dt .

(14)

N −1

∑ j,k =1 δ (t j+k − t j − t ).

(15)

Inserting Eq. (15) into Eq. (14) yields

The pdf of the ISI is

p (tn ) = Pn/Δt .

hN (t ) e−iωt dt +

where

(10)

Pn = Fn − Fn − 1 = Fn − 1 (1 − qn ).

∫0

In this way, we can get the pdf of the ISI with a specific initial phase, potential and threshold. In a firing train of I-F neuron, the next fire is only concerned with the last one, the whole firing train is a typical Markov chain. Using the Markov analysis [35], we can get the ISI distribution p (t , ϕ) under different last firing phaseϕ and the global firing phase distribution pφ (ϕ) . Fig. 3A and B are examples of the ISI distribution and the fire phase distribution obtained by this method (line) and the numerical simulation (dots).

N −1

N⎡ 1+ L⎢ ⎣

S (ω) =

(12)

∑ m, n = 1(n ≠ m)

1⎡ N+2 L⎢ ⎣

=

⎤ cosω (tm − tn ) ⎥ ⎦

N −1

N −1

i=1

i=1

∑ cosω (ti − ti +1) + 2 ∑ cosω (ti − ti +2) + ⋯⋯⎤⎥

⎦ (16)

As a stochastic series, when the length of the spike train is long − N−1 N 1 enough, L ≈ L = − (ISI is the average of ISI ), and the expectation ISI

of PSD of the spike train with length L is

S (ω)

Output power spectrum

=

With the ISI distribution p (t , ϕ) under different last firing phaseϕ and the global firing phase distribution pφ (ϕ) , we can obtain the power

1 2 + 〈ISI 〉L 〈ISI 〉L

∫0

2π

∫0

2π

∫0

⎡ ⎣

L

2 R1 (t , ϕ)cosωtdt⎤ dϕ + 〈ISI 〉L ⎦

L

R2 (t , ϕ)cosωtdt⎤ dϕ + ⋯ ⎦ ∞ 2π L 1 2 ∑ ∫ ⎡∫ Rk (t , ϕ)cosωtdt⎤ dϕ = + 〈ISI 〉L 〈ISI 〉L k = 1 0 ⎣ 0 ⎦ =

⎡ ⎣

∫0

1 2 + 〈ISI 〉L 〈ISI 〉L

∫0

2π

∞

L ∫0 ⎛⎜ ∑ Rk (t, ϕ) ⎞⎟ cosωtdt⎤⎥ dϕ

⎡ ⎢ ⎣

⎝ k=1

⎠

⎦

(17)

In the Eq. (17), 〈ISI 〉L is the expectation of ISI of firing train with lengthL , which can be obtained by ISI distribution p (t , ϕ) and firing phase distribution pφ (ϕ) .

〈ISI 〉L =

∫0

2π

∫0

⎡ ⎣

L

p (t , ϕ) dt⎤ pφ (ϕ) dϕ ⎦

In Eq. (17), Rk (t , ϕ) is distribution of time interval t between two spikes apart k spikes, when the phase of the first spike among them is ϕ .

R1 (t , ϕ) = p (t , ϕ)

R2 (t , ϕ) =

∫0

t

Rk (t , ϕ) =

∫0

t

R1 (τ , ϕ) p (t − τ , φ) dτ Rk − 1 (τ , ϕ) p (t − τ , φ) dτ

In the equations above, φ = ϕ + (t − τ )/ ω0 , is the phase of the spike, which is between, and neighbors the last one of, two spikes with time interval t . Using R (t , ϕ) ,

Fig. 3. Example of the ISI distribution (A) and firing phase distribution pφ (ϕ) (B) obtained by stochastic analysis method in this work (solid line) and numerical −

simulation (dots). Neuron parameter values are τ = 0.02 , h = 0.15, σT = 0.02 , λ = 50000 , ε = 223.6 . Input is s (t ) = 10 + 10cos100πt + σs ξ (t ) and the noise intensity σs = 0.05.

R (t , ϕ) =

∞

∑k =1 Rk (t , ϕ),

(18)

the expectation PSD of a firing train with length L can be expressed 4

Results in Physics 15 (2019) 102615

T. Shen, et al.

subthreshold. When there is only internal noise (Fig. 5B), the output SNR decreases non-monotonously with internal noise intensity, but it is always smaller than the output SNR without internal noise. When both external noise and internal noise exist simultaneously, external noise, as usual, cannot improve quality of output signal (Fig. 5C), but in some range of intensity, internal noise can improve quality of output signal (Fig. 5D). Fig. 5D shows that there is an optimal internal noise intensity which makes suprathreshold noisy signal transmitted best in a neuron. To demonstrate the effect of internal noise on the signal transmission of neuron, we use SNR gain to reflect the change of output signal quality with internal noise, which is defined as the ratio of output SNR of neuron with internal noise to that without internal noise. Fig. 6A shows that without external noise, the SNR gain produced by internal noise is always below a unit (dash line), and the quality of the neuron output is always deteriorated by internal noise. When there is external noise added to input signal, the SNR can be gained by several dozen times, and the quality of the neuron output is significantly improved by internal noise. There is an optimal internal noise intensity which makes the output SNR gain reaches a peak. Fig. 6B shows that peak SNR gain is dependent on the intensity of external noise added to input signal. But this dependency is not monotonous. When the intensity of external noise is small, peak SNR gain rises as external noise intensity increases. There is a certain intensity of external noise at which peak SNR gain reaches a maximum, and the beneficial effect of internal noise is best. Fig. 7 demonstrates the mechanism underlying the internal noise enhanced noisy signal transmission. It shows that when internal noise induced mode lock (Fig. 7A), the background noise in the output of the neuron was depressed (Fig. 7B) and the signal amplitude of the neuron output was augmented at the same time (Fig. 7C). At the beginning of neuron entering mode lock state with internal noise intensity increasing, the signal amplitude rises to its maximum (Fig. 7B), the background noise lowers to its minimum (Fig. 7C). As internal noise intensity increases, signal amplitude decreases and background noise increases gradually (Fig. 7B, C). This makes the output SNR reaches its peak at the initial of increasing noise induced mode lock. The phenomena that Noise induced mode lock augments output signal and depresses output noise also appears in the situation without external noise. This also makes the output SNR increase with internal noise intensity increasing and exhibit a peak (Fig. 5B). But the peak SNR is still lower than the SNR without internal noise. To clarify whether this signal enhancing effect of internal noise revealed in this research is just a matching effect of spike rate with input signal frequency, we checked situations where mode lock ratio is not 1:1. Our research shows that, except 1:1 locked mode, in other mode lock ratio, noise induced mode lock also depresses output noise and enhances signal. Fig. 8 demonstrates examples of these cases, in which mode lock ratios are 1:2 and 2:1 respectively. These examples mean that when spike rate is not equal to input signal frequency, but half, or double of input signal frequency, internal noise can still suppress output noise, and augment output signal. So, the noise suppression and signal augment of internal noise is not an effect of spike rate matching with signal frequency, but an effect of noise induced mode lock.

in

1 2 S (ω) = + 〈ISI 〉L 〈ISI 〉L 1 2 = + 〈ISI 〉L 〈ISI 〉L

∫0

2π

∫0

⎡ ⎣

L

∫0 ⎡⎣∫0

L

2π

R (t , ϕ)cosωtdt⎤ dϕ ⎦ R (t , ϕ) dϕ⎤ cosωtdt . ⎦

(19)

To avoid summation of infinite terms in the calculation of R (t , ϕ) in Eq. (18), we deduced a recursive formula about R (τ , ϕ) as following. Multiply Eq. (18) by p (t − τ , φ) and integrate,

∫0

t

R (τ , ϕ) p (t − τ , φ) dτ = =

∞ t ∫0 ∑k=1 Rk (τ , ϕ) p (t − τ , φ) dτ ∞

∑k =2 Rk (t , ϕ) = R (t , ϕ) − R1 (t , φ),

we obtain

R (t , ϕ) = R1 (t , φ) +

∫0

t

R (τ , ϕ) p (t − τ , φ) dτ .

(20)

In our discrete method, the time step is Δt , t = nΔt . Consider p (0, φ) = 0 , the discrete expression of Eq. (20) is

R (nΔt , ϕ) = p (nΔt , ϕ) +

n−1

∑m =0 R (mΔt , ϕ) p ((n − m)Δt , φ)Δt.

(21)

In this discrete recursive relation, the initial value of R (nΔt , ϕ) is

R (0, ϕ) = p (0, ϕ) = 0 With this formula, we can obtained R (nΔt , ϕ) as R (t , ϕ) in Eq. (18), up to nΔt = L . By this discrete stochastic analysis method, we can obtain power spectral density of spike train with time length L . Fig. 4 is an example of this result (solid line) and the corresponding numerical simulation result (stars). With power spectral density, we can get signal amplitude SA , background noise amplitude NA around the signal peak, and SNR of output:

SNR =

SA . NA

Noise suppression and signal enhancement by internal noise Fig. 5 gives output SNR in different noisy situations. It can be seen that when there is only external signal noise, it will not improve the quality of output signal (Fig. 5A). This is due to that the precondition for stochastic resonance is not satisfied, that is, the input signal is not

Discussion In this research, we distinguished internal noise from external noise according to whether a noise is added to neuronal threshold or added to input, and revealed their difference in their effect on output noise and signal. But from the view of mathematics, the problem of neuron information transmission with internal and external noises dealt with in this research is the same as that with two external noises and a constant threshold, or that with threshold fluctuated by two internal noises and a determined input. The equivalents of both internal and external noises in our research reflected in membrane potential, or threshold fluctuation, are OU processes; the differences between them are the speeds of

Fig. 4. Example of the power spectral density of the neuron output spike train calculated by stochastic analysis method (solid line) and numerical simulation −

(stars). The neuron parameter values are τ = 0.02 , h = 0.15, σT = 0.01, λ = 50000 , ε=223.6. The input is s (t ) = 10 + 10cos100πt + σs ξ (t ) and the external noise strength σs = 0.1. 5

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T. Shen, et al.

Fig. 5. Signal to Noise Ratio (SNR) of neuron output changes with external and internal noise intensities. (A) Without internal noise, output SNR changes with external noise intensity. (B) Without external noise, output SNR changes with internal noise intensity. (C) With internal noise, σT = 0.01, output SNR changes with −

external noise intensity. (D) With external noise, σS = 0.01, output SNR changes with internal noise intensity. Other parameters are τ = 0.02 , h = 0.15, ε = 223.6 , λ = 50000 , A = 10 , B = 10 , ω0 = 100π , θ = 0 .

Fig. 6. Internal noise produced neuronal output SNR gain. (A) Output SNR gain variance with internal noise intensity in different external noise intensities. For a certain external noise, there is an optical internal noise which makes SNR gain reach a peak. (B) Peak SNR gain variance with external noise. The parameter

Fig. 7. Internal neuron noise produced 1:1 mode lock (A), output noise depression (B) and output signal augment (C). The input is s(t ) = 10 + 10cos100πt + σs ξ (t ) , σs = 0.02 , ξ (t ) is Gaussian white noise with unit

−

λ = 50000 , values areτ = 0.02 , nal.s (t ) = 10 + 10cos100πt + σS ξ (t )

ε = 223.6 ,

h = 0.15,

input

sig-

−

intensity. Other parameters are τ = 0.02 , h = 0.15, ε = 223.6 , λ = 50000 .

their variance, that is, their time constants. So, the essential difference between the two noises in this research is not whether they were added to input or neuron threshold, but that they have different time constants. The internal noise in this research, with time constant 1/ λ = 0.00002s , is faster than the external noise, with time constant of neuron τ = 0.02s . This raises a problem: why a fast noise with shorter time constant (1/ λ ) can introduced mode lock and suppresses output noise, but a slow noise with longer time constant (τ ) cannot do it? From the view that noise can increase the opportunity of membrane potential crossing threshold, either increasing intensity of external noise or internal noise can increase the firing rate of a neuron. It was

this effect of noise that resulted in noise induced change in ratio of − average inter-spike interval (ISI ) to signal period (T ), and mode lock (solid line in Fig. 2B) induced by suitable internal noise. But when external noise changes its intensity and consequently firing rate, it did − not introduce mode lock at any ratio of average inter-spike interval (ISI ) to signal period (T ). Fig. 9A shows this situation. It can be seen that suitable external noise can make ratio of average inter-spike interval − (ISI ) to signal period (T ) equals to 1:1, but there is no mode lock. − Fig. 9B show the firing phase distribution in the situation with ISI/T = 1 in Fig. 9A and that in the mode lock situation introduced by internal 6

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Fig. 8. Internal noise induced different mode lock and its noise depressing and signal augmenting effects. Input signal s (t ) = 10 + 10cos100πt + σS ξ (t ) and the noise −

−

intensity σS = 0.02 . In the left panel, the average threshold h = 0.205 and makes a 2:1 mode-lock in proper threshold noise strength. In the right panel, h = 0.085 and the mode-lock ratio is 1:2. The other parameters in both panels are τ = 0.02 , λ = 50000 . −

(Fig. 9B solid line), the firing is concentrated ata particular phase, and out of a narrow range around the phase, the firing possibility density is zero. In this situation, neuronal fire can only appear at certain moments, and neuron expresses information in firing temporal pattern. So, the prerequisite for noise introducing mode lock and suppressing another noise in a neuron is that it is in temporal code modality. Modeling neuronal fire into a membrane potential walker driven by input signal passing ceiling boundary fluctuated by noise, Thibaud

noise in Fig. 2B (solid line). It can be seen that though ISI/T = 1 is common, the characteristics of firing phase distributions in both situations are different. The firing phase distribution in the situation without mode lock is smooth function of signal phase (Fig. 9B dash line), which is similar with and reflects the variance of input with phase. In this situation, neuronal fire can appear at any moment, and neuron expresses information in firing rate. But in mode lock situation

−

−

Fig. 9. (A) Variance of ISI/T with signal noise intensity. Signal noise can change spike rate, and introduce status of ISI/T = 1. But mode lock did not appear. (B) Firing −

phase distribution in mode lock (solid line) and unlock (dash line) status. Firing phase distribution (dash line) in the status of ISI/T = 1 of Fig. 9A (where signal noise −

intensity σS = 0.7 ) is similar with input signal, reflects variance of input with phase. Firing phase distribution (solid line) in mode lock status with ISI/T = 1 of Fig. 2B (Internal noise intensity σT = 0.01) is concentrated a narrow range around a particular phase (0.25π ). At other phase, firing probability density is zero. 7

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Fig. 10. Holder exponents of threshold fluctuation and neuronal membrane potential responses to input signal and noise respectively. In this figure, the horizontal ordinate is the time window in the Holder exponent calculation. The membrane potential response of neuron to external Gaussian white noise is an OUP noise, the time constant of which is the time constant of the neuron, τ = 0.02s . The OUP with time constant τ = 0.00002s is the fluctuation added to the threshold as internal noise. Holder exponent of OUP increases with its time constant. The input signal s (t ) = 10 + 10cos100πt .

Fig. 11. Holder exponents of neuron membrane potential response to external noise and input signals with different frequencies. The external noise is Gaussian white noise. In this figure, the OUP is the membrane potential response of neuron to the Gaussian white noise. The time constant of the OUP is the time constant of the neuron, τ = 0.02s . Input signal s (t ) = 10 + 10cos 2πft , f = 5, 10, 25, 50Hz .

driven membrane potential walker decreases and gradually approaches the Holder exponent of the OUP of external noise, but would never be quite smaller than the Holder exponent of OUP of external noise. Fig. 11 demonstrates some samples of our calculation. The phenomena that the Holder exponent of OUP of external noise becomes the low limit of Holder exponent of signal driven membrane potential walker is due to the low pass filter effect of the neuron, the time constant of which is the time constant of ceiling boundary of external noise and determines the cutoff frequency of the filter at the same time. This phenomena means that extern noise cannot introduce mode lock to a neuron driven by high frequency input, but it may introduce mode lock to a neuron driven by low frequency input, the period of which is significant larger than the time constant of the neuron, and suppress another noise, internal or external noise. Holder exponent of an OUP is dependent on its time constant. To make a neuron in temporal code modality, internal noise should be with so quite shorter time constant than the neuron that its Holder exponent is quite smaller than the input driven membrane potential walker. So, it is not any internal noise, but internal noise with time constant much smaller than that of a neuron that can introduce the neuron into mode lock. In this work, the values of neuron constant and internal noise time constant are adopted according to physiological experiment observations [25]. So, our result that internal noise of a neuron can depress external noise and enhance output signal is significant in biology.

Taillefumier and Marcelo O. Magnasco for the first time explained the underlying mechanism of temporal code and rate code of a neuron by roughness difference between the walker and the boundary (see Fig. 1 in Ref. [37]). According to Taillefumier and Magnasco [37], when a walker is smoother, or rise slower than a ceiling boundary, the walker will cannot hit the boundary near behind a local minimum of ceiling boundary. This makes fires of a neuron concentrated at some particular moments with local minimum of boundary, and probability density of firing at other time is zero. The different effect of internal noise and external noise due to time constant difference in this research can be explained in the same way. We consider a neuron driven by sinusoidal input with only external or internal noise, the fire of which can be considered as membrane potential walker driven by sinusoidal input crossing a boundary of OUP with time constant equal to neuron time constant, or a boundary of internal OUP noise with shorter time constant than neuron. The smoothness or roughness of a function, especially a non-derivable function, can be indicated in Holder exponent [37,38] (refer to [38] for calculation of Holder exponent). A larger Holder exponent indicates more smoothness, less roughness and a slower local rising speed of a function [37,38]. Fig. 10 demonstrates Holder exponents of input signal driven membrane potential walker and OUPs corresponding to internal noise and external noise driven membrane potential respectively. We can see that internal OUP noise with shorter time constant is rougher (with smaller Holder exponent) than the OUP of external noise with longer time constant (with larger Holder exponent). The local Holder exponent of membrane potential walker vd (t ) , described in Eq. (4), is larger than that of internal OUP noise, but approximates to, and fluctuates around that of OUP of external noise as analyzing time window increases. So, the walker rises slower than the ceiling boundary of internal noise, and the neuron with only internal noise meets the prerequisite for temporal code modality. But the walker rises as fast as ceiling boundary of external noise, and the neuron with only external noise is in rate code modality. That is, the external noise cannot introduce the neuron into mode lock status. To explore whether an external noise can introduce neuron mode lock in some situation, we calculated and compared Holder exponents of neuron membrane potential driven by external noise and input signal with different frequencies respectively. Our calculation demonstrated that as frequency of input signal increases, Holder exponent of signal

Conclusion In the neuron systems, our analysis demonstrated that in a single neuron, fast changing intrinsic noise of the neuron can depress output noise originated from external input and augment output signal. The noise suppression and signal enhancement effects of internal noise are originated from internal noise induced output mode lock to noisy input signal. Experiments have observed mode locks in neural systems [39], which have been suggested in temporal pattern of spike train expressing or encoding sensory information, such as pitch of sound in auditory system [40]. The preservation of precise temporal information of neural spiking is crucial for coding and decoding, especially below ~2 kHz in mammal auditory perception. Previous researches have theoretically explained these biologically observed mode lock (or phase-lock) phenomena in a neuron without external noise in driving signal [32,33]. This work revealed that when there is external noise in driving signal, 8

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internal noise in a neuron can also introduce mode lock, and consequently depresses the external noise and makes the temporal information transmitted better. This work may help us to understand why the upper neural expression and the subject feeling of external stimulus are determined though peripheral sensory neuron response to stimulus and the transmission of the response are noisy and undetermined. The revealed positive effect of internal neuron noise only acts when signal to be processed is noisy. This means that the role of noises in interaction situations is different from that when they appear individually, which has been recognized previously. And the role of ubiquitous noises in neural system needs to be reconsidered in interacting situations. This work is the first of efforts to reveal the possible advantages of interaction of multiple neural noise sources in neural information transfer and code. The interaction of internal and external noise also occurs in biochemical systems such as gene expression models. In particular condition, one stochastic fluctuation could be suppressed by another one [42,43]. It is possible that in other systems, such as electronic system for information processing, the noise effect revealed in this work can be used through specific design to exert inevitable internal noise as a beneficial factor to suppress signal noise.

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