Statistics of a neuron model driven by asymmetric colored noisepeople.physik.hu-berlin.de/~lindner/PDF/MuellerHansen_PRE...PHYSICAL REVIEW E 91, 022718 (2015) Statistics of a neuron

PHYSICAL REVIEW E 91, 022718 (2015)

Statistics of a neuron model driven by asymmetric colored noise

Finn Müller-Hansen,1,2,* Felix Droste,1,3 and Benjamin Lindner1,31Bernstein Center for Computational Neuroscience, Haus 2, Philippstraße 13, 10115 Berlin, Germany

2Department of Physics, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany3Department of Physics, Humboldt Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany

(Received 25 September 2014; published 27 February 2015)

Irregular firing of neurons can be modeled as a stochastic process. Here we study the perfect integrate-and-fire neuron driven by dichotomous noise, a Markovian process that jumps between two states (i.e., possessesa non-Gaussian statistics) and exhibits nonvanishing temporal correlations (i.e., represents a colored noise).Specifically, we consider asymmetric dichotomous noise with two different transition rates. Using a first-passage-time formulation, we derive exact expressions for the probability density and the serial correlation coefficient ofthe interspike interval (time interval between two subsequent neural action potentials) and the power spectrumof the spike train. Furthermore, we extend the model by including additional Gaussian white noise, and we giveapproximations for the interspike interval (ISI) statistics in this case. Numerical simulations are used to validatethe exact analytical results for pure dichotomous noise, and to test the approximations of the ISI statistics whenGaussian white noise is included. The results may help to understand how correlations and asymmetry of noiseand signals in nerve cells shape neuronal firing statistics.

DOI: 10.1103/PhysRevE.91.022718 PACS number(s): 87.18.Tt, 87.19.lc, 05.40.−a

I. INTRODUCTION

Excitable systems driven by noise play an important rolein various fields, such as laser physics, biophysics, physicalchemistry, and neuroscience [1]. In essence, an excitablesystem remains silent after a weak stimulus but responds to asufficiently strong perturbation by a stereotypical pulse (spike).A main objective of theory is to develop methods to calculatethe statistics of these spikes for various cases of stochasticforcing.

In particular, stochasticity in excitable neural systems hasattracted a lot of attention because it may play a functionallyimportant role in the processing of information [1–4]. Onthe level of single neurons, fluctuations stem from intrinsicnoise in the processing of action potentials, such as synapticunreliability [5] and ion channel noise [6]. On the networklevel, cortical neurons receive input from several thousands ofother neurons, which can be mimicked by an effective randominput [7,8].

Most studies of stochastic neuron models have focusedon uncorrelated Gaussian input (for a review, see [9,10]).Different types of neuron models driven or perturbed byGaussian noise with short or vanishing correlation time havebeen thoroughly studied, and statistical properties have beenderived analytically [1,11,12]. However, neuronal input is ingeneral neither temporally uncorrelated (it is a colored andnot a white noise) nor does it possess Gaussian statistics. Pro-nounced input correlations in time arise because of temporallycorrelated input stimuli, presynaptic refractoriness or bursting,network up-down states, or short-term synaptic plasticity.Correlated and non-Gaussian random inputs are usually moredifficult to treat analytically. Most studies on correlated noisehave focused on low-pass-filtered Gaussian inputs [11–17] (foran exception with band-pass-filtered Gaussian noise, see [18]).Analytical studies on temporally uncorrelated non-Gaussian

*[email protected]

stimuli in the form of white shot noise can be found inRefs. [7,19–21].

To our knowledge, Markovian dichotomous noise is one ofthe rare analytically accessible cases of a temporally correlatednon-Gaussian neural input [16,22,23]. Such noise (also knownas the telegraph process) jumps between two discrete stateswith constant rates and has been used in physics for a longtime to study the effect of noise correlations on nonlineardynamical systems [24–29]. It can also be used to test theinfluence of an asymmetry of the input noise on the statisticsof the driven system.

The aim of this paper is to better understand the effectof non-Gaussian correlated noise on neuronal firing. We usethe perfect integrate-and-fire (PIF) neuron model driven bydichotomous noise to investigate the influence of correlationsand asymmetry in the input on the statistics of the output spiketrain. The PIF model adequately describes the statistics ofsome neurons in the tonically firing regime [18,30,31]. Dueto its simple dynamics, the PIF model allows us to deriveclosed expressions for its statistical properties, a feature thatintegrate-and-fire models do not have in general.

In the neurobiological context, dichotomous noise is partic-ularly suitable to mimic up-down state input. Up-down statesare activity patterns of neural populations that are found inthe cortices of many mammalian species [32–34]. They arecharacterized by alternating periods of neural spiking (upstate) and virtually no firing (down state). Of course, a puretwo-valued input is a gross oversimplification of any neuralinput. Hence, we also take weak uncorrelated fluctuationsaround the two states into account.

The output spike train of a neuron can be described byvarious statistics. In this paper, we focus on three measures:(i) The probability distribution of the interspike interval (ISI),(ii) the correlation coefficient among ISIs, and (iii) the powerspectrum of the spike train.

In Ref. [16], the probability density and correlation coef-ficient of the ISI was derived analytically for a PIF neurondriven by a symmetric dichotomous noise (equal transition

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http://dx.doi.org/10.1103/PhysRevE.91.022718

MÜLLER-HANSEN, DROSTE, AND LINDNER PHYSICAL REVIEW E 91, 022718 (2015)

rates). Here, we extend this analytical framework in two ways:First, we consider asymmetric dichotomous noise that can havetwo different transition rates. On average, it thus stays longer inone preferred state than in the other one. This generalization isof special interest because neural input such as that resultingfrom network up-down states is in general not symmetric.Second, we examine the effect of a mixture of dichotomousand Gaussian noise on the ISI statistics. The use of additionalGaussian white noise is motivated by the presence of intrinsicnoise and by deviations of real input from an ideal two-stateprocess.

This paper is organized as follows: In the second section, weintroduce the PIF neuron model and the statistical measures. Inthe following sections, we discuss the ISI distribution, the ISIcorrelations, and the power spectrum. Each of these sectionsstarts with an analytic derivation of the respective measure forthe case of driving with pure dichotomous noise. Subsequently,we discuss the results and illustrate characteristic features.For the ISI density and ISI correlations, we also introduceapproximations for additional white noise. All of our analyticalresults are compared with stochastic simulations. We concludewith a brief discussion of our results.

II. NEURON MODEL AND STATISTICAL MEASURES

This paper considers the perfect integrate-and-fire neuron.It belongs to the model class of integrate-and-fire neurons,which describe a neuron only by its membrane voltage v.Instead of an active spike generation mechanism, the modelhas a spike-and-reset rule. Accordingly, the model registers aspike whenever the membrane potential reaches a thresholdvoltage vT . Then, the voltage is reset to the reset potential vR .A first-order differential equation describes the dynamics ofthe membrane voltage. This equation becomes a stochasticdifferential equation if a noise process drives the neuronmodel.

To simplify notation, we set the membrane time constantτm = 1. The time is therefore measured in units of τm. Acombination of a constant input current μ, a dichotomousnoise process η(t), and white noise with intensity D drive thePIF model. For this setup, the stochastic differential equationdescribing the voltage dynamics reads

v̇(t) = μ + η(t) +√

2Dξ (t), (1)

where ξ (t) is Gaussian white noise with autocorrelation〈ξ (t)ξ (t ′)〉 = δ(t − t ′). To keep the equations simple, we setthe reset potential to zero. Because the right-hand side ofEq. (1) is independent of v, we can shift v such that vR = 0without reformulating the problem. The threshold voltage toocould be rescaled such that vT = 1. For the generality of ourfinal formulas, we keep the parameter vT in the followingcalculations.

The dichotomous noise process η(t) is characterized by twodiscrete states. Without loss of generality, we consider in thispaper a process that jumps between the values +σ and −σ ;two arbitrary values σ+ and σ− can always be shifted by aconstant c such that σ± = c ± σ , and c can be lumped intoμ. In the following, we limit the analysis to the case in which

μ > σ > 0, and thus

μ ± σ > 0. (2)This implies that the voltage can cross the threshold in bothstates of the dichotomous noise. The case in which the voltagecan pass the threshold only in the +σ state is tractable [23] butless interesting because serial correlations vanish (see below).

The dichotomous process jumps between the states withtwo different transition probabilities (rates). The transitionrates are labeled λ+ for jumps from the plus to the minusstate and λ− vice versa. If λ+ �= λ−, the mean residence timesτ± = 1/λ±, i.e., the mean time between a jump into a stateand the following jump out of it, differ. This results in anasymmetry between the two states. Therefore, this processis called asymmetric dichotomous noise. If the two rates areequal, the mean residence times are the same and we call theprocess symmetric dichotomous noise.

To quantify the asymmetry between the rates, we introducethe parameter u that can take values between −1 and 1 and isdefined as

u = λ− − λ+λ− + λ+ . (3)

This parameter helps to shorten notation and thus appears inthe following calculations. By setting u = 0, we can recoverthe special case of symmetric dichotomous noise.

The asymmetry u is closely related to the mean and thevariance of the asymmetric dichotomous process:

〈η〉 = uσ, 〈�η2〉 = σ 2(1 − u2). (4)Furthermore, the skewness of the dichotomous noise γDN isgiven by

γDN = −2u√1 − u2 . (5)

The stationary dichotomous process is characterized by anexponentially decaying autocorrelation function [24]

C(τ ) = 〈η(t)η(t + τ )〉 − 〈η(t)〉2 = σ 2(1 − u2)e−|τ |/τc . (6)However, the higher-order correlation functions of dichoto-mous noise differ from those of exponentially correlatedGaussian noise, i.e., the Ornstein-Uhlenbeck process. Thecorrelation time τc appearing in Eq. (6) is proportional to theinverse of the mean transition rate λ,

τc = 12λ

with λ = 12

(λ+ + λ−), (7)and, together with the variance, determines the intensity of thedichotomous noise defined as DDN = τc〈�η2〉.

Figure 1 shows a realization of dichotomous noise and thecorresponding voltage trajectories of the PIF model with andwithout white noise. In the absence of white noise [Fig. 1(b)],the voltage has only two different slopes corresponding tothe two noise states. Figure 1(c) shows the effect of additionalGaussian white noise on the voltage for the same realization ofdichotomous noise. Figure 2 illustrates different asymmetriesof the dichotomous noise and their effect on the voltage andthe spiking.

The statistics of the point process generated by the modelcan be characterized in different ways. Neuroscientists often

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STATISTICS OF A NEURON MODEL DRIVEN BY . . . PHYSICAL REVIEW E 91, 022718 (2015)

(a)

(b)

(c)

FIG. 1. (Color online) Illustration of the voltage dynamics of thePIF model. (a) Realization of dichotomous noise. (b) Simulation ofthe voltage driven by this realization. (c) As (b) but with additionalweak Gaussian white noise (D = 0.01). The spikes (horizontal linesexceeding vT , red) were added for a better illustration of spike times.Parameters: σ = 0.7,λ = μ = vT = 1,u = −0.5.

consider the statistics of the interspike intervals (ISIs) Ij =tj − tj−1, in which tj is the time instant of the j th spike. Moregenerally, we may also consider sums of n subsequent ISIs ornth-order intervals [7], defined as T jn =

∑n−1k=0 Ij+k . In the next

sections, we derive analytical expressions for the probabilitydistribution of the nth-order interval and its moments. Bysetting n = 1, we obtain the distribution and moments forthe ISI.

Another statistic of interest is the correlation among ISIsthat can be quantified by the serial correlation coefficient

ρk = 〈Ij+kIj 〉 − 〈Ij+k〉〈Ij 〉〈I 2j

〉 − 〈Ij 〉2 , (8)which measures correlations between two intervals with lag k,i.e., that are (k − 1) ISIs apart. The serial correlation coefficient(SCC) is normalized to values between −1 and 1. A positiveSCC indicates that on average long ISIs follow long onesand/or short ISIs short ones. A negative SCC points at a statisticwhere short ISIs succeed long ones and/or vice versa.

Instead of interval statistics, we may also consider thestatistics of the spike train, x(t) = ∑i δ(t − ti), in particularits power spectrum, which describes the frequency distributionof the variance,

S(ω) = limT →∞

〈x̃(ω)x̃∗(ω)〉T

. (9)

Here x̃(ω) = ∫ T0 dt x(t)eiωt is the Fourier transform of thespike train, and the angular brackets indicate an ensembleaverage over all noise processes involved.

(a)

(b)

(c)

FIG. 2. (Color online) Different asymmetries of the dichotomousnoise process. (a) Positive asymmetry u = 0.6. (b) Symmetricdichotomous noise u = 0. (c) Negative asymmetry u = −0.6.Remaining parameters as in Fig. 1(c).

In the following, we provide detailed derivations of analyt-ical expressions for the nth-order interval distribution, the ISIcorrelations, and the power spectrum.

III. PROBABILITY DENSITY OFTHE NTH-ORDER INTERVAL

For the simple PIF model, the determination of thenth-order interval distribution constitutes a first-passage-time(FPT) problem for the voltage variable. Put differently, thisdistribution equals the probability density of the time neededto reach the threshold for the first time after starting at the resetvalue. The nth-order interval density can be calculated becausethe dynamics of the PIF model [Eq. (1)] is independent of themembrane voltage v itself. Therefore, the time the voltageneeds to go n times from the reset vR = 0 to the thresholdv = vT with the reset mechanism in place is equivalent tothe time it takes to go one time from the reset to the thresholdv = nvT [16]. This property allows us to compute the nth-orderinterval distribution by solving the first-passage-time problemwith the threshold nvT . We start with the problem in theabsence of white noise (D = 0), for which we denote theprobability density of the nth-order interval by JD,n(Tn) (D inthe index stands for the dichotomous noise). At the end of thesection, we will also present an approximation of the densityJD+W,n(Tn) in the presence of both dichotomous and Gaussianwhite noise.

By P±(v,t) we denote the probability density to find thesystem at time t in the plus or minus state around the voltagev. Then, Eq. (1) without white noise (i.e., D = 0) has the

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FIG. 3. (Color online) Illustration of the first-passage-time prob-lem in state space at one point in time. The probability distributionsP− and P+ evolve in time according to Eqs. (10) and (11). Apart fromdrifting in the v direction, probability can also flow from P+ to P−and vice versa. The first-passage-time distribution corresponds to theprobability current at the threshold nvT .

following associated master equations [35]:

∂tP+ = λ−P− − λ+P+ − (μ + σ )∂vP+, (10)

∂tP− = λ+P+ − λ−P− − (μ − σ )∂vP−. (11)The resulting time evolution of the probability densities P+ andP− is illustrated in Fig. 3. The first-passage-time problem fromv = 0 to v = nvT is associated with the solution of the abovemaster equation in the presence of an absorbing boundary atv = nvT . More specifically, given the correct initial conditions,the nth-order interval distribution JD,n(Tn) is equivalent to theprobability current in the v direction,

J (v,t) = (μ + σ )P+(v,t) + (μ − σ )P−(v,t), (12)taken at the absorbing threshold v = nvT ,

JD,n(Tn) = J (nvT ,Tn). (13)The initial conditions for this problem have to fulfill two

characteristics: (i) At t = 0, all trajectories start at the resetvoltage vR = 0. (ii) For a stationary spike train, the probabilityfor the initial value of the noise has to be self-consistent, i.e.,the initial distribution of noise states at the reset has to beequal to the distribution of the noise at the threshold (noiseupon firing), pF (η).

The stationary probability pDN,± to find the dichotomousnoise in either the plus or the minus state is

pDN,± = 12 (1 ± u). (14)The probability to cross the threshold is higher when thedichotomous noise is in the plus state because in this statethe voltage moves faster toward the threshold. Thus, theprobability density of the noise upon firing is proportionalto the velocity of the voltage in the respective noise state. Weobtain after normalization

pF (±σ ) = μ ± σμ + uσ pDN,±. (15)

In line with the assumption Eq. (2), trajectories alwaysmove in the positive v direction. Hence, the voltage cannotcross the threshold multiple times, and the evolution of theprobability density in the presence of an absorbing boundaryis the same as that in the absence of such a boundary. Putdifferently, we may use the freely evolving solution of theequation.

Equations (10) and (11) can be reduced to a second-orderequation,[

∂2t + 2λ∂t + 2μ∂v∂t + 2λ(μ + uσ )∂v + (μ2 − σ 2)∂2v]P±

= 0, (16)where we have used for notational ease the mean transition rateλ and the asymmetry parameter u of the dichotomous noise[cf. Eq. (4)]. To find a solution of Eq. (16), we first transformthe equation via the Galilean transformation v → v − μt toa frame of reference where μ = 0. Second, as was donepreviously in Ref. [36], we remove the asymmetry by applyinga Lorentz transformation known from special relativity:

v′ = γ (v − uσ t), t ′ = γ(t − uv

σ

), (17)

λ′ = λγ

, γ = 1√1 − u2 =

λ√λ+λ−

. (18)

Ultimately, we arrive at the second-order differential equationknown as the telegrapher’s equation,(

∂2t ′ + 2λ′∂t ′ − σ 2∂2v′)P± = 0. (19)

The general solution of this equation for arbitrary initialconditions can be found in [37, p. 868]. For our purpose,we need the conditional probability densities Pη0,η(v,t |0,0)denoting the probability to find a trajectory at time t aroundv and in the noise state η = ±σ provided it was at time zeroat v = 0 and the dichotomous noise was in state η0. To thisend, we have to transform the solution of Eq. (19) back to theoriginal variables t and v. The normalized probability densitiesread

P±σ,±σ (v,t |0,0) = e−λ(t− uvσ )[δ(v ∓ σ t)

+ λ2γ σα(v,t)

(σ t ± v)I1(

λα(v,t)

γ σ

) ],

(20)

P±σ,∓σ (v,t |0,0) = e−λ(t− uvσ ) λ2σ

(1 ∓ u)I0(

λα(v,t)

γ σ

), (21)

where α(v,t) = √σ 2t2 − v2 and = θ (σ t − |v|) with theHeaviside step function θ (x). δ(x) is the Dirac delta functionand Ib(x) denotes the modified Bessel function of the first kindand order b.

Applying the Galilean back transformation simply replacesv by v − μt in these expressions. The initial states aredistributed according to the noise upon firing, so that

P±(v,t) =∑η0

pF (η0)Pη0,±σ (v − μt,t |0,0). (22)

Together with Eq. (12), the nth-order interval distribution isthen given by the following sum over initial and final noisestates:

JD,n(Tn) =∑η0,η

(μ+ η)Pη0η(nvT −μTn,Tn|0,0)pF (η0). (23)

022718-4


When plugging the probability densities Eqs. (20) and (21)into this expression, we finally obtain

JD,n(Tn) = vT λ2

σνexp{−λ [Tn − u (nvT − μTn)/σ ]}

×[σ

λ

(1 + uμ− σ δ(Tn − T

+n ) +

1 − uμ+ σ δ(Tn − T

−n )

)+

(nν

2

[1 + μ

μ + uσ(

1 + μuσ

)]− λTn

[1 + μu

σ

])I1[α(Tn)/γ ]

γ α(Tn)+ I0[α(Tn)/γ ]

γ 2

],

(24)

with α(Tn) = λ/σ√

σ 2T 2n − (nvT − μTn)2. This equation forthe nth-order interval is valid for T +n � Tn � T −n with T ±n =nvTμ±σ . The nth-order interval density is zero for shorter andlonger times Tn. To shorten notation, we have introduced theparameter

ν = 2λvT (μ + uσ )μ2 − σ 2 . (25)

If u = 0, Eq. (24) reduces to the expression found in [16].Although we assumed σ < μ for our calculations, it is

possible to take the limit σ → μ in the expression for the ISIdensity [Eq. (24)]. In this case, the increase of the voltage iszero when the dichotomous noise is in the minus state. Thus,the threshold is always reached in the plus state. The limityields an expression in which the second δ peak and the termproportional to I0 vanish.

In the following, we discuss the obtained result andcompare it with numerical simulations. Because higher-orderdistributions do not differ qualitatively, we restrict ourselvesto the illustration of the ISI distribution (corresponding tothe first-order interval, i.e., n = 1). Figure 4 displays the ISIdistribution [Eq. (24)] for different mean transition rates. Thedistribution consists of two δ peaks and a continuous part;in a binned version of the histogram as shown in Fig. 4, theδ peaks turn into peaks of finite height while the continuouscontribution changes only a little. The δ peaks correspond torealizations of the dichotomous noise in which no switchingbetween noise states occurs while the voltage rises from thereset to the threshold. A short calculation shows that thesepeaks are weighted with the exponential factor pF (±σ )e−λ±T ±n .Hence, the distribution is clearly bimodal if the transitionrates are small (λ < 1) and the asymmetry is not pronounced(|u| < 1/2). In the case of high transition rates, the exponentialweights become so small that only a negligible fraction of thetotal probability is contained in the δ peaks. The continuouspart becomes more and more peaked around the mean withincreasing transition rates. In the limit of high transitionrates, the distribution approaches an inverse Gaussian. Thiscorresponds to the ISI distribution if Gaussian white noisedrives the model (see below).

Figure 5 illustrates how the asymmetry u changes the ISIdistribution: We see that a positive asymmetry shifts the peak ofthe distribution to shorter ISIs and makes it more pointed. Thisresults from an increase in the mean input of the neuron model.A negative asymmetry has the contrary effect. Additionally, the

0.0 0.5 1.0 1.5 2.0 2.5T1

0

2

4

6

8

10

12

J(T

1)

λ =1.0λ =10.0

λ =100.0

simulationtheory

FIG. 4. (Color online) ISI distribution for different mean transi-tion rates λ and dichotomous input only (D = 0). The theory (solidlines) is compared to stochastic simulations (black circles). Note thatwe use a temporally discretized histogram version of Eq. (24) (binwidth equal to that of simulation data), in which δ peaks turn intopeaks of finite height. Parameters: μ = vT = 1.0, u = −0.4, σ = 0.5.

asymmetry increases the probability contained in one of theδ peaks while reducing it in the other. Thus, a pronouncedasymmetry leads to a more skewed distribution.

To focus only on the effect of asymmetry, we compareISI distributions with different u while keeping the mean andvariance of the total input constant. Figure 6 shows that theasymmetry of the noise changes the asymmetry of the ISIdistribution. The latter becomes more skewed to one direction.

0.0 0.5 1.0 1.5 2.0 2.5T1

0

2

4

6

8

10

12

J(T

1) u =-0.9

u =0.0

u =0.7

simulationtheory

FIG. 5. (Color online) As Fig. 4 but for different asymmetriesu and a mean transition rate λ = 10. When holding μ constant, adecreasing asymmetry shifts the distribution to the right.

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4T1

0.0

0.5

1.0

1.5

2.0

2.5

3.0

J(T

1)

u =-0.7u =0.0

u =0.5

simulationtheory

FIG. 6. (Color online) ISI distribution for different asymmetriesu and a mean transition rate λ = 10. To illustrate only the effect of theasymmetry, we keep the mean and the variance of the input currentconstant: μ + uσ = 1 and σ 2(1 − u2) = 0.25.

In the next section, we quantify this effect by the skewnessparameter of the ISI density.

Let us consider how the probability density of the nth-orderinterval changes as we include additional white noise input[D > 0 in Eq. (1)]. At the level of the density equation, thisleads to additional second-order derivatives of P+(v,t) andP−(v,t) in Eqs. (10) and (11) as well as to a different boundarycondition. An exact solution of this much more complicatedproblem (if possible at all) is beyond the scope of this paper,so we only discuss in the following an approximation of thenth-order interval density denoted by JW+D,n(T ).

Our approximation is based on the assumption that theeffects of the dichotomous noise and the white noise areseparable, which should be valid for weak white noise andslow dichotomous driving. If the PIF model is driven byGaussian white noise only, the solution for the probabilitydensity of the nth-order interval is equivalent to the first-passage-time density of an overdamped Brownian particle in aheat bath and subject to a constant force, which was solved bySchrödinger [38] and discussed in the neurobiological contextby Gerstein and Mandelbrot [30]. The solution is given by theso-called inverse Gaussian distribution with mean nth-orderinterval T̄n = nvT /μ and variance 2DnvT /μ3,

JW,n(t,T̄n) = nvT√4πDt3

exp

(− (nvT )

2(t − T̄n)24DT̄ 2n t

). (26)

This formula also applies to our setup with both dichoto-mous and Gaussian noise if the dichotomous noise remainsconstant during the entire nth-order interval; the state of thedichotomous noise controls then the value of T̄n but does notchange the variance of the interval. For a very slow switchingbetween the two states, the probability density could beapproximated by a weighted sum of the two contribution cor-responding to ±σ leading to the respective extremal intervals

FIG. 7. (Color online) ISI distribution for a PIF neuron driven byGaussian white noise and dichotomous noise. Stochastic simulationresults (black circles) are compared to the approximation of Eq. (27).The δ peaks broaden with increasing noise intensity D. The plotshows examples for D = 0.005 and 0.05 and the model parametersλ = 0.1, μ = vT = 1.0, σ = 0.5, and u = −0.6.

T̄n = T ±n = nvT /(μ ± σ ). The weight factors are exactlygiven by the probability density of the ISI in the absence ofwhite noise, i.e., the prefactors of the δ functions in JD,n(T ).Generalizing the idea of the weighted sum also to intervals inbetween the limits leads us to the following integral formula forthe probability density of the nth-order interval in the presenceof both dichotomous and Gaussian white noise,

JW+D,n(Tn) =∫ ∞

0JW,n(Tn,T̄n)JD,n(T̄n)dT̄n. (27)

By definition, this approximation is positive for every Tn > 0and normalized to 1. The integral cannot be solved analytically,but numerical integration shows that the ISI density actuallyapproximates the results of stochastic simulations very wellif the mean transition rate λ is small. Figure 7 showsthe approximation for a specific choice of parameters andcompares it to the driving with dichotomous noise only. Theδ peaks of the distribution without white noise broaden intoinverse-Gaussian-shaped peaks. The continuous part of thedistribution remains at the same height when adding weakwhite noise (D = 0.005). With stronger additional white noise(D = 0.05), both peaks show an increasing overlap such thatthe distribution loses its bimodal nature. For transition rates λin the order of 1 or larger, there are significant deviations ofthe approximation from the simulations (not shown). We notethat in both limit cases λ → ∞ and λ → 0, our approximationfor the nth-order interval density Eq. (27) becomes exact.

IV. MOMENTS OF THE ISI DISTRIBUTIONAND ISI CORRELATIONS

In this section, we derive the moments of the nth-orderISI distribution and use them to calculate the mean, variance,

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and skewness γs of the ISI. Furthermore, we derive anexpression for the serial correlation coefficient. Again, we startby considering dichotomous input only.

In principle, for D = 0 the ISI moments can be calculatedby integrals over the probability density Eq. (24). However, theBessel functions in this formula are difficult to integrate ana-lytically. Therefore, we derive the moments of the distributionfrom its Laplace transform,

Ĵ (v,s) =∫ ∞

0e−st J (v,t)dt. (28)

We recall that (i) the nth-order interval distribution isequivalent to the probability current Eq. (12) at v = nvT ;(ii) P+ and P− are solutions of the master equation (16).Because J (v,t) is just a weighted sum of P+ and P−, J (v,t) isa solution of Eq. (16) too. We can derive the following ordinarydifferential equation of second order by Laplace transformingthe master equation:

Ĵ ′′ + 2AĴ ′ + BĴ = Cδ(v) + δ′(v), (29)where we have used the initial condition for the current,

J (v,0) = [(μ + σ )pF (σ ) + (μ − σ )pF (σ )]δ(v). (30)In Eq. (29), the prime denotes the derivative with respectto v. A, B, and C are functions of the Laplace transformedvariable s:

A(s) = λ(μ + uσ ) + μsμ2 − σ 2 , B(s) =

s(s + 2λ)μ2 − σ 2 ,

C(s) = 2λ(μ + uσ )μ2 − σ 2 +

s(μ2 + σ 2 + 2μuσ )(μ + uσ )(μ2 − σ 2) . (31)

Equation (29) is solvable by standard methods. Requiringthat Ĵ (v < 0) = 0 and Ĵ ′(v < 0) = 0 fixes the integrationconstants. The final solution reads

Ĵ (v,s) = e−v(A+√

A2−B)(

1

2− C − A

2√

A2 − B

)θ (v)

+ e−v(A−√

A2−B)(

1

2+ C − A

2√

A2 − B

)θ (v). (32)

From this expression, we can derive the kth moment of theISI distribution by using the following property of the Laplacetransform: 〈

T kn〉 = (−1)k ∂k

∂skĴ (nvT ,s)

∣∣∣∣s=0

. (33)

Simplification of the formula for k = 1,2,3 leads to explicitexpressions for the mean, the variance, and the third centralmoment:

〈Tn〉 = nvTμ + uσ , (34)〈

�T 2n〉 = 〈T 2n 〉 − 〈Tn〉2= nvT σ

2(1 − u2)λ(μ + uσ )3

[e−νn − 1

νn+ 1

], (35)

〈�T 3n

〉 = 〈T 3n 〉 − 3〈Tn〉〈�T 2n 〉 − 〈Tn〉3= 3nvT σ

2(1 − u2)(σ 2 + μuσ )λ2(μ + uσ )5

×[

2

νn(e−νn − 1) + e−νn + 1

]. (36)

From these expressions, we can derive other statisticalmeasures of the neuron model, such as the stationary firingrate r0 and the coefficient of variation cv:

r0 = 1〈T1〉 =μ + uσ

vT, (37)

cv =√〈

�T 21〉

〈T1〉 =√

σ 2(1 − u2)λvT (μ + uσ )

[e−ν − 1

ν+ 1

]. (38)

A shape measure that quantifies the asymmetry of the ISIdistribution is the skewness γs . It is given by

γs =〈�T 31

〉〈�T 21

〉3/2 . (39)Figure 8 illustrates how a varying asymmetry u of the

dichotomous input changes the asymmetry of the ISI distri-bution. In the plot, we hold the mean and variance of theinput constant while changing u. Because the theory requiresμ > σ , there is a maximal u for which this condition still holdsif we keep the mean and variance constant. Figure 8 showsthe monotonic increase in the skewness with increasing u.Furthermore, we can see that the skewness is more pronouncedfor small mean transition rates λ than for large ones.

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4u

4

2

0

2

4

6

8

λ =100.0

λ =1.0

λ =0.01

simulationtheory

FIG. 8. (Color online) Skewness of the ISI distribution γs as afunction of the asymmetry u of the dichotomous noise. The theory(colored lines) is compared to simulations (black circles) for differentλ. u is varied while holding the mean μ + uσ = 1 and the varianceσ 2(1 − u2) = 0.25 of the input constant. Note that there is an upperlimit for u because of the constraint of the theory μ > σ .

022718-7


FIG. 9. (Color online) Skewness of the ISI distribution γs as afunction of the input skewness γDN and the effect of additional whitenoise on it (theory, continuous line; simulations, black circles forD = 0; black squares, otherwise). The mean μ + uσ = 1 and thevariance σ 2(1 − u2) = 0.25 of the input are kept constant. The theoryrequires μ > σ , which results in a minimal input skewness.

A comparison to the skewness of the ISI distribution of aPIF neuron driven by Gaussian white noise with equivalentnoise intensity D = DDN,

γ WNs = 3√

2D

μvT, (40)

reveals that for u = 0, γs is always smaller than γ WNs (notshown). With higher mean transition rates λ, the skewnessapproaches this value even for u �= 0 because the slope of theu dependence becomes smaller. From Eqs. (35) and (36) itcan be shown that the skewness diverges to ±∞ in the limitsu → ±1.

To plot the relationship between the asymmetries ofthe dichotomous input and of the resulting ISI distributiondifferently, we can relate u to the skewness of the dichotomousnoise γDN via Eq. (5). Figure 9 illustrates the dependenceof the skewness of the ISI distribution on the skewness ofthe dichotomous input. This time, there is a minimal valuecorresponding to the constraint that μ > σ . Clearly, the ISIskewness decreases monotonically with the skewness of thenoise. A bias toward smaller (larger) values of the dichotomousprocess is associated with a positive (negative) skewness of theinput noise and leads to a preference of longer (shorter) ISIs,which in turn results in a negative (positive) ISI skewness.Figure 9 also demonstrates the effect of additional white noise.In general, additional Gaussian white noise decreases the slopeof the function but does not change the monotonic decline ofthe ISI skewness with increasing noise skewness. The changein slope is more pronounced for the larger noise intensity.

To capture deviations of the skewness from a PIF modeldriven only by white noise, we rescale the skewness such thatit is 1 for an inverse Gaussian. The skewness of an inverse

0.5 1.0 1.5 2.0 2.5 3.03

2

1

0

1

2

3

4

5

6

u =-0.6

u =0.0u =0.6

theory for D=0simulation with D=0simulation with D=0.01

FIG. 10. (Color online) The rescaled skewness αs plotted overthe constant input current μ for different values of the noiseasymmetry u. The black circles indicate simulations without whitenoise, which are in perfect agreement with the theory. The blacksquares are simulations with additional white noise D = 0.01.Remaining parameters: λ = vT = 1.0, σ = 0.5.

Gaussian is equal to 3cv , so that the rescaled skewness αs isdefined as [39]

αs = γs3cv

= 〈�T3〉〈T 〉

3〈�T 2〉2 . (41)

In the absence of white noise, we can compute αs fromEqs. (34)–(36). Figure 10 shows the change in the rescaledskewness αs with increasing base current μ. The increase ordecrease in αs at large values of μ is directly related to theasymmetry u. In the limit of large μ, the curve approaches theasymptotic straight line,

αs ≈ 2λuσvT + 3σ2

9σ 2(1 − u2) +2u

σ 2(1 − u2)μ. (42)

In some parameter regimes, however, αs shows a nonmono-tonic dependency on μ at small values μ (not shown).Although additional white noise diminishes the effect ofdichotomous noise on the skewness (cf. squares in Fig. 10),the qualitative dependence remains the same even for D > 0.

In an experiment, one could inject a constant current intoa cortical cell receiving input from a presynaptic populationshowing up-down states. This corresponds to a change in theparameter μ of the model. A monotonic change in the rescaledskewness then may indicate an asymmetry in the presynapticinput.

We now turn to the calculation of the serial correlationcoefficient ρk for the output spike train. To do so, we make useof the following relation between the variance of the nth-orderinterval var(Tn) and the SCC [16]:

ρk = var(Tk−1) + var(Tk−1) − 2 var(Tk)2 var(T1)

. (43)

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0 20 40 60 80 100k

0.0

0.2

0.4

0.6

0.8

λ =0.5

λ =0.1 λ =0.02

λ =0.004

simulationtheory

FIG. 11. (Color online) Serial correlation coefficient obtained bysimulations (black circles) compared with the theory (colored lines)for pure dichotomous noise (D = 0) [Eq. (44)]. The curves showthe SCC for different mean transition rates λ. Remaining modelparameters: μ = vT = 1.0, σ = 0.5, and u = 0.8.

Using the variance Eq. (35), Eq. (43) yields, after somesimplifications, a simple expression for the SCC,

ρk = 2 sinh2(ν/2)

ν − 1 + e−ν e−kν . (44)

This expression has the same form as the SCC for symmetricdichotomous noise found in [16]. The only difference is thatthe asymmetry u appears in the parameter ν [cf. Eq. (25)].Figure 11 shows that the theory agrees with numericalsimulations. In the limit σ → μ discussed above, ν goes toinfinity and thus the correlations become zero. This resultagrees with the consideration that the noise cannot carrymemory from one ISI to the next if the threshold is only reachedin one of the two noise states [40].

To obtain an approximation for the ISI correlations inthe case of driving with both dichotomous and Gaussianwhite noise, we can make use of the approximation discussedin the preceding section. Additional white noise decreasesthe correlations. Because all correlations result from thedichotomous noise, it is reasonable to assume that the majorcontribution to the expected decorrelation comes from anincreased variance in the denominator of Eq. (43). Theapproximation Eq. (27) allows us to calculate the increaseof the variance due to additional white noise to linear orderin D:

var(TDN+WN) ≈〈�T 2DN

〉 + 2Dv2T

〈T 3DN

〉, (45)

where 〈T 3DN〉 is the third moment of the ISI distributionwith only dichotomous noise given in Eq. (36). Using thisapproximation, we obtain the following expression for theserial correlation coefficient:

ρDN+WNk ≈ρDNk

1 + βD , β =2〈T 3DN

〉v2T

〈�T 2DN

〉 . (46)

FIG. 12. (Color online) Serial correlation coefficients (for lagk = 1, 5, 10, and 20) normalized by the respective SCC withoutwhite noise and plotted over different white noise intensities. Thedots represent stochastic simulations, whereas the solid line showsthe approximation of Eq. (46) with model parameters λ = 0.01,μ = vT = 1.0, σ = 0.5, and u = 0.6.

Numerical simulations (Fig. 12) at small switching ratesconfirm the prediction of the correlation reduction by whitenoise quantitatively. In particular, the reduction seems to beindependent of the lag k: Fig. 12 shows the SCC for differentlags k normalized by the respective SCC without white noise,resulting in data points that are on top of each other and agreewell with the theoretical formula. The figure further suggeststhat, at least for strong ISI correlations induced by a slowdichotomous noise, the approximation is valid for quite largenoise intensities D.

V. POWER SPECTRUM

We now turn to the calculation of a second-order spike-train statistics, the power spectrum of x(t). The startingpoint of our consideration is the correlation function C(τ ) =〈x(t)x(t + τ )〉 − 〈x(t)〉2, which can be expressed [41] byC(τ ) = r0[δ(τ ) + m(τ )] − r20 . Here r0 is the stationary firingrate and m(τ ) denotes the spike-triggered rate, i.e., the condi-tional probability density to observe a spike at time t + τ giventhat there was (another) spike at time t . The power spectrum,defined in Eq. (9), is related to the correlation function by aFourier transformation (Wiener-Khinchin theorem), and henceit follows that [7,41,42]

S(ω) = r0(1 + 2 Re(m̃(ω))), (47)with the stationary firing rate r0 and the one-sided Fouriertransform of the spike-triggered rate

m̃(ω) =∫ ∞

0eiωtm(t)dt. (48)

This quantity is the transform of a real valued function andthus obeys m̃∗(ω) = m̃(−ω). In the following, we mark all

022718-9


quantities that are one-sided Fourier transforms with a tilde.For a shorter notation, we use angular frequencies ω, whichreadily transform to normal frequencies f = ω/(2π ).

In principle, we know m(t) already and could calculate thespectrum from Eqs. (48) and (47): the spike-triggered rate isgiven by the sum over all nth-order interval densities [7]

m(t) =∞∑

n=1JD,n(t). (49)

However, the summation over the Bessel functions and theFourier transformation of the result are difficult. Here we useanother method, which is based on the Fourier transformationof the master equations and is similar to the method outlinedin Ref. [42] for the case of a white-noise-driven IF model.

To calculate m̃(ω) for the case of purely dichotomous noise,we start with the master equations (10) and (11). To captureall spikes and not only the first spike, we have to include termsthat describe the absorption and reset of the voltage trajectory,

∂tP+ = λ−P− − λ+P+ − (μ + σ )∂vP+ + m+(t)f (v), (50)

∂tP− = λ+P+ − λ−P− − (μ − σ )∂vP− + m−(t)f (v), (51)

with f (v) = δ(v) − δ(v − vT ). Here, we have split m(t) =m+(t) + m−(t) into two parts, m+(t) and m−(t), correspondingto spikes that occur during the plus and minus states,respectively. Both functions are not yet known (in fact, ourconsideration only serves the purpose of calculating them),but their Fourier transforms can be found as outlined inthe following. Initial conditions for Eqs. (50) and (51) areas above for the first-passage-time problem, i.e., P±(v,0) =pF (±σ )δ(v).

From the master equations (50) and (51), we can obtain asecond-order partial differential equation for P± with the samehomogeneous part as Eq. (16). By virtue of the equation’slinearity, we may instead write the same type of equation forthe total probability current J defined in Eq. (12) and for thedifference between the currents in the plus and in the minusstate,

Q = (μ + σ )P+ − (μ − σ )P−. (52)

To determine the two quantities m̃+ and m̃−, we need tocompute the Fourier transforms of the two functions J andQ. To do so, we perform a one-sided Fourier transformationof the resulting differential equations for J and Q, whichreduces them to simpler ordinary differential equations:

L̃0J̃ = h−j m̃− + h+j m̃+ + hij , (53)

L̃0Q̃ = h−q m̃− + h+q m̃+ + hiq, (54)

where J̃ and Q̃ as well as the inhomogeneities h are functionsof ω and v, and L̃0 denotes the Fourier transform of the operatorof the resulting homogeneous equation,

L̃0 = d2

dv2+ 2A(ω) d

dv+ B(ω), (55)

with

A(ω) = λ(μ + uσ ) − iωμμ2 − σ 2 , B(ω) = −

ω2 + 2iωλμ2 − σ 2 . (56)

The inhomogeneities resulting from absorption and reset (h±)and from the initial conditions (hi) are given by

h±j = C±j (ω)f (v) + f ′(v), (57)

hij = Cij (ω)δ(v) + δ′(v), (58)

h±q = C±q (ω)f (v) + f ′(v), (59)

hiq = Ciq(ω)δ(v) + Diqδ′(v), (60)with the coefficients

C+j (ω) =−iω(μ + σ ) + 2λ(μ + uσ )

μ2 − σ 2 , (61)

C−j (ω) =−iω(μ − σ ) + 2λ(μ + uσ )

μ2 − σ 2 , (62)

Cij (ω) =2λ(μ + uσ ) − iω

μ+uσ (μ2 + σ 2 + 2μuσ )

μ2 − σ 2 , (63)

C+q (ω) =−iω(μ + σ ) + 2λσ (1 + μu

σ

)μ2 − σ 2 , (64)

C−q (ω) =iω(μ − σ ) + 2λσ (1 + μu

σ

)μ2 − σ 2 , (65)

Ciq(ω) =2λσ

(1 + μu

σ

) − iωμ+uσ [2μσ + u(μ2 + σ 2)]μ2 − σ 2 , (66)

Diq =σ

μ + uσ(

1 + μuσ

). (67)

Because Eqs. (53) and (54) are linear, we can construct thefull solutions from the solutions for only one inhomogeneity.Specifically, J̃ and Q̃ can be expressed as the sums

J̃ = m̃−J̃− + m̃+J̃+ + J̃ i , (68)

Q̃ = m̃−Q̃− + m̃+Q̃+ + Q̃i, (69)

where J̃ k and Q̃k are solutions of

L̃0J̃k = hkj , (70)

L̃0Q̃k = hkq, (71)

with k = +, − ,i. We obtain them with standard methods byusing the conditions J k(v < 0,ω) and Qk(v < 0,ω) = 0. Forthe inhomogeneities from the initial conditions, the solutionsare

J i(v,ω) = θ (v)e−vA(ω)

×[

cosh[vF (ω)] + Cij (ω) − A(ω)

F (ω)sinh[vF (ω)]

],

(72)

022718-10


Qi(v,ω) = θ (v)e−vA(ω)[Diq cosh[vF (ω)]

+ Ciq(ω) − DiqA(ω)

F (ω)sinh[vF (ω)]

], (73)

with

F (ω) =√

A(ω)2 − B(ω)

=√

λ2(μ + uσ )2 − 2iωλσ (σ + μu) − σ 2ω2μ2 − σ 2 (74)

[A(ω) and B(ω) were defined in Eq. (56)]. The differentialequations with inhomogeneities from absorption and resetyield

J̃±(v,ω) =(

1

2− C

±j (ω) − A(ω)

2F (ω)

)ϕ+(v)

+(

1

2+ C

±j (ω) − A(ω)

2F (ω)

)ϕ−(v), (75)

Q̃±(v,ω) =(

1

2− C

±q (ω) − A(ω)

2F (ω)

)ϕ+(v)

+(

1

2+ C

±q (ω) − A(ω)

2F (ω)

)ϕ−(v), (76)

where

ϕ±(v) = θ (v)e−v[A(ω)±F (ω)] − θ (v − vT )e−(v−vT )[A(ω)±F (ω)].(77)

Our aim is to calculate the Fourier transforms of thespike-triggered rate m̃ = m̃+ + m̃−. To do so, we can use theboundary condition that J̃ and Q̃ have to vanish for v > vTbecause all voltage trajectories are reset when they reach vT .This condition allows us to determine the spike-triggered rates:

m̃+ = J̃iQ̃− − J̃−Q̃i

J̃−Q̃+ − J̃+Q̃− , m̃− =J̃+Q̃i − J̃ iQ̃+J̃−Q̃+ − J̃+Q̃− , (78)

where the different J̃ ’s and Q̃’s are all taken at v > vT . In theexpressions for m̃− and m̃+, the v dependences cancel eachother out.

Using the above expressions for J̃ k and Q̃k , we find, aftersubstantial simplifications,

m̃(ω) = (2{cosh[vT A(ω)] − cosh[vT F (ω)]})−1

×[A(ω) + iω

μ+uσF (ω)

sinh[vT F (ω)]

+ cosh[vT F (ω)] − e−vT A(ω)]. (79)

This is the final result of this section, and together with Eq. (47)it permits the calculation of the spike-train power spectrum.Note that in general, F (ω) is a complex-valued function, whichas an argument of the hyperbolic functions introduces periodiccomponents into the power spectrum.

Figure 13 shows examples of the spike-train power spectraof a PIF neuron with dichotomous driving and compares the

FIG. 13. (Color online) The power spectrum S(ω) from simula-tions (blue dots) compared with the theory [red (light gray) solid line],Eqs. (47) and (79), for different mean transition rates λ and purelydichotomous input. Remaining parameters: μ = vT = 1.0, σ = 0.7,u = −0.4.

analytical result to those of numerical simulation. In particular,at small values of the switching rate λ, the shape is ratherunusual for a spike-train power spectrum (for some experi-mental examples of power spectra, see [43]): the spectrum hasa clearly periodic part; in particular, this periodicity dominatesat large frequencies. There are two characteristic frequenciesand multiples thereof at which peaks occur in the spectrum,

ω± = 2π μ ± σvT

. (80)

These frequencies correspond to the two values at which δfunctions occur in the ISI distribution, i.e., the realizationswhere no switching occurs between reset and threshold. Math-ematically, we can understand the occurrence of undampedoscillations in the power spectrum by looking at Eqs. (49)and (47). The power spectrum given in Eq. (47) containsthe Fourier transform of a sum, Eq. (49), that contains δpeaks according to Eq. (24). The Fourier transform of a δpeak yields an undamped oscillation, which explains whywe observe such unusual periodicity in the spectrum. Foran alternative explanation, one may consider that the powerspectrum of a Dirac comb (a perfectly regular spike train withrandom initial time) is again a Dirac comb [7], i.e., it exhibitsundampened periodicity. The spike train of our model canbe regarded as a sequence of time windows with perfectlyregularly spaced spikes, i.e., finite-duration “Dirac combs.”Because the durations of the time windows are stochastic,the resulting periodic structure in the power spectrum doesnot contain δ peaks as it would do for an infinite Dirac comb.Hence, the particular features of the power spectrum rely solelyon the discrete support of the dichotomous noise.

At small transition rates λ, the ratio between the height ofthe two different peaks corresponds to the square of the ratiobetween the probabilities of crossing the threshold in one of

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the noise states. Thus, the relative peak height is influencedby the asymmetry of the driving process. With increasing λ,the peaks broaden until they overlap such that the spectrum isalmost constant for high frequencies. However, even for highλ, there remains a small oscillating part, which is related to thesurvival of the δ peaks in the nth-order interval density for anyfinite value of λ. This behavior clearly differs from the powerspectra of model neurons driven by white noise [44], where alloscillations vanish at high frequencies and the power spectrumapproaches r0. In fact, even if the input noise is colored buthas a continuous support, we do not expect oscillations in thehigh-frequency part of the power spectrum, which is in linewith previous results for a PIF neuron driven by an Ornstein-Uhlenbeck process [15].

The limit ω → 0 of the power spectrum is linked to theserial correlation coefficient by the general relation [45]

limω→0

S(ω) = r0c2v(

1 +∞∑

k=1ρk

)= r0F∞, (81)

where F∞ is the Fano factor in the limit of large spike countwindows. The Fano factor is defined as the variance over themean of the spike count in a specified time window and thus isa relative measure of the count variability. Taking the limit aswell as performing the infinite sum yield the same expressionand thus shows that the results for the SCC and the powerspectrum are consistent. The resulting Fano factor is given bythe simple expression

F∞ = σ2(1 − u2)

vT λ(μ + uσ ) . (82)

Interestingly, this expression corresponds to the result for aPIF neuron driven by white noise, for which the Fano factor isgiven by F∞ = 2D/(vT μ). Here, one has to replace the whitenoise intensity D by the noise intensity of the dichotomousnoise, DDN = τc〈�η(t)2〉.

How much of the peculiar spectral structure survivesif the neuron is additionally driven by white noise? Forthe parameters from Fig. 13(b) we have inspected this bynumerical simulations (Fig. 14). With increasing white noise,the spectral multipeaked structure is replaced by one withfewer and broader peaks. When driving the model neuronwith both dichotomous and white noise, the periodicity ofthe power spectrum is damped toward high frequencies. Inthe limit ω → ∞, it saturates at the stationary firing rate. AsFig. 14 shows, the periodicity decays stronger with higherwhite noise intensities D and the peaks become wider. WithD = 0.1 there is hardly any periodic structure left.

VI. CONCLUSION

This paper explored the effect of correlations and asymme-try of neuronal input noise on the firing statistics of a singleneuron. To get some analytical insights, we derived differentspike-train statistics for a perfect integrate-and-fire neurondriven by asymmetric dichotomous noise: the ISI distribution,the serial correlation coefficient, and the power spectrum. Weverified the analytical expressions with stochastic simulations.We found a set of features that stem from the noise correlationsand the asymmetry in the dichotomous noise: In contrast to

FIG. 14. (Color online) The simulated power spectrum S(ω) ofa PIF neuron driven by dichotomous noise and Gaussian whitenoise for different white-noise intensities D as indicated. Remainingparameters as in Fig. 13(b).

the statistics of a neuron driven by uncorrelated noise, the ISIdistribution is clearly bimodal for small transition rates of theinput noise. In the limit of high transition rates, the distributionapproaches an inverse Gaussian. A high asymmetry betweenthe rates of the dichotomous noise leads to a strongly skeweddistribution of interspike intervals.

The ISI correlations decay exponentially with the lag, aspreviously found in [16] for symmetric dichotomous noise.In our case, both the decay constant and the prefactor arefunctions of the asymmetry, but asymmetric noise does notlead to qualitatively different behavior. The power spectrumshows periodic oscillations at high frequencies. In contrast tothe driving with Gaussian noise, the oscillations do not decayat large frequencies.

We also gave approximations of the ISI statistics for amodel neuron driven by both slow dichotomous and Gaussianwhite noise, which were confirmed by numerical simulations.Here we focused on the case of slow dichotomous noisebecause in this limit the effects of the non-Gaussian cor-related noise process are most pronounced. We found thatwhite noise decreases the positive ISI correlations caused bythe dichotomous input by a factor independent of the lag.Furthermore, numerical simulations of the power spectra showthat the periodic oscillations from the dichotomous noise aredamped due to additional white noise.

The theory can possibly be applied in experimental studiesof up-down states in the cortex. Because up-down statesare collective phenomena of neural populations, their mea-surement requires a lot of experimental effort. The outputstatistics of a neuron may allow us to infer properties ofthis joint activity of a presynaptic neural population from thefiring statistics of a neuron receiving such two-state-like inputcurrents. For example, the ISI density can show features suchas bimodality that hint at dichotomous input. Furthermore,

022718-12


exponentially decaying ISI correlations are likely to stem fromexponentially correlated input. By injecting constant inputcurrents of varying amplitude, the rescaled skewness of theISI distribution could allow us to infer information about theasymmetry of up-down state transitions via the parametricdependence shown in Fig. 10. One assumption that has to bemet to apply our results in such investigations would be thatthe considered neuron is in a mean-driven (tonically spiking)regime, for which an approximation by a perfect IF model canbe meaningful [18,31].

The neural dynamics of the perfect integrate-and-fire modelis simple, but more elaborate models can show similar features.Preliminary simulation results (not shown) suggest that theresults for the ISI distribution can approximate a leakyintegrate-and-fire (LIF) model in the tonically firing regime

with a small leak. By rescaling the parameters of the PIFmodel, we were able to approximate the ISI density of anLIF (for a similar method in the case of driving with whitenoise and adaptation current, see [46]). However, furtheranalysis is needed to investigate which of the features ofthe spike statistics, analytically revealed in our study, arepreserved in biophysically more realistic neuron models witha spike-generating mechanism.

ACKNOWLEDGMENTS

We acknowledge funding by the Bundesministerium fürBildung und Forschung (BMBF) (FKZ: 01GQ1001A) and theDeutsche Forschungsgemeinschaft (DFG) under the projectResearch Training Group GRK 1589/1.

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