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Modulated escape from a metastable state driven by colored noise

Many phenomena in nature are described by excitable systems driven by colored noise. The temporal correlations in the fluctuations hinder an analytical treatment. We here present a general method of reduction to a white-noise system, capturing the color of the noise by effective and time-dependent boundary conditions. We apply the formalism to a model of the excitability of neuronal membranes, the leaky integrate-and-fire neuron model, revealing an analytical expression for the linear response of the system valid up to moderate frequencies. The closed form analytical expression enables the characterization of the response properties of such excitable units and the assessment of oscillations emerging in networks thereof.

Many phenomena in nature are described by excitable systems driven by colored noise. For systems with noise fast compared to their intrinsic time scale, we here present a general method of reduction to a lower dimensional effective system, capturing the details of the noise by dynamic boundary conditions. We apply the formalism to the leaky integrate-and-fire neuron model, revealing an analytical expression for the transfer function valid up to moderate frequencies. This enables the characterization of the response properties of such excitable units and the assessment of the stability of networks thereof. In his pioneering work Kramers [1] investigated chemical reaction rates by considering the noise-activated escape from a metastable state as a problem of Brownian motion in a double well (Landau) potential. As in the original work, subsequent studies (see references in [2]) assume the noise statistics to have a white spectrum. This idealization simplifies the analytical treatment and is the limit of non-white processes for vanishing correlation time [3]. White noise cannot exist in real systems, which is obvious considering, for example, the voltage fluctuations generated by thermal agitation in a resistor [4,5]: if the fluctuations had a flat spectrum, an infinite amount of power would be dissipated over all frequencies. Colored noise characterized by a single time constant is a more realistic noise model, but comes along with considerable difficulties, since it adds a dimension to the governing Fokker-Planck equation. The effect of colored noise is relevant in fluctuation-induced transitions observed in dye lasers, chemical reactions, turbulent transition and optical systems (see [6,7] and references therein for applications). These transition phenomena have been studied in abstract terms as the mean first passage time (MFPT) of a particle in the Landau potential [8][9][10][11][12][13]. The effect of colored noise on the escape from a potential well is also relevant in modeling biological membranes: the leaky integrate-and-fire (LIF) neuron model with exponential synaptic currents can equivalently be described as a one-dimensional system driven by colored noise. Based on the work of Doering et al. [13] and Kłosek and Hagan [14], Brunel et al. [15,16] calculated the low-and high-frequency limits of its transfer function. Here we extend these works by presenting a general method to reduce a first order differential equation driven by additive fast colored noise to an effective one-dimensional system, for which standard methods are available [17]. The effective formulation implicitly contains the matching between outer and boundary layer solutions as well as the half-range expansion required to obtain the latter, thereby avoiding the difficulties in constructing explicit solutions for particular applications.
Consider a pair of coupled stochastic differential equations (SDE) with a slow component y with time scale τ , driven by a fast Ornstein-Uhlenbeck process z with time scale τ s . In dimensionless time s = t/τ and with k = τ s /τ relating the two time constants we have with a unit variance white noise ξ(s + u) ξ(s) = δ(u).
We are interested in the case τ s ≪ τ , where the slow component y effectively obeys the one-dimensional SDE ∂ s y = f (y, s) + ξ(s). Heuristically, y integrates z on a time scale τ . For finite τ s the integral of the autocorrelation of z is k 2 , identical to the limit of vanishing τ s , where z(s) = kξ(s) becomes a white noise. To formally derive an effective diffusion equation for y and to obtain a formulation in which we can include absorbing boundaries, we consider the Fokker-Planck equation [17] corresponding to (1) where P (y, z, s) denotes the probability density and we introduced the probability flux operator in y-direction as S y = f (y, s) + z/k. Factoring-off the stationary solution of the fast part of the Fokker-Planck operator, √ π , we observe the change of the differential operator ∂ z and we refer to Q as the outer solution, since initially we do not consider boundary conditions. The strategy is as follows: We show that the terms of first and second order in the small parameter k of the perturbation ansatz Q = 2 n=0 k n Q (n) + O(k 3 ) cause an effective flux acting on the z-marginalized solutioñ P (y, s) = dz e −z 2 √ π Q(y, z, s) that can be expressed as a one-dimensional Fokker-Planck equation correct up to linear order in k. To this end we need to know the first order correction to the marginalized probability flux ν y (y, s) ≡ dz e −z 2 √ π S y Q(y, z, s) = 1 n=0 k n ν (n) y (y, s) + O(k 2 ). Inserting the perturbation ansatz into (3) we have LQ (0) = 0 and obtain for the first and second orders With Lz = −z the particular solution for the first order leaves the freedom to choose a homogeneous (zindependent) solution Q Collecting all terms which contribute to ν y in orders k 2 and higher in ι, the only relevant part of the second order solution is −z∂ y Q (1) 0 (y, s), which leaves us with from which we obtain the marginalized flux We observe that f (y, s)− 1 2 ∂ y is the flux operator of a onedimensional system driven by unit variance white noise andP (y, s) ≡ Q (0) (y, s) + kQ (1) 0 (y, s) is the marginalization of (6) over z. Note that in (6) the higher order terms in k appear due to the operator kz∂ y in (3) that couples the z and y coordinate. Eq. (7) shows that these terms cause an effective flux that only depends on the z -marginalized solutionP (y, s). This allows us to obtain the time evolution by applying the continuity equation to the effective flux (7) yielding the effective Fokker-Planck equation Effective Fokker-Planck equations, to first order identical to (8), but also including higher order terms, have been derived earlier [8][9][10][11][12]. These approaches have been criticized for applying white noise boundary conditions to the effective system [14]. Further discussion evolved since an absorbing boundary either amounts to reaching a separatrix in the two dimensional domain [18] or to one component crossing a constant threshold value [19], depending on the physics of the system. Doering et al. [13] and Kłosek and Hagan [14] treated the latter by singular A B Figure 1. Boundary conditions. A: Flux and boundary condition in the two-dimensional colored noise system. On the negative half plane z < 0 the density must vanish at threshold θ. B: Density of the white noise system (dashed) vanishes at threshold. The density of the effective system has a finite value at threshold which is determined by the matching between the outer solution Q and the boundary layer solution Q B (right).
perturbation methods and boundary layer theory, showing that the O(k) correction to the static MFPT stems from colored noise boundary conditions for the marginalized density of the effective system. Extending this approach to the transient case requires a time-dependent boundary condition forP or, equivalently, for Q 0 , because we assume the one-dimensional problem to be exactly solvable and hence the boundary value of Q (0) to be known. Without loss of generality, we assume an absorbing boundary at the right end y = θ of the domain. The flux vanishes at this threshold for all points with negative velocity in y given by f (θ, s) + z k < 0. A change of coordinates z + kf (θ, s) → z simplifies the condition to z k Q(θ, z, s) = 0 for z < 0. The flux and this half range boundary condition are shown in Figure 1A. The resulting boundary layer at θ requires the transformation of the Fokker-Planck equation (3) to the shifted and scaled coordinate r = y−θ with Q B (r, z, s) ≡ Q(y(r), z, s) and the operator s)). The boundary condition then takes the form Q B (0, z, s) = 0 ∀z < 0. With the perturbation ansatz the solution of which must match the outer solution. The latter varies only weakly on the scale of r and therefore a first order Taylor expansion at the boundary yields the matching condition Q B (r, z, s) = Q(θ, z, s) + kr ∂ y Q(θ, z, s), illustrated in Figure 1B. We note with Q (0) (θ, s) = 0 that the zeroth order Q B(0) vanishes. Inserting (5) into the matching condition, with y (s) the instantaneous flux in the white-noise system, the first order takes the form The vanishing zeroth order implies with (10) for the first order LQ B(1) − z∂ r Q B(1) = 0. Appendix B of Kłosek and Hagan [14] states the solution of the latter equation satisfying the half-range boundary condition with α = √ 2|ζ( 1 2 )| given by Riemann's ζ-function and b n proportional to the n-th Hermitian polynomial. We equate (12) to (11) neglecting the exponential term decaying on a small length scale, so that the term proportional to z − r fixes the time dependent function C(s) = 2ν reducing the colored-noise problem to the solution of a one-dimensional Fokker-Planck equation. The boundary condition can be understood intuitively, as the filtered noise slows down the diffusion at the absorbing boundary with increasing k; the noise spectrum and the velocity of y are bounded, so in contrast to the white noise case, there can be a finite density at the boundary. Its magnitude results from a momentary equilibrium of the escape and the flow towards the boundary (∝ ν (0) y ). The qualitative difference to the static case studied in [14] is, apart from the time dependence, that our boundary condition is given at the same point in the domain, while the previous work found a displaced, but perfectly absorbing boundary. In systems where the dynamic variable is reset to a value R after each escape, an analog calculation [20] yields the additional boundary conditioñ P (R+, s) −P (R−, s) = kαν (0) y (s). We now apply the novel theory to the LIF model exposed to filtered synaptic noise to derive a long searched for analytical expression for the input-output transfer function bridging the earlier obtained limits for modulations at low and high frequencies [15,16]. The corresponding system of coupled differential equations τV = −V + I + µ and τ sİ = −I + σ √ τ ξ [16] describes the evolution of the membrane potential V and the synaptic current I driven by input with mean µ and variance σ 2 in diffusion approximation. Note that this system can be obtained from (1) by introducing the coordinates y = V −µ σ , z = k σ I, and the linear choice f (y, s) = −y. Using x = √ 2y, and the marginalized density ρ(x, s) ≡ 1 √ 2P (x/ √ 2, s), we get the reduced effective system (8) With the function u(x) = e − 1 4 x 2 the right hand side can be transformed to the Hermitian form ∂ s q(x, s) = −a † a q(x, s), where q(x, s) = u −1 (x)ρ(x, s) and we defined the operators a ≡ 1 2 x + ∂ x , a † ≡ 1 2 x − ∂ x satisfying [a, a † ] = 1 and a † a(a † ) n q 0 = n(a † ) n q 0 , as in the quantum harmonic oscillator. The stationary solution that has a constant flux ν between reset and threshold obeys, with the flux operator −ua, the ordinary inhomogeneous linear differential equation that can be solved by standard methods given the boundary conditions (13) 1 identical to the earlier found solution [15,16] showing how the correction term ∝ k reduces the firing rate compared to the well known result (τ ν 0 [21]. We now consider the response properties of the LIF neuron to a periodic perturbation, i.e. its transfer function. We first simplify the derivation for white noise [22,23] by exploiting the analogies to the quantum harmonic oscillator introduced above [20] and then study the effective system for colored noise. To linear order a periodic input with µ(t) = µ + ǫµ e iωt and σ 2 (t) = σ 2 +Hσ 2 e iωt modulates the firing rate ν 0 (t)/v 0 = 1 + n(ω)e iωt , proportional to the transfer function n(ω). With a perturbation ansatz for the modulated density ρ(x, s) = ρ 0 (x) + ρ 1 (x, s) and the separation of the time dependent part ρ 1 (x) e iωτ s follows a second order linear differential equation iωτ ρ 1 = L 0 ρ 1 + L 1 ρ 0 . Here x is the perturbation operator, the first term of which originates from the periodic modulation of the mean input with G = √ 2ǫµ/σ, the second from the modulation of the variance. In operator notation the perturbed Fokker-Planck equation transforms to (iωτ + a † a) q 1 = (G a † + H (a † ) 2 ) q 0 with the particular solution showing that the variation of µ contributes the first excited state, the modulation of σ 2 the second. The homogeneous solution q h can be expressed as a linear combination of parabolic cylinder functions U (iωτ − 1 2 , x), [23,24]. For white noise, the boundary condition on q 1 = q h + q p is The flux due to the perturbation can be expressed with the transfer function as τ ν 0 n(ω) = u −aq 1 + (G + H a † )q 0 {xR,x θ } . Knowing the particular solution yields four conditions, for the homogeneous solution q h | {xR,x θ } and its derivative ∂ x q h | {xR,x θ } that determine the homogeneous solution on the whole domain as well as the transfer function to obtain the known result [15,22,23]. We derive a novel analytical expression for the colored noise transfer function n cn by employing the general theory developed above. To lowest order in k the modulated firing rate is ν We obtain the colored noise transfer function by replacing the boundary condition (17) with (19) and then following the same derivation as in the white noise case. Note that we here only consider a modulation of the mean µ, which dominates the response properties. Treating a modulation of the variance σ 2 is difficult within our approach since there is no corresponding choice of f in (1). The explicit calculation is given in [20] and the result valid to linear order in k is n cn (ω) = n(ω)| H=0 (20) The first correction term ∝ k is similar to the H−term in (18), meaning that colored noise has a similar effect on the transfer function as a modulation of the variance [23] in the white noise case. For infinite frequencies this similarity was already found: modulation of the variance leads to finite transmission at infinite frequencies in the white noise system [23], the same is true for modulation of the mean in the presence of filtered noise [15]. However, for infinite frequencies our analytical expression behaves differently. The two correction terms in the second line of (20) cancel each other, since Φ ′′ ω → (iωτ ) 2 Φ ω and Φ ′ 2 ω → (iωτ ) 2 Φ 2 ω . Thus the transfer function decays to zero as in the white noise case. These deviations originate from neglecting the time derivative on the left hand side of (9), that is of order k 2 only up to moderate frequencies ωk ≪ 1. The high frequency limit ωk ≫ 1, however, is known explicitly [15].
The left column of Figure 2 shows the absolute value of the transfer function (20) and the right its phase for different firing rates controlled by noise levels σ and mean µ. The analytical predictions for the absolute value are in agreement with the simulations for the range of frequencies shown here. Above 100 Hz deviations occur as expected. These deviations are more pronounced and already observed at lower frequencies in the phase shift. At high firing rate and low noise, the neuron is mean driven and exhibits a resonance at its firing frequency, again well predicted by the analytical result. The lower row of Figure 2 shows results for different values of τ s including the white noise case τ s = 0. As expected the deviations at high frequencies increase with the synaptic time constant, since the neglected terms are ∝ ωk.
The analytical expression (20) has various applications: Transmission of correlated activity by pairs of neurons exposed to common input [25], can now be studied in a time-resolved manner. On the network level, the theory of correlated activity (reviewed in [26]) relies on the transfer function and also the stability of networks and emerging oscillations [22] become analytically tractable in the case of realistic synaptic filtering.