Fig 1.
Linear response of a neuron models spike timing.
The membrane potential of a tonically firing neuron with deterministic ISI T* is subject to an arbitrary perturbation u(ti + τ) (left panel, red line). Here ti is the time of a reference spike and τ ∈ [0, T*] is the relative time since the last spike, resembling a phase. How strongly this perturbation advances or delays the “phase” τ will in general depend on the “phase” itself at which the perturbation is applied. This sensitivity is quantified by the phase-response curve Z(τ) shown in blue in the middle panel. The term Z(τ)u(ti + τ) can thus be thought of a perturbation of the phase (middle panel, red line). In linear response these phase perturbations can be integrated to yield the cumulative phase shift or spike time derivation δTi+1 = ti+1 − (ti + T*) (right panel, red arrow). Note that separating the perturbation from the deterministic dynamics of the neuron model, i.e. finding u(ti + τ) is part of the problem that we address in this paper and solve in detail in the Methods section.
Fig 2.
Serial correlation coefficients for the adaptive QIF (Theta) model with colored noise.
Panel A shows the transformed membrane potential θ(t) = 2tan−1v(t), adaptation current a(t) and colored noise η(t) with spike times {ti}, ISIs {Ti} and peak adaptation values {ai}. Panel B displays the deterministic limit cycle (dashed line) and exemplary noisy trajectory (solid line) in the phase plane (θ, a). Note that the jump is of constant size Δ/τa, while the voltage or equivalently phase always resets to a fixed value θR. Panel C depicts the corresponding type I PRC quantifying the QIF model’s response characteristics. For a non-adaptive QIF model the PRC would be symmetric around T*/2; for the adaptive QIF, however, the maximum is shifted towards the right, i.e. the neuron is particularly sensitive to stimuli applied at the end of the ISI. Panel D shows the SCC with initial, slightly positive correlation coefficient due to positively correlated noise and subsequent negative correlations governed by adaptation. This pattern cannot be described by a single geometric sequence. Parameters: μ = 5, τa = 6, Δ = 18, τη = 4, σ2 = 0.5, D = 0 and resulting T* ≈ 4.0 and coefficient of variation CV ≈ 0.2.
Fig 3.
General correlation coefficient ρk of the adaptive LIF model subject to white and colored noise.
The specific SCCs ρk,a and ρk,η are obtained by considering one correlation inducing process at a time, i.e. adaptation and white noise or colored and white noise, respectively. Two qualitatively different cases are displayed distinguished by i) the base pattern exhibited by ρk,a that is exponentially decaying in A and oscillatory in B and ii) the sign of the first (k = 1) and every subsequent (k ≥ 1) SCC. For example consider A where ρ1 > 0 and ρk < 0 for k > 0. The inverse case is shown in B, i.e. negative correlations at lag 1 and positive ISI correlation for every subsequent lag. Such patterns have been reported in cats peripheral auditory fibers and the weakly electric fish electroreceptors [50, 51] and can not be explained by adaptation or colored noise alone. Note that the SCC of the full model is not bound by the specific SCCs. Parameters (A, B): γ = 1, μ = (5, 20), τa = (2, 1), Δ = (2, 10), τη = (0.5, 5), σ2 = 2 ⋅ 10−2, D = 10−3.
Fig 4.
Pattern of interval correlations for the adaptive LIF model.
The PRC Z(τ) and SCC ρk for two different cases that are strong ν < 0 A and weak adaptation 0 < ν < 1 B are shown. In both cases the colored noise correlation time τη is gradually increased. For small correlation times the SCC is governed by the adaption as the colored noise becomes essentially white (dark line and circles). In the other limit of long correlation times the SCCs are positive and governed by the colored noise (light line and circles). For intermediate time scales the SCCs are determined by both processes equally as shown in Fig 3. Parameters (A, B): γ = 1, μ = (20, 5), τa = 2, Δ = (20, 2), σ2 = 0.1, D = 0 and resulting T* = (0.67, 1.04).
Fig 5.
Pattern of interval correlations for the adaptive GIF model.
The PRC Z(τ) and SCC ρk for three different cases that corresponds to ν < 0 A, 0 < ν < 1 B and ν > 1 C are shown. The first two cases A, B resemble the previously discussed cases of the adaptive LIF model, see Fig 4. For the third case C both adaptation and correlated noise can have counter intuitive effects on the SCC if they act mainly on the proportion of the ISI where the PRC is negative. This can be ensured by appropriate choice of the time scales, here τa ≈ T*/2 and τη ≈ T*/2. Thus the adaptation can give rise to positive interval correlations and additional colored noise with varying correlation time initially decreases the SCCs for intermediate τη and eventually leads to enhanced positive correlations for large τη. Parameters (A, B, C): γ = (1, 1, −1), μ = (10, 20, 1), βw = (3, 1.5, 5), τw = (1.5, 1.5, 1.1), τa = (10, 10, 1), Δ = (10, 10, 2.3), σ2 = 10−3, D = 0 and resulting T* = (1.24, 0.57, 1.91).
Fig 6.
First serial correlation coefficient of the GIF model with respect to the correlated noise time constant τη.
Panel A shows the SCC for an adaptive GIF with parameters similar to those in Fig 5C. In panel B we consider a GIF model without adaptation and parameters chosen so that the PRCs in A and B qualitatively agree. The SCC at lag k = 1 can exhibit non-monotonic behavior with respect to the time constant τη due to the partially negative PRC. The PRC is shown in the upper left inset and is in both cases found to be negative until τ ≈ T*/2. If the correlation time matches this proportion of the ISI the SCC is significantly decreased compared to the case of short correlation times. For τη ≫ T* adjacent ISIs are positively correlated as they are similarly affected by the slow varying noise. Parameters (A, B): γ = −1, μ = 1, βw = 5, τw = (1.1, 1.1), wR = (0, 1), τa = (1, 0), Δ = (2.3, 0), σ2 = 10−3, D = 0 and resulting T* = (1.91, 1.76).
Fig 7.
Serial correlation coefficients for the LIF with adaptation-channel noise.
The SCC of adjacent intervals ρ1 (black lines and dots) for our model Eq (2) with identical time constants τη = τa = τc shows non-monotonic behavior with a minimum as a function of τc given that the white noise intensity D is sufficiently large. This is so because the kick amplitude scales with . The panels A and B correspond to weak and strong adaptation, respectively. The limiting cases are shown in orange (no adaptation or white noise, Δ = 0, D = 0) and blue (no colored noise, σ = 0, D = 0.1). However, only the limit case of vanishing colored noise can be attained by the full model through varying the white noise intensity. Parameters: (A, B) γ = 1 μ = (5, 20), Δ = (2, 20), σ2 = 0.1. Note that in contrast to the Figs 4–6, the deterministic period T* depends for some of the curves on τc. For the black and blue curves the deterministic period T* ∈ [0.56, 0.67] in A and T* ∈ [0.74, 1.08] in B and increases in both panels with τc. For the orange curve T* = 0.22 in A and T* = 0.05 in B.
Fig 8.
Serial correlation coefficients for the LIF with network-noise-like fluctuations.
Panel A shows the input power spectrum Sin with reduced power at low frequencies of a random process that is applied to a LIF model. The resulting SCC ρ1 for no adaptation (upper line) and weak adaptation (lower line) are displayed in panel B. The coefficient ρ1 has a minimum because for both long and short correlation times the noise becomes essentially white. The pattern of interval correlations are shown in C for no and D for weak adaptation. Even in the absence of an adaptation current the network-noise can generate negative ISI correlations. This is due to the lack of power at low frequency. For an adaptive LIF model negative correlations are enhanced. Parameters (C, D): γ = 1, μ = 5, τa = (0, 2), Δ = (0, 2), D = 0.18, τησ2 = 4.5 ⋅ 10−2 and resulting T* = (0.22, 0.67).
Fig 9.
Range of validity and effect of varying noise strengths.
Coefficient of variation CV (top) and SCC of adjacent intervals ρ1 (bottom) for the GIF model with weak adaptation. We test to which extent simulation results are well described by our weak-noise theory with respect to the two small parameters (D, σ2). In A the variance of the colored noise is fixed (σ2 = 0.1) and the white noise intensity D is varied; in B we fix D = 0.01 and vary σ2. We find quantitative agreement for CV < 0.3 in A and CV < 0.15 in B. In both cases qualitative agreement in terms of the shape of the SCCs (see insets) is found over the whole range of D and σ2, respectively. Parameters: γ = 1, μ = 20, βw = 1.5, τw = 1.5, τa = 10, Δ = 10, τη = 1 and resulting T* = 0.57.
Table 1.
Simulation parameters for the Traub-Miles model.
Fig 10.
Traub-Miles model with slow M current.
Panel A shows the membrane potential V(t) and the adaptation’s gating variable z(t). At t = 25ms a constant current I = 5μA/cm2 (red line panel) is applied so that the model undergoes a transition from the excitable to the tonic firing regime. Due to the slow build-up of the adaptation current the model shows a transient behavior where the firing-rate decreases until z(t) has reached its stationary value (doted line). The inset shows that z has two different phases, one during which z rapidly increases and another where z slowly decays. Panel B shows the deterministic limit cycle (dashed line) together with a noise trajectory of the tonically firing model with T* = 18.9ms. Parameters are as given in Table 1.
Fig 11.
Serial correlation coefficients for the Traub-Miles model with slow M current.
Panel A shows the Phase-response curve Z(τ) as well as the deterministic time-course of the membrane potential V(t) from which the PRC was obtained. Note that the PRC is always positive wherefore we expect to find SCCs that are qualitatively similar to those obtained from a LIF model. Panel B shows a comparison between numerically obtained and theoretically calculated SCCs. As in Fig 3 we compare specific and general SCCs ρk,a, ρk,η and ρk and find the same patterns as for a LIF model with likewise purely positive PRC.