Full population density model as a set of interacting Markovian systems.

We consider a network composed of N integrate-and-fire (IF) neurons, each receiving an input current I(t) resulting from the low-pass filtered linear combination of the spikes emitted at times t_j, with Poissonian distribution, by a subset of presynaptic cells in the network [9, 17]. For simplicity, here we assume a first-order dynamics for synaptic current I, with decay time τ_s and constant efficacy . The membrane potential V(t) of an IF neuron evolves according to the general equation , with membrane decay constant τ_m, neuron resistance R = τ_m/C_m, membrane capacitance C_m and a voltage-dependent leakage drift f(V), which depends on the model neuron type. For instance, f(V) is a constant drift for the perfect IF (PIF) neuron and f(V) = V for the leaky IF (LIF) [18, 19].

Neurons may receive an additional current I_ext modeling external sources, like incoming synaptic input from other networks. Once V crosses a threshold value v_thr, a spike is emitted and potential V is reset to v_res < v_thr (boundary conditions). Fig 1A shows a Poisson spike train (top panel) and the corresponding input current I(t) (middle panel) and membrane potential V(t) (bottom panel) for a LIF neuron with τ_s = 4 ms, τ_m = 10 ms and I_ext = 0.

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Fig 1. Spike input filtering by synaptic transmission.

(A, C) Presynaptic activity modeled by a Poissonian spike train (top) is low pass-filtered by the synaptic transmission to produce the input current I(t) (middle) to the membrane potential V(t) (bottom) of a leaky integrate-and-fire (LIF) neuron. Examples are shown for fast (A, τ_s = 4 ms) and slow (C, τ_s = 64 ms) synaptic filtering. (B, D) Stationary distribution in the plane (V, I) sampled from simulations of the same LIF neurons as in A and C with fast (B) and slow (D) synaptic transmission. Red arrows: mean fluxes of realizations (probability currents) at different (V, I). v_thr: absorbing barrier representing the spike emission threshold. v_res: reset membrane potential following the emission of a spike. Here I_ext = 0.

https://doi.org/10.1371/journal.pcbi.1007404.g001

Due to the boundary conditions, even under stationary regimes single-neuron dynamics is not analytically tractable. This is particularly apparent looking at the stationary probability distribution computed from the long time series partially shown in Fig 1A. The probability distribution of states (V, I) worked out numerically is shown in Fig 1B. The absorbing barrier in v_thr and the reentering flux of realizations in v_res modeling the emission of spikes make this distribution asymmetric [6, 8, 20] and correlation between V and I non-monotonic, a feature even more apparent for slower synaptic filtering, as shown in Fig 1C and 1D, obtained by increasing τ_s to 64 ms.

Under diffusion approximation, holding for large rate of incoming spikes each only mildly affecting V [18, 19], membrane potential dynamics is described by the following system of Langevin equations [2] (1) where RI_ext = μ_ext + σ_ext ξ_ext is a Gaussian white noise. We remark that the filtering effect of synaptic dynamics makes the noise colored, thus leading to finite correlation time constants. Moreover, V(t), individually, is not Markovian but V(t) and I(t), together, constitute a bi-variate Markov process [2].

Fig 2 provides an at-a-glance description of our method to obtain the final population dynamics and is used as a main reference throughout this section. The first intermediate Langevin model (1) corresponds to Fig 2A.

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Fig 2. Sketch of the main steps of the theoretical derivation.

(A) Diffusion approximation (right) of the single-neuron dynamics. (B) Probability density ρ(x, y) at fixed t, with the associated marginal distributions for both membrane potential (ρ_x, gray) and current (ρ_y, orange). Orange dashed line: instantaneous mean μ_y of the current (see main text for details). (C) “Slices” of the two-dimensional ρ(x, y) bounded by red dotted lines; each “slice” (which actually has infinitesimal thickness) corresponds to a one-dimensional subpopulation of neurons with similar y and is described by the expansion coefficients {a_n(y)} of a suited one-dimensional FP operator. (D) Partial firing rates ν_y (purple) are integrated over y to obtain the ν(t) of the whole network. (E) The instantaneous firing rate ν(t) is now the only state variable needed to be fed back in order to operate the moment closure and take into account only the “slice” centered around μ_y (the 0-th order approximation).

https://doi.org/10.1371/journal.pcbi.1007404.g002

When the mean driving force alone (i.e., the deterministic part of the input I + I_ext) is not enough to make the membrane potential V cross the threshold v_thr, the neurons are evolving in a noise-dominated (or subthreshold) regime and irregular firing occurs, due to the stochastic part of the input. In the opposite case, the emission of an action potential can occur also in the absence of noisy afferent currents, and the neurons are in a drift-dominated (or suprathreshold) regime of activity, characterized by the regular emission of spike trains [21–23].

The stochastic differential Eq (1) describe a single neuron where the time course of V(t) and I(t) is probabilistic. Therefore, the dynamics of V and I is fully described by the probability density ρ(V, I, t). Fig 1B and 1D are examples of distributions of (V, I) samples obtained from the numerical integration of the single-neuron dynamics under stationary conditions and before assuming the diffusion approximation. The probability density ρ(V, I, t) describes (under diffusion approximation) how this distribution evolves in time.

If we consider a network of statistically identical neurons, each with a different realization of the stochastic input, ρ(V, I, t) can be considered a population density. The population density approach [2, 24] provides a method to analyze the dynamics of ensembles of connected neurons, relating the firing properties of a network to the features of single neurons and of their synaptic connections. This is done by assuming the “mean-field” hypothesis, which allows to use the probability density for independent neurons in a “self-consistent” way. According to this theoretical ansatz, the infinitesimal moments of RI (2) characterize the white noise ξ driving the activity-dependent synaptic current [8, 25] as a function of the network firing rate ν(t) (i.e., the average number of spikes each neuron emits per time unit) and the average number of presynaptic contacts C (≤ N) per neuron. The probability density ρ(x, y, t) to find neurons at time t with membrane potential V(t) = x and synaptic current I(t) = y/R follows the FP equation [2] (see Fig 2B): (3) This is a continuity equation in which the density changes are determined by the divergence of the probability current (i.e., the flux of realizations) together with few specific boundary conditions (see Fig 1): i) spike emission as an absorbing barrier at x = v_thr (only for y > 0 if σ_ext = 0), ii) reentering flux of absorbed realizations at the reset potential v_res and iii) lower bound for the membrane potential implemented as a reflecting barrier at v_min ≤ v_res [8, 15, 26, 27].

In general, Eq (3) is hard to solve and only approximated solutions (mainly under stationary conditions) have been worked out [2, 6, 7, 28]. Here we derive a solution for this two-dimensional problem without making any assumption like stationarity in time and restricted ranges for the synaptic filtering timescale. We extend the spectral expansion approach exploited to study the one-dimensional case [15, 26], by describing ρ(x, y, t) as the superposition of an infinite set of one-dimensional interacting sub-populations; each sub-population collects the neurons that at time t receive a synaptic current y/R. Therefore we have iso-current sub-populations labeled by y and changing in time as the realizations move along the y-dimension due to the action of . In other words, we “slice” the two-dimensional diagrams of the probability current corresponding to ρ(x, y, t) along the y direction, thus obtaining probability densities , as sketched in Fig 2C.

The dynamics of each iso-current sub-population is studied, as in [15], by projecting ρ at fixed y on the eigenfunctions {|ϕ_m〉} of a suited one-dimensional FP operator, which here we set to be (4) with an additional diffusion term σ²(y) with respect to of Eq (3), which is the variance of an arbitrary Gaussian white noise depending on the state variable y.

As shown later, this additional term will play a crucial role in our dimensional reduction. Indeed, σ²(y) will be chosen self-consistently to take into account the statistical variability of the input currents received by all the iso-current subpopulations. This unconventional choice allows us to use an expansion in eigenfunctions of this FP operator, with coefficients , which now explicitly depend on the term y and where 〈ψ_n| is the n-th eigenfunction of the adjoint operator of . The stationary mode with eigenvalue λ₀ = 0 has 〈ψ₀| = 1, thus a₀(y, t) is the marginal probability density (Fig 2, orange distribution).

The dynamics of a_n can be worked out by deriving a_n with respect to time [15, 26], and rewriting : (5) where |ϕ_q〉 and 〈ψ_n| are the eigenfunctions of the operator and of its adjoint, respectively. For n = 0, the dynamics of ρ_y(y, t) = a₀(y, t) can be worked out, as both and {a_q} do not depend on x: The integral on the r.h.s. can be solved by parts and turns out to be 0 thanks to both the conservation of the flux of realizations exiting from v_thr and re-entering in v_res () and the reflecting barrier condition in v_min, provided that the reasonable assumption ϕ₀(v_min, y) = 0 holds (this is surely true for v_min → −∞, as ϕ₀ is a probability density). Thus, the ρ_y dynamics reduces to (6) Note that this expression can be derived directly from the FP Eq (3).

Relying on the same spectral expansion as in [15], each ‘partial’ rate (Fig 2, purple distribution) can be decomposed as follows: (7) where the stationary mode contribution is separated from the others. The network firing rate ν(t) is obtained by integrating the partial rates ν_y(y, t) over y, as sketched in Fig 2D: (8)

In this framework, is the firing rate of the stationary one-dimensional density ϕ₀ at fixed y, while the other nonstationary modes contribute with the fluxes . Both Φ and f_n depend on the total firing rate ν(t) through the input current moments μ_I and σ_I. The main advantage of having transformed the original two-dimensional description into a set of infinite one-dimensional FP equations is that the dynamics described by these equations for each iso-frequency population is known to be tractable for a wide range of network settings. This is crucial to derive an approximated dynamics of ν(t) valid for any synaptic time scale τ_s, neuron model and dynamical regime, as shown in the following.

Dimensional reduction of the firing rate dynamics.

Eqs (7) and (8) together are a reformulation of the original problem (3), not a solution. However, in this framework a perturbative approach can be envisaged. From Eq (6), synaptic current y results to have time-dependent mean μ_y(t) = 〈y〉 and variance (9) like an Ornstein-Uhlenbeck process with nonstationary input moments μ_I(t) and σ_I(t) [2]. We remark that should not be confused with the arbitrary and additional diffusion term σ²(y) introduced in the definition of : is the variance of the synaptic current y, whereas σ²(y) is the state-dependent variance of an unknown diffusion process included in , which will be chosen later to obtain a self-consistent equation for the firing rate ν(t).

Synaptic current displacement in time is and, for small σ_I,—i.e., for σ_I ≪ v_thr – we can assume a negligible role for a y distant enough from μ_y. This assumption of a narrow ρ_y centered around y = μ_y allows to simplify the integrals across the y domain weighted by the expansion coefficients {a_n(y, t)}. Indeed, approximations of these integrals can be obtained by expanding in Taylor’s series the y-dependent functions in the integrands, as detailed in the S1 Appendix and sketched in Fig 2E.

The next step is to find a self-consistent approximation (neglecting terms of order ) of ν(t) in terms of 0-th order functions Φ₀(ν) ≡ Φ(μ_y, ν) and f_0n(ν) ≡ f_n(μ_y, ν) and of corresponding coefficients: (10)

The dynamics of the n-th expansion coefficient is ruled by the following equation, neglecting again terms of order : (11) where λ_0n ≡ λ_n(μ_y) (see details in the S1 Appendix).

This is a dimensional reduction of Eq (5) as the coefficients do not depend on y, and it holds provided that y has a narrow distribution across neurons at each time t (i.e., small σ_I(t)), leaving τ_s unconstrained.

To obtain a self-consistent equation for the firing rate ν(t), we still need to choose a suited σ(y) in . To this purpose, the case of instantaneous synaptic transmission τ_s = 0 can be used as reference, as Eq (3) reduces to a one-dimensional FP equation with the only operator [15, 21, 27]. Hence, must tend to for vanishing τ_s, which is the case if σ²(y) = Jy. Indeed, in this limit , being μ_y = μ_I = τ_m J C ν from Eq (9). This implies that eigenvalues λ_0n and eigenfunctions in Eq (11) are the same as in the one-dimensional case with δ-correlated synaptic input, but with collective firing rate ν(t) = μ_y(t)/(τ_m JC).

To summarize, is the variance of the white noise driving the synaptic current and depends on ν(t) (see Eq (2)); is the variance of the Gaussian white noise representing the external input (see Eq (1)); is the variance of the synaptic current y (see Eq (9)); σ²(y) = Jy is the variance of the diffusion process appearing in (see Eq (4)), which is different for each iso-current “slice” being dependent on y. However, in the limit τ_s → 0 (where the decay time τ_s is the relaxation time of μ_y(t), see Eq (9)), the synaptic transmission becomes instantaneous and , differently from the variance σ²(y) which in the relevant iso-current “slice” at y = μ_y is .

In conclusion, this dimensional reduction extends the firing rate equation previously worked out for τ_s = 0 [15], and, by combining Eqs (9)–(11) and neglecting terms, it results to be (12) Here, the infinite vectors and are introduced together with the eigenvalue matrix Λ₀ = diag(λ_0n). Separating stationary from non-stationary modes, the synaptic coupling vector and matrix are also included, respectively. For the chosen σ(y), we remark that all these terms depend on the total firing rate ν(t) only through μ_y(t).

As in [15], Eq (12) expresses the firing rate ν(t) as a superposition of contributions due to both the stationary mode for a given μ_y(t) (i.e., the input-output gain function Φ₀), and the firing rate due to non-stationary modes () depicting how far the actual density is from the equilibrium. As a result, the farther the system from the equilibrium, the larger the contribution to ν(t) due to the non-stationary modes. Indeed, under these conditions the expansion coefficients are significantly different from 0, whereas under stationary conditions and ν = Φ₀.

The first row of Eq (12) is an infinite set of equations for the infinite expansion coefficients {a_0n}. It describes their relaxation dynamics towards an equilibrium point which can change with time, as the stationary one-dimensional density ϕ₀ depends on μ_y(t). Time scales of this relaxation are dictated by the eigenvalues Λ₀ and the coupling coefficients W₀, the latter being 0 as the synaptic efficacy J vanishes. Also Φ₀, , Λ₀ and W₀ depend on μ_y(t) and on the other changes in the input current received by the neurons. This means that Eq (12) incorporates both slow and fast time scales. The former are due to the slowest a_0n relaxation or to the synaptic low-pass filtering for large τ_s, as pointed out by the second row of Eq (12). On the other hand, fast time scales can be due to the fast changes of the input affecting directly the aforementioned equation coefficients, such as the gain function Φ₀. This multiscale nature of Eq (12) makes this theoretical representation of general applicability, as shown later.

Eq (12) is one of our main results, obtained following a procedure that can be summarized as follows (see Fig 2). We described the single-neuron dynamics as a two-dimensional Langevin Eq (1), relying on the diffusion approximation. From this, an extended mean-field approximation [25] led us to derive Eq (3), a two-dimensional continuity equation for the dynamics of the population density ρ(x, y, t). We then represented ρ as the time evolution of its projections onto the axes defined by the eigenfunctions of the FP operator, as shown in Eqs (7) and (8) – an approach similar to the Hartree-Fock method in quantum mechanics but with the main difference that here we take into account also the temporal dimension. Finally, ρ(x, y, t) is approximated assuming y ≃ μ_y(t), which is a low-pass filtered version of μ_I(t). This allows us to operate a central moment closure focusing on the instantaneous firing rate ν(t). We remark that this dimensional reduction aims at finding an effective model of the firing rate ν(t), not of the full probability density ρ(x, y, t). This means we are not expecting to recover accurately the full dynamics of the neuronal membrane potential V(t), as detailed later.

Equivalence of non-instantaneous synaptic transmission and distribution of axonal delays.

So far, we considered the filtering activity operated by local synapses transmitting incoming spikes as a post-synaptic potential non-instantaneous in time. Here, we recall the known dynamics of a network of spiking neurons where synaptic transmission is instantaneous but a distribution of axonal delays is taken into account [15, 27, 29]. In this one-dimensional case (τ_s = 0), the network dynamics can be simply worked out by replacing ν in the expressions of μ_I and (see Eq (2)) with the instantaneous rate of spikes received by neurons when a distribution ρ_d(δ) of axonal transmission delays δ is taken into account. Thus, the population density ρ_x(x, t) follows the one-dimensional FP equation (13) with instantaneous firing rate .

The spectral expansion of this continuity equation leads to a known firing rate equation with coefficients resulting to have a straightforward relationship with those in Eq (12). Indeed, the elements of the synaptic coupling vector and matrix now are for any n and m, since for what shown above. Due to this, the searched firing rate equation equivalent to the 1D FP Eq (13) reduces to (14) where the specific delay distribution (15) has been taken into account. This leads to be a version of the collective firing rate ν smoothed in time by a first-order low-pass filter with decay time τ_d. Θ(δ) is the Heaviside function, as only positive transmission delays are admitted.

A comparison between Eqs (12) and (14) remarkably points out the equivalence of having, in a spiking neuron network, a non-instantaneous synaptic transmission or a suited distribution of axonal delays. In Eq (14) the expansion coefficients in refer to the one-dimensional density ρ_x(x, t), while in Eq (12) is only an effective representation of the two-dimensional density ρ(x, y, t). Nevertheless, provided that τ_s = τ_d, both the dynamics seen from the perspective of ν(t) are the same, as the functions Φ₀, , , W₀ and Λ₀ are the same. In other words, under mean-field approximation and for not too large synaptic current fluctuations σ_I, having a local synaptic filtering with cut-off frequency 1/τ_s is equivalent to having random axonal delays with exponential distribution (15) and decay constant τ_d = τ_s. Although such equivalence might appear as not surprising, to the best of our knowledge this is the first proof of its general validity, i.e., holding for any IF neuron model, dynamical regime and synaptic timescale.

Match between input-output gain functions.

As a first evaluation of the effectiveness of the low-dimensional description derived above, we inspect the simplest condition given by the asymptotic firing rate of isolated neurons with or without a synaptic filtering of the incoming spike trains. In the absence of recurrent connectivity and under stationary condition, from Eq (12) the firing rate ν is given for any τ_s by the single-neuron input-output gain function Φ₀(μ, σ). In Fig 3, we show the results of this test by comparing the steady-state firing rates of LIF neurons measured from single-neuron simulations and their theoretical input-output gain function Φ₀(μ, σ) [27, 33, 34]: (16) where μ = μ_I + τ_mI_ext/C_m is the total mean input current and is the total variance. Each curve has been obtained by changing the rate ν of the external source of Poissonian spikes, such that both μ_I and σ_I change simultaneously, according to Eq (2).

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Fig 3. Match between theory and simulations in the input-output gain function of LIF neurons.

Gain functions Φ for N = 4000 LIF neurons (uncoupled and coupled), obtained by varying the total mean current μ = μ_I + τ_mI_ext/C_m for different τ_s and different size σ_I of the synaptic current fluctuations: (A) mV and (B) mV. In the explored range of μ, σ_I varies between 74% and 138% of its value at μ = v_thr. Left panels in (A-B): theoretical Φ₀ from 0-th order approximation (τ_s = 0, gray dashed curves) and measured for τ_s ∈ {0, 1, 4, 16} ms (reddish curves) from single-neuron simulations. Central and right panels in (A-B) show the differences , either including (right) or excluding (middle) an additional unfiltered white noise with . Only the three curves with τ_s > 0 are shown as the simulations with τ_s = 0 largely overlap the x-axis due to the remarkable agreement with the theoretical Φ₀. (C) Gain functions versus the mean external current μ_ext for a network of excitatory neurons. Network topology and neuron parameters as in Panel (B) with σ_ext = 2.0 mV, but with 50% of recurrent synapses (the others receive Poissonian spike trains as in (B)). Thick arrows: values of μ_ext = {15.2, 17.5, 20.5} mV driving the network from noise- to drift-dominated regimes, respectively. Bottom panel, steady-state firing rates (fulfilling the equilibrium condition Φ₀(ν*) = ν*) of the network with non-instantaneous synaptic transmission (blue plots) and distribution of axonal delays (red plots) at the highlighted μ_ext. Shaded strips in all panels: mean ± SEM from 10 independent simulations (see Methods for further details and parameters).

https://doi.org/10.1371/journal.pcbi.1007404.g003

The 0-th approximation provided by Eq (16) (gray dashed curves in Fig 3-left) perfectly overlaps with the simulation of a single LIF neuron incorporating instantaneous synaptic transmission (τ_s = 0, red curves), confirming the validity of the diffusion approximation. However, the output firing rates at fixed input μ change with increasing τ_s, as described in [8, 11]. Such discrepancy can be better appreciated from the central panels of Fig 3, where the differences ΔΦ between Φ₀ and the gain functions from simulations are shown to mainly increase with τ_s. This could seem in contradiction with the fact that our low-dimensional reduction is independent of the synaptic filtering timescales arising in the steady-state firing rate of isolated neurons, but it is not. Indeed, by comparing two examples at relatively small and large size σ_I of the synaptic current fluctuations (Fig 3A and 3B, respectively), the equivalence is recovered as long as σ_I ≪ v_thr, according to the assumption at the basis of our 0-th order approximation.

Note that the largest discrepancies are found in a relatively small range of the mean current μ around v_thr, where synaptic fluctuations have a major role in the spike emission process and this range shrinks with decreasing σ_I. Outside this range of μ, our 0-th order approximation is still valid to describe the steady-state firing rate of single-neurons both under noise- and drift-dominated regime.

Finally, we remark that the discrepancy between theory and simulations can be further reduced by adding the white noise with size σ_ext > 0. We introduced this additional input in the limit σ_ext → 0 in Eq (4) to safely manage the boundary conditions in the spectral expansion of the one-dimensional operator . However, it has been suggested to effectively model several neuronal features like the ionic channel stochasticity [35, 36] and the effect of thermal noise [37]. As a result, under this biologically plausible condition, the match between theoretical input-output gain function and simulations is recovered also for relatively large σ_I (Fig 3B-right).

As a second test, we consider the same network as above, with the same total number of average synaptic inputs per neuron, but with 50% of synaptic contacts from external Poisson-like excitatory neurons and 50% of recurrent synapses. Under stationary condition, we obtain results similar to those observed in the absence of recurrent connectivity: Fig 3C shows the gain functions for this network, plotted versus the mean external current μ_ext (upper panel) and the steady-state firing rates of the network, satisfying the equilibrium condition Φ(ν*) = ν* [25], plotted versus τ_s = τ_d (lower panel). The abscissa μ_ext is the sum of two mean currents: the white noise with drift and the exogenous Poissonian input. Notice that the average firing rate ν₀ when delay distributions are incorporated in simulation (red curves in lower panel) does not change by varying τ_d. This is an expected result, as under stationary conditions the input firing rate in Eq (14) coincides with the equilibrium value independently of τ_d. It is also apparent that the mean-field equilibrium point ν* (dashed lines) mildly overestimates the simulated equilibrium point ν₀ when both a delay distribution (red lines) or synaptic filters (blue lines) are incorporated. This is due to the assumptions underlying the diffusion and the mean-field approximations. Increasing the number of synaptic contacts per neuron and the network size N by keeping unchanged drift and diffusion coefficients of the related Langevin equation leads to reduce the difference |ν₀ − ν*|. Finally, due to the changes in the input-output gain function Φ₀ shown in the top panel, the measured mean firing rates ν₀ in the network with non-instantaneous synaptic transmission are different from those with distribution of axonal delays, and the difference increases with τ_s = τ_d [8, 10, 11]. This trend is particularly apparent around the critical value μ = v_thr (see the case μ_ext = 17.5 mV in the bottom panel), where our 0-th order perturbative approach is less accurate unless σ_I ≪ v_thr. We remark that the equivalence between delay distribution and synaptic filtering holds in both noise- (subthreshold) and drift-dominated (suprathreshold) regimes, and it is less accurate only at the transition between these two conditions.

Responses to a time-varying input.

In the comparison between steady-state firing rates, neither delay distribution nor out-of-equilibrium conditions have been explicitly tested as the input-output gain function is a single-neuron feature and the adopted input currents had stationary mean and variance.

To overcome this limited testing condition, here we further test the first-order statistics of the firing rate, by simulating the network of LIF neurons shown in Fig 7A with a time-varying input (Fig 4A). To this purpose, the rate of external spikes ν_ext(t) was modulated as a periodic wave with period T = 200 ms. Average responses of the firing rate ν(t) across a stimulation period are shown in Fig 4B for networks with the same mean-field parameters but different time scales τ_d = τ_s. Remarkably, in all conditions, the average firing rate of each network with synaptic filters (blue curves) widely overlaps with the one computed in the equivalent network incorporating a suited distribution of transmission delays (red curves) and also with the numerical integration (using the Python library described in [38]) of the one-dimensional FP Eq (13) equivalent to Eqs (12) and (14). We remark that very similar results can be obtained by using only the two slowest modes of the FP operator spectrum.

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Fig 4. Network response to a periodic step-wise input.

(A) Example of response of a network of LIF neurons to periodic modulation of ν_ext(t) = (1 + Δν_ext erf(2 sin(2πt/T))) 40 Hz with T = 200 ms and Δν_ext = 0.05 (top). Spiking activity and firing rate ν(t) (middle and bottom, respectively). Same network as in Fig 7A at drift-dominated regime with instantaneous synaptic transmission and a distribution of axonal delays with τ_d = 1 ms. (B) Average responses to a stimulation period (200 ms) of (blue curves) the same LIF neuron network with synaptic filters at different timescales τ_s = 1, 4, 16, 64 ms and (red curves) the equivalent networks with distributions of delays (τ_d = 1, 4, 16, 64 ms). White and gray shaded intervals mark the time when neurons in the network work in drift- (μ_I + μ_ext > v_thr) and noise-dominated (μ_I + μ_ext < v_thr) regime, respectively. Averages are computed across 300 periods, removing a first transient response (1 s). A grand average of this ν(t) is eventually computed across 10 networks with same mean-field parameters but randomly different connectivity matrices. Gray dashed curves: theoretical ν(t) resulting from the numerical integration of the one-dimensional FP Eq (13).

https://doi.org/10.1371/journal.pcbi.1007404.g004

The similarity of the response in the theoretical ν(t) and the two simulated networks is apparent for any τ_s = τ_d ranging from 1ms to 64 ms, during both the fast transient elicited by the sudden changes of ν_ext (at t = 0 and 100 ms), and the relaxation phases when the networks drift towards a new equilibrium point. The latter effect is not a surprise, being due to the expected low-pass filtering. However, notice the reduction of the time lag in crossing 40 Hz (firing rate of the unperturbed networks) as τ_d = τ_s increases. For networks with synaptic filtering, when the input suddenly changes, the non-instantaneous synaptic transmission leads to have a uniform shift of the probability density ρ_x(x, t) towards the emission threshold (towards right in Fig 1). As a consequence, firing rates can display arbitrarily fast reactions, which are even faster as τ_s increases [6, 9]. Conversely, networks with transmission delays do not express the same mechanism, as they have vanishing probability densities for V → v_thr. Here, the coexistence of slow and fast dynamics is likely due to the fact that, for τ_d long enough compared to T/2, ρ_x(x, t) has not enough time to approach its asymptotic value. As a consequence, just before the input has a step transition, a fraction of neurons (those with the longest pre-synaptic transmission delays) has still memory of the previous stimulation phase. This subset of neurons, whose size increases with τ_d, are the first to be primed, thus leading the whole network to rapidly react to ν_ext changes.

We remark that, as ν(t) changes across the stimulation period, not only μ_I, but also changes accordingly. This notwithstanding, the match of the three curves is remarkably good under both drift- and noise-dominated regimes (white and gray shaded intervals in Fig 4B, respectively), witnessing the quite general validity of the proposed approach.

Equivalence in the asynchronous state of finite-size networks.

As the equivalence between synaptic filter and delay distribution holds for the mean ν(t), here we evaluate the equivalence validity also for the second-order statistics of the firing rate around a stable equilibrium point. To this purpose, we inspect the activity in the presence of an endogenous noise in a network of a finite number N of neurons trapped into a stable asynchronous state. As previously done in [15], finite-size fluctuations can be incorporated into Eqs (12) and (14) as an additive forcing term to the expansion coefficient dynamics, giving rise to an equation for the firing rate ν_N(t) of a finite pool of neurons (see Methods for details). The resulting dynamics can be linearized around the equilibrium point ν* = Φ₀(ν*). This allows to compute the Fourier transform ν_N(ω), and from it the power spectral density P(ω) = |ν_N(ω)|², which turns out to be (17) Here, I is the identity matrix, all the coefficients depending on ν(t) are now constants computed at ν(t) = ν*, and r(iω) is the ratio between the Fourier transforms of μ_y and μ_I, which results to be r(iω) = 1/(1 + iωτ_s). Note that r(iω) coincides with the Fourier transform ρ_d(iω) of the delay distribution (15) with τ_d = τ_s. If we consider an additional fixed transmission delay δ, this transform generalizes to r(iω) = exp(−iωδ)/(1 + iωτ_s) [15, 29].

We compared this theoretical result with the P(ω) estimated from simulations of networks composed of simple IF neurons with synaptic filtering (see Methods for details), finding a remarkable agreement (Fig 5). Here we resorted to the VIF neuron model [21], an extended version of the widely used PIF neuron [39], for its amenability to analytical treatment [15] (see Methods for details).

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Fig 5. Match between theory and simulations in networks of VIF neurons working in drift-dominated regime with different synaptic decay times τ_s.

Power spectral densities P(ω) of the finite-size network firing rate ν_N(t) are shown for τ_s = {0, 4, 32} ms. All networks have a stable equilibrium point at ν* = 10 Hz, and are composed of N = 2000 excitatory VIF neurons. Synaptic matrices are random with connection probability C/N = 5% and average synaptic efficacy J = 7.510⁻³ v_thr. Spikes are delivered to the postsynaptic targets with a fixed transmission delay δ = 10 ms (see Table 1 for other parameters). Solid lines: theoretical P(ω) from Eq (17) computed relying on the first 4096 modes of the FP operator spectral expansion. Shaded strips: mean ±3 SEM (standard error mean) of P(ω) from 10 simulations with different random synaptic matrices and same mean-field parameters. Power spectra are normalized by N/ν₀, where ν₀ is the average in time of ν_N(t) and ν₀ = ν* for theoretical P(ω).

https://doi.org/10.1371/journal.pcbi.1007404.g005

As stated above, the theoretical power spectral density has an equilibrium point ν*. On the other side, the microscopic simulations provide a firing rate ν_N(t) whose mean value (averaged over 10 simulations with different random synaptic matrices and same mean-field parameters) is ν₀. Each power spectral density plot provides information about the frequencies at which the ν_N(t) variations are stronger.

The theoretical P(ω) from Eq (17) is computed on the first 4096 modes/eigenfunctions of the FP operator. The large number of modes is justified by the need of capturing the network behavior over a wide frequency range, characterized by the presence of many narrow resonant peaks.

Besides the good matching between theory (solid lines) and simulations (shaded strips), it is interesting to note how resonant peaks are differently affected by increasing τ_s. Not surprisingly, high-ω peaks are broadened due to the low-pass filtering of synaptic transmission. These resonances are related to the synaptic reverberation of spiking activity, which gives rise to the so-called transmission poles in the linearized dynamics [15, 40]. In an excitatory network working in drift-dominated regime as in Fig 5, these are expected to be found at frequencies multiple of the inverse of the average time needed by a presynaptic spike to affect postsynaptic potential, here roughly 1/(δ + τ_s). On the other hand, the low-ω peaks do not display any shift in frequency, although their power is broadened as well. This is because such peaks are due to the so-called diffusion poles in the linearized dynamics [15, 41]. They occur when neurons emit spikes at drift-dominated (suprathreshold) regime. In this regime, the distribution of the inter-spike intervals is narrow and resonant peaks emerge at ω/2π multiples of ν* (equal to 10 Hz in Fig 5). We remark that, due to the shifting at low-ω of the transmission peaks, a constructive interference with diffusion peaks may occur. This explains the non-monotonic change of power of the second diffusion peak at 20 Hz in Fig 5. Indeed, its power decreases when τ_s is increased from 0 ms to 4 ms, as expected, whereas due to the mentioned interference it is heightened for τ_s = 32 ms. This is because the lowest transmission peak in this case is expected to be found at ω/2π = 23.8 Hz, not too far from 2ν* = 20 Hz.

Equivalence under noise- and drift-dominated regime.

To further test the equivalence, we investigated whether this match worked well for networks of neurons not only under drift-dominated spiking regime, as shown in Fig 5, but also under noise-driven regimes. Indeed, synaptic filtering may have significantly different effects on the steady-state firing rate in these two regimes [9, 11], as shown in Fig 3.

Fig 6A shows for comparison the power spectral densities P(ω) of the firing rate ν(t) from an excitatory VIF neuron network working under noise-dominated regime by varying τ_s(= τ_d), in which either non-instantaneous synaptic transmission (blue curves) or a distribution of transmission delays (red curves) was incorporated. The theoretical P(ω) from Eq (17) (gray dashed curves) overlaps the simulation results for a wide range of ω. Notice that we used the first 256 modes of the FP operator spectrum, i.e., 16 times less modes than for the drift-dominated case: this points out the greater effectiveness of our approach in the noise-dominated regime. It is also important to remark that the discrepancy observable at low-ω is not due to a failure of our approximation, as it occurs also at τ_s = τ_d = 0 ms, when synaptic transmission is instantaneous. This indeed is related to the mean-field approximation, which overestimates the firing rate of the equilibrium point, as explained more in detail later in this subsection.

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Fig 6. Equivalence of non-instantaneous synaptic transmission and distribution of synaptic delays in a network of VIF neurons.

(A) Power spectral densities P(ω) of ν(t) obtained for τ_s(= τ_d) = {0, 4, 32} ms in an excitatory VIF neuron networks under noise-dominated regime with either non-instantaneous synaptic transmission (blue) or distribution of transmission delays (red). Gray dashed curves: theoretical P(ω) from Eq (17) computed relying on the first 256 modes of the FP operator spectral expansion. (B) Power spectral density P_s(ω) of the simulated network activity in the case of non-instantaneous synaptic transmission, varying the synaptic time constant τ_s. Dashed lines: P(ω) sections plotted in Fig 5 and in panel A. (C) P_d(ω) for networks with an exponential distribution of spike transmission delays and with instantaneous synaptic transmission, varying the delay time constant τ_d. In B and C, top and bottom panels show the results for the network set in an asynchronous state in which neurons are in a drift- (top) and noise-dominated (bottom) regime, respectively. (D) Iso-power curves from panels B (solid lines) and C (dashed lines). (E) Average firing rate ν₀ from the simulations used for the left panels with non-instantaneous synaptic transmission (blue plots) and distribution of axonal delays (red plots). Error bars representing SEM are not visible. The not specified network parameters are as in Fig 5 (see Table 1). Also here, power spectra are averages across 10 simulations with different random synaptic matrices and same mean-field parameters.

https://doi.org/10.1371/journal.pcbi.1007404.g006

Fig 6B (non-instantaneous synaptic transmission) and Fig 6C (distribution of transmission delays) show the same comparisons for a larger set of τ_s = τ_d values (in logarithmic scale), corresponding to networks working in noise- (bottom panels) or drift-dominated (top panels) regimes; in this case, the power spectrum amplitude is color-coded. The power spectral densities in panel A correspond to the cuts marked by dashed horizontal lines in (bottom) panels B and C. A remarkable agreement between simulations in the two tested conditions is confirmed by Fig 6D, showing iso-power curves from panels B (solid lines) and C (dashed lines): the equivalence of non-instantaneous synaptic transmission and distribution of axonal delays is witnessed by the tight overlapping of solid and dashed iso-power curves for a wide range of filter timescales.

The same good matching between these two network types is apparent also under noise-dominated regime (bottom panels). As expected in this regime, the diffusion poles become real numbers [15] and resonant peaks at frequencies multiple (by ω/2π) of the equilibrium firing rate ν* disappear, although the high-ω resonances corresponding to transmission poles remain almost unaffected by this change of regime. We point out that widening the filtering time window (i.e., increasing τ_s and τ_d) leads to almost flat power spectra (see Fig 5), compatibly with the flattening of the response amplitude of isolated neurons receiving colored noise as input currents [9, 10].

As shown in Fig 1B–1D, the section of ρ(x, y, t) close to firing threshold v_thr shrinks when τ_s increases from 4 ms to 64 ms. In our perturbative approach this dependence is completely neglected, as the distribution is assumed to be a Dirac’s δ.

We remark that the shown comparisons rely on normalized spectra, i.e. P(ω)N/ν₀. On the one hand, this means that power spectra shapes do not depend on the particular transmission protocol implemented in the network. On the other hand, this does not guarantee, as shown for the LIF neuron case, that average firing rates ν₀ are the same and do not change with the protocol-related timescales. To address this issue, in Fig 6E the measured ν₀ are compared, showing features completely similar to those observed in Fig 3 for the LIF neuron model.

Notice that the discrepancy between theory and simulations (ν₀ and ν*, respectively) explains the differences found at low-ω in Fig 6A. Indeed, such difference leads to have different slopes of the gain function (), which in turn affect differently the power spectrum at low-ω as .

Since we already checked the accuracy of the theoretical P(ω), in the rest of the paper we will focus on the power spectral densities estimated from microscopic simulations. Taking as reference these simulations instead of Eq (14) in the comparison with the supposedly equivalent networks with synaptic filtering has indeed the advantage of allowing to test our theoretical prediction also in neuron models for which analytical expressions for the eigenvalues and eigenfunctions of the FP operator are not available, as in the case of exponential IF (EIF, [42]) neurons. Moreover, this strategy allowed us to overcome the possible issue related to computing a large number of modes of the FP operator spectrum for more realistic single-cell models.

Independence from spiking neuron models.

The developed theory is of general applicability, i.e. it applies to a wide range of spiking neuron models. As a result, the equivalence proved for a network of simplified VIF neurons together with the theoretical expression for the power spectra of ν(t) are both expected to hold also for networks of more realistic single-cell models like the LIF and the EIF neurons.

To verify this expectation, we directly simulated networks of LIF and EIF neurons with a mean-field stable equilibrium point at ν* = 40 Hz. In Fig 7 the mean P(ω)N/ν₀ for these kinds of networks is compared both under drift-dominated (top panels) and noise-dominated (bottom) regimes. A remarkable agreement is apparent for both LIF and EIF neuron networks (Fig 7A and 7B, respectively), confirming the generality of the developed approach. To further test the absence of any bias due to the specific neuron model chosen, we computed the relative differences between the power spectral densities when non-instantaneous synaptic filtering (P_s) and transmission delay distribution (P_d) are taken into account. The cumulative distributions of these discrepancies across all tested time scales (τ_s = τ_d) and Fourier frequencies (ω/2π) are shown in Fig 7C. Interestingly, no significant differences are visible for the three chosen models (VIF, LIF, EIF), further proving that in all regimes (drift- and noise-dominated) our dimensional reduction does not depend on the specific single neuron dynamics.

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Fig 7. Equivalence between synaptic filtering and delay distribution in networks with different types of IF neurons.

Match between relative power spectral densities P(ω)N/ν₀ in networks of leaky (A) and exponential (B) integrate-and-fire excitatory neurons (LIF and EIF, respectively) with non-instantaneous synaptic transmission (solid lines) and distributions of synaptic delays (dashed lines). Network parameters are set in order to have an asynchronous state with average firing rate ν₀ = 40 Hz. Contour lines as in Fig 6D. P(ω) are averages across 10 simulations with different random synaptic matrices and same mean-field parameters. (C) Cumulative distribution of relative differences ΔP(ω) = P_d(ω)/P_s(ω) − 1 across different time scales (τ_s and τ_d) and Fourier frequencies ω from panels A and B for LIF (red) and EIF (orange) neuron networks, respectively. As a reference, also the same cumulative distribution for VIF neuron networks (obtained from P(ω) in Fig 6D) is plotted (black). ΔP(ω) are measured as z-scores, i.e. averages divided by standard deviations of values across simulations. Top and bottom panels: networks working under drift-dominated and noise-dominated regime, respectively (see parameters in Tables 2 and 3).

https://doi.org/10.1371/journal.pcbi.1007404.g007

Starting from this, it is not surprising to note here that spectral features similar to those highlighted in Fig 6 for VIF neuron networks are also displayed by LIF and EIF neuron networks. More specifically, resonant peaks at multiple frequencies of ν₀ under drift-dominated regimes and resonances at higher-ω due to the transmission poles—i.e., those related to the average spike transmission delay—are also visible in both Fig 7A and 7B in LIF and EIF neuron networks, respectively.

Equivalence beyond the asynchronous state.

So far, we tested the validity of the equivalence between non-instantaneous synaptic transmission and delay inhomogeneity in linearizable dynamical regimes like the asynchronous state and in purely excitatory networks. To further assess the generality of our theoretical results, we simulated a multi-modular network containing an inhibitory (I) and an excitatory (E) population of LIF neurons (Fig 8A, see Methods), with the purpose of analyzing its dynamical behaviors by changing a bifurcation parameter.

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Fig 8. Comparison between multi-modular networks of inhibitory and excitatory LIF neurons with either synaptic filtering or transmission delay distribution.

(A) Sketch of the multi-modular network. (B) Three examples of firing rate time evolution for each network stable state. Plum and orange curves represent inhibitory and excitatory neuronal populations, respectively, with synaptic filter. (C) Left plots: mean (ν_α0, α = E, I) and standard deviation (σ_ν) of the instantaneous firing rate across time (binning size of 0.1 ms, discarding the first second of the time series) for inhibitory (bottom panels) and excitatory (top panels) populations. Right plots: match between relative power spectral densities P(ω)N/ν_α0 for excitatory (top panel) and inhibitory (bottom panel) populations with non-instantaneous synaptic transmission (solid lines) and distributions of transmission delays (dashed lines). Contour lines as in Fig 6D. Different states are obtained by increasing the connection probability with pre-synaptic inhibitory neurons, namely c_αI ≡ c_EI = c_II (see text for details). The spectra P(ω) are averages across 10 simulations with different random synaptic matrix realizations. Dashed horizontal lines roughly mark the bifurcations. Rightmost arrows point out the values of c_αI corresponding to the examples displayed in panels B and D. (D) Comparisons between relative spectra from networks with delay distribution (red lines) and synaptic filtering (blue lines) for the three network states AL (dotted lines), AH (dashed-dotted lines), and GO (solid lines), for excitatory (top panel) and inhibitory (bottom panel) populations.

https://doi.org/10.1371/journal.pcbi.1007404.g008

In this specific example network, the bifurcation parameter is the percentage c_EI = c_II ≡ c_αI of outward inhibitory synaptic connections per neuron. To compensate for these changes in the inhibitory component, the synaptic efficacies of both recurrent and external excitatory neurons are progressively increased: this allows to keep an equilibrium point ν* = Φ₀(ν*) at the same relatively high firing rate Hz for both populations, regardless of its stability (see details in Methods). For low enough c_αI, this equilibrium point is unstable and only an asynchronous irregular state at low firing rate (denoted as AL) can be reached by both populations, as shown in Fig 8B-bottom for the E-I network with c_αI = 10% and non-instantaneous synaptic transmission. A suited increase of c_αI stabilizes the high-frequency asynchronous state (denoted as AH), in which the neurons fire at the equilibrium rate of 10 Hz (Fig 8B-middle, c_αI = 50%). A stronger synaptic inhibition eventually leads the E-I network to display synchronous firing with coherent global oscillations (GO) of the spiking activity (Fig 8B-top, c_αI = 90%).

We compared networks with both synaptic filtering and delay distribution across this repertoire of dynamical regimes, by making extensive simulations on a finer grid of c_αI values, by using the same mean-field parameters (Fig 8C). In all cases the networks undergo the same state transitions at the same critical c_αI values. More specifically, the value c_αI ≈ 12.5% marks a transcritical bifurcation where the low-frequency and the 10-Hz equilibrium points exchange their stability and the transition from AL to AH occurs. In both networks, the mean firing rate ν₀ of the AL state increases with its standard deviation σ_ν (Fig 8C-left) until the 10-Hz equilibrium point corresponding to the AH state becomes the lowest stable equilibrium point. This point undergoes a supercritical Hopf bifurcation at c_αI ≈ 70%, where the network has a transition from the AH to the GO state. The resulting synchronous oscillations of the firing rate have an amplitude that increases with c_αI. This is why σ_ν increases and ν₀ remains unchanged at 10 Hz. Even in this case, the first two moments of the firing rate in networks with both non-instantaneous synaptic transmission (blue) and distribution of axonal delays (red) match almost perfectly.

This close resemblance is also apparent in the relative power spectra P(ω)N/ν₀ shown in Fig 8C-right, although significant changes in the power distribution emerge when the network state has a transition. To better appreciate the comparison between non-instantaneous synaptic transmission (solid lines) and distributions of axonal delays (dashed lines), Fig 8D (where c_αI = {10, 50, 90}%, as in Fig 8B) shows three paradigmatic relative spectral densities. As expected, P(ω) for the network in the AL state close to the bifurcation (red and blue dotted curves) displays a large peak at low-ω due to the slow fluctuations of the firing rate determined by the weakening of the 10-Hz attractor stability. As (see Eq (17)), the small but significant differences between the spectra at low-ω for non-instantaneous synaptic transmission (blue) and distributions of axonal delays (red) have to be attributed to the different shape of the input-output gain functions Φ₀. Indeed, our approximated Φ₀ overestimates the steady-state firing rate when τ_s > 0 (see Fig 3) in a relatively narrow range of the mean input current μ around v_thr, thus affecting also the slope . This difference is also responsible for the small discrepancies between the measured ν₀ and ν* as in Fig 6E. Further increasing the recurrent inhibition c_αI, in the AH state (dashed-dotted lines) resonance peaks due to transmission poles at ω/2π ≈ 50 Hz pop up. In the GO state (solid lines) also higher-order harmonics start to be visible in both P_d(ω) and P_s(ω). Although the overlap between these spectra is remarkable also in the GO regime, a small unexpected shift of the resonant frequencies is visible, highlighting a mild increase of the oscillation frequency when the synaptic filter is incorporated.

Similar results were obtained for the same network topology by changing the filtering timescales, as shown in Fig 9. Fig 9A shows relative power spectral densities for the inhibitory pool with non-instantaneous synaptic transmission. As expected, increasing τ_s from 1 ms to 16 ms (from bottom to top panels, respectively) makes the network more stable, thus shifting at higher c_αI the critical value at which the transition to the GO state occurs and the peaks associated to higher-order harmonics pop up. Larger τ_s also imply a reduction of the pace of the damped and synchronous oscillations from ω/2π ≈ 90 Hz to ≈ 40 Hz.

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Fig 9. Match between the same multi-modular networks as in Fig 8 for different timescales.

(A) Relative power spectra P(ω) of the inhibitory population activity in the case of non-instantaneous synaptic transmission for three different synaptic time constants (τ_s = 1, 4, 16 ms). (B) Cumulative distribution (in terms of Z-scores) of relative differences between power spectra ΔP(ω) = P_d(ω)/P_s(ω) − 1 across Fourier frequencies ω for three timescales (τ_s = τ_d = 1, 4, 16 ms), as in Fig 7C. Orange and plum curves are associated to excitatory and inhibitory populations respectively, whereas different line styles mark different values of τ_s = τ_d (see legend). (C) Coefficient of variation c_v = std(ν)/mean(ν) of ν(t) versus c_EI = c_II for excitatory (top) and inhibitory (bottom) populations, with non-instantaneous synaptic filtering (blue lines) and distributions of transmission delays (red lines). Different line styles mark different values of τ_s = τ_d, as in (B).

https://doi.org/10.1371/journal.pcbi.1007404.g009

The cumulative distributions of the Z-score ΔP(ω) = P_d(ω)/P_s(ω) − 1 in Fig 9B show, for the three considered timescales (τ_s = τ_d), that the differences of spectra display always the same statistics for both excitatory (orange) and inhibitory (blue) populations, similarly to Fig 7C. This equivalence under these conditions is also apparent by comparing the coefficient of variation c_v = std(ν)/mean(ν) = σ_ν/ν₀ for the excitatory and inhibitory pools (top and bottom panels in Fig 9C, respectively), with non-instantaneous synaptic transmission (blue) and exponential delay distributions (red). These plots confirm and complete the results already described in Fig 8C for τ_s = τ_d = 4 ms: for low c_αI (network in the AL state) the relative large variability is mainly due to a reduction of ν₀; moreover, c_v remains at its lowest value until the transition from AH to GO occurs. After this bifurcation, it increases with the amplitude of the global oscillation. As remarked above, the Hopf bifurcation value increases with τ_s = τ_d. In all cases, these plots confirm the sequence of state transitions already pointed out.

Altogether, these results further prove the generality of the theoretical equivalence proposed in this paper, clearly pointing out that both communication protocols give rise to the same collective dynamics not only in a tameable condition like the linearizable asynchronous state, but also for multi-modular networks working in nonstationary states like the GO regime.

Limitations of the theoretical approximation.

To reduce the dimensionality of the population density dynamics, we introduced a forcing white noise current with an arbitrarily small size σ_ext, as in [43]. This allowed us to safely manage the boundary conditions in the spectral expansion. The intrinsic stochasticity of ionic channels of neuronal membrane potential can be the source of this additional white noise current [35, 36]. However, it is important to remark that, in the absence of this external noise, the spectral expansion we used is no longer valid, as it relies on the eigenfunctions of a FP operator with ρ(v_thr, y, t) = 0 [8, 10, 28, 44]. More specifically, if σ_ext = 0 a different basis for the probability density decomposition must be adopted.

To investigate this aspect more in depth we exploited the multi-modular architecture of the last analyzed network. Indeed, the inhibitory population receives two excitatory contributions: one memoryless from C_I,ext synaptic contacts with Poisson-like external neurons, and another recurrent from the neurons composing the excitatory population. Both kind of synapses share the same synaptic efficacy and all the pre-synaptic (external and recurrent) neurons emit spikes at the same rates. Under mean-field approximation, this implies that, even replacing external with internal synapses, the statistics of the excitatory input received by the inhibitory neurons does not change. As a result, the equilibrium point at 10 Hz persists across variations of the balance between internal and external excitatory input. This is not the case for the equilibrium stability, which is lost as the recurrent contribution increases, giving rise to global oscillations, as shown in Fig 10A. By gradually decreasing the fraction of the external synaptic contacts C_I,ext/(C_IE + C_I,ext), the peak in the marginal density ρ(I) shrinks, due to the increasing weight of the low-pass filtered component of the synaptic input I(t). This shrinking effect is highlighted by the white iso-density curves in Fig 10A-left. Under this condition, a more drift-driven membrane potential V is expected and, due to the global oscillations of the firing rate ν(t), the peak in the marginal density ρ(V) widens (Fig 10A-right). This spreading of V fluctuations can be better appreciated looking at the three selected cuts of ρ(V) shown in Fig 10B (blue curves). As expected, ρ(V) in networks with synaptic filters is significantly different from the marginal densities of V in the equivalent networks with a distribution of axonal delays (Fig 10B, red curves). This difference is even more apparent when V approaches v_thr, as the blue curve does not converge to 0 due to the progressive lack of a diffusive component in the synaptic current. Intriguingly, almost irrespective of these differences, the power spectral density P_d(ω) matches remarkably well with P_s(ω) (Fig 10C, red and blue, respectively). Even when the memoryless contribution to the synaptic current received by the inhibitory neurons is completely replaced by the recurrent one (Fig 10C-bottom), resonance peaks and the band-pass filtering features widely overlap.

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Fig 10. Test of the limitations of the theoretical approximation.

(A) Marginal densities of the synaptic current [left, lim_t→∞ ρ_y(I, t) = ρ(I) = ∫ρ(V, I)dV] and the membrane potential [right, lim_t→∞ ρ_x(V, t) = ρ(V) = ∫ρ(V, I)dI] represented also versus the balance C_I,ext/(C_IE + C_I,ext) between recurrent (synaptically filtered) and external (memoryless) excitatory connections. White lines, iso-density curves. Marginal densities are averaged across 20 randomly chosen inhibitory neurons and 10 simulations with same mean-field parameters but different synaptic matrix realizations. Multi-modular network configuration as in Fig 8, with c_EI = c_II = 1% and τ_s = τ_d = 16 ms. (B) Three examples of marginal densities ρ(V) with different balance between recurrent (filtered) and external (memoryless) excitatory input (see dashed arrows and lines in panel A for the balance value). Blue and red curves: marginal densities from networks with synaptic filtering (as in panel A) and delay distribution, respectively. Gray dashed curves, sections of the density ρ(V, I) centered around the mean synaptic current I = μ_I/R estimated from inhibitory neurons with synaptic filtering. (C) Average power spectral density P(ω) of the firing rate ν_I(t) of the inhibitory neurons for the same example networks shown in panel B. The inset network sketches the change in the balance of the recurrent external currents. Color and styles as in panel B. Horizontal dashed lines: asymptotic values (ν_I0/N_I) of the spectra.

https://doi.org/10.1371/journal.pcbi.1007404.g010

The reasons for the effectiveness of our approximated description are mainly two. Firstly, in networks with synaptic filtering, the marginal density ρ(V) and the related slice of the full probability density ρ(V, I) centered around the average synaptic current μ_I (dashed gray curves in Fig 10B) are almost indistinguishable, at least for fractions of external synaptic contacts above 20%. In other words, the “slice” at I = μ_I/R (ρ(V, μ_I/R)) is well representative of the two-dimensional ρ(V, I). We remark that Eq (12) results from a 0-th order approximation of the current fluctuations which takes into account only this slice. This explains why the equivalence between synaptic filters and delay distributions still holds even when σ_ext provides a minor contribution to the fluctuations of I(t).

Secondly, we remind that in our approach the effective population dynamics of the slice at I = μ_I/R is governed by the operator , which incorporates an additional diffusion term proportional to σ²(y). This means that the probability density of membrane potentials changes in time according to this one-dimensional FP operator. As a result, Eq (12) actually describes a system with approximately the same firing rate dynamics but with a different ρ(V, t). In other words, we derived an equivalent stochastic diffusion process V(t) capable to have approximately the same flux of realizations crossing v_thr (in part due to σ²(y)), but at the same time differing from the original system in the density of V. This occurs as σ²(y) and the slope are adjusted in such a way that the flux of realizations crossing the absorbing barrier at x = v_thr (given by their product, see Eq (13)) is the same as for the 2D case.

The power spectra in Fig 10C allow us to test the limitations of our approximation not only for second-order statistics of the firing activity ν(t), but also for the first-order statistics, i.e., the mean. Indeed, from Eq (17), the asymptotic power lim_ω→∞ P(ω) = ν_I0/N_I of the inhibitory population provides this information (blue dashed lines in Fig 10C). From this measure, it is rather apparent that no significant changes occur as the diffusion white noise term σ²(y) is progressively replaced by the equivalent filtered synaptic input provided by the excitatory population (σ²(y) ∝ C_I,ext → 0). At a first sight, this could seem to contradict what shown in Fig 3C, where the mean-field equilibrium point ν* predicted by our 0-th order approximation is the same for any mean delay τ_d and differs from the ones measured by incorporating different synaptic filters. Actually, the network displays global oscillations (GO regime) such that the time spent with μ ≃ v_thr is small and, outside a small interval around this critical value, Φ₀ is a good approximation of both under noise- and drift-dominated regimes. Therefore, this result is perfectly in line with the ones shown in Fig 3C.