Correlations uniquely determine effective connectivity: A simple example

In this section we give an intuitive one-dimensional example to show that effective connectivity determines the shapes of the average pairwise cross-covariances and vice versa. For the following, we first introduce a few basic quantities. Consider a binary or spiking network consisting of several excitatory and inhibitory populations with potentially source- and target-type-dependent connectivity. For the spiking networks, we assume leaky integrate-and-fire (LIF) dynamics with exponential synaptic currents. The dynamics of the binary and LIF networks are respectively introduced in “Binary network dynamics” and “Spiking network dynamics”. We assume irregular network activity, approximated as Poissonian for the spiking network, with population means ν_α. For the binary network, ν = ⟨n⟩ is the expectation value of the binary variable. For the spiking network, we absorb the membrane time constant into ν, defining ν = τ_m r where r is the firing rate of the population. The external drive can consist of both a DC component μ_α,ext and fluctuations with variance , provided either by Poisson spikes or by a Gaussian current. The working points of each population, characterized by mean μ_α and variance of the combined input from within and outside the network, are given by (3) (4) (5) where J_αβ is the synaptic strength from population β to population α, and K_αβ is the number of synapses per target neuron (the in-degree) for the corresponding projection (we use ≡ in the sense of “is defined as”). We call “external variance” in the following, and the remainder “internal variance”. The mean population activities are determined by μ_α and σ_α according to Eqs (39) and (67). Expressions for correlations in binary and LIF networks are given respectively in “First and second moments of activity in the binary network” and “First and second moments of activity in the spiking network”.

As a one-dimensional example, consider a binary network with a single population and vanishing transmission delays. The effective connectivity W is just a scalar, and the population-averaged autocovariance a and cross-covariance c are functions of the time lag Δ. We define the population-averaged effective connectivity as (6) where w(J, μ, σ) is an effective synaptic weight that depends on the mean μ [Eq (3)] and the variance σ² [Eq (4)] of the input. For LIF networks, w = ∂r_target/∂r_source is defined via Eq (68) and can be obtained as the derivative of Eq (67). Note that we treat the effective influence of individual inputs as independent. A more accurate definition of the population-level effective connectivity, beyond the scope of this paper, could be obtained by also considering combinations of inputs in the sense of a Volterra series [57]. When the dependence of w on J is linearized, the effective connectivity can be written as (7) where the susceptibility S(μ, σ) measures to linear order the effect of a unit input to a neuron on its outgoing activity. In our one-dimensional example, W quantifies the self-influence of an activity fluctuation back onto the population. Expressed in these measures, the differential equation for the covariance function [Eq (52)] takes the form (8) with initial condition [from Eq (41)] (9) which is solved by (10) Eq (10) shows that the effective connectivity W together with the time constant τ of the neuron (which we assume fixed under scaling) determines the temporal structure of the correlations. Furthermore, since a sum of exponentials cannot equal a sum of exponentials with a different set of exponents, the temporal structure of the correlations uniquely determines W. Hence we see that there is a one-to-one correspondence between W and the correlation structure if the time constant τ is fixed, which implies that preserving correlation structure under a reduction in the in-degrees K requires adjusting the effective synaptic weights w(J, μ, σ) such that the effective connectivity W is maintained. If, in addition, the mean activity ⟨n⟩ is kept constant this also fixes the variance a = ⟨n⟩(1 − ⟨n⟩). Eq (10) shows that, under these circumstances with W and a fixed, correlation sizes are determined by N.

Correlations uniquely determine effective connectivity: The general case

More generally, networks consist of several neural populations each with different dynamic properties and with population-dependent transmission delays dαβ. Since this setting does not introduce additional symmetries, intuitively the one-to-one relationship between the effective connectivity and the correlations should still hold. We here show that, under certain conditions, this is indeed the case.

Instead of considering the covariance matrix in the time domain, for population-dependent dynamic properties we find it convenient to stay in the frequency domain. The influence of a fluctuating input on the output of the neuron can to lowest order be described by the transfer function H(ω). This quantity measures the amplitude and phase of the modulation of the neuronal activity given that the neuron receives a small sinusoidal perturbation of frequency ω in its input. The transfer function depends on the mean μ [Eq (3)] and the variance σ² [Eq (4)] of the input to the neuron. We here first consider LIF networks; in the Supporting Information we show how the results carry over to the binary model.

In “First and second moments of activity in the spiking network”, we give the covariance matrix including the autocovariances in the frequency domain, , as (11) where M has elements H_αβ(ω)W_αβ. If is invertible, we can expand the inverse of Eq (11) to obtain (12) where we assumed the transfer function to have the form which is often a good approximation for the LIF model [45]. In the second step we distinguish terms that only contribute on the diagonal (α = β), those that only contribute off the diagonal (α ≠ β), and those that contribute in either case. For α = β, only the first and last terms contribute, and we get (13) If we want to preserve , this fixes A_α and thereby also W_αα, since it multiplies terms with unique ω-dependence. For α ≠ β, we obtain (14) With A_α fixed, this additionally fixes W_αβ, in view of the unique ω-dependence it multiplies.

Since , a constraint on A necessary for preserving may not translate into the same constraint when we only require the cross-covariances C(ω) to be preserved. However, C(ω) and have identical ω-dependence, as they differ only by constants on the diagonal (approximating autocorrelations as delta functions in the time domain [45]). To derive conditions for preserving C(ω), we therefore ignore the constraint on A but still require the ω-dependence to be unchanged. A potential transformation leaving the ω-dependent terms in both Eqs (13) and (14) unchanged is A_α → kA_α, W_αβ → kW_αβ, W_αα → kW_αα, but this only works if τ_α = τ_γ, d_αα − d_αβ = d_γα − d_γβ for some γ, and if the terms for the corresponding γ are also transformed to offset the change in ; or if some of the entries of W vanish. The ω-dependence of and C would otherwise change, showing that, at least in the absence of such symmetries in the delays or time constants, or zeros in the effective connectivity matrix (i.e., absent connections at the population level, or inactive populations), there is a one-to-one relationship between covariances and effective connectivity. Hence, preserving the covariances requires preserving A and W except in degenerate cases. Note that the autocovariances and hence the firing rates can be changed while affecting only the size but not the shape of the correlations, but that the correlation shapes determine W.

Even in case of identical transfer functions across populations, including in particular equal transmission delays and identical τ, the one-to-one correspondence between effective connectivity and correlations can be demonstrated except for a narrower set of degenerate cases. The argument for d = 0 proceeds in the time domain along the same lines as “Correlations uniquely determine effective connectivity: a simple example”, using the fact that for a population-independent transfer function, the correlations can be expressed in terms of the eigenvalues and eigenvectors of the effective connectivity matrix (cf. “First and second moments of activity in the binary network” and “First and second moments of activity in the spiking network”). For general delays, a derivation in the frequency domain can be used. Through these arguments, we show in the Supporting Information that the one-to-one correspondence holds at least if W is diagonalizable and has no eigenvalues that are zero or degenerate.

Correlation-preserving scaling

If the working point (μ, σ) is maintained, the one-to-one correspondence between the effective connectivity and the correlations implies that requiring unchanged average covariances leaves no freedom for network scaling except for a possible trade-off between in-degrees and synaptic weights. In the linear approximation W(J, μ, σ) = S(μ, σ)JK, this trade-off is J ∝ 1/K.

When this scaling is implemented naively without adjusting the external drive to recover the original working point, the covariances change, as illustrated in Fig 3B for a two-population binary network with parameters given in Table 1. The results of J ∝ 1/K scaling with appropriate adjustment of the external drive are shown in Fig 3C. The scaling shown in Fig 3B also increases the mean activities (E: from 0.16 to 0.23, I: from 0.07 to 0.11), whereas that in Fig 3C preserves them.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Correlations from theory and simulations for a two-population binary network with asymmetric connectivity.

A Average pairwise cross-covariances from simulations (solid curves) and Eq (55) (dashed curves). B Naive scaling with J ∝ 1/K but without adjustment of the external drive changes the correlation structure. C With an appropriate adjustment of the external drive (σ_ex = 53.4, σ_ix = 17.7), scaling synaptic weights as J ∝ 1/K is able to preserve correlation structure as long as N and K are reduced by comparable factors. D The same holds for (μ_ex = 43.3, μ_ix = 34.6, σ_ex = 46.2, σ_ix = 15.3), but the susceptibility S is increased by about 20% already for N = 0.75 N₀ in this case. In B, C, and D, results of simulations are shown. The curves in C and D are identical because internal inputs, the standard deviation of the external drive, and the distance to threshold due to the DC component of the drive in D are exactly times those in C. Hence, identical realizations of the random numbers for the connectivity and the Gaussian external drive cause the total inputs to the neurons to exceed the threshold at exactly the same points in time in the two simulations. The simulated time is 30 s, and the population activity is sampled at a resolution of 0.3 ms.

https://doi.org/10.1371/journal.pcbi.1004490.g003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Parameters of the asymmetric binary network.

https://doi.org/10.1371/journal.pcbi.1004490.t001

If one relaxes the constraint on the working point while still requiring mean activities to be preserved, the network does have additional symmetries due to the fact that only some combination of μ and σ needs to be fixed, rather than each of these separately. This combination is more easily determined for binary than for LIF networks, for which the mean firing rates depend on μ and σ in a complex manner [cf. Eq (67)]. When the derivative of the gain function is narrow (e.g., having zero width in the case of the Heaviside function used here) compared to the input distribution, the mean activities of binary networks depend only on (μ − θ)/σ [9]. Changing σ while preserving (μ − θ)/σ leads for a Heaviside gain function to a new susceptibility S′ = (σ/σ′)S [cf. Eq (43)]. For constant K, if the standard deviation of the external drive is changed proportionally to the internal standard deviation, we have σ ∝ J and thus J′ S′ = JS, implying an insensitivity of the covariances to the synaptic weights J [52]. In particular, this symmetry applies in the absence of an external drive. When K is altered, this choice for adjusting the external drive causes the covariances to change. However, adjusting the external drive such that σ′/σ = (J′ K′)/(JK), the change in S is countered to preserve W and correlations. This is illustrated in Fig 3D for , which is another natural choice, as it preserves the internal variance if one ignores the typically small contribution of the correlations to the input variance ([9] Fig 3D illustrates the smallness of this contribution for an example network). This is only one of a continuum of possible scalings preserving mean activities and covariances (within the bounds described in the following section) when the working point and hence the susceptibility are allowed to change.

Limit to in-degree scaling

We now show that both the scaling J ∝ 1/K for LIF networks (for which we do not consider changes to the working point, as analytic expressions for countering these changes are intractable), and correlation-preserving scalings for binary networks (where we allow changes to the working point that preserve mean activities) are applicable only up to a limit that depends on the external variance.

For the binary network, assume a generic scaling K′ = κK, J′ = ιJ and a Heaviside gain function. We denote variances due to inputs from within the network and due to the external drive respectively by and The preservation of the mean activities implies S′ = (σ/σ′)S as above, where . To keep SJK fixed we thus require (15) where we have used in the second line. For σ_ext = 0 this scaling only works for κ > 1, i.e., increasing instead of decreasing the in-degrees. More generally, the limit to downscaling occurs when or (16) independent of the scaling of the synaptic weights. Thus, larger external and smaller internal variance before scaling allow a greater reduction in the number of synapses. The in-degrees of the example network of Fig 3 could be maximally reduced to 73%. Note that ι could in principle be chosen in a κ-dependent manner such that is fixed or increased instead of decreased upon downscaling, namely . However, Eq (16) is still the limit beyond which this fails, as ι then diverges at that point.

Note that the limit to the in-degree scaling also implies a limit on the reduction in the number of neurons for which the scaling equations derived here allow the correlation structure to be preserved, as a greater reduction of N compared to K increases the number of common inputs neurons receive and thereby the deviation from the assumptions of the diffusion approximation. This is shown by the thin curves in Fig 3C,3D.

Now consider correlation-preserving scaling of LIF networks. Reduced K with constant JK does not affect mean inputs [cf. Eq (3)] but increases the internal variance according to Eq (4). To maintain the working point (μ, σ), it is therefore necessary to reduce the variance of the external drive. When the drive consists of excitatory Poisson input, one way of keeping the mean external drive constant while changing the variance is to add an inhibitory Poisson drive. With K′ = K/ι and J′ = ιJ, the change in internal variance is , where is the internal variance due to input currents in the full-scale model. This is canceled by an opposite change in by choosing excitatory and inhibitory Poisson rates (17) (18) where r_e,0 is the Poisson rate in the full-scale model, and the excitatory and inhibitory synapses have weights J_ext and −g J_ext, respectively. Eqs (17) and (18) match Eq (E.1) in [45] except for the 1 + g in the denominator, which was there erroneously given as 1+g². Since downscaling K implies ι > 1, it is seen that the required rate of the inhibitory inputs is negative. Therefore, this method only allows upscaling. An alternative is to use a balanced Poisson drive with weights J_ext and − J_ext, choosing the rate of both excitatory and inhibitory inputs to generate the desired variance, and adding a DC drive I_ext to recover the mean input, (19) (20) In this manner, the network can be downscaled up to the point where the variance of the external drive vanishes. Substituting this condition into Eq (15), the same expression for the minimal in-degree scaling factor Eq (16) is obtained as for the binary network.

Robustness of correlation-preserving scaling

In this section, we show that the scaling J ∝ 1/K, which maintains the population-level feedback quantified by the effective connectivity, can preserve correlations (within the bounds given in “Limit to in-degree scaling”) under fairly general conditions. To this end, we consider two types of networks: 1. a multi-layer cortical microcircuit model with distributed in- and out-degrees and lognormally distributed synaptic strengths (cf. “Network structure and notation”); 2. a two-population LIF network with different mean firing rates (parameters in Table 2). For both types of models, we contrast the scaling J ∝ 1/K with , in each case maintaining the working point given by Eqs (3) and (4). Fig 4 illustrates that the former closely preserves average pairwise cross-covariances in the cortical microcircuit model, whereas the latter changes both their size and temporal structure.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Full-scale parameters of the two-population spiking networks used to demonstrate the robustness of J ∝ 1/K scaling to mean firing rates.

The two networks are distinguished by their external drives.

https://doi.org/10.1371/journal.pcbi.1004490.t002

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Within the restrictive bounds imposed by Eq (16), preserving effective connectivity can preserve correlations also in a complex network.

Simulation results for the cortical microcircuit at full scale and with in-degrees reduced to 90%. Synaptic strengths are scaled as indicated, and the external drive is adjusted to restore the working point. Mean pairwise cross-covariances are shown for population 2/3E. Qualitatively identical results are obtained within and across other populations. The simulation duration is 30 s and covariances are determined with a resolution of 0.5 ms. To enable downscaling with J ∝ 1/K, the excitatory Poisson input of the original implementation of [58] is replaced by balanced inhibitory and excitatory Poisson input with a DC drive according to Eqs (19) and (20). A Scaling synaptic strengths as changes the mean covariance. Light green curve: stretching the covariance of the scaled network along the vertical axis to match the zero-lag correlation of the full-scale network shows that not only the size but also the temporal structure of the covariance is affected. B Scaling synaptic strengths as J ∝ 1/K closely preserves the covariance of the full-scale network. However, note that this scaling is only applicable down to the in-degree scaling factor given by Eq (16), which for this example is approximately 0.9.

https://doi.org/10.1371/journal.pcbi.1004490.g004

Fig 5 demonstrates the robustness of J ∝ 1/K scaling to the firing rate of the network. In this example, both the full-scale network and the downscaled networks receive a balanced Poisson drive producing the desired variance, while the mean input is provided by a DC drive. By changing the parameters of the external drive, we create two networks each with irregular spiking but with widely different mean rates (3.3 spikes/s and 29.6 spikes/s). Downscaling only the number of synapses but not the number of neurons, both the temporal structure and the size of the correlations are closely preserved. Reducing the in-degrees and the number of neurons N by the same factor, the correlations are scaled by 1/N. Hence, the correlations of the full-scale network of size N₀ can be estimated simply by multiplying those of the reduced network by N/N₀. In contrast, changes correlation sizes even when N is held constant, and combined scaling of N and K can therefore not simply be compensated for by the factor N/N₀. In the high-rate network, the spiking statistics of the neurons is non-Poissonian, as seen from the gap in the autocorrelations (insets in Fig 5B, 5D). Nevertheless, J ∝ 1/K preserves the correlations more closely than , showing that the predicted scaling properties hold beyond the strict domain of validity of the underlying theory.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Scaling synaptic strengths as J ∝ 1/K can preserve correlations in networks with widely different firing rates.

Results of simulations of a LIF network consisting of one excitatory and one inhibitory population (Table 2). Average cross-covariances are determined with a resolution of 0.1 ms and are shown for excitatory-inhibitory neuron pairs. Each network receives a balanced Poisson drive with excitatory and inhibitory rates both given by , where is chosen to maintain the working point of the full-scale network. The synaptic strengths for the external drive are 0.1 mV and −0.1 mV for excitatory and inhibitory synapses, respectively. A DC drive with strength μ_ext is similarly adjusted to maintain the full-scale working point. All networks are simulated for 100 s. For each population, cross-covariances are computed as averages over all neuron pairs across two disjoint groups of 𝓝 × 1000 neurons, where 𝓝 is the scaling factor for the number of neurons (a given pair has one neuron in each group). Autocovariances are computed as averages over 100 neurons in each population. A, B Reducing in-degrees K to 50% while the number of neurons N is held constant, J ∝ 1/K closely preserves both the size and the shape of the covariances, while diminishes their size. C, D Reducing both N and K to 50%, covariance sizes scale with 1/N for J ∝ 1/K but with a different factor for . Dashed curves represent theoretical predictions. The insets show mean autocovariances for time lags Δ ∈ (−30, 30) ms.

https://doi.org/10.1371/journal.pcbi.1004490.g005

Zero-lag correlations in binary network

Although it is not generally possible to keep mean activities and correlations invariant upon downscaling, transformations may be found when only one aspect of the correlations is important, such as their zero-lag values. We illustrate this using a simple, randomly connected binary network of N excitatory and γN inhibitory binary neurons, where each neuron receives K = pN excitatory and γK inhibitory inputs. The parameters are given in Table 3. The linearized effective connectivity matrix for this example is (21)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Parameters of the symmetric binary network.

https://doi.org/10.1371/journal.pcbi.1004490.t003

When the threshold θ is ≤ 0, the network is spontaneously active without external inputs. In the diffusion approximation and assuming stationarity, the mean zero-lag cross-covariances between pairs of neurons from each population can be estimated from Eq (41) (see also [52]) (22) where the subscripts e and i respectively denote excitatory and inhibitory populations. Moreover, W_e is the effective excitatory coupling, (23) with S the susceptibility as defined in Eq (43). Furthermore, a is the variance of the single-neuron activity, (24) which is identical for the excitatory and inhibitory populations. The mean input to each neuron is given by [cf. Eq (3)], (25) and, under the assumption of near-independence of the neurons, the variance of the inputs is well approximated by the sum of the variances from each sending neuron [cf. Eq (4)], (26) Finally, the mean activity can be obtained from the self-consistency relation Eq (39).

Eq (22) shows that, when excitatory and inhibitory synaptic weights are scaled equally, the covariances scale with 1/N as long as the network feedback is strong (W_e ≫ 1), (for this argument, we assume that ⟨n⟩ is held constant, which may be achieved by adjusting a combination of θ and the external drive). Hence, conventional downscaling of population sizes tends to increase covariances.

We use Eq (22) to perform a more sophisticated downscaling (cf. Fig 6). Let the new size of the excitatory population be N′. Eq (22) shows that the covariances can only be preserved when a combination of W_e, γ, and g is adjusted. We take γ constant, and apply the transformation (27) Solving Eq (22) for f and g′ yields (cf. Fig 6B) (28) (29) The change in W_e can be captured by K → f K as long as the working point (μ, σ) is maintained. This intuitively corresponds to a redistribution of the synapses so that a fraction f comes from inside the network, and 1 − f from outside (cf. Fig 6A). However, the external drive does not have the same mean and variance as the internal inputs, since it needs to make up for the change in g. The external input can be modeled as a Gaussian noise with parameters (30) (31) independent for each neuron.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Binary network scaling that approximately preserves both mean activities and zero-lag covariances.

A Increased covariances due to reduced network size can be countered by a change in the relative inhibitory synaptic weight combined with a redistribution of the synapses so that a fraction comes from outside the network. Adjusting a combination of the threshold and external drive restores the working point. B Scaling parameters versus relative network size for an example network. Since γ = 1 in this example, the scaling only works down to g = 1 (indicated by the horizontal and vertical dashed lines): Lower values of g only allow a silent or fully active network as steady states. C, E The mean activities are well preserved both by the conventional scaling in Eq (1) with an appropriate adjustment of θ (panel C), and by the method proposed here (panel E). D, F Conventional scaling increases the magnitude of zero-lag covariances in simulated data (panel D), while the proposed method preserves them (panel F). Dark colors: full-scale network. Light colors: downscaled network. Crosses and dots indicate zero-lag correlations in the full-scale and downscaled networks, respectively.

https://doi.org/10.1371/journal.pcbi.1004490.g006

An alternative is to perform the downscaling in two steps: First change the relative inhibitory weights according to Eq (29) but keep the connection probability constant. The mean activity can be preserved by solving Eq (39) for θ, but the covariances are changed. The second step, which restores the original covariances, then amounts to redistributing the synapses so that a fraction comes from inside the network, and from outside, where the external (non-modeled) neurons have the same mean activity as those inside the network. This mean activity is negative, as the balanced regime implies stronger inhibition than excitation. Note that , since W_e changes already in the first step.

The requirement that inhibition dominate excitation places a lower limit on the network size for which the scaling is effective. The reason is that g decreases with network size, so that a bifurcation occurs at g = 1/γ, beyond which the only steady states correspond to a silent network or a fully active one.

Symmetric two-population spiking network

We have seen that the one-to-one relationship between effective connectivity and correlations does not hold in certain degenerate cases. Here we consider such a degenerate case and perform a scaling that preserves mean activities as well as both the size and the temporal structure of the correlations under reductions in both the number of neurons and the number of synapses. The network consists of one excitatory and one inhibitory population of LIF neurons with a population-independent connection probability and vanishing transmission delays. Due to the appearance of the eigenvalues in the numerator of the expression for the correlations in LIF networks [cf. Eqs (70) and (71)], such networks are subject to a reduced number of constraints when W has a zero eigenvalue, as this leaves a freedom to change the corresponding eigenvectors. Furthermore, identically vanishing delays greatly simplify the equations for the covariances.

The single-neuron and network parameters are as in Table 2 except that, here, N = 10,000, J = 0.2 mV, and the external drive is chosen such that the mean and standard deviation of the total input to each neuron are μ = 15 mV, σ = 10 mV. Furthermore, the delay is chosen equal to the simulation time step to approximate d = 0, which we assume here. The effective connectivity matrix for this network is (32) where w = ∂r_target/∂r_source is the effective excitatory synaptic weight obtained as the derivative of Eq (67). Here, we take into account the dependence of w on J to quadratic order. The inhibitory weight is approximated as gw to allow an analytical expression for the relative inhibitory weight in the scaled network to be derived. The left and right eigenvectors are corresponding to eigenvalue L = w K(1 − γ g) and corresponding to eigenvalue 0. The normalization is chosen such that the bi-orthogonality condition Eq (47) is fulfilled.

A transformed connectivity matrix should have the same eigenvalues as W, and can thus be written as (33) (34) Denote the new population sizes by N₁ and N₂. Equating the covariances before and after the transformation yields using Eq (71) and A^jk = v^jT A v^k [cf. Eq (49)], (35) In Eq (35) we have assumed that the working points, and thus a₁ and a₂, are preserved, which may be achieved with an appropriate external drive as long as the corresponding variance remains positive. The four equations are simultaneously solved by (36) where w′ K′ may be chosen freely. Thus, the new connectivity matrix reads (37) which may also be cast into the form (38) where γ′ = N₂/N₁ and .

When the populations receive statistically identical external inputs, we have a₁ = a₂ = r, since the internal inputs are also equal. Fig 7 illustrates the network scaling for the choice w′ = w. Results are shown as a function of the relative size N₁/N of the excitatory population. External drive is provided at each network size to keep the mean and standard deviation of the total inputs to each neuron at the level indicated. The mean is supplied as a constant current input, while the variability is afforded by Poisson inputs according to Eqs (17) and (18) (Fig 7D). It is seen that the transformations (Fig 7B) are able to reduce both the total numbers of neurons and the total number of synapses (Fig 7C) while approximately preserving covariance sizes and shapes (Fig 7E,7F). Small fluctuations in the theoretical predictions in Fig 7E are due to the discreteness of numbers of neurons and synapses, and deviations of the effective inhibitory weight from the linear approximation g w. The fact that the theoretical prediction in Fig 7F misses the small dips around t = 0 may be due to the approximation of the autocorrelations by delta functions, eliminating the relative refractoriness due to the reset. The numbers of neurons and synapses increase again below some N₁/N, and diverge as g′ becomes zero. This limits the scalability despite the additional freedom provided by the symmetry.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Spiking network scaling that approximately preserves mean firing rates and covariances.

A Diagram illustrating the network and indicating the parameters that are adjusted. B Excitatory in-degrees K′, relative inhibitory synaptic weight g′, and relative number of inhibitory neurons γ′ versus scaling factor N₁/N. The dashed vertical line indicates the limit below which the scaling fails. C Total number of neurons N_total = (1+γ′)N₁ and total number of synapses N_syn = (1+γ′)² K′ N₁ versus scaling factor. D Rates of external excitatory and inhibitory Poisson inputs necessary for keeping firing rates constant. Average firing rates are between 23.1 and 23.5 spikes/s for both excitatory and inhibitory populations and all network sizes. E Integrated covariances, corresponding to zero-frequency components in the Fourier domain. Crosses: simulation results, dots: theoretical predictions. F Average covariance between excitatory-inhibitory neuron pairs for different network sizes. The dashed curve indicates the theoretical prediction for N = 10,000. Each network was simulated for 100 s.

https://doi.org/10.1371/journal.pcbi.1004490.g007

Software

We verify analytical results for networks of binary neurons and networks of spiking neurons using direct simulations performed with NEST [83] revisions 10711 and 11264 for the spiking networks and revision 11540 for the binary networks. For simulating the multi-layer microcircuit model, PyNN version 0.7.6 (revision 1312) [84] was used with NEST 2.6.0 as back end, single-threaded on 12 MPI processes on a high-performance cluster. All simulations have a time step of 0.1 ms. Spike times in the microcircuit model are constrained to the grid. The other spiking network simulations use precise spike timing [85]. In part, Sage was used for symbolic linear algebra [86]. Pre- and post-processing and numerical analysis were performed with Python.

Network structure and notation

For both the binary and the spiking networks, we derive analytical results where both the number of populations N_pop and the population-level connectivity are arbitrary. Specific examples are given of networks with a single, inhibitory population, or with two populations (one excitatory, one inhibitory) with either population-specific or population-independent connectivities. In addition, we discuss a multi-layer spiking cortical microcircuit model consisting of 77,169 neurons with approximately 3 × 10⁸ synapses, with eight populations (2/3E, 2/3I, 4E, 4I, 5E, 5I, 6E, 6I) and population-specific connection probabilities [58], slightly adjusted to enhance the asynchrony of the activity. The adjustments consist of replacing normally by lognormally distributed weights with the same mean and with coefficient of variation 3; and using 4.5 instead of 4 as the relative strength of synapses from 4I to 4E compared to excitatory synaptic strengths. Besides distributed synaptic strengths, the model has binomially distributed in- and out-degrees, and normally distributed delays (clipped at the simulation time step), thereby deviating from the assumptions of our analytic theory. It thus serves to evaluate the robustness of our analytical results to such deviations from the underlying assumptions.

In all cases, pairs of populations are randomly connected. In the binary and one- and two-population LIF network simulations, in-degrees are fixed and multiple directed connections between pairs of neurons (multapses) are disallowed. In the multi-layer microcircuit model, in-degrees are distributed and multapses are allowed. In case of population-specific connectivities, we denote the (unique or mean) in-degree for connections from population β to population α by K_αβ, and synaptic strengths by J_αβ. Population sizes are denoted by N_α. For the example networks with population-independent connection probability, we denote the size of the excitatory population by N, the in-degree from excitatory neurons by K = pN, and the size of the inhibitory relative to the excitatory population by γ, so that the inhibitory in-degree is γK. Synaptic strengths are also taken to only depend on the source population, and are written as J for excitatory and −gJ for inhibitory synapses.

Binary network dynamics

We denote the activity of neuron j by n_j(t). The state n_j(t) of a binary neuron is either 0 or 1, where 1 indicates activity, 0 inactivity [7, 42, 87]. The state of the network of N such neurons is described by a binary vector n = (n₁, …, n_N) ∈ {0,1}^N. We denote the mean activity by ⟨n_j(t)⟩_t, where the average ⟨⟩_t is over time and realizations of the stochastic activity. The neuron model shows stochastic transitions (at random points in time) between the two states 0 and 1. In each infinitesimal interval [t, t + δt), each neuron in the network has the probability to be chosen for update [88], where τ is the time constant of the neuronal dynamics. We use an equivalent implementation in which the time points of update are drawn independently for all neurons. For a particular neuron, the sequence of update points has exponentially distributed intervals with mean duration τ, i.e., update times form a Poisson process with rate τ⁻¹. The stochastic update constitutes a source of noise in the system. Given that the j-th neuron is selected for update, the probability to end in the up state (n_j = 1) is determined by the gain function F_j(n(t)) = Θ(∑_k J_jk n_k(t) − θ) which in general depends on the activity n of all other neurons. Here θ denotes the threshold of the neuron and Θ(x) the Heaviside function. The probability of ending in the down state (n_j = 0) is 1 − F_j(n). This model has been considered previously [42, 87, 89], and here we follow the notation introduced in [87] that we also employed in our earlier works. We skip details of the derivation here that are already contained in [9].

First and second moments of activity in the binary network

The combined distribution of large numbers of independent inputs can be approximated as a Gaussian 𝓝(μ, σ²) by the central limit theorem. The arguments μ and σ are the mean and standard deviation of the synaptic input noise, together referred to as the working point [cf. Eqs (3) and (4)]. The stationary mean activity of a given population of neurons then obeys [7, 9, 46, 52] (39) This equation needs to be solved self-consistently because ⟨n⟩ influences μ, σ through interactions within the population itself and with other populations.

When network activity is stationary, the covariance of the activities of a pair (j, k) of neurons is defined as c_jk(Δ) = ⟨δn_j(t + Δ)δn_k(t)⟩_t, where δn_j(t) = n_j(t) − ⟨n_j(t)⟩_t is the deviation of neuron j’s activity from expectation, and Δ is a time lag. Instead of the raw correlation ⟨n_j(t + Δ)n_k(t)⟩_t, here and for the spiking networks we measure the covariance, i.e., the second centralized moment, which is also identical to the second cumulant. To derive analytical expressions for the covariances in binary networks in the asynchronous regime, we follow the theory developed in [7, 9, 42, 52, 53]. We first consider the case of vanishing transmission delays d = 0 and then discuss networks with delays.

Let (40) be the covariance averaged over disjoint pairs of neurons in two (possibly identical) populations α, β, and the population-averaged single-neuron variance a_j(Δ) = ⟨δn_j(t + Δ)δn_j(t)⟩_t. Note that for α = β there are only N_α(N_α − 1) disjoint pairs of neurons, so c_αα differs from the average pairwise cross-correlation by a factor (N_α − 1)/N_α, but we choose this definition because it slightly simplifies the population-level equations. For sufficiently weak synapses and sufficiently high firing rates, and when higher-order correlations can be neglected, a linearized equation relating these quantities can be derived for the case d = 0 ([42] Eqs (9.14)–(9.16); [7] supplementary material Eq (36), [9] Eq (10)), (41) Here, we have assumed identical time constants across populations, and (42) is the linearized effective connectivity. The susceptibility S is defined as the slope of the gain function averaged over the noisy input to each neuron [9, 52, 53], reducing for a Heaviside gain function to (43)

With the definitions (44) (45) Eq (41) is recognized as a continuous Lyapunov equation (46) which can be solved using known methods. Let v^j,u^k be the left and right eigenvectors of W, with eigenvalues λ_j and λ_k, respectively. Choose the normalization such that the left and right eigenvectors are biorthogonal, (47) Then multiplying Eq (46) from the left with v^jT and from the right with v^k yields (48) Define (49) for . Then solving Eq (48) for gives (50) as can be verified using Eq (47). This provides an approximation of the population-averaged zero-lag correlations, including contributions from both auto- and cross-correlations.

To determine the temporal structure of the population-averaged cross-correlations, we start from the single-neuron level, for which the correlations approximately obey ([53] Eq (29)) (51) where w_ij is the neuron-level effective connectivity (w_ij = S_i J_ij if a connection exists and w_ij = 0 otherwise). This equation also holds on the diagonal, j = k. To obtain the population-level equation, we use Eqs (40) and (44) and count the numbers of connections, which yields a factor K_αβ for each projection. Eq (51) then becomes (52) This step from the single-neuron to the population level constitutes an approximation when the out-degrees are distributed, but is exact for fixed out-degree [8, 53]. The correlations for Δ < 0 are determined by . With the definition Eq (49), Eq (52) yields (53) Using the initial condition for from Eq (50) and multiplying Eq (53) by u^j u^kT, summing over j and k, we obtain the solution (54) The shape of the autocovariances is well approximated by that for isolated neurons, , with corrections due to interactions being O(1/N) [42]. Substituting this form in Eq (54) leads to (55) equivalent to [42] Eq (6.20). Note that this equation still needs to be solved self-consistently, because the variance of the inputs to the neurons, which goes into S(μ, σ), depends on the correlations. However, correlations tend to contribute only a small fraction of the input variance in the asynchronous regime (cf. [9] Fig 3D). The accuracy of the result Eq (55) is illustrated in Fig 3A for a network with parameters given in Table 1 by comparison with a direct simulation. Note that the delays were not zero but equal to the simulation time step of 0.1 ms, sufficiently small for the correlations to be well approximated by Eq (55).

Now consider arbitrary transmission delay d > 0, and let both d and the input statistics be population-independent. This case is most easily approached from the Fourier domain, where the population-averaged covariances including autocovariances can be approximated as [53] (56) Here, H(ω) is the transfer function (57) which is equal for all populations under the assumptions made. The transfer function is the Fourier transform of the impulse response, which is a jump followed by an exponential relaxation, (58) where Θ is the Heaviside step function.

For the case of population-independent H(ω), Fourier back transformation to the time domain is feasible, and was performed in [53] for symmetric connectivity matrices. Here, we consider generic connectivity (insofar as consistent with equal H(ω)), and again use projection onto the eigenspaces of W to obtain a form similar to Eq (55), i.e., insert the identity matrix (59) both on the left and on the right of Eq (56), and Fourier transform to obtain (60) In the third line of Eq (60), we used A^jk = v^jT A v^k and collected the frequency-dependent terms for clarity. The exponential e^iωΔ does not have any poles, so the only poles stem from f_jk, which we denote by z_l(λ_j) and the corresponding residues by Res_j,k[z_l(λ_j)]. We only need to consider Δ ≥ 0, since the solution for negative lags follows from . The equation can then be solved by contour integration over the upper half of the complex plane, as the integrand vanishes at ω → +i∞. Stability requires that the poles of the first term of Eq (60) lie only in the upper half plane (note that the linear approximation we have employed only applies in the stable regime). The poles of the second term correspondingly lie in the lower half plane and hence need not be considered. For d > 0, the locations of the poles are given by [53] Eq (12), (61) where W_l is the l^th of the infinitely many branches of the Lambert-W function defined by x = W(x)e^W(x) [90]. For d = 0, the poles are . Using the residue theorem thus brings Eq (60) into the form (62) where I(γ) = 1 is the winding number of the contour γ around the poles. To see that Eq (62) reduces to Eq (55) when d = 0, substitute the poles in the upper half plane with residues [iτ(2 − λ_j − λ_k)]⁻¹ and note that .

When the input statistics and hence transfer functions are population-specific, Eq (56) becomes (63) (64) where M_αβ(ω) = H_αβ(ω)W_αβ.

Spiking network dynamics

The spiking networks consist of single-compartment leaky integrate-and-fire neurons with exponential current-based synapses. The subthreshold dynamics of neuron i is given by (65) where we have set the resting potential to zero without loss of generality, and absorbed the membrane resistance into the synaptic current I_i, in line with previous works [45, 91]. Bringing back the corresponding parameters, the dynamics reads (66) Thus, our scaled synaptic amplitudes J_ij in terms of the amplitudes of the synaptic current due to a single spike are . Here, τ_m and τ_s are membrane and synaptic time constants, E_L is the leak or resting potential, R_m is the membrane resistance, d is the transmission delay, is the total synaptic current, and are the incoming spike trains. When V_i reaches a threshold θ, a spike is assumed, and the membrane potential is clamped to a level V_r for a refractory period τ_ref. Threshold and reset potential in physical units are shifted by the leak potential (, ), showing that the assumption E_L = 0 in Eq (65) does not limit generality. The intrinsic dynamics of the neurons in the different populations are taken to be identical, so that population differences are only expressed in the couplings.

First and second moments of activity in the spiking network

An approximation of the stationary mean firing rate of LIF networks with exponential current-based synapses was derived in [91], (67) where the summed synaptic input is characterized by a Gaussian noise with first moment μ and second moment σ² based on the diffusion approximation, and ζ is the Riemann zeta function.

For the covariances, we follow and extend the theory developed in [45, 53], starting with the average influence of a single synapse. Assuming that the network is in the asynchronous state, and that synaptic amplitudes are small, the synaptic influences can be averaged around the mean activity r_j of each neuron j. These influences are characterized by linear response kernels h_jk(t, t′) defined as the derivative of the density of spikes of spike train s_j(t) of neuron j with respect to an incoming spike train s_k(t′), averaged over realizations of the remaining incoming spike trains s\s_k that act as noise. In the stationary state, the kernel only depends on the time difference t − t′, giving (68) where δs_j ≡ s_j − r_j is the j-th centralized (zero-mean) spike train. Here, w_jk is the integral of h_jk(t − t′), and h(t − t′) is a normalized function capturing its time dependence, which may be source- and target-specific. The dimensionless effective weights w_jk are determined nonlinearly by the synaptic strengths J_jk, the single-neuron parameters, and the working point (μ_j,σ_j) (cf. [45] Eq (A.3) but note that β as given there has a spurious factor J). We approximate the impulse response by the form Eq (58), where τ is now an effective time constant depending on the working point (μ_j,σ_j) and the parameters of the target neurons. This form of the impulse response, corresponding to a low-pass filter, appears to be a good approximation in the noisy regime when the neuron fires irregularly. In the mean-driven regime (μ ≫ σ) the transfer function of the LIF neuron is known to exhibit resonant behavior with a peak close to its firing rate. In this regime a single exponential response kernel is expected to be a poor approximation (see, e.g., [92] Fig 1). In general, the source population dependence of Eq (58) comes in through the delay d, and the target population dependence through both τ and d.

As for binary networks with delays, the average pairwise covariance functions c_ij(Δ) ≡ ⟨δs_i(t + Δ)δs_j(t)⟩_t are most conveniently derived starting from the frequency domain. In case of identical transfer functions for all populations, the matrix of average cross-covariances is given by [53] Eq (16) minus the autocovariance contribution, (69) Here, W contains the effective weights of single synapses from population β to population α times the corresponding in-degrees, w_αβ K_αβ; and A contains the population-averaged autocovariances, which we approximate as , with r_α the mean firing rate, as also done in [45]. In [53], Eq (69) was written using a more general diagonal matrix instead of A, to help clarify close similarities between binary and LIF networks and Ornstein-Uhlenbeck processes or linear rate models; however, for LIF networks, this diagonal matrix corresponds precisely to the autocovariance matrix. We chose the form Eq (69) because it separates terms that vanish at either ω → i∞ or ω → −i∞ depending on Δ. This facilitates Fourier back transformation, as contour integration with an appropriate contour can be used for each term.

To perform the Fourier back transformation, we apply the same method as used for the binary network. Let v^j,u^j be the left and right eigenvectors of the connectivity matrix W, and λ_j the corresponding eigenvalues. Insert ∑_j u^j v^jT = 𝟙 into Eq (69) on the left and right, and Fourier transform, (70) As for the binary case, we only need to consider Δ ≥ 0, as the solution for Δ < 0 is given by c(Δ) = c^T(−Δ). The contour can then be closed over the upper half plane, where the term containing only H(−ω) has no poles due to the stability condition. When Δ < d, the contour for the term containing only H(ω) can also be closed in the lower half plane where it has no poles, so that the corresponding integral vanishes. Analogously, the integral of the term with only H(−ω) vanishes when 0 > Δ > −d. Therefore, the second and third terms represent ‘echoes’ of spikes arriving after one transmission delay [53]. For Δ = 0 and d > 0, only the first term contributes, and the contour can be closed in either half plane. As before, the poles are given by Eq (61) for d > 0, and by for d = 0. The residue theorem yields a solution of the form Eq (62), the only difference being the precise form of the residues, and the fact that we here consider c as opposed to .

In the absence of delays, an explicit solution can again be derived. For Δ > 0, the poles inside the contour are corresponding to the terms with H(ω)⁻¹. The residue corresponding to is , and the term is finite and evaluates at the pole to . Using A^jk = v^jT A v^k we get (71) which is reminiscent of but not identical to Eq (55) for the binary network. Note that Eq (71) for the LIF network corresponds to spike train covariances with the dimensionality of 1/t² due to [A^jk] = [1/t] and the factor 1/τ, whereas the covariances for the binary network are dimensionless.

The population-specific generalization of Eq (69) reads (72) where M(ω) has elements H_αβ(ω)K_αβ w_αβ, as before. The covariance matrix including autocovariances can be more simply written as (73) The only difference compared to the expression Eq (63) for the binary network is the form of the diagonal matrix, here analogous to white output noise in a linear rate model, whereas the binary network resembles a linear rate model with white noise on the input side, which is passed through the transfer function before affecting the correlations [53].

Fluctuating rate equation and stability condition

An alternative description of the spiking dynamics can be obtained by considering a system of linear coupled rate equations that produces the same moments to second order as the spiking dynamics [53]. The convolution equation (74) with pairwise uncorrelated white noises x_j and the response kernel h_jk given by Eq (68) can be shown to yield a cross-covariance matrix of the form Eq (69) by considering the Fourier transform of Eq (74), written in matrix notation as (75) We can expand the latter equation into eigenmodes by multiplying from the left with the left-sided eigenvector v^k of W and by writing the general solution as a linear combination of right-sided eigenmodes Y(ω) = ∑_j η_j(ω) u^j to obtain (with the bi-orthogonality relation v^kT u^j = δ_kj) (76) The latter equation shows that the same poles z(λ_k) that appear in the covariance function Eq (70) also determine the evolution of the effective rate equation. Moreover, transforming Eq (76) back to the time domain, we see with that the eigenmodes have a time evolution determined by e^iz(λ_k)t. Hence the imaginary part of the pole z(λ_k) controls whether the mode is exponentially growing (Im(z) < 0) or decaying (Im(z) > 0), while the real part determines the oscillation frequency.