Statistically inferred neuronal connections in subsampled neural networks strongly correlate with spike train covariances

Statistically inferred neuronal connections from observed spike train data are often skewed from ground truth by factors such as model mismatch, unobserved neurons, and limited data. Spike train covariances, sometimes referred to as “functional connections,” are often used as a proxy for the connections between pairs of neurons, but reflect statistical relationships between neurons, not anatomical connections. Moreover, covariances are not causal: spiking activity is correlated in both the past and the future, whereas neurons respond only to synaptic inputs in the past. Connections inferred by maximum likelihood inference, however, can be constrained to be causal. However, we show in this work that the inferred connections in spontaneously active networks modeled by stochastic leaky integrate-and-fire networks strongly correlate with the covariances between neurons, and may reflect noncausal relationships, when many neurons are unobserved or when neurons are weakly coupled. This phenomenon occurs across different network structures, including random networks and balanced excitatory-inhibitory networks. We use a combination of simulations and a mean-field analysis with fluctuation corrections to elucidate the relationships between spike train covariances, inferred synaptic filters, and ground-truth connections in partially observed networks.

Figure 1

Schematics of the hidden neuron problem and effective neuronal connection inference. Recording techniques are often only able to record from a subset of neurons in a neural circuit, especially in the tissue of a living organism. In this schematic three neurons' spike trains are shown to be recorded, while the activity of other nearby neurons remains unobserved. Statistical inference techniques applied to this activity data can predict effective connections between neurons, but these do not necessarily reflect the true anatomical connections.

Figure 2

Maximum-likelihood estimates (MLE) of coupling filters $J_{ij}$ for $N_{\mathrm{obs}}=3$ observed neurons out of 64 in a circuit with $50\%$ sparsity and Gaussian weights. The ground-truth synaptic filters of the generative model are shown in blue dashed lines (values on left axis). Two types of inferred filters are shown: for each time point (red points) and using a basis expansion (red solid line). We also compare the filters to the empirical covariances, shown in grey bars (scaled up to be visible on the same plots as the filters; true values on right axes). The covariances correlate strongly with the filters inferred without using a basis expansion.

Figure 3

Directed magnitudes $\left|\right|C_{ij}\left|\right|=\sqrt{{\int }_{0^{+}}^{\infty }dt{C}_{ij}{\left(t\right)}^2}$ of the empirical spike train covariances estimated with the simulated spike trains. (a) (Log-)Magnitudes for a sparse random network. Top: Directed magnitude of the positive-lag of the autocovariance with zero-lag excluded; bottom: directed magnitudes of the crosscovariances. (b) (Log-)Magnitudes for a balanced excitatory-inhibitory network. Left-most column: directed magnitudes for the autocovariances with zero-lag excluded, separated by cell type (51 excitatory, 13 inhibitory). Middle- and right-most columns: directed magnitudes for the crosscovariances, separated by which pairs of cell types are connected. For the crosscovariances the histograms are separated also by whether the ground-truth connection is zero or not, showing that there is overlap between the covariances of connected and unconnected pairs of neurons.

Figure 4

Pearson correlation coefficient between the empirical spike train covariances and ground-truth synaptic filters. (a) Correlations for a sparse random network. Top: correlation between the positive-lag of the autocovariance and the autocoupling (self-history) filter; bottom: correlation between the crosscovariances and the crosscoupling filters between pairs of neurons. (b) Correlations for a balanced excitatory-inhibitory network. Left-most column: correlations between autocovariances and autocoupling, separated by cell type (51 excitatory, 13 inhibitory). Middle- and right-most columns: correlations between the crosscovariances and crosscouplings, separated by which pairs of cell types are connected. For the crosscovariances/couplings correlations are only computed for synaptic connections which are nonzero.

Figure 5

Pearson correlation between the spike train covariances and inferred coupling filters in strongly coupled random networks and balanced E-I networks of 64 neurons. (a) Violin plots show how the correlations change when different fractions of neurons are observed in a network with sparse Gaussian synaptic connections, $3\%, 6\%, 12\%, 25\%, 50\%, 75\%$, and $100\%$. In this 64 neuron network these fractions corresponds to 2, 4, 8, 16, 32, 48, and 64 observed neurons. The spike train covariance functions strongly correlate with the inferred filters inferred for each time point when fewer neurons are observed and the correlation decreases as more neurons are observed. The left panel shows the distribution of the Pearson correlation coefficients between the autocovariance and autocoupling (self-history) filters, while the right panel shows the distribution of the Pearson correlation coefficients between the crosscovariance and coupling filters between neuron pairs. (b) Same as panel (a) but for the balanced E-I networks.

Figure 6

Pearson correlation between the inferred spike train filters and the ground-truth filters in strongly coupled random networks and balanced E-I networks of 64 neurons. (a) For sparse random networks. (b) For balanced E-I networks. Plot details are the same as in Fig. 5. The correlation between ground-truth and inferred filters are highest when the network is fully observed, and are relatively high for the E-I network, but can be negative in the random network. The left panels show the distribution of the Pearson correlation coefficients between the autocouplings (self-connections implementing the soft reset), while the right panels show the distribution of correlation coefficients between the crosscouplings between neuron pairs.

Figure 7

Correlations are high for both random and balanced excitatory-inhibitory networks in weak coupling regimes. (a) Random network. When the weight matrix coefficient $J_0$ decreased from 3 to 1, the network transitioned into a noise-driven regime, and the mean firing rates of the neurons varied less and were driven by the same baseline drive in the generative model, as shown in the left panel. As shown in the right panel, in the weak coupling regime, such as when $J_0=1$, high correlations between the MLE inferred filters and spike train covariances were observed even when all the neurons in the network are observed, as compared to the network in a strong coupling regime where the high correlation between the MLE inferred filters and spike train covariances only happen in the sub-sampled network. (b) Similar results hold for a balanced excitatory-inhibitory network, where $20\%$ of the neurons are inhibitory and $80\%$ of them are excitatory. The weight matrix coefficient $J_0$ is tuned from 1 to 7, with $J_0=7$ the largest possible integer value for which the spiking process is still stable.

Figure 8

Directed magnitudes $\left|\right|C_{ij}\left|\right|=\sqrt{{{\int }_{0^{+}}^{\infty }dt{C}_{ij}{\left(t\right)}^2}^{\phantom{\int }}}$ of the spike train covariances calculated using mean-field theory. (a) (Log-)Magnitudes for a sparse random network. Top: Autocovariance magnitudes; bottom: crosscovariance magnitudes. (b) (Log-)Magnitudes for a balanced excitatory-inhibitory network. Left-most column: Autocovariance magnitudes separated by cell type. Middle- and right-most columns: crosscovariance magnitudes separated by which pairs of cell types are connected. For the crosscovariances the histograms are separated also by whether the ground-truth connection is zero or not, showing that there is overlap between the covariances of connected and unconnected pairs of neurons.

Figure 9

Pearson correlation coefficient between the mean-field spike train covariance estimates and ground-truth synaptic filters. (a) Correlations for a sparse random network. Top: correlation between the positive-lag of the autocovariance and the autocoupling (self-history filter); bottom: correlation between the crosscovariances and the crosscoupling filters between pairs of neurons. (b) Correlations for a balanced excitatory-inhibitory network. Left-most column: correlations between autocovariances and autocoupling, separated by cell type. Middle- and right-most columns: correlations between the crosscovariances and crosscouplings, separated by which pairs of cell types are connected. For the crosscovariances/couplings correlations are only computed for synaptic connections which are nonzero in ground truth.

Figure 10

Pearson correlation between the mean-field theory predictions of the spike train covariances and inferred coupling filters in strongly coupled random networks and balanced E-I networks of 64 neurons. (a) Results for sparse random networks. (b) Results for balanced E-I networks. The left panels show the correlation coefficients between the autocovariance and autocoupling (self-history) filters, while the right panels show the distribution of correlation coefficients between the crosscovariance and coupling filters. For both types of networks we use the same synaptic weight matrices as the simulations shown in Fig. 5. We estimate the covariances using Eq. (7) and the inferred filters using Eq. (15). Violin plots show how the correlations change when different numbers of neurons are observed in a network with sparse Gaussian synaptic connections; the percentages shown for this 64 neuron network correspond to 2, 4, 8, 16, 32, and 48, observed. For small $N_{\mathrm{obs}}$ several different pairs are randomly sampled. In the fully observed case, $N_{\mathrm{obs}}=64$, the inferred and ground-truth filters are perfectly correlated, and the corresponding correlation is shown in Fig. 9. The mean-field estimates qualitatively reproduce the trends seen in simulations, shown in Fig. 5, displaying large correlations when only a small fraction of the network is observed. The vertical axis has been truncated to [0,1] to show details more clearly, cutting off the long tails of the crosscovariance distributions, which extend close to $-1$. i.e., the mean-field theory estimates that several inferred filters are anticorrelated with the spike train covariances.

Figure 11

Pearson correlation between the mean-field estimates of the inferred spike train filters and the ground-truth filters in strongly coupled random networks and balanced E-I networks of 64 neurons. (a) Results for sparse random networks. (b) Results for balanced E-I networks. The left panels show the correlation coefficients between the autocovariance and autocoupling (self-history) filters, while the right panels show the distribution of correlation coefficients between the crosscovariance and coupling filters. Other details are the same as Fig. 10. The results qualitatively reproduce the trends seen in simulations, shown in Fig. 6, albeit with a quantitatively weaker agreement, particularly for the sparse random network.

Figure 12

Dependence of fits on the number of observed neurons in a homogeneous network. (a) The empirical covariance looks similar to the ground-truth filter, but shows clear differences in both simulations and the mean-field approximation (inset), especially for stronger networks. Simulation parameters: $N=64, J=0.00925,0.0185,0.037, \mu =-2, {\lambda }_0=1, 4\times {10}^6$ time points. (b) For the strongest synaptic weight ($J=0.037$), the inferred filters decrease in amplitude as the number of observed neurons $N_{\mathrm{obs}}$ increases to the total number of neurons in the network $N$, observed in fits to simulated data and our mean-field analysis (inset). In the ground-truth model the autocoupling (self-history) filters ${\widehat{J}}_{\mathrm{auto}}\left(t\right)$ and crosscoupling filters ${\widehat{J}}_{\mathrm{cross}}\left(t\right)$ are identical, as is the theoretical prediction from solving Eq. (14). The inferred auto- and crosscoupling filters differ due to finite data effects, but are close. (c) The predicted overlap ${\rho }_{C\widehat{J}}$ drops as the synaptic weight $J$ increases (i.e., $NJ{\lambda }_0{e}^{1+\mu }$ increases), with varying speed depending on the ratio $N_{\mathrm{obs}}/N$. $N_{\mathrm{obs}}/N=1$ corresponds to the fully observed case. The inset shows an enlarged view of the drop. Compare to Fig. 5. (d) The predicted overlap between inferred and ground-truth filters drops as the synaptic weight $J$ increases (i.e., $NJ{\lambda }_0{e}^{1+\mu }$ increases), approaching a finite limit of $\sim {(N_{\mathrm{obs}}/N)}^{1/4}$. The correlation between covariance and ground-truth filters, ${\rho }_{CJ}$, is included for comparison in both panels (c) and (d), being a limiting boundary in both cases.