Table 1.
Notation used in the text.
Figure 1.
(A) The input to the post-synaptic cell is a fixed spike train which is convolved with a synaptic kernel. (B) A sample voltage path for the post-synaptic cell receiving the input shown in A) in the presence of background noise. (C) Raster plot of 100 realizations of output spike trains of the post-synaptic cell. (D) The output firing rate, , obtained by averaging over realizations of the output spike trains in C). The rate obtained using Monte Carlo simulations (shaded in gray) matches predictions of linear response theory obtained using Eq. (4) (black).
Figure 2.
Iterative construction of the linear approximation to network activity.
(A) An example recurrent network. (B)–(D) A sequence of graphs determines the successive approximations to the output of neuron 1. Processes defined by the same iteration of Eq. (11) have equal color. (B) In the first iteration of Eq. (11), the output of neuron 1 is approximated using the unperturbed outputs of its neighbors. (C) In the second iteration the results of the first iteration are used to define the inputs to the neuron. For instance, the process depends on the base process
which represents the unperturbed output of neuron 1. Neuron 4 receives no inputs from the rest of the network, and all approximations involve only its unperturbed output,
. (D) Cells 3 and 4 are not part of recurrent paths, and their contributions to the approximation are fixed after the second iteration. However, the recurrent connection between cells 1 and 2 implies that subsequent approximations involve contributions of directed chains of increasing length.
Figure 3.
The relation between correlation structure and response statistics in a feed-forward inhibitory microcircuit.
(A) The FFI circuit (left) can be decomposed into three submotifs. Equation (18) shows that each submotif provides a specific contribution to the cross-correlation between cells and
. (B) Comparison of the theoretical prediction with the numerically computed cross-correlation between cells
and
. Results are shown for two different values of the inhibitory time constant,
(
ms, solid line,
ms, dashed line). (C) The contributions of the different submotifs in panel A are shown for both
ms (solid) and
ms (dashed). Inset shows the corresponding change in the inhibitory synaptic filter. The present color scheme is used in subsequent figures. Connection strengths were
for excitatory and inhibitory connections. In each case, the long window correlation coefficient
between the two cells was
.
Figure 4.
The relation between correlation structure and response statistics for two bidirectionally coupled, excitatory cells.
(A) The cross-correlation between the two cells can be represented in terms of contributions from an infinite sequence of submotifs (See Eq. (20)). Though we show only a few “chain” motifs in one direction, one should note that there will also be contributions to the cross-correlation from chain motifs in the reverse direction in addition to indirect common input motifs (See the discussion of Figure 5). (B), (E) Linear response kernels in the excitable (B) and oscillatory (E) regimes. (C), (F) The cross-correlation function computed from simulations and theoretical predictions with first and third order contributions computed using Eq. (19) in the excitable (C) and oscillatory (F) regimes. (D), (G) The auto-correlation function computed from simulations and theoretical predictions with zeroth and second order contributions computed using Eq. (19) in the excitable (D) and oscillatory (G) regimes. In the oscillatory regime, higher order contributions were small relative to first order contributions and are therefore not shown. The network's symmetry implies that cross-correlations are symmetric, and we only show them for positive times. Connection strengths were . The long window correlation coefficient
between the two cells was
in the excitable regime and
in the oscillatory regime. The ISI CV was approximately 0.98 for neurons in the excitable regime and 0.31 for neurons in the oscillatory regime.
Figure 5.
The motifs giving rise to terms in the expansion of Eq. (15).
(A) Terms containing only unconjugated (or only conjugated) interaction kernels correspond to directed chains. (B) Terms containing both unconjugated and conjugated interaction kernels
correspond to direct or indirect common input motifs.
Figure 6.
All–to–all networks and the importance of higher order motifs.
(A) Some of the submotifs contributing to correlations in the all–to–all network. (B) Cross-correlations between two excitatory cells in an all–to-all network () obtained using Eq. (21) (Solid – precisely tuned network with
[
], dashed – non-precisely tuned network with
[
]). (C) Comparison of first and second order contributions to the cross-correlation function in panel A in the precisely tuned (left) and non-precisely tuned (right) network. In both cases, the long window correlation coefficient
was 0.05.
Figure 7.
Correlations in random, fixed in-degree networks.
(A) A comparison of numerically obtained excitatory-inhibitory cross-correlations to the approximation given by Eq. (26). (B) Mean and standard deviation for the distribution of correlation functions for excitatory-inhibitory pairs of cells. (Solid line – mean cross-correlation, shaded area – one standard deviation from the mean, calculated using bootstrapping in a single network realization). (C) Mean and standard deviation for the distribution of cross-correlation functions conditioned on cell type and first order connectivity for a reciprocally coupled excitatory-inhibitory pair of cells. (Solid line – mean cross-correlation function, shaded area – one standard deviation from the mean found by bootstrapping). (D) Average reduction in error between cross-correlation functions and their respective first-order conditioned averages, relative to the error between the cross-correlations and their cell-type averages. Blue circles give results for a precisely tuned network, and red squares for a network with stronger, faster inhibition. Error bars indicate two standard errors above and below the mean.
for panels A-C are as in the precisely tuned network of Figure 6, and the two networks of panel D are as in the networks of the same figure.