Functional connectivity in a rhythmic inhibitory circuit using Granger causality

Abstract

Background: Understanding circuit function would be greatly facilitated by methods that allow the simultaneous estimation of the functional strengths of all of the synapses in the network during ongoing network activity. Towards that end, we used Granger causality analysis on electrical recordings from the pyloric network of the crab Cancer borealis, a small rhythmic circuit with known connectivity, and known neuronal intrinsic properties.

Results: Granger causality analysis reported a causal relationship where there is no anatomical correlate because of the strong oscillatory behavior of the pyloric circuit. Additionally, we failed to find a direct relationship between synaptic strength and Granger causality in a set of pyloric circuit models.

Conclusions: We conclude that the lack of a relationship between synaptic strength and functional connectivity occurs because Granger causality essentially collapses the direct contribution of the synapse with the intrinsic properties of the postsynaptic neuron. We suggest that the richness of the dynamical properties of most biological neurons complicates the simple interpretation of the results of functional connectivity analyses using Granger causality.

Figure 1

Canonical pyloric triphasic rhythm. (A) Simultaneous intracellular recordings from the core pyloric circuit neurons (top three traces) and an extracellular recording of a motor nerve (lvn) during an ongoing pyloric circuit oscillation. Cell abbreviations are: AB/PD (anterior burster/pyloric dilator, top trace, blue), LP (lateral pyloric neuron, middle trace, green), PY (pyloric neuron, bottom trace, purple). Spiking in all three cells can be measured concurrently on the lateral ventricular nerve (lvn, bottom row, colors demarcate spikes from each individual neuron and illustrate the temporal segregation of each neuron's spike times). (B) Simplified connectivity diagram of the pyloric circuit showing the individual neurons color coded as in (A) and the major synaptic connections between the cells. Notably, there is no synapse from the PY to the PD neuron. Synapses are also color coded as in (A) with the presynaptic cell determining color. All chemical synapses are inhibitory.

Figure 2

Granger Causality analysis of biological stomatogastric ganglion (STG) circuits. (A-D). Extracellular recordings show the activity of the PD neurons (blue), the LP (green), and PY (purple) neurons from four different animals. Variability in period is visible between animals; but the phase relationships (relative times of firing) are maintained. Estimations of the functional relationships between the neurons using Granger causality (GC) are shown in the right panels and revealed a circular pattern of causal relationships in the network. Neurons are represented as circles and functional connections, as predicted by GC analysis, as lines between them with the diamond tips indicating the directionality of the connection. Each functional connection shown fell below the threshold for statistical significance of 0.05 (adjusted for multiple comparisons to 0.008). All parameters were kept constant during analysis across different data sets except for the total length of the input trace, ranging in these examples between 3 - 10 seconds of data. The auto-regressive model time lag parameter was fixed at 400 ms for all networks

Figure 3

Granger Causality analysis correctly identifies the relationship between correlated noise. (A - E). Left panels show randomly generated correlated noise (A - E, black and gray lines, S1 and S2 respectively). Correlation values ranged between perfectly anti-correlated (-100%) to perfectly correlated (+100%). In all cases (A - E, panels plot traces with correlation values of -80%, -20%, 0%, 20% and 80%), an artificial delay of 200 ms is introduced into the correlation from the black trace (S1) to the gray trace (S2). The cross-correlation (A - E, top right panels) shows a peak proportional to the amount of correlation used to generate the noise and a peak offset by the delay value of 200 ms. For each value of correlation, we compute the Granger causality (A - E, bottom right panels). Schematic network diagrams summarize the predicted Granger causality (GC) values by the thickness of the lines from S1 to S2 and diamond tips indicate the direction of the causal relationship. In all cases where a significant GC value is computed, S2 is Granger caused by S1 as expected by the time lag relationship. Note that for 0% correlation (for example, two independent noise traces), the computed GC value does not fall above the threshold for significance (C, bottom right panel, P = 0.33, dashed line). For either positive or negative correlation at 20% the computed GC is significant (B: -20%, P = 1.22 × 10^-6; D: +20%, P = 1.63 × 10^-4). For 80% correlated noise GC is highly significant (A: -80%, P < 1 × 10^-6; E: +80%, P < 1 × 10^-6).

Figure 4

Granger Causality analysis correctly identifies the inhibitory relationship between model neurons. (A) Three quadratic integrate-and-fire Izhikevitch neurons [48] were coupled in a cyclically inhibitory fashion (network topology schematic in panel C) and made to fire slowly and with variability (rate = 2.5 Hz with CV = 0.6, black, dark gray, light gray traces correspond to individual model cells). (B) Traces show an average spike (top panel) and average voltage waveforms in each cell in a 60 ms time window triggered on a spike occurring in the presynaptic neuron (bottom three panels). Trace gray scales correspond to the legend in A. (C) Network layout includes three cells that inhibit each other in a cyclical fashion. Synaptic maximal conductances are all uniformly set to 0.1 nS. Filled circles indicate inhibition. Gray scales of cells (numbered 1, 2, 3) correspond to the legend in A. (D) Granger causality analysis predicts a causal relationship closely matching synaptic architecture. Filled triangles indicate the directionality of the statistically significant Granger causal relationships between cells. Granger values and P-values are indicated in the figure. (E) Pyloric-like network topology comparable to that shown in Figure 1B. Synaptic coupling between cells 1 and 2 is reciprocal as is the synaptic coupling between cell 2 and cell 3. Cell 1 additionally inhibits cell 3 but cell 3 does not synapse onto cell 1. (F) Granger causality analysis succeeded in reconstructing this network architecture. Statistically significant Granger values were found for all synaptic connections between cells 1 and 2 as well as for cells 2 and 3. For the synapse from cell 1 to cell 3 the Granger value came close to the threshold for statistical significance (line drawn as dashed to indicate lack of significance). The Granger value for the non-existent synapse from cell 3 to cell 1 (not drawn) was non-significant (P = 0.33).

Figure 5

Granger causality values do not correlate with synaptic strengths in a set of pyloric oscillator model networks. We selected a set of 84 model networks all of which displayed representative pyloric-like outputs and which had diverse underlying biophysical implementations in terms of their ionic and synaptic conductances. For each synapse between the three model cells (PD, pyloric dilator; LP, lateral pyloric; PY, pyloric neuron) we computed a Granger causality (GC) value (panels A - E). We then plotted this value against the synaptic strength given to that connection in that network (black dots each represent the values from a single network). We found no significant linear relationship between synaptic strength and GC value for any of the five synapses (r² values ranged from 0.00 - 0.05). Panel F shows the GC values predicted for the PY to PD synapse which is known to have no anatomical analog (maximal conductance for that synapse was 0.0 in all model networks). We separated the networks into two groups: pyloric (black dots) and pyloric-like (gray dots) and found that the pyloric group had a significantly higher GC values on average (Kruskal-Wallis test, P = 0.0044, n_pyloric = 29, n_pyloric-like = 55, mean_pyloric = 0.22, mean_pyloric-like = 0.15, error bars are standard error of the mean (SEM)).

Figure 6

Convolution of binary spike trains with a half-Gaussian is used to prepare data for analysis with Granger causality. Illustration of the transform applied to spiking data before running Granger causality analysis. Several example spike trains are shown in gray (A: regular spiking, B: slow bursting, C: fast bursting). In general, raw voltage traces were rasterized (that is, converted to a series of delta functions at the time of each spike) and then convolved with a half-Gaussian function. This converts a point process into a smoother rate function (black traces) that is better suited to Granger causality analysis.