Nonlinear dynamic causal models for fMRI

Abstract

Models of effective connectivity characterize the influence that neuronal populations exert over each other. Additionally, some approaches, for example Dynamic Causal Modelling (DCM) and variants of Structural Equation Modelling, describe how effective connectivity is modulated by experimental manipulations. Mathematically, both are based on bilinear equations, where the bilinear term models the effect of experimental manipulations on neuronal interactions. The bilinear framework, however, precludes an important aspect of neuronal interactions that has been established with invasive electrophysiological recording studies; i.e., how the connection between two neuronal units is enabled or gated by activity in other units. These gating processes are critical for controlling the gain of neuronal populations and are mediated through interactions between synaptic inputs (e.g. by means of voltage-sensitive ion channels). They represent a key mechanism for various neurobiological processes, including top-down (e.g. attentional) modulation, learning and neuromodulation. This paper presents a nonlinear extension of DCM that models such processes (to second order) at the neuronal population level. In this way, the modulation of network interactions can be assigned to an explicit neuronal population. We present simulations and empirical results that demonstrate the validity and usefulness of this model. Analyses of synthetic data showed that nonlinear and bilinear mechanisms can be distinguished by our extended DCM. When applying the model to empirical fMRI data from a blocked attention to motion paradigm, we found that attention-induced increases in V5 responses could be best explained as a gating of the V1-->V5 connection by activity in posterior parietal cortex. Furthermore, we analysed fMRI data from an event-related binocular rivalry paradigm and found that interactions amongst percept-selective visual areas were modulated by activity in the middle frontal gyrus. In both practical examples, Bayesian model selection favoured the nonlinear models over corresponding bilinear ones.

Figure 1. comparison of bilinear and nonlinear DCMs

This figure shows schematic examples of bilinear (A) and nonlinear (B) DCMs, which describe the dynamics of a neuronal state vector x. In both equations, the matrix A represents the fixed (context-independent or endogenous) strength of connections between the modelled regions, the matrices B⁽ⁱ⁾ represent the context-dependent modulation of these connections, induced by the i^th input u_i, as an additive change, and the C matrix represents the influence of direct (exogenous) inputs to the system (e.g. sensory stimuli). The new component in the nonlinear equations are the D^(j) matrices, which encode how the n regions gate connections in the system. Specifically, any non-zero entry $D_{\mathit{kl}}^{\left(j\right)}$ indicates that the responses of region k to inputs from region l depend on activity in region j.

Figure 2. example of a simple nonlinear DCM

The right panel shows synthetic neuronal and BOLD time-series that were generated using the nonlinear DCM shown on the left. In this model, neuronal population activity x₁ (blue) is driven by irregularly spaced random events (delta-functions). Activity in x₂ (green) is driven through a connection from x₁; critically, the strength of this connection depends on activity in a third population, x₃ (red), which receives a connection from x₂ but also receives a direct input from a box-car input. The effect of nonlinear modulation can be seen easily: responses of x₂ to x₁ become negligible when x₃ activity is low. Conversely, x₂ responds vigorously to x₁ inputs when the x₁→x₂ connection is gated by x₃ activity. Strengths of connections are indicated by symbols (−: negative; +: weakly positive; +++: strongly positive).

Figure 3. nonlinear and bilinear DCMs used for simulating data

The nonlinear (A) and bilinear (B) DCM used for generation of synthetic data. As in Fig. 2, the first input (u₁) comprises an irregular sequence of random events (delta-functions), whereas the second input (u₂) corresponds to a box-car function. Strengths of connections are annotated as in Fig. 2.

Figure 4. BMS of correct and incorrect DCMs for synthetic data sets

This figure summarises the results of the Bayesian model comparisons between correct and incorrect models applied to synthetic data generated by nonlinear and bilinear models (shown by Fig. 3), under two levels of noise. The first two plots (A, B) contain the results for data with low signal-to-noise (SNR=2), and the last two plots (C, D) show the results for data with high signal-to-noise (SNR=5). Plots A and C show the log evidence differences between the nonlinear (NL) model and the bilinear (BL) model for 20 synthetic data sets generated by a nonlinear model. Conversely, plots B and D show the log evidence differences between the NL model and the BL model for data generated by a BL model. The dashed horizontal lines indicate a log evidence difference of ≈1.1, corresponding to a Bayes factor of ≈3 which is classically regarded as “positive” evidence for one model over another (Kass & Raftery 1995). The solid horizontal lines denote a log evidence difference of ≈3, corresponding to a Bayes factor of ≈20 which is conventionally considered to represent “strong” evidence for one model over another (Kass & Raftery 1995). It can be seen that in the majority of cases the correct model is identified as superior with at least positive evidence.

Figure 5. MAP estimates for synthetic data sets

This figure plots the results of our analysis how well the true values of nonlinear and bilinear modulatory parameters can be estimated in the presence of noise. We assessed this by checking if the true parameter values fell within the 95% confidence interval based on the sample density of the maximum a posteriori (MAP) parameter estimates over 20 synthetic data sets. The four plots in this figure show the MAP estimates of modulatory parameters, obtained from fitting nonlinear models (upper row) or bilinear models (lower row) to synthetic data generated by the same type of model. The left column contains the results for data with low signal-to-noise (SNR=2), whereas the right column contains the results for high signal-to-noise data (SNR=5). The true values of the modulatory parameters are indicated by the dashed lines (nonlinear models: D = 1; bilinear models: B = 0.3). The dotted lines indicated the average MAP estimates across data sets. In the low SNR case the true values of both the nonlinear and bilinear modulatory parameters were not contained in the 95% confidence interval of the respective estimates (nonlinear: 0.806 ± 0.063; bilinear: 0.261 ± 0.022). This overly conservative estimation of the modulatory parameters is due to the effect of the zero-mean shrinkage priors in DCM and has been observed in previous simulations (Kiebel et al. 2007). In contrast, in the high SNR case the true values of both the nonlinear and bilinear modulatory parameters fell within the 95% confidence interval of the respective estimates (nonlinear: 0.957 ± 0.072; bilinear: 0.286 ± 0.016).

Figure 6. BMS of different DCMs for the attention to motion data set

Summary of the model comparison results for the attention to motion data set. For reasons of clarity, we do not display a bilinear modulation of the V1→V5 connection by motion, which is present in every model. The best model was a nonlinear one (model M4), in which attention-driven activity in posterior parietal cortex (PPC) was allowed to modulate the V1→V5 connection. BF = Bayes factor.

Figure 7. posterior density of the nonlinear modulation of the V1→V5 connection in model M4

(A) Maximum a posteriori estimates of all parameters in the optimal model for the attention to motion data (model M4, see Fig. 6). PPC = posterior parietal cortex. (B) Posterior density of the estimate for the nonlinear modulation parameter for the V1→V5 connection. Given the mean and variance of this posterior density, we have 99.1% confidence that the true parameter value is larger than zero or, in other words, that there is an increase in gain of V5 responses to V1 inputs that is mediated by PPC activity.

Figure 8. fit of the nonlinear model M4 to the data

Fit of the nonlinear model to the attention to motion data (model M4, see Figs. 4 and 5). Dotted lines represent the observed data, solid lines the responses predicted by the nonlinear DCM. The increase in the gain of V5 responses to V1 inputs during attention is clearly visible.

Figure 9. nonlinear DCM results for binocular rivalry data

(A) The structure of the nonlinear DCM fitted to the binocular rivalry data, along with the maximum a posteriori estimates of all parameters. The intrinsic connections between FFA and PPA are negative in both directions; i.e. FFA and PPA mutually inhibited each other. This may be seen as an expression, at the neurophysiological level, of the perceptual competition between the face and house stimuli. This competitive interaction between FFA and PPA is modulated nonlinearly by activity in the middle frontal gyrus (MFG), which showed higher activity during rivalry vs. non-rivalry conditions. (B) Our confidence about the presence of this nonlinear modulation is very high (99.9%), for both connections.

Figure 10. observed and fitted responses of nonlinear DCM for binocular rivalry data

Fit of the nonlinear model in Fig. 9A to the binocular rivalry data. Dotted lines represent the observed data, solid lines the responses predicted by the nonlinear DCM. The upper panel shows the entire time series. The lower panel zooms in on the first half of the data (dotted box). One can see that the functional coupling between FFA (blue) and PPA (green) depends on the activity level in MFG (red): when MFG activity is high during binocular rivalry blocks (BR; short black arrows), FFA and PPA are strongly coupled and their responses are difficult to disambiguate. In contrast, when MFG activity is low, during non-rivalry blocks (nBR; long grey arrows), FFA and PPA are less coupled, and their activities evolve more independently.