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.

Introduction

Models of effective connectivity, i.e. the causal influences that system elements exert over another, are essential for studying the functional integration of neuronal populations and for understanding the mechanisms that underlie neuronal dynamics (Friston 2002; Horwitz et al. 1999). In the past, a variety of models have been proposed for inferring effective connectivity from neuroimaging data, including regression-based models like psycho-physiological interactions (PPI; Friston et al. 1997), structural equation modelling (SEM; McIntosh & Gonzalez-Lima 1994; Büchel & Friston 1997; Bullmore et al. 2000), multivariate autoregressive models (MAR; Harrison et al. 2003; Roebroeck et al. 2005) and dynamic causal modelling (DCM; Friston et al. 2003).

DCM is a general framework for inferring processes and mechanisms at the neuronal level from measurements of brain activity with different techniques, including fMRI (Friston et al. 2003), EEG/MEG (David et al. 2006) and frequency spectra based on local field potentials (Moran et al. 2008). In contrast to other models of effective connectivity, DCM does not operate on the measured time-series directly. Instead, it combines a model of the hidden neuronal dynamics with a forward model that translates neuronal states into predicted measurements. For fMRI, DCM is based on bilinear differential equations describing neuronal population dynamics, which are combined with a hemodynamic forward model. Since its original description (Friston et al. 2003), a number of methodological developments have improved and extended DCM for fMRI, e.g. Bayesian model selection amongst alternative DCMs (Penny et al. 2004a), precise sampling from predicted responses (Kiebel et al. 2007), additional states at the neuronal level (Marreiros et al. 2008) and a refined hemodynamic model (Stephan et al. 2007a). In this paper, we describe a novel DCM for fMRI that enables one to model a class of nonlinear neuronal processes, which are important for a variety of cognitive processes, including learning and attention.

Effective connectivity is inherently context-dependent and dynamic: there is a lot of evidence that the functional coupling amongst neuronal populations changes as a function of processing demands (for reviews, see McIntosh 2000; Stephan 2004). Therefore, models of effective connectivity are typically used to infer whether the functional coupling is modulated by experimental manipulations; e.g. task demands (Mechelli et al. 2003; McIntosh et al. 1994; Sonty et al. 2007; Stephan et al. 2007b), stimulus properties (Fairhall & Ishai 2007; Haynes et al. 2005), learning (Büchel et al. 1999; McIntosh et al. 1998), drugs (Honey et al. 2003) or TMS (Lee et al. 2003). As discussed by Penny et al. (2004b), to characterize context-dependent changes in coupling, early models of effective connectivity divided the data into condition-specific subsets and applied separate linear models to each subset; later developments used bilinear equations, allowing known input functions (which represent the experimentally controlled context variable) to change connection strengths (see Fig. 1).

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.

The bilinear model has two important limitations. First, the neuronal origin of the modulatory influence is not specified. Second, it may not be the most appropriate framework for modelling fast changes in effective connectivity, which are mediated by nonlinear effects at the level of single neurons. These mechanisms are instances of “short-term synaptic plasticity” (STP), an umbrella term for a range of processes which alter synaptic strengths with time constants in the range of milliseconds to minutes; e.g. NMDA-controlled rapid trafficking of AMPA receptors (Malinow & Malenka 2002), synaptic depression/facilitation (Zucker & Regehr 2002) or “early LTP” (Frey & Morris 1998) ¹. All these processes are driven by the history of prior synaptic activity and are thus nonlinear (Zucker & Regehr 2002).

A particularly interesting mechanism, which relies on STP is “neuronal gain control” (Freeman 1979). This is a general and fundamental mechanism for a large range of processes, including eye and limb movements, spatial perception and, perhaps most significantly, attention (Salinas & Thier 2000). Neuronal gain, i.e. the response of a given neuron N₁ to presynaptic input from a second neuron N_2, depends on the history of inputs that N₁ receives from other neurons, e.g. a third neuron N₃. Such a nonlinear modulation or “gating” of the N₂→ N₁ connection by N₃ has been shown to have the same mathematical form across a large number of experiments (e.g. Chance et al. 2002; McAdams & Maunsell 1999a, 1999b; Larkum et al. 2004; for review, see Salinas & Sejnowski 2001): the change in the gain of N₁ results from a multiplicative interaction among the synaptic inputs from N₂ and N₃, i.e. a second-order nonlinear effect. Biophysically, neuronal gain control can arise through various mechanisms that mediate interactions among synaptic inputs, occurring close in time but not necessarily in the same dendritic compartment. These neurophysiological mechanisms are described in more detail in the Discussion.

Critically, the bilinear framework precludes a representation, at the neuronal level, of the mechanisms described above. As stated in the original DCM paper (Friston et al. 2003), in order to model processes like neuronal gain control and synaptic plasticity properly, one needs “to go beyond bilinear approximations to allow for interactions among the states. This is important when trying to model modulatory or nonlinear connections such as those mediated by backward afferents that terminate predominantly in the supragranular layers and possibly on NMDA receptors.”

One might wonder, however, whether these nonlinearities can be neglected in models of fMRI data, due to the fact that (i) fMRI records the responses of large neuronal populations, whose ensemble activity can often be well characterised by linear approximations (despite the highly nonlinear behaviour of individual neurons; Deco et al. 2007), and that (ii) the hemodynamic transfer function, which has low-pass filtering properties, may destroy most or all of the nonlinearities (that occur over short periods of time). Therefore, an important question is whether nonlinear and bilinear modulatory processes can be distinguished reliably in fMRI data and, if so, how much can be gained in practice by using nonlinear, as opposed to bilinear, models of effective connectivity. In this technical paper, which establishes the nonlinear framework for DCM, we perform analyses of both synthetic and empirical data to address these questions. First, we use synthetic data and Bayesian model selection (BMS) to demonstrate that nonlinear and bilinear mechanisms of generating fMRI data can be reliably distinguished, even at reasonably high levels of observation noise. Second, we apply both nonlinear and bilinear DCMs to two empirical fMRI studies. Theses studies look at attention and binocular rivalry; processes for which nonlinear mechanisms have been proposed on the basis of electrophysiological recordings. Using BMS, we demonstrate that, in both cases, nonlinear DCMs are superior to corresponding bilinear DCMs.

Methods

Bilinear Dynamic Causal Modelling (DCM)

DCM for fMRI is based on an input-state-output model of deterministic neuronal dynamics in a system of n interacting brain regions. In this model, neuronal population activity of each region is represented by a single state variable and is perturbed by experimentally controlled (and therefore known) inputs u. DCM models the temporal change of the neuronal state vector x around the system's resting state (i.e., x₀=0, u₀=0), using a bilinear Taylor series approximation to any nonlinear function f(x,u) that governs the dynamics of the system:

$$\begin{array}{cc}\hfill & f(x,u)=\frac{\mathit{dx}}{\mathit{dt}}\hfill \\ \hfill & \approx f(0,0)+\frac{\partial f}{\partial x}x+\frac{\partial f}{\partial u}u+\frac{{\partial }^{2}f}{\partial x\partial u}\mathit{xu}\hfill \end{array}$$ (1)

Importantly, this series is truncated to include only a second-order (bilinear) term describing interactions between neuronal states x and inputs u. Given m known inputs, one can parameterise this equation with A = ∂f/∂x|_u=0, B⁽ⁱ⁾ = ∂² f/∂x∂u_i, and C = ∂f/∂u|_x=0 to obtain a form that lends itself to a direct neurophysiological interpretation:

$$f(x,u)=\frac{\mathit{dx}}{\mathit{dt}}=\left(A+\underset{i=1}{\overset{m}{\Sigma }}{u}_i{B}^{\left(i\right)}\right)x+\mathit{Cu}$$ (2)

In this bilinear differential equation, the matrix A represents the fixed (context-independent or endogenous) strength of connections between the modelled regions, and the matrices B⁽ⁱ⁾ represent the modulation of these connections (e.g. due to learning, attention, etc.) induced by the i^th input u_i as an additive change. Finally, the C matrix represents the influence of direct (exogenous) inputs to the system (e.g. sensory stimuli).

Nonlinear DCM

To model nonlinear interactions amongst the n states of the system, one can extend the Taylor series in Eq. 1 to be second order in the states, i.e.

$$\begin{array}{cc}\hfill & f(x,u)=\frac{\mathit{dx}}{\mathit{dt}}\hfill \\ \hfill & \approx f(0,0)+\frac{\partial f}{\partial x}x+\frac{\partial f}{\partial u}u+\frac{{\partial }^{2}f}{\partial x\partial u}\mathit{xu}+\frac{1}{2}\frac{{\partial }^{2}f}{\partial x^2}\frac{x^2}{2}\hfill \end{array}$$ (3)

Setting $D^{\left(j\right)}={\phantom{\mid }\frac{1}{2}\frac{{\partial }^{2}f}{\partial x_j^2}\mid }_{u=0}(1\le j\le n)$ makes Eq. 3 equivalent to:

$$f(x,u)=\frac{\mathit{dx}}{\mathit{dt}}=\left(A+\underset{i=1}{\overset{m}{\Sigma }}{u}_i{B}^{\left(i\right)}+\underset{j=1}{\overset{m}{\Sigma }}{x}_j{D}^{\left(j\right)}\right)x+\mathit{Cu}$$ 4)

Here, the D^(j) matrices encode which of the n regions gate which connections in the system. Specifically, any non-zero entry $D_{\mathit{kl}}^{\left(j\right)}$ indicates that responses of region k to inputs from region l depend on activity in region j. Figure 1 schematically juxtaposes bilinear and nonlinear DCMs, and Figure 2 shows a simple example of a nonlinear DCM, illustrating the sort of dynamics, both at the neuronal and hemodynamic level this sort of model exhibits.

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).

To explain regional BOLD responses, DCM for fMRI combines the models of neuronal dynamics described above with a hemodynamic model. This model, which was originally described by Buxton et al. (1998) and extended by Friston et al. (2000), comprises a set of differential equations linking changes in neuronal population activity to changes in vasodilatation, blood flow, blood volume v and deoxyhemoglobin content q; the predicted BOLD signal is a nonlinear function of the last two state variables. In this work, we use the most recent formulation of this hemodynamic model as described in Stephan et al. (2007a).

Together, the neuronal and hemodynamic state equations yield a deterministic forward model with hidden states. For any given combination of parameters θ ⊇ A,B,C,D,K and inputs u, the measured BOLD response y is modelled as the predicted BOLD signal h(u,θ) plus a linear mixture of confounds Xβ (e.g. signal drift) and Gaussian observation error e:

$$y=h(u,\theta )+X\beta +e$$ (5)

Parameter estimation and stability

In DCM, parameter estimation employs a Bayesian scheme, with empirical priors for the hemodynamic parameters and zero-mean shrinkage priors for the coupling parameters (see Friston 2002 and Friston et al. 2003 for details). The Gaussian observation error in Eq. 5 is modelled as a linear combination of covariance components Q controlled by hyperparameters λ, i.e., e ∼ N(0,Σexp(λ_i)Q_i). Briefly, the posterior moments of the parameters are updated iteratively using variational Bayes under a fixed-form Laplace approximation, q(θ), to the conditional density p(θ | y); similarly for p(λ | y) These updates are achieved through gradient ascent on a free-energy bound on the log-evidence, F ≤ p(y | m) for any model m, specified by a priori constraints on which connections exist (see below and Friston et al. 2007). Two aspects are particularly important. First, the use of informed priors condition the objective function by suppressing local minima that are far away from the prior mean and facilitates identification of its global maximum by gradient ascent schemes. Second, once the estimation scheme moves into a domain of parameter space in which dynamics become unstable, i.e. runaway excitation, the value of the objective function will necessarily decrease. The gradient ascent in DCM, however, rejects any updates that decrease the objective function: in this case, the algorithm will return to the previous estimate and reduce its step-size, using temporal regularisation (see Friston et al. 2007 for details); this regularisation scheme is similar to (but more robust than) a Levenberg-Marquardt algorithm. This is repeated until the update yields parameter estimates in a stable domain of parameter space and the objective function starts to increase again. This ensures that one obtains parameter estimates for which the modelled system dynamics are stable.

Integration of the nonlinear state equations

Even though stability is guaranteed, there may be parts of parameter space for which evaluation of the state equations during integration encounters numerical problems. For example, some of the state equations in the hemodynamic model contain roots (see appendix); for negative values of hemodynamic states like blood volume and deoxyhemoglobin content², evaluating these roots will result in complex numbers. In order to prevent such cases, it is useful to transform the hemodynamic states into log space; this guarantees that they will always have positive values and places formal constraints on the support of these non-negative states. This transformation is described in the appendix.

In the original bilinear DCM for fMRI, the state equations are integrated using a computationally efficient integration scheme based on the matrix exponential of the system's Jacobian, J(x) = ∂f/∂x (for details, see Friston 2002; Friston et al. 2003). This scheme exploits the sparse structure of the known input functions, making an evaluation of the Jacobian necessary only when inputs change. In contrast, the nonlinear model proposed in this paper requires the Jacobian to be evaluated at each time step. This is because the second-order terms in Eq. 3 render the Jacobian a function of the states, which are changing continuously. For integrating nonlinear differential equations, it is standard to use numerical integration procedures based on local schemes with an update interval τ that is typically five to ten times smaller than the characteristic time-constant of the system (Wilson 1999). Given that the characteristic time-constant in our system is determined by the self-connections, implementing decay of induced activity with an a priori time constant of 1s, all analyses in this paper are based on an update interval of τ = 200 ms.³ In this paper, we used the local linearisation approach proposed by Ozaki (Ozaki 1992; see also Friston et al. 2007 for an independent derivation within a Variational Bayes framework). Ozaki (1992) showed that this approach to integrating nonlinear differential equations is superior to other methods like Euler or Runge-Kutta schemes, in terms of both numerical stability and mathematical consistency between the time-continuous differential equation and its discrete approximation. The local linearisation approach has been used successfully in various nonlinear system identification settings (e.g. Jimenez & Ozaki 2003; Riera et al. 2004; Friston et al. 2007; Valdes et al. 1999). This method evaluates the nonlinear state equation f(x,u) (c.f. Eq. 3) explicitly and uses the system's Jacobian J, which is re-evaluated at every time step τ, to give the update:

$$x(t+\tau )=x\left(t\right)+[\exp \left(J\left(x\left(t\right)\right)\tau \right)-I]J{\left(x\left(t\right)\right)}^{-1}f(x\left(t\right),u\left(t\right))$$ (6)

This corresponds to a nonlinear or generalised convolution of the inputs and is used to form the prediction h(u,θ) in Eq. 5 above.

Bayesian model selection (BMS)

In this paper, we use BMS to compare nonlinear and bilinear DCMs that are applied to both synthetic and empirical fMRI data. A decision about which of several competing models is optimal cannot be based only on the relative fit to the data but also needs to consider differences in model complexity; i.e., the number of free parameters and the functional form of the generative model (Pitt & Myung 2002). Penalizing for model complexity is important because as complexity increases, model fit increases monotonically, but at some point the model will start fitting noise that is specific to the particular data (i.e., “over-fitting”). Models that are too complex are less generalisable across multiple realizations of the same underlying generative process. Therefore, under the condition that all models are equally likely a priori, the question “what is the optimal model?” can be reformulated as “what is the model that represents the best balance between fit and complexity?” This is the model that maximizes the model evidence:

$$p(y\mid m)=\int p(y\mid \theta ,m)p(\theta \mid m)d\theta$$ (7)

Here, the integration subsumes the number and conditional dependencies among free parameters as well as the functional form f(x,u,θ) of the generative model. Unfortunately, this integral cannot usually be solved analytically, therefore approximations must be used (Penny et al. 2004a; Friston et al. 2007). In this study, we use the variational free-energy, F. As detailed elsewhere (Stephan et al., in preparation), this has the advantage over other approximations that its complexity term not only accounts properly for the effective degrees of freedom of the model but also for posterior covariance (or dependency) among the parameters. This is important when comparing models whose likelihood functions have different functional forms (e.g. nonlinear vs. bilinear). F is a lower bound on the log model evidence such that

$$F=\ln p(y\mid m)-\mathit{KL}[q\left(\theta \right),p(\theta \mid y,m)]$$ (8)

Here, KL denotes the Kulback-Leibler divergence (Kullback & Leibler 1951) between an approximating posterior density q(θ) and the true posterior, p(θ | y,m). F is the free-energy bound on the log-evidence above and serves as the objective function for inversion. After convergence of the estimation, the divergence is minimised and F ≈ ln p(y|m). An equivalent decomposition of F is in terms of accuracy and complexity:

$$F={\langle \log p(y\mid \theta ,m)\rangle }_q-\mathit{KL}[q\left(\theta \right),p\left(\theta \right)]$$ (9)

where p(θ) and q(θ) represent the prior and approximate posterior densities, respectively. This demonstrates that F embodies the two opposing requirements of a good model: that it explains the data accurately (i.e., its log likelihood is high) and is as simply as possible (i.e., uses a minimal number of parameters whose posterior densities deviate minimally from their prior; see Penny et al. 2007; Stephan et al. 2007a).

Finally, to quantify the relative goodness of two models m_i and m_j, one can either report the differences in their log-evidences or their Bayes factor (BF):

$${\mathit{BF}}_{\mathit{ij}}=\frac{p(y\mid m_i)}{p(y\mid m_j)}\approx \exp (F_i-F_j)$$ (10)

Synthetic data

We assessed the sensitivity of our nonlinear model using simulated data with known properties. In particular, we were interested in assessing its ability to distinguish nonlinear from bilinear processes. For this purpose, we generated synthetic fMRI data, using a three-area model based on either nonlinear (Fig. 3A) or bilinear state equations (Fig. 3B), and adding observation noise such that the resulting time-series had either a high or low signal-to-noise ratio (SNR = 5 or 2, respectively)⁴. We then used both nonlinear and bilinear models to estimate the parameters from each noisy data set. This resulted in four sets of fitted models, under which the models used to generate the data and to estimate the parameters were identical (i.e., nonlinear models fitted to time-series generated from nonlinear models and bilinear models fitted to time-series generated from bilinear models) and another four sets of fitted models in which they were not (i.e., bilinear models fitted to time-series generated from nonlinear models and nonlinear models fitted to time-series generated from bilinear models). Each set of models comprised 20 synthetic BOLD time-series for each of the three areas; overall, 160 models were fitted and evaluated. Notably, all numerical procedures, including the integration schemes, were identical for generation and inversion of all models.

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.

The nonlinear model is shown in Fig. 3A. An irregular sequence of 25 delta functions or events (randomly located within a 100 second time window) served as driving input u₁ to region x₁ and a box-car function (two blocks with 25s duration) as driving input u₂ to region x₃. The output from x₃ modulated the strength of the x₁→x₂ connection. We sampled the resulting BOLD time-series with a sampling frequency of 1Hz over a period of 100 seconds and added Gaussian observation noise as described above. The bilinear model (Fig. 3B) was identical in terms of connectivity structure and inputs, except that the nonlinear modulation of the x₁→x₂ connection was omitted and replaced by a bilinear modulation of the same connection by input u₂. As a consequence, the modulatory processes represented by nonlinear and bilinear models were qualitatively similar, and both models contained the same number of free parameters; the key difference was the modulation of a connection by an exogenous input (bilinear) or a hidden neuronal state (nonlinear).

With this factorial simulation set-up, we asked two questions. First, can nonlinear and bilinear mechanisms underlying the modulation of connectivity be differentiated reliably on the basis of BOLD time series? This question was addressed by Bayesian model selection, comparing the evidence for the correct generative model against the evidence for the incorrect one. Second, how well are the true values of the modulatory parameters 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 the 20 realisations. This is a quite severe test, because the true value could easily lie within the 95% posterior confidence interval of each realisation but the mode of the posterior density (the MAP estimate) might be systematically smaller than the true value (due to the effects of shrinkage priors).

Analyses of empirical fMRI data

Attention to visual motion

To demonstrate the face validity of our nonlinear DCM, we analysed a single subject fMRI dataset from an experiment on attention to visual motion (Büchel et al. 1998). These data have been used in previous analyses of effective connectivity (Büchel & Friston 1997; Friston & Büchel 2000; Friston et al. 2003; Harrison et al. 2003; Marreiros et al. 2008; Penny et al. 2004a,b); a full description of the experimental paradigm can be found in Büchel & Friston (1997). Both this study, and the study below, had local ethics approval and participants gave informed consent. Briefly, subjects were studied with an fMRI block design under four different conditions: fixating centrally (F), passively viewing stationary dots (S), passively viewing radially moving dots (N) and attending to radially moving dots (A), trying to detect putative velocity changes that actually never occurred, thus keeping physical stimulation identical. Echo planar imaging data were acquired at 2 Tesla using a Siemens Magnetom Vision whole body MRI system (TE = 40ms, TR = 3.22 seconds, matrix size = 64×64×32, voxel size 3×3×3mm). Omitting dummy conditions (to allow for magnetic saturation effects) and concatenating the data across four sessions, the dataset comprises 360 whole-brain volumes. As in Friston et al. (2003), the conventional SPM analysis included three regressors: “photic” (conditions S + N + A), “motion” (conditions N + A), and “attention” (condition A). Regional time-series representing primary visual cortex (V1), motion-sensitive area V5 and posterior parietal cortex (PPC) were extracted by computing the principal eigenvariates from all voxels within spheres of 8 mm radius. These spheres were centred on local maxima of suitable contrasts in a conventional SPM analysis (V1: photic; V5: motion, masked inclusively by attention; PPC: attention).

We inverted a series of three-area DCMs representing either bilinear or nonlinear mechanisms and compared these models using Bayesian model selection. Each of these models encoded a specific mechanism for the attention-induced increase in V5 response that was observed in the SPM analysis (see Fig. 6 for a summary of all models). We then used Bayesian model selection to investigate whether there was sufficient information in the measured fMRI data to enable reliable differentiation between bilinear and nonlinear models of attentional modulation. Additionally, we performed a posterior density analysis of the modulatory parameters in the optimal model to quantify our certainty that an attention-induced increase in V5 activity was mediated by modulation of afferent connections to V5.

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.

Binocular rivalry

To illustrate the use of nonlinear DCMs in a different empirical setting, we analysed a binocular rivalry fMRI data set. This experiment was a 2×2×2 factorial generalisation of the binocular rivalry experiment by Tong et al. (1998), the three factors being percept (face vs. house), rivalry (binocular rivalry vs. non-rivalry), and motion (rocking vs. stationary stimuli). Specifically, subjects wore red-blue dichromatic glasses and viewed a red house and blue face (or vice versa) in 30 sec blocks, separated by 16 sec fixation intervals. In each block the stimuli were presented on a TFT-screen, viewed via a mirror, either in a superimposed fashion (inducing binocular rivalry) or in a sequential (non-rivalling) manner. During rivalry blocks, subjects were asked to indicate by button press when they experienced a transition from a face to a house percept or vice versa. In each non-rivalry block, the stimuli were presented sequentially with the timings as reported in a previous rivalry block (i.e. replay); again subject reported perceptual transitions, which were yoked to the rivalry blocks. As motion is known to influence the duration of stable percepts during binocular rivalry, our third experimental factor concerned the use of stationary and rocking stimuli. The latter stimuli were rocked continuously between extremes of 72 degrees from the vertical meridian (one cycle per second). In the analysis presented in this methodological paper, however, we entirely focus on the rivalry × percept interaction and ignore any effects of motion.

Echo planar imaging data were acquired using a 3 Tesla Philips Achieva whole body MRI scanner (TE = 30ms, TR = 2.014s, matrix size 112×100×26, zero-filled to 128×128×26, voxel size 1.875×1.875×4.5 mm³) in four sessions. Omitting dummy scans (to allow for magnetic saturation effects) and concatenating the data across four sessions, this dataset comprises 1136 whole-brain volumes. Here, we report DCM results from a 22 year old female subject. A complete analysis of the group data will be reported in a future paper.

Data were realigned to the first image, co-registered and normalized to the MNI template in SPM5. Conventional SPM analysis used a general linear model with regressors for each of the eight conditions in the factorial design, using two basis functions per regressor (a canonical hemodynamic response function and its temporal derivative to account for slice-timing errors). Regions of interest were identified using the appropriate contrasts from our factorial design and included the parahippocampal place area (PPA), the fusiform face area (FFA) and the middle frontal gyrus (MFG). The DCMs we considered allowed for full reciprocal connectivity among all three regions with modulation of the lateral connections between the PPA and FFA. As above, these were based on experimental (exogenous) inputs (bilinear) or top-down effects form MFG (nonlinear).

Results

Simulated data

As described above, we assessed the sensitivity of our nonlinear model to the difference between bilinear and nonlinear effects, using simulated data with known properties. We ran four sets of simulations, i.e. for each combination of SNR and model type. This resulted in 80 synthetic datasets to which we fitted both the correct model type (which had been used to generate the data) and the incorrect model type. The results are summarized in Fig. 4 and Fig. 5: Amongst all 80 model comparisons, there were only five cases in which there was higher evidence for the wrong model (Fig. 4), and in each of these cases the superiority of the wrong model was marginal, not even reaching the conventional threshold for “positive” evidence, i.e. BF ≥ 3 (Kass & Raftery 1995). In contrast, positive evidence for the correct model was obtained for 13 out of 20 datasets in the worst case (nonlinear model, low SNR; Fig. 4A) and for 20 out of 20 datasets in the best case (bilinear model, high SNR; Fig. 4D). Moreover, the overall (pooled) evidence for the correct model was very strong in all four sets of simulations: group Bayes factors (GBF) ranged between 10¹⁴ and 10⁷⁵ in favour of the correct model, and average Bayes factors (ABF; the geometric mean of GBF) ranged between 5.6 and 6170 in favour of the correct model (Fig. 4; note that for scaling reasons this figure shows the log-transformed Bayes factors, i.e. the relative log-evidence between models). Altogether, these results demonstrate that, under the levels of SNR used in this analysis, bilinear and nonlinear mechanisms underlying modulation of connection strengths can be differentiated.

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).

Second, for moderate noise levels (SNR = 5), the 95% confidence intervals for both the nonlinear (D) and bilinear (B) modulatory parameters contained the true values, thus demonstrating the robustness of our estimates (Fig. 5). For high noise levels (SNR = 2), however, we observed a significant deviation of both nonlinear and bilinear modulatory parameter estimates from their true values, generally shrinking towards their prior expectation of zero (p<0.05). This result is due to the shrinkage priors on the modulatory parameters, p(d_ij) = N(0,1), whose influence on the posterior estimates increases with signal noise. The same shrinkage effect has been observed in previous simulation studies of bilinear DCMs (Kiebel et al. 2007). This means that for noisy data, as expected, inversion of both nonlinear and bilinear DCMs will yield conservative estimates of modulatory parameters.

Empirical data

Attention to motion

Next, we applied our model to a single-subject data set from a blocked fMRI study of attention to visual motion (Büchel & Friston 1997). We inverted and compared four different models (see Fig. 6), each of which embodied a different explanation for the empirical finding that V5 responses increased during attention to motion, compared to unattended motion. Again all numerical procedures, including the integration scheme (Eq. 6), were identical for the inversion of all models, nonlinear and bilinear. Our first model, M1, allowed for a bilinear attentional modulation of the backward connection from the posterior parietal cortex (PPC) to V5. Comparing this model to a second bilinear model, M2, in which attention modulated the forward connection from V1 to V5, we found positive evidence in favour of M2 (BF = 2966). This replicated previous results, based on a different integration scheme for the state equations, which showed that modulation of the forward connection results in a better model than modulation of the backward connection (Penny et al. 2004; Marreiros et al. 2008). The next step was to augment this model such that attention also exerted a driving influence on PPC directly, in addition to its modulatory effect on the V1→V5 connection. This extended (but still bilinear) model M3, in which attention both exerted stimulus-dependent and stimulus-independent effects on PPC, proved to be better than M2 (BF = 12). The critical question was now whether the attentional enhancement of V5 activity could be better explained by a nonlinear mechanism, for example, by allowing PPC activity to modulate the V1→V5 connection. This nonlinear model, M4, assumed a direct effect of attention on PPC whose activity then increased the gain of the V1→V5 connection. This model was found to be superior to all previous models; comparing it to the previously best model M3 resulted in a Bayes factor of 23 in favour of M4 (see Fig. 6). Analysis of the posterior density under M4 indicated that nonlinear gating of the V1→V5 connection by attention could be inferred with 99.1% confidence (see Fig. 7). Notably, the fixed V1→V5 connection strength was fairly low (A₂₁ = 0.12), suggesting that top-down, nonlinear PPC-dependent modulation was quantitatively essential to explain V5 responses. Fig. 8 shows the observed and fitted time-series of all areas and highlights the attentional gating effect on V5 activity, such that V5 activity was higher when subjects attended the moving stimuli.

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.

Binocular rivalry

As a second demonstration of nonlinear DCMs, we present an analysis of a single subject fMRI data set acquired during an event-related binocular rivalry paradigm. Binocular rivalry arises when two different stimuli are projected separately to the two eyes; the subject then experiences a single percept at a time, and this percept fluctuates between the two competing stimuli with a time constant in the order of a few seconds. While there is no clear consensus about the mechanisms that underlie this phenomenon, it has been suggested that binocular rivalry (i) depends on nonlinear mechanisms and (ii) may arise from modulation of connections amongst neuronal representations of the competing stimuli by feedback connections from higher areas (see Blake & Logothetis 2002).

We acquired fMRI data during a factorial paradigm in which face and house stimuli were presented either during binocular rivalry or during a matched non-rivalry (i.e. replay) condition. For the subject studied here, the conventional SPM analysis showed a rivalry × percept interaction in both the right fusiform face area (FFA) and the right parahippocampal place area (PPA): in FFA, the face vs. house contrast was higher during non-rivalry than during rivalry; conversely, in PPA the house vs. face contrast was higher during non-rivalry than during rivalry (both p<0.05, small-volume corrected).⁵ Additionally, testing for a main effect of rivalry, we replicated previous findings (Lumer et al. 1997) that several prefrontal regions, including the right middle frontal gyrus (MFG), showed higher activity during rivalry than during non-rivalry conditions.

These SPM results motivated a nonlinear DCM in which the connections between face- and house-selective regions (i.e. FFA and PPA) were modulated by the activity in a source that was sensitive to the degree of rivalry in the visual input (i.e. MFG). The structure of the resulting DCM (along with the MAP estimates for all parameters) is shown in Fig. 9A. First, the fixed (intrinsic) connection strengths between FFA and PPA are negative in both directions, i.e. FFA and PPA exert a mutual negative influence on each other, when the system is not perturbed by inputs (i.e. during fixation); this could be regarded as a “tonic” or “baseline” reciprocal inhibition. Much more important, however, is that during the presentation of visual stimuli this competitive interaction between FFA and PPA is modulated by activity in the middle frontal gyrus (MFG), which showed higher activity during rivalry vs. non-rivalry conditions. As shown in Fig. 9B, our confidence about the presence of this nonlinear modulation is very high (99.9%) for both connections. All parameter estimates are shown in Fig. 9A; they provide a straightforward mechanistic explanation for the rivalry × percept interaction that was found in both FFA and PPA in the SPM analysis. According to the model, activity levels in the MFG determine the magnitude of the face vs. house activity differences in FFA and PPA by controlling the influence that face-elicited activations and house-elicited deactivations of FFA have on PPA (and vice versa). For example, the positive MAP estimate (2.43) for the nonlinear modulation of the FFA→PPA connection by MFG activity (see Fig. 9A) means that during face-perception under rivalry conditions (which elicit positive activity in the FFA and MFG, respectively) there is a positive influence of FFA on PPA, overriding the “baseline” inhibition. This means that during binocular rivalry FFA and PPA become more tightly coupled which destroys their stimulus selectivity: their activity becomes very similar, regardless of whether a face or a house is being perceived. In contrast, deactivation of MFG during non-rivalry conditions decreases the influence that FFA has on PPA during house perception; therefore responses in FFA and PPA become less coupled and their relative selectivity for face and house percepts is restored. This dynamic coupling and uncoupling, leading to less selectivity of FFA and PPA during rivalry and higher selectivity during non-rivalry, is clearly visible in Fig. 10 which plots the observed and fitted responses of all three areas. Here, the short black arrows indicate blocks with binocular rivalry (when FFA and PPA show very similar time courses) and the long grey arrows denote non-rivalry blocks (when FFA and PPA activities evolve more independently). These changes in effective connectivity over time, which are controlled by the activity level in MFG, provide a nice explanation for the rivalry × percept interaction identified by the SPM analysis, where regional selectivity for faces and houses, respectively, was found to be more pronounced during non-rivalry than rivalry.

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.

Finally, we compared this nonlinear DCM to similar bilinear models. First, we constructed a bilinear model that was capable of expressing the same range of modulatory effects as the nonlinear model, i.e. both the FFA→PPA and PPA→FFA connections were allowed to vary during both rivalry and non-rivalry conditions. Bayesian model comparison showed that this bilinear model, which had two more parameters than the nonlinear one, was an inferior model: the Bayes factor in favour of the nonlinear model was larger than 10¹⁸. In another bilinear model we removed the driving inputs into MFG and introduced rivalry effects solely through modulation of the connections between FFA and PPA in order to give it the same number of parameters as the nonlinear model. This bilinear model performed even worse; here, the Bayes factor in favour of the nonlinear model was larger than 10³².

Discussion

The nonlinear DCM presented in this paper enables one to model activity-dependent gating of connections or, equivalently, changes in the gain of interacting neuronal populations. This is a critical mechanism in various neurobiological processes, including top-down modulation (e.g. by attention), learning and effects exerted by neuromodulatory transmitters.

Biophysically, neuronal gain control can arise through various mechanisms of short-term synaptic plasticity (STP) that result from interactions among synaptic inputs arriving close in time, but not necessarily at the same dendritic compartment. For example, two major mechanisms are known to induce very fast changes in connection strengths, without inducing lasting structural alterations of synapses. The first one is non-linear dendritic integration of inputs due to voltage-dependent ion channels, e.g. non-inactivating dendritic sodium conductances (Schwindt & Crill 1995). The second mechanism is synaptic depression/facilitation (Abbott et al. 1997; Abbott & Regehr 2004). Other mechanisms, although not relying on lasting structural synaptic changes, are likely to induce them; e.g. activation of dendritic calcium conductances by back-propagating action potentials (Larkum et al. 2004) and amplification of neuronal responses by activation of NMDA conductances (Fox et al. 1990). Finally, gain control is also affected strongly by various neuromodulatory transmitters that are known to regulate synaptic plasticity (Gu 2002; Katz 2003), including noradrenalin (Ego-Stengel et al. 2002), serotonin (Hurley & Pollak 2001) and acetylcholine (De Bruyn et al. 1986).

A few previous studies of effective connectivity have modelled changes in connection strength as a function of activity in a different region (Friston et al. 1995; Büchel & Friston 1997: Friston et al. 1997; Friston & Büchel 2000). However, all of these studies differed in two crucial points to the approach presented here. First, they operated directly on the measured BOLD time-series and could not disambiguate whether nonlinearities arose from neuronal or from hemodynamic causes. In contrast, the present model distinguishes between nonlinearities in the BOLD signal that are due to neuronal and hemodynamic processes, respectively (c.f. Friston et al. 2003 and Stephan et al. 2007a). A partial exception is the approach suggested by Gitelman et al. (2003), which can be used to compute interaction terms used in psycho-physiological and physio-physiological interaction analyses of fMRI data. In this approach, a deconvolution procedure is applied to BOLD data prior to Hadamard multiplication. However, neither the deconvolution procedure nor the regression-based model of effective connectivity in this approach affords the same flexibility and realism as combining nonlinear neuronal state equations with a hemodynamic forward model as presented in this article. A second critical difference is that all previous models were essentially variants of the general linear model and thus remained linear in the parameters; nonlinearities were only accounted for by including regressors or predictor variables that resulted from multiplication of two time-series.

Since its introduction a few years ago (Friston et al. 2003), DCM has already enjoyed widespread application to fMRI data, resulting in more than thirty published studies to date. We expect that nonlinear DCMs will further extend the practical applications of DCM. As exemplified by the two examples in this paper, nonlinear mechanisms can, at least sometimes, better explain empirically measured fMRI responses than linear ones. We would like to emphasise though that neither example in this methodological paper is meant to provide an exhaustive treatment of questions on neuronal gain modulation during attention or on nonlinear mechanisms during binocular rivalry. These examples are only meant to lend face validity to our approach and provide anecdotal illustrations of the sort of insights that can be gained with nonlinear DCMs. Similarly, the hierarchical sequence of comparisons we employed for the attention-to-motion dataset is very useful for that particular analysis but does not necessarily represent a blueprint for other DCM analyses; instead, the exact model comparison strategy should always be tailored to the specific questions entailed by the model space examined.

A particularly exciting prospect is that nonlinear DCMs provide a good starting point for directly embedding computational models (e.g. of learning processes) into physiological models (c.f. Stephan 2004). This is addressed by ongoing work in our group, with a focus on how neuromodulatory transmitters shape synaptic plasticity during learning.

Appendix

Log-transformation of hemodynamic state equations

Classically, the hemodynamic model in DCM consists of the following differential equations in which x is the neuronal state vector, κ is the rate constant of vasodilatory signal decay, γ is the rate constant of flow-induced feedback regulation, τ is the mean transit time of venous blood, α is the resistance of the venous balloon, and E₀ is the resting oxygen extraction fraction (see Friston et al. 2000 and Stephan et al. 2007a for details):

$$\text{Changes in vasodilatory signalling}s:\frac{\mathit{ds}}{\mathit{dt}}=x-\kappa s-\gamma (f-1)$$ (A1)

$$\text{Changes in blood flow}f:\frac{\mathit{df}}{\mathit{dt}}=s$$ (A2)

$$\text{Changes in venous blood volume}v:\tau \frac{\mathit{dv}}{\mathit{dt}}=f-v^{1∕\alpha }$$ (A3)

$$\text{Changes in deoxyhemoglobin content}q:\tau \frac{\mathit{dq}}{\mathit{dt}}=f\frac{1-{(1-E_0)}^{1∕f}}{E_0}-v^{1∕\alpha }\frac{q}{v}$$ (A4)

To ensure positive values of the hemodynamic states and thus numerical stability of the parameter optimization scheme (see main text), we convert these equations, such that all hemodynamic states z = {s, f, v, q} are in log space by applying the chain rule after a change of variables, z̃ = ln z. That is, for any given state variable z with the state equation $\frac{\mathit{dz}}{\mathit{dt}}=F\left(z\right)$:

$$\begin{array}{cc}\hfill & \tilde{z}=\ln z\iff z=\exp \left(\tilde{z}\right)\Rightarrow \hfill \\ \hfill & \frac{d\tilde{z}}{\mathit{dt}}=\frac{d\ln \left(z\right)}{\mathit{dz}}\frac{\mathit{dz}}{\mathit{dt}}=\frac{F\left(z\right)}{z}\hfill \end{array}$$ (A5)

This means that z(t) = exp(z̃(t)) is always positive, ensuring a proper support for these non-negative states and numerical stability when evaluating the state equations during parameter estimation. Applied to the four hemodynamic state equations in DCM (Eqs. A1 to A4), this transformation gives:

$$\frac{d\tilde{s}}{\mathit{dt}}=\frac{x-\kappa s-\gamma (f-1)}{s}$$ (A6)

$$\frac{d\tilde{f}}{\mathit{dt}}=\frac{s}{f}$$ (A7)

$$\frac{d\tilde{v}}{\mathit{dt}}=\frac{f-v^{1∕\alpha }}{\tau v}$$ (A8)

$$\frac{d\tilde{q}}{\mathit{dt}}=\frac{f\frac{1-{(1-E_0)}^{1∕f}}{E_0}-v^{1∕\alpha }\frac{q}{v}}{\tau q}$$ (A9)

It is important to note that this log-transformation does not affect the model parameters, only the hemodynamic states (which must have positive values due to their physical nature; e.g. there is no such thing as a negative blood volume). This is because it is only the optimisation scheme that operates on the log transformed states. In contrast, when evaluating the BOLD output equation (c.f. Stephan et al. 2007a), the log-hemodynamic states are exponentiated. In other words, after each step of the optimisation scheme, we use z(t) to compute the predicted BOLD signal at time t, not z̃(t).