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Conceived and designed the experiments: NM PJM PB. Performed the experiments: NM PJM. Analyzed the data: RJM VL PJB RJD KJF PB. Contributed reagents/materials/analysis tools: RJM KJF. Wrote the paper: RJM NM VL RJD PJM KJF PB.

The authors have declared that no competing interests exist.

Cortico-basal ganglia-thalamocortical circuits are severely disrupted by the dopamine depletion of Parkinson's disease (PD), leading to pathologically exaggerated beta oscillations. Abnormal rhythms, found in several circuit nodes are correlated with movement impairments but their neural basis remains unclear. Here, we used dynamic causal modelling (DCM) and the 6-hydroxydopamine-lesioned rat model of PD to examine the effective connectivity underlying these spectral abnormalities. We acquired auto-spectral and cross-spectral measures of beta oscillations (10–35 Hz) from local field potential recordings made simultaneously in the frontal cortex, striatum, external globus pallidus (GPe) and subthalamic nucleus (STN), and used these data to optimise neurobiologically plausible models. Chronic dopamine depletion reorganised the cortico-basal ganglia-thalamocortical circuit, with increased effective connectivity in the pathway from cortex to STN and decreased connectivity from STN to GPe. Moreover, a contribution analysis of the Parkinsonian circuit distinguished between pathogenic and compensatory processes and revealed how effective connectivity along the indirect pathway acquired a strategic importance that underpins beta oscillations. In modelling excessive beta synchrony in PD, these findings provide a novel perspective on how altered connectivity in basal ganglia-thalamocortical circuits reflects a balance between pathogenesis and compensation, and predicts potential new therapeutic targets to overcome dysfunctional oscillations.

Parkinson's disease is a progressive age-related neurodegenerative disorder that severely disrupts movement. The major pathology in Parkinson's disease is the degeneration of a group of neurons that contain a chemical known as dopamine. Treatment of Parkinsonism includes pharmacological interventions that aim to replace dopamine and more recently, implanted devices that aim to restore movement through electrical stimulation of the brain's movement circuits. Understanding the electrical properties that emerge as a result of depleted dopamine may reveal new avenues for developing these technologies. By combining a novel model-based approach with multi-site electrophysiological recordings from an animal model of Parkinson's disease we provide empirical evidence for a link between abnormal electrical activity in the Parkinsonian brain and its physiological basis. We have examined the connections along the brain's motor circuits, and found an abnormality in inter-area connections in a particular neural pathway, a pathway critically dependent on dopamine. The scheme makes strong and testable predictions about which neural pathways are significantly altered in the pathological state and so represent empirically motivated therapeutic targets.

In Parkinson's disease (PD), degeneration of midbrain dopamine neurons severely disrupts neuronal activity in looping circuits formed by cortico-basal ganglia (BG)-thalamocortical connections [1,2,3]. Studies have shown that excessive oscillations at beta frequencies (13–30 Hz) are a key pathophysiological feature of these Parkinsonian circuits, when recorded at the level of unit activity and/or local field potentials (LFPs) in several key circuit nodes. These nodes include the frontal cortex, subthalamic nucleus (STN), external globus pallidus (GPe) and internal globus pallidus (GPi) [4,5,6,7,8,9]. Suppression of pathological beta-activity is achieved by dopamine replacement therapies [10] and surgical treatments e.g. high-frequency, deep brain stimulation (DBS) of the STN; where prolonged attenuation after stimulation is observed [11,12]. Bradykinesia and rigidity are the primary motor impairments associated with beta activity and, following dopamine replacement therapies, improvements in these motor deficits correlate with reductions in beta power [13,14,15,16]. Moreover, a recent report has shown that stimulating the STN at beta frequencies exacerbates motor impairments in Parkinsonian rodents [17], in line with similar findings in PD patients [18,19].

Precisely how dopamine depletion leads to abnormal beta power is unknown. Recent work in rodents has revealed that excessive beta-activity emerges in cortex and STN after chronic dopamine loss but not after acute dopamine receptor blockade [5,8]. Here, we examine whether changes in effective connectivity between the nodes of the cortico-basal ganglia-thalamocortical network can account for enhanced beta oscillations following chronic dopamine loss. To test this hypothesis we used dynamic causal modelling (DCM). This approach allows one to characterise the distributed neuronal architectures underlying spectral activity in LFPs. DCM is a framework for fitting differential equations to brain imaging data and making inferences about parameters and models using a Bayesian approach. A range of differential equation models have been developed for various imaging modalities and output data features. The current library of DCMs includes DCM for fMRI, DCM for event related potentials and DCM for steady state responses (DCM-SSR). The current paper is based on DCM-SSR, designed to fit spectral data features [20,21].

Using spectral data, recorded simultaneously from multiple basal ganglia nuclei and the somatic sensory-motor cortex, we asked whether systematic changes in re-entrant neural circuits produce the excessive beta oscillations observed in LFPs recorded from the 6-hydroxydopamine (6-OHDA)-lesioned rat model of PD [2,5,22]. We inverted the models (i.e., optimised the model parameters or “fit” the data) using LFP data collected simultaneously from electrodes implanted in frontal cortex, striatum, GPe and STN. Specifically, we used neural mass models that characterise the main projection cell types at each circuit node as glutamatergic or GABAergic. Neural mass models describe neuronal dynamics in terms of the average neurophysiological states (e.g., depolarisation) over populations of neurons. Inference on effective connectivity differences observed between the Parkinsonian and control cases was based on

Measures of functional connectivity have been applied previously to examine frequency-specific signal correlations between nodes in the cortico-basal ganglia-thalamocortical network. These measures have highlighted excessive coupling between the cortex and STN [22] and between STN and GPe [5,22] in animal models of PD. While functional connectivity and effective connectivity measures share some technical aspects e.g. likelihood models [23] or Bayesian estimators [24], the underlying concepts are fundamentally different [25]. The distinction between functional connectivity (a descriptive characterisation of the statistical dependence between two time series) and effective connectivity (a model-based characterisation of causal influences) emerged from the analysis of electrophysiological time series: Aertsen et al. [26], used the term effective connectivity to define the neuronal interactions that could explain observed spike trains using a “minimum simple neuronal model”. In what follows, we employ such a minimum model approach, using the key elements of known cortico-basal-ganglia-thalamocortical interactions. Our model predicts the output of this loop circuit

Dynamic causal models for LFP data typically comprise connected cortical sources, where each source is described by a neural mass [27]. This neural mass ascribes point estimators to hidden neuronal states (ensemble depolarisation and firing rates), capturing the average activity of a population of neurons [28], i.e. a mean-field approximation. These dynamics depend on model parameters that encode, for example, inter-regional connectivity, the amplitude of postsynaptic responses and/or synaptic rate constants. Here, three (layered) populations were used to model the cortical source, while a single population of neurons, (either glutamatergic [excitatory] or GABAergic [inhibitory]) was used for distinct BG nuclei (see

(A) The structure of the DCM encompasses the principal nodes and connections in the rodent cortico-basal ganglia-thalamocortical loop. The nodes include a cortical source,

This figure summarises the generative model as a Bayesian Network (upper left insert), which has been unpacked to show the form of the conditional dependencies in terms of the equations of the generative model (lower right). This model provides a probabilistic description of data in terms of the parameters that cause them. These include parameters controlling the spectral density of neuronal and channel noise and parameters controlling the variance of observation error on the cross-spectra predicted. The parameters of interest pertain to a neuronal model that is cast as a continuous-time state-space model. It is this neuronal model that constrains the mapping from neuronal noise or innovations to observed cross-spectra. The details of the neuronal (mass) model are provided in the next figure. Please see the main text for a detailed explanation of the equations that define the conditional dependencies.

These are the differential equations modelling the hidden neuronal states in the subpopulations comprising the nodes of the circuit model. These equations take the form of Equation 7. For simplicity, we have dropped the dependency on time and have therefore omitted the delays (see Equation 7). In the cortex, the parameters

The connections in our standard DCM were based on the well characterised re-entrant circuits linking the cortex, basal ganglia and thalamus in rodents and primates (

Recent work has emphasized the significance of the hyperdirect pathway for the functional organisation of cortico-basal ganglia-thalamocortical circuits [30] and, importantly, this pathway has been shown to be crucial for the expression of abnormal slow oscillations in the STN-GPe network in Parkinsonism [33]. In short, our standard model architecture incorporates the major glutamatergic and GABAergic connections between the six key components of the cortico-basal ganglia-thalamocortical circuit. In accommodating the core elements of the loop circuit, the model also adheres to the established organisational principles embodied in the direct, indirect and hyperdirect pathways. Note that our standard model does not include all known connections. However, the addition of more connections does not necessarily improve the ability of the loop circuit (and model) to sustain beta oscillations. To test this, we tried adding two less well-studied, but potentially important, pallidofugal connections, either from GPe to EPN or that from GPe to striatum [34]. The addition of either connection did not improve the performance of (evidence for) the standard model, which provided the optimum balance of accuracy and complexity for our given data set (see Figure S2 in

A time-frequency analysis of resting state LFPs at 10–35 Hz revealed consistent and long-lasting high-amplitude beta oscillations in all cortical and BG recordings from the 6-OHDA-lesioned animals (

In

(A) Observed and modelled cross-spectral densities. Densities for frequencies from 10 Hz to 35 Hz were extracted from time domain data using a vector autoregressive model.

(A) MAP estimates of extrinsic connectivity parameters (comparable to

Using the posterior estimates from the DCM of the grand averaged spectra, we simulated the system's response for a wide band of frequencies. This allowed us to identify the predominant changes in spectral activity, associated with our optimised models of control and Parkinsonian animals: For linear systems, the frequency response can be illustrated as poles and zeros on the unit circle. This involves a reformulation of the system (i.e. the differential equations used by the DCM) using a z-transform [21]. This transform produces a transfer function, which summarises the model system's input-output spectral properties.

In

The Parkinsonian circuit we have described represents the effects of chronic dopamine depletion and, potentially, a balance between primarily pathogenic changes and compensatory or adaptive changes. Consequently, we asked which connection strengths in the reorganised Parkinsonian circuit could contribute to (or attenuate) beta activity. This entailed using the optimised DCM from the lesioned animals to predict spectral responses to changes in the model circuit architecture: To do this, we quantified the degree to which a change in each connection affected beta activity throughout the circuit (power summed over 16–18 Hz and channels). This provided a measure of ‘beta contribution’ per animal, per connection; in terms of the partial derivative of summed beta activity, with respect to each connection. Note that beta contributions (derivatives) could be either positive, whereby small changes exacerbate beta, or negative whereby small changes ameliorate beta (e.g. see Figures S4 and S6 in

(A) Using the DCMs from the Parkinsonian animals, we quantified changes in beta power with respect to changes in connectivity parameters (averaged over channels). The measure of beta power was the height of the beta peak centred at 17 Hz (summed over 16–18 Hz). Increasing connections from striatum to GPe and from GPe to STN exacerbated beta oscillations (positive derivatives), where the striatum to GPe connectivity was the most effective (*

These simulations suggest that if connections along the indirect pathway could be weakened in the Parkinsonian state, then excessive beta-activity might be attenuated, thereby providing important candidate therapeutic targets. An apparent paradox is that while we find an increase in effective connectivity in the hyperdirect pathway from cortex to STN, our contribution analysis shows that manipulating the strength of this connection did not change beta activity significantly. Conversely, the remaining two connections (striatum to GPe and GPe to STN) that profoundly influenced beta activity in our simulations did not differ in their mean strength between the control and Parkinsonian groups. This apparent paradox can be resolved by considering the extended network that incorporates all nine connections. The strength of a connection need not necessarily change between the healthy and diseased states in order for that connection to have a different effect on oscillatory strength within the new network; in other words, perturbations to a connection with a given strength will have a different functional effect, depending on the network in which it is embedded. Moreover since the conditional densities of the parameters of interest in the DCM are identifiable (see Figure S7 in

We have clear evidence that the extended network differs between the control and Parkinsonian states with increases in the effective connection strength of the hyperdirect cortex to STN pathway and reduction in that from STN to GPe. Above, we identified that several connections in the indirect pathway have the capacity to dynamically modulate beta. Thus we compared beta contribution between the control and Parkinsonian animals for the ‘beta critical’ indirect pathway connections. This showed that these connections engendered much higher beta activity when embedded in the Parkinsonian network (two sample t-test;

Classical models of connectivity within the cortico-basal ganglia-thalamocortical loop circuit explain PD symptoms in terms of altered firing rates along the direct/indirect pathways [2]. More recent research has highlighted the oscillatory nature of excessive neuronal synchronisation in the Parkinsonian state [7]. Here, we present a new model of effective connectivity in the cortico-basal ganglia-thalamocortical loop, which emphasizes neuronal synchrony and oscillatory dynamics over rate coding. Our scheme is based on dynamic causal modelling of multisite LFPs, which, in order to be detected, necessitate spatiotemporal summation and hence, synchronisation of population activity. Importantly, the LFP data were derived from a rat model of PD that recapitulates clinical pathophysiology, most notably the dominance of beta oscillations in the untreated Parkinsonian state. Using neural mass models that comprise ensemble firing output and membrane potential inputs [27], our model can generate low-frequency broadband activity or Parkinsonian excessive beta activity by increasing and decreasing particular extrinsic connections. In effect, operating from a stationary equilibrium, the Parkinsonian cortico basal-ganglia-thalamocortical circuit has a modulation transfer function that peaks at beta frequencies. Similar frequency tuning is seen in the cortico-basal ganglia circuit of untreated PD patients in response to phasic inputs to STN [39].

We found specific differences between control and Parkinsonian groups at two pathways along re-entrant circuits. The effective connection strength of the cortical ‘hyperdirect’ input to the STN was dramatically increased in the Parkinsonian animals, compared to control animals, a finding in accord with current views of subthalamic hyperactivity in PD [5,6,33,40], and also with optogenetic circuit perturbations that point to the cortex as a key driver of this hyperactivity [17]. In contrast, the STN input to the GPe decreased in the Parkinsonian animals. The latter represents a potentially important and novel finding. Nevertheless, as discussed below, it builds on a literature implicating the reciprocally-connected STN-GPe network in the generation and dissemination of abnormally synchronized oscillations in Parkinsonism [5,41,42].

Our standard model architecture might not be the only one that can sustain exaggerated beta oscillations. However, our model does incorporate the major glutamatergic and GABAergic connections between the six key components of the loop circuit, thus capturing the core elements of the direct, indirect and hyperdirect pathways and placing it within established frameworks. The architecture was chosen as the simplest that could support an answer to our questions about inter-regional connectivity. The large parameter space and the effective data size (via the AR spectral decomposition) would normally predispose to ‘over fitting’ in a maximum likelihood setting. However, Bayesian inference finesses this problem because it optimises the marginal likelihood (model evidence), which includes a complexity term. This complexity rests on the use of priors or constraints on the underlying neurobiology (e.g., synaptic time constants). This effectively limits the degrees of freedom in the model to those parameters with relatively uninformative priors; i.e., the connectivity parameters of interest. Moreover, the standard model performed no better when two additional pallidofugal connections were incorporated (see Figure S2 in

The documented changes in steady-state effective connectivity may indicate primary pathological changes and/or secondary compensatory mechanisms (see Figure S1 in

The decrease in STN input to the GPe in Parkinsonian animals is also interesting. Here, contribution analyses demonstrated that increasing the strength of the reciprocal connection exacerbates the beta oscillations that characterise the Parkinsonian state. Recent experimental data also suggest that interactions between GPe and STN could both support and actively promote the emergence of excessively synchronized oscillations at the network level [5], while a recent computational model also emphasises the importance of strong excitatory connections from STN to GPe in the promotion of beta-frequency activity [42]. The reduction in STN to GPe connectivity that we observed here may instantiate compensatory neural plasticity (see Figure S1 in

The above findings underscore the importance of our contribution analysis in interpreting the nature of steady-state changes. It is important to note that while the effects found in our contribution analysis are dependent on the model inversion as a whole, i.e. including intrinsic parameters, it is the change in the extrinsic connections themselves that promote beta oscillations. This approach was key to identifying the indirect pathway connections as influencing abnormal activity in the chronicly reorganised circuit, even though their connection strengths remained relatively unchanged between control and Parkinsonian states. The importance of these connections could not have been suspected from simple contrasts of control and Parkinsonian steady-state networks, and it was their potency in promoting beta oscillations that was very much greater when embedded in the Parkinsonian network. This means that connections of the indirect pathway have a new strategic role in the re-organised circuit, and provide potential therapeutic targets, in line with the recent finding that selective excitation of striatal neurons in the indirect pathway elicits a Parkinsonian behavioral state [43]. Indeed, it is likely that D2-mediated suppression of striatal input to GPe might explain the attenuation of beta oscillations in patients with PD following therapy with apomorphine [44] or L-Dopa [10].

Our results should encourage exploration of the therapeutic potential of suppressing the GPe to STN connection in PD. The directionality of this prediction is important given the different neurotransmitters in reciprocal STN-GPe connections. We have assumed that the network changes occurring under anaesthesia are also relevant in the awake, behaving animal. There is good evidence to support an extrapolation of our findings to the un-anesthetized state. First, the excessive beta oscillations we have modelled occur in both anesthetized and awake 6-OHDA-lesioned rats [5,6,8,22], moreover the gamma activity shift for the control animals reveals how broadband spectral activity is preserved under anaesthesia. Second, at least some of the network alterations defined here are likely the result of chronic plasticity as they only appear several days after dopamine neurons are lesioned [5,8]. Thus, some of the critical features identified in the model fitting from the anesthetized state are likely to represent underlying changes in the microcircuit, and therefore may still be relevant in the awake state.

In summary, our analyses lead to a new view of connectivity in the cortico-basal ganglia-thalamocortical circuit, which acknowledges the importance of synchrony in the pathophysiology of Parkinson's disease [7]. Our scheme makes strong and testable inferences about what are essentially permissive vs. compensatory changes as well as which connections have altered strategic contributions to the pathological state. These connections represent candidate therapeutic targets (see Figure S1 in

Experimental procedures were carried out on adult male Sprague-Dawley rats (Charles River, Margate, UK), and were conducted in accordance with the Animals (Scientific Procedures) Act, 1986 (UK). Recordings were made in eight dopamine-intact control rats (288–412 g) and nine 6-OHDA-lesioned rats (285–428 g at the time of recording), as described previously [5,6,46]. Briefly, anaesthesia was induced with 4% v/v isoflurane (Isoflo™, Schering-Plough Ltd., Welwyn Garden City, UK) in O_{2}, and maintained with urethane (1.3 g/kg, i.p.; ethyl carbamate, Sigma, Poole, UK), and supplemental doses of ketamine (30 mg/kg, i.p.; Ketaset™, Willows Francis, Crawley, UK) and xylazine (3 mg/kg, i.p.; Rompun™, Bayer, Germany). The electrocorticogram (ECoG), a type of cortical local field potential, was recorded via a 1 mm diameter steel screw juxtaposed to the dura mater above the right frontal (somatic sensory-motor) cortex (4.5 mm anterior and 2.0 mm lateral of bregma [47] and was referenced against another screw implanted in the skull above the ipsilateral cerebellar hemisphere. Raw ECoG was band-pass filtered (0.3–1500 Hz, −3 dB limits) and amplified (2000×; DPA-2FS filter/amplifier: Scientifica Ltd., Harpenden, UK) before acquisition. Extracellular recordings of LFPs in the striatum, GPe and STN were simultaneously made in each animal using ‘silicon probes’ (NeuroNexus Technologies, Ann Arbor, MI). Each probe had one or two vertical arrays of recording contacts (impedance of 0.9–1.3 MΩ measured at 1000 Hz; area of ∼400 µm^{2}). The same probe was used throughout these experiments but it was cleaned after each experiment in a proteolytic enzyme solution to ensure that contact impedances and recording performance were not altered by probe use and re-use [33]. Monopolar probe signals were recorded using high-impedance unity-gain operational amplifiers (Advanced LinCMOS: Texas Instruments, Dallas, TX) and were referenced against a screw implanted above the contralateral cerebellar hemisphere. After initial amplification, extracellular signals were further amplified (1000×) and low-pass filtered at 6000 Hz using programmable differential amplifiers (Lynx-8: Neuralynx, Tucson, AZ). The ECoG and probe signals were each sampled at 17.9 kHz using a Power1401 Analog-Digital converter and a PC running Spike2 acquisition and analysis software (Cambridge Electronic Design Ltd., Cambridge, UK). Neuronal activity was recorded during episodes of spontaneous ‘cortical activation’, which contain patterns of activity that are similar to those observed during the awake, behaving state [48]. Cortical activation was defined according to ECoG activity [5,6]. Neuronal activity patterns present under this anaesthetic regime may only be qualitatively similar to those present in the unanesthetized brain. However, the urethane-anesthetized animal still serves as a useful model for assessing ensemble dynamics within the basal ganglia [46]. Indeed, in 6-OHDA-lesioned animals, exaggerated beta oscillations emerge in cortico-basal ganglia circuits during activated brain states [5,6] thus accurately mimicking the oscillatory activity recorded in awake, unmedicated PD patients [10].

Unilateral 6-OHDA lesions were carried out on 200–250 g rats, as described previously [5,6]. Twenty five minutes before the injection of 6-OHDA, all animals received a bolus of desipramine (25 mg/kg, i.p.; Sigma) to minimize the uptake of 6-OHDA by noradrenergic neurons [49]. Anaesthesia was induced and maintained with 4% v/v isoflurane (see above). The neurotoxin 6-OHDA (hydrochloride salt; Sigma) was dissolved immediately before use in ice-cold 0.9% w/v NaCl solution containing 0.02% w/v ascorbate to a final concentration of 4 mg/ml. Then 3 µl of 6-OHDA solution was injected into the region adjacent to the medial substantia nigra (4.5 mm posterior and 1.2 mm lateral of bregma, and 7.9 mm ventral to the dura [47]. The extent of the dopamine lesion was assessed 14–16 days after 6-OHDA injection by challenge with apomorphine (0.05 mg/kg, s.c.; Sigma [50]). The lesion was considered successful in those animals that made >80 net contraversive rotations in 20 min. Note that the emergence of exaggerated beta oscillations after 6-OHDA lesions is not dependent on apomorphine [6,22]). Electrophysiological recordings were carried out ipsilateral to 6-OHDA lesions in anesthetized rats 21–42 days after surgery, when pathophysiological changes in the basal ganglia are likely to have levelled out near their maxima [6].

We restricted our analysis to the ECoG and LFP activities present during a spontaneous activated brain state, which mimics that accompanying alert behaviours in which beta oscillations are most prominent in PD patients and 6-OHDA-lesioned rats [5,6,22,51]. Thirty second epochs of cortical activation were analysed from one contact in each BG nucleus and the contemporaneous ECoG from each animal. Cross-spectral density

Data from the four channels (Cortex, Striatum, GPe and STN) were summarised using an autoregressive (AR) process [52] of order

DCM is a model comparison framework for the inversion and comparison of generative (forward) models based on differential equations. DCM allows data from multiple recording sites to be analysed as a distributed system. Originally developed for fMRI [54], the framework uses a generative model of the neural processes (usually neural mass and mean field models) [55,56] that cause observed data. Bayesian model inversion furnishes estimates of coupling or effective connectivity between regions and how this coupling is changed by experimental context [57]. For LFP and ECoG data, the generative model contains details about the structure and synaptic properties within neuronal sources, as well as the synaptic input that each source receives [21]. In what follows, we describe the precise form of the dynamic causal model we used in this application. The underlying neural mass model has been described and validated in a series of previous papers

In DCM for steady-state responses, one is trying to predict or explain observed spectral activity. Effectively, this entails modelling the mapping between the spectral density of neuronal fluctuations or innovations and the resulting responses. This mapping is parameterised in terms of the (synaptic) parameters of a neural mass model. The neural mass model is used to determine how random neuronal fluctuations are filtered to produce observed cross-spectra. Model inversion involves optimising model parameters to explain empirical cross-spectra. In what follows, we will describe the generative model in terms of the kernels or transfer functions that couple the spectra of neuronal innovations to observed cross-spectra and then describe the neural mass model that determines how the transfer functions (kernels) are parameterised.

We require the model to predict cross-spectra corresponding to the data features described above. This prediction is based on the parameters

The transfer functions are the Fourier transforms of the corresponding first-order kernels,

Equation 5 allows us to predict the systems kernels and spectral behaviour given the equations of motion (differential equations) that constitute our model of hidden neuronal states. The effect of neuronal noise or fluctuations is modelled here in terms of their expression as cross spectra in channel space. The neural model is composed of subpopulations, where each subpopulation can have different synaptic rate constants and amplitudes. Subpopulations are grouped into sources and coupled with intrinsic connections, while sources are connected by extrinsic connections between specific subpopulations in different sources. Our model comprised a cortical source, four basal-ganglia nuclei and a thalamic source (

The dynamics of hidden neuronal states and the ensuing observation are assumed to have the form

The architecture depicted in

Because the model parameters are non-negative they are treated as scale-parameters with Gaussian priors on the log of their values. These are specified in terms of a prior mean

The priors, together with the likelihood model (Equation 3), constitute a generative model of observed cross spectral densities based on a neuronal state-space model formulated in continuous time. This generative model is summarised in

Variational Laplace appeals to a mean field approximation and factorises an approximate joint posterior into two densities over the covariance and model parameters: ^{−2}. Note that Variational Laplace generalises previous variational schemes based upon expectation maximisation, which ignore condition uncertainty about the parameters optimised in the maximisation step: e.g., [69]. For dynamic models of the sort used in this paper, this approach has been evaluated in relation to Monte Carlo Markov chain (MCMC) techniques and has been found to be robust, providing accurate posterior estimates while being far superior to sampling schemes in terms of computational efficiency [70].

Given the empirical data, either from each animal separately (spectra from 10 to 35 Hz, comprising the auto-spectra of cortex, striatum, GPe, STN and their cross-spectra;

We used the MAP estimates above to analyse the response of the networks to small changes in circuit connectivity. Our goal was to see if particular connections (along the cortico-basal ganglia-thalamocortical pathways) contribute to beta frequencies more than others. This analysis assumes that the generative model of the spectra has been optimised and omits channel noise; i.e. using

As outlined above, the free energy is a lower bound on the model evidence [71] and is used for model selection, when testing a series of possible neural architectures using Bayes Factors (see Figure S2 in

A DCM can be seen as an input-state-output model of neuronal responses, where the white and pink noise inputs at each of sources (and four channels), renders the system a MIMO (

Additional model comparison, sensitivity analyses and robustness estimates. This explores a possible model space using Bayesian model comparison and presents additional sensitivity and robustness analyses that support our main conclusions.

(DOC)

We thank Dr William D. Penny for his helpful input.