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, 55 (4), 1694-708

Consistent Spectral Predictors for Dynamic Causal Models of Steady-State Responses

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Consistent Spectral Predictors for Dynamic Causal Models of Steady-State Responses

Rosalyn J Moran et al. Neuroimage.

Abstract

Dynamic causal modelling (DCM) for steady-state responses (SSR) is a framework for inferring the mechanisms that underlie observed electrophysiological spectra, using biologically plausible generative models of neuronal dynamics. In this paper, we examine the dynamic repertoires of nonlinear conductance-based neural population models and propose a generative model of their power spectra. Our model comprises an ensemble of interconnected excitatory and inhibitory cells, where synaptic currents are mediated by fast, glutamatergic and GABAergic receptors and slower voltage-gated NMDA receptors. We explore two formulations of how hidden neuronal states (depolarisation and conductances) interact: through their mean and variance (mean-field model) or through their mean alone (neural-mass model). Both rest on a nonlinear Fokker-Planck description of population dynamics, which can exhibit bifurcations (phase transitions). We first characterise these phase transitions numerically: by varying critical model parameters, we elicit both fixed points and quasiperiodic dynamics that reproduce the spectral characteristics (~2-100 Hz) of real electrophysiological data. We then introduce a predictor of spectral activity using centre manifold theory and linear stability analysis. This predictor is based on sampling the system's Jacobian over the orbits of hidden neuronal states. This predictor behaves consistently and smoothly in the region of phase transitions, which permits the use of gradient descent methods for model inversion. We demonstrate this by inverting generative models (DCMs) of SSRs, using simulated data that entails phase transitions.

Figures

Fig. 1
Fig. 1
Source model. (A) Neuronal architecture for each cortical source comprising an input layer of spiny stellate cells, and supra/infra granular regions continuing inhibitory interneurons and pyramidal cells. Intrinsic connections between the subpopulations are drawn with arrows; red arrows indicate the presence of NMDA receptors, postsynaptically. (B) NMDA switch function (Eq. (3)) illustrated for increasing values of parameter α. As α increases, a voltage-dependent magnesium switch becomes highly nonlinear. (C) Stochastic equations describing the dynamical system with states comprising voltages and conductances.
Fig. 2
Fig. 2
Spectral response of the neural-mass model. (A) Left: Spectral phase-diagram illustrating the maximal frequency band over the u and γ31E dimensions. Right: phase-diagram illustrating the maximum frequency with 1 Hz resolution, which illustrates the highest maximum frequency of 21 Hz. Note that in these grey scale images, one can observe changes in frequency within a particular band, where here at γ31E = 0.25, for example, we see an increasing frequency in the alpha limit cycle regime before it returns to a beta fixed point. (B) Bifurcation diagram illustrating the maximum and minimum pyramidal cell membrane potential along parameter u for γ31E = 0.5. At u ~ 3.06 mA the system bifurcates and enters a limit-cycle attractor. Then at u ~ 10.6 mA, the system undergoes a second phase transition back to a fixed-point. (C) Time domain and phase domain portraits for one region of the 2-D parameter plane (u = 0.06 mA; γ31E = 0.25), where the system lies at an alpha fixed-point. (D) Time domain and phase domain portraits for one region of the 2-D parameter plane (u = 4 mA; γ31E = 0.5), where the system enters an alpha limit cycle. (E) A second region in parameter space, here at a beta fixed-point, when (u = 16 mA; γ31E = 0.25).
Fig. 3
Fig. 3
Spectral response of the neural-mass model. (A) Left: Spectral phase-diagram illustrating the maximal frequency band over the u and α dimensions. Right: phase-diagram illustrating the maximum frequency with 1 Hz resolution, which illustrates the highest maximum frequency of 22 Hz. (B) Bifurcation diagram illustrating maximum and minimum pyramidal cell membrane potential along parameter u for α = 0.279. At u ~ 3.06 mA the system bifurcates and enters a limit-cycle attractor. Then at u ~ 10.6 mA, the system undergoes a second phase transition back to a fixed-point. (C) Time domain and phase domain portraits for one region of the 2-D parameter plane (u = 0.0625 mA; α = 0.279), where the system oscillates with theta frequencies around a fixed-point attractor. (D) A second region in parameter space, here at an alpha limit cycle.
Fig. 4
Fig. 4
Spectral response of the mean-field model. (A) Left: Spectral phase-diagram illustrating the maximal frequency band over the u and γ31E dimensions. Right: phase-diagram illustrating the maximum frequency with 1 Hz resolution. This mean-field formulation produces high frequency gamma oscillations, with the highest maximum frequency of 89 Hz. (B) Bifurcation diagram illustrating maximum and minimum pyramidal cell membrane potential along parameter u for γ31E = 1.5. For this particular value of γ31E, the system remains at fixed points. (C) Time domain and phase domain portraits for a region of parameter space exhibiting a low frequency quasiperiodic attractor. (D) For high input current and high levels of forward excitatory connectivity (i.e. from the stellate cell input layer to the pyramidal cells) the system exhibits gamma resonance (u = 100 mA; γ31E = 5).
Fig. 5
Fig. 5
Spectral response of the mean-field model. (A) Left: Spectral phase-diagram illustrating the maximal frequency band over the u and α dimensions. Right: phase-diagram illustrating the maximum frequency with 1 Hz resolution, which illustrates the highest maximum frequency of 65 Hz. (B) Bifurcation diagram illustrating irregular bifurcation structure depending on both u and α. (C) Time domain and phase domain portraits for a region of parameter space exhibiting a low frequency quasiperiodic attractor. (u = 6.25 mA; α = 0.062). (D) A second region in parameter space, exhibiting at a beta limit cycle.
Fig. 6
Fig. 6
Diffusion on a limit cycle. (A) The mean-field model as shown in Fig. 4C (u = 5.065 mA; γ31E = 0.5), exhibits low frequency, theta limit-cycle oscillations (dashed line). Adding state noise to the system by increasing the diffusion coefficient causes the orbit to speed up at Vpyram ~ -40 mV, while decreasing the strength of the diffusion coefficient slow it down. (B) Phase domain plots showing greater symmetry (resp. asymmetry) in the orbit at D = 150% (resp. 50%) baseline. (C) Frequency spectra show a higher peak frequency for D = 150%.
Fig. 7
Fig. 7
Sampling phase-space. (A) Six possible sampling sets for an orbit. Samples are distributed for delta, theta, alpha, beta, gamma or high gamma (colour coded as per Figs. 2–5). The time series is differentiated at the sample points in each set. In parentheses, we show the number of derivatives greater than zero. The set with greatest sign balance (i.e. number positive derivative = 4) is chosen as the sampling set. (B) Alpha limit cycle of NMM as per Fig. 2C, sampled using the set algorithm.
Fig. 8
Fig. 8
Sampled spectral responses. (A) Top: Spectral phase-diagram illustrating the maximal frequency band over the u and γ31E dimensions for the neural-mass model using the modulation transfer function sampled from the centre manifold. The diagram shows a good comparison between this and the full (non-sampled) estimates shown in Fig. 2A. Bottom: The spectral prediction at u = 6.25 mA and γ31E = 0.25 (white circle) contains a peak at 14 Hz, in the alpha range and a secondary peak at 17 Hz, in the beta range. This illustrates the ability of the model to generate multimodal spectra. (B) Top: Spectral phase-diagram illustrating the maximal frequency band over the u and α dimensions for the neural-mass model. The diagram shows smoother transitions than in Fig. 3A. Bottom: The spectral prediction at u = 5.06 mA and α = 0.279 (white circle) contains peaks in the alpha (11 Hz) and theta (7 Hz) range. (C) Top: Spectral phase-diagram illustrating the maximal frequency band over the u and γ31E dimensions for the mean-field model. These responses have a similar profile to Fig. 4A. Bottom: For the MFM, the sampling scheme also produces multimodal spectra, with beta (19 Hz) and gamma (39 Hz) peaks for u = 5.06 mA and γ31E = 0 (white circle). (D) Top: Spectral phase-diagram illustrating the maximal frequency band over the u and α dimensions for the mean-field model. These responses have a similar profile to Fig. 5A. Bottom: For u = 9 mA and α = 0.56 (white circle), the prediction contains two peaks, one in the beta range at 16 Hz, and one in the theta range at 4 Hz.
Fig. 9
Fig. 9
Simulations. (A) Two trials with different spectral profiles generated from the NMM. At (u = 0. 25 mA; α = 0.031) we generated an “alpha trial” from fixed-point resonance (hashed red circle & spectrum) and at (u = 0. 25 mA; α = 0.248) we generated a “theta trial” from fixed-point resonance (hashed green circle and spectrum). Model fits and MAP estimates are shown as full circles in the phase-diagram (reproduced from Fig. 3) and full lines in the spectra plots. (B) Two trials with different spectral profiles generated from the NMM. At (u = 4 mA; γ31E = 0.25) we generated an “alpha trial” from an alpha limit cycle (hashed red circle and spectrum) and at (u = 14.063 mA; γ31E = 0.25), we generated a “beta trial” from fixed-point resonance (hashed grey circle and spectrum). All other parameters are as per Table 1. Model fits and MAP estimates are shown as full circles in the phase-diagram (from Fig. 2) and full lines in the spectra plots. The colours of the lines in the plots correspond to the different trials coloured according to their predominant frequency.

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