Temporally correlated fluctuations drive epileptiform dynamics

Abstract

Macroscopic models of brain networks typically incorporate assumptions regarding the characteristics of afferent noise, which is used to represent input from distal brain regions or ongoing fluctuations in non-modelled parts of the brain. Such inputs are often modelled by Gaussian white noise which has a flat power spectrum. In contrast, macroscopic fluctuations in the brain typically follow a 1/f^b spectrum. It is therefore important to understand the effect on brain dynamics of deviations from the assumption of white noise. In particular, we wish to understand the role that noise might play in eliciting aberrant rhythms in the epileptic brain. To address this question we study the response of a neural mass model to driving by stochastic, temporally correlated input. We characterise the model in terms of whether it generates "healthy" or "epileptiform" dynamics and observe which of these dynamics predominate under different choices of temporal correlation and amplitude of an Ornstein-Uhlenbeck process. We find that certain temporal correlations are prone to eliciting epileptiform dynamics, and that these correlations produce noise with maximal power in the δ and θ bands. Crucially, these are rhythms that are found to be enhanced prior to seizures in humans and animal models of epilepsy. In order to understand why these rhythms can generate epileptiform dynamics, we analyse the response of the model to sinusoidal driving and explain how the bifurcation structure of the model gives rise to these findings. Our results provide insight into how ongoing fluctuations in brain dynamics can facilitate the onset and propagation of epileptiform rhythms in brain networks. Furthermore, we highlight the need to combine large-scale models with noise of a variety of different types in order to understand brain (dys-)function.

Keywords: Epilepsy; Ictogenesis; Jansen-Rit model; Neural mass models; Nonlinear dynamics; Ornstein‐Uhlenbeck noise; Stochastic effects.

Fig. 1

A scheme of the Jansen-Rit model of a cortical column that comprises three neuronal populations. A population of pyramidal neurons is marked with green, and populations of excitatory and inhibitory interneurons with blue and red, respectively. Somata are depicted with the triangle, hexagon and circle. Continuous lines stand for dendritic processing and dashed ones for axonal processing. A dot means multiplication and a star operator denotes convolution. Cyan indicates lumped external input from sub-cortical and cortico-cortical structures. The black circuit depicts an analytic description of the underlying structure of a cortical column. See text for details.

Fig. 2

Bifurcation diagram of the Jansen-Rit model defined in Eqs. (6)–(8). Parameters of the model were set to biologically plausible values proposed in Jansen and Rit (1995). The X axis shows external, constant input to the pyramidal population $I_{\mathrm{ex}}=p$. The Y axis shows net postsynaptic potential on this population: $y_1-y_2$. Continuous (dashed) lines represent stable (unstable) solutions. Cyan and blue denote a node and a focus, respectively, and green and red indicate alpha and epileptiform limit cycles, respectively. Vertical, grey dotted lines divide the diagram to six regimes (denoted by roman numerals) of qualitatively distinct dynamical properties. See text for details.

Fig. 3

Methodology for classification of dynamics. Panel A shows $y_1-y_2$ obtained from 10 s of stochastic simulation for $p=89{\mathrm{s}}^{-1}$, $\tau ={10}^{-0.5}\mathrm{s}$, ${\sigma }_{\mathrm{ou}}=50{\mathrm{s}}^{-1}$. Background colours indicate the type of activity assigned with the classification algorithm. Red stands for epileptiform dynamics, green for alpha oscillations, and blue for random fluctuations around the node. Panel B shows a smoothed version of the $y_1-y_2$ signal from panel A, obtained with a running mean computed within a 0.4-second-long sliding window. The dashed line denotes the ${\mathrm{Th}}_{\mathrm{A}}=5\mathrm{mV}$ threshold, which is used to discriminate between stochastic fluctuations around the node (smoothed $y_1-y_2\le {\mathrm{Th}}_{\mathrm{A}}$) and alpha oscillations (smoothed $y_1-y_2>{\mathrm{Th}}_{\mathrm{A}}$). Grey marks root mean square of $y_1-y_2$ around its smoothed version (${\mathrm{RMS}}_{(y_1-y_2)}$). This value is shown in panel C in grey along with the ${\mathrm{Th}}_{\mathrm{B}}=2.25\mathrm{mV}$ threshold, which is used to identify epileptiform dynamics (when ${\mathrm{RMS}}_{(y_1-y_2)}>{\mathrm{Th}}_{\mathrm{B}}$).

Fig. 4

Response of the Jansen-Rit model to driving with the Ornstein-Uhlenbeck (OU) noise. The left panels of the figure show example outputs (time courses of $y_1-y_2$) produced by the model under driving with the OU noise characterised with correlation time τ equal to 10⁻³ s (panel A), ${10}^{-1.5}\mathrm{s}$ (panel B) and 10⁰ s (panel C). Background colours mark periods of random fluctuations around the node (blue), epileptiform dynamics (red) and alpha activity (green). In all these cases stationary standard deviation of the noise ${\sigma }_{\mathrm{ou}}$ was equal to $50{\mathrm{s}}^{-1}$ and p was set to $89{\mathrm{s}}^{-1}$. Panel D shows the fraction of time that the system spent in epileptiform dynamics as a function of the noise correlation time τ (varied along the X axis in logarithmic scale) and the noise stationary standard deviation ${\sigma }_{\mathrm{ou}}$ (varied along the Y axis). Locations of the red letters A,B and C mark settings in which time traces shown in panels A,B and C were obtained. The white lines denote points of equal values of noise intensity D: the dashed line marks $D=\sqrt{1000}{\mathrm{s}}^{-1}$ and the dotted one marks $D=100{\mathrm{s}}^{-1}$. In all cases initial conditions corresponded exactly to the node.

Fig. 5

Distribution of spectral power in frequency bands of standard brain rhythms and dependence of location of maximum power on noise correlation time τ. Evaluation of the $E(\tau ,f_{\min },f_{\max })$ function (see Eq. (13)) of an Ornstein-Uhlenbeck noise characterised with correlation time τ within a frequency range $f_{\min },f_{\max }$ is plotted for fixed frequency ranges that correspond to distinct brain rhythms: δ (2–4 Hz, magenta), θ (4–8 Hz, cyan), α (8–12 Hz, brown), β (12–30 Hz, grey), γ (30–100 Hz, yellow) and combined δ+θ (2–8 Hz, black). Units on the Y axis express fraction of the spectral power of the noise characterised with τ contained within the $f_{\min },f_{\max }$ range. Correlation time of the noise τ varies along the X axis. The inset illustrates the meaning of $E(\tau ,f_{\min },f_{\max })$. It shows an example theoretical power spectrum of the Ornstein-Uhlenbeck noise calculated for $\tau ={10}^{-3.0}\mathrm{s}$ (green), $\tau ={10}^{-1.4}\mathrm{s}$ (blue) and $\tau ={10}^0\mathrm{s}$ (red). In each case stationary variance $\frac{D}{\tau }$ was set to an arbitrary value 1 s⁻². Dashed vertical lines mark the $f_{\min }=2\mathrm{Hz}$, $f_{\max }=8\mathrm{Hz}$ range, for which the black plot shown in the main panel was derived from Eq. (13). Green, blue and red arrows on the main plot indicate values of the $E(\tau ,f_{\min },f_{\max })$ function that correspond to these spectra. The value indicated by the blue arrow is highest (in this case it corresponds to the maximum), which follows from the fact that the area below the blue curve, limited by $f_{\min }$ and $f_{\max }$ in the inset is greater that area set by either red, or green curves. Coloured circles on the X axis indicate values of τ corresponding to maxima of $E(\tau ,f_{\min },f_{\max })$: $\tau ={10}^{-2.54}\mathrm{s}$ for γ, $\tau ={10}^{-2.08}\mathrm{s}$ for β, $\tau ={10}^{-1.79}\mathrm{s}$ for α, $\tau ={10}^{-1.55}\mathrm{s}$ for θ, $\tau ={10}^{-1.25}\mathrm{s}$ for δ, and $\tau ={10}^{-1.4}\mathrm{s}$ for δ + θ.

Fig. 6

Phase diagram showing the different dynamical regimes resulting from oscillatory driving with varying amplitude and period. The response of the Jansen-Rit model under harmonic driving was classified as either a node (oblique stripes from top-right to bottom-left), alpha activity (oblique stripes from top-left to bottom-right), or epileptiform dynamics (grey). This classification was conducted for varying driving amplitude $\tilde{A}$, displayed on Y axes, and driving period T, displayed on X axes in logarithmic (bottom) and linear (top) scales. Ranges and names of typical brain rhythms are denoted on the linear scale. In general, different dynamical regimes might coexist, therefore patterns overlap. Panel A corresponds to initial conditions set exactly to the node and panel B to initial conditions set exactly to alpha oscillations. In both cases $p=89{\mathrm{s}}^{-1}$. Black lines divide the diagram into distinct regimes, annotated with letters.