Alpha-Delta Transitions in Cortical Rhythms as grazing bifurcations

TL;DR

The paper addresses how alpha and delta cortical rhythms emerge and transition within the Jansen–Rit neural mass framework. It develops a singular perturbation analysis by nondimensionalising the model and introducing a small parameter $ obreakoldsymbol{\e}$, enabling a piecewise-smooth, grazing mechanism to explain the alpha–delta transition. The main result is that the alpha-to-delta boundary corresponds to a grazing bifurcation of alpha-type periodic orbits in the singular limit, with explicit two-parameter mappings in $(b^*,G)$ that align with Hopf and SNIC structures. This provides a mechanistic, analytic explanation for rhythm switching in cortical activity, clarifying the nature of previously labeled “false bifurcations” and offering predictive boundaries for transitions under parameter variations and nonzero inputs.

Abstract

The Jansen-Rit model of a cortical column in the cerebral cortex is widely used to simulate spontaneous brain activity (EEG) and event-related potentials. It couples a pyramidal cell population with two interneuron populations, of which one is fast and excitatory and the other slow and inhibitory. Our paper studies the transition between alpha and delta oscillations produced by the model. Delta oscillations are slower than alpha oscillations and have a more complex relaxation-type time profile. In the context of neuronal population activation dynamics, a small threshold means that neurons begin to activate with small input or stimulus, indicating high sensitivity to incoming signals. A steep slope signifies that activation increases sharply as input crosses the threshold. Accordingly in the model the excitatory activation thresholds are small and the slopes are steep. Hence, a singular limit replacing the excitatory activation function with all-or-nothing switches, eg. a Heaviside function, is appropriate. In this limit we identify the transition between alpha and delta oscillations as a discontinuity-induced grazing bifurcation. At the grazing the minimum of the pyramidal-cell output equals the threshold for switching off the excitatory interneuron population, leading to a collapse in excitatory feedback.

Figures (7)

• Figure 1: (a) Interactions between $3$ neuronal populations in local circuit of a single cortical column in cerebral cortex of brain modelled by \ref{['jansenrit']}: pyramidal cells ($Y_1$) and excitatory ($Y_3$, EINs) and inhibitory ($Y_2$, IINs) interneurons. (b) Blue axis: Sigmoid profile of $\mathop{\mathrm{Sigm}}\nolimits$, given in \ref{['orginsigm']}, with slope $r=0.56$ at threshold $y_0=6$ mV (vertical dotted line), at which half maximum $e_0$ of $\mathop{\mathrm{Sigm}}\nolimits$ is achieved. (b) Black axis: dimensionless sigmoid $\mathop{\mathrm{S}}\nolimits$, given in \ref{['newsigwitheps']}, for maximal slope value$1/(4\epsilon)=0.25/0.024\approx10.5$ at activation threshold $y_{0,1}=0.08$ (non-dimensionalised threshold $y_{0,1}$ for excitatory populations indicated by black vertical dotted line). See Table \ref{['tab:paramter model']} and Table \ref{['tab:nondimparameters']} for other parameters of $\mathop{\mathrm{Sigm}}\nolimits$ and $\mathop{\mathrm{S}}\nolimits$.
• Figure 2: (a) Bifurcation diagram of Jansen-Rit model \ref{['jansenrit']} for varying ${A}$; (b) frequency of oscillations with input ${p=0}$ (black) and input ${p=120}$ (yellow) for varying ${A}$, indicating alpha, theta and delta frequency ranges. (c) Same bifurcation diagram as (a) but using nondimensionalised quantities $G=B/A$ and $y_3$. (d),(e) Time profiles of oscillations in alpha (${A=11}$) and delta (${A=10}$) rhythm regimes corresponding to vertical dashed lines in panel b. See Tables \ref{['tab:paramter model']} and \ref{['tab:nondimparameters']} for parameters. Computation performed with cocodankowicz2013recipes.
• Figure 3: (a-g) Intersections between left- (red) and right-hand (blue) side of \ref{['y1piecewise']} for different bifurcation parameters $G$ (plots used $\epsilon=0.001$, $y_{0,1},y_{0,3}=0.08$ and $y_{0,2}=0.3$). (h) Numerical bifurcation diagram of equilibria in $(G,y_1)$-plane obtained by cocodankowicz2013recipes. $\mathop{\mathrm{S}}\nolimits_\epsilon/a_1=\mathop{\mathrm{S}}\nolimits_\epsilon(y_3-y_2- y_{0,1})$ for $y_1$ in equation \ref{['xeqy1']}. For other parameters see Table \ref{['tab:nondimparameters']}.
• Figure 4: Equilibria and periodic orbits branching off Hopf bifurcation (HB1) for system \ref{['nondimwitheps']} for varying $G$ and small $\epsilon=0.001$, showing equilibrium values, or maxima and minima of the neural activities $y_1,y_2,y_3$, respectively. Green vertical line corresponds to canard orbit shown in Fig \ref{['canard']}. Grey vertical line is at parameter $G=1.7$, used in singular limit in Fig. \ref{['figsamllepstime']}(a). Other parameters are listed in Table \ref{['tab:nondimparameters']}.
• Figure 5: Time profile of alpha-type piecewise exponential periodic orbits given in \ref{['pworbit:crossing']},representing solutions of the piecewise linear ODE \ref{['y1dot']}, \ref{['y2dot']} for $y_{0,1},y_{0,3}=0.08$, $y_{0,2}=0.3$, and $y_3$ fixed at its equilibrium value $2\alpha_2/G$. Panel (a): orbit for $(b^*,G)=(0.5,1.7)$; $t_\mathrm{s_2,off}$, $t_\mathrm{s_2,on}$: threshold crossing times of $y_1$ (where $y_1=y_{0,2}$); $t_\mathrm{s_1,off}$, $t_\mathrm{s_1,on}$: threshold crossing times of $y_2$ (where $y_2=y_3-y_{0,1}$). Dashed blue line (legend entry $S_0$(pc)) is activation switch $\mathop{\mathrm{S}}\nolimits_{0}\left(y_3-y_2(t)-y_{0,1}\right)$ for $y_1$ with $\epsilon=0.001$. Dashed red line (legend entry $S_0$(inh)) is activation switch $\mathop{\mathrm{S}}\nolimits_{0}\left(y_1(t)-y_{0,2}\right)$ for $y_2$. Panel (b): same as panel (a) but for $(b^*,G)=(0.44,1.7)$, where orbit grazes ($y_{1,\min}=y_{0,3}$, green star on grey horizontal line $\{y=y_{0,3}\}$). For other parameters see Table \ref{['tab:nondimparameters']}.