Mean-Field Models for EEG/MEG: From Oscillations to Waves

Abstract

Neural mass models have been used since the 1970s to model the coarse-grained activity of large populations of neurons. They have proven especially fruitful for understanding brain rhythms. However, although motivated by neurobiological considerations they are phenomenological in nature, and cannot hope to recreate some of the rich repertoire of responses seen in real neuronal tissue. Here we consider a simple spiking neuron network model that has recently been shown to admit an exact mean-field description for both synaptic and gap-junction interactions. The mean-field model takes a similar form to a standard neural mass model, with an additional dynamical equation to describe the evolution of within-population synchrony. As well as reviewing the origins of this next generation mass model we discuss its extension to describe an idealised spatially extended planar cortex. To emphasise the usefulness of this model for EEG/MEG modelling we show how it can be used to uncover the role of local gap-junction coupling in shaping large scale synaptic waves.

Keywords: Brain rhythms; Gap-junction coupling; Neural field; Neural mass; Synaptic coupling; Synchrony; Waves.

Fig. 1

Model schematic. At each point in a two-dimensional spatial continuum there resides a density of QIF neurons whose mean-field dynamics are described by the triple (R, V, U), where R represents population firing rate, V the average membrane potential, and U the synaptic activity. The non-local interactions are described by a kernel w, taken to be a function of the distance between two points. The space-dependent delays arising from signal propagation along axonal fibres are determined in terms of the speed of the action potential c

Fig. 2

Validity of the mean-field reduction. A comparison of the mean-field dynamics (red) with the corresponding network of spiking neurons (blue). The top panel shows a raster plot for a sample of 100 of the 1000 neurons in the network of synaptically and electrically coupled QIF neurons. Below are comparisons of the mean firing rate R, average membrane potential V and within population synchrony |Z| for the spiking network and mean field model. Parameter values: ${\eta }_0=2$, ${\kappa }_v=1$, ${\kappa }_s=1$, $\tau =16$, $\alpha =0.5$, $\gamma =0.5$. For the spiking network simulations $v_{\text{r}}=-1000$ and $v_{\text{th}}=1000$, while in the mean field limit these are assumed to be $-\infty$ and $\infty$, respectively

Fig. 3

Beta rebound. Time course of the within population synchrony and synaptic current (${\kappa }_{s}U$)) and a time-frequency spectrogram of the synaptic current for different gap-junction coupling strengths when a temporally filtered square pulse of length 400 ms and magnitude 3 μA was applied to the model (3)–(4). a Weak gap-junction coupling ${\kappa }_v=0.5$. b Intermediate gap-junction coupling ${\kappa }_v=1$. c Strong gap-junction coupling ${\kappa }_v=1.5$. Parameter values: ${\eta }_0=1$, ${\kappa }_s=1$, $\tau =15$, $\alpha =0.1$, $\gamma =0.5$

Fig. 4

Single population dynamics. a Oscillations in the population firing rate R (teal) and average membrane voltage V (yellow), b Corresponding oscillations in the complex Kuramoto order parameter $Z=|Z|{\mathrm{e}}^{i\theta }$, where |Z| reflects the degree of within-population synchrony (green), and $\theta$ a corresponding phase (red). Parameter values: ${\eta }_0=1$, ${\kappa }_v=1.2$, ${\kappa }_s=1$, $\tau =15$, $\alpha =0.5$, $\gamma =0.5$

Fig. 5

Single population bifurcation diagrams. a A Hopf bifurcation is found with an increase in the strength of gap-junction coupling ${\kappa }_v$, giving rise to limit cycle oscillations. Red (black) lines denote the stable (unstable) fixed point, while the green lines show the minimum and maximum of the oscillation. b A two parameter bifurcation diagram in the $({\kappa }_v,{\eta }_0)$-plane tracing the locus of Hopf bifurcations for different values of $\gamma$. Oscillations emerge to the right of each curve. Parameter values: ${\eta }_0=1$, ${\kappa }_s=1$, $\tau =15$, $\alpha =0.5$, $\gamma =0.5$

Fig. 6

Excitatory-inhibitory network dynamics: Oscillations in a the excitatory population firing rate $R_E$ (teal), and b in the average membrane potential $V_E$ (yellow). Corresponding oscillations for the inhibitory population, c $R_I$ and d $V_I$. Kuramoto order parameters for the excitatory population e $Z_E=|Z_E|{\mathrm{e}}^{i{\theta }_E}$, $|Z_E|$ (green) and f ${\theta }_E$ (red). Corresponding traces for g $|Z_I|$ and h ${\theta }_I$ of the inhibitory population. Parameter values: ${\eta }_0^E=5$, ${\eta }_0^I=-3$, ${\kappa }_v^E={\kappa }_v^I=0.5$, ${\kappa }_s^{\mathit{EE}}=15$, ${\kappa }_s^{\mathit{IE}}=25$, ${\kappa }_s^{\mathit{EI}}=-15$, ${\kappa }_s^{\mathit{II}}=-15$, ${\tau }_E=1,{\tau }_I=1$, ${\alpha }_{\mathit{EE}}=0.2$, ${\alpha }_{\mathit{IE}}=0.1$, ${\alpha }_{\mathit{EI}}=0.07$, ${\alpha }_{\mathit{II}}=0.06$, ${\gamma }_E={\gamma }_I=0.5$

Fig. 7

Two population bifurcation diagrams: Continuations in the median background drive to the inhibitory population ${\eta }_0^I$ for different combinations of gap-junction coupling strengths ${\kappa }_v^E$ and ${\kappa }_v^I$. Red (black) lines denote the stable (unstable) fixed point, while the green (blue) lines show the minimum and maximum of the stable (unstable) oscillation a No gap-junction coupling, ${\kappa }_v^E=0$, ${\kappa }_v^I=0$, b Gap junctions in the inhibitory population only, ${\kappa }_v^E=0$, ${\kappa }_v^I=0.5$, c Gap junctions in the excitatory population only, ${\kappa }_v^E=0.5$, ${\kappa }_v^I=0$, d Gap junction coupling in both populations, ${\kappa }_v^E=0.5$, ${\kappa }_v^I=0.5$. Other parameters: ${\eta }_0^E=5$, ${\kappa }_s^{\mathit{EE}}=15$, ${\kappa }_s^{\mathit{IE}}=25$, ${\kappa }_s^{\mathit{EI}}=-15$, ${\kappa }_s^{\mathit{II}}=-15$, ${\tau }_E=1,{\tau }_I=1$, ${\alpha }_{\mathit{EE}}=0.2$, ${\alpha }_{\mathit{IE}}=0.1$, ${\alpha }_{\mathit{EI}}=0.07$, ${\alpha }_{\mathit{II}}=0.06$, ${\gamma }_E={\gamma }_I=0.5$

Fig. 8

Turing instability analysis for the one-dimensional neural field model. The left panel shows the Hopf and Turing–Hopf curves as a function of the action potential speed c and gap-junction coupling strength ${\kappa }_v$. Above these curves patterned states emerge. The three right hand panels show simulations near Hopf, and two Turing–Hopf points: (I) Bulk oscillation with $c=0.1$, ${\kappa }_v=0.85$, (II) Standing wave with $c=0.11$, ${\kappa }_v=0.855$, (III) Periodic travelling wave with $c=1.0$, ${\kappa }_v=0.88$. Other parameter values: ${\eta }_0=1$, ${\kappa }_s=10$, $\tau =15$, $\alpha =0.5$, $\gamma =0.5$

Fig. 9

Simulations of the one-dimensional neural field model under variation in ${\kappa }_v$: (I) Standing wave with ${\kappa }_v=0.86$, (II) Bulk oscillations with ${\kappa }_v=1.0$, (III) Mixed dynamics with ${\kappa }_v=1.2$. Other parameters $c=0.11$, ${\eta }_0=1$, ${\kappa }_s=10$, $\tau =15$, $\alpha =0.5$, $\gamma =0.5$

Fig. 10

Simulations of the two-dimensional neural field model showing that, beyond a dynamic Turing instability, rotating waves with source and sink dynamics may emerge. Top: a snapshot of a patterned state in the (R, V) and $(|Z|,\theta )$ variables. Bottom: the corresponding time-series for the point marked by the small green circle in the top panel. A movie illustrating how this pattern evolves in time is given in Supplementary material 7. Parameter values: $c=1$, ${\eta }_0=2$, ${\kappa }_v=0.695$, ${\kappa }_s=12$, $\tau =20$, $\alpha =0.5$, $\gamma =0.5$

Fig. 11

Simulations of the two-dimensional neural field model with moderate gap-junction coupling strength. In this case robust spiral waves emerge at the centre of rotating cores. The spiral is tightly wound with a diffused tail of high amplitude activity that propagates into the rest of the domain and interacts with the other rotating waves. Top: a snapshot of a patterned state in the (R, V) and $(|Z|,\theta )$ variables. Bottom: the corresponding time-series for the point marked by the small green circle in the top panel. The full spatio-temporal can be seen in Supplementary material 8. Parameter values: $c=1$, ${\eta }_0=2$, ${\kappa }_v=0.8$, ${\kappa }_s=12$, $\tau =20$, $\alpha =0.5$, $\gamma =0.5$

Fig. 12

Simulations of the two-dimensional neural field model with short-range excitation and long-range inhibition, showing the emergence of a spatially localised spot solution (top panel). Note that the core of the spot has a rich temporal dynamics, as indicated in the bottom panel showing the time course for a point within the core (green dot in top panel). A movie showing the full spatio-temporal can be found in Supplementary material 9. Parameters values: $c=10.0$, ${\eta }_0=0.1$, ${\kappa }_v=1.0$, ${\kappa }_s=-25$, $\tau =1$, $\alpha =5$, $\gamma =0.5$