# A Computational Study of a Spatiotemporal Mean Field Model Capturing the Emergence of Alpha and Gamma Rhythmic Activity in the Neocortex

## Abstract

**:**

## 1. Introduction

## 2. A Continuum Mean Field Model of Electrocortical Activity

^{®}(Stockholm, Sweden).

## 3. Computational Framework

^{®}to solve (1)–(8) with periodic boundary conditions and with the initial values and input variables as specified in the following sections.

## 4. Alpha Rhythms in the Resting State

## 5. Emergence of Gamma Rhythms

## 6. Conclusions and Future Research Directions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Neocortex structure with intracortical and corticocortical connections [18].

**Figure 5.**Comparison of the random input ${g}_{\mathrm{E}\mathrm{E}}$ and the membrane potential ${v}_{\mathrm{E}}$ at the resting state. (

**a**) The random input ${g}_{\mathrm{E}\mathrm{E}}$ at $t=500$ ms. (

**b**) The membrane potential ${v}_{\mathrm{E}}$ at $t=500$ ms.

**Figure 6.**Time and frequency analysis of the $\alpha $ oscillations. Left: Measurements of the random input ${g}_{\mathrm{E}\mathrm{E}}$ and the membrane potential ${v}_{\mathrm{E}}$ at a randomly chosen probe location. Right: Power spectral density of the measurements of ten probes located randomly over the domain of the neocortex $\mathsf{\Omega}$. The solid black curve is the average of the power densities of the ten measurements. The zero frequency components (mean value) of all signals are removed.

**Figure 7.**The bifurcation diagram associated with the space homogenous ordinary differential equations (ODE) version of (1)–(8). The bifurcation parameter $\eta $ indicates the percentage of the deviation of ${\mathrm{N}}_{\mathrm{I}\mathrm{I}}$ from its nominal value. The curve of equilibria is shown in blue, and the curves of the maximum and minimum values of the limit cycles are shown in red. Solid lines denote stable equilibria and limit cycles, and dashed lines denote unstable equilibrai and limit cycles. The two Hopf bifurcation points are marked by H.

**Figure 8.**(

**a**) The initial value of ${v}_{\mathrm{E}}$ in mV. (

**b**) The locations of the measurement probes used to extract signals for the time and frequency analysis of the $\gamma $-rhythms.

**Figure 9.**Emergence of $\gamma $-band rhythmic activity. Snapshots are taken from ${v}_{\mathrm{E}}$ at every 50 ms.

**Figure 10.**Measurements of F1–F8 probes at the locations shown in Figure 8b.

**Figure 11.**Measurements of B1–B8 probes at the locations shown in Figure 8b.

**Figure 12.**Power spectral density of the measurements of F1, F2, and B1–B8 probes shown in Figure 10 and Figure 11. The zero frequency components (mean value) of all signals are removed. Power densities are calculated based on the last 256 ms measurements of F1 and F2 probes and the last 400 ms measurements of the B1–B8 probes to remove the effect of the initial period of transitions on the spectrum.

**Table 1.**Definition and range of values for the biophysical parameters of the mean field model (1)–(8). All electric potentials are given with respect to the mean resting soma membrane potential ${v}_{\mathrm{rest}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}-70\phantom{\rule{4.pt}{0ex}}\mathrm{mV}$ [8].

Parameter | Definition | Range | Unit |
---|---|---|---|

${\tau}_{\mathrm{E}}$ | Passive excitatory membrane decay time constant | $[0.005,0.15]$ | s |

${\tau}_{\mathrm{I}}$ | Passive inhibitory membrane decay time constant | $[0.005,0.15]$ | s |

${\mathrm{V}}_{\mathrm{E}\mathrm{E}}$, ${\mathrm{V}}_{\mathrm{E}\mathrm{I}}$ | Mean excitatory Nernst potentials | $[50,80]$ | mV |

${\mathrm{V}}_{\mathrm{I}\mathrm{E}}$, ${\mathrm{V}}_{\mathrm{I}\mathrm{I}}$ | Mean inhibitory Nernst potentials | [–$20,$–$5]$ | mV |

${\gamma}_{\mathrm{E}\mathrm{E}}$, ${\gamma}_{\mathrm{E}\mathrm{I}}$ | Excitatory postsynaptic potential rate constants | $[100,1000]$ | s${}^{-1}$ |

${\gamma}_{\mathrm{I}\mathrm{E}}$, ${\gamma}_{\mathrm{I}\mathrm{I}}$ | Inhibitory postsynaptic potential rate constants | $[10,500]$ | s${}^{-1}$ |

${{\rm Y}}_{\mathrm{E}\mathrm{E}}$, ${{\rm Y}}_{\mathrm{E}\mathrm{I}}$ | Amplitude of excitatory postsynaptic potentials | $[0.1,2.0]$ | mV |

${{\rm Y}}_{\mathrm{I}\mathrm{E}}$, ${{\rm Y}}_{\mathrm{I}\mathrm{I}}$ | Amplitude of inhibitory postsynaptic potentials | $[0.1,2.0]$ | mV |

${\mathrm{N}}_{\mathrm{E}\mathrm{E}}$, ${\mathrm{N}}_{\mathrm{E}\mathrm{I}}$ | Number of intracortical excitatory connections | $[2000,5000]$ | — |

${\mathrm{N}}_{\mathrm{I}\mathrm{E}}$, ${\mathrm{N}}_{\mathrm{I}\mathrm{I}}$ | Number of intracortical inhibitory connections | $[100,1000]$ | — |

$\nu $ | Corticocortical conduction velocity | $[100,1000]$ | cm/s |

${\mathsf{\Lambda}}_{\mathrm{E}\mathrm{E}}$, ${\mathsf{\Lambda}}_{\mathrm{E}\mathrm{I}}$ | Decay scale of corticocortical excitatory connectivities | $[0.1,1.0]$ | cm${}^{-1}$ |

${\mathrm{M}}_{\mathrm{E}\mathrm{E}}$, ${\mathrm{M}}_{\mathrm{E}\mathrm{I}}$ | Number of corticocortical excitatory connections | $[2000,5000]$ | — |

${\mathrm{F}}_{\mathrm{E}}$ | Maximum mean excitatory firing rate | $[50,500]$ | s${}^{-1}$ |

${\mathrm{F}}_{\mathrm{I}}$ | Maximum mean inhibitory firing rate | $[50,500]$ | s${}^{-1}$ |

${\mu}_{\mathrm{E}}$ | Excitatory firing threshold potential | $[15,30]$ | mV |

${\mu}_{\mathrm{I}}$ | Inhibitory firing threshold potential | $[15,30]$ | mV |

${\sigma}_{\mathrm{E}}$ | Standard deviation of excitatory firing threshold potential | $[2,7]$ | mV |

${\sigma}_{\mathrm{I}}$ | Standard deviation of inhibitory firing threshold potential | $[2,7]$ | mV |

**Table 2.**The set of biophysically plausible parameter values used for the computational analysis of the model (1)–(8) ([8], Table V, col. 11). The parameters ${\overline{g}}_{\mathrm{E}\mathrm{E}}$, ${\overline{g}}_{\mathrm{E}\mathrm{I}}$, ${\overline{g}}_{\mathrm{I}\mathrm{E}}$, and ${\overline{g}}_{\mathrm{I}\mathrm{I}}$ are, respectively, the mean values of the physiologically shaped random inputs ${g}_{\mathrm{E}\mathrm{E}}$, ${g}_{\mathrm{E}\mathrm{I}}$, ${g}_{\mathrm{I}\mathrm{E}}$, and ${g}_{\mathrm{I}\mathrm{I}}$ used in [8].

Parameter | ${\tau}_{\mathrm{E}}$ | ${\tau}_{\mathrm{I}}$ | ${\mathrm{V}}_{\mathrm{E}\mathrm{E}}$ | ${\mathrm{V}}_{\mathrm{E}\mathrm{I}}$ | ${\mathrm{V}}_{\mathrm{I}\mathrm{E}}$ | ${\mathrm{V}}_{\mathrm{I}\mathrm{I}}$ | ${\gamma}_{\mathrm{E}\mathrm{E}}$ | ${\gamma}_{\mathrm{E}\mathrm{I}}$ |

Value | $32.209$$\times {10}^{-3}$ | $92.26$$\times {10}^{-3}$ | $79.551$ | $77.097$ | $-8.404$ | $-9.413$ | $122.68$ | $982.51$ |

Parameter | ${\gamma}_{\mathrm{I}\mathrm{E}}$ | ${\gamma}_{\mathrm{I}\mathrm{I}}$ | ${{\rm Y}}_{\mathrm{E}\mathrm{E}}$ | ${{\rm Y}}_{\mathrm{E}\mathrm{I}}$ | ${{\rm Y}}_{\mathrm{I}\mathrm{E}}$ | ${{\rm Y}}_{\mathrm{I}\mathrm{I}}$ | ${\mathrm{N}}_{\mathrm{E}\mathrm{E}}$ | ${\mathrm{N}}_{\mathrm{E}\mathrm{I}}$ |

Value | $293.1$ | $111.4$ | $0.29835$ | $1.1465$ | $1.2615$ | $0.20143$ | $4202.4$ | $3602.9$ |

Parameter | ${\mathrm{N}}_{\mathrm{I}\mathrm{E}}$ | ${\mathrm{N}}_{\mathrm{I}\mathrm{I}}$ | $\nu $ | ${\mathsf{\Lambda}}_{\mathrm{E}\mathrm{E}},{\mathsf{\Lambda}}_{\mathrm{E}\mathrm{I}}$ | ${\mathrm{M}}_{\mathrm{E}\mathrm{E}}$ | ${\mathrm{M}}_{\mathrm{E}\mathrm{I}}$ | ${\mathrm{F}}_{\mathrm{E}}$ | ${\mathrm{F}}_{\mathrm{I}}$ |

Value | $443.71$ | $386.43$ | $116.12$ | $0.6089$ | 3228 | $2956.9$ | $66.433$ | $393.29$ |

Parameter | ${\mathrm{\mu}}_{\mathrm{E}}$ | ${\mathrm{\mu}}_{\mathrm{I}}$ | ${\mathrm{\sigma}}_{\mathrm{E}}$ | ${\mathrm{\sigma}}_{\mathrm{I}}$ | ${\overline{g}}_{\mathrm{E}\mathrm{E}}$ | ${\overline{g}}_{\mathrm{E}\mathrm{I}}$ | ${\overline{g}}_{\mathrm{I}\mathrm{E}}$ | ${\overline{g}}_{\mathrm{I}\mathrm{I}}$ |

Value | $27.771$ | $24.175$ | $4.7068$ | $2.9644$ | $2250.6$ | $4363.4$ | 0 | 0 |

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**MDPI and ACS Style**

Haddad, W.M.
A Computational Study of a Spatiotemporal Mean Field Model Capturing the Emergence of Alpha and Gamma Rhythmic Activity in the Neocortex. *Symmetry* **2018**, *10*, 568.
https://doi.org/10.3390/sym10110568

**AMA Style**

Haddad WM.
A Computational Study of a Spatiotemporal Mean Field Model Capturing the Emergence of Alpha and Gamma Rhythmic Activity in the Neocortex. *Symmetry*. 2018; 10(11):568.
https://doi.org/10.3390/sym10110568

**Chicago/Turabian Style**

Haddad, Wassim M.
2018. "A Computational Study of a Spatiotemporal Mean Field Model Capturing the Emergence of Alpha and Gamma Rhythmic Activity in the Neocortex" *Symmetry* 10, no. 11: 568.
https://doi.org/10.3390/sym10110568