Here is a summary of neual mass models used in literatures.
Wilson-Cowan Model
Muldoon 2016 Stimulation-Based Control of Dynamic Brain Networks
https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1005076
$$\tau \frac{dE_j}{dt} = -E_j(t) + (S_{e_max}-E_j(t))S_e\lgroup c_1E_j(t) - c_2I_j(t) + c_5\sum_k A_{jk}E_k(t-\tau_d^k)+P_j(t)\rgroup + \sigma w_j(t)$$
$$\tau \frac{dI_j}{dt} = -I_j(t) +(S_{i_max}-I_j(t))S_i(c_3E_j(t)-c_4I_j(t))+\sigma v_j(t)$$
- $E(t)/I(t)$: the firing rate of the excitatory/inhibitory population respectively
- $\tau$: time constant
- $P(t)$: an external stimulus parameter
- $w_j(t)$, $v_j(t)$: drawn from a standard normal distribution and act as additive noise to the system with σ = 0.00001.
- $A$: connectivity matrix
- $\tau_d$: delays between regions.(are calculated as a function of physical distance between identified brain regions, assuming a transmission velocity of 10 m/s (range $\tau_d$ = 0.8 − 14.8 ms))
- $S_{e/i}(x)$: transfer function(given by the sigmoidal function)
$$S_{e/i}(x) = \frac{1}{1+e^{(-a_{e/i}(x-\theta_{e/i}))}} - \frac{1}{1+e^{a_{e/i}\theta_{e/i}}} $$
Parameters used in paper:
$c_1 = 16, c_2 = 12, c_3 = 15, c_4 = 3, a_e = 1.3, a_i = 2, θ_e = 4, θ_i = 3.7$
To simulate brain activity, set $P(t) = 0$ for all regions
Model stimulation to single brain region, $j$, by setting $P_j(t) = 1.25$.
Hlinka 2012 Using computational models to relate structural and functional brain connectivity
https://onlinelibrary.wiley.com/doi/full/10.1111/j.1460-9568.2012.08081.x
$$\dot u_i = -u_i +f(c_1u_i - c_2v_i + P + \epsilon \sum_j w_{ij}u_j)$$
$$\dot v_i = -v_i + f(c_3u_i - c_4v_i + Q)$$
$$f(x) = \frac{1}{1+exp(-x)}$$
- $f(x)$: a population firing rate function
Wong-Wang Model
Deco 2013 Resting-State Functional Connectivity Emerges from Structurally and Dynamically Shaped Slow Linear Fluctuations
http://www.jneurosci.org/content/33/27/11239
$$\frac{dS_i(t)}{dt} = -\frac{S_i}{\tau_S} +(1-S_i)\gamma H(x_i) +\sigma v_i(t) $$
$$H(x_i) = \frac{ax_i - b}{1-exp(-d(ax_i-b))}$$
$$x_i = wJ_NS_i + GJ_N\sum_j C_{ij}S_j + I_0 $$
- $H(x_i)$: population firing rate
- $S_i$: the average synaptic gating variable at the local cortical area i
- $w$: the local excitatory recurrence, = 0.9
- $C_{ij}$: the structural connectivity matrix expressing the neuroanatomical links between the areas i and j.
- $J_N$: the synaptic couplings, = 0.2609 (nA)
- $I_0$: the overall effective external input, = 0.3 (nA).
- $υ_i(t)$: uncorrelated standard Gaussian noise, and the noise amplitude at each node is $σ = 0.001 (nA)$.
Parameters used in the paper:
$a = 270 (n/C), b = 108 (Hz), d = 0.154 (s), \gamma = 0.641/1000$ (the factor 1000 is for expressing everything in ms), $\tau S = 100 ms$
Hansen 2014 Functional connectivity dynamics: Modeling the switching behavior of the resting state
https://www.sciencedirect.com/science/article/pii/S1053811914009033
$$\frac{dS_i}{dt} = -\frac{S_i}{\tau_S} +(1-S_i)\gamma R_i +\sigma \eta_i(t) $$
$$R_i = \frac{ax_i - b}{1-exp(-d(ax_i-b))}$$
$$x_i = wJ_NS_i + GJ_N\sum_j C_{ij}S_j + I_0 $$
- $S_i$: NMDA synaptic input currents
- $\tau_S$: NMDA decay time constant
- $R_i$: collective firing rates
- $\gamma$: kinetic parameter, = 0.641
- $x_i$: total synaptic inputs to a region
- $N$: an intensity scale for synaptic currents, = 0.2609 nA
- $w$: relative strength of recurrent connections within the region
- $C_{ij}$: entries of the SC matrix globally reweighted by a single scalar G adjusted as a control parameter
- $σ$: noise amplitude
- $η$: a stochastic Gaussian variable with a zero mean and unit variance.
- $I_0$: external input and sets the level of regional excitability.
Parameters used in the paper:
$a = 270(V·nC)^{−1}, b = 108 Hz, d = 0.154, G = 2.4, σ = 0.001, w = 0.9, I_0 = 0.3 nA$
Schirner 2018 Inferring multi-scale neural mechanisms with brain network modelling
Deco 2014
https://elifesciences.org/articles/28927
$$ I_i^{(E)} = W_EI_0 + G\sum_j C_{ij}S_j^{(E)} - J_iS_i^{(I)} + w_{BG}^{(E)}I_{BG} $$
$$I_i^{(I)} = W_II_0 - S_i^{(I)} +w_{BG}^{(I)}I_{BG} $$
$$r_i^{(E)} = \frac{a_EI_i^{(E)}-b_E}{1-exp(-d_E(a_EI_i^{(E)}-b_E))}$$
$$r_i^{(I)} = \frac{a_II_i^{(I)}-b_I}{1-exp(-d_I(a_II_i^{(I)}-b_I))}$$
$$\frac{dS_i^{(E)}(t)}{dt} = -\frac{S_i^{(E)}}{\tau_E} +(1-S_i^{(E)})\gamma_E r_i^{(E)} $$
$$\frac{dS_i^{(I)}(t)}{dt} = -\frac{S_i^{(I)}}{\tau_I} +\gamma_I r_i^{(I)} $$
- $r_i^{(E,I)}$: population firing rate of the excitatory (E) and inhibitory (I) population of brain area i.
- $S_i^{(E,I)}$: the average excitatory or inhibitory synaptic gating variables of each brain area, while their input currents are given by $I_i^{(E,I)}$
- $I_{BG}$: excitatory postsynaptic currents using region-wise aggregated EEG source activity that is added to the sum of input currents $I_i^{(E,I)}$.
- $ω_{BG}^{(E,I)}$: weight parameters, rescale the z-score normalized EEG source activity independently for excitatory and inhibitory populations.
- $G$: long-range coupling strength scaling factor that rescales the structural connectivity matrix $C_{ij}$ that denotes the strength of interaction for each region pair i and j.
- $w_+$: Local excitatory recurrence, = 1.4
- $C_{ij}$: Structural connectivity matrix
- $γ_E, γ_I$: Kinetic parameters, = $6.41×10^{−4}, 1.0×10^{−3}$
- $a_E, b_E, d_E, τ_E, W_E$: Excitatory gating variables, = $310 (nC^{−1}), 125(Hz), 0.16(s), 100(ms), 1$
- $a_I, b_I, d_I, τ_I, W_I$: Inhibitory gating variables, $615 (nC^{−1}), 177 (Hz), 0.087 (s), 10 (ms), 0.7$
- $J_{NMDA}$: Excitatory synaptic coupling, = 0.15 (nA)
- $J_I$: Feedback inhibitory synaptic coupling, Obtained by FIC heuristic (nA)
- $I_0$: overall effective external input, = 0.382 (nA)
- $G$: global coupling scaling factor, obtained from model fitting
- $w^{(E)}_{BG}, w^{(I)}_{BG}$: weights for scaling EEG-derived input currents, obtained from model fitting