Transitions to spike-wave oscillations and epileptic dynamics in a human cortico-thalamic mean-field model

Abstract

In this paper we present a detailed theoretical analysis of the onset of spike-wave activity in a model of human electroencephalogram (EEG) activity, relating this to clinical recordings from patients with absence seizures. We present a complete explanation of the transition from inter-ictal activity to spike and wave using a combination of bifurcation theory, numerical continuation and techniques for detecting the occurrence of inflection points in systems of delay differential equations (DDEs). We demonstrate that the initial transition to oscillatory behaviour occurs as a result of a Hopf bifurcation, whereas the addition of spikes arises as a result of an inflection point of the vector field. Strikingly these findings are consistent with EEG data recorded from patients with absence seizures and we present a discussion of the clinical significance of these results, suggesting potential new techniques for detection and anticipation of seizures.

Appendix A

1.1 A.1 Mean-field cortico-thalamic model

The global(spatially invariant) mean-field delayed cortico-thalamic model is expressed as follows:

$$ \left\{\begin{array}{ll} \displaystyle\frac {d}{dt}\phi_{e}(t)= y(t),\\[6pt] \displaystyle\frac {d}{dt}y(t)= \gamma_e^2\left[-\phi_{e}(t) + \varsigma(V_e(t))\right] - 2\gamma_ey(t),\\[6pt] \displaystyle\frac {d}{dt}V_{e}(t)= z(t),\\[6pt] \displaystyle\frac {d}{dt}z(t)= \alpha\beta[-V_{e}(t) + \nu_{ee}\phi_{e}(t) + \nu_{ei}\varsigma(V_e(t))\\[6pt] \nu_{es}\varsigma(V_s(t - \tau))] - (\alpha + \beta)z(t),\\[6pt] \displaystyle\frac {d}{dt}V_{s}(t)= w(t),\\[6pt] \displaystyle\frac {d}{dt}w(t)= \alpha\beta[-V_{s}(t) + \nu_{sn}\phi_n + \nu_{se}\phi_{e}(t - \tau)\\[6pt] \nu_{sr}\varsigma(V_{r}(t))] - (\alpha+ \beta)w(t),\\[6pt] \displaystyle\frac {d}{dt}V_{r}(t)= v(t),\\[6pt] \displaystyle\frac {d}{dt}v(t)= \alpha\beta\left[-V_{r}(t) + \nu_{re}\phi_{e}(t - \tau) + \nu_{rs}\varsigma(v_s(t))\right]\\[6pt] (\alpha+ \beta)v(t). \end{array} \right. $$

(11)

The principal state variables are ϕ _e (mean excitatory cortical field), V _s (averaged membrane voltage of the thalamic specific neurons) and V _r (membrane voltage of thalamic relay neurons). External sensory inputs or noise is projected onto the specific relay neurons.

1.2 A.2 Initial conditions employed in the model

These initial conditions are defined through linear stability analysis (Robinson et al. 2002), which allows to place the system (11) in some unstable region of attraction where seizure can occur. Denoting the cortico-thalamic model (11) by Eq. (8), then the initial condition φ(t) is

$$ \varphi\!:\![-\tau, 0] \!\rightarrow \!\mathbb{R}^{8} \!:\! \left\{\!\begin{array}{ll} \phi_{e}(\cdot)= <{\rm random}>,\\[2pt] y(\cdot)= 0,\\[2pt] V_{e}(\cdot)= \varsigma^{-1}(<{\rm random}>),\\[2pt] z(\cdot)= 0,\\[2pt] V_{s}(\cdot)= VS(\varsigma^{-1}(<{\rm random}>)),\\[2pt] w(\cdot)= 0,\\[2pt] V_{r}(\cdot)\!=\!\nu_{re}\varsigma(\varsigma^{-1}(11))\\[2pt]\nu_{rs}\varsigma(\nu_{rs}\varsigma (\varsigma^{-1}(\!<\!{\rm random}\!>\!))),\\[2pt] v(\cdot)=0,\\[2pt] \end{array} \right. $$

(12)

where < random > is a continuous random variable sampled from a uniform probability density function and the inverse sigmoidal is defined in the following way

$$ \varsigma^{-1}(<{\rm random}>)=\theta+ \frac{\sigma \sqrt{3}}{\pi}log\left(\frac{Q^{\rm max}}{y} -1\right). $$

(13)

and the pseudo-code for the Vs(.) function follows the Newton’s iteration scheme defined as

• function Vs(double a)[

    integer i;

    integer namx=10;

    double root=0;

    for(i=0; i< namx; i++)[

      x=nu_re*varsigma(a) + nu_rs*varsigma(root);

      ff=-root+*nu_sn*phi_n + nu_se*varsigma(a) + nu_sr*varsigma(x);

      fp=-1+nu_sr*nu_rs*varsigma^{’}(x)*varsigma^{’}(root);

      root=root-ff/fp;

    ]

    return (root);

  ]

Note that we run our simulations a constant input signal. That is the variance of the noise that projects to the specific relay cells is zeros, ϕ _n  = 1. ς ^′ is the derivative of ς with respect to voltage, which is defined as follows

$$ \varsigma^{'}=\frac{d \varsigma(V)}{dV}=\frac{\frac{Q^{\rm max}\pi}{\sqrt{3}\sigma}e^{\frac{-\pi}{\sqrt{3}}(\frac{V-\theta}{\sigma})}}{\bigg[1+ e^{\frac{-\pi}{\sqrt{3}}(\frac{V-\theta}{\sigma})}\bigg]^2} $$

(14)

1.3 A.3 Parameters values employed in the model

The parameters values used to simulate absence seizures are given in the following table:

Table 1 Parameter values for absence seizures

1.4 A.4 Periodic boundary value problem

Mathematically, the periodic boundary value problem with the inflection point condition is defined as follows

$${\kern-3pt}\left\{\begin{array}{ll} {\kern-2pt}\dot {x}= T f(x(t), x(t-\tau/T \mod T), \nu),\\[3pt] {\kern-2pt}x(0)=x(1),\\[3pt] {\kern-3pt}\displaystyle\frac{\partial f_1(x(1), x(1\!-\!\tau/T),\nu)}{\partial t}\! =\! \frac{\partial^2\! f_1(x(1), x(1\!-\!\tau/T), \nu)}{\partial t^2} \!=\!0.\\[3pt] \end{array} \right. $$

(15)

Here we have rescaled the vector field F(.) of the model (11), so that its time scale, corresponds to that of period T, giving the term x(t − τ/T). ν is the set of model’s parameters and the parameter, T, is the period of oscillation as determined by the boundary value problem. The periodicity condition is given by x(0) = x(1), that is periodic oscillations are rescaled to the time interval [0, 1]. The final equation defines the conditions for an inflection to occur on the first component of the vector field, f ₁. To numerically solve (15), a descretisation scheme is required for which we use the collocation method (Engelborghs et al. 2001b). The exact implementation in DDE-BIFTOOL of the periodic boundary condition can be obtained at the following website (Rodrigues et al. 2008).