Modeling of the Neurovascular Coupling in Epileptic Discharges

Abstract

Despite the interest in simultaneous EEG-fMRI studies of epileptic spikes, the link between epileptic discharges and their corresponding hemodynamic responses is poorly understood. In this context, biophysical models are promising tools for investigating the mechanisms underlying observed signals. Here, we apply a metabolic-hemodynamic model to simulated epileptic discharges, in part generated by a neural mass model. We analyze the effect of features specific to epileptic neuronal activity on the blood oxygen level dependent (BOLD) response, focusing on the issues of linearity in neurovascular coupling and on the origin of negative BOLD signals. We found both sub- and supra-linearity in simulated BOLD signals, depending on whether one observes the early or the late part of the BOLD response. The size of these non-linear effects is determined by the spike frequency, as well as by the amplitude of the excitatory activity. Our results additionally indicate a minor deviation from linearity at the neuronal level. According to a phase space analysis, the possibility to obtain a negative BOLD response to an epileptic spike depends on the existence of a long and strong excitatory undershoot. Moreover, we strongly suggest that a combined EEG-fMRI modeling approach should include spatial assumptions. The present study is a step towards an increased understanding of the link between epileptic spikes and their BOLD responses, aiming to improve the interpretation of simultaneous EEG-fMRI recordings in epilepsy.

Appendix

The Metabolic Hemodynamic Model

We first present a brief overview of the metabolic hemodynamic model suggested in Sotero and Trujillo-Barreto (2007), see Fig. 10, right.³ Changes in exc. and inh. neuronal activities induce changes in the glucose consumption g _e (t), g _i (t) (normalized to baseline) by means of linear differential equations where s _e (t), s _i (t) are glucose consumption inducing signals:

$$ \begin{aligned} \dot{s_e}(t) &= \frac{a_e}{\tau_e} (N_{exc} (t-\delta_e)-1) - \frac{2}{\tau_e} s(t) - \frac{1}{\tau_{e}^2} (g_e(t)-1) \\ \dot{g_e(}t) &= s_e(t). \end{aligned} $$

$\delta_e$ describes the delay between N _exc and g _e after stimulus onset, a _e represents the efficiency of the glucose consumption response to excitation (amplitude of the impulse response), and τ _e gives the time constant (i.e., the width) the impulse response.

Fig. 10

Schematic representations of the two models used in this study. Left: generic version of the NMM suggested in Wendling et al. (2000, 2005). Right: MHM introduced in Sotero and Trujillo-Barreto (2007), adapted from their Fig. 1. Note the parallel arrangement of CBF and metabolic processes, and that the CBF dynamics depend only on excitation

The glucose variables were then directly related to the normalized metabolic rates of oxygen for exc. and inh activities m _e (t), m _i (t), as well as to the total oxygen consumption:

$$ m_e(t) = \frac{2-x(t)}{2-x_0} g_e(t)\hbox { and } m_i(t) = g_i(t) $$

For exc synapses, a fraction x of lactate is lost to the oxidative metabolism occurring in the neuron. This lost is modeled as a sigmoid function with threshold x ₀.

The cerebral blood flow f, however, depends only on the exc. neuronal activity:

$$ \begin{aligned} \dot{s_f}(t) &= \epsilon (N_{exc} (t-\delta_f)-1) - \frac{1}{\tau_s} s_f(t) - \frac{f(t)-1}{\tau_f} \\ \dot{f}(t) &= s_f(t), \end{aligned} $$

where s _f is some flow-inducing signal, $\epsilon$ is the efficacy with which N _exc causes an increase, τ _s is the time constant for signal decay or elimination and τ _f is the time constant for autoregulatory feedback from blood flow, while the delay between N _exc and CBF responses is given by $\delta_f.$

Both the CBF and the total glucose consumption g(t) enter the Balloon model (Buxton et al. 2004). Here, CBF and g(t) are linked (via differential equations) to the normalized (to their values at rest) cerebral blood volume v and its deoxyhemoglobin content q. Knowing the latter two, the BOLD signal is calculated:

$$ \hbox {BOLD}(t) = V_0 (a_1(1-q)-a_2(1-v)) $$

(2)

where V ₀ describes the resting venous blood volume fraction, while the parameters a ₁, a ₂ depend on several experimental and physiological parameters.

The Neural Mass Model

Secondly, let us give a brief overview of the neural mass model in Wendling et al. (2000, 2005), see Fig. 10, left. In “Neural Mass Model as Input for the MHM” section we explained that one population of the NMM consists of one exc. and one inh. sub-population, coupled to each other (via exc. and inh. synapses). Each sub-population is characterized by (i) one (or two) dynamic linear transfer functions that transform average action potential densities into EPSPs and IPSPs, respectively, and by (ii) a static non-linear function (sigmoid function S) that relates the average membrane potential of a sub-population into an average action potential density. The following set of six differential equations governs the model dynamics:

$$ \begin{aligned} \dot{y_0}(t) &= y_3(t) \\ \dot{y_3}(t) &= A\cdot a\cdot S(y_1-y_2) -2\cdot a\cdot y_3(t) - a^2\cdot y_0(t) \\ \dot{y_1}(t) &= y_4(t) \\ \dot{y_4}(t) &= A\cdot a\cdot \{p(t)+C_{ee}\cdot S[C_{ee}\cdot y_0(t)]\} - 2\cdot a\cdot y_4(t)-a^2\cdot y_1(t) \\ \dot{y_2}(t) &= y_5(t) \\ \dot{y_5}(t) &= B\cdot b\cdot \{C_{ei}\cdot S(C_{ie}\cdot y_0(t)\}-2\cdot b\cdot y_5(t)-b^2\cdot y_2(t). \\ \end{aligned} $$

The parameters A and B describe the synaptic gain of exc. and inh. synapses, respectively, while a and b describe their time constants (cf. Table 3). p(t) is the exc. input, representing the average density of afferent action potentials (for C _ee etc. see “Neural Mass Model as Input for the MHM” section)

Table 3 List of parameters used in the NMM in order to generate the variable spike shapes and trains S1–S5 in see “Neural Mass Model as Input for the MHM” section (C _ee , C _ei , C _ie describe the internal connectivity)