Modeling the effect of sleep regulation on a neural mass model

Abstract

In mammals, sleep is categorized by two main sleep stages, rapid eye movement (REM) and non-REM (NREM) sleep that are known to fulfill different functional roles, the most notable being the consolidation of memory. While REM sleep is characterized by brain activity similar to wakefulness, the EEG activity changes drastically with the emergence of K-complexes, sleep spindles and slow oscillations during NREM sleep. These changes are regulated by circadian and ultradian rhythms, which emerge from an intricate interplay between multiple neuronal populations in the brainstem, forebrain and hypothalamus and the resulting varying levels of neuromodulators. Recently, there has been progress in the understanding of those rhythms both from a physiological as well as theoretical perspective. However, how these neuromodulators affect the generation of the different EEG patterns and their temporal dynamics is poorly understood. Here, we build upon previous work on a neural mass model of the sleeping cortex and investigate the effect of those neuromodulators on the dynamics of the cortex and the corresponding transition between wakefulness and the different sleep stages. We show that our simplified model is sufficient to generate the essential features of human EEG over a full day. This approach builds a bridge between sleep regulatory networks and EEG generating neural mass models and provides a valuable tool for model validation.

Appendix

1.1 A Model equations

The cortex model is given by the following set of equations:

$$\begin{array}{@{}rcl@{}} \tau_{p} \dot V_{p} &=& - I^{p}_{\mathrm{L}} - I_{\text{AMPA}}(s_{ep}) - I_{\text{GABA}}(s_{gp}) - \tau_{p} C_{m}^{-1} I_{\text{KNa}}, \\ \tau_{i} \dot V_{i} &=& - I^{i}_{\mathrm{L}} - I_{\text{AMPA}}(s_{ei}) - I_{\text{GABA}}(s_{gi}), \\ \ddot s_{ep} &=& {\gamma_{e}^{2}}\left( N_{pp} Q_{p}(V_{p})+\phi_{n} - s_{ep}\right) - 2\gamma_{e} \dot s_{ep}, \\ \ddot s_{gp} &=& {\gamma_{g}^{2}}\left( N_{pi} Q_{i}(V_{i})\hspace{2.6em} - s_{gp}\right) - 2\gamma_{g} \dot s_{gp}, \\ \ddot s_{ei} &=& {\gamma_{e}^{2}}\left( N_{ip} Q_{p}(V_{p})\,+\phi^{\prime}_{n} - s_{ei}\right) - 2\gamma_{e} \dot s_{ei}, \\ \ddot s_{gi} &= &{\gamma_{g}^{2}}\left( N_{ii}\,Q_{i}(V_{i})\hspace{2.6em} - s_{gi}\right) - 2\gamma_{g} \dot s_{gi}, \\ \left[\dot {\text{Na}}\right] &=& (\alpha_{\text{Na}}Q_{p}(V_{p})-\text{Na}_{\text{pump}}([\text{Na}]))/\tau_{\text{Na}}. \\ \tau_{g}\dot{g}_{\text{KNa}} &=& \bar{g}_{\text{KNa}}(1-0.95C_{A})(1-0.75C_{E})(1+0.85C_{G}) -g_{\text{KNa}}, \\ \tau_{\sigma}\dot{\sigma}_{p} &=& \bar{\sigma}_{p}-(4 C_{E} + 2 C_{A}) - \sigma_{p}. \end{array} $$

With the currents defined by:

$$\begin{array}{@{}rcl@{}} I_{\mathrm{L}} &=& \bar{g}_{L} (V_{k} - E_{\mathrm{L}}), \\ I_{\text{AMPA}}&=& \bar{g}_{\text{AMPA}} s_{ek} (V_{k} - E_{\text{AMPA}}), \\ I_{\text{GABA}}&=& \bar{g}_{\text{GABA}} s_{gk}(V_{k} - E_{\text{GABA}}), \\ I_{\text{KNa}} &=& g_{\text{KNa}} w_{\text{KNa}} (V_{p} - E_{\mathrm{K}}), \\ w_{\text{KNa}} &=& \frac{0.37}{1+\left( \frac{38.7}{[\text{Na}]}\right)^{3.5}}. \end{array} $$

The sodium pump and firing rate functions are given by:

$$\begin{array}{@{}rcl@{}} \text{Na}_{\text{pump}}([\text{Na}]) &=&R_{\text{pump}}\left( \frac{[\text{Na}]^{3}}{[\text{Na}]^{3}+3375}- \frac{[\text{Na}]_{\text{eq}}^{3}}{[\text{Na}]_{\text{eq}}^{3}+3375}\right), \\ Q_{k}(V_{k}) &=& \frac{Q_{k}^{\max}}{1 + \exp (-(V_{k} - \theta_{k})/\sigma_{k})}, \\ Q_{k}^{SR}(Y) &=& \frac{F_{k}^{\max}}{1 + \exp (-(Y - \beta_{k})/\alpha_{k})}. \end{array} $$

The sleep regulatory network is described by:

$$\begin{array}{@{}rcl@{}} \tau_{W} \dot{F}_{W} &=& Q_{W}\left( g_{GW}C_{G} + g_{AW}C_{A}\right) - F_{W}, \\ \tau_{N} \dot{F}_{N} &=& Q_{N}\left( g_{EN}C_{N}\right) - F_{N}, \\ \tau_{R} \dot{F}_{R} &=& Q_{R}\left( g_{ER}C_{E} + g_{GR}C_{G} + g_{AR} C_{A}\right) - F_{R}, \\ \tau_{E}\dot{C}_{E} &=& \tanh(F_{W}/\gamma_{E}) -C_{E}, \\ \tau_{G}\dot{C}_{G} &=& \tanh(F_{N}/\gamma_{G}) -C_{G}, \\ \tau_{A}\dot{C}_{A} &=& \tanh(F_{R}/\gamma_{A}) -C_{A}, \\ \dot{h} &=& \frac{h^{\max}-h}{{\tau^{w}_{h}}}\mathcal{H}(F_{W} - \theta_{h}) - \frac{h}{{\tau^{s}_{h}}}\mathcal{H}(\theta_{h}-F_{W}). \end{array} $$

1.2 B Parameter values

Here, we describe the different symbols used in the cortex and sleep regulation module and give their values.

It should be noted, that in the original manuscript by Diniz Behn and Both the parameter values for the sleep regulatory network are given in seconds or hours. However, since we combine the cortical and the sleep regulatory model, we have to decide on one time unit.