# Achermann model

## Sleep homeostat

### Process S

The Process S is the colloquial sleepiness, or sleep pressure,
physiologically defined and quantified numerically in EEG spectral
power units. It is intimately reciprocally related to the
instantaneous SWA, the latter being the means to dissipate the
former, whilst the former builds up in absence of the latter.
Acting together in a self-regulating fashion, these processes
constitute an hourglass mechanism: the **sleep homeostat**.

Elaborating on the original BorbĂ©ly et al's assumptions, Achermann's Process S needs not be a mathematically pure exponential line; rather, the function of S(t) depends on both S(t-1), SWA(t-1) as well as on the current vigilance state. Thus the model can capture the process of sleep pressure dissipation in a more refined way, less abstract way.

### Simulation of Achermann's model with Aghermann

The equations used to compute the S and SWA in Aghermann are those originally proposed in the 1993 paper, except that:

- Noise is not used;
- Three (optional) 'amendments' can be enforced.

One prerequisite for running the simulations is that your episodes must be sufficiently scored. This is mainly to differentiate between principal vigilance states (that is, NREM, REM and Wake).

The table below shows the most interesting parameters of the model:
*gc* is the precious one.

Parameter | Unit | Description |
---|---|---|

\(rc\) | min^{-1} | SWA rise constant |

\(fc_{\mathrm{R}}\) | min^{-1} | SWA fall constant triggered by REM |

\(fc_{\mathrm{W}}\) | min^{-1} | SWA fall constant triggered by Wake |

\(gc\) | min^{-1} | Gain constant, x10^{-2} |

\(S_0\) | % | Level of S at sleep onset |

\(S_U\) | % | Upper asymptote of S |

\(t_{\mathrm{a}}\) | min | Anticipated effect of REM on SWA |

\(t_{\mathrm{p}}\) | min | Extension of effect of REM on SWA |

\(rs\) | min^{-1} | Rise rate of S, 10^{-3} |

(*) \(S_{\mathrm{U}}\) is tied to \(rs\) and not tuned independently: \(S_{\mathrm{U}} = (S_{0} - S_{\mathrm{baseline_end}} \cdot \mathrm{exp}(-t \cdot rs)) / (1 - \mathrm{exp}( -t \cdot rs))\), \(t\) being here 24~h less the duration of the baseline night.